Abstract:
A method for engaging a target missile includes sensing the position of the target and of an interceptor missile, and determining time-to-go to intercept and direction of thrust of the interceptor. A one-step intercept solution is determined based on position estimates of the target and the interceptor and is used to iteratively estimate at least two components of a three-dimensional unit thrust vector, and apply updated guidance commands to the interceptor. A system for thrust vector control of an interceptor against a target missile includes a processor for receiving sensed target signals, determining a one-step initial solution to produce time-to-go and current direction of thrust of the interceptor, iteratively estimating at least two components of a three-dimensional unit thrust vector, and producing a guidance vector for application to the interceptor.

Description:
This application claims priority to provisional application No. 60/962,065 filed Jul. 26, 2007. 
    
    
     This invention was made with Government support under contract number N00024-03-C-6110 awarded by the Department of the Navy. The Government has certain rights in this invention. 
    
    
     FIELD OF THE INVENTION 
     This invention relates to generation of guidance control commands for an interceptor missile attack on a target missile. 
     BACKGROUND OF THE INVENTION 
     Currently used state-of-the-art exoatmospheric antimissile guidance algorithms are generally limited to engagements in which the target missile is ballistic, in that it has no acceleration attributable to a rocket motor. This is true of a system and algorithm known as Burnout Reference Guidance (BRG) currently used for thrust vector control (TVC) of the SM-1 interceptor during exoatmospheric portions of flight. BRG works, in general, by proportional navigation that attempts to null out the line-of-sight rate. Interest has recently been directed toward launching interceptor missiles and intercepting target missiles during the boost phase of target missile flight. Analysis of BRG guidance, even when modified to include target missile acceleration (and renamed “modBRG”), suggests that it may not be optimal against boosting target missiles, in that guidance errors may result in missing of the target. Amended algorithms applied to modBRG have not sufficiently decreased guidance errors. 
     Improved thrust control guidance control of antimissiles is desired for action against target missiles in both their boost and ballistic states. 
     SUMMARY OF THE INVENTION 
     In general, a guidance system according to an aspect of the invention attempts to generate an exact solution to the intercept point of an interceptor missile with a target missile, based on nonlinear iterative algorithms in which approximations are reduced or eliminated. More particularly, a “one-step” or “bootstrap” solution to the intercept point is generated by determining time-to-go to intercept and the direction of the thrust vector of the interceptor missile, and using this one-step solution as the basis or state vector as a starting point for an iterative solution. The iterative solution generates the commands for the interceptor missile. 
     The logic of one-step initial intercept solution is aided by the following analysis. Let the initial position and velocity at time t 0  of a target T, such as a missile, be denoted by p T (0), v T (0) respectively. The motion of the target due to the effect of acceleration a n   T  from nature (e.g., acceleration due to gravity, centripetal acceleration, Coriolis acceleration) and thrust a t   T  is given by
 
 {umlaut over (p)}   T   =a   n   T   +a   t   T   (1)
 
Let the displacement of the target from its initial position due to the effect of its thrust be denoted by p t   T  and the corresponding velocity of the target be denoted by v t   T . Integrating (2), one has for the velocity of the target at time t k . This intercept solution is obtained in a non-rotating inertial frame. The displacement vector between interceptor and target at any arbitrary time is given by using a simplification for gravity, and one has an approximate one-step bootstrap solution to begin from. The squared error between the interceptor and the target is used to determine the two components of the unit vector û 1 . The one-step solution involves obtaining the initial time-to-go and thrust vector direction unit vector û 1 . Once the time-to-intercept or time-to-go t go  is determined in the one-step solution, the vector û 1  defining the direction of the interceptor thrust can be determined. Thus, the one-step solution includes determination of the time-to-go t go  and of the direction of thrust û 1 . Three unknown quantities: (1) the time t, and (2) two components of the unit vector û 1  are solved for during the following iterative process to find the unknown solution to be denoted by the 3-tuple
 
     
       
         
           
