Abstract:
A hostile missile engager senses the missile and supplies kinematic data to an interceptor missile fire control processor, which predicts the target&#39;s future location with the aid of a powered/unpowered identifier. The identifier passes the kinematic data through filters having lags for powered and unpowered operation, to produce residuals and residual covariances. The probabilities of powered and unpowered motion are determined, and thresholded. If the probability of powered motion or unpowered motion exceeds its threshold, the motion is deemed known. If the target is deemed to be powered, a set of the three-dimensional kinematic features is applied to a nine-state Kalman filter to produce an optimal state. If the target missile is unpowered, a set of the three-dimensional kinematic features corresponding to the second lag is applied to a six-state Kalman filter to produce an optimal state for the unpowered motion. The optimal states control the interceptor toward the target.

Description:
STATEMENT OF GOVERNMENTAL INTEREST 
     This invention was made with Government support under Contract/Grant N00024-03-C-6110 awarded by the Department of the Navy. The United States Government has certain rights in this invention. 
    
    
     BACKGROUND OF THE INVENTION 
     Modern transportation and military systems place great reliance on systems for tracking airborne or flying objects. In the context of transportation, air traffic control requires that the locations of aircraft be known, preferably with great precision. U.S. Pat. Nos. 7,009,554 and 7,180,443, issued Mar. 7, 2006 and Feb. 20, 2007, respectively, in the name of Mookerjee et al. describe tracking systems generally. In the context of military systems, there is a need to track missiles, both friendly and hostile, and which may be either unpowered (ballistic) or powered (boost). U.S. Pat. No. 7,026,980, issued Apr. 11, 2006 in the name of Mavroudakis et al. describes a system for identifying and tracking a target missile. A missile tracking system described in U.S. patent application Ser. No. 11/879,538, filed Jul. 18, 2007 in the name of Luu et al. describes a system which estimates the location of a missile. Luu et al. make use of a “staging logic and early thrust termination” function for determining the boost or ballistic state, to improve the accuracy of their estimator. A system and method for tracking and engaging a hostile missile is described in U.S. patent application Ser. No. 11/868,554, filed Oct. 8, 2007 in the name of Luu, et al. This system and method also attempts to determine the boost or ballistic state of the missile being tracked. U.S. patent application Ser. No. 11/253,309, filed Oct. 19, 2005 in the name of Wolf, now U.S. Pat. No. 7,295,149, describes a system including a radar sensor, a hostile missile, an interceptor missile, a boost/ballistic determination arrangement, and an interceptor controller using the boost/ballistic determination to aid in interceptor guidance. 
     It will be clear that an accurate estimate of the location and path of a target missile to be engaged by an interceptor missile requires some knowledge of the boost and ballistic operating condition of the target missile. Improved and or alternative boost/ballistic estimating arrangements are desired. 
     SUMMARY OF THE INVENTION 
     A method according to an aspect of the invention is for engaging a target missile with an interceptor missile. The method comprises the steps of sensing three-dimensional or full kinematic features, including at least position of the target missile, to thereby generate sensed kinematic data. The kinematic data includes position, which is at least one of slant range, bearing and elevation. The current value of the three-dimensional kinematic features is or are stored, together with a set of previous values of the three-dimensional kinematic features. At least a portion of the current value of the sensed kinematic data is passed through a first filter having a fixed first filter lag. The first filter is suited for maneuver motion of the missile, and generates first residuals and first residual covariances. At least a portion of the current value of the sensed kinematic data is passed through a second filter having a fixed second filter lag. The second filter is suited for non-maneuver motion of the missile, to generate second residuals and second residual covariances. From the first residuals, the first residual covariances, the second residuals, and the second residual covariances, the probability of being in maneuvering motion and the probability of being in non-maneuvering motion are determined. The probability of being in maneuvering motion tends toward unity when the target missile is in maneuvering motion and tends toward zero when the target missile is in non-maneuvering motion. The probability of being in non-maneuvering motion tends toward unity when the target missile is in non-maneuvering motion and tends toward zero when the target missile is in maneuvering motion. The probability of being in maneuvering motion and the probability of being in non-maneuvering motion are compared with at least one threshold value The target missile is deemed to be in maneuvering motion if the probability of being in maneuvering motion exceeds the threshold value, and the target missile is deemed to be in non-maneuvering motion if the probability of being in non-maneuvering motion exceeds the threshold value. If the target missile is deemed to be in maneuvering motion, a set of the three-dimensional kinematic features corresponding to the first lag is applied to a nine-state Kalman filter to produce an optimal state for the maneuvering motion. The nine states of the nine-state Kalman filter correspond with three position states, three velocity states, and three acceleration states. If the target missile is deemed to be in non-maneuvering motion, a set of the three-dimensional kinematic features corresponding to the second lag is applied to a six-state Kalman filter to produce an optimal state for the non-maneuvering motion. The six states of the six-state Kalman filter correspond to three position states and three velocity states. The optimal states are applied to at least (a) a missile engagement planner for initialization of interceptor guidance and (b) to an interceptor for guidance of the interceptor toward the target missile. 
