Abstract:
A constrained four dimensional grid-based filter (CGBF) used to provide state estimates for a target maneuvering in two dimensions. Optimal grid and sampling sizes or chosen and the kinematic constraints of the target a y used to restrict the possible predicted states resulting in quality target estimates.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
       [0001]    This application claims the benefit of priority from U.S. Provisional Patent Application Ser. No. 61/708,673, filed on Oct. 2, 2012. 
     
    
     STATEMENT OF GOVERNMENT INTEREST 
       [0002]    The invention described herein may be manufactured and used by or for the Government of the United States of America for governmental purposes without payment of any royalties thereon or therefore. 
     
    
     BACKGROUND 
       [0003]    The present invention relates to an improved system for using a grid based filter to track a maneuvering target. More specifically, but without limitation, the present invention is a constrained multi-dimensional grid based filter used to provide state estimates for a maneuvering target. 
         [0004]    Over the years, many systems have been developed to address the target tracking problem. However, tracking multiple maneuvering targets, as shown, in  FIG. 1 , remains a notable weakness of these systems. At the heart of a tracking system is the target track filter. The track filter computes the most likely target state given all the measurements collected on that target. This most likely state is typically based on the one that minimizes the mean squared error (MSE). In the early 1960s, Rudolf Kalman published a paper entitled “New Approach to Linear Filtering and Prediction Problems.” (Transactions of the ASME—Journal of Basic Engineering, 82 (Series D): 35-45.) Kalman described a filter that yields minimum MSE state estimates for systems that exhibit linear dynamics. This filter became known as the Kalman filter and since its debut, it has been extensively studied and has emerged as the premier method for target track filtering due to its computational efficiency and its filtering effectiveness. 
         [0005]    The Kalman filter is now used in many applications from GPS tracking to air traffic control. If the motion is linear and the measurement errors are Gaussian, a Kalman filter can provide accurate estimates of a target state efficiently. 
         [0006]    Nevertheless, although the Kalman filter is efficient and effective for computing state estimates of a moving target, it produces poor estimates when tracking a maneuvering target. The reason for these poor estimates is that the Kalman filter is a linear filter. It assumes the dynamics of the tracked target can be modeled as a linear system, but maneuvering targets do not generally exhibit linear motion. The reason the target motion needs to be linear is due to a property of the normal distribution. A normal random variable remains normal through a linear transform. If the transform is not linear, there is no guarantee that the normal variable will remain normal. 
         [0007]    The Kalman filter includes a process noise parameter which provides a means to capture the uncertainty in the target motion. Thus, a straightforward way to allow the Kalman filter to track maneuvering targets would be to simply employ a large process noise. The idea is to treat the non-linear target motion simply as a larger uncertainty in its linear motion. However, increasing the process noise parameter causes large errors in the computed state estimates. These larger errors cause more ambiguity making the tracking process more difficult. 
         [0008]    Since the Kalman filter was so effective for linear targets, researchers began looking for enhancements and/or extensions to the Kalman filter. One of the early extensions was the extended Kalman filter (EKF). Radar was (and still is) the dominant sensor used for target tracking. One of the early refinements to radar systems was the ability to measure the closing rate of the target. This closing rate is known as the range rate. Actually, the range rate is a closing rate only when the target is moving radially towards, or away from the radar. If the target is moving along some other path, the range rate is measuring only that component of the target&#39;s velocity that is moving radially towards (or away from) the radar. This radial component is proportional to the cosine of the difference of the angles between the target&#39;s course and the radial angle from the target to the radar. This radial angle to the target is often called the azimuth angle. Cosine is a non-linear function and since the range rate is a function of cosine, the (linear) Kalman filter had to be extended to address this non-linear quantity. The resulting Extended Kalman Filter (EKF), includes this range rate measurement into the calculations by performing a first-order approximation of its additional information. Since the Kalman filter is inherently a linear filter, the intent is to linearize the range rate information. However, the cosine cannot always be estimated well with only a first-order approximation, so the EKF is a suboptimal filter. As a result, an EKF will at times perform no better than a Kalman filter and could even perform worse by failing to properly track the target. 
         [0009]    Several other extensions to the Kalman filter subsequently emerged including iterated extended Kalman filters (IEKF), unscented Kalman filters (UKFs), and interacting motion model Kalman filters (IMMs). All these methods have the advantage that they are based on the Kalman filter and have closed-form solutions. This makes them computationally efficient and well-grounded in a mathematical formulization. However, all these methods make assumptions about the maneuvering target that are often not true, which leads to erroneous estimates. 
         [0010]    Vast improvements to radar, computers and computational power have led researchers to pursue more numerical (and thus, more computational) methods for target tracking. Two methods that emerged were grid-based filters (GBFs) and particle filters (PFs). These methods were broadly classified as sequential Monte Carlo (SMC) techniques. A key feature of these methods is that they allow many of the restricted assumptions placed on the closed-form methods to be relaxed. It then became possible to consider more general, non-linear target dynamics. Any presumed motion of the target could be modeled without being limited to linear dynamics. In addition, the measurement errors could be modeled more realistically, unlike the Kalman filter-like methods, which assume Gaussian error on the measurements even though radar measurements are generally not Gaussian (at least not in rectilinear space). 
         [0011]    The idea behind a GBF is to first discretize the region where the targets are presumed to reside into a set of cells. The next step is to model how the targets would transition from cell to cell over time. Finally, measurements are used to increase the probability of the target being in a cell. Unfortunately, even with faster computers, this approach is far too computational to execute in real-time. Although the grid-based filter approach is an attractive solution, due to its computational burden, it is difficult to implement and, thus, has not gained widespread use. What is needed is a computationally economical method capable of using a four dimensional grid based filter for tracking a target maneuvering in two dimensions. 
       SUMMARY 
       [0012]    The present invention is directed to a multi-dimensional grid based filter for tracking a moving target. The filter provides state estimates for a maneuvering target using a grid-based filter by initially forming a first grid of cells to capture an initial target state comprising an estimate of the target&#39;s position, course and speed using a first measurement. A second grid of cells, of a finite size and scale, is formed wherein each cell comprises possible estimated states to which the target could have transitioned over a time between the first and a second measurement. Using a sampling technique, the prediction distribution is estimated for each cell in the grid using the various maneuvering probabilities of the target. Each cell corresponds to a particular rectangular region in an x and y coordinate plane. An initial estimate is made of the probability that the target is in the corresponding rectangular region. There is also an estimate of the expected initial speed of the target and the variance of the speed in the corresponding rectangular region. Similarly, there is an estimate of the expected initial course of the target and the variance of the course in the corresponding rectangular region. A discretized probability distribution over all the cells in the second grid is formed using Monte Carlo sampling. The position distribution is assumed to be uniformly distributed over the cell and each cell forms a normal distribution for the possible speeds for that region. Also, each cell forms a normal distribution of the possible courses for that region. This process repeats for each measurement update where the second grid becomes the first grid for the next update. 
     
    
     
       DRAWINGS 
         [0013]    These and other features, aspects and advantages of the present invention will become better understood with reference to the following description and appended claims, and accompanying drawings wherein: 
           [0014]      FIG. 1  is a picture of an aircraft tracking various surface targets via radar. 
           [0015]      FIG. 2  is an algorithm for computing the destination grid (i.e., the second grid) for each update. 
           [0016]      FIG. 3  is a graph of true predicted positional containment region. 
           [0017]      FIG. 4  is a table comparing the Constrained Grid-Based Filter predicted state and covariance to the Kalman Filter as the maximum turn a of the target is varied. 
           [0018]      FIG. 5  is an illustration showing the grid cell update using Monte Carlo sampling. 
           [0019]      FIG. 6  is a graph shoving the target motion symmetry, mirrored state calculation. 
       
    
    
     DESCRIPTION 
       [0020]    In the following description of the present invention, reference will be made to various embodiments which are not meant to be all inclusive. The current invention can be implemented using various forms of software for execution on a variety of computer systems. The preferred embodiments of the present invention are illustrated by way of example below and in the referenced Figures. 
         [0021]    The power of a grid based filter (GBF) can be exploited with two-dimensional grid. The solution assumes that the grid discretizes the x,y locations, and then extends what is stored in each cell of the grid. Instead of merely storing the probability of the target being at a given x,y location, four other values are also included: 1) the expected speed of the target given it is at that location, 2) the variance of speeds of the target at that location 3) the expected course of the target at the given location and 4) the variance of the targets expected course at the given location. 
         [0022]    Because the kinematic constraints on a target are usually specified in terms of speeds and turn rates, these values are used instead of component velocities. This ensures that the targets kinematic constraints are enforced and exploited in the analysis. It is not necessary to store the velocity as speed and course. It could be stored as x and y velocity components. The purpose of storing the course and speed is to more easily and directly exploit the speed, acceleration, and turn rate constraints of the target. However, it should be noted that working with course angle brings up different computation difficulties due to its circular nature. 
         [0023]    There are two grids, an originating grid and a destination grid. These grids can be thought of as the From and To grids. Let the From grid be labeled F and the To grid be labeled T. Then F(x,y) is the cell at location x,y in the From grid. ‘Dot notation’ will be used to select each of the five values within a cell where F(x,y).prob, F(x,y).spd, F(x,y).svar, F(x,y).crs, and F(x,y).cvar are the probability, mean speed, speed variance, mean course, and course variance, respectively. The algorithm in  FIG. 2  shows how the probability, mean speed and speed variance values are computed. The computation of the mean course and its variance require additional steps to deal with the circular nature of the course parameter, but are not further described. 
         [0024]    There are three separate loops in  FIG. 2 . The first loop forms the weighted sums (probability) based on Monte Carlo samples. It is also the loop that determines to which track a new measurement should be assigned in multi-target scenarios. The measurement would be assigned to the grid which yields the highest cumulative probability. After the sampling is complete, the second loop converts these weighted sums into mean and variance, using the equations 1-3 shown below. The final loop re-normalizes the mass in the cells. 
         [0000]    
       
         
           
             
               
                 
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       The 2D Target Motion Model  
       [0025]    The algorithm in  FIG. 2  also references three external functions: Normal, RandomMove, and CellOverlapFunction. The ‘Normal’ function creates normally distributed random variables. The RandomMove function creates random, but kinematically feasible predicted states over the time interval. The function allows any target motion model to be used for a CGBF. The motion model can be developed as follows. Assume that a target maneuvers at times t k , k=1, 2, 3, . . . These times are independent of the measurement update times and are within an update interval. At each time t k , the target undergoes a constant acceleration and/or constant turn maneuver that persists until the next maneuver time. Assume, at time t k-1 , the target is at location (x k-1 , y k-1 ) with speed and course, s k-1 , θ k-1 , respectively. Let {dot over (s)} be a (randomly) selected acceleration (along the direction of travel) and {dot over (θ)} be a (randomly) selected turn rate that the target will undergo for the duration of the next movement. Let τ=t k −t k-1  be the duration of the next maneuver. Then the target state as a function of time over the interval 0 to τ would be defined by the equations below: 
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         [0000]    Even though the motion equations are mathematically sound, there could be computational problems when the maximum turn rate is (nearly) zero. To avoid these problems, let ε θ &gt;0 be the smallest turn rate value at which smaller values are equivalent to the larger moving along a constant straight line course. Then the equations 8 and 9 become. 
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                                         . 
                                       
                                       τ 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           
                             otherwise 
                           
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   10 
                 
               
             
             
               
                 
                   
                     y 
                      
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       y 
                       
                         k 
                         - 
                         1 
                       
                     
                     + 
                     
                       { 
                       
                         
                           
                             
                               
                                 1 
                                 
                                   
                                     θ 
                                     . 
                                   
                                   2 
                                 
                               
                                
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         
                                           
                                             s 
                                             . 
                                           
                                            
                                           
                                               
                                           
                                            
                                           cos 
                                            
                                           
                                               
                                           
                                            
                                           
                                             θ 
                                              
                                             
                                               ( 
                                               t 
                                               ) 
                                             
                                           
                                         
                                         + 
                                         
                                           
                                             s 
                                              
                                             
                                               ( 
                                               t 
                                               ) 
                                             
                                           
                                            
                                           
                                             θ 
                                             . 
                                           
                                            
                                           
                                               
                                           
                                            
                                           sin 
                                            
                                           
                                               
                                           
                                            
                                           
                                             θ 
                                              
                                             
                                               ( 
                                               t 
                                               ) 
                                             
                                           
                                         
                                          
                                         
                                             
                                         
                                         - 
                                       
                                     
                                   
                                   
                                     
                                       
                                         
                                           
                                             s 
                                             . 
                                           
                                            
                                           
                                               
                                           
                                            
                                           cos 
                                            
                                           
                                               
                                           
                                            
                                           
                                             θ 
                                             
                                               k 
                                               - 
                                               1 
                                             
                                           
                                         
                                         - 
                                         
                                           
                                             s 
                                             
                                               k 
                                               - 
                                               1 
                                             
                                           
                                            
                                           
                                             θ 
                                             . 
                                           
                                            
                                           
                                               
                                           
                                            
                                           sin 
                                            
                                           
                                               
                                           
                                            
                                           
                                             θ 
                                             
                                               k 
                                               - 
                                               1 
                                             
                                           
                                         
                                       
                                     
                                   
                                 
                                 ] 
                               
                             
                           
                           
                             
                               
                                 if 
                                  
                                 
                                     
                                 
                                  
                                 
                                    
                                   
                                     θ 
                                     . 
                                   
                                    
                                 
                               
                               &gt; 
                               
                                 ε 
                                 
                                   θ 
                                   . 
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 ( 
                                 
                                   
                                     
                                       s 
                                       
                                         k 
                                         - 
                                         1 
                                       
                                     
                                      
                                     t 
                                   
                                   + 
                                   
                                     
                                       1 
                                       2 
                                     
                                      
                                     
                                       s 
                                       . 
                                     
                                      
                                     
                                       t 
                                       2 
                                     
                                   
                                 
                                 ) 
                               
                                
                               
                                 cos 
                                  
                                 
                                   ( 
                                   
                                     
                                       θ 
                                       
                                         k 
                                         - 
                                         1 
                                       
                                     
                                     + 
                                     
                                       
                                         θ 
                                         . 
                                       
                                       τ 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           
                             otherwise 
                           
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   11 
                 
               
             
           
         
       
     
       The Mean and Variance of the Transition Density (Single Maneuver)  
       [0026]    Assume there is only one maneuver over the update time interval, i.e., T=τ. Thus, the maneuver duration, τ, will be used to emphasize this fact. Refer to equations 4-9. The state at the end of the maneuver is found using t=τ. Recall that the subscript k refers to the (true) state of the target after the k th  maneuver within an update time interval. A subscript of zero is the true state of the target at the beginning of the time interval. So: 
         [0000]    
       
         
           
             
               
                 
                   
                       
                   
                    
                   
                     
                       s 
                       1 
                     
                     = 
                     
                       
                         s 
                          
                         
                           ( 
                           τ 
                           ) 
                         
                       
                       = 
                       
                         
                           s 
                           0 
                         
                         + 
                         
                           
                             s 
                             . 
                           
                            
                           τ 
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   12 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       θ 
                       1 
                     
                     = 
                     
                       
                         θ 
                          
                         
                           ( 
                           τ 
                           ) 
                         
                       
                       = 
                       
                         
                           θ 
                           0 
                         
                         + 
                         
                           
                             θ 
                             . 
                           
                            
                           τ 
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   13 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         x 
                         . 
                       
                       1 
                     
                     = 
                     
                       
                         
                           x 
                           . 
                         
                          
                         
                           ( 
                           τ 
                           ) 
                         
                       
                       = 
                       
                         
                           
                             s 
                              
                             
                               ( 
                               τ 
                               ) 
                             
                           
                            
                           sin 
                            
                           
                               
                           
                            
                           
                             θ 
                              
                             
                               ( 
                               τ 
                               ) 
                             
                           
                         
                         = 
                         
                           
                             s 
                             1 
                           
                            
                           sin 
                            
                           
                               
                           
                            
                           
                             θ 
                             1 
                           
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   14 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         y 
                         . 
                       
                       1 
                     
                     = 
                     
                       
                         
                           y 
                           . 
                         
                          
                         
                           ( 
                           τ 
                           ) 
                         
                       
                       = 
                       
                         
                           
                             s 
                              
                             
                               ( 
                               τ 
                               ) 
                             
                           
                            
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                              
                             
                               ( 
                               τ 
                               ) 
                             
                           
                         
                         = 
                         
                           
                             s 
                             1 
                           
                            
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             1 
                           
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   15 
                 
               
             
             
               
                 
                   
                     x 
                     1 
                   
                   = 
                   
                     
                       x 
                        
                       
                         ( 
                         τ 
                         ) 
                       
                     
                     = 
                     
                       
                         x 
                         0 
                       
                       + 
                       
                         
                           1 
                           
                             
                               θ 
                               . 
                             