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     Thus, the algorithm for solution of the intercept can be summarized as follows: 
     (a) Obtain the one-step initial t go    
     (b) obtain one-step initial û 1    
     (c) iteratively solve 
             x   ⁢     =   def     ⁢     [             u   ^     1   1             u   ^     1   2               t   go     ]     ′                   
The solution of the iteration is deemed complete when conditions are met based on the difference between successive computations of
 
             x   ⁢     =   def     ⁢     [             u   ^     1   1             u   ^     1   2               t   go     ]     ′                   
being arbitrarily small.
 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWING 
         FIG. 1  is a simplified scenario of sensing of information relating to a target missile which may be in a boost phase or a ballistic phase, processing of the sensed information together with information relating to an antimissile or interceptor missile, and guidance control of the interceptor missile; 
         FIG. 2  is a simplified logic flow chart or diagram illustrating processing according to an aspect of the invention; 
         FIG. 3  is a constant target acceleration profile; 
         FIG. 4  is a depiction of the kinematical propagation of a target object; 
         FIGS. 5A ,  5 B, and  5 C together represent the kinematic components of an accelerating target object,  FIG. 5A  represents position,  FIG. 5B  represents velocity, and  FIG. 5   c  represents acceleration; 
         FIG. 6  is an acceleration profile of a rocket motor based upon a rocket equation; and 
         FIG. 7  is a gravity model for space-borne object near a spherical rotating earth with varying gravity. 
     
    
    
     DESCRIPTION OF THE INVENTION 
       FIG. 1  illustrates a scenario  10  in which a target missile  12  follows a path or track  14  including an earlier first position  14   1  and a second, later, position  14   2 . Target missile  12  is in a boost phase, suggested by the presence of a plume  12   p  at position  14   1 , and in a ballistic phase at position  14   2 , at which position the target missile may split into plural portions, such as decoy and active. Somewhere between locations  14   1  and  14   2 , the target missile makes a transition between boost phase and ballistic phase. One or more sensors  16 , suggested by a radar system  16   r , produce signals indicative of the moment-to-moment location of the target missile  12 . The sensor may be a camera or sensor suite rather than a simple radar system. Radar system (or other sensor)  16  transmits and receives electromagnetic signals, suggested by “lightning bolt” symbols  18   a  and  18   b , and generates sensed signals representing at least the location of the target missile. 
     The sensed signals from sensor  16  are applied to processing illustrated as a block  22  in FIG.  1 . The processing of block  22  estimates the current target missile position and velocity. The current target missile estimated position and velocity information is applied to an interceptor missile  30  controller, illustrated as a block  24 . Controller  24  commands the launching of the interceptor missile  30  generally toward the target missile  12 . The current target missile estimated position and velocity information is also applied from estimating block  22  to a processing block  26  according to an aspect of the invention. Processing block  26  generates thrust vector commands for interceptor missile  30 , for vectoring the interceptor missile  30  to an intercept with the target missile  12 , regardless of the boost or ballistic state of the target missile. The thrust vector commands are made available by way of a path  27  to the interceptor missile control block  24 . The thrust vector commands cause the interceptor missile  30  to close with and intercept the target missile. 
     In general, a guidance system according to an aspect of the invention attempts to generate an exact solution to the intercept point of an interceptor missile with a target missile, based on nonlinear iterative algorithms in which approximations are reduced or eliminated. More particularly, a “one-step” or “bootstrap” solution to the intercept point is generated by determining time-to-go to intercept and the direction of the thrust vector of the interceptor missile, and using this one-step solution as the basis or state vector as a starting point for an iterative solution. The iterative solution generates the commands for the interceptor missile. 
       FIG. 2  illustrates a simplified logic flow chart or diagram illustrating processing  210  according to an aspect of the invention. The processing may be performed by computers associated with the sensor or radar  16  of  FIG. 1 , with processing blocks  22 ,  24 , or  26 , or possibly in computers associated with interceptor missile  30 , or the processing may be distributed among a plurality of processors, wherever located. The processing or logic  210  of  FIG. 2  starts at a START block  212 , and flows to a block  214 . Block  214  represents the sensing of information about the target missile ( 12  of  FIG. 1 ), as might be performed by sensor  16 . Block  214  represents the sensing of information about at least the moment-to-moment position of the target missile, from which the target missile velocity can be determined. Alternatively, the target missile velocity can be directly sensed, as by use of Doppler information. The interceptor missile is launched in a direction at least nominally toward the missile, as suggested by block  216 . From block  216 , the logic of  FIG. 2  proceeds to a block  218 , which represents the estimation of the position and velocity of the target missile from the sensed information. Block  220  represents the sensing of the position and velocity of the interceptor missile. From block  220 , the logic of  FIG. 2  flows to a block  222 , which represents the determination of a one-step initial intercept (bootstrap) solution, including time-to-go (to intercept) and the three-dimensional interceptor missile thrust vector associated, with the time-to-go. 
     The logic of one-step initial intercept solution block  222  is aided by the following analysis. Let the initial position and velocity at time t 0  of a target T, such as a missile, be denoted by p T (0), v T (0) respectively. The motion of the target due to the effect of acceleration a n   T  from nature (e.g., acceleration due to gravity, centripetal acceleration, Coriolis acceleration) and thrust a t   T  is given by
 
