Abstract:
Methods and apparatus to point a payload at a target are disclosed herein. An example machine accessible medium having instructions stored thereon that, when executed, causes a machine to at least command a first actuator to move to a first corrected stroke position and a second actuator to move to a second corrected stroke position to point a payload at a target along a line of sight vector without verifying a target pointing direction of a base when the first and second actuators are positioned to the respective first and second corrected stroke positions and without using a feedback to verify the base being at the target pointing direction when the first and second actuators are positioned to the respective first and second corrected stroke positions.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
       [0001]    The patent arises from a continuation of U.S. patent application Ser. No. 13/665,857, filed on Oct. 31, 2012, titled Methods and Apparatus to Point a Payload at a Target, and is hereby incorporated herein by reference in its entirety. 
     
    
     FIELD 
       [0002]    The present disclosure relates generally to payloads and, more particularly, to methods and apparatus to point a payload at a target. 
       BACKGROUND 
       [0003]    Generally, a payload is to be substantially pointed at a target. For example, an antenna is to be substantially pointed at a target to enable communication between the antenna and the target. If the antenna points away from the target, the communication between the antenna and the target is affected. The antenna may be disposed on a satellite in orbit around Earth. Due to the distance between the satellite and a target on Earth, an alignment error of the antenna influenced by thermal distortion, machining tolerances, etc. may cause the antenna to point away from the target. 
       SUMMARY 
       [0004]    An example machine accessible medium having instructions stored thereon that, when executed, causes a machine to at least: estimate a target pointing direction of a base supporting a payload to point the payload at a target, where the base is movable relative to a pivot about an azimuth angle and an elevation angle via a first actuator and a second actuator directly coupled to the base; determine a first estimated stroke position of the first actuator and a second estimated stroke position of a second actuator to position the base at the target pointing direction. The instructions further cause the machine to obtain a base orientation error. The instructions further cause the machine to determine a first stroke position error of the first actuator and a second stroke position error of the second actuator based on the base orientation error. The instructions further cause the machine to determine a first corrected stroke position of the first actuator based on a difference between the first stroke position error and the first estimated stroke position. The instructions further cause the machine to determine a second corrected stroke position of the second actuator based on a difference between the second stroke position error and the second estimated stroke position. Also, the instructions cause the machine to and command the first actuator to move to the first corrected stroke position and the second actuator to move to the second corrected stroke position to point the payload at the target along a line of sight vector without verifying the target pointing direction of the base when the first and second actuators are positioned to the respective first and second corrected stroke positions and without using a feedback to verify the base being at the target pointing direction when the first and second actuators are positioned to the respective first and second corrected stroke positions. 
         [0005]    Another example tangible machine readable storage medium disclosed herein includes instructions that, when executed, cause a machine to at least receive a first command to move a payload relative to a first target. The instructions also cause the machine to estimate a base orientation of a base coupled to the payload to point the payload to the first target along a first line of sight vector. The instructions also cause the machine to determine a first estimated stroke position of a first linear actuator and a second estimated stroke position of a second linear actuator to position the base to the estimated base orientation. The instructions further cause the machine to modify the first estimated stroke position to a first corrected stroke position based on a base orientation error. Also, the instructions cause the machine to modify the second estimated stroke position to a second corrected stroke position based on the base orientation error. The instructions further cause the machine to command the first linear actuator to actuate to the first corrected stroke position and commanding the second linear actuator to actuate to the second corrected stroke position without verifying a final position of the payload after the first linear actuator is actuated to the first corrected stroke position and the second linear actuator is actuated to the second corrected stroke position. 
         [0006]    Another example tangible machine readable storage medium disclosed herein includes instructions that, when executed, cause a machine to at least receive a command to actuate a first linear actuator and a second linear actuator operatively coupled to a base of a payload to orient the base at a first azimuth angle and a first elevation angle to point the payload at a target, the base being pivotably coupled to the payload via a joint and having only the first and second linear actuators to move the base about the joint relative to the azimuth angle and the elevation angle; determine a pointing error based on an experimentally determined second azimuth angle and an experimentally determined second elevation angle of the base to point the payload at the target; determine a first stroke position error of the first linear actuator and a second stroke position error of the second linear actuator based on the pointing error; determine a first corrected stroke position of the first linear actuator and a second corrected stroke position of the second linear actuator; and command the first linear actuator to move to the first corrected stroke position and the second linear actuator to move to the second corrected stroke position to point the payload at the target without verifying a final position of the payload after the first linear actuator is actuated to the first corrected stroke position and the second linear actuator is actuated to the second corrected stroke position 
         [0007]    An example apparatus disclosed herein includes an instruction processor to receive a command to point a payload coupled to a base at a target, the base being movable relative to a pivot about an azimuth angle and an elevation angle via only a first linear actuator and a second linear actuator, the pivot, the first linear actuator and the second linear actuator being directly coupled to the base. In some examples, the apparatus includes a line of sight determiner to determine a line of sight vector between the payload and the target. In some examples, the apparatus includes a base orientation determiner to determine an estimated base orientation to point the payload at the target. In some examples, the apparatus includes an estimated stroke position determiner to determine a first estimated stroke position of a first linear actuator and a second estimated stroke position of a second linear actuator to orient the base at the estimated base orientation. In some examples, the apparatus includes a base orientation error determiner to determine a base orientation error. In some examples, the apparatus includes a stroke position error determiner to determine a first stroke positioner error of the first linear actuator based on the base orientation error and a second stroke position error of the second linear actuator based on the base orientation error. In some examples the apparatus includes a corrected stroke position determiner to determine a first corrected stroke position of the first actuator and a second corrected stroke position of the second actuator. In some examples, the apparatus includes an actuator controller to command the first linear actuator to move to the first corrected stroke position and the second linear actuator to move to the second corrected stroke position to point the payload at the target along a line of sight vector without verifying the target pointing direction of the base when the first and second linear actuators are positioned to the respective first and second corrected stroke positions and without using a feedback to verify the base being at the target pointing direction when the first and second linear actuators are positioned to the respective first and second corrected stroke positions. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0008]      FIG. 1  illustrates an example satellite in communication with a target in accordance with the teachings of this disclosure. 
           [0009]      FIG. 2  is a perspective view of an example payload assembly disclosed herein. 
           [0010]      FIG. 3  is a rear view of the example payload assembly of  FIG. 2 . 
           [0011]      FIG. 4  is a top view of the example payload assembly of  FIGS. 2-3 . 
           [0012]      FIG. 5  is a two-dimensional plot of payload signal power levels detected at the example target of  FIG. 1 . 
           [0013]      FIG. 6  is a three-dimensional plot of the payload signal power levels detected at the example target of  FIG. 1 . 
           [0014]      FIG. 7  is a block diagram of an example controller in accordance with the teachings of this disclosure. 
           [0015]      FIG. 8  is a flow diagram of an example process disclosed herein. 
           [0016]      FIG. 9  is a flow diagram of another example process disclosed herein. 
           [0017]      FIG. 10  is a block diagram of an example processing platform capable of executing machine readable instructions to implement the example controller of  FIG. 7 . 
       
    
    
