Abstract:
This invention provides a system and method for autonomously tracking a moving target from unmanned aerial vehicles (UAVs) with a variety of airframe and sensor payload capabilities so that the target remains within the vehicle&#39;s sensor field of view regardless of the specific target motion patterns. The invention uses information about target location, UAV platform type and states, sensor payload capability, and ratio of target-to-UAV speeds to select from a suite of sub-algorithms, each of which generates desired platform positions (in form of waypoints) and/or sensor orientation commands to keep the target in view.

Description:
GOVERNMENT LICENSE RIGHTS 
       [0001]    The U.S. Government may have certain rights in the present invention as provided for by the terms of Contract No. FA8650-04-C-7142 with the Defense Advanced Research Projects Agency. 
     
    
     BACKGROUND TECHNOLOGY 
       [0002]    Unmanned aerial vehicles (UAVs) are remotely piloted or self-piloted aircraft that can carry cameras, sensors, communications equipment, or other payloads. They have been used in a reconnaissance and intelligence-gathering role for many years. More recently, UAVs have been developed for the purpose of surveillance and target tracking. 
         [0003]    Autonomous surveillance and target tracking performed by UAVs in either military or civilian environments is becoming an important aspect of intelligence-gathering. Typically, when a target is being tracked from aerial vehicles (e.g. a UAV), human operators must closely monitor imagery streamed from the aircraft to assess target behavior and ensure that the target continues to be in view. 
       SUMMARY 
       [0004]    This invention provides a system and method for autonomously tracking a moving target from UAVs with a variety of airframe and sensor payload capabilities so that the target remains within the vehicle&#39;s sensor field of view regardless of the specific target motion patterns. The invention uses information about target location, UAV platform type and states, sensor payload capability, and ratio of target-to-UAV speeds to select from a suite of sub-algorithms, each, of which generates desired platform positions (in form of waypoints) and/or sensor orientation commands to keep the target in view. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0005]    Features of the present invention will become apparent to those skilled in the art from the following description with reference to the drawings. Understanding that the drawings depict only typical embodiments of the invention and are not therefore to be considered limiting in scope, the invention will be described with additional specificity and detail through the use of the accompanying drawings, in which: 
           [0006]      FIG. 1  is a schematic diagram depicting a system for aerial tracking of a ground vehicle according to one embodiment of the invention. 
           [0007]      FIG. 2  is a simplified block diagram of an entity arranged to implement aspects of the exemplary embodiment. 
           [0008]      FIG. 3  is a flow chart depicting functions that can be carried out in accordance with the exemplary embodiment. 
           [0009]      FIG. 4  is a diagram depicting an example of a sensor footprint. 
           [0010]      FIG. 5  is a diagram of a footprint that illustrates variables needed to determine how close target  116  is from leaving sensor  114 &#39;s field of view. 
           [0011]      FIG. 6  is a diagram depicting SIN tracking. 
           [0012]      FIG. 7  is an illustration depicting the parameters for adjusting the loitering orbit when target  116  is in motion. 
           [0013]      FIG. 8  depicts a forward-looking sensor footprint that has been normalized. 
       
    
    