 {umlaut over (p)}   T   =a   n   T   +a   t   T   (2)
 
Let the displacement of the target from its initial position due to the effect of its thrust be denoted by p t   T  and the corresponding velocity of the target be denoted by v t   T . Integrating (2), one has for the velocity of the target at time t k 
 
                       v   T     ⁡     (     t   k     )       =         ∫   0     t   k       ⁢         a   n   T     ⁡     (   τ   )       ⁢           ⁢     ⅆ   τ         +       v   T     ⁡     (   0   )       +     v   t   T     +     Ω   ×     p   g   T                 (   3   )               
This intercept solution is obtained in a non-rotating inertial frame. Consequently, the terms Ω×p g   T  and Ω×p g   M  are included in the solution, where Ω is angular velocity relative to an inertial frame, p g   T  is position of the target missile due to gravity, and p g   T  is position of the interceptor missile due to gravity. Integrating equation (3), one has for the position of the target at time t k 
 
                       p   T     ⁡     (     t   k     )       =         ∫   0     t   k       ⁢       (       ∫   0     t   k       ⁢         a   n   T     ⁡     (   τ   )       ⁢           ⁢     ⅆ   τ         )     ⁢     ⅆ   t         +       [         v   T     ⁡     (   0   )       +     Ω   ×     p   g   T         ]     ⁢     t   k       +       p   T     ⁡     (   0   )       +     p   t   T               (   4   )               
Let the initial position and velocity at time t 0  of the interceptor be denoted by p M (0), v M (0) respectively. The motion of the interceptor due to the effect of acceleration a n   M  from nature (e.g., acceleration due to gravity, centripetal acceleration, Coriolis acceleration) and thrust a t   M  is given by
 
 {umlaut over (p)}   M   +a   n   M   +a   t   M   (5)
 
Let the displacement of the interceptor from its initial position due to the effect of its thrust be denoted by p t   M  and the velocity of the interceptor due to the effect of its thrust be denoted by v t   M . Integrating (5), one has for the velocity of the interceptor
 
                       v   M     ⁡     (   t   )       =     {               ∫   0   t     ⁢         a   n   M     ⁡     (   τ   )       ⁢     ⅆ   τ         +       v   M     ⁡     (   0   )       +     Ω   ×     p   g   M             if         t   ≤     T   1                     ∫   0   t     ⁢         a   n   M     ⁡     (   τ   )       ⁢     ⅆ   τ         +       v   M     ⁡     (   0   )       +       v   t   M     ⁢       u   ^     1       +     Ω   ×     p   g   M             if         t   &gt;     T   2                       (   6   )               
where û 1  is the direction of the thrust. Integrating (6), one has for the position of the interceptor at time t
 
                       p   M     ⁡     (   t   )       =     {               ∫   0   t     ⁢       (       ∫   0   t     ⁢         a   n   M     ⁡     (   τ   )       ⁢     ⅆ   τ         )     ⁢     ⅆ   t         +       [         v   M     ⁡     (   0   )       +     Ω   ×     p   g   M         ]     ⁢   t     +       p   M     ⁡     (   0   )             if         t   ≤     T   1                     ∫   0   t     ⁢       (       ∫   0   t     ⁢         a   n   M     ⁡     (   τ   )       ⁢     ⅆ   τ         )     ⁢     ⅆ   t         +       [         v   M     ⁡     (   0   )       +     Ω   ×     p   g   M         ]     ⁢   t     +       p   M     ⁡     (   0   )       +         if         t   &gt;     T   2                   {       p   t   M     +       v   t   M     ⁡     (     t   -     T   2       )         }     ⁢       u   ^     1                                               (   7   )               
The displacement vector ε MT (t) between interceptor and target at any arbitrary time t&gt;T 2  is given by
 