                             2 
                           
                         
                          
                         
                           [ 
                           
                             
                               
                                 s 
                                 . 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 1 
                               
                             
                             - 
                             
                               
                                 s 
                                 1 
                               
                                
                               
                                 θ 
                                 . 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 1 
                               
                             
                             - 
                             
                               
                                 s 
                                 . 
                               
                                
                               
                                   
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 0 
                               
                             
                             + 
                             
                               
                                 s 
                                 0 
                               
                                
                               
                                 θ 
                                 . 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 0 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   16 
                 
               
             
             
               
                 
                   
                     y 
                     1 
                   
                   = 
                   
                     
                       y 
                        
                       
                         ( 
                         τ 
                         ) 
                       
                     
                     = 
                     
                       
                         y 
                         0 
                       
                       + 
                       
                         
                           1 
                           
                             
                               θ 
                               . 
                             
                             2 
                           
                         
                          
                         
                           [ 
                           
                             
                               
                                 s 
                                 . 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 1 
                               
                             
                             + 
                             
                               
                                 s 
                                 1 
                               
                                
                               
                                 θ 
                                 . 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 1 
                               
                             
                             - 
                             
                               
                                 s 
                                 . 
                               
                                
                               
                                   
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 0 
                               
                             
                             - 
                             
                               
                                 s 
                                 0 
                               
                                
                               
                                 θ 
                                 . 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 0 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   17 
                 
               
             
           
         
       
     
       The Mean Variance of the Speed and Course  
       [0027]    The mean and variance of the speed and course is determined using the equations below: 
         [0000]        s   1   =s (τ)= s   0   +{dot over (s)} τ and θ 1 =θ(τ)=θ 0 +{dot over (θ)}τ  Eq. 18
 
         [0000]        E[s   1   ]=E[s   0   +{dot over (s)}τ]=E[s   0   ]+E[{dot over (s)}τ]=E[s   0   ]+τE[{dot over (s)}]=s   0    Eq. 19
 
         [0000]    And similarly from equations 13, 
         [0000]        E[θ   1   ]=E[θ   o +{dot over (θ)}τ ]=E[θ   0   ]+τE[{dot over (θ)}]=θ   0    Eq. 20
 
         [0000]    The expected speed and course after the target maneuvers are identical to what they were prior to the maneuver. Calculation of the variances for the speed and course are calculated as follows: 
         [0000]      Var[ s   1 ]=Var[ s   0 +{dot over ( s )}τ]=Var[s 0 ]+Var[{dot over (s)}τ]=Var[ s   0 ]+τ 2 Var[{dot over ( s )}]=τ 2 σ s   2    Eq. 21
 
         [0000]      Var[θ 1 ]=Var[θ 0 +{dot over (θ)}τ]=Var[θ 0 ]+τ 2 Var[{dot over (θ)}]=τ 2 σ θ   2    Eq. 22
 
         [0000]    The selection of accelerations and turn rates were picked from a uniform distribution over the range of possible rates. As a result, 
         [0000]    
       
         
           
             
               
                 σ 
                 
                   s 
                   . 
                 
                 2 
               
               = 
               
                 
                   
                     
                       ( 
                       
                         2 
                          
                         
                             
                         
                          
                         A 
                       
                       ) 
                     
                     2 
                   
                   12 
                 
                 = 
                 
                   
                     
                       
                         A 
                         2 
                       
                       3 
                     
                      
                     
                         
                     
                      
                     and 
                      
                     
                         
                     
                      
                     
                       σ 
                       
                         θ 
                         . 
                       
                       2 
                     
                   
                   = 
                   
                     
                       
                         
                           ( 
                           
                             2 
                              
                             
                               Θ 
                               . 
                             
                           
                           ) 
                         
                         2 
                       
                       12 
                     
                     = 
                     
                       
                         
                           Θ 
                           . 
                         
                         2 
                       
                       3 
                     
                   
                 
               
             
             , 
           
         
       
     
         [0000]    where A is the maximum acceleration and {dot over (Θ)} is the maximum turn rate. The variance of the velocity is quadratic with respect to time, or equivalently, the standard deviation of the velocity is linear in time. 
       Mean and Variance of the Component Velocities  
       [0028]    The mean and variance of the component velocities can also be determined. In doing so, assume the distribution is in polar space and the acceleration change is independent of the course change. Therefore, the following calculations are used. 
         [0000]        E[{dot over (x)}   1   ]=E[s   1  sin θ 1 ]  Eq. 23
 
         [0000]        E[{dot over (y)}   1   ]=E[s   1  cos θ 1 ]  Eq. 24
 
         [0000]    Since it is assumed that the acceleration change is independent of the course change, 
         [0000]        E[{dot over (x)}   1   ]=E[s   1  sin θ 1   ]=E[s   1   ]E [sin θ 1 ]  Eq. 25
 
         [0000]        E[{dot over (y)}   1   ]=E[s   1  cos θ 1   ]=E[s   1   ]E [cos θ 1 ]  Eq. 26
 
         [0000]    the E[s 1 ] was already found in equation 19. Using the properties of sine and cosine, 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           E 
                            
                           
                             [ 
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 1 
                               
                             
                             ] 
                           
                         
                         = 
                           
                          
                         
                           E 
                            
                           
                             [ 
                             
                               sin 
                               ( 
                               
                                 
                                   θ 
                                   0 
                                 
                                 + 
                                 
                                   
                                     θ 
                                     . 
                                   
                                    
                                   τ 
                                 
                               
                               ) 
                             
                             ] 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           E 
                            
                           
                             [ 
                             
                               
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 
                                   θ 
                                   0 
                                 
                                  
                                 cos 
                                  
                                 
                                     
                                 
                                  
                                 
                                   θ 
                                   . 
                                 
                                  
                                 τ 
                               
                               + 
                               
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 
                                   θ 
                                   . 
                                 
                                  
                                 τcos 
                                  
                                 
                                     
                                 
                                  
                                 
                                   θ 
                                   0 
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               0 
                             
                              
                             
                               E 
                                
                               
                                 [ 
                                 
                                   cos 
                                    
                                   
                                       
                                   
                                    
                                   
                                     θ 
                                     . 
                                   
                                    
                                   τ 
                                 
                                 ] 
                               
                             
                           
                           + 
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               0 
                             
                              
                             
                               E 
                                
                               
                                 [ 
                                 
                                   sin 
                                    
                                   
                                       
                                   
                                    
                                   
                                     θ 
                                     . 
                                   
                                    
                                   τ 
                                 
                                 ] 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   27 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           E 
                            
                           
                             [ 
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 1 
                               
                             
                             ] 
                           
                         
                         = 
                           
                          
                         
                           E 
                            
                           
                             [ 
                             
                               cos 
                               ( 
                               
                                 
                                   θ 
                                   0 
                                 
                                 + 
                                 
                                   
                                     θ 
                                     . 
                                   
                                    
                                   τ 
                                 
                               
                               ) 
                             
                             ] 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           E 
                            
                           
                             [ 
                             
                               
                                 cos 
                                  
                                 
                                     
                                 
                                  
                                 
                                   θ 
                                   0 
                                 
                                  
                                 cos 
                                  
                                 
                                     
                                 
                                  
                                 
                                   θ 
                                   . 
                                 
                                  
                                 τ 
                               
                               - 
                               
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 
                                   θ 
                                   0 
                                 
                                  
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 
                                   θ 
                                   . 
                                 
                                  
                                 τ 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               0 
                             
                              
                             
                               E 
                                
                               
                                 [ 
                                 
                                   cos 
                                    
                                   
                                       
                                   
                                    
                                   
                                     θ 
                                     . 
                                   
                                    
                                   τ 
                                 
                                 ] 
                               
                             
                           
                           - 
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               0 
                             
                              
                             
                               E 
                                
                               
                                 [ 
                                 
                                   sin 
                                    
                                   
                                       
                                   
                                    
                                   
                                     θ 
                                     . 
                                   
                                    
                                   τ 
                                 
                                 ] 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   28 
                 
               
             
           
         
       
     
         [0029]    The remaining expected values can be found using the definition of expected value. Assume, as before, that {dot over (θ)} is a uniform random variable on the interval [−{dot over (Θ)}, +{dot over (Θ)}]. Therefore, 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           E 
                            
                           
                             [ 
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 . 
                               
                                
                               τ 
                             
                             ] 
                           
                         
                         = 
                           
                          
                         
                           
                             ∫ 
                             
                               - 
                               
                                 Θ 
                                 . 
                               
                             
                             
                               + 
                               
                                 Θ 
                                 . 
                               
                             
                           
                            
                           
                             
                               ( 
                               
                                 1 
                                 
                                   2 
                                    
                                   
                                       
                                   
                                    
                                   
                                     Θ 
                                     . 
                                   
                                 
                               
                               ) 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               . 
                             
                              
                             τ 
                              
                             
                                
                               
                                 θ 
                                 . 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               - 
                               
                                 ( 
                                 
                                   1 
                                   
                                     2 
                                      
                                     
                                       Θ 
                                       . 
                                     
                                      
                                     τ 
                                   
                                 
                                 ) 
                               
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               . 
                             
                              
                             τ 
                           
                            
                           
                             | 
                             
                               - 
                               
                                 Θ 
                                 . 
                               
                             
                             
                               + 
                               
                                 Θ 
                                 . 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             - 
                             
                               ( 
                               
                                 1 
                                 
                                   2 
                                    
                                   
                                     Θτ 
                                     . 
                                   
                                 
                               
                               ) 
                             
                           
                            
                           
                             ( 
                             
                               
                                 cos 
                                  
                                 
                                     
                                 
                                  
                                 
                                   Θ 
                                   . 
                                 
                                  
                                 τ 
                               
                               - 
                               
                                 cos 
                                  
                                 
                                     
                                 
                                  
                                 
                                   Θ 
                                   . 
                                 
                                  
                                 τ 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         0 
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   29 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             E 
                              
                             
                               [ 
                               
                                 cos 
                                  
                                 
                                     
                                 
                                  
                                 
                                   θ 
                                   . 
                                 
                                  
                                 τ 
                               
                               ] 
                             
                           
                           = 
                             
                            
                           
                             
                               ∫ 
                               
                                 - 
                                 
                                   Θ 
                                   . 
                                 
                               
                               
                                 + 
                                 
                                   Θ 
                                   . 
                                 
                               
                             
                              
                             
                               
                                 ( 
                                 
                                   1 
                                   
                                     2 
                                      
                                     
                                         
                                     
                                      
                                     
                                       Θ 
                                       . 
                                     
                                   
                                 
                                 ) 
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 . 
                               
                                
                               τ 
                                
                               
                                  
                                 
                                   θ 
                                   . 
                                 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                             
                            
                           
                             
                               
                                 ( 
                                 
                                   1 
                                   
                                     2 
                                      
                                     
                                       Θ 
                                       . 
                                     
                                      
                                     τ 
                                   
                                 
                                 ) 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 . 
                               
                                
                               τ 
                             
                              
                             
                               | 
                               
                                 - 
                                 
                                   Θ 
                                   . 
                                 
                               
                               
                                 + 
                                 
                                   Θ 
                                   . 
                                 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                             
                            
                           
                             
                               ( 
                               
                                 1 
                                 
                                   2 
                                    
                                   
                                     Θτ 
                                     . 
                                   
                                 
                               
                               ) 
                             
                              
                             
                               ( 
                               
                                 
                                   sin 
                                    
                                   
                                       
                                   
                                    
                                   
                                     Θ 
                                     . 
                                   
                                    
                                   τ 
                                 
                                 + 
                                 
                                   sin 
                                    
                                   
                                       
                                   
                                    
                                   
                                     Θ 
                                     . 
                                   
                                    
                                   τ 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                             
                            
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 Θ 
                                 . 
                               
                                
                               τ 
                             
                             
                               
                                 Θ 
                                 . 
                               
                                
                               τ 
                             
                           
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     Thus 
                     , 
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   30 
                 
               
             
             
               
                 
                   
                     E 
                      
                     
                       [ 
                       
                         sin 
                          
                         
                             
                         
                          
                         
                           θ 
                           1 
                         
                       
                       ] 
                     
                   
                   = 
                   
                     sin 
                      
                     
                         
                     
                      
                     
                       θ 
                       0 
                     
                      
                     
                       
                         sin 
                          
                         
                             
                         
                          
                         
                           Θ 
                           . 
                         
                          
                         τ 
                       
                       
                         
                           Θ 
                           . 
                         
                          
                         τ 
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   31 
                 
               
             
             
               
                 
                   
                     E 
                      
                     
                       [ 
                       
                         cos 
                          
                         
                             
                         
                          
                         
                           θ 
                           1 
                         
                       
                       ] 
                     
                   
                   = 
                   
                     cos 
                      
                     
                         
                     
                      
                     
                       θ 
                       0 
                     
                      
                     
                       
                         sin 
                          
                         
                             
                         
                          
                         
                           Θ 
                           . 
                         
                          
                         τ 
                       
                       
                         
                           Θ 
                           . 
                         
                          
                         τ 
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   32 
                 
               
             
           
         
       
     
         [0000]    From equations 27, 28, 29, 31, and 32, 
         [0000]    
       
         
           
             
               
                 
                   
                     E 
                      
                     
                       [ 
                       
                         
                           x 
                           . 
                         
                         1 
                       
                       ] 
                     
                   
                   = 
                   
                     
                       s 
                       0 
                     
                      
                     sin 
                      
                     
                         
                     
                      
                     
                       θ 
                       0 
                     
                      
                     
                       
                         sin 
                          
                         
                             
                         
                          
                         
                           Θ 
                           . 
                         
                          
                         τ 
                       
                       
                         
                           Θ 
                           . 
                         
                          
                         τ 
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   33 
                 
               
             
             
               
                 
                   
                     E 
                      
                     
                       [ 
                       
                         
                           y 
                           . 
                         
                         1 
                       
                       ] 
                     
                   
                   = 
                   
                     
                       s 
                       0 
                     
                      
                     cos 
                      
                     
                         
                     
                      
                     
                       θ 
                       0 
                     
                      
                     
                       
                         sin 
                          
                         
                             
                         
                          
                         
                           Θ 
                           . 
                         
                          
                         τ 
                       
                       
                         
                           Θ 
                           . 
                         
                          
                         τ 
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   34 
                 
               
             
           
         
       
     
         [0030]    Where {dot over (Θ)}τ is small, 
         [0000]    
       
         
           
             
               
                 sin 
                  
                 
                     
                 
                  
                 
                   Θ 
                   . 
                 
                  
                 τ 
               
               
                 
                   Θ 
                   . 
                 
                  
                 τ 
               
             
             ≈ 
             1. 
           
         
       
     
         [0000]    The variances of the component velocities can now be found. 
         [0000]      Var[X]=E[X 2 ]−( E[X] ) 2    Eq. 35
 
         [0000]    Thus, from equations 27, 28, 33, 34, 
         [0000]    
       
         
           
             
               
                 
                   
                     Var 
                      
                     
                       [ 
                       
                         
                           x 
                           . 
                         
                         1 
                       
                       ] 
                     
                   
                   = 
                   
                     
                       
                         E 
                          
                         
                           [ 
                           
                             
                               x 
                               . 
                             
                             1 
                             2 
                           
                           ] 
                         
                       
                       - 
                       
                         
                           ( 
                           
                             E 
                              
                             
                               [ 
                               
                                 
                                   x 
                                   . 
                                 
                                 1 
                               
                               ] 
                             
                           
                           ) 
                         
                         2 
                       
                     
                     = 
                     
                       
                         
                           E 
                            
                           
                             [ 
                             
                               s 
                               1 
                               2 
                             
                             ] 
                           
                         
                          
                         
                           E 
                            
                           
                             [ 
                             
                               
                                 sin 
                                 2 
                               
                                
                               
                                 θ 
                                 1 
                               
                             
                             ] 
                           
                         
                       
                       - 
                       
                         
                           ( 
                           
                             
                               s 
                               0 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               0 
                             
                              
                             
                               
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 
                                   Θ 
                                   . 
                                 
                                  
                                 τ 
                               
                               
                                 
                                   Θ 
                                   . 
                                 
                                  
                                 τ 
                               
                             
                           
                           ) 
                         
                         2 
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   36 
                 
               
             
             
               
                 
                   
                     Var 
                      
                     
                       [ 
                       
                         
                           y 
                           . 
                         
                         1 
                       
                       ] 
                     
                   
                   = 
                   
                     
                       
                         E 
                          
                         
                           [ 
                           
                             
                               y 
                               . 
                             
                             1 
                             2 
                           
                           ] 
                         
                       
                       - 
                       
                         
                           ( 
                           
                             E 
                              
                             
                               [ 
                               
                                 
                                   y 
                                   . 
                                 