     A method according to an aspect of the invention is for estimating or determining transitions between powered and unpowered motion of a target missile. The method comprises the steps of sensing three-dimensional kinematic features of the target. The three-dimensional kinematic features include position, which includes at least one of slant range, bearing and elevation. The sensing of the three-dimensional kinematic features generates sensed kinematic data. The current value of the full set of three-dimensional kinematic features is stored in a memory, which may be a computer memory, together with a set of previous values of the full three-dimensional kinematic features. At least a portion of the current value of the sensed kinematic data is passed through a first filter having a fixed first filter lag, the first filter being suited for or matching powered motion of the missile, to thereby generate first residuals and first residual covariances. At least a portion of the current value of the sensed kinematic data is passed through a second filter having a fixed second filter lag. The second filter is suited for or matching non-powered motion of the missile, to generate second residuals and second residual covariances. From the first residuals, the first residual covariances, the second residuals, and the second residual covariances, determining the probability of being in powered motion and the probability of being in non-powered motion, which probability of being in powered motion tends toward unity when the missile is in powered motion and tends toward zero when the missile is in non-powered motion, and which probability of being in non-powered motion tends toward unity when the missile is in non-powered motion and tends toward zero when the missile is in powered motion. The probability of being in powered motion and the probability of being in non-powered motion are compared with at least one threshold value, to thereby deem the missile to be in powered motion if the probability of being in powered motion exceeds the threshold value, and to deem the missile to be in non-powered motion if the probability of being in non-powered motion exceeds the threshold value. If the missile is deemed to be in powered motion, a set of the full or three-dimensional kinematic features corresponding to the first lag are applied to a nine-state (position velocity acceleration) Kalman filter to produce an optimal state for the powered motion. If the missile is deemed to be in non-powered motion, a set of the three-dimensional or full kinematic features corresponding to the second lag are applied to a six-state Kalman filter (three position states and three velocity states) to produce an optimal state for the non-powered motion. 
     A method according to another aspect of the invention is for estimating or determining transitions between ACM (Attitude Control Module) and non-ACM motion of an airborne vehicle. The method comprises the steps of sensing three-dimensional kinematic features (including position, which is at least one of slant range, bearing and elevation) of the target, to thereby generate sensed kinematic data. The method includes the steps of storing, as in computer memory, the current value of the three-dimensional or full) kinematic features, and a set of previous values of the three-dimensional kinematic features. At least a portion of the current value of the sensed kinematic data is passed through a first filter having a fixed first filter lag. The first filter is suited for ACM motion of the vehicle to generate first residuals and first residual covariances. At least a portion of the current value of the sensed kinematic data is passed through a second filter having a fixed second filter lag. The second filter is suited for non-ACM motion of the vehicle to generate second residuals and second residual covariances. From (a) the first residuals, the (b) first residual covariances, (c) the second residuals, and (d) the second residual covariances, the probability of being in ACM motion and the probability of being in non-ACM motion are determined. The probability of being in ACM motion tends toward unity when the vehicle is in ACM motion and tends toward zero when the vehicle is in non-ACM motion. The probability of being in non-ACM motion tends toward unity when the vehicle is in non-ACM motion and tends toward zero when the vehicle is in ACM motion. The probability of being in ACM motion and the probability of being in non-ACM motion are compared with at least one threshold value. The vehicle is deemed to be in ACM motion if the probability of being in ACM motion exceeds the threshold value, and is deemed to be in non-ACM motion if the probability of being in non-ACM motion exceeds the threshold value. If the vehicle is deemed to be in ACM motion, a set of the three-dimensional (full) kinematic features corresponding to the first lag is applied to a nine-state Kalman filter to thereby produce an optimal state for the ACM motion. If the vehicle is deemed to be in non-ACM motion, a set of the three-dimensional (full) kinematic features corresponding to the second lag is applied to a six-state Kalman filter (three position states and three velocity states) to produce an optimal state for the non-ACM motion. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWING 
         FIG. 1  is a simplified block diagram of a radar arrangement for sensing the kinematic features of a target which may be powered or unpowered, for processing the kinematic features to ultimately produce estimated states of the target, and for using the estimated states of the target to engage the target with an interceptor missile; 
         FIG. 2  is a simplified block diagram of a fire control system as described in U.S. patent application Ser. No. 11/430,535, now U.S. Pat. No. 7,511,252, which may be used in the arrangement of  FIG. 1 ; and 
         FIG. 3  is a simplified block diagram of a target missile boost/ballistic condition determination arrangement according to an aspect of the invention, which may be used in the target missile tracking system of  FIG. 2 . 
     