       [0018]    Wherever possible, the same reference numbers will be used throughout the drawing(s) and accompanying written description to refer to the same or like parts. As used in this disclosure, stating that any part (e.g., a layer, film, area, or plate) is in any way positioned on (e.g., positioned on, located on, disposed on, or formed on, etc.) another part, means that the referenced part is either in contact with the other part, or that the referenced part is above the other part with one or more intermediate part(s) located therebetween. Stating that any part is in contact with another part means that there is no intermediate part between the two parts. 
       DESCRIPTION 
       [0019]    Methods and apparatus to point a payload at a target are disclosed herein. An example apparatus disclosed herein includes a payload such as, for example, an antenna, a transmitter, a sensor (e.g., an infrared sensor), an optical device, a camera, and/or any other suitable payload coupled to a base that is rotatable about a joint. In some examples, the base is operatively coupled to a first actuator (e.g., a linear actuator such as, for example, a jackscrew) and a second actuator (e.g., a linear actuator such as, for example, a jackscrew). The first actuator and the second actuator may enable rotation of the base about the joint to adjust an azimuth angle and/or an elevation angle of the base. A controller may be in communication with the first actuator and the second actuator to control a first stroke position of the first actuator and a second stroke position of the second actuator. 
         [0020]    To point the payload at a target, the controller may determine and compensate for a first stroke position error corresponding to the first actuator and a second stroke position error corresponding to the second actuator. The first stroke position error and/or the second stroke position error may be influenced by thermal distortion, machining tolerances, alignment errors, etc. In some examples, the first stroke position error and the second stroke position error are constant relative to an orientation of the base and/or an amount of rotation of the base. As such, the base may be rotated over a wide distance range (e.g., ten degrees of rotation or more) via the first actuator and the second actuator to point the payload at the target. 
         [0021]      FIG. 1  illustrates an example satellite  100  in accordance with the teachings of this disclosure. In the illustrated example, the satellite  100  is in orbit around Earth  102 . However, in other examples, the satellite  100  may be in orbit around another celestial body such as, for example, Earth&#39;s moon. In the illustrated example, the satellite  100  is in communication with a ground station  104  located on Earth  102 . In some examples, the satellite  100  is in communication with one or more other ground stations, satellites, and/or other targets. As described in greater detail below, the example satellite  100  includes a payload ( FIG. 2 ), which may be employed to communicate information to the ground station  104 , receive signals, generate images, etc. 
         [0022]      FIG. 2  is a perspective view of an example payload assembly  200  disclosed herein, which may be used to communicate information from the example satellite  100  of  FIG. 1  to a target (e.g., the ground station  104 ), receive signals (e.g., optical signals) from the target, generate images of the target, etc. In the illustrated example, the payload assembly  200  includes a base  202 , a first actuator  204 , a second actuator  300  ( FIG. 3 ), and a payload  206  such as, for example, an antenna, a sensor (e.g., an infrared sensor), a camera, an optical device, etc. In the illustrated example, the base  202  is a triangular plate. Thus, the base  202  includes a first corner  208 , a second corner  210  and a third corner  212 . The base  202  defines a substantially planar surface  214  onto which the payload  206  is coupled. In the illustrated example, the planar surface  214  is to substantially face the ground station  104  during operation of the example satellite  100 . In other examples, the base  202  is other shapes (e.g., rectangular, circular, a shape that defines a rounded or uneven (e.g., stepped) surface facing the ground station  104 , etc.). In the illustrated example, the payload  206  is disposed on the base  202  substantially normal to the planar surface  214  (i.e., the payload  206  is substantially perpendicular to the base  202 ). 
         [0023]    In the illustrated example, the base  202  is rotatably coupled to the satellite  100  via a pivot joint  216  (e.g., a ball joint) disposed at or adjacent the first corner  208 . The example joint  216  enables the base  202  to rotate at point A about a first axis (X-axis)  218  and a second axis (Y-axis)  220 . The example first axis  218  and the example second axis  220  intersect at point A. In the illustrated example, a third axis (Z-axis)  222  intersects the first axis  218  and the second axis  220  at point A. Thus, point A corresponds to coordinates (0, 0, 0). The above-noted axes  218 ,  220  and  222  are merely examples and, thus, other axes may be employed in other examples. In the illustrated example, an elevation angle, ε, of the base  202  is an amount of rotation of the base  202  from coordinates (0, 0, 0) about the first axis  218 . An azimuth angle, α, of the example base  202  is an amount of rotation of the base  202  from coordinates (0, 0, 0) about the second axis  220 . Thus, a position or orientation of the example base  202  may be defined by the azimuth angle and the elevation angle of the base  202 . 
         [0024]      FIG. 3  is a rear view of the example payload assembly  200  of  FIG. 2 . In the illustrated example, the base  202  is operatively coupled to the satellite  100  via the first actuator  204  and a second actuator  300 , respectively. The example first actuator  204  is coupled to the base  202  at point B, which is adjacent the second corner  210  of the base  202 . The example second actuator  300  is coupled to the base  202  at point C, which is adjacent the third corner  212  of the base  202 . In the illustrated example, the first actuator  204  and the second actuator  300  are jackscrews. However, the first actuator  204  and the second actuator  300  may be implemented using any type of linear actuator. In the illustrated example, strokes of the first actuator  204  and the second actuator  300  are substantially parallel to the third axis  222 . For example, when actuated, a first arm  302  of the first actuator  204  and a second arm  304  of the second actuator  300  move substantially parallel to the third axis  222 , thereby moving the second corner  210  and/or the third corner  212 , respectively, toward or away from the satellite  100 ). The example first actuator  204  and/or the example second actuator  300  may be actuated to rotate the base  202  approximately ten degrees about an axis of rotation (e.g., the first axis  218  and/or the second axis  220 ). Other examples rotate other amounts (e.g., five degrees, forty five degrees, etc.) and/or about other axes. 
         [0025]    In the illustrated example, the first actuator  204  and the second actuator  300  are in communication with a controller  306 . The example controller  306  may reside in the satellite  100 , within the ground station  104 , and/or in any other suitable location. In the illustrated example, the controller  306  controls a first stroke position, h 1 , of the first actuator  204  and a second stroke position, h 2 , of the second actuator  300 . In other examples, the first actuator  204  and the second actuator  300  are controlled via separate controllers. As described in greater detail below, the controller  306  determines and compensates for a first stroke position error corresponding to the first actuator  204  and a second stroke position error corresponding to the second actuator  300  to point the payload  206  at the ground station  104 . 
         [0026]      FIG. 4  is a top view of the example payload assembly  200  of  FIGS. 2-3 . In the illustrated example, point B (i.e., where the example first actuator  204  is coupled to the base  202 ) is a first distance L 1  from point A. Point C (i.e., where the example second actuator  300  is coupled to the base  202 ) is a second distance L 2  from point A. In the illustrated example, the first distance L 1  and the second distance L 2  are substantially equal. In other examples, the first distance L 1  and the second distance L 2  may be different. The example first actuator  204  and the example second actuator  300  are spaced apart from each other such that the first actuator  204  and the second actuator  300  are substantially ninety degrees apart relative to the joint  216  (i.e., a separation angle φ between the first actuator  204  and the second actuator  300  is substantially ninety degrees). In other examples, the first actuator  204  and/or the second actuator  300  are in other positions relative to the joint  216  (e.g., the separation angle φ is greater than or less than ninety degrees, etc.). 
         [0027]    In the illustrated example, to communicate information from the example satellite  100  to the ground station  104 , the example satellite  100  transmits one or more signals to the ground station  104  via the payload  206 . To facilitate transmission of the signal, the example payload  206  is pointed at or toward the ground station  104 . In other examples, the payload  206  is pointed at a target other than the ground station  104  to, for example, generate images of the target, receive signals from the target, take measurements (e.g., via a sensor), etc. Thus, while the following examples are described in conjunction with the example ground station  104 , the base may be oriented to point the payload  206  at any other suitable target in accordance with the teachings of this disclosure. In the illustrated example, the payload  206  is pointed at the ground station  104  by aligning the payload  206  with a line of sight (LOS) vector extending from the ground station  104  to the satellite  100  or from the satellite  100  to the ground station  104  (e.g., such that the signals transmitted via the payload  206  substantially propagate along the LOS vector). To align the payload  206  with the LOS vector, the controller  306  actuates the first actuator  204  and/or the second actuator  300  to adjust an orientation of the base  202  (i.e., the azimuth angle and the elevation angle of the base  202 ) and, thus, a pointing direction of the payload  206 . 
         [0028]    In some examples, the controller  306  and/or the ground station  104  determine the LOS vector from the satellite  100  to the ground station  104  and/or from the ground station  104  to the satellite  100 . Based on the LOS vector, an estimated base orientation (i.e., an estimated azimuth angle and an estimated elevation angle of the base  202 ) to point the payload  206  along the LOS vector (e.g., such that a signal transmitted via the payload propagates substantially along the LOS vector to the ground station  104 ) may be determined. To move (i.e., orient) the base  202  to the estimated base orientation, a first estimated stroke position and a second estimated stroke position of the first actuator  204  and the second actuator  300 , respectively, are determined, and the first actuator  204  and the second actuator  300  are actuated to the first estimated stroke position and the second estimated stroke position, respectively. 
         [0029]    In some examples, the LOS vector is determined based on a position of the ground station  104  and a position of the satellite  100 . The position of the ground station  104  may be determined based on a position vector of the ground station  104  in an inertial frame such as, for example, an Earth-Centered Earth Fixed frame based on an Earth-Centered Inertial Frame. The position of the satellite  100  may be determined in a frame of the satellite  100  via an orbit frame. In some examples, the orbit frame is determined based on an orbit position vector of the satellite  100  in the Earth-Centered Frame. The LOS vector may then be determined based on a difference vector between the position of the ground station  104  and the position of the satellite  100  in the frame of the satellite  100 . Based on the LOS vector, the estimated base orientation to point the payload  206  at the ground station  104  may be determined. 
         [0030]    In the illustrated example, once the estimated base orientation is determined, the first estimated stroke position of the first actuator  204  and the second estimated stroke position of the second actuator  300  to orient the base  202  at the estimated base orientation are determined. The first estimated stroke position, h 1  of the first actuator  204  and the second estimated stroke position, h 2 , of the second actuator  300  are a function of the estimated azimuth angle, α, and the estimated elevation angle, ε, of the base  202  as shown in the following equations: 
         [0000]    
       
         
           
             
               
                 
                     
                   
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         [0031]    In Equations 1 and 2, φ is the separation angle, and L is the first distance L 1  (i.e., the distance from point A to B) or the second distance L 2  (i.e., the distance from point A to point C). In the illustrated examples disclosed herein, L 1 =L 2 . In other examples, when L 1  is not equal to L 2 , calculations and/or equations described herein account for other variables. As a result, for example, Equations 1 and 2 become more complex when L 1  is not equal to L 2 . However, the calculations and/or equations disclosed herein may be configured or obtained when L 1  is not equal to L 2 . Equations 1 and 2 may be used for each of the first and second actuators to determine the estimated azimuth angle, α, and the estimated elevation angle, ε, of the base  202 . In the illustrated example, because the separation angle φ is substantially ninety degrees, Equation 1 and Equation 2 simplify as shown in Equations 3 and 4 below: 
         [0000]    
       
         
           
             
               
                 
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         [0032]    Based on Equations 3 and 4, the first estimated stroke position, h 1 , and the second estimated stroke position, h 2 , may be determined using the following equations: 
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                             h 
                             2 
                           
                           = 
                           
                             
                               1 
                               2 
                             
                              
                             
                               
                                 
                                   
                                     
                                       ( 
                                       
                                         
                                           
                                             2 
                                              
                                             L 
                                              
                                             
                                                 
                                             
                                              
                                             tan 
                                              
                                             
                                                 
                                             
                                              
                                             α 
                                           
                                           
                                             2 
                                           
                                         
                                         + 
                                         
                                           tan 
                                            
                                           
                                               
                                           
                                            
                                           ɛ 
                                            
                                           
                                             
                                               
                                                 2 
                                                  
                                                 
                                                   L 
                                                   2 
                                                 
                                               
                                               + 
                                               
                                                 
                                                   ( 
                                                   
                                                     
                                                       2 
                                                        
                                                       L 
                                                        
                                                       
                                                           
                                                       
                                                        
                                                       tan 
                                                        
                                                       
                                                           
                                                       
                                                        
                                                       α 
                                                     
                                                     
                                                       2 
                                                     
                                                   
                                                   ) 
                                                 
                                                 2 
                                               
                                             
                                           
                                         
                                       
                                       ) 
                                     
                                     = 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       L 
                                       
                                         2 
                                       
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           tan 
                                            
                                           
                                               
                                           
                                            
                                           α 
                                         
                                         + 
                                         
                                           
                                             tan 
                                              
                                             
                                                 
                                             
                                              
                                             ɛ 
                                           
                                           
                                             cos 
                                              
                                             
                                                 
                                             
                                              
                                             α 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   s 
                    
                   
                       
                   
                    
                   7 
                    
                   
                       
                   
                    
                   and 
                    
                   
                       
                   
                    
                   8 
                 
               
             
           
         
       
     
         [0000]    Equations 7 and 8, for example, provide the actuators&#39; travel computation. In some examples in which the separation angle φ is not ninety degrees, the first estimated stroke position and the second estimated stroke position may be determined using Equations 9-12 below: 
         [0000]    
       
         
           
               
             
               { 
               
                 
                   
                     
                       
                         
                           ( 
                           
                             
                               h 
                               1 
                             
                             + 
                             
                               h 
                               2 
                             
                           
                           ) 
                         
                         = 
                         
                           2 
                            
                           L 
                            
                           
                               
                           
                            
                           
                             cos 
                              
                             
                               ( 
                               
                                 φ 
                                  
                                 
                                   / 
                                 
                                  
                                 2 
                               
                               ) 
                             
                           
                            
                           
                             tan 
                              
                             
                               ( 
                               α 
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           ( 
                           
                             
                               h 
                               1 
                             
                             - 
                             
                               h 
                               2 
                             
                           
                           ) 
                         
                         = 
                         
                           
                             - 
                             
                               
                                 
                                   tan 
                                    
                                   
                                     ( 
                                     ɛ 
                                     ) 
                                   
                                 
                                  
                                 
                                   
                                     
                                       
                                         
                                           ( 
                                           
                                             1 
                                             - 
                                             
                                               cos 
                                                
                                               
                                                 ( 
                                                 φ 
                                                 ) 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                     
                                       
                                         
                                           ( 
                                           
                                             
                                               
                                                 
                                                   
                                                     2 
                                                      
                                                     
                                                       L 
                                                       2 
                                                     
                                                   
                                                   + 
                                                   
                                                     2 
                                                      
                                                     
                                                       L 
                                                       2 
                                                     
                                                      
                                                     cos 
                                                      
                                                     
                                                       ( 
                                                       φ 
                                                       ) 
                                                     
                                                   
                                                   + 
                                                 
                                               
                                             
                                             
                                               
                                                 
                                                   
                                                     ( 
                                                     
                                                       2 
                                                        
                                                       L 
                                                        
                                                       
                                                           
                                                       
                                                        
                                                       
                                                         cos 
                                                          
                                                         
                                                           ( 
                                                           
                                                             φ 
                                                              
                                                             
                                                             / 
                                                             
                                                              
                                                             2 
                                                           
                                                           ) 
                                                         
                                                       
                                                        
                                                       
                                                         tan 
                                                          
                                                         
                                                           ( 
                                                           α 
                                                           ) 
                                                         
