# Patent

## Title
Ballistic missile phase categorization technique

## Abstract

A computer-implemented analysis method is provided for identifying a flight trajectory of a bogey relative to earth's surface. The method includes a first step of obtaining first and second altitudes and velocities of the bogey separated by a first time interval. The second step calculates a first difference between the first and second velocities divided by the first time interval to obtain an acceleration vector. The third step calculates an acceleration magnitude from the acceleration vector. The remaining steps characterize the vectors and magnitude to report whether the bogey represents a ballistic projectile in a particular phase of Boost, Apogee or Descent.

## Background
CROSS REFERENCE TO RELATED APPLICATION
The invention is a Continuation-in-Part, claims priority to and incorporates by reference in its entirety U.S. patent application Ser. No. 12/930,168 filed Dec. 8, 2010 and assigned Navy Case 99970, which claims the benefit of priority, pursuant to 35 U.S.C. §119, from provisional application 61/336,131 filed Jan. 15, 2010.


STATEMENT OF GOVERNMENT INTEREST
The invention described was made in the performance of official duties by one or more employees of the Department of the Navy, and thus, the invention herein may be manufactured, used or licensed by or for the Government of the United States of America for governmental purposes without the payment of any royalties thereon or therefor.


BACKGROUND
The invention relates generally to categorization of aerial bogeys. In particular, the invention relates to techniques to distinguish between ballistic and flat trajectory airborne vehicles and categorize the ballistic vehicles accordingly for appropriate interception response.
An unidentified aerial object or “bogey” represents a prospective threat to a defended asset. To intercept the threat as a target, the defender must categorize ballistic trajectories on the basis of intermittent observation of an object's trajectory. This has represented a difficult problem by conventional analysis techniques using observably available data.
SUMMARY
Conventional techniques for categorizing intermittently observed object trajectories yield disadvantages addressed by various exemplary embodiments of the present invention. In particular, various exemplary embodiments provide an indication of apparent acceleration direction with respect to an object's local horizontal plane associated with the intermittently observed object trajectory.
Various exemplary embodiments provide a computer-implemented analysis method to identify a flight trajectory of a bogey relative to earth's surface. The method includes a first step of obtaining first and second altitudes and velocities of the bogey separated by a first time interval. The second step calculates a first difference between the first and second velocities divided by the first time interval to obtain an acceleration vector. The third step calculates an acceleration magnitude from the acceleration vector. The remaining steps characterize the vectors and magnitude to report whether the bogey represents a ballistic projectile in a particular phase of Boost, Apogee or Descent. Various additional embodiments provide transformations to aid in processing the vector information.



BRIEF DESCRIPTION OF THE DRAWINGS
These and various other features and aspects of various exemplary embodiments will be readily understood with reference to the following detailed description taken in conjunction with the accompanying drawings, in which like or similar numbers are used throughout, and in which:
FIG. 1 is a graphical diagram view of a ballistic trajectory;
FIG. 2 is a tabular view of comparing flight path characteristics;
FIG. 3 is an elevation vector diagram view of relative coordinates;
FIG. 4 is a vector diagram view of an intermediate coordinate system;
FIG. 5 is an elevation vector diagram view of coordinate transformation;
FIGS. 6A and 6B are text views of MATLAB code of variable definitions and associated calculations;
FIGS. 7A through 7E are text views of annotated MATLAB code for the coordinate conversion process;
FIG. 8 is a graphical interface view of angle deviation profiles;
FIGS. 9A and 9B are graphical plot views of standard deviation variation with signal-to-noise; and
FIG. 10 is a flowchart identifying the procedural process for ballistic missile determination.