     DETAILED DESCRIPTION 
       [0014]    In the following detailed description, embodiments are described in sufficient detail to enable those skilled in the art to practice the invention. It is to be understood that other embodiments may be utilized without departing from the scope of the present invention. The following detailed description is, therefore, not to be taken in a limiting sense. 
         [0015]      FIG. 1  is a simplified diagram depicting a system  100  for automatically tracking a target from a UAV. As shown in  FIG. 1 , the system includes (1) a UAV  112  equipped with at least one sensor  114 , (3) a target  116  and (4) a remote operator  118 . It should be understood that while remote operator  118  is shown in  FIG. 1 , remote operator  118  may be many miles away from UAV  112  and target  116 . 
         [0016]    The UAV  112  can either be a hover-capable aerial vehicle or a fixed-wing aerial vehicle. Sensor  114  may be any device capable of imaging a target, such as a camera or radar. Target  116  may be anything being monitored by UAV  112 . For example, target  116  may be a ground-based vehicle, an air-based vehicle, a roadway, or a person. To acquire a target, UAV  112  typically sends images from sensor  114  to remote operator  118 . Remote operator  118  then defines an area in the image as the target, and sends the target to UAV  112 . 
         [0017]    Remote operator  118  may be any device capable of communicating with UAV  112 . In addition, remote operator  118  may be configured to remotely control UAV  112 . Remote operator  118  may be a device such as a desktop computer, laptop, or a personal data assistant (“PDA”), for example. 
         [0018]    Aspects of the present invention may be carried out by UAV  112  and/or remote operator  118  (or any other entity capable of controlling UAV  112 ).  FIG. 2  depicts functional components that may be included in UAV  112  and/or remote operator  118  to carry out various aspects of the invention. As shown in  FIG. 2 , the components include a communication interface  200 , a processing unit  202 , and data storage  206 , all of which may be coupled together by a system bus, network, or other mechanism  210 . 
         [0019]    Communication interface  200  comprises a mechanism for communicating over an air interface, so as to facilitate communication between UAV  112  and remote operator  118 . Further, communication interface  200  may include one or more antennas to facilitate air interface communication. 
         [0020]    Processing unit  202  comprises one or more general purpose processors (e.g., INTEL microprocessors) and/or one or more special purpose processors (e.g., digital signal processors). Data storage  204 , in turn, comprises one or more volatile and/or non-volatile storage mechanisms, such as memory and/or disc-drive storage for instance, which may be integrated in whole or in part with processing unit  202 . 
         [0021]    As shown, data storage  204  includes program logic  206  and reference data  208 . Program logic  206  comprises one or more logic modules (applications), and preferably includes machine language instructions executable by processing unit  204  to carry out various functions described herein, such as (1) identifying the coordinates of the footprint of sensor  114 , (2) determining whether the target is close to leaving the field of view of the sensor, and (3) causing UAV  112  to fly in a track mode that keeps target  116  in the field of view of sensor  114 . Reference data  208 , in turn, may include data such as imaging data acquired by sensor  114 . 
         [0022]      FIG. 3  is a flow chart depicting identifying various coordinates in a sensor footprint used to autonomously track target  116 . In particular,  FIG. 3  depicts (1) identifying the coordinates of the footprint of sensor  114 , (2) determining whether the target is close to leaving the field of view of the sensor, and (3) causing UAV  112  to fly in a track mode that keeps target  116  in the field of view of sensor  114 . 
         [0023]    As shown in  FIG. 3 , at step  302 , UAV  112  identifies the coordinates of the vertices and center of the footprint (i.e., the viewing window) of sensor  114 . Examples of sensor footprints are depicted in  FIG. 4 . As shown in  FIG. 4 , UAV  112  is equipped with forward and side looking sensors. Forward looking sensor footprint  402  includes vertices {a, b, c, d}. The center of footprint  402  is identified as {i}. Side-looking sensor footprint  404  includes vertices {e, f, g, h}. The center of side-looking sensor footprint is identified as {/}. 
         [0024]      FIG. 8  depicts a forward-looking sensor footprint that has been normalized (i.e., displayed as a rectangle). As shown in  FIG. 8 , the footprint includes vertices {a, b, c, d}, center {i}, midpoints {ad c , ab c , bc c , dc c }, and angles 
         [0000]    
       
         
           
             
               { 
               
                 
                   
                     α 
                     v 
                   
                   2 
                 
                 , 
                 
                   
                     α 
                     h 
                   
                   2 
                 
               
               } 
             
             , 
           
         
       
     
         [0000]    where α h  and α v  are the horizontal and vertical field of view angles for sensor  114 . 
         [0025]    Returning to  FIG. 3 , the coordinates of the vertices and center of the sensor footprint may be computed using the following data: 
         [0026]    [α h , α v ], the horizontal and vertical field of view for sensor  114 ; 
         [0027]    [θ, φ, ψ], the attitude angles of UAV  112 , where θ, is the pitch, φ is the roll, and ψ is the yaw. In this example climb requires a positive pitch, the right wing down is a positive roll and clockwise from the top of the vehicle is a positive yaw; 
         [0028]    [θ c , φ c , ψ c ], the attitude angles of sensor  114 , where θ, is the pitch, φ is the roll, and ψ is the yaw. In this example, pitch is measured between 0 and 90 degrees measured from straight down. The Camera lookdown angle is (1-θ c ), the roll angle is positive right and the yaw angle is positive in the clockwise direction. Consequently, a forward facing sensor  114  has ψ c =0, while a left-pointing camera has a ψ c =−90 degrees; and 
         [0029]    [N, E, h], the position coordinates of UAV  112  where N=north, E=east, and h=height from some reference point (such as UTM northings, eastings and altitude). 
         [0030]    The local coordinates of the vertices and center of the footprint are identified as follows: 
         [0000]    
       
         
           
             
               [ 
               
                 
                   
                     a 
                   
                 
                 
                   
                     b 
                   
                 
                 
                   
                     c 
                   
                 
                 
                   
                     d 
                   
                 
                 
                   
                     i 
                   
                 
               
               ] 
             
             = 
             
               [ 
               
                 
                   
                     
                       tan 
                        
                       
                         ( 
                         
                           
                             α 
                             v 
                           
                           2 
                         
                         ) 
                       