                       ɛ   MT     ⁡     (   t   )       =           ∫   0   t     ⁢       (       ∫   0   t     ⁢         a   n   M     ⁡     (   τ   )       ⁢     ⅆ   τ         )     ⁢     ⅆ   t         +       [         v   M     ⁡     (   0   )       +     Ω   ×     p   g   M         ]     ⁢   t     +       p   M     ⁡     (   0   )       +       {       p   t   M     +       v   t   M     ⁡     (     t   -     T   2       )         }     ⁢       u   ^     1       -       ∫   0   t     ⁢       (       ∫   0   t     ⁢         a   n   T     ⁡     (   τ   )       ⁢     ⅆ   τ         )     ⁢     ⅆ   t         -       [         v   T     ⁡     (   0   )       +     Ω   ×     p   g   T         ]     ⁢   t     -       p   T     ⁡     (   0   )       -     p   t   T       =       [       {         p   M     ⁡     (   0   )       -       p   T     ⁡     (   0   )         }     +       {       p   t   M     -       v   t   M     ⁢     T   2         }     ⁢       u   ^     1       -     p   t   T       ]     +       {         v   M     ⁡     (   0   )       +     Ω   ×     p   g   M       -       v   T     ⁡     (   0   )       -     Ω   ×     p   g   T       +       v   t   M     ⁢       u   ^     1         }     ⁢   t     +       ∫   0   t     ⁢       (       ∫   0   t     ⁢         a   n   M     ⁡     (   τ   )       ⁢     ⅆ   τ         )     ⁢     ⅆ   t         -       ∫   0   t     ⁢       (       ∫   0   t     ⁢         a   n   T     ⁡     (   τ   )       ⁢     ⅆ   τ         )     ⁢     ⅆ   t                     (   8   )               
Using a simplification for gravity, one has
 
                         ∫   0   t     ⁢       (       ∫   0   t     ⁢         a   n   M     ⁡     (   τ   )       ⁢     ⅆ   τ         )     ⁢     ⅆ   t         -       ∫   0   t     ⁢       (       ∫   0   t     ⁢         a   n   T     ⁡     (   τ   )       ⁢     ⅆ   τ         )     ⁢     ⅆ   t           =       1   3     ⁢   Δ   ⁢           ⁢     g   ⁡     (   0   )       ⁢     t   2               (   9   )               
Defining
 
     
       
         
           
             
               
                 
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                   B 
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                     def 
                   
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                         ⁡ 
                         
                           ( 
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                           M 
                         
                       
                       - 
                       
                         
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                   ( 
                   11 
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                   C   ⁢     =   def     ⁢     [       {         p   M     ⁡     (   0   )       -       p   T     ⁡     (   0   )         }     +       {       p   t   M     -       v   t   M     ⁢     T   2         }     ⁢       u   ^     1       -     p   t   T       ]             (   12   )               
equation (8) can be rewritten as
 
ε MT ( t )= C+Bt+At   2   (13)
 
The squared error J between the interceptor and the target is given by
 
 J=[ε   MT ( t )] t [ε MT ( t )]=[ C+Bt+At   2 ] t   [C+Bt+At   2 ]  (14)
 
where the primes associated with the matrices represent the transpose. Note that J in equation (14) is a scalar function of three unknown quantities. These are: (1) the time t, and (2) two components of the unit vector û 1  in the direction of thrust of the interceptor. Note that the third component of a unit vector û 1  is known if two of its components are known. A simultaneous nonlinear solution for these quantities is desired for block  222  of  FIG. 2 .
 