    
    
     DESCRIPTION OF THE INVENTION 
     In  FIG. 1 , a system  10  includes a radar  12  with an antenna  12   a . Radar  12  generates electromagnetic signals which are transmitted by antenna  12   a , as suggested by “lightning bolt”  14 . If there is a target within range of the radar  12 , a portion of the transmitted signals will be reflected back toward the radar. In this case, a target illustrated as a missile  16  is present. Missile  16  may be in powered (boost) flight, or it may be in unpowered (ballistic) flight. In any case, target  16  causes reflected or return signals return along the same path illustrated as  14  to the radar system. The radar system  12  processes the reflected signals to produce information relating to at least target slant range, and possibly bearing and elevation. These are referred to as “kinematic” features of the target. The kinematic features appear at an output port  12   o  of radar  12 , and are coupled to an interceptor missile fire control system  94 . Fire control system  94  estimates the actual current location of the target missile  16 , and estimates its trajectory or path  16   p . Fire control system  94  generates fire control solutions, initiates the interceptor missile  322  while it is still at the interceptor launcher  92 . Fire control system  94  commands a launch of the interceptor missile, and also tracks the location of the interceptor missile  322 . Fire control system  94  guides the interceptor  322  toward the expected intercept point  88  after launch. 
     In  FIG. 2 , the sensor measurements block  310  represents the receipt of signals by way of path  3102  from radar  12  of  FIG. 1  and by way of a path  310 , from another source, designated Overhead Non-Imaging InfraRed (ONIR). The sensor measurements are applied to a track filter  312 , which multihypothesis track filter combines a-priori knowledge of the characteristics of various possible types of threat missiles with the track data from the sensors, to establish separate estimates of the target hypotheses&#39; states (at least position and velocity) for n possible hypotheses. A multihypothesis track filter maintains separate filters for each hypothesis and uses a priori information about each hypothesis to improve the state estimates for each hypothesis over (by comparison with) a filter that uses no a priori information. One implementation of a multihypothesis filter is found in U.S. patent application Ser. No. 11/072,902, filed Mar. 4, 2005 in the name of Mavroudakis et al., and entitled “Missile Identification and Tracking System and Method,” now U.S. Pat. No. 7,026,980. The separate estimates consisting of n possible threat/stage hypotheses appear at output ports  1 ,  2 ,  3 , . . . , n of the track filter  312 , and are made available to a typing processor represented as a block  314  and to a target propagator represented as a block  316 . Typing algorithm or block  314  uses multiple metrics to determine the likelihood that each hypothesis is correct. These likelihoods sum to unity or one, and this summing characteristic is later used to generate weighting factors for each threat hypothesis. The track filter states from track filter block  312  of  FIG. 3  and the likelihoods from typing block  314  are applied to a multihypothesis target propagator block  316 . Target propagator block  316  propagates into the future all those threat hypotheses having likelihoods above a threshold level. In one version of the target propagator, the threshold level is fixed. As illustrated in  FIG. 3 , target propagator block  316  produces n-m propagated threat states, where m is the number of hypotheses with likelihoods falling below the threshold. Significant computational resource savings arise from not propagating the low-likelihood threat states. Early in an engagement little track data is available from which the track filter can produce estimated track states, so there is little information on which to base the discarding of hypotheses, so most or all hypotheses are propagated. As more data becomes available from the sensors the typing block  314  is expected to be able to discount certain hypotheses and to assign low probabilities to other hypotheses. The propagated multihypothesis threat states from target propagator block  316  of  FIG. 3  are applied to an engagement planner illustrated as a block  318  together with the likelihood information from typing block  314 . Engagement planner  318  also receives interceptor missile flyout data or tables from a block illustrated as  319 . Engagement planner  318  uses a subset of the hypotheses to determine an optimal way to engage the target. In order to conserve computational resources, the engagement planner  318  ignores those hypotheses having likelihoods below the threshold. The optimal engagement information is coupled to initialize an interceptor guidance processing illustrated as a block  320 . Interceptor guidance block  320  of  FIG. 3  receives optimal engagement information from block  318 , propagated threat missile states from  316 , likelihood information from block  314 , and interceptor state information from a block  324 . The interceptor state information contains information describing the state of the interceptor (such as position, velocity, and acceleration) at the time that the interceptor guidance function block  320  is entered from block  324 . Interceptor guidance block  320  estimates the location of the target at the time of intercept. Launch and intercept guidance commands are applied from interceptor guidance block  320  to the interceptor  322 . The intercept guidance commands are preferably based on a boost/ballistic phase guidance algorithm which uses the propagated hypothesis positions and likelihoods to estimate the location of the target at intercept. During guidance a composite track is formed or generated that represents a weighted-average position of the hypotheses. The weighted-average position is a function of each threat hypothesis&#39; position and its associated likelihood. During the course of an engagement, the target state and hypothesis likelihoods are updated as more information becomes available. The track filter  312 , the target propagator  316 , the engagement planner  318 , and the interceptor guidance  320  all take advantage of the new data to refine their solutions. 