DETAILED DESCRIPTION
In the following detailed description of exemplary embodiments of the invention, reference is made to the accompanying drawings that form a part hereof, and in which is shown by way of illustration specific exemplary embodiments in which the invention may be practiced. These embodiments are described in sufficient detail to enable those skilled in the art to practice the invention. Other embodiments may be utilized, and logical, mechanical, and other changes may be made without departing from the spirit or scope of the present invention. The following detailed description is, therefore, not to be taken in a limiting sense, and the scope of the present invention is defined only by the appended claims.
In accordance with a presently preferred embodiment of the present invention, the components, process steps, and/or data structures may be implemented using various types of operating systems, computing platforms, computer programs, and/or general purpose machines.
In addition, those of ordinary skill in the art will readily recognize that devices of a less general purpose nature, such as hardwired devices, or the like, may also be used without departing from the scope and spirit of the inventive concepts disclosed herewith. General purpose machines include devices that execute instruction code. A hardwired device may constitute an application specific integrated circuit (ASIC) or a floating point gate array (FPGA) or other related component.
Conventional techniques for bogey identification rely on additional information beyond observations of the object's trajectory. Conventional techniques include independent indicators that the object's trajectory is accelerating or that the object's altitude is above earth's surface. Such techniques potentially assume that the object's behavior is associated with the object's position. This condition is not always valid, which can lead to interception failure. The instant disclosure describes a technique for categorizing ballistic tracks based on a sequence of system reports of a target's velocity.
FIG. 1 shows a diagram view 100 of a notional ballistic trajectory for an object such as a missile. For such an object, the overall trajectory can be divided into three phases labeled as BOOST, ASCENT, and DESCENT. Time represents the abscissa 110 and altitude represents the ordinate 120. An inverse parabolic dash line 130 represents the notional trajectory for an exemplary rocket launched missile. While the rocket fires, the missile operates in boost mode (or interval) 140 until motor cutoff 145. The missile continues to climb in altitude due to inertia during the coast ascent mode 150 until reaching apogee 155. Thereafter, gravity overcomes the initial launching force and the missile proceeds downward in altitude during descent mode 160.
For the notional diagram view 100, the ASCENT and DESCENT phases correspond to ballistic motion phases of the trajectory. Observed velocity and acceleration vector quantities have distinct characteristics during each phase:
BOOST—During boost mode 140 until burnout 145, acceleration points up, close to the direction of the velocity, which may approach vertical in the upward direction.
ASCENT—During coast 150 (after boost) and through apogee 155, acceleration points down, while velocity gradually changes from up to horizontal.
DESCENT—During descent 160, both quantities point down, with velocity changing from horizontal to potentially near vertical down.
In the diagram view 100, lower arrows 170 on the V-line indicate the velocity direction, while upper arrows 180 on the A-line indicate the acceleration direction. For aircraft trajectories having constant velocity, the acceleration is substantially zero (for all directions, in comparison to a ballistic missile). These considerations enable development of a procedure to distinguish between ballistic and non-ballistic type trajectories.
The following information describes an algorithmic process for calculating the velocity and acceleration directions of a bogey, referred to herein as a “target” generally. The process starts with an observed (or calculated) velocity data stream. The process then tests those values to determine transition from BOOST phase to ASCENT intervals for purpose of categorizing the velocity data stream as representative of a ballistic-type trajectory. The analysis assumes that input velocity stream is or can be represented in the coordinate system horizontal to the earth surface at the target's position.
An observed target's velocity components can be transformed from the observed position and velocity coordinates to the target's coordinate system using supplementary formulae described subsequently. Within the target's frame, both the Flight Path Angle (FPA) and the Acceleration Path Angle (APA) formulae indicated herein may be applied to the transformed velocity coordinates.
The relative processes considered are described herein. Velocity vector components are provided in the axial (x), lateral (y) and vertical (z) directions for time increments denoting previous (i−1) and instant (i) described as arrays:




  
    (
    
      
        
          
            v
            
              x
              ,
              
                i
                -
                1
              
            
          
        
      
      
        
          
            v
            
              y
              ,
              
                i
                -
                1
              
            
          
        
      
      
        
          
            v
            
              z
              ,
              
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                -
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    )
  
  ⁢
  
    
  
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  and
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  ⁢
  
    (
    
      
        
          
            v
            
              x
              ,
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            v
            
              y
              ,
              i
            
          
        
      
      
        
          
            v
            
              z
              ,
              i
            
          
        
      
    
    )
  




for time values ti-1 and ti for which velocity values of the target are available, in addition to altitude. Thus, at the instant interval i, the velocity components in the axial, lateral and vertical directions are respectively vx,i, vy,i and vz,i. The relative acceleration can be expressed as |ai−g|, being the absolute difference between instant acceleration and gravity.