                                                       
                                                     
                                                     ) 
                                                   
                                                   2 
                                                 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   2 
                                 
                                  
                                 
                                   cos 
                                    
                                   
                                     ( 
                                     
                                       φ 
                                        
                                       
                                         / 
                                       
                                        
                                       2 
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                           = 
                           
                             - 
                             
                               
                                 L 
                                  
                                 
                                     
                                 
                                  
                                 
                                   tan 
                                    
                                   
                                     ( 
                                     ɛ 
                                     ) 
                                   
                                 
                                  
                                 
                                   sin 
                                    
                                   
                                     ( 
                                     φ 
                                     ) 
                                   
                                 
                               
                               
                                 
                                   cos 
                                    
                                   
                                     ( 
                                     
                                       φ 
                                        
                                       
                                         / 
                                       
                                        
                                       2 
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   cos 
                                    
                                   
                                     ( 
                                     α 
                                     ) 
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
                  
                 
                   
 
                 
                  
                 
                   { 
                   
                     
                       
                         
                           
                             
                               h 
                               1 
                             
                             = 
                             
                               
                                 
                                   1 
                                   2 
                                 
                                  
                                 
                                   L 
                                    
                                   
                                     ( 
                                     
                                       
                                         
                                           
                                             
                                               2 
                                                
                                               
                                                   
                                               
                                                
                                               
                                                 cos 
                                                  
                                                 
                                                   ( 
                                                   
                                                     φ 
                                                      
                                                     
                                                       / 
                                                     
                                                      
                                                     2 
                                                   
                                                   ) 
                                                 
                                               
                                                
                                               tan 
                                                
                                               
                                                   
                                               
                                                
                                               
                                                 ( 
                                                 α 
                                                 ) 
                                               
                                             
                                             - 
                                           
                                         
                                       
                                       
                                         
                                           
                                             
                                               
                                                 tan 
                                                  
                                                 
                                                   ( 
                                                   ɛ 
                                                   ) 
                                                 
                                               
                                                
                                               
                                                 sin 
                                                  
                                                 
                                                   ( 
                                                   φ 
                                                   ) 
                                                 
                                               
                                             
                                             
                                               
                                                   
                                               
                                                
                                               
                                                 
                                                   cos 
                                                    
                                                   
                                                     ( 
                                                     
                                                       φ 
                                                        
                                                       
                                                         / 
                                                       
                                                        
                                                       2 
                                                     
                                                     ) 
                                                   
                                                 
                                                  
                                                 
                                                   cos 
                                                    
                                                   
                                                     ( 
                                                     α 
                                                     ) 
                                                   
                                                 
                                               
                                             
                                           
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                               = 
                               
                                 L 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         
                                           
                                               
                                           
                                            
                                           
                                             
                                               
                                                 cos 
                                                  
                                                 
                                                   ( 
                                                   
                                                     φ 
                                                      
                                                     
                                                       / 
                                                     
                                                      
                                                     2 
                                                   
                                                   ) 
                                                 
                                               
                                                
                                               tan 
                                                
                                               
                                                   
                                               
                                                
                                               
                                                 ( 
                                                 α 
                                                 ) 
                                               
                                             
                                             - 
                                           
                                         
                                       
                                     
                                     
                                       
                                         
                                           
                                             
                                               tan 
                                                
                                               
                                                 ( 
                                                 ɛ 
                                                 ) 
                                               
                                             
                                              
                                             
                                               sin 
                                                
                                               
                                                 ( 
                                                 
                                                   φ 
                                                    
                                                   
                                                     / 
                                                   
                                                    
                                                   2 
                                                 
                                                 ) 
                                               
                                             
                                           
                                           
                                             
                                                 
                                             
                                              
                                             
                                               cos 
                                                
                                               
                                                 ( 
                                                 α 
                                                 ) 
                                               
                                             
                                           
                                         
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               h 
                               2 
                             
                             = 
                             
                               
                                 
                                   1 
                                   2 
                                 
                                  
                                 
                                   L 
                                    
                                   
                                     ( 
                                     
                                       
                                         
                                           
                                             
                                               2 
                                                
                                               
                                                   
                                               
                                                
                                               
                                                 cos 
                                                  
                                                 
                                                   ( 
                                                   
                                                     φ 
                                                      
                                                     
                                                       / 
                                                     
                                                      
                                                     2 
                                                   
                                                   ) 
                                                 
                                               
                                                
                                               tan 
                                                
                                               
                                                   
                                               
                                                
                                               
                                                 ( 
                                                 α 
                                                 ) 
                                               
                                             
                                             + 
                                           
                                         
                                       
                                       
                                         
                                           
                                             
                                               
                                                 tan 
                                                  
                                                 
                                                   ( 
                                                   ɛ 
                                                   ) 
                                                 
                                               
                                                
                                               
                                                 sin 
                                                  
                                                 
                                                   ( 
                                                   φ 
                                                   ) 
                                                 
                                               
                                             
                                             
                                               
                                                   
                                               
                                                
                                               
                                                 
                                                   cos 
                                                    
                                                   
                                                     ( 
                                                     
                                                       φ 
                                                        
                                                       
                                                         / 
                                                       
                                                        
                                                       2 
                                                     
                                                     ) 
                                                   
                                                 
                                                  
                                                 
                                                   cos 
                                                    
                                                   
                                                     ( 
                                                     α 
                                                     ) 
                                                   
                                                 
                                               
                                             
                                           
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                               = 
                               
                                 L 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         
                                           
                                               
                                           
                                            
                                           
                                             
                                               
                                                 cos 
                                                  
                                                 
                                                   ( 
                                                   
                                                     φ 
                                                      
                                                     
                                                       / 
                                                     
                                                      
                                                     2 
                                                   
                                                   ) 
                                                 
                                               
                                                
                                               tan 
                                                
                                               
                                                   
                                               
                                                
                                               
                                                 ( 
                                                 α 
                                                 ) 
                                               
                                             
                                             + 
                                           
                                         
                                       
                                     
                                     
                                       
                                         
                                           
                                             
                                               tan 
                                                
                                               
                                                 ( 
                                                 ɛ 
                                                 ) 
                                               
                                             
                                              
                                             
                                               sin 
                                                
                                               
                                                 ( 
                                                 
                                                   φ 
                                                    
                                                   
                                                     / 
                                                   
                                                    
                                                   2 
                                                 
                                                 ) 
                                               
                                             
                                           
                                           
                                             
                                                 
                                             
                                              
                                             
                                               cos 
                                                
                                               
                                                 ( 
                                                 α 
                                                 ) 
                                               
                                             
                                           
                                         
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                     . 
                   
                 
               
             
           
         
       
     
         [0033]    Using the first estimated stroke position and the second estimated stroke position, the controller  306  communicates a command to the first actuator  204  and the second actuator  300  to actuate to the first estimated stroke position and the second estimated stroke position, respectively. In some examples, the payload  206  may not point at the ground station  104  when the first actuator  204  and the second actuator  300  actuate to the first estimate stroke position and the second estimated stroke position, respectively. Instead, the base  202  may be oriented at a resultant base orientation different than the base orientation at which the payload  206  points at the ground station  104 . A difference between the resultant base orientation and the base orientation at which the payload  206  points to the ground station  104  is a base orientation or pointing error. If not compensated for, the base orientation error may affect communication between the satellite  100  and the ground station  104 . 
         [0034]    In the illustrated example, the base orientation error, 
         [0000]    
       
         
           
             
               [ 
               
                 
                   
                     
                       d 
                        
                       
                           
                       
                        
                       α 
                     
                   
                 
                 
                   
                     
                       d 
                        
                       
                           
                       
                        
                       ɛ 
                     
                   
                 
               
               ] 
             
             , 
           
         
       
     
         [0000]    is a function of a first stroke position error, dh 1 , corresponding to the first actuator  204  and a second stroke position error, dh 2 , corresponding to the second actuator  300  as shown in the following equation: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         [ 
                         
                           
                             
                               
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 α 
                               
                             
                           
                           
                             
                               
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 ɛ 
                               
                             
                           
                         
                         ] 
                       
                       = 
                       
                         
                           M 
                            
                           
                             [ 
                             
                               
                                 
                                   
                                     dh 
                                     1 
                                   
                                 
                               
                               
                                 
                                   
                                     dh 
                                     2 
                                   
                                 
                               
                             
                             ] 
                           
                         
                         = 
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     m 
                                     11 
                                   
                                 
                                 
                                   
                                     m 
                                     12 
                                   
                                 
                               
                               
                                 
                                   
                                     m 
                                     21 
                                   
                                 
                                 
                                   
                                     m 
                                     22 
                                   
                                 
                               
                             
                             ] 
                           
                            
                           
                             [ 
                             
                               
                                 
                                   
                                     dh 
                                     1 
                                   
                                 
                               
                               
                                 
                                   
                                     dh 
                                     2 
                                   
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                     ; 
                     where 
                   
                    
                   
                     
 
                   
                    
                   
                     
                       m 
                       11 
                     
                     = 
                     
                       
                         m 
                         12 
                       
                       = 
                       
                         
                           
                             2 
                           
                            
                           L 
                         
                         
                           
                             
                               ( 
                               
                                 
                                   h 
                                   1 
                                 
                                 + 
                                 
                                   h 
                                   2 
                                 
                               
                               ) 
                             
                             2 
                           
                           + 
                           
                             2 
                              
                             
                               L 
                               2 
                             
                           
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       m 
                       21 
                     
                     = 
                     
                       
                         - 
                         
                           m 
                           22 
                         
                       
                       = 
                       
                         - 
                         
                           
                             
                               
                                 
                                   L 
                                   3 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       L 
                                       2 
                                     
                                     + 
                                     
                                       h 
                                       2 
                                       2 
                                     
                                     + 
                                     
                                       
                                         h 
                                         1 
                                       
                                        
                                       
                                         h 
                                         2 
                                       
                                     
                                   
                                   ) 
                                 
                               
                                
                               
                                 ( 
                                 
                                   
                                     2 
                                      
                                     
                                       L 
                                       2 
                                     
                                   
                                   + 
                                   
                                     
                                       ( 
                                       
                                         
                                           h 
                                           1 
                                         
                                         + 
                                         
                                           h 
                                           2 
                                         
                                       
                                       ) 
                                     
                                     2 
                                   
                                 
                                 ) 
                               
                             
                             
                               
                                 ( 
                                 
                                   
                                     L 
                                     2 
                                   
                                   + 
                                   
                                     h 
                                     1 
                                     2 
                                   
                                   + 
                                   
                                     h 
                                     2 
                                     2 
                                   
                                 
                                 ) 
                               
                                
                               
                                 
                                   ( 
                                   
                                     
                                       
                                         L 
                                         2 
                                       
                                        
                                       
                                         ( 
                                         
                                           
                                             h 
                                             1 
                                           
                                           + 
                                           
                                             h 
                                             2 
                                           
                                         
                                         ) 
                                       
                                     
                                     + 
                                     
                                       2 
                                        
                                       
                                         L 
                                         4 
                                       
                                     
                                   
                                   ) 
                                 
                                 
                                   3 
                                   / 
                                   2 
                                 
                               
                             
                           
                           . 
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   13 
                 
               
             
           
         
       
     