       FIG. 3  is a simplified block diagram illustrating a boost/ballistic or powered/unpowered estimator according to an aspect of the invention, which may be used in the arrangement of  FIG. 2  or in any scenario in which the boost or ballistic state of a missile is to be determined. In  FIG. 3 , the threat missile hypothesis state estimates are coupled by way of paths illustrated as  18  to a memory  20 , and to first and second kinematic detection filters illustrated as blocks  22  and  24 , and referred to jointly as a Power Detector  96 . First filter block  22  has its lag or delay selected for the case of powered motion of the target, and second filter block  24  has its lag or delay selected for the case of unpowered motion of the target When data is outputted from the filters it represents the state of affairs a short duration prior to the current state of affairs. This short duration is the “lag” of the filter. 
     The output at port  22   o  of first filter block  22  of  FIG. 3  represents the first residuals and the first residual covariances of the filtered data and these are coupled by way of a path  23  to a first input port  26   i   1  of a processor illustrated as a block  26 . The output at port  24   o  of second filter block  24  of  FIG. 1  represents the second residuals and the second residual covariances of the filtered data, and these are coupled by way of a path  25  to a second input port  26   i   2  of processor  26 . 
     Processor  26  of  FIG. 1  receives the first residuals and first residual covariances at input port  26   i   1  and receives the second residuals and the second residual covariances at input port  26   i   2 . Processor  26  processes the information and produces (a) at an output port  26   o   1  a measure or indication of the probability that the target missile is in a powered state and (b) at an output port  26   o   2  a measure or indication of the probability that the target missile is in an unpowered state. The signal appearing at output port  26   o   1 , representing the probability of being in powered motion, tends toward unity when the missile is in powered motion and tends toward zero when the missile is in non-powered motion. The signal appearing at output port  26   o   2 , representing the probability of being in unpowered motion, tends toward unity when the missile is in non-powered motion and tends toward zero when the missile is in powered motion. 
     The signal representing the probability that the missile is in powered flight which appears at output port  260   1  of block  26  of  FIG. 1  is coupled to a threshold function illustrated as a block  28 . Threshold function  28  compares the signal with a threshold value to make a determination (PWR) as to whether the missile is in powered flight or not. Similarly, the signal representing the probability that the missile is in unpowered flight which appears at output port  260   2  is coupled to a threshold function illustrated as a block  28 . Threshold function  28  compares the signal with a threshold value to make a determination (  PWR ) as to whether the missile is in unpowered flight or not. By nature of the processor, the probability that the missile is in powered flight is one (1) minus the probability that the missile is in unpowered flight. Similarly the probability that the missile is in unpowered flight is one (1) minus the probability that the missile is in powered flight. 
     Regardless of whether the missile is in powered or unpowered flight, both a 9-State Kalman “powered” filter designated  32  and a 6-State Kalman “unpowered” filter designated  34  begin track filtering by initializing with the kinematic data accessed via paths  98  and  27 , respectively, from data memory  20 . Filtered data from the 9-State Kalman powered motion filter  32  represents position, velocity and acceleration in the x direction, y direction, and z direction (x,{dot over (x)},{umlaut over (x)},y,{dot over (y)},ÿ,z,ż,{umlaut over (z)}). The filtered data from the 9-State Kalman “powered motion” filter  32  is outputted or coupled by way of a path  29  to a 9-State filtered data memory designated  36  and by way of a path  31  to an end user designated  40  if the threshold test determines the missile to be in powered flight. If the threshold test determines that the missile is not in powered flight, the filtered data from powered flight filter  32  is not stored or used. Similarly, the filtered data from the 6-State Kalman “unpowered motion” data filter which represents position and velocity in the x direction, y direction, and z direction (x,{dot over (x)},y,{dot over (y)},z,ż), is sent by way of a path  33  to a 6-state filtered data memory  38  and to the end user  40  if the threshold test designated  28  determines the missile to be in unpowered flight, and is not stored or used if the threshold test determines that the missile is not in unpowered flight. 
     First “powered motion” detector  22  and second “unpowered motion” detector  24  of  FIG. 1  continually filter the raw data provided by radar  12   o . Prior to a time at which the missile  16  changes from unpowered flight to powered flight, 6-state filter  34  will have been filtering the data and storing information in 6-state memory  38 . When missile  16  changes from unpowered flight to powered flight, threshold test block  28  alerts the 9-state Kalman “powered motion” filter  32  via path  37 . The 9-state Kalman filter for powered motion will access the 6-state filtered data memory  38  via path  39  for the filtered data provided at t-lag, where t-lag represents the current time minus the lag duration. This data initializes the 9-State “powered flight” filter  32  with the most accurate available data. The raw data in memory  20  will be accessed by the 9-state Kalman filter,  32 , for the data points in the interval from t-lag to the current time and filters the data and continually sends the filtered data to the end user  40 . This condition continues until the missile switches from powered motion to unpowered motion and the threshold test  28  alerts the change. The 6-State Kalman filter for unpowered motion,  34 , will then access the t-lag data stored in the 9-state filtered data memory  36  via link  41 , and initialize using the t-lag stored data. Six-state Kalman filter  34  then continues to filter the data in the interval from t-lag to the current time data which is stored in memory  20 , and provides filtered data to end user  40 . 
     In  FIG. 3 , the Power Detector  96  includes two range filters; a range (α,β) detector filter  24  and a range (α,β,γ) detector filter  22 . A fixed lag is used in filtering. Likelihood functions then utilize the filter outputs to calculate probabilities. The Power Detector  96  predictor equations for α,β detector filter  24  range direction are
 