When starting with position and velocity coordinates, the altitude can be determined directly from the position components and supplementary knowledge of the local earth's radius. Many different algorithms exist for doing this. In practice, the accuracy of altitude determination may affect the performance of the algorithmic process provided in the various exemplary embodiments when the velocity data are not initially provided in the target local horizontal coordinate system.
Based on these velocity vectors, the following steps can be performed:
(a) calculate flight path angle (also denoted as variable θ for under transform) in eqn. (1) for the instant time step (i) as:




  
    
      
        
          FPA
          i
        
        =
        
          
            sign
            ⁡
            
              (
              
                v
                
                  z
                  ,
                  i
                
              
              )
            
          
          ⁢
          
            
              arccos
              ⁡
              
                (
                
                  
                    
                      
                        v
                        
                          x
                          ,
                          i
                        
                        2
                      
                      +
                      
                        v
                        
                          y
                          ,
                          i
                        
                        2
                      
                    
                  
                  
                    
                      
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                          i
                        
                        2
                      
                      +
                      
                        v
                        
                          y
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                        2
                      
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                        v
                        
                          z
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                          i
                        
                        2
                      
                    
                  
                
                )
              
            
            .
          
        
      
    
    
      
        (
        1
        )
      
    
  




(b) calculate acceleration in eqns. (2), (3) and (4) for three directions as:




  
    
      
        
          a
          
            x
            ,
            i
          
        
        =
        
          
            
              v
              
                x
                ,
                i
              
            
            -
            
              v
              
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                ,
                
                  i
                  -
                  1
                
              
            
          
          
            
              t
              i
            
            -
            
              t
              
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                1
              
            
          
        
      
    
    
      
        (
        2
        )
      
    
  
  
    
      
        
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        (
        3
        )
      
    
  
  
    
      
        
          a
          
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            -
            
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              i
            
            -
            
              t
              
                i
                -
                1
              
            
          
        
      
    
    
      
        (
        4
        )
      
    
  




thereby defining the acceleration vector at each step i by the orthogonal components.

(c) based on the previous step, calculate acceleration path angle in eqn. (5) for the instant time step (i) as:




  
    
      
        
          APA
          i
        
        =
        
          
            sign
            ⁡
            
              (
              
                a
                
                  z
                  ,
                  i
                
              
              )
            
          
          ⁢
          
            
              arccos
              ⁡
              
                (
                
                  
                    
                      
                        a
                        
                          x
                          ,
                          i
                        
                        2
                      
                      +
                      
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                          y
                          ,
                          i
                        
                        2
                      
                    
                  
                  
                    
                      
                        a
                        
                          x
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                          i
                        
                        2
                      
                      +
                      
                        a
                        
                          y
                          ,
                          i
                        
                        2
                      
                      +
                      
                        a
                        
                          z
                          ,
                          i
                        
                        2
                      
                    
                  
                
                )
              
            
            .
          
        
      
    
    
      
        (
        5
        )
      
    
  




(d) calculate the magnitude of acceleration in eqn. (5) for the instant time step (i) as total root sum square:

ai=√{square root over (ax,i2+ay,i2+az,i2)}  (6)