                                 1 
                               
                               ] 
                             
                           
                           ) 
                         
                         2 
                       
                     
                     = 
                     
                       
                         
                           E 
                            
                           
                             [ 
                             
                               s 
                               1 
                               2 
                             
                             ] 
                           
                         
                          
                         
                           E 
                            
                           
                             [ 
                             
                               
                                 cos 
                                 2 
                               
                                
                               
                                 θ 
                                 1 
                               
                             
                             ] 
                           
                         
                       
                       - 
                       
                         
                           ( 
                           
                             
                               s 
                               0 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               0 
                             
                              
                             
                               
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 
                                   Θ 
                                   . 
                                 
                                  
                                 τ 
                               
                               
                                 
                                   Θ 
                                   . 
                                 
                                  
                                 τ 
                               
                             
                           
                           ) 
                         
                         2 
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   37 
                 
               
             
           
         
       
     
         [0000]    Using equation E[s 1   2 ]=Var[s 1 ]+(E[s 1 ]) 2 , so from equation 19 and 21, 
         [0000]        E[s   1   2 ]=τ 2 σ s   2   +s   0   2    Eq. 38
 
         [0000]    E[sin 2  θ 1 ], and E[cos 2  θ 1 ] can be found using the definition of expected value and Wolfram Mathematica Online Integrator. 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           E 
                            
                           
                             [ 
                             
                               
                                 sin 
                                 2 
                               
                                
                               
                                 θ 
                                 1 
                               
                             
                             ] 
                           
                         
                         = 
                           
                          
                         
                           
                             ∫ 
                             
                               - 
                               
                                 Θ 
                                 . 
                               
                             
                             
                               + 
                               
                                 Θ 
                                 . 
                               
                             
                           
                            
                           
                             
                               ( 
                               
                                 1 
                                 
                                   2 
                                    
                                   
                                       
                                   
                                    
                                   
                                     Θ 
                                     . 
                                   
                                 
                               
                               ) 
                             
                              
                             
                               
                                 sin 
                                 2 
                               
                                
                               
                                 ( 
                                 
                                   
                                     θ 
                                     0 
                                   
                                   + 
                                   
                                     
                                       θ 
                                       . 
                                     
                                      
                                     τ 
                                   
                                 
                                 ) 
                               
                             
                              
                             
                                
                               
                                 θ 
                                 . 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               
                                 2 
                                  
                                 
                                   ( 
                                   
                                     
                                       θ 
                                       0 
                                     
                                     + 
                                     
                                       
                                         θ 
                                         . 
                                       
                                        
                                       τ 
                                     
                                   
                                   ) 
                                 
                               
                               - 
                               
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 2 
                                  
                                 
                                   ( 
                                   
                                     
                                       θ 
                                       0 
                                     
                                     + 
                                     
                                       
                                         θ 
                                         . 
                                       
                                        
                                       τ 
                                     
                                   
                                   ) 
                                 
                               
                             
                             
                               8 
                                
                               
                                   
                               
                                
                               
                                 Θ 
                                 . 
                               
                                
                               τ 
                             
                           
                            
                           
                             | 
                             
                               - 
                               
                                 Θ 
                                 . 
                               
                             
                             
                               + 
                               
                                 Θ 
                                 . 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               
                                 
                                   
                                     2 
                                      
                                     
                                       ( 
                                       
                                         
                                           θ 
                                           0 
                                         
                                         + 
                                         
                                           
                                             Θ 
                                             . 
                                           
                                            
                                           τ 
                                         
                                       
                                       ) 
                                     
                                   
                                   - 
                                   
                                     sin 
                                      
                                     
                                         
                                     
                                      
                                     2 
                                      
                                     
                                       ( 
                                       
                                         
                                           θ 
                                           0 
                                         
                                         + 
                                         
                                           
                                             Θ 
                                             . 
                                           
                                            
                                           τ 
                                         
                                       
                                       ) 
                                     
                                   
                                   - 
                                   
                                     2 
                                      
                                     
                                       ( 
                                       
                                         
                                           θ 
                                           0 
                                         
                                         - 
                                         
                                           
                                             Θ 
                                             . 
                                           
                                            
                                           τ 
                                         
                                       
                                       ) 
                                     
                                   
                                   + 
                                 
                               
                             
                             
                               
                                 
                                   sin 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                    
                                   
                                     ( 
                                     
                                       
                                         θ 
                                         0 
                                       
                                       - 
                                       
                                         
                                           Θ 
                                           . 
                                         
                                          
                                         τ 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                           
                             8 
                              
                             
                                 
                             
                              
                             
                               Θ 
                               . 
                             
                              
                             τ 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             1 
                             
                               8 
                                
                               
                                   
                               
                                
                               
                                 Θ 
                                 . 
                               
                                
                               τ 
                             
                           
                            
                           
                             [ 
                             
                               
                                 4 
                                  
                                 
                                   Θ 
                                   . 
                                 
                                  
                                 τ 
                               
                               - 
                               
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 2 
                                  
                                 
                                   ( 
                                   
                                     
                                       θ 
                                       0 
                                     
                                     + 
                                     
                                       
                                         Θ 
                                         . 
                                       
                                        
                                       τ 
                                     
                                   
                                   ) 
                                 
                               
                               + 
                               
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 2 
                                  
                                 
                                   ( 
                                   
                                     
                                       θ 
                                       0 
                                     
                                     - 
                                     
                                       
                                         Θ 
                                         . 
                                       
                                        
                                       τ 
                                     
                                   
                                   ) 
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             1 
                             2 
                           
                           + 
                           
                             
                               
                                 
                                   
                                     
                                       
                                         - 
                                         sin 
                                       
                                        
                                       
                                           
                                       
                                        
                                       2 
                                        
                                       
                                         θ 
                                         0 
                                       
                                        
                                       cos 
                                        
                                       
                                           
                                       
                                        
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       
                                         Θ 
                                         . 
                                       
                                        
                                       τ 
                                     
                                     - 
                                     
                                       cos 
                                        
                                       
                                           
                                       
                                        
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       
                                         θ 
                                         0 
                                       
                                        
                                       sin 
                                        
                                       
                                           
                                       
                                        
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       
                                         Θ 
                                         . 
                                       
                                        
                                       τ 
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       sin 
                                        
                                       
                                           
                                       
                                        
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       
                                         θ 
                                         0 
                                       
                                        
                                       cos 
                                        
                                       
                                           
                                       
                                        
                                       2 
                                        
                                       
                                         Θ 
                                         . 
                                       
                                        
                                       τ 
                                     
                                     - 
                                     
                                       cos 
                                        
                                       
                                           
                                       
                                        
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       
                                         θ 
                                         0 
                                       
                                        
                                       sin 
                                        
                                       
                                           
                                       
                                        
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       
                                         Θ 
                                         . 
                                       
                                        
                                       τ 
                                     
                                   
                                 
                               
                             
                             
                               8 
                                
                               
                                   
                               
                                
                               
                                 Θ 
                                 . 
                               
                                
                               τ 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             1 
                             2 
                           
                           - 
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               2 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 0 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               2 
                                
                               
                                 Θ 
                                 . 
                               
                                
                               τ 
                             
                             
                               4 
                                
                               
                                   
                               
                                
                               
                                 Θτ 
                                 . 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             1 
                             2 
                           
                            
                           
                             ( 
                             
                               1 
                               - 
                               
                                 
                                   
                                     sin 
                                      
                                     
                                         
                                     
                                      
                                     2 
                                      
                                     
                                         
                                     
                                      
                                     
                                       Θ 
                                       . 
                                     
                                      
                                     τ 
                                   
                                   
                                     2 
                                      
                                     
                                       Θ 
                                       . 
                                     
                                      
                                     τ 
                                   
                                 
                                  
                                 cos 
                                  
                                 
                                     
                                 
                                  
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 
                                   θ 
                                   0 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   39 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           E 
                            
                           
                             [ 
                             
                               
                                 cos 
                                 2 
                               
                                
                               
                                 θ 
                                 1 
                               
                             
                             ] 
                           
                         
                         = 
                           
                          
                         
                           
                             ∫ 
                             
                               - 
                               
                                 Θ 
                                 . 
                               
                             
                             
                               + 
                               
                                 Θ 
                                 . 
                               
                             
                           
                            
                           
                             
                               ( 
                               
                                 1 
                                 
                                   2 
                                    
                                   
                                       
                                   
                                    
                                   
                                     Θ 
                                     . 
                                   
                                 
                               
                               ) 
                             
                              
                             
                               
                                 cos 
                                 2 
                               
                                
                               
                                 ( 
                                 
                                   
                                     θ 
                                     0 
                                   
                                   + 
                                   
                                     
                                       θ 
                                       . 
                                     
                                      
                                     τ 
                                   
                                 
                                 ) 
                               
                             
                              
                             
                                
                               
                                 θ 
                                 . 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               
                                 2 
                                  
                                 
                                   ( 
                                   
                                     
                                       θ 
                                       0 
                                     
                                     + 
                                     
                                       
                                         θ 
                                         . 
                                       
                                        
                                       τ 
                                     
                                   
                                   ) 
                                 
                               
                               + 
                               
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 2 
                                  
                                 
                                   ( 
                                   
                                     
                                       θ 
                                       0 
                                     
                                     + 
                                     
                                       
                                         θ 
                                         . 
                                       
                                        
                                       τ 
                                     
                                   
                                   ) 
                                 
                               
                             
                             
                               8 
                                
                               
                                   
                               
                                
                               
                                 Θ 
                                 . 
                               
                                
                               τ 
                             
                           
                            
                           
                             | 
                             
                               - 
                               
                                 Θ 
                                 . 
                               
                             
                             
                               + 
                               
                                 Θ 
                                 . 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     
                                       2 
                                        
                                       
                                         ( 
                                         
                                           
                                             θ 
                                             0 
                                           
                                           + 
                                           
                                             
                                               Θ 
                                               . 
                                             
                                              
                                             τ 
                                           
                                         
                                         ) 
                                       
                                     
                                     + 
                                     
                                       sin 
                                        
                                       
                                           
                                       
                                        
                                       2 
                                        
                                       
                                         ( 
                                         
                                           
                                             θ 
                                             0 
                                           
                                           + 
                                           
                                             
                                               Θ 
                                               . 
                                             
                                              
                                             τ 
                                           
                                         
                                         ) 
                                       
                                     
                                     - 
                                     
                                       2 
                                        
                                       
                                         ( 
                                         
                                           
                                             θ 
                                             0 
                                           
                                           - 
                                           
                                             
                                               Θ 
                                               . 
                                             
                                              
                                             τ 
                                           
                                         
                                         ) 
                                       
                                     
                                     - 
                                   
                                 
                               
                               
                                 
                                   
                                     sin 
                                      
                                     
                                         
                                     
                                      
                                     2 
                                      
                                     
                                       ( 
                                       
                                         
                                           θ 
                                           0 
                                         
                                         - 
                                         
                                           
                                             Θ 
                                             . 
                                           
                                            
                                           τ 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                             ] 
                           
                           
                             8 
                              
                             
                                 
                             
                              
                             
                               Θ 
                               . 
                             
                              
                             τ 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             1 
                             
                               8 
                                
                               
                                   
                               
                                
                               
                                 Θ 
                                 . 
                               
                                
                               τ 
                             
                           
                            
                           
                             [ 
                             
                               
                                 4 
                                  
                                 
                                   Θ 
                                   . 
                                 
                                  
                                 τ 
                               
                               + 
                               
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 2 
                                  
                                 
                                   ( 
                                   
                                     
                                       θ 
                                       0 
                                     
                                     + 
                                     
                                       
                                         Θ 
                                         . 
                                       
                                        
                                       τ 
                                     
                                   
                                   ) 
                                 
                               
                               - 
                               
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 2 
                                  
                                 
                                   ( 
                                   
                                     
                                       θ 
                                       0 
                                     
                                     - 
                                     
                                       
                                         Θ 
                                         . 
                                       
                                        
                                       τ 
                                     
                                   
                                   ) 
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             1 
                             2 
                           
                           + 
                           
                             
                               
                                 
                                   
                                     
                                       sin 
                                        
                                       
                                           
                                       
                                        
                                       2 
                                        
                                       
                                         θ 
                                         0 
                                       
                                        
                                       cos 
                                        
                                       
                                           
                                       
                                        
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       
                                         Θ 
                                         . 
                                       
                                        
                                       τ 
                                     
                                     + 
                                     
                                       cos 
                                        
                                       
                                           
                                       
                                        
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       
                                         θ 
                                         0 
                                       
                                        
                                       sin 
                                        
                                       
                                           
                                       
                                        
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       
                                         Θ 
                                         . 
                                       
                                        
                                       τ 
                                     
                                     - 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       sin 
                                        
                                       
                                           
                                       
                                        
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       
                                         θ 
                                         0 
                                       
                                        
                                       cos 
                                        
                                       
                                           
                                       
                                        
                                       2 
                                        
                                       
                                         Θ 
                                         . 
                                       
                                        
                                       τ 
                                     
                                     + 
                                     
                                       cos 
                                        
                                       
                                           
                                       
                                        
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       
                                         θ 
                                         0 
                                       
                                        
                                       sin 
                                        
                                       
                                           
                                       
                                        
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       
                                         Θ 
                                         . 
                                       
                                        
                                       τ 
                                     
                                   
                                 
                               
                             
                             
                               8 
                                
                               
                                   
                               
                                
                               
                                 Θ 
                                 . 
                               
                                
                               τ 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             1 
                             2 
                           
                           + 
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               2 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 0 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               2 
                                
                               
                                 Θ 
                                 . 
                               
                                
                               τ 
                             
                             
                               4 
                                
                               
                                   
                               
                                
                               
                                 Θτ 
                                 . 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             1 
                             2 
                           
                            
                           
                             ( 
                             
                               1 
                               + 
                               
                                 
                                   
                                     sin 
                                      
                                     
                                         
                                     
                                      
                                     2 
                                      
                                     
                                         
                                     
                                      
                                     
                                       Θ 
                                       . 
                                     
                                      
                                     τ 
                                   
                                   
                                     2 
                                      
                                     
                                       Θ 
                                       . 
                                     
                                      
                                     τ 
                                   
                                 
                                  
                                 cos 
                                  
                                 
                                     
                                 
                                  
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 
                                   θ 
                                   0 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   40 
                 
               
             
           
         
       
     
         [0000]    Therefore, using equations 36-40, 
         [0000]    
       
         
           
             
               
                 
                   
                     Var 
                      
                     
                       [ 
                       
                         
                           x 
                           . 
                         
                         1 
                       
                       ] 
                     
                   
                   = 
                   
                     
                       
                         ( 
                         
                           
                             
                               τ 
                               2 
                             
                              
                             
                               σ 
                               
                                 s 
                                 . 
                               
                               2 
                             
                           
                           + 
                           
                             s 
                             0 
                             2 
                           
                         
                         ) 
                       
                        
                       
                         ( 
                         
                           
                             1 
                             2 
                           
                           - 
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               2 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 0 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               2 
                                
                               
                                   
                               
                                
                               
                                 Θ 
                                 . 
                               
                                
                               τ 
                             
                             
                               4 
                                
                               
                                 Θ 
                                 . 
                               
                                
                               τ 
                             
                           
                         
                         ) 
                       
                     
                     - 
                     
                       
                         ( 
                         
                           
                             s 
                             0 
                           
                            
                           sin 
                            
                           
                               
                           
                            
                           
                             θ 
                             0 
                           
                            
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 Θ 
                                 . 
                               
                                
                               τ 
                             
                             
                               
                                 Θ 
                                 . 
                               
                                
                               τ 
                             
                           
                         
                         ) 
                       
                       2 
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   41 
                 
               
             
             
               
                 
                   
                     Var 
                      
                     
                       [ 
                       
                         
                           y 
                           . 
                         
                         1 
                       
                       ] 
                     
                   
                   = 
                   
                     
                       
                         ( 
                         
                           
                             
                               τ 
                               2 
                             
                              
                             
                               σ 
                               
                                 s 
                                 . 
                               
                               2 
                             
                           
                           + 
                           
                             s 
                             0 
                             2 
                           
                         
                         ) 
                       
                        
                       
                         ( 
                         
                           
                             1 
                             2 
                           
                           + 
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               2 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 0 
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               2 
                                
                               
                                   
                               
                                
                               
                                 Θ 
                                 . 
                               
                                
                               τ 
                             
                             
                               4 
                                
                               
                                 Θ 
                                 . 
                               
                                
                               τ 
                             
                           
                         
                         ) 
                       
                     
                     - 
                     
                       
                         ( 
                         
                           
                             s 
                             0 
                           
                            
                           cos 
                            
                           
                               
                           
                            
                           
                             θ 
                             0 
                           
                            
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 Θ 
                                 . 
                               
                                
                               τ 
                             
                             
                               
                                 Θ 
                                 . 
                               