 r   k   p   =r   k-1   s   +{dot over (r)}   k-1   s Δ
 
 {dot over (r)}   k   p   ={dot over (r)}   k-1   s  
 
where:
 
r k   p  is the current predicted position data point;
 
r k-1   s  is the most recent corrected position data point;
 
{dot over (r)} k-1   s Δ is the most recent corrected velocity data point;
 
The power detector  96  α-β detector filter  24  gain equations in the x direction are given by
 
               α   r     =     1   -       (     τ     T   +   τ       )     2                     β   r     =     2   ⁢       (     T     T   +   τ       )     2             
where:
 
α is the filter position gain;
 
τ is the fixed lag; and
 
β is the filter velocity gain.
 
The power detector  96  α-β detector filter  24  corrector equations in the x direction are
   r   k   s   =r   k   p +α r ( r   k   m   −r   k   p ) 
     
       
         
           
             
               
                 r 
                 . 
               
               k 
               s 
             
             = 
             
               
                 
                   r 
                   . 
                 
                 k 
                 p 
               
               + 
               
                 
                   
                     β 
                     r 
                   
                   Δ 
                 
                 ⁢ 
                 
                   ( 
                   
                     
                       r 
                       k 
                       m 
                     
                     - 
                     
                       r 
                       k 
                       p 
                     
                   
                   ) 
                 
               
             
           
         
       
     
     The power detector  96  α-β-γ detector filter  22 } predictor equations in the x direction are 
               r   k   p     =       r     k   -   1     s     +     Δ   ⁢       r   .       k   -   1     s       +       1   2     ⁢       r   ¨       k   -   1     s     ⁢     Δ   2                         r   .     k   p     =         r   .       k   -   1     s     +     Δ   ⁢       r   ¨       k   -   1     s                         r   ¨     k   p     =       r   ¨       k   -   1     s           
The power detector  96  α-β-γ detector filter  22  gain equations in the x direction are
 
               α   r     =     1   -       (     τ     T   +   τ       )     3                     β   r     =       3   ⁢     (       2   ⁢   τ     +   T     )     ⁢     T   2           (     τ   +   T     )     3                     γ   r     =       6   ⁢     T   3           (     τ   +   T     )     3             
The power detector  96  α-β-γ detector filter  22  corrector equations in the x direction are
 
     
       
         
           
             
               r 
               k 
               s 
             
             = 
             
               
                 x 
                 k 
                 p 
               
               + 
               
                 
                   α 
                   r 
                 
                 ⁡ 
                 
                   ( 
                   
                     
                       r 
                       k 
                       m 
                     
                     - 
                     
                       r 
                       k 
                       p 
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 r 
                 . 
               
               k 
               s 
             
             = 
             
               
                 
                   r 
                   . 
                 
                 k 
                 p 
               
               + 
               
                 
                   