(e) apply logic check to determine ballistic phase for a bogey that represents a missile (rather than an alternate aerial platform), initially for identifying the interval immediately following launch: For example, if the acceleration magnitude ai at the relevant time step exceeds a boost acceleration threshold to overcome gravity such that |ai−g|>g, APA exceeds acceleration-up angle, and FPA exceeds a flying-up angle threshold, then the target is in candidate BOOST phase for parameters having such values that reach those relevant thresholds. This can be coded as: IF (ai>boost_acceleration) AND (APA>acceleration_up_angle) AND (FPA>flying_up_angle), THEN the conclusion can be rendered target_is_candidate_boost_phase for these conditions.
(f) continue to apply logic check, for determining maximum altitude: For example, if the acceleration difference is less than the gravitational threshold such that absolute acceleration approximates gravity |ai| ̃g (or alternatively, |ai−g|<coast_tolerance) at the relevant time step, APA is less than near vertical down (i.e., approaching negative ninety degrees (−90°), meaning vertically downward), and the absolute value of FPA is less than the horizontal flight threshold, then the target is in candidate APOGEE state. This may be represented in logic as statement: IF (|ai| ̃g) AND (APA>near_vertical_down) AND (|FPA|<horizontal_flight_threshold), THEN the conclusion can be rendered target_is_candidate_apogee_phase for parameters having such values that reach those relevant thresholds.
(g) continue to apply logic check, for determining fall: For example, if the acceleration difference is less than the gravitational threshold such that |ai| ̃g at the relevant time step, APA is less than near vertical down, and FPA is less than the horizontal flight threshold, then the target is in candidate DESCENT phase. This can be represented as: IF (|ai| ̃g) AND (APA>near_vertical_down) AND (FRA<horizontal_flight_threshold), THEN the conclusion can be rendered target_is_candidate_descent_phase for parameters having such values that reach those relevant thresholds. Artisans of ordinary skill will recognize that altitude does not constitute a distinguishing characteristic for any of these phase identifications in logic checks (e), (f) and (g).
This logic represents an expansion over that reported in application Ser. No. 12/930,168, in which a determination can be made that the bogey represents a ballistic missile for the following conditions (at the relevant time step): FPA significantly exceeds zero, APA approaches negative ninety degrees (−90°, meaning vertically downward), and altitude corresponds to being above regular commercial airliner traffic. For purposes of this disclosure, a determination can be made that the bogey represents a ballistic missile for the following three concurrent conditions: FPA significantly exceeds flying_up_angle, APA is less than near_vertical_down, and altitude is above regular commercial traffic.
This logic can be distinguished from application Ser. No. 12/930,168 in which logic indicators identify the ballistic flight phase as ASCENT (to distinguish from non-ballistic targets, such as aircraft), whereas the instant application distinguishes additional ballistic flight phases as BOOST, APOGEE and DESCENT. To summarize application Ser. No. 12/930,168: a determination can be made that the bogey represents a ballistic missile for the following conditions: FPA significantly exceeds zero, APA approaches negative ninety degrees (−90°, meaning vertically downward), and altitude corresponds to being above regular commercial airliner traffic. Thus, for the prior described condition IF (FPA>flying_up_angle) AND (APA<near_vertical_down) AND (alt>normal_commercial_traffic_altitude), THEN the conclusion can be rendered target_is_candidate_ballistic_missile for parameters having such values that reach that relevant threshold. By comparison, the instant disclosure provides further phase distinctions.
For example as an illustration for the BOOST phase, for an acceleration a that exceeds a boost_acceleration_threshold, an Acceleration Path Angle APA that exceeds a threshold acceleration_up_angle, and a Flight Path Angle FPA that exceeds a threshold flying_up_angle, then the target is classified as being in a candidate_boost_phase of a ballistic missile. For acceleration difference |ai−g| being less than 1 g_threshold, and APA being less than near_vertical_down, then the target can be described as one of two possible states: APOGEE or DESCENT phases. For the first possibility, when the absolute value of FPA is less than the horizontal_flight_threshold, then the target is at candidate_apogee state. Otherwise, when the FPA is less than negative value of the horizontal_flight_threshold, then the candidate is at candidate_descent state.
During the BOOST phase, acceleration points up, close to the direction of the velocity, which may be approaching vertical_up. During the BOOST phase an engine provides acceleration force for the object (e.g., a missile). During BOOST phase, the net acceleration is the acceleration provided by the rocket motor, minus the acceleration provided by gravity. As understood by artisans of ordinary skill, gravitational acceleration is represented by a vector that, to a reasonable degree of accuracy, points to the center of earth's mass. The ASCENT phase begins after the BOOST phase and continues up to the APOGEE state as the missile's inertia continues following motor burnout.
During the ASCENT phase, the acceleration points down (because the motor provides no further thrust for acceleration, such that gravity becomes the dominant force affecting the target's motion), while the velocity gradually changes from upward (or upward angularly) to horizontal. At APOGEE (or the highest point in the trajectory in the gravitational field), acceleration points down, while velocity is predominately horizontal. During the DESCENT phase, both acceleration and velocity point down, with velocity changing from horizontal to potentially near vertical down. In FIG. 1, the direction of acceleration is indicated with arrows on the line labeled “A”, whereas the direction of the velocity is indicated with arrows on the line labeled “V” as shown.
For normal constant velocity aircraft trajectories the apparent acceleration will be zero. More generally, the accelerations observed for aircraft will in general be significantly less than the accelerations observed for ballistic missiles, and the trajectories of the two types of objects will also be significantly different. These considerations allow development of simple processes to distinguish between the various phases of ballistic trajectories and non-ballistic type trajectories.
The process for calculating the velocity and acceleration directions can be described starting with an observed (or calculated) velocity data stream. The computations are performed in a general purpose programmable computer having instructions recorded on a machine-readable medium. The system includes an observation apparatus, such as an optical observation system, an infrared observation system, a radar based observation system and the like, which can monitor and observe the trajectory of an object as a function of increments of time ti. Information that can be observed includes position as a function of time, and velocity as a function of time.
These values are then evaluated to determine which phase of ballistic flight is represented by the velocity data stream. In some embodiments, the input velocity stream is represented in the coordinate system horizontal to the earth's surface at the target's position. Generally, an observed target's velocity components should be transformed from the observed position and velocity coordinates to the target's coordinate system using supplementary formulae described in U.S. Ser. No. 12/930,168. Once in the target's coordinate system, both the Flight Path Angle (FPA) and the Acceleration Path Angle (APA) formulae may be applied to the transformed velocity coordinates.
Various exemplary embodiments provide systems and methods that distinguish between various phases of ballistic trajectories on the basis of intermittent observation of an object as it executes a trajectory in a gravitational field, such as that of earth. The systems and methods operate to distinguish the various phases of ballistic tracks on the basis of a sequence of system reports of a target's velocity. The velocity can be used to compute acceleration according to the relation:

v(t)=a(t)·t,  (7)

where v(t) represents velocity, a(t) represents acceleration and t represents time. As used herein, the symbol g represents an acceleration due to gravity, such as the acceleration due to earth's mass at sea level.

An exemplary comparison can be described for explanatory purposes. The examples include a bottle rocket and a balsa-wood toy airplane. The high-performance bottle rocket produces thrust against the force of gravity, causing the rocket to move upwards. During this BOOST phase 140, both gravity and thrust forces affect the rocket's motion. After exhaustion of its propellant at cutoff 145, the rocket continues upward until the gravity overcomes its inertia so as to reduce vertical ascent speed. This is labeled as the BALLISTIC ASCENT phase 150, during which gravity gradually overcomes the upward velocity until the rocket reaches its maximum height on an inverted parabolic trajectory. Upon the rocket reaching its peak at apogee 155, it begins to descend in the mode labeled as BALLISTIC DESCENT 160.
As a simplified numerical example, a massless bottle rocket exerting thrust to accelerate at 20 m/s2 (meters-per-second-squared), with an acceleration angle of 75° from horizontal, and subject to 9.8 m/s2 gravity, achieves a velocity vector (x, y, z) of




  
    (
    
      
        
          46.6
        
      
      
        
          0
        
      
      
        
          85.7
        
      
    
    )
  
   




m/s after 9 seconds, and a velocity vector of





  
    (
    
      
        
          51.8
        
      
      
        
          0
        
      
      
        
          95.2
        
      
    
    )
  
   




m/s after 10 seconds, assuming launch in the direction of the axial x-coordinate axis. The apparent FPA and APA both exceed 61° at end of boost (at 9 seconds). Immediately after BOOST, FPA slowly decreases, and APA reaches −90°. This contrasts with a toy airplane having continuous velocity vector of





  
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          0
        
      
      
        
          0
        
      
    
    )
  
  ⁢
  
     
    ,
  




indicating a flat non-accelerating horizontal flight path.