                                
                               τ 
                             
                           
                         
                         ) 
                       
                       2 
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   42 
                 
               
             
           
         
       
     
         [0031]    These equations emphasize how different the CGBF prediction model is from the one in the Kalman filter. To gain some appreciation for these equations, consider the case when θ 0 =0, i.e., the target&#39;s course is due north. Then, (41) and (42) become: 
         [0000]    
       
         
           
             
               
                 
                   
                     Var 
                      
                     
                       [ 
                       
                         
                           x 
                           . 
                         
                         1 
                       
                       ] 
                     
                   
                   = 
                   
                     
                       ( 
                       
                         
                           
                             τ 
                             2 
                           
                            
                           
                             σ 
                             
                               s 
                               . 
                             
                             2 
                           
                         
                         + 
                         
                           s 
                           0 
                           2 
                         
                       
                       ) 
                     
                      
                     
                       ( 
                       
                         
                           1 
                           2 
                         
                         - 
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             2 
                              
                             
                               Θ 
                               . 
                             
                              
                             τ 
                           
                           
                             4 
                              
                             
                                 
                             
                              
                             
                               Θτ 
                               . 
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   43 
                 
               
             
             
               
                 
                   
                     Var 
                      
                     
                       [ 
                       
                         
                           y 
                           . 
                         
                         1 
                       
                       ] 
                     
                   
                   = 
                   
                     
                       
                         ( 
                         
                           
                             
                               τ 
                               2 
                             
                              
                             
                               σ 
                               
                                 s 
                                 . 
                               
                               2 
                             
                           
                           + 
                           
                             s 
                             0 
                             2 
                           
                         
                         ) 
                       
                        
                       
                         ( 
                         
                           
                             1 
                             2 
                           
                           + 
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               2 
                                
                               
                                 Θ 
                                 . 
                               
                                
                               τ 
                             
                             
                               4 
                                
                               
                                   
                               
                                
                               
                                 Θτ 
                                 . 
                               
                             
                           
                         
                         ) 
                       
                     
                     - 
                     
                       
                         ( 
                         
                           
                             s 
                             0 
                           
                            
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 Θ 
                                 . 
                               
                                
                               τ 
                             
                             
                               
                                 Θ 
                                 . 
                               
                                
                               τ 
                             
                           
                         
                         ) 
                       
                       2 
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   44 
                 
               
             
           
         
       
     
         [0000]    As {dot over (Θ)}τ goes to zero, i.e., {dot over (Θ)}τ→0, the ration 
         [0000]    
       
         
           
             
               
                 
                   sin 
                    
                   
                       
                   
                    
                   
                     Θ 
                     . 
                   
                    
                   τ 
                 
                 
                   
                     Θ 
                     . 
                   
                    
                   τ 
                 
               
               -&gt; 
               1 
             
             , 
           
         
       
     
         [0000]    so: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       Var 
                        
                       
                         [ 
                         
                           
                             x 
                             . 
                           
                           1 
                         
                         ] 
                       
                     
                     -&gt; 
                     
                       
                         ( 
                         
                           
                             
                               τ 
                               2 
                             
                              
                             
                               σ 
                               
                                 s 
                                 . 
                               
                               2 
                             
                           
                           + 
                           
                             s 
                             0 
                             2 
                           
                         
                         ) 
                       
                        
                       
                         ( 
                         
                           
                             1 
                             2 
                           
                           - 
                           
                             1 
                             2 
                           
                         
                         ) 
                       
                     
                   
                   = 
                   0 
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   45 
                 
               
             
             
               
                 
                   
                     
                       Var 
                        
                       
                         [ 
                         
                           
                             y 
                             . 
                           
                           1 
                         
                         ] 
                       
                     
                     -&gt; 
                     
                       
                         
                           ( 
                           
                             
                               
                                 τ 
                                 2 
                               
                                
                               
                                 σ 
                                 
                                   s 
                                   . 
                                 
                                 2 
                               
                             
                             + 
                             
                               s 
                               0 
                               2 
                             
                           
                           ) 
                         
                          
                         
                           ( 
                           
                             
                               1 
                               2 
                             
                             + 
                             
                               1 
                               2 
                             
                           
                           ) 
                         
                       
                       - 
                       
                         
                           ( 
                           
                             s 
                             0 
                           
                           ) 
                         
                         2 
                       
                     
                   
                   = 
                   
                     
                       τ 
                       2 
                     
                      
                     
                       σ 
                       
                         s 
                         . 
                       
                       2 
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   46 
                 
               
             
           
         
       
     
         [0032]    The variance in the x direction should go to zero because when {dot over (Θ)}τ=0, the target is (only) moving exactly along its initial course, so there is no uncertainty in course. The resulting variance in the y direction (i.e., the variance in position uncertainty along the target&#39;s initial course) agrees with the process model for the Kalman filter. Thus, the CGBF uses a directional process model. 
       The Mean and Variance of the Position  
       [0033]    The mean and variance of the position can also be determined. The mean of the position is considered first. Without any loss of generality, assume the target is initially moving due north. This assumption does not limit the generality because the distribution is being computed along the initial course of the target. Thus, any other initial course is just a rotation of the mean position to align with the target&#39;s initial course. With this assumption, 
         [0000]      θ 0 =0.
 
         [0034]    The distribution must be symmetric about the initial course since the target can turn left just as easily as turning right. Since the target is assumed to have an initial course of zero, i.e., due north, the distribution must be symmetric about the y-axis. Therefore, E[x 1 ]=x 0 . However, this result will be proven. The E[x 1 ] is computed from the modified position Eq. 10 and 11. 
         [0000]    
       
         
           
             
               
                 
                   
                     E 
                      
                     
                       [ 
                       
                         x 
                         1 
                       
                       ] 
                     
                   
                   = 
                   
                     E 
                     [ 
                     
                       
                         x 
                         0 
                       
                       + 
                       
                         { 
                         
                           
                             
                               
                                 
                                   1 
                                   
                                     
                                       θ 
                                       . 
                                     
                                     2 
                                   
                                 
                                  
                                 
                                   [ 
                                   
                                     
                                       
                                         s 
                                         . 
                                       
                                        
                                       sin 
                                        
                                       
                                           
                                       
                                        
                                       
                                         θ 
                                         1 
                                       
                                     
                                     - 
                                     
                                       
                                         s 
                                         1 
                                       
                                        
                                       
                                         θ 
                                         . 
                                       
                                        
                                       cos 
                                        
                                       
                                           
                                       
                                        
                                       
                                         θ 
                                         1 
                                       
                                     
                                     - 
                                     
                                       
                                         s 
                                         . 
                                       
                                        
                                       sin 
                                        
                                       
                                           
                                       
                                        
                                       
                                         θ 
                                         0 
                                       
                                     
                                     + 
                                     
                                       
                                         s 
                                         0 
                                       
                                        
                                       
                                         θ 
                                         . 
                                       
                                        
                                       cos 
                                        
                                       
                                           
                                       
                                        
                                       
                                         θ 
                                         0 
                                       
                                     
                                   
                                   ] 
                                 
                               
                             
                             
                               
                                 
                                   if 
                                    
                                   
                                       
                                   
                                    
                                   
                                      
                                     
                                       θ 
                                       . 
                                     
                                      
                                   
                                 
                                 &gt; 
                                 
                                   ε 
                                   
                                     θ 
                                     . 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   ( 
                                   
                                     
                                       
                                         s 
                                         0 
                                       
                                        
                                       τ 
                                     
                                     + 
                                     
                                       
                                         1 
                                         2 
                                       
                                        
                                       
                                         s 
                                         . 
                                       
                                        
                                       
                                         τ 
                                         2 
                                       
                                     
                                   
                                   ) 
                                 
                                  
                                 
                                   sin 
                                    
                                   
                                     ( 
                                     
                                       
                                         θ 
                                         0 
                                       
                                       + 
                                       
                                         
                                           θ 
                                           . 
                                         
                                          
                                         τ 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                             
                               otherwise 
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   47 
                 
               
             
             
               
                 
                   
                     E 
                      
                     
                       [ 
                       
                         y 
                         1 
                       
                       ] 
                     
                   
                   = 
                   
                     E 
                     [ 
                     
                       
                         y 
                         0 
                       
                       + 
                       
                         { 
                         
                           
                             
                               
                                 
                                   1 
                                   
                                     
                                       θ 
                                       . 
                                     
                                     2 
                                   
                                 
                                  
                                 
                                   [ 
                                   
                                     
                                       
                                         s 
                                         . 
                                       
                                        
                                       cos 
                                        
                                       
                                           
                                       
                                        
                                       
                                         θ 
                                         1 
                                       
                                     
                                     + 
                                     
                                       
                                         s 
                                         1 
                                       
                                        
                                       
                                         θ 
                                         . 
                                       
                                        
                                       sin 
                                        
                                       
                                           
                                       
                                        
                                       
                                         θ 
                                         1 
                                       
                                     
                                     - 
                                     
                                       
                                         s 
                                         . 
                                       
                                        
                                       cos 
                                        
                                       
                                           
                                       
                                        
                                       
                                         θ 
                                         0 
                                       
                                     
                                     - 
                                     
                                       
                                         s 
                                         0 
                                       
                                        
                                       
                                         θ 
                                         . 
                                       
                                        
                                       sin 
                                        
                                       
                                           
                                       
                                        
                                       
                                         θ 
                                         0 
                                       
                                     
                                   
                                   ] 
                                 
                               
                             
                             
                               
                                 
                                   if 
                                    
                                   
                                       
                                   
                                    
                                   
                                      
                                     
                                       θ 
                                       . 
                                     
                                      
                                   
                                 
                                 &gt; 
                                 
                                   ε 
                                   
                                     θ 
                                     . 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   ( 
                                   
                                     
                                       
                                         s 
                                         0 
                                       
                                        
                                       τ 
                                     
                                     + 
                                     
                                       
                                         1 
                                         2 
                                       
                                        
                                       
                                         s 
                                         . 
                                       
                                        
                                       
                                         τ 
                                         2 
                                       
                                     
                                   
                                   ) 
                                 
                                  
                                 
                                   cos 
                                    
                                   
                                     ( 
                                     
                                       
                                         θ 
                                         0 
                                       
                                       + 
                                       
                                         
                                           θ 
                                           . 
                                         
                                          
                                         τ 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                             
                               otherwise 
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   48 
                 
               
             
           
         
       
     
         [0000]    As before, assume the target is initially moving due north, so θ 0 =0. With this simplification: 
         [0000]    
       
         
           
             
               
                 
                   
                     E 
                      
                     
                       [ 
                       
                         x 
                         1 
                       
                       ] 
                     
                   
                   = 
                   
                     E 
                     [ 
                     
                       
                         x 
                         0 
                       
                       + 
                       
                         { 
                         
                           
                             
                               
                                 
                                   1 
                                   
                                     
                                       θ 
                                       . 
                                     
                                     2 
                                   
                                 
                                  
                                 
                                   [ 
                                   
                                     
                                       
                                         s 
                                         . 
                                       
                                        
                                       sin 
                                        
                                       
                                           
                                       
                                        
                                       
                                         θ 
                                         . 
                                       
                                        
                                       τ 
                                     
                                     - 
                                     
                                       
                                         s 
                                         1 
                                       
                                        
                                       
                                         θ 
                                         . 
                                       
                                        
                                       cos 
                                        
                                       
                                           
                                       
                                        
                                       
                                         θ 
                                         . 
                                       
                                        
                                       τ 
                                     
                                     + 
                                     
                                       
                                         s 
                                         0 
                                       
                                        
                                       
                                         θ 
                                         . 
                                       
                                     
                                   
                                   ] 
                                 
                               
                             
                             
                               
                                 
                                   if 
                                    
                                   
                                       
                                   
                                    
                                   
                                      
                                     
                                       θ 
                                       . 
                                     
                                      
                                   
                                 
                                 &gt; 
                                 
                                   ε 
                                   
                                     θ 
                                     . 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   ( 
                                   
                                     
                                       
                                         s 
                                         0 
                                       
                                        
                                       τ 
                                     
                                     + 
                                     
                                       
                                         1 
                                         2 
                                       
                                        
                                       
                                         s 
                                         . 
                                       
                                        
                                       
                                         τ 
                                         2 
                                       
                                     
                                   
                                   ) 
                                 
                                  
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 
                                   θ 
                                   . 
                                 
                                  
                                 τ 
                               
                             
                             
                               otherwise 
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   49 
                 
               
             
             
               
                 
                   
                     E 
                      
                     
                       [ 
                       
                         y 
                         1 
                       
                       ] 
                     
                   
                   = 
                   
                     E 
                     [ 
                     
                       
                         y 
                         0 
                       
                       + 
                       
                         { 
                         
                           
                             
                               
                                 
                                   1 
                                   
                                     
                                       θ 
                                       . 
                                     
                                     2 
                                   
                                 
                                  
                                 
                                   [ 
                                   
                                     
                                       
                                         s 
                                         . 
                                       
                                        
                                       cos 
                                        
                                       
                                           
                                       
                                        
                                       
                                         θ 
                                         . 
                                       
                                        
                                       τ 
                                     
                                     + 
                                     
                                       
                                         s 
                                         1 
                                       
                                        
                                       
                                         θ 
                                         . 
                                       
                                        
                                       sin 
                                        
                                       
                                           
                                       
                                        
                                       
                                         θ 
                                         . 
                                       
                                        
                                       τ 
                                     
                                     - 
                                     
                                       s 
                                       . 
                                     
                                   
                                   ] 
                                 
                               
                             
                             
                               
                                 
                                   if 
                                    
                                   
                                       
                                   
                                    
                                   
                                      
                                     
                                       θ 
                                       . 
                                     
                                      
                                   
                                 
                                 &gt; 
                                 
                                   ε 
                                   
                                     θ 
                                     . 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   ( 
                                   
                                     
                                       
                                         s 
                                         0 
                                       
                                        
                                       τ 
                                     
                                     + 
                                     
                                       
                                         1 
                                         2 
                                       
                                        
                                       
                                         s 
                                         . 
                                       
                                        
                                       
                                         τ 
                                         2 
                                       
                                     
                                   
                                   ) 
                                 
                                  
                                 cos 
                                  
                                 
                                     
                                 
                                  
                                 
                                   θ 
                                   . 
                                 
                                  
                                 τ 
                               
                             
                             
                               otherwise 
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   50 
                 
               
             
           
         
       
     
         [0000]    And again assuming {dot over (s)} and {dot over (θ)} are independent random variables, 
         [0000]    
       
         
           
             
               
                 
                   
                     E 
                      
                     
                       [ 
                       
                         x 
                         1 
                       
                       ] 
                     
                   
                   = 
                   
                     
                       x 
                       0 
                     
                     + 
                     
                       { 
                       
                         
                           
                             
                               
                                 
                                   E 
                                    
                                   
                                     [ 
                                     
                                       s 
                                       . 
                                     
                                     ] 
                                   
                                 
                                  
                                 
                                   E 
                                    
                                   
                                     [ 
                                     
                                       
                                         sin 
                                          
                                         
                                             
                                         
                                          
                                         
                                           θ 
                                           . 
                                         
                                          
                                         τ 
                                       
                                       
                                         
                                           θ 
                                           . 
                                         
                                         2 
                                       
                                     
                                     ] 
                                   
                                 
                               
                               - 
                               
                                 
                                   E 
                                    
                                   
                                     [ 
                                     
                                       s 
                                       1 
                                     
                                     ] 
                                   
                                 
                                  
                                 
                                   E 
                                    
                                   
                                     [ 
                                     
                                       
                                         cos 
                                          
                                         
                                             
                                         
                                          
                                         
                                           θ 
                                           . 
                                         
                                          
                                         τ 
                                       
                                       
                                         θ 
                                         . 
                                       
                                     
                                     ] 
                                   
                                 
                               
                               + 
                               
                                 
                                   s 
                                   0 
                                 
                                  
                                 
                                   E 
                                    
                                   
                                     [ 
                                     
                                       1 
                                       
                                         θ 
                                         . 
                                       
                                     
                                     ] 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 if 
                                  
                                 
                                     
                                 
                                  
                                 
                                    
                                   
                                     θ 
                                     . 
                                   