                     β 
                     r 
                   
                   Δ 
                 
                 ⁢ 
                 
                   ( 
                   
                     
                       r 
                       k 
                       m 
                     
                     - 
                     
                       r 
                       k 
                       p 
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 r 
                 ¨ 
               
               k 
               s 
             
             = 
             
               
                 
                   r 
                   ¨ 
                 
                 k 
                 p 
               
               + 
               
                 
                   
                     γ 
                     r 
                   
                   
                     Δ 
                     2 
                   
                 
                 ⁢ 
                 
                   ( 
                   
                     
                       r 
                       k 
                       m 
                     
                     - 
                     
                       r 
                       k 
                       p 
                     
                   
                   ) 
                 
               
             
           
         
       
     
     A sliding window memory is maintained in block  20  consisting of current data values as well as a set number of historical points. Specifically, the measurement values for position and velocity (x,y,z and {dot over (x)},{dot over (y)},ż) are maintained. Current range values are also found in the memory. 
     The Optimal tracking filters  94  include two optimal tracking filters  32  and  34 . Filter  34  is an optimal α,β filter for x,y, and z measurements and the other 32 filter is an optimal α,β,γ filter for {dot over (x)},{dot over (y)}, and z measurements. The (α,β) filter  34  accesses the data from the 9-State Filtered Data Memory  36  and the α,β,γ filter  32  accessed the data from the 6-State Filtered Data Memory  38  once the powered detector has detected that the vehicle has transitioned between the states of powered and unpowered. 
     The Optimal filter  94  α-β optimal filter  34  predictor equations in the x direction are
 
 x   k   p   =x   k-1   s   +{dot over (x)}   k-1   s Δ
 
 {dot over (x)}   k   p   ={dot over (x)}   k-1   s ,
 
the Optimal tracking filter  94  α,β filter  34  gain equations in the x direction are
 
               α   x     =     1   -       (     τ     T   +   τ       )     2                       β   x     =     2   ⁢       (     T     T   +   τ       )     2         ,           ⁢   and         
the Optimal tracking filter  94  α-β filter  34  corrector equations in the x direction are
 
               x   k   s     =       x   k   p     +       α   x     ⁡     (       x   k   m     -     x   k   p       )                         x   .     k   s     =         x   .     k   p     +         β   x     Δ     ⁢       (       r   k   m     -     r   k   p       )     .               
The Optimal tracking filter  94  α-β filter  34  predictor equations in the y direction are
   y   k   p   =y   k-1   s   +{dot over (y)}   k-1   s Δ   {dot over (y)}   k   p   ={dot over (y)}   k-1   s , 
the Optimal tracking filter  94  α-β filter  34  gain equations in the y direction are
 
               α   y     =     1   -       (     τ     T   +   τ       )     2                       β   y     =     2   ⁢       (     T     T   +   τ       )     2         ,           ⁢   and         
the Optimal tracking filter  94  α-β filter  34  corrector equations in the y direction are
 
               y   k   s     =       y   k   p     +       α   y     ⁡     (       r   k   m     -     r   k   p       )                         y   .     k   s     =         y   .     k   p     +         β   y     Δ     ⁢       (       y   k   m     -     y   k   p       )     .               
The Optimal tracking filter  94  α-β filter  34  predictor equations in the z direction are
   z   k   p   =z   k-1   s   +ż   k-1   s Δ   ż   k   p   =ż   k-1   s , 
the Optimal tracking filter  94  α-β filter  34  gain equations in the z direction are
 
               α   z     =     1   -       (     τ     T   +   τ       )     2                       β   z     =     2   ⁢       (     T     T   +   τ       )     2         ,           ⁢   and         
the Optimal tracking filter  94  α-β filter  34  corrector equations in the z direction are
 
     
       
         
           
             
               z 
               k 
               s 
             
             = 
             
               
                 z 
                 k 
                 p 
               
               + 
               
                 
                   α 
                   z 
                 
                 ⁡ 
                 
                   ( 
                   
                     
                       z 
                       k 
                       m 
                     
                     - 
                     
                       z 
                       k 
                       p 
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 z 
                 . 
               
               k 
               s 
             
             = 
             
               
                 
                   z 
                   . 
                 
                 k 
                 p 
               
               + 
               
                 
                   
                     β 
                     z 
                   
                   Δ 
                 
                 ⁢ 
                 
                   ( 
                   
                     
                       z 
                       k 
                       m 
                     
                     - 
                     
                       z 
                       k 
                       p 
                     
                   
                   ) 
                 
               
             
           
         
       
     
     The Optimal tracking filter  94  α-β-γ filter filter equations in the x direction are 
     
       
         
           
             
               x 
               k 
               p 
             
             = 
             
               
                 x 
                 
                   k 
                   - 
                   1 
                 
                 s 
               
               + 
               
                 Δ 
                 ⁢ 
                 
                   
                     x 
                     . 
                   