                     
                   
                   
                     
                       - 
                       
                         tan 
                          
                         
                           ( 
                           
                             
                               α 
                               h 
                             
                             2 
                           
                           ) 
                         
                       
                     
                   
                   
                     1 
                   
                 
                 
                   
                     
                       tan 
                        
                       
                         ( 
                         
                           
                             α 
                             v 
                           
                           2 
                         
                         ) 
                       
                     
                   
                   
                     
                       tan 
                        
                       
                         ( 
                         
                           
                             α 
                             h 
                           
                           2 
                         
                         ) 
                       
                     
                   
                   
                     1 
                   
                 
                 
                   
                     
                       - 
                       
                         tan 
                          
                         
                           ( 
                           
                             
                               α 
                               v 
                             
                             2 
                           
                           ) 
                         
                       
                     
                   
                   
                     
                       tan 
                        
                       
                         ( 
                         
                           
                             α 
                             h 
                           
                           2 
                         
                         ) 
                       
                     
                   
                   
                     1 
                   
                 
                 
                   
                     
                       - 
                       
                         tan 
                          
                         
                           ( 
                           
                             
                               α 
                               v 
                             
                             2 
                           
                           ) 
                         
                       
                     
                   
                   
                     
                       - 
                       
                         tan 
                          
                         
                           ( 
                           
                             
                               α 
                               h 
                             
                             2 
                           
                           ) 
                         
                       
                     
                   
                   
                     1 
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
               
               ] 
             
           
         
       
     
         [0031]    At step  304 , the local coordinates of the midpoints for each side of the sensor footprint are identified as follows: 
         [0000]    
       
         
           
             
               [ 
               
                 
                   
                     
                       ab 
                       c 
                     
                   
                 
                 
                   
                     
                       bc 
                       c 
                     
                   
                 
                 
                   
                     
                       dc 
                       c 
                     
                   
                 
                 
                   
                     
                       ad 
                       c 
                     
                   
                 
               
               ] 
             
             = 
             
               [ 
               
                 
                   
                     
                       tan 
                        
                       
                         ( 
                         
                           
                             α 
                             v 
                           
                           / 
                           2 
                         
                         ) 
                       
                     
                   
                   
                     0 
                   
                   
                     1 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       tan 
                        
                       
                         ( 
                         
                           
                             α 
                             h 
                           
                           / 
                           2 
                         
                         ) 
                       
                     
                   
                   
                     1 
                   
                 
                 
                   
                     
                       - 
                       
                         tan 
                          
                         
                           ( 
                           
                             
                               α 
                               v 
                             
                             / 
                             2 
                           
                           ) 
                         
                       
                     
                   
                   
                     0 
                   
                   
                     1 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       - 
                       
                         tan 
                          
                         
                           ( 
                           
                             
                               α 
                               h 
                             
                             / 
                             2 
                           
                           ) 
                         
                       
                     
                   
                   
                     1 
                   
                 
               
               ] 
             
           
         
       
     
         [0032]    At step  306 , the radii of the ellipse that circumscribes the frame is identified as follows: 
         [0000]    
       
         
           
             
               r 
               h 
             
             = 
             
               tan 
                
               
                 
                   α 
                   h 
                 
                 2 
               
             
           
         
       
       
         
           
             
               r 
               v 
             
             = 
             
               tan 
                
               
                 
                   α 
                   v 
                 
                 2 
               
             
           
         
       
     
         [0033]    where r h  is the radius in the horizontal direction and r v  is the radius in the vertical direction. The smaller of these two radii corresponds to the length of the semi-minor axis of the ellipse. 
         [0034]    At step  308 , each coordinate is transformed to inertial coordinates my multiplying the coordinate by pitch-roll-yaw rotation matrices [R] and [R c ], where 
         [0000]      [ R]=[R (θ)][ R (φ)][ R (ψ)]; and 
         [0000]      [ R   c   ]=[R (θ c )][ R (φ c )][ R (ψ c )]. 
         [0000]      Thus, 
         [0000]      A=a[R][R c ] 
         [0000]      B=b[R][R c ] 
         [0000]      C=c[R][R c ] 
         [0000]      D=d[R][R c ] 
         [0000]      I=i[R][R c ] 
         [0000]      AB c =ab c [R][R c ] 
         [0000]      BC c =bc c [R][R c ] 
         [0000]      DC c =dc c [R][R c ] 
         [0000]      AD c =ad c [R][R c ] 
         [0035]    Rotational matrices are well known in the art, and are not described in detail here. 
         [0036]    At step  310  the scaled coordinates of the sensor footprint are computed by scaling the inertial coordinates by the height (h) that UAV  112  is flying above the ground (if target  116  is a ground target), or the height of UAV  112  is flying above the target  116  (if target  116  is not necessarily on the ground). The footprint is calculated as follows: 
         [0000]    
       