                                    
                                 
                               
                               &gt; 
                               
                                 ε 
                                 
                                   θ 
                                   . 
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 E 
                                  
                                 
                                   [ 
                                   
                                     
                                       
                                         s 
                                         0 
                                       
                                        
                                       τ 
                                     
                                     + 
                                     
                                       
                                         1 
                                         2 
                                       
                                        
                                       
                                         s 
                                         . 
                                       
                                        
                                       
                                         τ 
                                         2 
                                       
                                     
                                   
                                   ] 
                                 
                               
                                
                               
                                 E 
                                  
                                 
                                   [ 
                                   
                                     sin 
                                      
                                     
                                         
                                     
                                      
                                     
                                       θ 
                                       . 
                                     
                                      
                                     τ 
                                   
                                   ] 
                                 
                               
                             
                           
                           
                             otherwise 
                           
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   51 
                 
               
             
             
               
                 
                   
                     E 
                      
                     
                       [ 
                       
                         y 
                         1 
                       
                       ] 
                     
                   
                   = 
                   
                     
                       y 
                       0 
                     
                     + 
                     
                       { 
                       
                         
                           
                             
                               
                                 
                                   E 
                                    
                                   
                                     [ 
                                     
                                       s 
                                       . 
                                     
                                     ] 
                                   
                                 
                                  
                                 
                                   E 
                                    
                                   
                                     [ 
                                     
                                       
                                         cos 
                                          
                                         
                                             
                                         
                                          
                                         
                                           θ 
                                           . 
                                         
                                          
                                         τ 
                                       
                                       
                                         
                                           θ 
                                           . 
                                         
                                         2 
                                       
                                     
                                     ] 
                                   
                                 
                               
                               + 
                               
                                 
                                   E 
                                    
                                   
                                     [ 
                                     
                                       s 
                                       1 
                                     
                                     ] 
                                   
                                 
                                  
                                 
                                   E 
                                    
                                   
                                     [ 
                                     
                                       
                                         sin 
                                          
                                         
                                             
                                         
                                          
                                         
                                           θ 
                                           . 
                                         
                                          
                                         τ 
                                       
                                       
                                         θ 
                                         . 
                                       
                                     
                                     ] 
                                   
                                 
                               
                               - 
                               
                                 
                                   E 
                                    
                                   
                                     [ 
                                     
                                       s 
                                       . 
                                     
                                     ] 
                                   
                                 
                                  
                                 
                                   E 
                                    
                                   
                                     [ 
                                     
                                       1 
                                       
                                         
                                           θ 
                                           . 
                                         
                                         2 
                                       
                                     
                                     ] 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 if 
                                  
                                 
                                     
                                 
                                  
                                 
                                    
                                   
                                     θ 
                                     . 
                                   
                                    
                                 
                               
                               &gt; 
                               
                                 ε 
                                 
                                   θ 
                                   . 
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 E 
                                  
                                 
                                   [ 
                                   
                                     
                                       
                                         s 
                                         0 
                                       
                                        
                                       τ 
                                     
                                     + 
                                     
                                       
                                         1 
                                         2 
                                       
                                        
                                       
                                         s 
                                         . 
                                       
                                        
                                       
                                         τ 
                                         2 
                                       
                                     
                                   
                                   ] 
                                 
                               
                                
                               
                                 E 
                                  
                                 
                                   [ 
                                   
                                     cos 
                                      
                                     
                                         
                                     
                                      
                                     
                                       θ 
                                       . 
                                     
                                      
                                     τ 
                                   
                                   ] 
                                 
                               
                             
                           
                           
                             otherwise 
                           
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   52 
                 
               
             
           
         
       
     
         [0000]    Since E[{dot over (s)}]=0, 
         [0000]    
       
         
           
             
               
                 
                   
                     E 
                      
                     
                       [ 
                       
                         x 
                         1 
                       
                       ] 
                     
                   
                   = 
                   
                     
                       x 
                       0 
                     
                     + 
                     
                       { 
                       
                         
                           
                             
                               
                                 
                                   
                                     - 
                                     
                                       s 
                                       0 
                                     
                                   
                                    
                                   
                                     E 
                                      
                                     
                                       [ 
                                       
                                         
                                           cos 
                                            
                                           
                                               
                                           
                                            
                                           
                                             θ 
                                             . 
                                           
                                            
                                           τ 
                                         
                                         
                                           θ 
                                           . 
                                         
                                       
                                       ] 
                                     
                                   
                                 
                                 + 
                                 
                                   
                                     s 
                                     0 
                                   
                                    
                                   
                                     E 
                                      
                                     
                                       [ 
                                       
                                         1 
                                         
                                           θ 
                                           . 
                                         
                                       
                                       ] 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   if 
                                    
                                   
                                      
                                     
                                       θ 
                                       . 
                                     
                                      
                                   
                                 
                                 &gt; 
                                 
                                   ε 
                                   
                                     θ 
                                     . 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   s 
                                   0 
                                 
                                  
                                 τ 
                                  
                                 
                                     
                                 
                                  
                                 
                                   E 
                                    
                                   
                                     [ 
                                     
                                       sin 
                                        
                                       
                                           
                                       
                                        
                                       
                                         θ 
                                         . 
                                       
                                        
                                       τ 
                                     
                                     ] 
                                   
                                 
                               
                             
                             
                               otherwise 
                             
                           
                         
                         = 
                         
                           
                             x 
                             0 
                           
                           + 
                           
                             { 
                             
                               
                                 
                                   
                                     
                                       s 
                                       0 
                                     
                                      
                                     
                                       E 
                                        
                                       
                                         [ 
                                         
                                           
                                             1 
                                             - 
                                             
                                               cos 
                                                
                                               
                                                   
                                               
                                                
                                               
                                                 θ 
                                                 . 
                                               
                                                
                                               τ 
                                             
                                           
                                           
                                             θ 
                                             . 
                                           
                                         
                                         ] 
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       if 
                                        
                                       
                                           
                                       
                                        
                                       
                                          
                                         
                                           θ 
                                           . 
                                         
                                          
                                       
                                     
                                     &gt; 
                                     
                                       ε 
                                       
                                         θ 
                                         . 
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       s 
                                       0 
                                     
                                      
                                     τ 
                                      
                                     
                                         
                                     
                                      
                                     
                                       E 
                                        
                                       
                                         [ 
                                         
                                           sin 
                                            
                                           
                                               
                                           
                                            
                                           
                                             θ 
                                             . 
                                           
                                            
                                           τ 
                                         
                                         ] 
                                       
                                     
                                   
                                 
                                 
                                   otherwise 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   53 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       E 
                        
                       
                         [ 
                         
                           y 
                           1 
                         
                         ] 
                       
                     
                     = 
                     
                       
                         y 
                         0 
                       
                       + 
                       
                         { 
                         
                           
                             
                               
                                 
                                   s 
                                   0 
                                 
                                  
                                 
                                   E 
                                    
                                   
                                     [ 
                                     
                                       
                                         sin 
                                          
                                         
                                             
                                         
                                          
                                         
                                           θ 
                                           . 
                                         
                                          
                                         τ 
                                       
                                       
                                         θ 
                                         . 
                                       
                                     
                                     ] 
                                   
                                 
                               
                             
                             
                               
                                 
                                   if 
                                    
                                   
                                      
                                     
                                       θ 
                                       . 
                                     
                                      
                                   
                                 
                                 &gt; 
                                 
                                   ε 
                                   
                                     θ 
                                     . 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   s 
                                   0 
                                 
                                  
                                 τ 
                                  
                                 
                                     
                                 
                                  
                                 
                                   E 
                                    
                                   
                                     [ 
                                     
                                       cos 
                                        
                                       
                                           
                                       
                                        
                                       
                                         θ 
                                         . 
                                       
                                        
                                       τ 
                                     
                                     ] 
                                   
                                 
                               
                             
                             
                               otherwise 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   54 
                 
               
             
           
         
       
     
         [0000]    The expected values in Eq. 51 and Eq. 52 can be written as: 
         [0000]    
       
         
           
             
               
                 
                   
                     E 
                      
                     
                       [ 
                       
                         x 
                         1 
                       
                       ] 
                     
                   
                   = 
                   
                     
                       x 
                       0 
                     
                     + 
                     
                       
                         s 
                         0 
                       
                        
                       
                         [ 
                         
                           
                             
                               
                                 
                                   
                                     ∫ 
                                     
                                       - 
                                       
                                         Θ 
                                         . 
                                       
                                     
                                     
                                       - 
                                       
                                         ε 
                                         
                                           θ 
                                           . 
                                         
                                       
                                     
                                   
                                    
                                   
                                     
                                       ( 
                                       
                                         1 
                                         
                                           2 
                                            
                                           
                                             Θ 
                                             . 
                                           
                                         
                                       
                                       ) 
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           1 
                                           - 
                                           
                                             cos 
                                              
                                             
                                                 
                                             
                                              
                                             
                                               θ 
                                               . 
                                             
                                              
                                             τ 
                                           
                                         
                                         
                                           θ 
                                           . 
                                         
                                       
                                       ) 
                                     
                                      
                                     
                                        
                                       
                                         θ 
                                         . 
                                       
                                     
                                   
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 
                                   τ 
                                    
                                   
                                     
                                       ∫ 
                                       
                                         - 
                                         
                                           ε 
                                           
                                             θ 
                                             . 
                                           
                                         
                                       
                                       
                                         + 
                                         
                                           ε 
                                           
                                             θ 
                                             . 
                                           
                                         
                                       
                                     
                                      
                                     
                                       
                                         ( 
                                         
                                           1 
                                           
                                             2 
                                              
                                             
                                                 
                                             
                                              
                                             
                                               Θ 
                                               . 
                                             
                                           
                                         
                                         ) 
                                       
                                        
                                       sin 
                                        
                                       
                                           
                                       
                                        
                                       
                                         θ 
                                         . 
                                       
                                        
                                       τ 
                                        
                                       
                                          
                                         
                                           θ 
                                           . 
                                         
                                       
                                     
                                   
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 
                                   ∫ 
                                   
                                     + 
                                     
                                       ε 
                                       
                                         θ 
                                         . 
                                       
                                     
                                   
                                   
                                     + 
                                     
                                       Θ 
                                       . 
                                     
                                   
                                 
                                  
                                 
                                   
                                     ( 
                                     
                                       1 
                                       
                                         2 
                                          
                                         
                                           Θ 
                                           . 
                                         
                                       
                                     
                                     ) 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         1 
                                         - 
                                         
                                           cos 
                                            
                                           
                                               
                                           
                                            
                                           
                                             θ 
                                             . 
                                           
                                            
                                           τ 
                                         
                                       
                                       
                                         θ 
                                         . 
                                       
                                     
                                     ) 
                                   
                                    
                                   
                                      
                                     
                                       θ 
                                       . 
                                     
                                   
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   55 
                 
               
             
             
               
                 
                   
                     E 
                      
                     
                       [ 
                       
                         y 
                         1 
                       
                       ] 
                     
                   
                   = 
                   
                     
                       y 
                       0 
                     
                     + 
                     
                       
                         s 
                         0 
                       
                        
                       
                         [ 
                         
                           
                             
                               
                                 
                                   
                                     ∫ 
                                     
                                       - 
                                       
                                         Θ 
                                         . 
                                       
                                     
                                     
                                       - 
                                       
                                         ε 
                                         
                                           θ 
                                           . 
                                         
                                       
                                     
                                   
                                    
                                   
                                     
                                       ( 
                                       
                                         
                                           sin 
                                            
                                           
                                               
                                           
                                            
                                           
                                             θ 
                                             . 
                                           
                                            
                                           τ 
                                         
                                         
                                           θ 
                                           . 
                                         
                                       
                                       ) 
                                     
                                      
                                     
                                       ( 
                                       
                                         1 
                                         
                                           2 
                                            
                                           
                                               
                                           
                                            
                                           
                                             Θ 
                                             . 
                                           
                                         
                                       
                                       ) 
                                     
                                      
                                     
                                        
                                       
                                         θ 
                                         . 
                                       
                                     
                                   
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     ∫ 
                                     
                                       - 
                                       
                                         ε 
                                         
                                           θ 
                                           . 
                                         
                                       
                                     
                                     
                                       + 
                                       
                                         ε 
                                         
                                           θ 
                                           . 
                                         
                                       
                                     
                                   
                                    
                                   
                                     
                                       ( 
                                       
                                         1 
                                         
                                           
                                             2 
                                              
                                             
                                               Θ 
                                               . 
                                             
                                           
                                            
                                           
                                               
                                           
                                         
                                       
                                       ) 
                                     
                                      
                                     cos 
                                      
                                     
                                         
                                     
                                      
                                     
                                       θ 
                                       . 
                                     
                                      
                                     τ 
                                      
                                     
                                        
                                       
                                         θ 
                                         . 
                                       
                                     
                                   
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 τ 
                                  
                                 
                                   
                                     ∫ 
                                     
                                       - 
                                       
                                         ε 
                                         
                                           θ 
                                           . 
                                         
                                       
                                     
                                     
                                       + 
                                       
                                         Θ 
                                         . 
                                       
                                     
                                   
                                    
                                   
                                     
                                       ( 
                                       
                                         
                                           sin 
                                            
                                           
                                               
                                           
                                            
                                           
                                             θ 
                                             . 
                                           
                                            
                                           τ 
                                         
                                         
                                           θ 
                                           . 
                                         
                                       
                                       ) 
                                     
                                      
                                     
                                       ( 
                                       
                                         1 
                                         
                                           2 
                                            
                                           
                                             Θ 
                                             . 
                                           
                                         
                                       
                                       ) 
                                     
                                      
                                     
                                        
                                       
                                         θ 
                                         . 
                                       
                                     
                                   
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   56 
                 
               
             
           
         
       
     
         [0035]    These integrals are identical to the expected values for E[sin {dot over (θ)}τ], E]cos {dot over (θ)}τ], 
         [0000]    
       
         
           
             
               E 
                
               
                 [ 
                 
                   
                     sin 
                      
                     
                         
                     
                      
                     
                       θ 
                       . 
                     
                      
                     τ 
                   
                   
                     θ 
                     . 
                   
                 
                 ] 
               
             
             , 
             
               
 
             
              
             and 
           
         
       
       
         
           
             E 
              
             
               [ 
               
                 
                   1 
                   - 
                   
                     cos 
                      
                     
                         
                     
                      
                     
                       θ 
                       . 
                     
                      
                     τ 
                   
                 
                 
                   θ 
                   . 
                 
               
               ] 
             
           
         
       
     
         [0000]    that were found earlier, except now with different limits of integration. Within the limits now defined, each function is well-behaved with no infinities in the range. First consider E[x 1 ]. All three integrals in Eq. 55 do not need to be evaluated. Since the integrands are odd functions and symmetrical area is being integrated, the area must sum to zero. Therefore, all the integrals for E[x 1 ] are zero. Now consider E[y 1 ]. The second integral in Eq. 56 is identical to E[cos {dot over (θ)}τ] that was already evaluated in (Eq. 30) except with different limits: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ∫ 
                       
                         - 
                         
                           ε 
                           
                             θ 
                             . 
                           
                         
                       
                       
                         + 
                         
                           ε 
                           
                             θ 
                             . 
                           
                         
                       
                     
                      
                     
                       
                         ( 
                         
                           1 
                           
                             2 
                              
                             
                               Θ 
                               . 
                             
                           
                         
                         ) 
                       
                        
                       cos 
                        
                       
                           
                       
                        
                       
                         θ 
                         . 
                       
                        
                       τ 
                        
                       
                          
                         
                           θ 
                           . 
                         
                       
                     
                   
                   = 
                   
                     
                       
                         
                           ( 
                           
                             1 
                             
                               2 
                                
                               
                                 Θ 
                                 . 
                               
                                
                               τ 
                             
                           
                           ) 
                         
                          
                         sin 
                          
                         
                             
                         
                          
                         
                           θ 
                           . 
                         
                          
                         τ 
                       
                        
                       
                         | 
                         
                           - 
                           
                             ε 
                             
                               θ 
                               . 
                             
                           
                         
                         
                           + 
                           
                             ε 
                             
                               θ 
                               . 
                             
                           
                         
                       
                     
                     = 
                     
                       
                         
                           ( 
                           
                             1 
                             
                               2 
                                
                               
                                 Θ 
                                 . 
                               
                                
                               τ 
                             
                           
                           ) 
                         
                          
                         
                           ( 
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 ε 
                                 
                                   θ 
                                   . 
                                 
                               
                                
                               τ 
                             
                             + 
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 ε 
                                 
                                   θ 
                                   . 
                                 
                               
                                
                               τ 
                             
                           
                           ) 
                         
                       
                       = 
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             ε 
                             
                               θ 
                               . 
                             
                           
                            
                           τ 
                         
                         
                           
                             Θ 
                             . 
                           
                            
                           τ 
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   57 
                 
               
             
           
         
       
     
         [0000]    The first and last integrals in Eq. 56 are similar to evaluating E[sin {dot over (θ)}τ/{dot over (θ)}]. 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         ∫ 
                         
                           - 
                           
                             Θ 
                             . 
                           
                         
                         
                           - 
                           
                             ε 
                             
                               θ 
                               . 
                             
                           
                         
                       
                        
                       
                         
                           ( 
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 . 
                               
                                
                               τ 
                             
                             
                               θ 
                               . 
                             
                           
                           ) 
                         
                          
                         
                           ( 
                           
                             1 
                             
                               2 
                                
                               
                                   
                               
                                
                               
                                 Θ 
                                 . 
                               
                             
                           
                           ) 
                         
                          
                         
                            
                           
                             θ 
                             . 
                           