                   
                     k 
                     - 
                     1 
                   
                   s 
                 
               
               + 
               
                 
                   1 
                   2 
                 
                 ⁢ 
                 
                   
                     x 
                     ¨ 
                   
                   
                     x 
                     - 
                     1 
                   
                   s 
                 
                 ⁢ 
                 
                   Δ 
                   2 
                 
               
             
           
         
       
       
         
           
             
               
                 x 
                 . 
               
               x 
               p 
             
             = 
             
               
                 
                   x 
                   . 
                 
                 
                   k 
                   - 
                   1 
                 
                 s 
               
               + 
               
                 Δ 
                 ⁢ 
                 
                   
                     x 
                     ¨ 
                   
                   
                     k 
                     - 
                     1 
                   
                   s 
                 
               
             
           
         
       
       
         
           
             
               
                 x 
                 ¨ 
               
               k 
               p 
             
             = 
             
               
                 
                   x 
                   ¨ 
                 
                 
                   k 
                   - 
                   1 
                 
                 s 
               
               . 
             
           
         
       
     
     The Optimal tracking filter  94  α-β-γ filter  32  gain equations in the x direction are 
               α   x     =     1   -       (     τ     T   +   τ       )     3                     β   x     =       3   ⁢     (       2   ⁢   τ     +   T     )     ⁢     T   2           (     τ   +   T     )     3                     γ   x     =       6   ⁢     T   3           (     τ   +   T     )     3             
The Optimal tracking filter  94  α-β-γ filter  32  corrector equations in the x direction are
 
               x   k   s     =       x   k   p     +       α   x     ⁡     (       x   x   m     -     x   k   p       )                         x   .     k   s     =         x   .     k   p     +         β   x     Δ     ⁢     (       x   k   m     -     x   k   p       )                         x   ¨     k   s     =         x   ¨     k   p     +         γ   x       Δ   2       ⁢     (       x   k   m     -     x   k   p       )               
The Optimal tracking filter  94  α-β-γ filter  32  predictor equations in the y direction are
 
               y   k   p     =       y     k   -   1     s     +     Δ   ⁢       y   .       k   -   1     s       +       1   2     ⁢       y   ¨       k   -   1     s     ⁢     Δ   2                         y   .     k   p     =         y   .       k   -   1     s     +     Δ   ⁢       y   ¨       k   -   1     s                           y   ¨     k   p     =       y   ¨       k   -   1     s       ,         
the Optimal tracking filter  94  α-β-γ filter  32  gain equations in the y direction are
 
               α   y     =     1   -       (     τ     T   +   τ       )     3                     β   y     =       3   ⁢     (       2   ⁢   τ     +   T     )     ⁢     T   2           (     τ   +   T     )     3                       γ   y     =       6   ⁢     T   3           (     τ   +   T     )     3         ,           ⁢   and         
the Optimal tracking filter  94  α-β-γ filter  32  corrector equations in the y direction are
   y   k   s   =y   k   p +α y ( y   k   m   −y   k   p ) 
                 y   .     k   s     =         y   .     k   p     +       β   Δ     ⁢     (       y   k   m     -     y   k   p       )                         y   ¨     k   s     =         y   ¨     k   p     +         γ   y       Δ   2       ⁢     (       y   k   m     -     y   k   p       )               
The Optimal tracking filter  94  α-β-γ filter  32  predictor equations in the z direction are
 
               z   k   p     =       z     k   -   1     s     +     Δ   ⁢       z   .       k   -   1     s       +       1   2     ⁢       z   ¨       k   -   1     s     ⁢     Δ   2                         z   .     k   p     =         z   .       k   -   1     s     +     Δ   ⁢       z   ¨       k   -   1     s                           z   ¨     k   p     =       z   ¨       k   -   1     s       ,         
the Optimal tracking filter  94  α-β-γ filter  32  gain equations in the z direction are
 
               α   z     =     1   -       (     τ     T   +   τ       )     3                     β   z     =       3   ⁢     (       2   ⁢   τ     +   T     )     ⁢     T   2           (     τ   +   T     )     3                       γ   z     =       6   ⁢     T   3           (     τ   +   T     )     3         ,           ⁢   and         
the Optimal tracking filter  94  α-β-γ filter  32  corrector equations in the z direction are
 
     
       
         
           
             
               z 
               k 
               s 
             
             = 
             
               
                 z 
                 k 
                 p 
               
               + 
               
                 
                   α 
                   z 
                 
                 ⁡ 
                 
                   ( 
                   
                     
                       z 
                       k 
                       m 
                     
                     - 
                     
                       z 
                       k 
                       p 
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 z 
                 . 
               