     An approximate one-step bootstrap solution is sought for this nonlinear solution to begin from. The squared error J between the interceptor and the target in equation (14) is more dependent on the time t than on the two components of the unit vector û 1 . Consider a preliminary unit vector û 1  defining a direction. The one-step solution involves obtaining the time t that minimizes squared error J, and subsequently using this value of t to solve for û 1 . The minimization of time t is formulated as 
                     t   go     =     arg   ⁢                     ⁢   min     t     ⁡     [     C   +   Bt   +     At   2       ]       ′     ⁡     [     C   +   Bt   +     At   2       ]                 (   15   )               
Minimizing J in (14) with respect to time t
 
                             ∂   J       ∂   t       =     2   ⁢       (     ɛ   MT     )     ′     ⁢     (       ∂     ɛ   MT         ∂   t       )                   =         2   ⁡     [     C   +   Bt   +     At   2       ]       ′     ⁡     [     B   +     2   ⁢   At       ]                   =     2   ⁡     [         C   ′     ⁢   B     +       (         B   ′     ⁢   B     +     2   ⁢     C   ′     ⁢   A       )     ⁢   t     +       (         A   ′     ⁢   B     +     2   ⁢     B   ′     ⁢   A       )     ⁢     t   2       +     2   ⁢     A   ′     ⁢     At   3         ]                     (   16   )               
Note that the term A (from equation 9) is usually small. Therefore, one can neglect the A′At 3  term, and solve (16) as a quadratic as follows
 
 C′B +( B′B+ 2 C′A ) t +( A′B+ 2 B′A ) t   2 =0  (17)
 
or
 
 a  t     2   +b  t +c= 0  (18)
 
where
 
 a=C′B   (19)
 
 b=B′B+ 2 C′A   (20)
 
 c=A′B+ 2 B′A   (21)
 
                     t   _     =     1   t             (   22   )               
Note that, if A is small, the term c is also small. This formulation, if A is small, avoids any difficulty of the quadratic solution.
 
Solving equation (18) yields
 
                     t   _     =         -   b     ±         b   2     -     4   ⁢   a   ⁢           ⁢   c             2   ⁢   a               (   23   )               
and
 
                   t   =     1     t   _               (   24   )               
time-to-go t go  is deemed to be equal to the value of t determined in equation (23).
 
     This first part of the one-step solution of block  222  of  FIG. 2  can alternately be expressed as determining time-to-go by 
                     t   go     =     1     t   _               (     23   ⁢   A     )               
where:
 
                     t   _     =         -   b     ±         b   2     -     4   ⁢           ⁢   a   ⁢           ⁢   c             2   ⁢   a               (   22   )               
where:
 
 a=C′B   (18)
 
 b=B′B+ 2 C′A   (19)
 
 c=A′B+ 2 B′A   (20)
 
where:
 
     
       
         
           
             
               
                 
                   A 
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                     def 
                   
                   ⁢ 
                   
                     
                       1 
                       3 
                     
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                     Δ 
                     ⁢ 
                     
                         
                     
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                       g 
                       ⁡ 
                       
                         ( 
                         0 
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                   ( 
                   9 
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                   B 
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                     def 
                   
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                           M 
                         
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                           ( 
                           0 
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                           M 
                         
                       
                       - 
                       
                         
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                           T 
                         
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                   ( 
                   10 
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                   C   ⁢     =   def     ⁢     [       {         p   M     ⁡     (   0   )       -       p   T     ⁡     (   0   )         }     +       {       p   t   M     -       v   t   M     ⁢     T   2         }     ⁢       u   ^     1       -     P   t   T       ]             (   11   )               
and:
 