FIG. 2 provides a table of values 200 to distinguish the above exemplary target types of bottle rocket and toy airplane in target type column 210. Velocity time-step columns 220 list intervals of mode shifts at time steps i and i+1. Similarly, FPA columns 230 list similar intervals in degrees for flight path, and APA column 240 lists similar degrees for acceleration. The table 200 provides progression of FPA and APA for the rocket and airplane at their respective operative modes.
For example, the rocket's BOOST mode 250 has initially zero velocity vectors, whereas these vectors are non-zero for end-of-boost 260, ASCENT 270 and DESCENT 280. Thus, the rocket's FPA shifts from positive to slight negative after APOGEE 155, and its APA shifts from positive to negative vertical after cutoff 145. In addition, the airplane 290 exhibits constant velocity and no change in altitude, so that the airplane's FPA remains zero and its APA is indefinite.
The exemplary discriminator has the advantage of being applicable on the basis of a target velocity data stream. Additional target position data and knowledge of earth radius may be incorporated to support conversion of observed target data to local target coordinate system.
Conventional interpretation assumes that acceleration components derive from measured velocity values. However, these have large noise characteristics, and calculations based on observer based information have low reliability. Additionally, concern over ballistic threats has only recent origins as compared to airplanes, such as bomber aircraft, with design choices possibly influenced by computational capabilities of prior generation computers (comparatively slow), leading to reluctance to perform calculations that may have questionable reliability.
Results of investigation in support of exemplary embodiments indicate that despite high noise characteristics for acceleration quantities, the angle computed from these can be suitable to support categorization for ballistic type motion (even when noisy), resulting in the technique presented herein.
Conventional system results indicate less than 15% of a specific set of targets can be correctly categorized as ballistic. Based on desktop analysis using the same data stream applied to the exemplary algorithm, various exemplary embodiments improve this performance to over 90% being correctly categorized.
FIG. 3 illustrates an elevation view 300 of frame comparison between local and remote coordinate systems. The local coordinates can be defined with respect to an observer's position 305 as a local horizontal 310 and a local vertical 320, with a target (or other object of interest) being initially located the intersection of lines parallel to these at position 330. The observer's position 305 can be described as radially distant from the earth's center by displacement vector Re, and the target position 330 is angularly displaced from that vector by a transition angle αT. The target travels along its velocity vector {right arrow over (v)} from an initial position 330 to a subsequent position.
Based on the earth's spheroidal surface 340, coordinates can be translated for the observer's position 305 to a remote horizontal 350 and a remote vertical 360. The elevation distance of the target's position 330 from the surface 340 can be expressed as altitude vector 370 from the surface 340. The remote vertical vector 370 is approximately collinear to the remote vertical 360, passing between the earth's center and the target position 330, and offset from the displacement vector Re by the transition angle αT.
From the observer's position 305, the local horizontal 310 and the local vertical 320 coordinates can translate to the remote horizontal 350 and the remote vertical 360 coordinates. Gravitational acceleration operates perpendicular to the surface 340 and thus along the local and remote verticals 320 and 370. The target's position 330 can be described relative to the observer's position as a distance vector {right arrow over (r)} as a horizon distance vector {right arrow over (r)}H as line 380 and a distance {right arrow over (r)}Z above local horizontal as line 390.
Formulae can be sought for the velocity angle αV of velocity vector {right arrow over (v)} with respect to the remote horizontal 350 to the spheroid surface 340 at the target position 330. Components of the velocity can be solved for the target horizontal coordinate system (THCS), labeled as Remote Horizontal. The following parameters can be defined:
{right arrow over (r)} position vector coordinates of a target position 330, e.g., in local frame coordinates.
{right arrow over (v)} velocity vector coordinates of the target position 330, e.g., in local frame coordinates.
alt altitude 370 above reference spheroid surface 340 at the target position 330.
The position vector {right arrow over (r)} is provided in local frame coordinates: horizontal 310 and vertical 320 as (rx ry rz)T, in which superscript T denotes matrix transpose. Altitude 370 is provided at target position 330. Expectant angle between the local horizontal 310 and the target remote horizontal 350 is given by eqn. (8) as:




  
    
      
        
          
            α
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          =
          
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              )
            
          
        
        ,
      
    
    
      
        (
        8
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where aT is the transition angle that denotes tilt between local and remote vertical axes, and Re is the earth's radius.

FIG. 4 illustrates vector diagram view 400 of frame comparison between local and intermediate coordinate systems. An observer position 405 corresponds to the position 305 in the elevation view 300. The horizontal x axis 410 and the vertical y axis 420 define the local coordinate system. A target position 430 corresponds to the horizontal components of the position 330 in the elevation view 300. A position vector {right arrow over (r)}H as line 440 denotes displacement to the target position 430 of the remote coordinate system from a local origin that corresponds to the observer's position 405, which corresponds to local coordinates rx and ry. The vector line 440 corresponds to the horizontal components of the vector line 380 in elevation view 300. A velocity vector {right arrow over (v)}H, as line 450, identifies the direction of travel from the remote target 430 in the horizontal plane.
A first local transition unit vector ê1L, as line 460, aligns along the {right arrow over (r)}H position vector 440, associated with a second local transition unit ê2L vector, as line 470, orthogonal to the first within the plane of the x and y axes 410, 420. The unit vectors ê1L and ê2L define the horizontal portion of an intermediate coordinate system 480. The first unit vector ê1L as 460 lies collinear to the position vector {right arrow over (r)}H as 440 from the observer position 405 to the target position 430. The transition vector ê2L, as line 470, is perpendicular to ê1L, as line 460, but also lies within the original local horizontal plane.
FIG. 5 illustrates an elevation view 500 of frame comparison between local and remote coordinate systems. A local system 510 includes the first transition unit vector ê1L, tangent along the spheroidal surface 340 and a third transition unit vector ê3L vertically orthogonal to the first (as opposed to the second unit vector ê2L as line 460 being in the same plane as the surface 340). Consequently, these first, second and third local unit vectors correspond to local axial, lateral and vertical directions.
A remote system 520 includes a horizontal transition unit vector ê1R also tangent along the surface 340, and a vertical transition unit vector ê3R parallel to earth radius Re as line 530. A local target velocity vector {right arrow over (v)}T as line 540 represents a velocity whose components are known within the local coordinate system, but whose components have not yet been determined within the remote coordinate system.
The intermediate coordinate system 480 transforms the local coordinate system 510 to the (non-standard) remote coordinate system 520. Construction of the intermediate coordinate system 480 provides for the ê1L axis to run from the origin of the local coordinate system 510 in the direction of the target position 330, while both unit vectors ê1L and ê2L remain in the original local horizontal plane. The remote coordinate system 520 represents a subsequent portion of the transformation involving transformation using the transition angle αT, described subsequently.
Generally, obtaining the flight path angle θ with respect to horizontal plane involves determining the components of the velocity vector {right arrow over (v)} in the plane of the local horizontal 310, then performing the calculation of eqn. (9):




  
    
      
        
          θ
          =
          
            arccos
            (
            
              
                
                
                  
                    v
                    →
                  
                  H
                
                
              
              
                
                
                  v
                  →
                
                
              
            
            )
          
        
        ,
      
    
    
      
        (
        9
        )
      