         
           
               
             
               
                 
                   
                     
                       A 
                       g 
                     
                     = 
                       
                      
                     
                       A 
                       × 
                       
                         h 
                         
                           A 
                            
                           
                             ( 
                             3 
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   
                     
                       B 
                       g 
                     
                     = 
                       
                      
                     
                       B 
                       × 
                       
                         h 
                         
                           B 
                            
                           
                             ( 
                             3 
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   
                     
                       C 
                       g 
                     
                     = 
                       
                      
                     
                       C 
                       × 
                       
                         h 
                         
                           C 
                            
                           
                             ( 
                             3 
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   
                     
                       D 
                       g 
                     
                     = 
                       
                      
                     
                       D 
                       × 
                       
                         h 
                         
                           D 
                            
                           
                             ( 
                             3 
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   
                     
                       I 
                       g 
                     
                     = 
                       
                      
                     
                       I 
                       × 
                       
                         h 
                         
                           I 
                            
                           
                             ( 
                             3 
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   
                     
                       AB 
                       cg 
                     
                     = 
                       
                      
                     
                       
                         AB 
                         c 
                       
                       × 
                       
                         h 
                         
                           
                             AB 
                             c 
                           
                            
                           
                             ( 
                             3 
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   
                     
                       BC 
                       cg 
                     
                     = 
                       
                      
                     
                       
                         BC 
                         c 
                       
                       × 
                       
                         h 
                         
                           
                             BC 
                             c 
                           
                            
                           
                             ( 
                             3 
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   
                     
                       DC 
                       cg 
                     
                     = 
                       
                      
                     
                       
                         DC 
                         c 
                       
                       × 
                       
                         h 
                         
                           
                             DC 
                             c 
                           
                            
                           
                             ( 
                             3 
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   
                     
                       AD 
                       cg 
                     
                     = 
                       
                      
                     
                       
                         AD 
                         c 
                       
                       × 
                       
                         h 
                         
                           
                             AD 
                             c 
                           
                            
                           
                             ( 
                             3 
                             ) 
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
         [0037]    Similarly, the length of the semi-minor axis may be scaled to account for the height UAV  112  is above target  116 : 
         [0000]    
       
         
           
             
               r 
               SM 
             
             = 
             
               
                 min 
                  
                 
                   ( 
                   
                     
                       r 
                       h 
                     
                     , 
                     
                       r 
                       v 
                     
                   
                   ) 
                 
               
               × 
               
                 h 
                 
                   I 
                    
                   
                     ( 
                     3 
                     ) 
                   
                 
               
             
           
         
       
     
         [0038]    After computing the various coordinates of sensor  114 &#39;s sensor footprint, at step  310 , the target&#39;s (1) position relative to the center of the camera footprint (r target ) on the ground, (2) the distance from the center of the camera footprint to the side (e.g., [AB cg  BC cg  DC cg  AD cg ] of the footprint that is closest to the target (r side ), and (3) the distance from the center of the frame to the target (r t ) are used to determine how close target  116  is from leaving sensor  114 &#39;s field of view. These positions are illustrated in  FIG. 5 , which is a diagram of a footprint that illustrates variables need to determine how close target  116  is from leaving sensor  114 &#39;s field of view. As shown in  FIG. 5 , the frame includes a center point, a target, r target , r t , r side , and the direction of the frame of motion. 
         [0039]    In order to determine whether the target is about to leave the target&#39;s field of view, the value of r target  and r side  is first calculated. r target  is calculated by using the following equation: 
         [0000]        r   target =( ê   ct   −ê   ce ) r   t    
         [0040]    where ê ct  and ê ce  are unit vectors along a line from the target to the center of the footprint and from the mid-point of the closest side to the center respectively. That is, ê ct  is the unit vector along r t , while ê ce  is the unit vector along r side . 
         [0041]    r side  is calculated using the following equation: 
         [0000]    
       
         
           
             
               r 
               side 
             
             = 
             
               arg 
                
               
                   
               
                
               min 
                
               
                  
                 
                   
                     
                       
                         
                           r 
                           
                             AB 
                             cg 
                           
                         
                         - 
                         
                           r 
                           target 
                         
                       
                     
                   
                   
                     
                       
                         
                           r 
                           
                             BC 
                             cg 
                           
                         
                         - 
                         
                           r 
                           target 
                         
                       
                     
                   
                   
                     
                       
                         
                           r 
                           
                             DC 
                             cg 
                           
                         
                         - 
                         
                           r 
                           target 
                         
                       
                     