                         
                       
                     
                     + 
                     
                       
                         ∫ 
                         
                           + 
                           
                             ε 
                             
                               θ 
                               . 
                             
                           
                         
                         
                           + 
                           
                             Θ 
                             . 
                           
                         
                       
                        
                       
                         
                           ( 
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 θ 
                                 . 
                               
                                
                               τ 
                             
                             
                               θ 
                               . 
                             
                           
                           ) 
                         
                          
                         
                           ( 
                           
                             1 
                             
                               2 
                                
                               
                                 Θ 
                                 . 
                               
                             
                           
                           ) 
                         
                          
                         
                            
                           
                             θ 
                             . 
                           
                         
                       
                     
                   
                   = 
                   
                     
                       
                         
                           Si 
                            
                           
                             ( 
                             
                               
                                 θ 
                                 . 
                               
                                
                               τ 
                             
                             ) 
                           
                         
                         
                           2 
                            
                           
                             Θ 
                             . 
                           
                         
                       
                        
                       
                         
                           
                              
                             
                               - 
                               
                                 Θ 
                                 . 
                               
                             
                             
                               - 
                               
                                 ε 
                                 
                                   θ 
                                   . 
                                 
                               
                             
                           
                            
                           
                             + 
                             
                               
                                 Si 
                                  
                                 
                                   ( 
                                   
                                     
                                       θ 
                                       . 
                                     
                                      
                                     τ 
                                   
                                   ) 
                                 
                               
                               
                                 2 
                                  
                                 
                                   Θ 
                                   . 
                                 
                               
                             
                           
                            
                         
                         
                           + 
                           
                             ε 
                             
                               θ 
                               . 
                             
                           
                         
                         
                           + 
                           
                             Θ 
                             . 
                           
                         
                       
                     
                     = 
                     
                       
                         
                           1 
                           
                             2 
                              
                             
                               Θ 
                               . 
                             
                           
                         
                          
                         
                           [ 
                           
                             
                               
                                 
                                   
                                     Si 
                                      
                                     
                                       ( 
                                       
                                         
                                           - 
                                           
                                             ε 
                                             
                                               θ 
                                               . 
                                             
                                           
                                         
                                          
                                         τ 
                                       
                                       ) 
                                     
                                   
                                   - 
                                   
                                     Si 
                                      
                                     
                                       ( 
                                       
                                         
                                           - 
                                           
                                             Θ 
                                             . 
                                           
                                         
                                          
                                         τ 
                                       
                                       ) 
                                     
                                   
                                   + 
                                 
                               
                             
                             
                               
                                 
                                   
                                     Si 
                                      
                                     
                                       ( 
                                       
                                         
                                           Θ 
                                           . 
                                         
                                          
                                         τ 
                                       
                                       ) 
                                     
                                   
                                   - 
                                   
                                     Si 
                                      
                                     
                                       ( 
                                       
                                         
                                           ε 
                                           
                                             θ 
                                             . 
                                           
                                         
                                          
                                         τ 
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                       = 
                       
                         
                           
                             1 
                             
                               2 
                                
                               
                                 Θ 
                                 . 
                               
                             
                           
                            
                           
                             [ 
                             
                               
                                 
                                   
                                     
                                       - 
                                       
                                         Si 
                                          
                                         
                                           ( 
                                           
                                             
                                               ε 
                                               
                                                 θ 
                                                 . 
                                               
                                             
                                              
                                             τ 
                                           
                                           ) 
                                         
                                       
                                     
                                     + 
                                     
                                       Si 
                                        
                                       
                                         ( 
                                         
                                           
                                             Θ 
                                             . 
                                           
                                            
                                           τ 
                                         
                                         ) 
                                       
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       Si 
                                        
                                       
                                         ( 
                                         
                                           
                                             Θ 
                                             . 
                                           
                                            
                                           τ 
                                         
                                         ) 
                                       
                                     
                                     - 
                                     
                                       Si 
                                        
                                       
                                         ( 
                                         
                                           
                                             ε 
                                             
                                               θ 
                                               . 
                                             
                                           
                                            
                                           τ 
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                             
                             ] 
                           
                         
                         = 
                         
                           
                             1 
                             
                               Θ 
                               . 
                             
                           
                            
                           
                             [ 
                             
                               
                                 Si 
                                  
                                 
                                   ( 
                                   
                                     
                                       Θ 
                                       . 
                                     
                                      
                                     τ 
                                   
                                   ) 
                                 
                               
                               - 
                               
                                 Si 
                                  
                                 
                                   ( 
                                   
                                     
                                       ε 
                                       
                                         θ 
                                         . 
                                       
                                     
                                      
                                     τ 
                                   
                                   ) 
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   58 
                 
               
             
           
         
       
     
         [0000]    Using these results in Eq. 55 and Eq. 56 the expected position using the modified motion equation is: 
         [0000]    
       
         
           
             
               
                 
                   
                     E 
                      
                     
                       [ 
                       
                         x 
                         1 
                       
                       ] 
                     
                   
                   = 
                   
                     x 
                     0 
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   59 
                 
               
             
             
               
                 
                   
                     E 
                      
                     
                       [ 
                       
                         y 
                         1 
                       
                       ] 
                     
                   
                   = 
                   
                     
                       y 
                       0 
                     
                     + 
                     
                       
                         
                           S 
                           
                             k 
                             - 
                             1 
                           
                         
                         
                           Θ 
                           . 
                         
                       
                        
                       
                         [ 
                         
                           
                             Si 
                              
                             
                               ( 
                               
                                 
                                   Θ 
                                   . 
                                 
                                  
                                 τ 
                               
                               ) 
                             
                           
                           - 
                           
                             Si 
                              
                             
                               ( 
                               
                                 
                                   ε 
                                   
                                     θ 
                                     . 
                                   
                                 
                                  
                                 τ 
                               
                               ) 
                             
                           
                           + 
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               ε 
                               
                                 θ 
                                 . 
                               
                             
                              
                             τ 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   60 
                 
               
             
           
         
       
     
         [0000]    Since ∈ {dot over (θ)}  is small, both Si(∈ {dot over (θ)} τ)≈0 and sin ∈ {dot over (θ)} τ≈0, so 
         [0000]    
       
         
           
             
               E 
                
               
                 [ 
                 
                   y 
                   1 
                 
                 ] 
               
             
             ≈ 
             
               
                 y 
                 0 
               
               + 
               
                 
                   s 
                   0 
                 
                  
                 
                   
                     
                       Si 
                        
                       
                         ( 
                         
                           
                             Θ 
                             . 
                           
                            
                           τ 
                         
                         ) 
                       
                     
                     
                       Θ 
                       . 
                     
                   
                   . 
                 
               
             
           
         
       
     
         [0036]    Now consider the variance of the position. To avoid the issue of possible division by zero, the modification, Eq. 10 and 11 or 49 and 50, as well will be used to find the variance. 
         [0000]    
       
         
           
             
               
                 
                   
                     E 
                      
                     
                       [ 
                       
                         x 
                         1 
                         2 
                       
                       ] 
                     
                   
                   = 
                   
                     E 
                     [ 
                     
                       ( 
                       
                         
                           x 
                           0 
                         
                         + 
                         
                           
                             { 
                             
                               
                                 
                                   
                                     
                                       1 
                                       
                                         
                                           θ 
                                           . 
                                         
                                         2 
                                       
                                     
                                      
                                     
                                       [ 
                                       
                                         
                                           
                                             
                                               
                                                 
                                                   s 
                                                   . 
                                                 
                                                  
                                                 sin 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 
                                                   θ 
                                                   1 
                                                 
                                               
                                               - 
                                               
                                                 
                                                   s 
                                                   1 
                                                 
                                                  
                                                 
                                                   θ 
                                                   . 
                                                 
                                                  
                                                 cos 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 
                                                   θ 
                                                   1 
                                                 
                                               
                                               - 
                                             
                                           
                                         
                                         
                                           
                                             
                                               
                                                 
                                                   s 
                                                   . 
                                                 
                                                  
                                                 sin 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 
                                                   θ 
                                                   0 
                                                 
                                               
                                               + 
                                               
                                                 
                                                   s 
                                                   0 
                                                 
                                                  
                                                 
                                                   θ 
                                                   . 
                                                 
                                                  
                                                 cos 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 
                                                   θ 
                                                   0 
                                                 
                                               
                                             
                                           
                                         
                                       
                                       ] 
                                     
                                   
                                 
                                 
                                   
                                     
                                       if 
                                        
                                       
                                           
                                       
                                        
                                       
                                          
                                         
                                           θ 
                                           . 
                                         
                                          
                                       
                                     
                                     &gt; 
                                     
                                       ε 
                                       
                                         θ 
                                         . 
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       ( 
                                       
                                         
                                           
                                             s 
                                             0 
                                           
                                            
                                           τ 
                                         
                                         + 
                                         
                                           
                                             1 
                                             2 
                                           
                                            
                                           
                                             s 
                                             . 
                                           
                                            
                                           
                                             τ 
                                             2 
                                           
                                         
                                       
                                       ) 
                                     
                                      
                                     
                                       sin 
                                        
                                       
                                         ( 
                                         
                                           
                                             θ 
                                             0 
                                           
                                           + 
                                           
                                             
                                               θ 
                                               . 
                                             
                                              
                                             τ 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                                 
                                   otherwise 
                                 
                               
                             
                             ) 
                           
                           2 
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   61 
                 
               
             
             
               
                 
                   
                     E 
                      
                     
                       [ 
                       
                         y 
                         1 
                         2 
                       
                       ] 
                     
                   
                   = 
                   
                     E 
                     [ 
                     
                       ( 
                       
                         
                           y 
                           0 
                         
                         + 
                         
                           
                             { 
                             
                               
                                 
                                   
                                     
                                       1 
                                       
                                         
                                           θ 
                                           . 
                                         
                                         2 
                                       
                                     
                                      
                                     
                                       [ 
                                       
                                         
                                           
                                             
                                               
                                                 
                                                   s 
                                                   . 
                                                 
                                                  
                                                 cos 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 
                                                   θ 
                                                   1 
                                                 
                                               
                                               - 
                                               
                                                 
                                                   s 
                                                   1 
                                                 
                                                  
                                                 
                                                   θ 
                                                   . 
                                                 
                                                  
                                                 sin 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 
                                                   θ 
                                                   1 
                                                 
                                               
                                               - 
                                             
                                           
                                         
                                         
                                           
                                             
                                               
                                                 
                                                   s 
                                                   . 
                                                 
                                                  
                                                 cos 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 
                                                   θ 
                                                   0 
                                                 
                                               
                                               + 
                                               
                                                 
                                                   s 
                                                   0 
                                                 
                                                  
                                                 
                                                   θ 
                                                   . 
                                                 
                                                  
                                                 sin 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 
                                                   θ 
                                                   0 
                                                 
                                               
                                             
                                           
                                         
                                       
                                       ] 
                                     
                                   
                                 
                                 
                                   
                                     
                                       if 
                                        
                                       
                                           
                                       
                                        
                                       
                                          
                                         
                                           θ 
                                           . 
                                         
                                          
                                       
                                     
                                     &gt; 
                                     
                                       ε 
                                       
                                         θ 
                                         . 
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       ( 
                                       
                                         
                                           
                                             s 
                                             0 
                                           
                                            
                                           τ 
                                         
                                         + 
                                         
                                           
                                             1 
                                             2 
                                           
                                            
                                           
                                             s 
                                             . 
                                           
                                            
                                           
                                             τ 
                                             2 
                                           
                                         
                                       
                                       ) 
                                     
                                      
                                     
                                       sin 
                                        
                                       
                                         ( 
                                         
                                           
                                             θ 
                                             0 
                                           
                                           + 
                                           
                                             
                                               θ 
                                               . 
                                             
                                              
                                             τ 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                                 
                                   otherwise 
                                 
                               
                             
                             ) 
                           
                           2 
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   62 
                 
               
             
           
         
       
     
         [0037]    All of the integrals arising from these expected values are solvable but produce “messy” solutions. Thus, although a closed form solution can be found for the variance of the position, it is much too complicated to be useful. Therefore, the predicted state and covariance will be estimated using numerical methods as described next. 
       The Mean and Variance of the Transition Density (Double Maneuver)  
       [0038]    Now that the distribution parameters have been determined for the case when there is one maneuver over the update time interval, the case when there are two maneuvers is considered next. For this two-step case, assume the target starts with some randomly selected acceleration and turn rate and executes that maneuver for τ time, just as was assumed for the one-step case. But at time τ, the target randomly selects another acceleration and turn rate and executes that maneuver for the remainder of the update time interval, T−τ. This assumed process now makes τ a random variable as well. Maintaining consistency with the rest of the analysis, assume τ is a uniform random variable such that 0≦τ≦T. Therefore, 
         [0000]    
       
         
           
             
               
                 
                   
                     E 
                      
                     
                       [ 
                       τ 
                       ] 
                     
                   
                   = 
                   
                     
                       1 
                       2 
                     
                      
                     T 
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   63 
                 
               
             
             
               
                 
                   
                     Var 
                      
                     
                       [ 
                       τ 
                       ] 
                     
                   
                   = 
                   
                     
                       σ 
                       τ 
                       2 
                     
                     = 
                     
                       
                         1 
                         12 
                       
                        
                       
                         T 
                         2 
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   64 
                 
               
             
           
         
       
     
         [0039]    To deal with the multiple maneuvers, the (randomly) selected accelerations and turn rates need to be subscripted as well. As before, assume at a specified update time, the target is at (x 0 , y 0 ). Let this update time be at time zero. The (predicted) state of the target is desired for the next update time, T. Let τ be the time when the target finishes its first maneuver and starts its second (and last) maneuver over the time update interval. As before, the subscripts reflect target states within an update time interval. So (x 0 , y 0 ) is the position of the target at the begin inning of the update time interval having initial speed s 0  and initial course θ 0 . An acceleration {dot over (s)} 1  and turn rate, {dot over (θ)} 1 , is (randomly) selected. The target would then have position (x 1 , y 1 ) τ time later with speed s 1  and course θ 1 . At this point in time, the target (randomly) selects a new acceleration and turn rate, {dot over (s)} 2  and {dot over (θ)} 2 , respectively. The target executes that maneuver for the remainder of the update time interval, T−τ. At the end of the update time period, T, the target would have position (x 2 , y 2 ) with speed and course s 2  and θ 2 , respectively. Using this notation, 
         [0000]    
       
         
           
             
               
                 
                   
                     s 
                     1 
                   
                   = 
                   
                     
                       s 
                       0 
                     
                     + 
                     
                       
                         
                           s 
                           . 
                         
                         1 
                       
                        
                       τ 
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   65 
                 
               
             
             
               
                 
                   
                     s 
                     2 
                   
                   = 
                   
                     
                       
                         s 
                         1 
                       
                       + 
                       
                         
                           
                             s 
                             . 
                           
                           2 
                         
                          
                         
                           ( 
                           
                             T 
                             - 
                             τ 
                           
                           ) 
                         
                       
                     
                     = 
                     
                       
                         s 
                         0 
                       
                       + 
                       
                         
                           
                             s 
                             . 
                           
                           1 
                         
                          
                         τ 
                       
                       + 
                       
                         
                           
                             s 
                             . 
                           
                           2 
                         
                          
                         
                           ( 
                           
                             T 
                             - 
                             τ 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   66 
                 
               
             
             
               
                 
                   
                     θ 
                     1 
                   
                   = 
                   
                     
                       θ 
                       0 
                     
                     + 
                     
                       
                         
                           θ 
                           . 
                         
                         1 
                       
                        
                       τ 
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   67 
                 
               
             
             
               
                 
                   
                     θ 
                     2 
                   
                   = 
                   
                     
                       
                         θ 
                         1 
                       
                       + 
                       
                         
                           
                             θ 
                             . 
                           
                           2 
                         
                          
                         
                           ( 
                           
                             T 
                             - 
                             τ 
                           
                           ) 
                         
                       
                     
                     = 
                     
                       
                         θ 
                         0 
                       
                       + 
                       
                         
                           
                             θ 
                             . 
                           
                           1 
                         
                          
                         τ 
                       
                       + 
                       
                         
                           
                             θ 
                             . 
                           
                           2 
                         
                          
                         
                           ( 
                           
                             T 
                             - 
                             τ 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   68 
                 
               
             
             
               
                 
                   
                     
                       x 
                       . 
                     
                     1 
                   
                   = 
                   
                     
                       s 
                       1 
                     
                      
                     sin 
                      
                     
                         
                     
                      
                     
                       θ 
                       1 
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   69 
                 
               
             
             
               
                 
                   
                     
                       x 
                       . 
                     
                     2 
                   
                   = 
                   
                     
                       s 
                       2 
                     
                      
                     sin 
                      
                     
                         
                     
                      
                     
                       θ 
                       2 
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   70 
                 
               
             
             
               
                 
                   
                     
                       y 
                       . 
                     