               k 
               s 
             
             = 
             
               
                 
                   z 
                   . 
                 
                 k 
                 p 
               
               + 
               
                 
                   
                     β 
                     z 
                   
                   Δ 
                 
                 ⁢ 
                 
                   ( 
                   
                     
                       z 
                       k 
                       m 
                     
                     - 
                     
                       z 
                       k 
                       p 
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 z 
                 ¨ 
               
               k 
               s 
             
             = 
             
               
                 
                   z 
                   ¨ 
                 
                 k 
                 p 
               
               + 
               
                 
                   
                     γ 
                     z 
                   
                   
                     Δ 
                     2 
                   
                 
                 ⁢ 
                 
                   ( 
                   
                     
                       z 
                       k 
                       m 
                     
                     - 
                     
                       z 
                       k 
                       p 
                     
                   
                   ) 
                 
               
             
           
         
       
     
     In order to determine the Mode Probability, let μ i   n  be the mode probability for the modes i=1, 2 
     where i=1 for the α-β filter  34  for mode, i=2 for the α-β-γ filter  32  mode, and n=observation time. Then μ i   n  can be described by 
                 μ   i   n     =         Λ   i   n     ⁢     μ   i     n   -   1               Λ   1   n     ⁢     μ   1     n   -   1         +       Λ   2   n     ⁢     μ   2     n   -   1               ,     i   =   1     ,   2         
where Λ i   n  is the likelihood function given by
 
               Λ   i   n     =       e     [       -     1   2       ⁢       (     v   i   n     )     T     ⁢       (     S   i   n     )       -   1       ⁢     (     v   i   n     )       ]             (     2   ⁢   π     )       3   2       ⁢            S   i   n            1   2                 
where ν i   n  is the filter residual for mode i, and S i   n  is the filter residual covariance for mode i given by
 
                 S   i   n     =     (         σ   ⁡     (   n   )       mi   2       (     1   -     α   i   n       )       )       ,         
where |S 1   n | is the determinant of S i   n , α i   n  is the filter gain for mode i, and
 
               σ   ⁡     (   n   )         m   i     2         
is the measurement error variance.
 
     For the case of optimal gains in optimal gains track filtering, the lag τ must be evaluated using the standard deviation of measurement error variance denoted here by σ. In the α-β filter  34 , the optimal lag τ may be equated as 
             τ   =         T     1   /   5       ⁡     (       3   ⁢   n   ⁢           ⁢     σ   m         2   ⁢   A       )         2   /   5             
where σ represents the measurement error, and the subscript m may be written as x, y, or z depending on the coordinate being filtered. The term A represents the powered acceleration, T the data rate, and n the confidence factor.
 
     For the lag calculation in the α-β-γ filter of  32 , the optimal lag τ may be equated as 
             τ   =         T     1   /   7       ⁡     (       15   ⁢   n   ⁢           ⁢     σ   m         2   ⁢   J       )         2   /   7             
where σ represents the measurement error, and the subscript m may be written as x, y, or z depending on the coordinate being filtered. The term A represents the powered acceleration, T the data rate, and n the confidence factor.
 
     In the case of the optimal filters  94 , measurement error must be calculated in order to determine the lag. The measurement error in Cartesian coordinates is 
               R   m     =       J   ⁡     [           σ   r   2                                                   r   ⁢           ⁢     σ   AZ   2                                                     r   ⁢           ⁢     σ   EL   2             ]       ⁢     J   T                 with             J   =     [             cos   ⁡     (   EL   )       ⁢     sin   ⁡     (   AZ   )                 cos   ⁡     (   EL   )       ⁢     sin   ⁡     (   AZ   )                 -     sin   ⁡     (   EL   )         ⁢     sin   ⁡     (   AZ   )                     cos   ⁡     (   EL   )       ⁢     cas   ⁡     (   AZ   )                 -     cos   ⁡     (   EL   )         ⁢     sin   ⁡     (   AZ   )                 -     sin   ⁡     (   EL   )         ⁢     cos   ⁡     (   AZ   )                   sin   ⁡     (   EL   )           0         cos   ⁡     (   EL   )             ]           
Thus,
 σ m     x     2   =R   m (1,1)           σ m     x   =√{square root over ( R   m (1,1))} σ m     y     2   =R   m (2,2)         σ m     y   =√{square root over ( R   m (2,2))} σ m     z     2   =R   m (3,3)         σ m     z   =√{square root over ( R   m (3,3))}
     σ m     x   , σ m     y   , σ m     z    are used in calculating optimal lag τ, which in turn is used to calculate the optimal filter gains (α-β-γ) for (α-β) filter and (α-β-γ) filter.