     Δg(0) is the differential gravity between the missile and the interceptor at time t 0 ; 
     v M (0) is the velocity of the interceptor or countermeasure missile at time t 0 ; 
     Ω is angular velocity relative to an inertial frame; 
     p g   M  is position of the interceptor missile due to gravity; 
     v T (0) is the initial velocity of the target missile at time t 0 ; 
     p g   T  is position of the target missile due to gravity; 
     û 1  is a unit vector in the direction of interceptor thrust; 
     p M (0) is the initial position of the interceptor at time t 0 ; 
     p T (0) is the initial position of the target missile at time t 0 ; 
     p t   M  is the displacement of the interceptor missile due to the effect of its thrust; 
     v t   M  is the velocity of the interceptor due to the effect of its thrust; 
     T 2  is the end of acceleration of the interceptor missile; and 
     p t   T  is displacement of the target missile due to its thrust. 
     As mentioned, once the time to intercept or time-to-go t go  is determined in the one-step solution performed in block  222  of  FIG. 2 , the vector û 1  defining the direction of the interceptor thrust can be determined. Thus, the one-step solution of block  222  includes determination of the time-to-go t go  and of the direction of thrust û 1 . Observe from equation (13) that C′B&lt;0, C′A&lt;0, A′B&gt;0 for Δ MT (t) to decrease. Thus in equation (23), a&lt;0, b&gt;0, c&gt;0. For real solutions, b 2 −4ac&gt;0. Also, one should choose the negative sign of the radical so that  t  takes the largest value and t is the least value. Having obtained one returns to equation (13), 
     in which the displacement vector ε MT (t) between the interceptor missile and the target missile can be rewritten as 
                             ɛ   MT     ⁡     (   t   )       =       ⁢       [       {         p   M     ⁡     (   0   )       -       p   T     ⁡     (   0   )         }     +       {       p   t   M     -       v   t   M     ⁢     T   2         }     ⁢       u   ^     1       -     p   t   T       ]     +                     ⁢         {         v   M     ⁡     (   0   )       -       v   T     ⁡     (   0   )       +       v   t   M     ⁢       u   ^     1         }     ⁢   t     +       1   3     ⁢   Δ   ⁢           ⁢     g   ⁡     (   0   )       ⁢     t   2                     =       ⁢         [       {       p   t   M     -       v   t   M     ⁢     T   2         }     +       v   t   M     ⁢   t       ]     ⁢       u   ^     1       +     {         p   M     ⁡     (   0   )       -       p   T     ⁡     (   0   )         }     +                     ⁢         {         v   M     ⁡     (   0   )       -       v   T     ⁡     (   0   )         }     ⁢   t     -     p   t   T     +       1   3     ⁢   Δ   ⁢           ⁢     g   ⁡     (   0   )       ⁢     t   2                       (   25   )               
Note that (25) is a three dimensional vector equation; however, the coefficient of û 1  is a scalar quantity. Solving equation (25) for zero yields
 
                             u   ^     1     =     -         {         p   M     ⁡     (   0   )       -       p   T     ⁡     (   0   )         }     -     p   t   T     +       {         v   M     ⁡     (   0   )       -       v   T     ⁡     (   0   )         }     ⁢   t     +       1   3     ⁢   Δ   ⁢           ⁢     g   ⁡     (   0   )       ⁢     t   2           [       {       p   t   M     -       v   t   M     ⁢     T   2         }     +       v   t   M     ⁢   t       ]                     =     -         {         p   M     ⁡     (   0   )       -       p   T     ⁡     (   0   )         }     -     p   t   T     +       {         v   M     ⁡     (   0   )       -       v   T     ⁡     (   0   )         }     ⁢   t     +       1   3     ⁢   Δ   ⁢           ⁢     g   ⁡     (   0   )       ⁢     t   2           [       p   t   M     +       v   t   M     ⁡     (     t   -     T   2       )         ]                       (   26   )               
The time-to-go, defined as t go , is set equal to the solution of t obtained in equation (24).
 