                     1 
                   
                   = 
                   
                     
                       s 
                       1 
                     
                      
                     cos 
                      
                     
                         
                     
                      
                     
                       θ 
                       1 
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   71 
                 
               
             
             
               
                 
                   
                     
                       y 
                       . 
                     
                     2 
                   
                   = 
                   
                     
                       s 
                       2 
                     
                      
                     cos 
                      
                     
                         
                     
                      
                     
                       θ 
                       2 
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   72 
                 
               
             
             
               
                 
                   
                     x 
                     1 
                   
                   = 
                   
                     
                       x 
                       0 
                     
                     + 
                     
                       
                         1 
                         
                           
                             θ 
                             . 
                           
                           1 
                           2 
                         
                       
                        
                       
                         [ 
                         
                           
                             
                               
                                 s 
                                 . 
                               
                               1 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               1 
                             
                           
                           - 
                           
                             
                               s 
                               1 
                             
                              
                             
                               
                                 θ 
                                 . 
                               
                               1 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               1 
                             
                           
                           - 
                           
                             
                               
                                 s 
                                 . 
                               
                               1 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               0 
                             
                           
                           + 
                           
                             
                               s 
                               0 
                             
                              
                             
                               
                                 θ 
                                 . 
                               
                               1 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               0 
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   73 
                 
               
             
             
               
                 
                   
                     x 
                     2 
                   
                   = 
                   
                     
                       x 
                       1 
                     
                     + 
                     
                       
                         1 
                         
                           
                             θ 
                             . 
                           
                           2 
                           2 
                         
                       
                        
                       
                         [ 
                         
                           
                             
                               
                                 s 
                                 . 
                               
                               2 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               2 
                             
                           
                           - 
                           
                             
                               s 
                               2 
                             
                              
                             
                               
                                 θ 
                                 . 
                               
                               2 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               2 
                             
                           
                           - 
                           
                             
                               
                                 s 
                                 . 
                               
                               2 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               1 
                             
                           
                           + 
                           
                             
                               s 
                               1 
                             
                              
                             
                               
                                 θ 
                                 . 
                               
                               2 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               1 
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   74 
                 
               
             
             
               
                 
                   
                     y 
                     1 
                   
                   = 
                   
                     
                       y 
                       0 
                     
                     + 
                     
                       
                         1 
                         
                           
                             θ 
                             . 
                           
                           1 
                           2 
                         
                       
                        
                       
                         [ 
                         
                           
                             
                               
                                 s 
                                 . 
                               
                               1 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               1 
                             
                           
                           - 
                           
                             
                               s 
                               1 
                             
                              
                             
                               
                                 θ 
                                 . 
                               
                               1 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               1 
                             
                           
                           - 
                           
                             
                               
                                 s 
                                 . 
                               
                               1 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               0 
                             
                           
                           + 
                           
                             
                               s 
                               0 
                             
                              
                             
                               
                                 θ 
                                 . 
                               
                               1 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               0 
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   75 
                 
               
             
             
               
                 
                   
                     y 
                     2 
                   
                   = 
                   
                     
                       y 
                       1 
                     
                     + 
                     
                       
                         1 
                         
                           
                             θ 
                             . 
                           
                           2 
                           2 
                         
                       
                        
                       
                         [ 
                         
                           
                             
                               
                                 s 
                                 . 
                               
                               2 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               2 
                             
                           
                           - 
                           
                             
                               s 
                               2 
                             
                              
                             
                               
                                 θ 
                                 . 
                               
                               2 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               2 
                             
                           
                           - 
                           
                             
                               
                                 s 
                                 . 
                               
                               2 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               1 
                             
                           
                           + 
                           
                             
                               s 
                               1 
                             
                              
                             
                               
                                 θ 
                                 . 
                               
                               2 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               1 
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   76 
                 
               
             
           
         
       
     
       The Mean and Variance of the Speed and Course  
       [0040]    To find the mean and variance of the speed and course for the two-maneuver case, note that it is necessary to deal with a product of independent random variables. It is known that the variance of a product of independent random variables, A, B is given by: 
         [0000]      Var[ AB]= ( E[A] ) 2 Var[ B ]+( E[B ]) 2 Var[ A ]+Var[ A ]Var[ B]   Eq. 77
 
         [0041]    Using this property along with 
         [0000]    
       
         
           
             
               
                 E 
                  
                 
                   [ 
                   
                     
                       s 
                       . 
                     
                     1 
                   
                   ] 
                 
               
               = 
               
                 
                   E 
                    
                   
                     [ 
                     
                       
                         s 
                         . 
                       
                       2 
                     
                     ] 
                   
                 
                 = 
                 
                   
                     E 
                      
                     
                       [ 
                       
                         
                           θ 
                           . 
                         
                         1 
                       
                       ] 
                     
                   
                   = 
                   
                     
                       E 
                        
                       
                         [ 
                         
                           
                             θ 
                             . 
                           
                           2 
                         
                         ] 
                       
                     
                     = 
                     0 
                   
                 
               
             
             , 
             
               
 
             
              
             
               
                 E 
                  
                 
                   [ 
                   
                     T 
                     - 
                     τ 
                   
                   ] 
                 
               
               = 
               
                 
                   E 
                    
                   
                     [ 
                     τ 
                     ] 
                   
                 
                 = 
                 
                   
                     1 
                     2 
                   
                    
                   T 
                 
               
             
             , 
             
               
 
             
              
             
               
                 Var 
                  
                 
                   [ 
                   
                     
                       s 
                       . 
                     
                     1 
                   
                   ] 
                 
               
               = 
               
                 
                   Var 
                    
                   
                     [ 
                     
                       
                         s 
                         . 
                       
                       2 
                     
                     ] 
                   
                 
                 = 
                 
                   σ 
                   
                     s 
                     . 
                   
                   2 
                 
               
             
             , 
             
               
 
             
              
             and 
           
         
       
       
         
           
             
               
                 Var 
                  
                 
                   [ 
                   
                     T 
                     - 
                     τ 
                   
                   ] 
                 
               
               = 
               
                 
                   Var 
                    
                   
                     [ 
                     τ 
                     ] 
                   
                 
                 = 
                 
                   
                     σ 
                     τ 
                     2 
                   
                   = 
                   
                     
                       1 
                       12 
                     
                      
                     
                       T 
                       2 
                     
                   
                 
               
             
             , 
             
               
 
             
              
             then 
           
         
       
       
         
           
             
               
                 
                   
                     
                       
                         
                           E 
                            
                           
                             [ 
                             
                               s 
                               2 
                             
                             ] 
                           
                         
                         = 
                           
                          
                         
                           
                             E 
                              
                             
                               [ 
                               
                                 s 
                                 0 
                               
                               ] 
                             
                           
                           + 
                           
                             E 
                              
                             
                               [ 
                               
                                 
                                   
                                     s 
                                     . 
                                   
                                   1 
                                 
                                  
                                 τ 
                               
                               ] 
                             
                           
                           + 
                           
                             E 
                              
                             
                               [ 
                               
                                 
                                   
                                     s 
                                     . 
                                   
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     T 
                                     - 
                                     τ 
                                   
                                   ) 
                                 
                               
                               ] 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             s 
                             0 
                           
                           + 
                           
                             
                               E 
                                
                               
                                 [ 
                                 
                                   
                                     s 
                                     . 
                                   
                                   1 
                                 
                                 ] 
                               
                             
                              
                             
                               E 
                                
                               
                                 [ 
                                 τ 
                                 ] 
                               
                             
                           
                           + 
                           
                             
                               E 
                                
                               
                                 [ 
                                 
                                   
                                     s 
                                     . 
                                   
                                   2 
                                 
                                 ] 
                               
                             
                              
                             
                               E 
                                
                               
                                 [ 
                                 
                                   T 
                                   - 
                                   τ 
                                 
                                 ] 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           s 
                           0 
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   78 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           Var 
                            
                           
                             [ 
                             
                               s 
                               2 
                             
                             ] 
                           
                         
                         = 
                           
                          
                         
                           
                             Var 
                              
                             
                               [ 
                               
                                 s 
                                 0 
                               
                               ] 
                             
                           
                           + 
                           
                             Var 
                              
                             
                               [ 
                               
                                 
                                   
                                     s 
                                     . 
                                   
                                   1 
                                 
                                  
                                 τ 
                               
                               ] 
                             
                           
                           + 
                           
                             Var 
                              
                             
                               [ 
                               
                                 
                                   
                                     s 
                                     . 
                                   
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     T 
                                     - 
                                     τ 
                                   
                                   ) 
                                 
                               
                               ] 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             { 
                             
                               
                                 
                                   
                                     
                                       
                                         
                                           ( 
                                           
                                             E 
                                              
                                             
                                               [ 
                                               
                                                 
                                                   s 
                                                   . 
                                                 
                                                 2 
                                               
                                               ] 
                                             
                                           
                                           ) 
                                         
                                         2 
                                       
                                        
                                       
                                         Var 
                                          
                                         
                                           [ 
                                           τ 
                                           ] 
                                         
                                       
                                     
                                     + 
                                     
                                       
                                         
                                           ( 
                                           
                                             E 
                                              
                                             
                                               [ 
                                               τ 
                                               ] 
                                             
                                           
                                           ) 
                                         
                                         2 
                                       
                                        
                                       
                                         Var 
                                          
                                         
                                           [ 
                                           
                                             
                                               s 
                                               . 
                                             
                                             1 
                                           
                                           ] 
                                         
                                       
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       Var 
                                        
                                       
                                         [ 
                                         
                                           
                                             s 
                                             . 
                                           
                                           1 
                                         
                                         ] 
                                       
                                     
                                      
                                     
                                       Var 
                                        
                                       
                                         [ 
                                         τ 
                                         ] 
                                       
                                     
                                   
                                 
                               
                             
                             } 
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           { 
                           
                             
                               
                                 
                                   
                                     
                                       
                                         ( 
                                         
                                           E 
                                            
                                           
                                             [ 
                                             
                                               
                                                 s 
                                                 . 
                                               
                                               2 
                                             
                                             ] 
                                           
                                         
                                         ) 
                                       
                                       2 
                                     
                                      
                                     
                                       Var 
                                        
                                       
                                         [ 
                                         
                                           T 
                                           - 
                                           τ 
                                         
                                         ] 
                                       
                                     
                                   
                                   + 
                                 
                               
                             
                             
                               
                                 
                                   
                                     
                                       
                                         ( 
                                         
                                           E 
                                            
                                           
                                             [ 
                                             
                                               T 
                                               - 
                                               τ 
                                             
                                             ] 
                                           
                                         
                                         ) 
                                       
                                       2 
                                     
                                      
                                     
                                       Var 
                                        
                                       
                                         [ 
                                         
                                           
                                             s 
                                             . 
                                           
                                           2 
                                         
                                         ] 
                                       
                                     
                                   
                                   + 
                                   
                                     
                                       Var 
                                        
                                       
                                         [ 
                                         
                                           
                                             s 
                                             . 
                                           
                                           2 
                                         
                                         ] 
                                       
                                     
                                      
                                     
                                       Var 
                                        
                                       
                                         [ 
                                         
                                           T 
                                           - 
                                           τ 
                                         
                                         ] 
                                       
                                     
                                   
                                 
                               
                             
                           
                           } 
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             2 
                             [ 
                             
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         1 
                                         2 
                                       
                                        
                                       T 
                                     
                                     ) 
                                   
                                   2 
                                 
                                  
                                 
                                   σ 
                                   
                                     s 
                                     . 
                                   
                                   2 
                                 
                               
                               + 
                               
                                 
                                   σ 
                                   
                                     s 
                                     . 
                                   
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       1 
                                       12 
                                     
                                      
                                     
                                       T 
                                       2 
                                     
                                   
                                   ) 
                                 
                               
                             
                              
                             
                                 
                             
                             ] 
                           
                           = 
                           
                             
                               2 
                               3 
                             
                              
                             
                               T 
                               2 
                             
                              
                             
                               σ 
                               
                                 s 
                                 . 
                               
                               2 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   79 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           E 
                            
                           
                             [ 
                             
                               θ 
                               2 
                             
                             ] 
                           
                         
                         = 
                           
                          
                         
                           
                             E 
                              
                             
                               [ 
                               
                                 θ 
                                 0 
                               
                               ] 
                             
                           
                           + 
                           
                             E 
                              
                             
                               [ 
                               
                                 
                                   
                                     θ 
                                     . 
                                   
                                   1 
                                 
                                  
                                 τ 
                               
                               ] 
                             
                           
                           + 
                           
                             E 
                              
                             
                               [ 
                               
                                 
                                   
                                     θ 
                                     . 
                                   
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     T 
                                     - 
                                     τ 
                                   
                                   ) 
                                 
                               
                               ] 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             θ 
                             0 
                           
                           + 
                           
                             
                               E 
                                
                               
                                 [ 
                                 
                                   
                                     θ 
                                     . 
                                   
                                   1 
                                 
                                 ] 
                               
                             
                              
                             
                               E 
                                
                               
                                 [ 
                                 τ 
                                 ] 
                               
                             
                           
                           + 
                           
                             
                               E 
                                
                               
                                 [ 
                                 
                                   
                                     θ 
                                     . 
                                   
                                   2 
                                 
                                 ] 
                               
                             
                              
                             
                               E 
                                
                               
                                 [ 
                                 
                                   T 
                                   - 
                                   τ 
                                 
                                 ] 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           θ 
                           0 
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   80 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           Var 
                            
                           
                             [ 
                             
                               θ 
                               2 
                             
                             ] 
                           
                         
                         = 
                           
                          
                         
                           
                             Var 
                              
                             
                               [ 
                               
                                 θ 
                                 0 
                               
                               ] 
                             
                           
                           + 
                           
                             Var 
                              
                             
                               [ 
                               
                                 
                                   
                                     θ 
                                     . 
                                   
                                   1 
                                 
                                  
                                 τ 
                               
                               ] 
                             
                           
                           + 
                           
                             Var 
                              
                             
                               [ 
                               
                                 
                                   
                                     θ 
                                     . 
                                   
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     T 
                                     - 
                                     τ 
                                   
                                   ) 
                                 
                               
                               ] 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             { 
                             
                               
                                 
                                   
                                     
                                       
                                         
                                           ( 
                                           
                                             E 
                                              
                                             
                                               [ 
                                               
                                                 
                                                   θ 
                                                   . 
                                                 
                                                 1 
                                               
                                               ] 
                                             
                                           
                                           ) 
                                         
                                         2 
                                       
                                        
                                       
                                         Var 
                                          
                                         
                                           [ 
                                           τ 
                                           ] 
                                         
                                       
                                     
                                     + 
                                     
                                       
                                         
                                           ( 
                                           
                                             E 
                                              
                                             
                                               [ 
                                               τ 
                                               ] 
                                             
                                           
                                           ) 
                                         
                                         2 
                                       
                                        
                                       
                                         Var 
                                          
                                         
                                           [ 
                                           
                                             
                                               θ 
                                               . 
                                             
                                             1 
                                           
                                           ] 
                                         
                                       
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       Var 
                                        
                                       
                                         [ 
                                         
                                           
                                             θ 
                                             . 
                                           
                                           1 
                                         
                                         ] 
                                       
                                     
                                      
                                     
                                       Var 
                                        
                                       
                                         [ 
                                         τ 
                                         ] 
                                       
                                     
                                   
                                 
                               
                             
                             } 
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           { 
                           
                             
                               
                                 
                                   
                                     
                                       
                                         ( 
                                         
                                           E 
                                            
                                           
                                             [ 
                                             
                                               
                                                 θ 
                                                 . 
                                               
                                               2 
                                             
                                             ] 
                                           
                                         
                                         ) 
                                       
                                       2 
                                     
                                      
                                     
                                       Var 
                                        
                                       
                                         [ 
                                         
                                           T 
                                           - 
                                           τ 
                                         
                                         ] 
                                       
                                     
                                   
                                   + 
                                 
                               
                             
                             
                               
                                 
                                   
                                     
                                       
                                         ( 
                                         
                                           E 
                                            
                                           
                                             [ 
                                             
                                               T 
                                               - 
                                               τ 
                                             
                                             ] 
                                           
                                         
                                         ) 
                                       
                                       2 
                                     
                                      
                                     
                                       Var 
                                        
                                       
                                         [ 
                                         
                                           
                                             θ 
                                             . 
                                           
                                           2 
                                         
                                         ] 
                                       
                                     
                                   
                                   + 
                                   
                                     
                                       Var 
                                        
                                       
                                         [ 
                                         
                                           
                                             θ 
                                             . 
                                           
                                           2 
                                         
                                         ] 
                                       
                                     
                                      
                                     
                                       Var 
                                        
                                       
                                         [ 
                                         
                                           T 
                                           - 
                                           τ 
                                         
                                         ] 
                                       
                                     
                                   
                                 
                               
                             
                           
                           } 
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             2 
                             [ 
                             
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         1 
                                         2 
                                       
                                        
                                       T 
                                     
                                     ) 
                                   
                                   2 
                                 
                                  
                                 
                                   σ 
                                   
                                     θ 
                                     . 
                                   