                   
                   
                     
                       
                         
                           r 
                           
                             AD 
                             cg 
                           
                         
                         - 
                         
                           r 
                           target 
                         
                       
                     
                   
                 
                  
               
             
           
         
       
     
         [0042]    where r AB     cg   -r AD     cg    is the distance from the center of the frame to the side of the frame. 
         [0043]    By calculating these values over time, UAV  112  (or remote operator  118 ) can determine if and when target  118  will leave the field of view of sensor  114 . 
         [0044]    At step  312  the location and speed of target  116  are used to cause UAV  112  to, depending on how fast target  116  is moving relative to the speed UAV  112  is capable of flying, fly in different track modes. These modes include, all at suitable altitude and stand-off ranges, (1) a straight following of the target (SFO), (2) flying orbits around the target (ORB), or (3) doing S-shapes (i.e. sinusoids) (SIN). This enables UAV  112  to maintain target  116  in the sensor footprint of sensor  114 . UAV  112  flies in these modes by receiving waypoint commands and flying to the waypoints. 
         [0045]    To determine which tracking mode UAV should use, the ratio (σ) of the speed of UAV  112  (v p ) to the speed of target  114  (v t ) is identified: 
         [0000]    
       
         
           
             σ 
             = 
             
               
                 v 
                 p 
               
               
                 v 
                 t 
               
             
           
         
       
     
         [0046]    If σ is around 1, UAV  112  and target  114  are traveling at similar speeds, and UAV  112  should employ SFO tracking. If σ is greater than 1 (i.e., is travelling faster than target  114 ), UAV  112  can slow down to match the speed of target  114  and maintain SFO tracking. However, if UAV  112  is unable to travel at such a slow speed (because it would stall), UAV  112  should employ either SIN or ORB tracking. Consequently, UAV  112  should employ SIN tracking. If UAV  112  would be unable to maintain the target in its sensor footprint using SIN tracking (i.e., the amplitude required is too large), then UAV  112  should employ ORB tracking. The value of σ that triggers UAV  112  to engage in a different mode is aircraft specific, as different aircraft have different properties (such as being a hovercraft vs. a fixed-wing aircraft). UAV  112  may automatically switch track modes depending on the value of &lt;J, or may be instructed by remote entity  118  to enter a different track mode depending on the value of σ. 
         [0047]    SFO Tracking 
         [0048]    If UAV  112  is flying in the SFO track mode, the waypoint (WPT SFO ) is calculated as follows: 
         [0000]        WPT   SFO =tgtPosn− I   g    
         [0049]    Where tgtPosn is the target&#39;s position in the inertial frame, and I g  (calculated above) is the offset between the aircraft and the center of the footprint. 
         [0050]    SIN Tracking 
         [0051]      FIG. 6  is an illustration depicting the variables used to calculate waypoints for SIN tracking. As shown in  FIG. 6 , UAV  112  is traveling at a fixed airspeed v p , and is tracking target  116 . Target  116  is moving along a straight line with a speed v t . In addition, a pair of aligned coordinate systems is shown. One system (x 1 , y 1 ) is fixed to the target, and the other (x 2 , y 2 ) is fixed to the beginning of the sinusoid. In order for UAV  112  to maintain track of target  116 , the distance (D) that UAV  112  travels in one period (J) is equal to the distance traveled by the target in the same direction: 
         [0000]    
       
      
       D=v 
       t 
       ×T  
      
     
         [0052]    The period is a free parameter that may be selected as desired. However, longer periods are more desirable because it decreases the size of the amplitude (A), making it easier for UAV  112  to track target  116 . The amplitude is determined by considering which direction of the footprint displacement and the distance from the target to the mid-point of the closest side of the footprint: 
         [0000]    
       
      
       A=k 
       tr 
       |r 
       side 
       −r 
       target  
      
     
         [0053]    where r side  and r target  were calculated above, and k tr  is a parameter used to tune the sensitivity of the algorithm. k tr  greater than zero and less than or equal to one and is a multiplicative factor based on the distance from the target to the closest side of the footprint. Generally 
         [0000]    
       
         
           
             
               
                 k 
                 tr 
               
               = 
               
                 1 
                 - 
                 
                   
                     r 
                     target 
                   
                   
                     r 
                     side 
                   
                 
               
             
             , 
           
         
       
     
         [0000]    although other values may be used. 
         [0054]    (x p , y p ) is the desired position of UAV  112  in the x 2 , y 2  coordinate system: 
         [0000]    
       
         
           
             
               y 
               p 
             
             - 
             
               A 
                
               
                   
               
                
               
                 sin 
                  
                 
                   ( 
                   
                     
                       2 
                        
                       
                           