         [0000]    In some examples, the base orientation error is determined experimentally by, for example, communicating signals from the satellite  100  to the ground station  104  and/or by communicating signals from the ground station  104  to the satellite  100 . 
         [0035]      FIGS. 5-6  are plots  500  and  600  illustrating power levels of a payload signal detected at the ground station  104 .  FIG. 5  is a two-dimensional plot  500  of the power levels detected at the ground station  104 , and  FIG. 6  is a three-dimensional plot  600  of the power levels detected at the ground station  104 . In the illustrated example, the base orientation error is determined based on the power levels of the payload signal detected at the ground station  104  while the payload  206  is moved to scan (e.g., raster scan) a given area. When the example payload  206  is pointed at the ground station  104 , the ground station  104  detects a maximum power level. As the payload  206  points away from the ground station  104 , the ground station  104  detects a lesser power level. In the illustrated example, a first axis of each of the plots  500  and  600  corresponds to the azimuth angle of the base  202 , and a second axis of each of the plots  500  and  600  corresponds to the elevation of the base  202 . In the example plot  500  of  FIG. 5 , the power levels are illustrated by rings  502  having sizes corresponding to an amount of power detected at the ground station  104 . In the example plot of  FIG. 5 , a smallest ring  504  corresponds to a maximum power level detected at the ground station  104 , and a largest ring  506  corresponds to a lowest power level detected at the ground station  104 . In the example plot  600  of  FIG. 6 , a third axis corresponds to the power levels detected at the ground station  104 . 
         [0036]    In the example plots  500  and  600  of  FIGS. 5 and 6 , coordinates (0, 0) and (0, 0, 0), respectively, correspond to the resultant base orientation (i.e., an orientation of the base  202  when the first actuator  204  and the second actuator  300  are actuated based on the first estimated stroke position and the second estimated stroke position, respectively). When the base  202  is oriented at the resultant base orientation, the ground station  104  detects the power levels of the signal communicated by the payload  206 . Then, the payload  206  is scanned (e.g., raster scanned) or moved to point to some or all positions illustrated in the plots  500  and  600  of  FIGS. 5-6  while the power levels are detected (e.g., continuously, at predetermined intervals of time, etc.) at the ground station  104 . In the illustrated example, the ground station  104  detects the maximum power level when the base  202  is positioned at coordinates (0.02, 0.04). Thus, the payload  206  substantially points at the ground station  104  when the base  202  is positioned at coordinates (0.02, 0.04). The base orientation error is a distance between the orientation of the base  202  at which the ground station  104  measures the maximum amount of power (e.g., coordinates (0.02, 0.04)) and the resultant base orientation (e.g., (0, 0)) as shown in the following equation: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       [ 
                       
                         
                           
                             
                               d 
                                
                               
                                   
                               
                                
                               α 
                             
                           
                         
                         
                           
                             
                               d 
                                
                               
                                   
                               
                                
                               ɛ 
                             
                           
                         
                       
                       ] 
                     
                     measure 
                   
                   = 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   
                                       
                                   
                                    
                                   α 
                                 
                               
                             
                             
                               
                                 
                                   
                                       
                                   
                                    
                                   ɛ 
                                 
                               
                             
                           
                           ] 
                         
                         measure 
                       
                       - 
                       
                         
                           [ 
                           
                             
                               
                                 
                                   
                                       
                                   
                                    
                                   α 
                                 
                               
                             
                             
                               
                                 
                                   
                                       
                                   
                                    
                                   ɛ 
                                 
                               
                             
                           
                           ] 
                         
                         cmd 
                       
                     
                     : 
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   14 
                 
               
             
           
         
       
     
         [0037]    where the first term on the right labeled with subscript “measure” represents measured values of the base angles, the second term on the right labeled with subscript “cmd” represents the commanded values (e.g., estimated values) of the base angle, and the term on the left is the difference between the commanded based angles and the measured base angles, or the base orientation error. 
         [0038]    In some examples, signals (e.g., radio frequency (RF) signals, etc.) are communicated from the ground station  104  (e.g., via ground-based beacon systems) to the satellite  100  to determine the base orientation error. When the satellite  100  receives the signals, the satellite  100  decodes and/or demodulates the signals to determine the base orientation error. Other examples employ other techniques to determine the base orientation error. 
         [0039]    Based on the base orientation error and using, for example, the following equation, a first stroke position error of the first actuator  204  and a second stroke position error of the second actuator  300  may be determined: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         [ 
                         
                           
                             
                               
                                 dh 
                                 1 
                               
                             
                           
                           
                             
                               
                                 dh 
                                 2 
                               
                             
                           
                         
                         ] 
                       
                       est 
                     
                     = 
                     
                       
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     m 
                                     11 
                                   
                                 
                                 
                                   
                                     m 
                                     12 
                                   
                                 
                               
                               
                                 
                                   
                                     m 
                                     21 
                                   
                                 
                                 
                                   
                                     m 
                                     22 
                                   
                                 
                               
                             
                             ] 
                           
                           
                             - 
                             1 
                           
                         
                          
                         
                           [ 
                           
                             
                               
                                 
                                   d 
                                    
                                   
                                       
                                   
                                    
                                   α 
                                 
                               
                             
                             
                               
                                 
                                   d 
                                    
                                   
                                       
                                   
                                    
                                   ɛ 
                                 
                               
                             
                           
                           ] 
                         
                       
                       measure 
                     
                   
                   ; 
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   15 
                 
               
             
           
         
       
     
         [0040]    where the term on the left represents estimated first and second stroke position errors and where 
         [0000]    
       
         
           
             
               m 
               11 
             
             = 
             
               
                 m 
                 12 
               
               = 
               
                 
                   
                     2 
                   
                    
                   L 
                 
                 
                   
                     
                       ( 
                       
                         
                           h 
                           1 
                         
                         + 
                         
                           h 
                           2 
                         
                       
                       ) 
                     
                     2 
                   
                   + 
                   
                     2 
                      
                     
                       L 
                       2 
                     
                   
                 
               
             
           
         
       
       
         
           
             
               m 
               21 
             
             = 
             
               
                 - 
                 
                   m 
                   22 
                 
               
               = 
               
                 - 
                 
                   
                     
                       
                         L 
                         3 
                       
                        
                       
                         ( 
                         
                           
                             L 
                             2 
                           
                           + 
                           
                             h 
                             2 
                             2 
                           
                           + 
                           
                             
                               h 
                               1 
                             
                              
                             
                               h 
                               2 
                             
                           
                         
                         ) 
                       
                     
                      
                     
                       ( 
                       
                         
                           2 
                            
                           
                             L 
                             2 
                           
                         
                         + 
                         
                           
                             ( 
                             
                               
                                 h 
                                 1 
                               
                               + 
                               
                                 h 
                                 2 
                               
                             
                             ) 
                           
                           2 
                         
                       
                       ) 
                     
                   
                   
                     
                       ( 
                       
                         
                           L 
                           2 
                         
                         + 
                         
                           h 
                           1 
                           2 
                         
                         + 
                         
                           h 
                           2 
                           2 
                         
                       
                       ) 
                     
                      
                     
                       
                         ( 
                         
                           
                             
                               L 
                               2 
                             
                              
                             
                               ( 
                               
                                 
                                   h 
                                   1 
                                 
                                 + 
                                 
                                   h 
                                   2 
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             2 
                              
                             
                               L 
                               4 
                             
                           
                         
                         ) 
                       
                       
                         3 
                         / 
                         2 
                       
                     
                   
                 
               
             
           
         
       
     
         [0000]    when assuming L 1  equals L 2 . 
         [0041]    In the illustrated example, the first and second stroke position errors, dh 1  and dh 2 , respectively, are inherent in the payload assembly  200  (i.e., the stroke position errors are constant or invariable relative to the orientation of the base  202  and/or an amount of rotation of the base  202  via the first actuator  204  and/or the second actuator  300 ). Thus, the controller  306  may compensate for the first and second stroke position errors to point the payload  206  at the target irrespective of a range of movement of the base  202 . In the illustrated example, the controller  306  compensates for the first and second stroke position errors by commanding the first actuator  204  and the second actuator  300  to actuate to the first and second corrected stroke positions. To determine the first and second corrected stroke positions, the first and second stroke position errors are subtracted from the first and second estimated stroke positions, respectively, as shown in the following equation: 
         [0000]    
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             h 
                             1 
                           
                         
                       
                       
                         
                           
                             h 
                             2 
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     h 
                                     1 
                                   
                                 
                               
                               
                                 
                                   
                                     h 
                                     2 
                                   
                                 
                               
                             
                             ] 
                           
                           nonlinear 
                         
                          
                       
                       
                         
                           
                             
                               
                                 
                                   
                                     Nominally 
                                      
                                     
                                         
                                     
                                      
                                     calculated 
                                      
                                     
                                         
                                     
                                      
                                     actuator 
                                   
                                 
                               
                               
                                 
                                   
                                     travel 
                                      
                                     
                                         
                                     
                                      
                                     based 
                                      
                                     
                                         
                                     
                                      
                                     on 
                                      
                                     
                                         
                                     
                                      
                                     the 
                                      
                                     
                                         
                                     
                                      
                                     desired 
                                      
                                     
                                         
                                     
                                      
                                     or 
                                   
                                 
                               
                             
                           
                         
                         
                           
                             
                               commanded 
                                
                               
                                   
                               
                                
                               base 
                                
                               
                                   
                               
                                
                               angles 
                             
                           
                         
                       
                     
                     - 
                     
                       
                         [ 
                         
                           
                             
                               
                                 dh 
                                 1 
                               
                             
                           
                           
                             
                               
                                 dh 
                                 2 
                               
                             
                           
                         
                         ] 
                       
                       est 
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   16 
                 
               
             
           
         
       
     
         [0042]    where the first term on the right labeled with subscript “nonlinear” is the nominally calculated actuator travel based on the desired (or commanded or estimated) base angles and the second term on the right with subscript “est” estimate the actuator errors. When the first actuator  204  is actuated to the first corrected stroke position and the second actuator  300  is actuated to the second corrected stroke position, the base  202  is oriented such that the payload  206  points at the ground station  104 . 
         [0043]    The pointing direction of the payload  206  may be subsequently adjusted to keep the payload  206  pointed at the ground station  104  (e.g., if the satellite  100  moves relative to the ground station  104 ), point the payload  206  at another target, etc. In such examples, the controller  306  determines an updated estimated base orientation to point the payload  206  at the ground station  104  (or other target) based on an updated LOS vector from the satellite  100  to the ground station  104  (or the other target) and/or from the ground station  104  to the satellite  100 . The controller  306  then determines a first updated estimated stroke position and a second updated estimated stroke position based on the updated estimated base orientation. To compensate for the first and second stroke position errors, the controller  306  determines first and second updated corrected stroke positions. To determine the first and second updated corrected stroke positions, the controller  306  subtracts the first and second stroke position errors from the first and second updated estimated stroke positions, respectively. The controller  306  communicates a command to the first actuator  204  and the second actuator  300  to move to the first and second updated corrected stroke positions, and the first actuator  204  and the second actuator  300  actuate to point the payload  206  at the ground station  104  (or the other target). 
         [0044]    In some examples, alignment errors such as, for example, actuator alignment errors, actuator assembly parameter errors influenced by, for example, thermal expansion, machining tolerances, etc. influence the first stroke position error and/or the second stroke position error. In some such examples, the example controller  306  may compensate for the alignment errors. The example alignment errors determined below are related to the separation angle φ and the first distance L 1  from point A to point B and/or the second distance L 2  from point A to point C. However, additionally or alternatively, other alignment errors may be determined in other examples. The example alignment errors may be determined using, for example, the following equation: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         [ 
                         