     Equations (23) and (25) of the one-step initial intercept solution are solved in block  222  of  FIG. 2 . The information flowing from logic block  222  includes initial time-to-go t go  and the direction of the initial interceptor thrust vector. From block  222 , the logic of  FIG. 2  flows by a path  223  to a block  224 . Block  224  represents an iterative estimation of time-to-go and of two components of the thrust vector, and determination of the third component from the estimated components. The two thrust vector components that are estimated are preferably the two smallest. 
     The displacement vector ε MT (t, û 1 ) between interceptor and target at any arbitrary time t&gt;T 2  is restated as 
                       ɛ   MT     ⁡     (     t   ,       u   ^     1       )       =       [       {         p   M     ⁡     (   0   )       -       p   T     ⁡     (   0   )         }     +       {       p   t   M     -       v   t   M     ⁢     T   2         }     ⁢       u   ^     1       -     p   t   T       ]     +       {         v   M     ⁡     (   0   )       -       v   T     ⁡     (   0   )       +       v   t   M     ⁢       u   ^     1         }     ⁢   t     +       1   3     ⁢   Δ   ⁢           ⁢     g   ⁡     (   0   )       ⁢     t   2                 (   27   )               
The displacement vector ε MT (t, û 1 ) in equation (27) is a nonlinear vector function of three unknown quantities. These three unknown quantities are: (1) the time t, and (2) two components of the unit vector û 1 . Consider the unknown solution to be denoted by the 3-tuple
 
             x   ⁢     =   def     ⁢         [             u   ^     1   1             u   ^     1   2         t         ]     ′     .           
A simultaneous nonlinear solution for ε MT (x)=0 is possible. The solution of x for ε MT (x)=0 is obtained by Newton-Raphson&#39;s formula as
 
 x ( k+ 1)= x ( k )−Δ x ( k )  (28)
 
                                     Δ   ⁢           ⁢     x   ⁡     (   k   )         =       [       ∂       ɛ   MT     ⁡     (   x   )           ∂   x       ]       -   1                x   =     x   ⁡     (   k   )           ⁢       ɛ   MT     ⁡     (   x   )                x   =     x   ⁡     (   k   )                     =         [         ∂       ɛ   MT     ⁡     (   x   )           ∂       u   ^     1   1         ⁢       ∂       ɛ   MT     ⁡     (   x   )           ∂       u   ^     1   2         ⁢       ∂       ɛ   MT     ⁡     (   x   )           ∂   t         ]       -   1       ⁡     [                   ɛ   1   MT     ⁡     (   x   )                   ɛ   2   MT     ⁡     (   x   )                         ɛ   3   MT     ⁡     (   x   )             ]                   =         [                     ∂       ɛ   1   MT     ⁡     (   x   )           ∂       u   ^     1   1         ⁢       ∂       ɛ   1   MT     ⁡     (   x   )           ∂       u   ^     1   2         ⁢       ∂       ɛ   1   MT     ⁡     (   x   )           ∂   t                       ∂       ɛ   2   MT     ⁡     (   x   )           ∂       u   ^     1   1         ⁢       ∂       ɛ   2   MT     ⁡     (   x   )           ∂       u   ^     1   2         ⁢       ∂       ɛ   2   MT     ⁡     (   x   )           ∂   t                             ∂       ɛ   3   MT     ⁡     (   x   )           ∂       u   ^     1   1         ⁢       ∂       ɛ   3   MT     ⁡     (   x   )           ∂       u   ^     1   2         ⁢       ∂       ɛ   3   MT     ⁡     (   x   )           ∂   t               ]       -   1       ⁡     [                   ɛ   1   MT     ⁡     (   x   )                   ɛ   2   MT     ⁡     (   x   )                         ɛ   3   MT     ⁡     (   x   )             ]                     (   29   )               
is evaluated at x=x(k). The expression for the first column
 
                 δɛ   MT     ⁡     (   x   )         δ   ⁢           ⁢       μ   ^     1   1             
is
 
                       ∂       ɛ   MT     ⁡     (   x   )           ∂       u   ^     1   1         =       {       p   t   M     -       v   t   M     ⁢     T   2       +       v   t   M     ⁢   t       }     ⁡     [         0           1             -       u   1         1   -     u   1   2     -     u   2   2                   ]               (   30   )               
and the expression for the second column
 
                 δɛ   MT     ⁡     (   x   )         δ   ⁢           ⁢       μ   ^     1   2             
is
 
                       ∂       ɛ   MT     ⁡     (   x   )           ∂       u   ^     1   2         =       {       p   t   M     -       v   t   M     ⁢     T   2       +       v   t   M     ⁢   t       }     ⁡     [         0           1             -       u   2         1   -     u   1   2     -     u   2   2                   ]               (   31   )               
Equations (30) and (31) can be combined as
 