                       
                        
                       π 
                        
                       
                           
                       
                        
                       
                         x 
                         p 
                       
                     
                     D 
                   
                   ) 
                 
               
             
           
         
       
       
         
           and 
         
       
       
         
           
             
               
                 y 
                 . 
               
               p 
             
             - 
             
               
                 A 
                 ′ 
               
                
               
                   
               
                
               
                 
                   cos 
                    
                   
                     ( 
                     
                       
                         2 
                          
                         
                             
                         
                          
                         π 
                          
                         
                             
                         
                          
                         
                           x 
                           p 
                         
                       
                       D 
                     
                     ) 
                   
                 
                 · 
                 
                   
                     x 
                     . 
                   
                   p 
                 
               
             
           
         
       
       
         
           
             
               
                 where 
                  
                 
                     
                 
                  
                 
                   A 
                   ′ 
                 
               
               = 
               
                 
                   2 
                    
                   
                       
                   
                    
                   π 
                    
                   
                       
                   
                    
                   A 
                 
                 D 
               
             
             , 
             and 
           
         
       
       
         
           
             
               
                 x 
                 . 
               
               p 
             
             = 
             
               
                 v 
                 p 
               
               
                 
                   1 
                   + 
                   
                     
                       A 
                       ′2 
                     
                      
                     
                       
                         cos 
                         2 
                       
                        
                       
                         ( 
                         
                           
                             2 
                              
                             
                                 
                             
                              
                             π 
                              
                             
                                 
                             
                              
                             
                               x 
                               p 
                             
                           
                           D 
                         
                         ) 
                       
                     
                   
                 
               
             
           
         
       
     
         [0055]    The waypoints (WPT SIN ) for UAV  112  when it is operating in SIN track mode are calculated as follows: 
         [0000]    
       
         
           
             
               WPT 
               SIN 
             
             = 
             
               
                 
                   [ 
                   
                     R 
                      
                     
                       ( 
                       
                         ψ 
                         tgt 
                       
                       ) 
                     
                   
                   ] 
                 
                  
                 
                   ∫ 
                   
                     
                       [ 
                       
                         
                           
                             
                               
                                 x 
                                 . 
                               
                               p 
                             
                           
                         
                         
                           
                             
                               
                                 y 
                                 . 
                               
                               p 
                             
                           
                         
                       
                       ] 
                     
                      
                     
                        
                       t 
                     
                   
                 
               
               - 
               
                 I 
                 g 
               
             
           
         
       
     
         [0056]    where (ψ tgt ) is the target&#39;s heading and └R(ψ tgt )┘ is the rotation matrix that transforms x p  and y p  back into the inertial frame. 
         [0057]    ORB Tracking 
         [0058]    ORB tracking enables UAV  112  to loiter over target  116 . The orbiting track waypoints (WPT ORB ) are given in N, E, h (north, east, height), and are calculated for a UAV having inertial coordinates N c  and E c  as follows: 
         [0000]    
       
         
           
             
               WPT 
               ORB 
             
             = 
             
               ( 
               
                 
                   
                     
                       N 
                       t 
                     
                   
                   
                     
                       E 
                       t 
                     
                   
                   
                     h 
                   
                 
                 
                   
                     
                       
                         N 
                         t 
                       
                       + 
                       
                         
                           r 
                           c 
                         
                          
                         sin 
                          
                         
                             
                         
                          
                         β 
                       
                     
                   
                   
                     
                       
                         E 
                         t 
                       
                       + 
                       
                         
                           r 
                           c 
                         
                          
                         cos 
                          
                         
                             
                         
                          
                         β 
                       
                     
                   
                   
                     h 
                   
                 
                 
                   
                     
                       
                         N 
                         t 
                       
                       + 
                       
                         r 
                         c 
                       
                     
                   
                   
                     
                       E 
                       t 
                     
                   
                   
                     h 
                   
                 
                 
                   
                     
                       
                         N 
                         t 
                       
                       + 
                       
                         
                           r 
                           c 
                         
                          
                         sin 
                          
                         
                             
                         
                          
                         β 
                       
                     
                   
                   
                     
                       
                         E 
                         t 
                       
                       - 
                       
                         
                           r 
                           c 
                         
                          
                         cos 
                          
                         
                             
                         
                          
                         β 
                       
                     
                   
                   
                     h 
                   
                 
                 
                   
                     
                       N 
                       t 
                     
                   
                   
                     
                       
                         E 
                         t 
                       
                       - 
                       
                         r 
                         c 
                       
                     
                   
                   
                     h 
                   
                 
                 
                   
                     
                       
                         N 
                         t 
                       
                       - 
                       
                         
                           r 
                           c 
                         
                          
                         sin 
                          
                         
                             