                           
                             
                               
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 α 
                               
                             
                           
                           
                             
                               
                                 d 
                                  
                                 
                                     
                                 
                                  
                                 ɛ 
                               
                             
                           
                         
                         ] 
                       
                       = 
                       
                         
                           
                             [ 
                             
                               M 
                                
                               
                                   
                               
                                
                               G 
                             
                             ] 
                           
                            
                           
                             [ 
                             
                               
                                 
                                   
                                     dh 
                                     1 
                                   
                                 
                               
                               
                                 
                                   
                                     dh 
                                     1 
                                   
                                 
                               
                               
                                 
                                   
                                     d 
                                      
                                     
                                         
                                     
                                      
                                     φ 
                                   
                                 
                               
                               
                                 
                                   
                                     d 
                                      
                                     
                                         
                                     
                                      
                                     L 
                                   
                                 
                               
                             
                             ] 
                           
                         
                         = 
                         
                           
                             
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         m 
                                         11 
                                       
                                     
                                     
                                       
                                         m 
                                         12 
                                       
                                     
                                     
                                       
                                         g 
                                         11 
                                       
                                     
                                     
                                       
                                         g 
                                         12 
                                       
                                     
                                   
                                   
                                     
                                       
                                         m 
                                         21 
                                       
                                     
                                     
                                       
                                         m 
                                         22 
                                       
                                     
                                     
                                       
                                         g 
                                         21 
                                       
                                     
                                     
                                       
                                         g 
                                         22 
                                       
                                     
                                   
                                 
                                 ] 
                               
                               
                                  
                                 C 
                               
                             
                              
                             
                               [ 
                               
                                 
                                   
                                     
                                       dh 
                                       1 
                                     
                                   
                                 
                                 
                                   
                                     
                                       dh 
                                       1 
                                     
                                   
                                 
                                 
                                   
                                     
                                       d 
                                        
                                       
                                           
                                       
                                        
                                       φ 
                                     
                                   
                                 
                                 
                                   
                                     
                                       d 
                                        
                                       
                                           
                                       
                                        
                                       L 
                                     
                                   
                                 
                               
                               ] 
                             
                           
                           = 
                           C 
                         
                       
                     
                     , 
                     
                       
 
                     
                      
                     
                       
                         where 
                          
                         
                             
                         
                          
                         
                           m 
                           11 
                         
                       
                       = 
                       
                         
                           m 
                           12 
                         
                         = 
                         
                           
                             
                               2 
                             
                              
                             L 
                           
                           
                             
                               
                                 ( 
                                 
                                   
                                     h 
                                     1 
                                   
                                   + 
                                   
                                     h 
                                     2 
                                   
                                 
                                 ) 
                               
                               2 
                             
                             + 
                             
                               2 
                                
                               
                                 L 
                                 2 
                               
                             
                           
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       m 
                       21 
                     
                     = 
                     
                       
                         - 
                         
                           m 
                           22 
                         
                       
                       = 
                       
                         - 
                         
                           
                             
                               
                                 L 
                                 3 
                               
                                
                               
                                 ( 
                                 
                                   
                                     L 
                                     2 
                                   
                                   + 
                                   
                                     h 
                                     2 
                                     2 
                                   
                                   + 
                                   
                                     
                                       h 
                                       1 
                                     
                                      
                                     
                                       h 
                                       2 
                                     
                                   
                                 
                                 ) 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   2 
                                    
                                   
                                     L 
                                     2 
                                   
                                 
                                 + 
                                 
                                   
                                     ( 
                                     
                                       
                                         h 
                                         1 
                                       
                                       + 
                                       
                                         h 
                                         2 
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                               
                               ) 
                             
                           
                           
                             
                               ( 
                               
                                 
                                   L 
                                   2 
                                 
                                 + 
                                 
                                   h 
                                   1 
                                   2 
                                 
                                 + 
                                 
                                   h 
                                   2 
                                   2 
                                 
                               
                               ) 
                             
                              
                             
                               
                                 ( 
                                 
                                   
                                     
                                       L 
                                       2 
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           h 
                                           1 
                                         
                                         + 
                                         
                                           h 
                                           2 
                                         
                                       
                                       ) 
                                     
                                   
                                   + 
                                   
                                     2 
                                      
                                     
                                       L 
                                       4 
                                     
                                   
                                 
                                 ) 
                               
                               
                                 3 
                                 / 
                                 2 
                               
                             
                           
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       g 
                       11 
                     
                     = 
                     
                       
                         
                           sin 
                            
                           
                             ( 
                             
                               α 
                               2 
                             
                             ) 
                           
                         
                         × 
                         
                           ( 
                           
                             
                               h 
                               1 
                             
                             + 
                             
                               h 
                               2 
                             
                           
                           ) 
                         
                       
                       
                         4 
                          
                         L 
                         × 
                         
                           
                             cos 
                             2 
                           
                            
                           
                             ( 
                             
                               α 
                               2 
                             
                             ) 
                           
                         
                         × 
                         
                           ( 
                           
                             
                               
                                 
                                   ( 
                                   
                                     
                                       h 
                                       1 
                                     
                                     + 
                                     
                                       h 
                                       2 
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                               
                                 4 
                                  
                                 
                                   L 
                                   2 
                                 
                                 × 
                                 
                                   
                                     cos 
                                     2 
                                   
                                    
                                   
                                     ( 
                                     
                                       α 
                                       2 
                                     
                                     ) 
                                   
                                 
                               
                             
                             + 
                             1 
                           
                           ) 
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       g 
                       12 
                     
                     = 
                     
                       - 
                       
                         
                           ( 
                           
                             
                               h 
                               1 
                             
                             + 
                             
                               h 
                               2 
                             
                           
                           ) 
                         
                         
                           2 
                            
                           
                             L 
                             2 
                           
                           × 
                           
                             cos 
                              
                             
                               ( 
                               
                                 α 
                                 2 
                               
                               ) 
                             
                           
                           × 
                           
                             ( 
                             
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         h 
                                         1 
                                       
                                       + 
                                       
                                         h 
                                         2 
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                                 
                                   4 
                                    
                                   
                                     L 
                                     2 
                                   
                                   × 
                                   
                                     
                                       cos 
                                       2 
                                     
                                      
                                     
                                       ( 
                                       
                                         α 
                                         2 
                                       
                                       ) 
                                     
                                   
                                 
                               
                               + 
                               1 
                             
                             ) 
                           
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       g 
                       21 
                     
                     = 
                     
                       - 
                       
                         
                           
                             
                               
                                 
                                   
                                     
                                       2 
                                     
                                      
                                     L 
                                      
                                     
                                         
                                     
                                      
                                     
                                       cos 
                                        
                                       
                                         ( 
                                         
                                           α 
                                           2 
                                         
                                         ) 
                                       
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           h 
                                           1 
                                         
                                         - 
                                         
                                           h 
                                           2 
                                         
                                       
                                       ) 
                                     
                                   
                                   
                                     
                                       2 
                                        
                                       
                                         [ 
                                         
                                           
                                             
                                               
                                                 
                                                   L 
                                                   4 
                                                 
                                                 - 
                                                 
                                                   
                                                     L 
                                                     4 
                                                   
                                                    
                                                   
                                                     cos 
                                                      
                                                     
                                                       ( 
                                                       
                                                         2 
                                                          
                                                         
                                                             
                                                         
                                                          
                                                         α 
                                                       
                                                       ) 
                                                     
                                                   
                                                 
                                                 - 
                                               
                                             
                                           
                                           
                                             
                                               
                                                 2 
                                                  
                                                 
                                                   
                                                     
                                                       L 
                                                       2 
                                                     
                                                      
                                                     
                                                       ( 
                                                       
                                                         
                                                           h 
                                                           1 
                                                         
                                                         + 
                                                         
                                                           h 
                                                           2 
                                                         
                                                       
                                                       ) 
                                                     
                                                   
                                                   2 
                                                 
                                                  
                                                 
                                                   ( 
                                                   
                                                     
                                                       
                                                         cos 
                                                          
                                                         
                                                           ( 
                                                           α 
                                                           ) 
                                                         
                                                       
                                                       2 
                                                     
                                                     - 
                                                     
                                                       1 
                                                       2 
                                                     
                                                   
                                                   ) 
                                                 
                                               
                                             
                                           
                                         
                                         ] 
                                       
                                     
                                     
                                       1 
                                       / 
                                       2 
                                     
                                   
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     2 
                                   
                                    
                                   L 
                                    
                                   
                                       
                                   
                                    
                                   
                                     cos 
                                      
                                     
                                       ( 
                                       
                                         α 
                                         2 
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     
                                       ( 
                                       
                                         
                                           h 
                                           1 
                                         
                                         - 
                                         
                                           h 
                                           2 
                                         
                                       
                                       ) 
                                     
                                      
                                     
                                       [ 
                                       
                                         
                                           
                                             
                                               
                                                 2 
                                                  
                                                 
                                                   L 
                                                   2 
                                                 
                                                  
                                                 
                                                   sin 
                                                    
                                                   
                                                     ( 
                                                     
                                                       2 
                                                        
                                                       α 
                                                     
                                                     ) 
                                                   
                                                 
                                               
                                               + 
                                             
                                           
                                         
                                         
                                           
                                             
                                               
                                                 L 
                                                 2 
                                               
                                                
                                               
                                                 sin 
                                                  
                                                 
                                                   ( 
                                                   α 
                                                   ) 
                                                 
                                               
                                                
                                               
                                                 
                                                   ( 
                                                   
                                                     
                                                       h 
                                                       1 
                                                     
                                                     + 
                                                     
                                                       h 
                                                       2 
                                                     
                                                   
                                                   ) 
                                                 
                                                 2 
                                               
                                             
                                           
                                         
                                       
                                       ] 
                                     
                                   
                                 
                                 
                                   
                                     2 
                                      
                                     
                                       [ 
                                       
                                         
                                           
                                             
                                               
                                                 L 
                                                 4 
                                               
                                               - 
                                               
                                                 
                                                   L 
                                                   4 
                                                 
                                                  
                                                 
                                                   cos 
                                                    
                                                   
                                                     ( 
                                                     
                                                       2 
                                                        
                                                       
                                                           
                                                       
                                                        
                                                       α 
                                                     
                                                     ) 
                                                   
                                                 
                                               
                                               - 
                                             
                                           
                                         
                                         
                                           
                                             
                                               2 
                                                
                                               
                                                 
                                                   
                                                     L 
                                                     2 
                                                   
                                                    
                                                   
                                                     ( 
                                                     
                                                       
                                                         h 
                                                         1 
                                                       
                                                       + 
                                                       
                                                         h 
                                                         2 
                                                       
                                                     
                                                     ) 
                                                   
                                                 
                                                 2 
                                               
                                                
                                               
                                                 ( 
                                                 
                                                   
                                                     
                                                       cos 
                                                        
                                                       
                                                         ( 
                                                         α 
                                                         ) 
                                                       
                                                     
                                                     2 
                                                   
                                                   - 
                                                   
                                                     1 
                                                     2 
                                                   
                                                 
                                                 ) 
                                               