                                   2 
                                 
                               
                               + 
                               
                                 
                                   σ 
                                   
                                     θ 
                                     . 
                                   
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       1 
                                       12 
                                     
                                      
                                     
                                       T 
                                       2 
                                     
                                   
                                   ) 
                                 
                               
                             
                              
                             
                                 
                             
                             ] 
                           
                           = 
                           
                             
                               2 
                               3 
                             
                              
                             
                               T 
                               2 
                             
                              
                             
                               σ 
                               
                                 θ 
                                 . 
                               
                               2 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   81 
                 
               
             
           
         
       
     
         [0042]    Comparing Eq. 78-80 to Eq. 19-22 shows the means for the speed and course for the two-maneuver case are the same as those for the one-maneuver case, but the variances for the two-maneuver case are smaller by two thirds. (Recall that τ=T for the one-maneuver case.) 
       The Man and Variance of the State  
       [0043]    Determining the mean and variance of the state when the target undergoes two (kinematically-constrained) maneuvers is difficult to do in closed-form. However, these parameters can be approximated using Monte Carlos techniques. For the analysis, 50,000,000 samples were used. For all the cases, the target was initially at the origin moving due North at 20 m/s. The update time interval was T=10 s and the maximum acceleration was A=2 m/s 2 . Thus, (x 0 , y 0 )=(0,0), s 0 =20, and θ 0 =0. The table in  FIG. 4  compares the predicted state and covariance from the CGBF against the Kalman filter. Since the Kalman filter does not use turn rates, its predicted state and covariance do not change as the maximum turn rate is varied. However, as the table shows, these values vary dramatically depending on the maximum turn rate assumed for the target. For example, when the maximum turn rate is small, the CGBF predicted position is nearly identical to Kalman. But as the maximum turn rate increases to 15 deg/s, the y position diminishes from 200 m to ˜151 m, a nearly 25% reduction. The y component of the velocity reduces from 20 m/s to ˜8 m/s, which is more than 50% reduction. The variance terms show an interesting difference as well. The Kalman filter assumes the distribution is circular; no initial direction of the target is considered. As a result, it grossly overestimates the uncertainty perpendicular to the initial course of the target when it has a small turn rate. The target needs a turn rate over 10 deg/s for the magnitude of the uncertainty to be as large as the Kalman filter assumes. Finally, the covariance terms are very different. The Kalman filter only specifies two covariance terms: x position with the x velocity and y position with the y velocity. The other four covariance terms are assumed to be zero. As shown in the table, this is a reasonable assumption. The last four covariance terms are all near, or possibly equal to zero regardless of the maximum rate. However, the table shows that for the two covariance terms that the Kalman filter computes, they are quite different from the CGBF. The Kalman filters uses ˜667 m 2 /s for both x and y. The CGBF estimates the x{dot over (x)} covariance to be as small as ˜0.2 m 2 /s for a small maximum turn rate target to as large as ˜1565 m 2 /s for a high maximum turn rate target. Interestingly, the y{dot over (y)} covariance does not vary much relative to the maximum turn rate. For most turn rates, it stays around 400 m 2 /s, which is about 35% small than the corresponding Kalman value. 
         [0044]    The state estimates are directly influenced by the predicted covariance. Sine the predicted covariance from the CGBF is tighter and more aligned to the real target state, it suggests that the CGBF should yield tighter and more accurate state estimates than the Kalman filter. 
       Motion Update  
       [0045]    Performing a motion predict is particularly computational for a GBF. Each cell in the grid must be propagated to all the other cells that the target could have transitioned to during the time to the next measurement update. These propagations are found by generating a large number of Monte Carlo samples for each cell in the grid. Since the grids tend to be large, an enormous number of Monte Carlo samples are required for each cell to form a good approximation of the cell&#39;s transition distribution. 
         [0046]    To form the target starting position for each cell, there are two common approaches: 1) use the center position of the cell, or 2) randomly sample within the cell positional limits. Randomly sampling within the cell limits is usually a better approach because the true target position could be near an edge of the cell. If the cells are large compared to the distance the target moves in an update, then the errors introduced by only using the cell center could be significant. However, random sampling over the cell may require even more Monte Carlo sample per cell to get a good distribution over the entire cell. 
         [0047]    With this invention, an alternative motion update procedure has been developed that has two key refinements over these two approaches. The first refinement is how the mass in the cells is transitioned. Current GBF approaches move the mass in a cell treating the cell state as a “point”, similar to a particle in a particle filter. In the CGBF, instead of moving a point within the cell, the entire cell region is moved per Monte Carlo sample. This is illustrated in  FIG. 5 . By moving the entire cell for each sample, there is more assurance of obtaining a good distribution over each origination cell, without the need to randomly sample within the cell limits. 
         [0048]    Note from  FIG. 5  that a transitioned cell from the origination grid will generally land across boundaries of cells within the destination grid. The probability in the transitioned origination cell is apportioned out to the destination cells based on the fractional coverage it receives (from the transitioned cell). The function CellOverlapFraction in  FIG. 2  determines this fractional coverage. The cells do not need to be the same size from the origination grid to the destination grid. Each destination cell only gets the percentage of mass based on the portion of overlap. The second refinement exploits the motion symmetry to obtain two Monte Carlo samples from each sample made. This final refinement is detailed in the next section. 
       Exploiting the Motion Symmetry  
       [0049]    As was pointed out earlier, the transition distribution is symmetric with respect to the target&#39;s initial course. As a result, this symmetry can be exploited in the generation of the random samples. Each time the target is randomly moved throughout the time interval, the mirrored state can also be used as a sample. Thus, two random samples are generated for each random walk which results in 2 m samples being generated for the computational cost of generating m samples. 
         [0050]    Let (x 0 , y 0 ) be the (estimated) initial state of the target at the beginning of the update time interval and (x k , y k ) be the final predicted state at the end of the time interval (by undergoing k maneuvers). Similarly, let s 0  and θ 0  be the initial speed and course and s k  and θ k  be the predicted speed and course at the end of the interval. The mirrored state is defined as the resulting target state if it had simply reversed all its turns such that left turns become right turns, and vice versa. This mirrored state can be used as a second Monte Carlo sample for each sample actually computed. The mirrored state is straightforward to determine directly from the final state of the target after it completed all its maneuvers for the time interval. 
         [0051]    Referring to  FIG. 6 , the resulting target state is (x k , y k ) at the end of time interval. The mirrored target state is (x′ k , y′ k ) with speed s′ k  and course θ′ k . Let β be the angle between the initial course of the target and the angle to the predicted state, (x k , y k ). Let α=θ 0 +β. Therefore, the mirrored position can be found as: 
         [0000]    
       
         
           
             
               
                 
                   
                     Δ 
                      
                     
                         
                     
                      
                     x 
                   
                   = 
                   
                     
                       x 
                       k 
                     
                     - 
                     
                       x 
                       0 
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   82 
                 
               
             
             
               
                 
                   
                     Δ 
                      
                     
                         
                     
                      
                     y 
                   
                   = 
                   
                     
                       y 
                       k 
                     
                     - 
                     
                       y 
                       0 
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   83 
                 
               
             
             
               
                 
                   d 
                   = 
                   
                     
                       
                         Δ 
                          
                         
                             
                         
                          
                         
                           x 
                           2 
                         
                       
                       + 
                       
                         Δ 
                          
                         
                             
                         
                          
                         
                           y 
                           2 
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   84 
                 
               
             
             
               
                 
                   
                     sin 
                      
                     
                         
                     
                      
                     α 
                   
                   = 
                   
                     
                       Δ 
                        
                       
                           
                       
                        
                       x 
                     
                     d 
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   85 
                 
               
             
             
               
                 
                   
                     cos 
                      
                     
                         
                     
                      
                     α 
                   
                   = 
                   
                     
                       Δ 
                        
                       
                           
                       
                        
                       y 
                     
                     d 
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   86 
                 
               
             
             
               
                 
                   
                     x 
                     k 
                     ′ 
                   
                   = 
                   
                     
                       x 
                       0 
                     
                     + 
                     
                       d 
                        
                       
                           
                       
                        
                       
                         sin 
                          
                         
                           ( 
                           
                             
                               θ 
                               0 
                             
                             - 
                             β 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   87 
                 
               
             
             
               
                 
                   
                     y 
                     k 
                     ′ 
                   
                   = 
                   
                     
                       y 
                       0 
                     
                     + 
                     
                       d 
                        
                       
                           
                       
                        
                       
                         cos 
                          
                         
                           ( 
                           
                             
                               θ 
                               0 
                             
                             - 
                             β 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   88 
                 
               
             
           
         
       
     
       But θ 0 &#39; 2 β=θ 0 −(θ 0 +β)+θ 0 =2θ 0 −α. So,  
       [0052]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           x 
                           k 
                           ′ 
                         
                         = 
                           
                          
                         
                           
                             x 
                             0 
                           
                           + 
                           
                             d 
                              
                             
                                 
                             
                              
                             
                               sin 
                                
                               
                                 ( 
                                 
                                   
                                     2 
                                      
                                     
                                       θ 
                                       0 
                                     
                                   
                                   - 
                                   α 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             x 
                             0 
                           
                           + 
                           
                             d 
                              
                             
                               [ 
                               
                                 
                                   sin 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                    
                                   
                                     θ 
                                     0 
                                   
                                    
                                   cos 
                                    
                                   
                                       
                                   
                                    
                                   α 
                                 
                                 - 
                                 
                                   cos 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                    
                                   
                                     θ 
                                     0 
                                   
                                    
                                   sin 
                                    
                                   
                                       
                                   
                                    
                                   α 
                                 
                               
                               ] 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             x 
                             0 
                           
                           + 
                           
                             d 
                              
                             
                               [ 
                               
                                 
                                   sin 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                    
                                   
                                     θ 
                                     0 
                                   
                                    
                                   
                                     
                                       Δ 
                                        
                                       
                                           
                                       
                                        
                                       y 
                                     
                                     d 
                                   
                                 
                                 - 
                                 
                                   cos 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                    
                                   
                                     θ 
                                     0 
                                   
                                    
                                   
                                     
                                       Δ 
                                        
                                       
                                           
                                       
                                        
                                       x 
                                     
                                     d 
                                   
                                 
                               
                               ] 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             x 
                             0 
                           
                           + 
                           
                             Δ 
                              
                             
                                 
                             
                              
                             y 
                              
                             
                                 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             2 
                              
                             
                               θ 
                               0 
                             
                           
                           - 
                           
                             Δ 
                              
                             
                                 
                             
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                             x 
                              
                             
                                 
                             
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                             cos 
                              
                             
                                 
                             
                              
                             2 
                              
                             
                               θ 
                               0 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   89 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           y 
                           k 
                           ′ 
                         
                         = 
                           
                          
                         
                           
                             y 
                             0 
                           
                           + 
                           
                             d 
                              
                             
                                 
                             
                              
                             
                               cos 
                                
                               
                                 ( 
                                 
                                   
                                     2 
                                      
                                     
                                       θ 
                                       0 
                                     
                                   
                                   - 
                                   α 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             y 
                             0 
                           
                           + 
                           
                             d 
                              
                             
                               [ 
                               
                                 
                                   cos 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                    
                                   
                                     θ 
                                     0 
                                   
                                    
                                   cos 
                                    
                                   
                                       
                                   
                                    
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                                 - 
                                 
                                   sin 
                                    
                                   
                                       
                                   
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                                     θ 
                                     0 
                                   
                                    
                                   sin 
                                    
                                   
                                       
                                   
                                    
                                   α 
                                 
                               
                               ] 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             y 
                             0 
                           
                           + 
                           
                             d 
                              
                             
                               [ 
                               
                                 
                                   cos 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                    
                                   
                                     θ 
                                     0 
                                   
                                    
                                   
                                     
                                       Δ 
                                        
                                       
                                           
                                       
                                        
                                       y 
                                     
                                     d 
                                   
                                 
                                 - 
                                 
                                   sin 
                                    
                                   
                                       
                                   
                                    
                                   2 
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                                     θ 
                                     0 
                                   
                                    
                                   
                                     
                                       Δ 
                                        
                                       
                                           
                                       
                                        
                                       x 
                                     
                                     d 
                                   
                                 
                               
                               ] 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             y 
                             0 
                           
                           + 
                           
                             Δ 
                              
                             
                                 
                             
                              
                             y 
                              
                             
                                 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             2 
                              
                             
                               θ 
                               0 
                             
                           
                           - 
                           
                             Δ 
                              
                             
                                 
                             
                              
                             x 
                              
                             
                                 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             2 
                              
                             
                               θ 
                               0 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   90 
                 
               
             
           
         
       
     
         [0053]    Computing the mirrored speed and course is starightforward as well. By definition, the mirrored speed does not change; only the target&#39;s course. Let Δθ=θ k &#39; 1 θ 0  be the course difference from the initial course to the final course. Thus, the final course can be thought of as θ k =θ 0 +Δθ. The mirrored course must be θ k =θ 0   31  Δθ. Therefore the mirrored state is: 
         [0000]        x ′   k   =x   0 +( y   k   −y   0 )sin 2θ 0 −( x   k   −x   0 )cos 2θ 0    Eq. 91
 
         [0000]        y′   k   =y   0 +( y   k   −y   0 )cos 7θ 0 +( x   k   −x   0 )sin 2θ 0    Eq. 93
 
         [0000]      s′ k =s k    Eq. 93
 
         [0000]      θ′ k =θ 0 −Δθ=θ 0 −(θ k −θ 0 )=2θ 0 −θ k    Eq. -b 94
 
       Using Uniform RVs to Approximate Normal RVs  
       [0054]    Although many computer systems provide the ability to generate normal random variables, they typically require many more computations than the generation of uniform random variables. When performing the Monte Carlo sampling, the random variables must be drawn from a normal distribution (see  FIG. 2 ). Fortunately, the CGBF sampling process does not need “true” normal random variables; they only need to be approximately normal. A simple approximation was developed for the CGBF to exploit this idea. The normal-like random variables are generated using the following procedure. Suppose a total of m random samples are needed where each random sample is drawn from a normal distribution with a specified standard deviation. The number of random samples is divided into thirds. Let k=m/3. Then:
       generate k uniform random samples that are within one sigma,   generate k uniform random samples that are within two signma, and finally,   generate k uniform random samples that are within three sigma.       
 
         [0058]    This method is called the thirds procedure. For a “true” (1D) normal random variable, ˜68.2% of the samples would be within one sigma, ˜95.4% are within two sigma, and ˜99.7% are within three sigma[12]. Using the approximating procedure just described, 
         [0000]    
       
         
           
             
               k 
               + 
               
                 
                   1 
                   2 
                 
                  
                 k 
               
               + 
               
                 
                   1 
                   3 
                 
                  
                 k 
               
             
             = 
             
               
                 
                   k 
                   6 
                 
                  
                 
                   ( 
                   
                     6 
                     + 
                     3 
                     + 
                     2 
                   
                   ) 
                 
               
               = 
               
                 
                   
                     11 
                     6 
                   
                    
                   
                     ( 
                     
                       m 
                       3 
                     
                     ) 
                   
                 
                 = 
                 
                   
                     ( 
                     
                       61.1 
                        
                       % 
                     
                     ) 
                   
                    
                   m 
                 
               
             
           
         
       
     
         [0000]    samples will be within one sigma, 
         [0000]    
       
         
           
             k 
             = 
             
               
                 k 
                 + 
                 
                   
                     2 
                     3 
                   
                    
                   k 
                 
               
               = 
               
                 
                   
                     k 
                     3 
                   
                    
                   
                     ( 
                     
                       3 
                       + 
                       3 
                       + 
                       2 
                     
                     ) 
                   
                 
                 = 
                 
                   
                     
                       8 
                       3 
                     
                      
                     
                       ( 
                       
                         m 
                         3 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       ( 
                       
                         88.8 
                          
                         % 
                       
                       ) 
                     
                      
                     m 
                   
                 
               
             
           
         
       
     
         [0000]    samples will be within two sigma, and of course, 100% of the samples wil be within three sigma. These percentages are close enough to get the proper sampling. Thus, using this thirds procedure with uniform random variables, the needed normal random variables are efficiently obtained. 
         [0059]    Although the invention has been described in detail with particular reference to these preferred embodiments, other embodiments can achieve the same results. Variations and modifications of the present invention will be obvious to those skilled in the art and it is the indent of this application to cover, in the appended claims, all such modifications and equivalents. The entire disclosure and all references, applications, patents and publications cited above are hereby incorporated by reference.