                         
                          
                         β 
                       
                     
                   
                   
                     
                       
                         E 
                         t 
                       
                       - 
                       
                         
                           r 
                           c 
                         
                          
                         cos 
                          
                         
                             
                         
                          
                         β 
                       
                     
                   
                   
                     h 
                   
                 
                 
                   
                     
                       
                         N 
                         t 
                       
                       - 
                       
                         r 
                         c 
                       
                     
                   
                   
                     
                       E 
                       t 
                     
                   
                   
                     h 
                   
                 
                 
                   
                     
                       
                         N 
                         t 
                       
                       - 
                       
                         
                           r 
                           c 
                         
                          
                         sin 
                          
                         
                             
                         
                          
                         β 
                       
                     
                   
                   
                     
                       
                         E 
                         t 
                       
                       + 
                       
                         
                           r 
                           c 
                         
                          
                         cos 
                          
                         
                             
                         
                          
                         β 
                       
                     
                   
                   
                     h 
                   
                 
               
               ) 
             
           
         
       
     
         [0059]    where (N t , E t ) is the position of target  116 , r c =√{square root over (N c   2 +E c   2 )}, and β is an angle dependent on how many waypoints you want to produce, and where waypoints should be relative to the target. For example, if there are 12 waypoints for UAV  112  to visit, 
         [0000]    
       
         
           
             β 
             = 
             
               
                 π 
                 6 
               
               . 
             
           
         
       
     
         [0060]    It should be understood that the first row of the WPT ORB  is not necessarily the first waypoint visited by UAV  112 . Instead, a cost function may be utilized that determines the order in which waypoints are visited by UAV  112 . The cost function uses the range to all the waypoints and the required change in heading needed for UAV  112  to get to each waypoint. 
         [0061]    In addition, the position of the orbital waypoints relative to the target&#39;s frame of motion may be adjusted in order to achieve acceptable tracking performance at smaller speed ratios (higher target speeds). This results in “stretching” UAV  112 &#39;s orbit ahead of target  116  and can reduce the amount that UAV  112  falls behind. 
         [0062]      FIG. 7  is an illustration depicting the parameters for adjusting the loitering orbit when target  116  is in motion. As shown in  FIG. 6 , the seven parameters include radii r 1 -r 4 , diameters d 1  and d 2 , angle θ, which is the angle between the motion of the target and d 1 . The shape of the orbit is adjusted by separating the circular orbit into four arcs (quarter-circles). Each arc is described by an elliptic section, constrained at their connection points. This is achieved by specifying the length of the four “halfaxes” of the elliptic sections. The circular sections are then special cases of the elliptic sections in which the length of all of the half-axes are equivalent. Additionally, the principal axes of the elliptic sections may be rotated and the center of the orbit may be displaced from the target position. These seven parameters can be computed analytically or chosen using numerical optimization techniques and stored in a look-up table as a function of the speed ratio. For generality, the piecewise elliptical orbit is also normalized. That is, the orbit is defined relative to a unit circle orbit and during application the actual radius of the circular orbit is used to scale the piecewise elliptical orbit. 
         [0063]    Once the piecewise elliptical orbit parameters have been chosen, a modified waypoint can be calculated as follows: 
         [0000]        WPT′   ORB =tgtPosn+ r   norm ×offset(tgtVel)+α( WPT   ORB ,tgtPosn,tgtVel)×( WPT   ORB −tgtPosn) 
         [0064]    where tgtPosn is the position of target  116  in the inertial frame, tgtVel is the velocity of target  116  in the inertial frame, r norm  is the radius of the nominal (circular) orbit, and: 
         [0000]    
       
         
           
             
               offset 
                
               
                 ( 
                 tgtVel 
                 ) 
               
             
             = 
             
               
                 
                   
                     d 
                     1 
                   
                    
                   
                     ( 
                     tgtVel 
                     ) 
                   
                 
                  
                 
                   v 
                   ^ 
                 
               
               + 
               
                 
                   
                     d 
                     2 
                   
                    
                   
                     ( 
                     tgtVel 
                     ) 
                   
                 
                  
                 
                   n 
                   ^ 
                 
               
             
           
         
       
       
         
           
             
               v 
               ^ 
             
             = 
             
               tgtVel 
               
                  
                 tgtVel 
                  
               
             
           
         
       
       
         
           
             
               n 
               ^ 
             
             = 
             
               
                 [ 
                 
                   
                     
                       0 
                     
                     
                       1 
                     
                   
                   
                     
                       
                         - 
                         1 
                       
                     
                     
                       0 
                     
                   
                 
                 ] 
               
                
               
                 v 
                 ^ 
               
             
           