                                             
                                           
                                         
                                       
                                       ] 
                                     
                                   
                                   
                                     3 
                                     / 
                                     2 
                                   
                                 
                               
                             
                           
                         
                         
                           
                             
                               2 
                                
                               
                                 L 
                                 2 
                               
                                
                               
                                 
                                   cos 
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     α 
                                     2 
                                   
                                   ) 
                                 
                               
                                
                               
                                 
                                   ( 
                                   
                                     
                                       h 
                                       1 
                                     
                                     - 
                                     
                                       h 
                                       2 
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                             
                             
                               
                                 
                                   
                                     
                                       
                                         L 
                                         4 
                                       
                                        
                                       
                                         cos 
                                          
                                         
                                           ( 
                                           
                                             2 
                                              
                                             
                                                 
                                             
                                              
                                             α 
                                           
                                           ) 
                                         
                                       
                                     
                                     - 
                                     
                                       L 
                                       4 
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     2 
                                      
                                     
                                       
                                         
                                           L 
                                           2 
                                         
                                          
                                         
                                           ( 
                                           
                                             
                                               h 
                                               1 
                                             
                                             + 
                                             
                                               h 
                                               2 
                                             
                                           
                                           ) 
                                         
                                       
                                       2 
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           
                                             cos 
                                              
                                             
                                               ( 
                                               α 
                                               ) 
                                             
                                           
                                           2 
                                         
                                         - 
                                         
                                           1 
                                           2 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                           - 
                           1 
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       g 
                       22 
                     
                     = 
                     
                       - 
                       
                         
                           
                             
                               
                                 
                                   
                                     
                                       2 
                                     
                                      
                                     L 
                                      
                                     
                                         
                                     
                                      
                                     
                                       cos 
                                        
                                       
                                         ( 
                                         
                                           α 
                                           2 
                                         
                                         ) 
                                       
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           h 
                                           1 
                                         
                                         - 
                                         
                                           h 
                                           2 
                                         
                                       
                                       ) 
                                     
                                   
                                   
                                     
                                       [ 
                                       
                                         
                                           
                                             
                                               
                                                 L 
                                                 4 
                                               
                                               - 
                                               
                                                 
                                                   L 
                                                   4 
                                                 
                                                  
                                                 
                                                   cos 
                                                    
                                                   
                                                     ( 
                                                     
                                                       2 
                                                        
                                                       
                                                           
                                                       
                                                        
                                                       α 
                                                     
                                                     ) 
                                                   
                                                 
                                               
                                               - 
                                             
                                           
                                         
                                         
                                           
                                             
                                               2 
                                                
                                               
                                                 
                                                   
                                                     L 
                                                     2 
                                                   
                                                    
                                                   
                                                     ( 
                                                     
                                                       
                                                         h 
                                                         1 
                                                       
                                                       + 
                                                       
                                                         h 
                                                         2 
                                                       
                                                     
                                                     ) 
                                                   
                                                 
                                                 2 
                                               
                                                
                                               
                                                 ( 
                                                 
                                                   
                                                     
                                                       cos 
                                                        
                                                       
                                                         ( 
                                                         α 
                                                         ) 
                                                       
                                                     
                                                     2 
                                                   
                                                   - 
                                                   
                                                     1 
                                                     2 
                                                   
                                                 
                                                 ) 
                                               
                                             
                                           
                                         
                                       
                                       ] 
                                     
                                     
                                       1 
                                       / 
                                       2 
                                     
                                   
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     2 
                                   
                                    
                                   L 
                                    
                                   
                                       
                                   
                                    
                                   
                                     cos 
                                      
                                     
                                       ( 
                                       
                                         α 
                                         2 
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     
                                       ( 
                                       
                                         
                                           h 
                                           1 
                                         
                                         - 
                                         
                                           h 
                                           2 
                                         
                                       
                                       ) 
                                     
                                      
                                     
                                       [ 
                                       
                                         
                                           
                                             
                                               
                                                 4 
                                                  
                                                 
                                                   L 
                                                   3 
                                                 
                                                  
                                                 
                                                   cos 
                                                    
                                                   
                                                     ( 
                                                     
                                                       2 
                                                        
                                                       α 
                                                     
                                                     ) 
                                                   
                                                 
                                               
                                               - 
                                             
                                           
                                         
                                         
                                           
                                             
                                               
                                                 4 
                                                  
                                                 
                                                   L 
                                                   3 
                                                 
                                               
                                               + 
                                               
                                                 4 
                                                  
                                                 
                                                   L 
                                                   3 
                                                 
                                               
                                               + 
                                               
                                                 4 
                                                  
                                                 
                                                   
                                                     L 
                                                      
                                                     
                                                       ( 
                                                       
                                                         
                                                           h 
                                                           1 
                                                         
                                                         + 
                                                         
                                                           h 
                                                           2 
                                                         
                                                       
                                                       ) 
                                                     
                                                   
                                                   2 
                                                 
                                               
                                             
                                           
                                         
                                         
                                           
                                             
                                               ( 
                                               
                                                 
                                                   
                                                     cos 
                                                      
                                                     
                                                       ( 
                                                       α 
                                                       ) 
                                                     
                                                   
                                                   2 
                                                 
                                                 - 
                                                 
                                                   1 
                                                   2 
                                                 
                                               
                                               ) 
                                             
                                           
                                         
                                       
                                       ] 
                                     
                                   
                                 
                                 
                                   
                                     2 
                                      
                                     
                                       [ 
                                       
                                         
                                           
                                             
                                               
                                                 L 
                                                 4 
                                               
                                               - 
                                               
                                                 
                                                   L 
                                                   4 
                                                 
                                                  
                                                 
                                                   cos 
                                                    
                                                   
                                                     ( 
                                                     
                                                       2 
                                                        
                                                       
                                                           
                                                       
                                                        
                                                       α 
                                                     
                                                     ) 
                                                   
                                                 
                                               
                                               - 
                                             
                                           
                                         
                                         
                                           
                                             
                                               2 
                                                
                                               
                                                 
                                                   
                                                     L 
                                                     2 
                                                   
                                                    
                                                   
                                                     ( 
                                                     
                                                       
                                                         h 
                                                         1 
                                                       
                                                       + 
                                                       
                                                         h 
                                                         2 
                                                       
                                                     
                                                     ) 
                                                   
                                                 
                                                 2 
                                               
                                                
                                               
                                                 ( 
                                                 
                                                   
                                                     
                                                       cos 
                                                        
                                                       
                                                         ( 
                                                         α 
                                                         ) 
                                                       
                                                     
                                                     2 
                                                   
                                                   - 
                                                   
                                                     1 
                                                     2 
                                                   
                                                 
                                                 ) 
                                               
                                             
                                           
                                         
                                       
                                       ] 
                                     
                                   
                                   
                                     3 
                                     / 
                                     2 
                                   
                                 
                               
                             
                           
                         
                         
                           
                             
                               2 
                                
                               
                                 L 
                                 2 
                               
                                
                               
                                 
                                   cos 
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     α 
                                     2 
                                   
                                   ) 
                                 
                               
                                
                               
                                 
                                   ( 
                                   
                                     
                                       h 
                                       1 
                                     
                                     - 
                                     
                                       h 
                                       2 
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                             
                             
                               
                                 
                                   
                                     
                                       
                                         L 
                                         4 
                                       
                                        
                                       
                                         cos 
                                          
                                         
                                           ( 
                                           
                                             2 
                                              
                                             
                                                 
                                             
                                              
                                             α 
                                           
                                           ) 
                                         
                                       
                                     
                                     - 
                                     
                                       L 
                                       4 
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     2 
                                      
                                     
                                       
                                         
                                           L 
                                           2 
                                         
                                          
                                         
                                           ( 
                                           
                                             
                                               h 
                                               1 
                                             
                                             + 
                                             
                                               h 
                                               2 
                                             
                                           
                                           ) 
                                         
                                       
                                       2 
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           
                                             cos 
                                              
                                             
                                               ( 
                                               α 
                                               ) 
                                             
                                           
                                           2 
                                         
                                         - 
                                         
                                           1 
                                           2 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                           - 
                           1 
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   17 
                 
               
             
           
         
       
     
         [0045]    In Equation 17, L is the first distance L 1  or the second distance L 2  when L 1 =L 2 . In some examples, when L 1  is not equal to L 2 , equation  17  becomes more complex and is not further described herein for simplicity. Thus, in some examples, a variation of Equation  17  may be used when L 1  is not equal to L 2 . Using, for example, Equation 18 below, a sequential estimator may be used to estimate the alignment errors in Equation 17: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       [ 
                       
                         
                           
                             
                               dh 
                               1 
                             
                           
                         
                         
                           
                             
                               dh 
                               1 
                             
                           
                         
                         
                           
                             
                               d 
                                
                               
                                   
                               
                                
                               φ 
                             
                           
                         
                         
                           
                             
                               d 
                                
                               
                                   
                               
                                
                               L 
                             
                           
                         
                       
                       ] 
                     
                     
                       est 
                       , 
                       k 
                     
                   
                   = 
                   
                     
                       
                         x 
                         _ 
                       
                       k 
                     
                     = 
                     
                       
                         
                           x 
                           _ 
                         
                         
                           k 
                           - 
                           1 
                         
                       
                       + 
                       
                         
                           K 
                           k 
                         
                          
                         
                           ( 
                           
                             
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         d 
                                          
                                         
                                             
                                         
                                          
                                         α 
                                       
                                     
                                   
                                   
                                     
                                       
                                         d 
                                          
                                         
                                             
                                         
                                          
                                         ɛ 
                                       
                                     
                                   
                                 
                                 ] 
                               
                               measure 
                             
                             - 
                             
                               C 
                                
                               
                                   
                               
                                
                               
                                 
                                   x 
                                   _ 
                                 
                                 
                                   k 
                                   - 
                                   1 
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   18 
                 
               
             
           
         
       
     
         [0000]    where the subscript “est” represent the estimated or commanded values and the subscript “measure” represent the measured values. In other examples, other estimators are used. In Equation 18,K is an update gain matrix at a k-th step. The gain matrix may be designed using any suitable method(s) such as minimum-variance, Kalman filter, fixed gain observers, etc. Thus, in the illustrated example, an amount of thermal distortion along the first distance (or any other suitable portion of the base  202 ) may be determined. Based on the alignment errors determined using Equations 17 and 18, corrected actuator assembly parameter values (e.g., the separation angle φ, the first distance L 1 , etc.) may be determined as follows: 
         [0000]    
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             φ 
                             est 
                           
                           = 
                           
                             φ 
                             + 
                             
                               d 
                                
                               
                                   
                               
                                
                               
                                 φ 
                                 est 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             L 
                             est 
                           
                           = 
                           
                             L 
                             + 
                             
                               d 
                                
                               
                                   
                               
                                
                               
                                 L 
                                 est 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   19 
                 
               
             
           
         
       
     
         [0000]    where values with subscript “est” represent estimated values. Because the alignment errors affect a determination of the estimated stroke positions, corrected estimated stroke positions may be determined based on the corrected actuator assembly parameters values as show in Equation 20 below: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       [ 
                       
                         
                           
                             
                               h 
                               1 
                             
                           
                         
                         
                           
                             
                               h 
                               2 
                             
                           
                         
                       
                       ] 
                     
                     
                       cmd 
                        
                       
                           
                       
                        