                     [         ∂       ɛ   MT     ⁡     (   x   )           ∂       u   ^     1   1         ⁢       ∂       ɛ   MT     ⁡     (   x   )           ∂       u   ^     1   2           ]     =       {       p   t   M     -       v   t   M     ⁢     T   2       +       v   t   M     ⁢   t       }     ⁡     [         1       0           0       1             -       u   1         1   -     u   1   2     -     u   2   2                   -       u   2         1   -     u   1   2     -     u   2   2                   ]               (   32   )               
The expression for the third column
 
                 δɛ   MT     ⁡     (   x   )         δ   ⁢           ⁢   t           
is
 
     
       
         
           
             
               
                 
                   
                     
                       ∂ 
                       
                         
                           ɛ 
                           MT 
                         
                         ⁡ 
                         
                           ( 
                           x 
                           ) 
                         
                       
                     
                     
                       ∂ 
                       t 
                     
                   
                   = 
                   
                     
                       { 
                       
                         
                           
                             v 
                             M 
                           
                           ⁡ 
                           
                             ( 
                             0 
                             ) 
                           
                         
                         - 
                         
                           
                             v 
                             T 
                           
                           ⁡ 
                           
                             ( 
                             0 
                             ) 
                           
                         
                         + 
                         
                           
                             v 
                             t 
                             M 
                           
                           ⁢ 
                           
                             
                               u 
                               ^ 
                             
                             1 
                           
                         
                       
                       } 
                     
                     + 
                     
                       
                         2 
                         3 
                       
                       ⁢ 
                       Δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         g 
                         ⁡ 
                         
                           ( 
                           0 
                           ) 
                         
                       
                       ⁢ 
                       t 
                     
                   
                 
               
               
                 
                   ( 
                   32 
                   ) 
                 
               
             
           
         
       
     
     Thus, the algorithm for solution of the one-step initial intercept, performed in blocks  222  and  224  of  FIG. 2 , can be summarized as follows: 
     (a) Obtain the one-step initial t go  using equations (10), (11, (12), (18), (19), (20), (22), and (23); 
     (b) obtain one-step initial û 1  using equation (26); and 
     (c) iteratively solve 
     
       
         
           
             x 
             ⁢ 
             
               = 
               def 
             
             ⁢ 
             
               
                 [ 
                 
                   
                     
                       
                         
                           u 
                           ^ 
                         
                         1 
                         1 
                       
                     
                     
                       
                         
                           u 
                           ^ 
                         
                         1 
                         2 
                       
                     
                     
                       
                         t 
                         go 
                       
                     
                   
                 
                 ] 
               
               ′ 
             
           
         
       
     
     using equations (28) until the condition for loop termination conditions are met. These conditions may be based on the difference between successive computations of 
             x   ⁢     =   def     ⁢       [             u   ^     1   1             u   ^     1   2           t   go           ]     ′           
becoming arbitrarily small. This produces on logic path  225  of  FIG. 2  the initial state guidance for the interceptor missile  30  of  FIG. 1 .
 
     Block  226  of  FIG. 2  receives the initial state guidance from block  224 , and represents application of the initial state guidance to the interceptor missile  30  of  FIG. 1 . The initial state guidance commands of the interceptor missile are followed by additional guidance commands. From logic block  226  of  FIG. 2 , the logic flows to a block  228 , which represents the recurrent estimation of two components of the thrust vector, and determination of the third component from the two estimated components, and estimation of time-to-go. The recurrently-generated guidance state vectors are produced on a logic path  229 . Block  230  represents the application of the recurrently-produced guidance state vector to the interceptor missile. 
     From block  230  of  FIG. 2 , the logic flows to a decision block  232 , which determines if the logic has converged on a solution. If the logic has not converged, the logic leaves decision block  232  by the NO path and returns to path  227  and the input of block  228  to perform another estimation of the two components of the thrust vector, and determination of the third component, and the estimation of time-to-go. The iteration around the loop including blocks  228 ,  230 , and  232  continues until decision block  232  determines that convergence on a solution has occurred, whereupon the logic leaves decision block  232  by the YES output and either returns by a path  236  to the START block  212  in readiness for control of another interceptor missile, or ends (not illustrated).