         
       
     
         [0065]    The scale factor, α, is computed using the equation of an ellipse. The principal axes of the ellipse are defined as follows: 
         [0000]        p   a =cos(θ) {circumflex over (v)} +sin(θ) {circumflex over (n)}   
         [0000]        p   b =−sin(θ) {circumflex over (v)}+cos(θ)   {circumflex over (n)}   
         [0066]    The lengths of the axes are defined as follows: 
         [0000]    
       
         
           
             
               r 
               a 
             
             = 
             
               { 
               
                 
                   
                     
                       r 
                       1 
                     
                   
                   
                     if 
                   
                   
                     
                       
                         
                           w 
                           ^ 
                         
                         · 
                         
                           p 
                           a 
                         
                       
                       &gt; 
                       0 
                     
                   
                 
                 
                   
                     
                       r 
                       3 
                     
                   
                   
                     
                         
                     
                   
                   
                     otherwise 
                   
                 
               
               } 
             
           
         
       
       
         
           
             
               r 
               b 
             
             = 
             
               { 
               
                 
                   
                     
                       r 
                       2 
                     
                   
                   
                     if 
                   
                   
                     
                       
                         
                           w 
                           ^ 
                         
                         · 
                         
                           p 
                           b 
                         
                       
                       &gt; 
                       0 
                     
                   
                 
                 
                   
                     
                       r 
                       4 
                     
                   
                   
                     
                         
                     
                   
                   
                     otherwise 
                   
                 
               
               } 
             
           
         
       
       
         
           
             
               w 
               ^ 
             
             = 
             
               
                 ( 
                 
                   
                     WPT 
                     ORB 
                   
                   - 
                   tgtPosn 
                 
                 ) 
               
               
                  
                 
                   ( 
                   
                     
                       WPT 
                       ORB 
                     
                     - 
                     tgtPosn 
                   
                   ) 
                 
                  
               
             
           
         
       
     
         [0067]    The equation of the ellipse is: 
         [0000]    
       
         
           
             
               
                 x 
                 T 
               
                
               Mx 
             
             = 
             1 
           
         
       
       
         
           
             M 
             = 
             
               YY 
               T 
             
           
         
       
       
         
           
             Y 
             = 
             
               [ 
               
                 
                   
                     p 
                     a 
                   
                   
                     r 
                     a 
                   
                 
                  
                 
                   
                     p 
                     b 
                   
                   
                     r 
                     b 
                   
                 
               
               ] 
             
           
         
       
     
         [0068]    α is the scalar multiplier such that αŵ lies on the ellipse, α may be computed by plugging x=αŵ into the equation x T Mx= 1 . Thus: 
         [0000]    
       
         
           
             α 
             = 
             
               
                 ( 
                 
                   1 
                   
                     
                       
                         w 
                         ^ 
                       
                       T 
                     
                      
                     M 
                      
                     
                       w 
                       ^ 
                     
                   
                 
                 ) 
               
               
                 1 
                 2 
               
             
           
         
       
     
         [0069]    Height Adjustment 
         [0070]    Regardless of whether UAV  112  is using SFO, SIN or ORB track modes, the height of UAV  112  may be adjusted to maintain target track. Height adjustment may be done using a gradient-descent law to minimize the cost function 
         [0000]    
       
         
           
             J 
             = 
             
               2 
                
               
                 
                   r 
                   target 
                   2 
                 
                 
                   
                      
                     
                       r 
                       SM 
                     
                      
                   
                   2 
                 
               
             
           
         
       
     
         [0000]    with respect to h. As noted above 
         [0000]    
       
         
           
             
               r 
               SM 
             
             = 
             
               
                 min 
                  
                 
                   ( 
                   
                     
                       r 
                       h 
                     
                     , 
                     
                       r 
                       v 
                     
                   
                   ) 
                 
               
               × 
               
                 
                   h 
                   
                     I 
                      
                     
                       ( 
                       3 
                       ) 
                     
                   
                 
                 . 
               
             
           
         
       
     
       Then, 
       [0071]    
       
         
           
             
               h 
               . 
             
             = 
             
               γ 
                
               
                 
                   δ 
                    
                   
                       
                   
                    
                   J 
                 
                 
                   δ 
                    
                   
                       
                   
                    
                   h 
                 
               
             
           
         
       
     
         [0072]    where γ is the gain of the gradient scheme. 
         [0073]    The present invention may be embodied in other specific forms without departing from its essential characteristics. The described embodiments are to be considered in all respects only as illustrative and not restrictive. The scope of the invention is therefore indicated by the appended claims rather than by the foregoing description. All changes that come within the meaning and range of equivalency of the claims are to be embraced within their scope.