                       _ 
                        
                       
                           
                       
                        
                       nom 
                     
                   
                   = 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   
                                     h 
                                     1 
                                   
                                    
                                   
                                     ( 
                                     
                                       α 
                                       , 
                                       ɛ 
                                       , 
                                       
                                         φ 
                                         est 
                                       
                                       , 
                                       
                                         L 
                                         est 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     h 
                                     2 
                                   
                                    
                                   
                                     ( 
                                     
                                       α 
                                       , 
                                       ɛ 
                                       , 
                                       
                                         φ 
                                         est 
                                       
                                       , 
                                       
                                         L 
                                         est 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                           ] 
                         
                         nonlinear 
                       
                        
                     
                     
                       
                         
                           
                             
                               
                                 
                                   Calculated 
                                    
                                   
                                       
                                   
                                    
                                   actuator 
                                    
                                   
                                       
                                   
                                    
                                   steps 
                                    
                                   
                                       
                                   
                                    
                                   from 
                                    
                                   
                                       
                                   
                                    
                                   nonlinear 
                                    
                                   
                                       
                                   
                                    
                                   relation 
                                 
                               
                             
                             
                               
                                 
                                   of 
                                    
                                   
                                       
                                   
                                    
                                   required 
                                    
                                   
                                       
                                   
                                    
                                   base 
                                    
                                   
                                       
                                   
                                    
                                   angles 
                                    
                                   
                                       
                                   
                                    
                                   and 
                                    
                                   
                                       
                                   
                                    
                                   the 
                                    
                                   
                                       
                                   
                                    
                                   estimated 
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                             actuator 
                              
                             
                                 
                             
                              
                             geometry 
                              
                             
                                 
                             
                              
                             parameters 
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   20 
                 
               
             
           
         
       
     
         [0000]    Based on the corrected estimated stroke positions and the stroke position errors, the corrected stroke positions may be determined using the following equation: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       [ 
                       
                         
                           
                             
                               h 
                               1 
                             
                           
                         
                         
                           
                             
                               h 
                               2 
                             
                           
                         
                       
                       ] 
                     
                     cmd 
                   
                   = 
                   
                     
                       
                         [ 
                         
                           
                             
                               
                                 
                                   h 
                                   1 
                                 
                                  
                                 
                                   ( 
                                   
                                     α 
                                     , 
                                     ɛ 
                                     , 
                                     
                                       φ 
                                       est 
                                     
                                     , 
                                     
                                       L 
                                       est 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   h 
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     α 
                                     , 
                                     ɛ 
                                     , 
                                     
                                       φ 
                                       est 
                                     
                                     , 
                                     
                                       L 
                                       est 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                         ] 
                       
                       nonlinear 
                     
                     - 
                     
                       
                         [ 
                         
                           
                             
                               
                                 dh 
                                 1 
                               
                             
                           
                           
                             
                               
                                 dh 
                                 2 
                               
                             
                           
                         
                         ] 
                       
                       est 
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   21 
                 
               
             
           
         
       
     
         [0000]    Thus, once the corrected stroke positions are determined, the controller  306  may communicate with the first actuator  204  and the second actuator  300  to actuate to the corrected stroke positions to point the payload  206  at the ground station  104 . In such examples, the corrected stroke positions compensate for the stroke position errors and the alignment errors. 
         [0046]      FIG. 7  is a block diagram of the example controller  306  of  FIG. 3 . In the illustrated example, the controller  306  includes an instruction processor  700 , a LOS information determiner  702 , a base orientation determiner  704 , a memory  706 , an estimated stroke position determiner  708 , a base orientation error determiner  710 , a stroke position error determiner  712 , an alignment error determiner  714 , a corrected stroke position determiner  716 , and an actuator controller  718 . 
         [0047]    The example instruction processor  700  of  FIG. 7  receives instructions from the ground station  104  and/or a flight computer  720  disposed on and/or in communication with the satellite  100 . In some examples, the instructions include a command to determine a, LOS vector to a target, adjust a pointing direction of the payload  206 , actuate the first actuator  204  and/or the second actuator  300 , transmit a signal to the ground station  104  via the payload  206 , decode and/or demodulate signals received via the satellite  100 , etc. These instructions may be executed via the example controller  306 , stored in the memory  706 , communicated to one or more components of the satellite  100  and/or the ground station  104 , etc. 
         [0048]    The example LOS determiner of  FIG. 7  determines the LOS vector between the satellite  100  and a target (e.g., the ground station  104 ). In some examples, the LOS vector is based on the position of the satellite  100  and the position of the target in one or more frames of reference. The LOS information determiner  702  may access and/or utilize position information related to the satellite  100  and/or the target from the memory  706  and/or the flight computer  720 . In some examples, in response to the flight computer  720  indicating relative movement between the satellite  100  and the ground station  104 , instructions to point the payload  206  at a different target, etc., the LOS information determiner  702  determines an updated LOS vector from the satellite  100  to the target and/or from the target to the satellite  100 . In some examples, some or all of the LOS information is determined and/or provided via the flight computer and/or the target. 
         [0049]    The example base orientation determiner  704  of  FIG. 7  determines the estimated base orientation to point the payload  206  at the target (e.g., orient the payload  206  to enable signals transmitted and/or received via the payload  206  to propagate substantially along the LOS vector to the target). In the illustrated example, the base orientation determiner  704  determines an estimated azimuth angle and an estimated elevation angle of the base  202  to point the payload  206  at the target. 
         [0050]    The estimated stroke position determiner  708  determines a first estimated stroke position and a second estimated stroke position of the first actuator  204  and the second actuator  300 , respectively, to orient the base  202  at the estimated base orientation. In some examples, the estimated stroke position determiner  708  determines the estimated stroke positions based on the estimated base orientation by, for example, using Equations 1-8 and/or Equations 9-12 above. If the example alignment error determiner  714  determines that one or alignment errors are present, the estimated stroke position determiner  708  determines corrected estimated stroke positions to compensate for the alignment errors. 
         [0051]    The example base orientation error determiner  710  of  FIG. 7  determines the base orientation error. In some examples, the base orientation error determiner  710  determines the base orientation error experimentally. For example, the base orientation error determiner  710  may instruct a transmitter  722  (e.g., an antenna transmitter) to transmit a signal when the first actuator  204  is in the first estimated stroke position and the second actuator  300  is in the second estimated stroke position. At the ground station  104 , a power level of the signal is detected, and the base orientation error determiner  710  may instruct the actuator controller  718  to move (e.g., span) the payload  206 . At the ground station  104 , power levels of the signal are detected while the payload  206  is moved. Based on the power levels detected at the ground station  104 , the base orientation error is determined by the base orientation error determiner  710 . In some examples, the base orientation error is determined via the ground station  104  and communicated to satellite  100 . In other examples, signals (e.g., radio frequency (RF) signals, etc.) are communicated from the ground station  104  (e.g., via ground-based beacon systems) to the satellite  100 , and the base orientation error determiner  710  determines the base orientation error by decoding and/or demodulating the signals. 
         [0052]    The example stroke position error determiner  712  of  FIG. 7  determines a first stroke position error of the first actuator  204  and a second stroke position error of the second actuator  300 . In the illustrated example, the stroke position error determiner  712  determines the first and second stroke position errors based on the base orientation error. In the illustrated example, the stroke position errors are inherent in the payload assembly  200  (i.e., the stroke position errors are constant or invariable relative to the orientation of the base  202  and/or an amount of rotation of the base  202  via the first actuator  204  and/or the second actuator  300 ). Thus, the controller  306  may compensate for the first and second stroke position errors irrespective of an amount of movement of the base  202  to point the payload  206  at the target. 
         [0053]    The example alignment error determiner  714  of  FIG. 7  determines alignment errors of the example actuator assembly such as, for example, an alignment error of the first actuator  204  and/or the second actuator  300 , actuator assembly parameter value errors influenced by thermal distortion (e.g., thermal expansion, thermal contraction, bending influenced by a temperature gradient, etc.), machining tolerances, etc, and/or other alignment errors, etc. In some examples, the alignment error determiner  714  determines corrected actuator assembly parameter values such as, for example, a corrected separation angle, a corrected distance between point A and point C (i.e., a length between a first point about which the base  202  rotates via the joint  216  and a second point where the second actuator  300  is coupled to the base  202 ). The corrected actuator assembly parameter values may be used by the estimated stroke position determiner  708  to determine corrected estimated stroke positions. 
         [0054]    The example corrected stroke position determiner  716  of  FIG. 7  determines a first corrected stroke position of the first actuator  204  and a second corrected stroke position of the second actuator  300  based on the first and second stroke position errors, respectively. In some examples, the first corrected stroke position and the second corrected stroke position compensate for the first and second stroke position errors, respectively, and the alignment errors. When the first actuator  204  and the second actuator  300  are actuated to the first corrected stroke position and the second corrected stroke position, respectively, the payload  206  points at the target. 
         [0055]    The actuator controller  718  controls the first actuator  204  and/or the second actuator  300 . In the illustrated example, the actuator controller  718  instructs the first actuator  204  to actuate to a given stroke position such as, for example, the first estimated stroke position, the first corrected stroke position, and/or any other stroke position. The example actuator controller  718  may instruct the second actuator  300  to actuate to a given stroke position such as, for example, the second estimated stroke position, the second corrected stroke position, and/or any other stroke position. In some examples, the actuator controller  718  instructs the first actuator  204  and/or the second actuator  300  to move the payload  206  to scan (e.g., raster scan) an area. 
         [0056]    The example memory  706  (e.g., volatile memory, non-volatile memory, etc.) stores information such as, for example, a position of the ground station  104 , a position of the satellite  100 , actuator assembly parameter values, corrected actuator assembly parameter values, estimated stroke positions, stroke position errors, payload alignment errors, and/or any other information. The information stored in the example memory  706  may be accessed by one or more components of the example controller  306 , the example satellite  100 , the example ground station  104 , etc. 
         [0057]    While an example manner of implementing the controller  306  of  FIG. 3  has been illustrated in  FIG. 7 , one or more of the elements, processes and/or devices illustrated in  FIG. 7  may be combined, divided, re-arranged, omitted, eliminated and/or implemented in any other way. Further, the instruction processor  700 , the LOS information determiner  702 , the base orientation determiner  704 , the memory  706 , the estimated stroke position determiner  708 , the base orientation error determiner  710 , the stroke position error determiner  712 , the alignment error determiner  714 , the corrected stroke position determiner  716 , the actuator controller  718 , the ground station  104 , the flight computer  720 , the transmitter  722 , the first actuator  204 , the second actuator  300  and/or, more generally, the example controller  306  of  FIG. 7  may be implemented by hardware, software, firmware and/or any combination of hardware, software and/or firmware. Thus, for example, any of the instruction processor  700 , the LOS information determiner  702 , the base orientation determiner  704 , the memory  706 , the estimated stroke position determiner  708 , the base orientation error determiner  710 , the stroke position error determiner  712 , the alignment error determiner  714 , the corrected stroke position determiner  716 , the actuator controller  718 , the ground station  104 , the flight computer  720 , the transmitter  722 , the first actuator  204 , the second actuator  300  and/or, more generally, the example controller  306  of  FIG. 7  could be implemented by one or more circuit(s), programmable processor(s), application specific integrated circuit(s) (ASIC(s)), programmable logic device(s) (PLD(s)) and/or field programmable logic device(s) (FPLD(s)), etc. When any of the apparatus or system claims of this patent are read to cover a purely software and/or firmware implementation, at least one of the instruction processor  700 , the LOS information determiner  702 , the base orientation determiner  704 , the memory  706 , the estimated stroke position determiner  708 , the base orientation error determiner  710 , the stroke position error determiner  712 , the alignment error determiner  714 , the corrected stroke position determiner  716 , the actuator controller  718 , the ground station  104 , the flight computer  720 , the transmitter  722 , the first actuator  204 , the second actuator  300  and/or, more generally, the example controller  306  of  FIG. 7  are hereby expressly defined to include a tangible computer readable medium such as a memory, DVD, CD, Blu-ray, etc. storing the software and/or firmware. Further still, the example controller  306  of  FIG. 7  may include one or more elements, processes and/or devices in addition to, or instead of, those illustrated in  FIG. 7 , and/or may include more than one of any or all of the illustrated elements, processes and devices. 
         [0058]      FIGS. 8-9  depict example flow diagrams representative of methods or processes that may be implemented using, for example, computer readable instructions. The example processes of  FIGS. 8-9  may be performed using a processor, a controller (e.g., the example controller  306  of  FIG. 7 ) and/or any other suitable processing device. For example, the example processes of  FIGS. 8-9  may be implemented using coded instructions (e.g., computer readable instructions) stored on a tangible computer readable medium such as a flash memory, a read-only memory (ROM), and/or a random-access memory (RAM). As used herein, the term tangible computer readable medium is expressly defined to include any type of computer readable storage and to exclude propagating signals. Additionally or alternatively, the example process of  FIGS. 8-9  may be implemented using coded instructions (e.g., computer readable instructions) stored on a non-transitory computer readable medium such as a flash memory, a read-only memory (ROM), a random-access memory (RAM), a cache, or any other storage media in which information is stored for any duration (e.g., for extended time periods, permanently, brief instances, for temporarily buffering, and/or for caching of the information). As used herein, the term non- transitory computer readable medium is expressly defined to include any type of computer readable medium and to exclude propagating signals. 
         [0059]    Alternatively, some or all of the example processes of  FIGS. 8-9  may be implemented using any combination(s) of application specific integrated circuit(s) (ASIC(s)), programmable logic device(s) (PLD(s)), field programmable logic device(s) (FPLD(s)), discrete logic, hardware, firmware, etc. Also, one or more operations depicted in  FIGS. 8-9  may be implemented manually or as any combination(s) of any of the foregoing techniques, for example, any combination of firmware, software, discrete logic and/or hardware. 
         [0060]    Further, although the example processes of  FIGS. 8-9  are described with reference to the flow diagrams of  FIG. 8-9 , respectively, other methods of implementing the processes of  FIGS. 8-9  may be employed. For example, the order of execution of the blocks may be changed, and/or some of the blocks described may be changed, eliminated, sub-divided, or combined. Additionally, one or more of the operations depicted in  FIGS. 8-9  may be performed sequentially and/or in parallel by, for example, separate processing threads, processors, devices, discrete logic, circuits, etc. 
         [0061]      FIG. 8  is a flowchart representative of an example method  800  that can be performed to point the payload  206  at a target such as, for example, the ground station of  FIG. 1 . The example method  800  of  FIG. 8  begins by the LOS information determiner  702  of the controller  306  determining line of sight information (block  802 ). In some examples, determining the line of sight information includes determining a line of sight vector from the target (e.g., the ground station  104 ) to the satellite  100  and/or from the satellite  100  to the target. In some examples, the ground station  104  and/or the flight computer  720  determines and/or provides some or all of the LOS information. Based on the line of sight information, the base orientation determiner  704  determines an estimated base orientation to point the payload  206  at the target (e.g., the ground station  104 ) (block  804 ). In some examples, the estimated base orientation is defined by an estimated azimuth angle of the base  202  and an estimated elevation angle of the base  202  to point the payload  206  at the target. 
         [0062]    At block  806 , the estimated stroke position determiner  708  determines a first estimated stroke position of the first actuator  204  and a second estimated stroke position of the second actuator  300  to orient the base  202  at the estimated base orientation. At block  808 , the actuator controller  718  communicates a command to the first actuator  204  and the second actuator  300  to actuate to the first estimated stroke position and the second estimated stroke position, respectively. In some examples, as a result of thermal distortion, machining tolerances, etc., the payload  206  does not point at the target when the first actuator  204  and the second actuator  300  actuate to the first estimated stroke position and the second estimated stroke position, respectively. In such examples, the base  202  moves to a resultant base orientation. At block  810 , the base orientation error determiner  710  determines a base orientation error. The base orientation error may be a distance from the resultant base orientation to the position at which the payload  206  points at the target. In some examples, the base orientation error is determined by detecting power levels of a payload signal (e.g., transmitted via the transmitter  722 ) at the target. In other examples, the base orientation error is determined by decoding and/or demodulating signals received via the satellite  100  and/or via any other suitable technique. 
         [0063]    At block  812 , the alignment error determiner  714  determines an alignment error of the base  202 . In some examples, the alignment error includes an error in an alignment of the first actuator  204  and/or the second actuator  300 , an error in an actuator assembly parameter value such as, for example, the separation angle φ, the first distance L 1 , the second distance L 2 , etc. In some examples, the alignment error determiner  714  determines one or more corrected actuator assembly parameter values based on the alignment error. Based on the base orientation error and the alignment error, the stroke position error determiner  712  determines a first stroke position error and a second stroke position error corresponding to the first actuator  204  and the second actuator  300 , respectively (block  814 ). At block  816 , the corrected stroke position determiner  716  determines a first corrected stroke position and a second corrected stroke position based on the first stroke position error and the second stroke position error, respectively. In the illustrated example, the first and second corrected stroke positions compensate for the alignment error and the stroke position errors. At block  818 , the actuator controller  718  communicates a command to the first actuator  204  and the second actuator  300  to actuate to the first corrected stroke position and the second corrected stroke position, respectively. When the first actuator  204  and the second actuator  300  actuate to the first corrected stroke position and the second corrected stroke position, respectively, the first actuator  204  and the second actuator  300  orient the base  202  such that the payload  206  points at the target. 
         [0064]      FIG. 9  is a flowchart representative of an example method  900  that can be performed to point the payload  206  at a target. The example method  900  of  FIG. 9  begins by determining if the payload pointing direction is to be adjusted (block  902 ). In some examples, the flight computer  720  monitors a position of the satellite  100  relative to the target. If the flight computer  720  senses relative movement of the satellite  100 , the flight computer  720  communicates instructions to the instruction processor  700  to adjust a pointing direction of the payload  206  to point the payload  206  at the target. In some examples, the controller  306  may be instructed to adjust the pointing direction of the payload  206  to point the payload  206  at a different target. If the payload pointing direction is to be adjusted, the LOS information determiner  702  determines line of sight information (block  904 ). In some examples, the flight computer  720  and/or the target provides and/or determines some or all of the LOS information. In some examples, the line of sight information includes a line of sight vector from the target to the satellite  100  and/or from the satellite  100  to the target. Based on the light of sight information, the base orientation determiner  704  determines the estimated base orientation to point the payload  206  at the target (block  906 ). At block  908 , the estimated stroke position determiner  708  determines a first estimated stroke position of the first actuator  204  and a second estimated stroke position of the second actuator  300  to orient the base  202  at the estimated base orientation. In some examples, the first estimated stroke position and/or the second estimated stroke position may be determined based on previously determined corrected actuator assembly parameter values such as, for example, a corrected separation angle, etc. The previously determined corrected actuator assembly parameter values accessed via the memory  706 . 
         [0065]    At block  910 , the corrected stroke position determiner  716  determines a first corrected stroke position and a second corrected stroke position of the first actuator  204  and the second actuator  300 , respectively. To determine the first corrected stroke position, the example corrected stroke position determiner  716  compensates for a previously determined first stroke position error (e.g., by subtracting the previously determined first stroke position error from the first estimated stroke position). To determine the second corrected stroke position, the example corrected stroke position determiner  716  compensates for a previously determined second stroke position error (e.g., by subtracting the previously determined second stroke position error from the second estimated stroke position). 
         [0066]    At block  912 , the actuator controller  718  communicates a command to the first actuator  204  and the second actuator  300  to actuate to the first corrected stroke position and the second corrected stroke position, respectively. When the first actuator  204  and the second actuator  300  are actuated to the first corrected stroke position and the second corrected stroke position, respectively, the base  202  is oriented such that the payload  206  points at the target. 
         [0067]      FIG. 10  is a block diagram of an example computer  1000  capable of executing the instructions of  FIGS. 8-9  to implement the controller  306  of  FIG. 7 . The computer  1000  can be any suitable type of computing device. 
         [0068]    The computer  1000  of the instant example includes a processor  1012 . For example, the processor  1012  can be implemented by one or more microprocessors or controllers from any desired family or manufacturer. 
         [0069]    The processor  1012  includes a local memory  1013  (e.g., a cache) and is in communication with a main memory including a volatile memory  1014  and a non-volatile memory  1016  via a bus  1018 . The volatile memory  1014  may be implemented by Synchronous Dynamic Random Access Memory (SDRAM), Dynamic Random Access Memory (DRAM), RAMBUS Dynamic Random Access Memory (RDRAM) and/or any other type of random access memory device. The non-volatile memory  1016  may be implemented by flash memory and/or any other desired type of memory device. Access to the main memory  1014 ,  1016  is controlled by a memory controller. 
         [0070]    The computer  1000  also includes an interface circuit  1020 . The interface circuit  1020  may be implemented by any type of interface standard, such as an Ethernet interface, a universal serial bus (USB), and/or a PCI express interface. 
         [0071]    One or more input devices  1022  are connected to the interface circuit  1020 . The input device(s)  1022  permit a user to enter data and commands into the processor  1012 . 
         [0072]    One or more output devices  1024  are also connected to the interface circuit  1020 . The output devices  1024  can be implemented, for example, by a transmitter (e.g., the transmitter  722 ). The interface circuit  1020 , thus, may include a graphics driver card. 
         [0073]    The interface circuit  1020  also includes a communication device (e.g., communication device  56 ) such as a modem or network interface card to facilitate exchange of data with external computers via a network  1026  (e.g., a bus, coaxial cable, RF signal transmitter, etc.). 
         [0074]    The computer  1000  also includes one or more mass storage devices  1028  for storing software and data. Examples of such mass storage devices  1028  include hard drive disks, compact disk drives and digital versatile disk (DVD) drives. The mass storage device  1028  may implement the local storage device  62 . 
         [0075]    The coded instructions  1032  of  FIGS. 10  may be stored in the mass storage device  1028 , in the volatile memory  1014 , in the non-volatile memory  1016 , and/or on a removable storage medium such as a CD or DVD. 
         [0076]    Although certain example methods, apparatus and articles of manufacture have been described herein, the scope of coverage of this disclosure is not limited thereto. On the contrary, this disclosure covers all methods, apparatus and articles of manufacture fairly falling within the scope of the claims. 
         [0077]    The Abstract at the end of this disclosure is provided to comply with 37 C.F.R. §1.72(b) to allow the reader to quickly ascertain the nature of the technical disclosure. It is submitted with the understanding that it will not be used to interpret or limit the scope or meaning of the claims.