    
  




where |{right arrow over (v)}| and |{right arrow over (v)}H| represent respective absolute magnitudes of the velocity and vertical velocity component. The flight path angle θ corresponds to the vector velocity angle αV.
This calculation can be used in either local frame coordinates 510 or remote frame coordinates 520, with the former employed in this example. These can be transformed to the remote frame coordinates 520 (through the intermediate coordinate system 480). New coordinates can be developed by defining an intermediate set of coordinates 480 that are aligned with the radial vector between the two coordinate systems (in this case target position vector {right arrow over (r)}), and apply the transformation from local to target frame using the transition angle αT. The process for accomplishing this is described below.
First, the local intermediate frame coordinate axes êiL (where subscript i represents the three orthogonal coordinates 1, 2, 3) can be developed from the position vector coordinates for all three directions, given by eqn. (10):




  
    
      
        
          
            
              e
              ^
            
            
              1
              ⁢
              
                
              
              ⁢
              L
            
          
          =
          
            
              (
              
                
                  
                    
                      
                        r
                        x
                      
                      
                        r
                        h
                      
                    
                  
                  
                    
                      
                        r
                        y
                      
                      
                        r
                        h
                      
                    
                  
                  
                    0
                  
                
              
              )
            
            T
          
        
        ,
        
          
            
              e
              ^
            
            
              2
              ⁢
              
                
              
              ⁢
              L
            
          
          =
          
            
              (
              
                -
                
                  
                    
                      
                        
                          r
                          y
                        
                        
                          r
                          h
                        
                      
                    
                    
                      
                        
                          r
                          x
                        
                        
                          r
                          h
                        
                      
                    
                    
                      0
                    
                  
                
              
              )
            
            T
          
        
        ,
        
          
            
              e
              ^
            
            
              3
              ⁢
              
                
              
              ⁢
              L
            
          
          =
          
            
              (
              
                
                  
                    0
                  
                  
                    0
                  
                  
                    1
                  
                
              
              )
            
            T
          
        
        ,
      
    
    
      
        (
        10
        )
      
    
  




where radial distance is defined by rh=√{square root over (rx2+ry2)}.

This defines a local intermediate coordinate system 520 whose coordinates are represented in the local frame coordinate system 510. In this system, the unit vector ê1L lies in the horizontal plane in the direction of the target. The unit vector ê3L lies along the local tangent plane vertical axis. The unit vector ê2L also lies in the horizontal plane and is perpendicular to the other two vectors (ê1L and ê3L).
The target velocity vector {right arrow over (v)}T 540 can be denoted in the local radial aligned system 520. The calculation for intermediate velocity vector {right arrow over (v)}I is given by eqn. (11):




  
    
      
        
          
            
              v
              →
            
            I
          
          =
          
            
              (
              
                
                  
                    
                      
                        v
                        →
                      
                      ·
                      
                        
                          e
                          ^
                        
                        
                          1
                          ⁢
                          
                            
                          
                          ⁢
                          L
                        
                      
                    
                  
                
                
                  
                    
                      
                        v
                        →
                      
                      ·
                      
                        
                          e
                          ^
                        
                        
                          2
                          ⁢
                          
                            
                          
                          ⁢
                          L
                        
                      
                    
                  
                
                
                  
                    
                      
                        v
                        →
                      
                      ·
                      
                        
                          e
                          ^
                        
                        
                          3
                          ⁢
                          
                            
                          
                          ⁢
                          L
                        
                      
                    
                  
                
              
              )
            
            ·
          
        
        ,
      
    
    
      
        (
        11
        )
      
    
  




such that the velocity vector {right arrow over (v)}T constitutes the locally expressed object velocity {right arrow over (v)} (in the local horizontal frame) dot multiplied for each component along each local unit vector.

The transformation through the transition angle αT occurs along the radial about the unit vector ê2L. The transformation can be written as eqn (10):




  
    
      
        
          M
          =
          
            (
            
              
                
                  
                    cos
                    ⁡
                    
                      (
                      
                        -
                        
                          α
                          T
                        
                      
                      )
                    
                  
                
                
                  0
                
                
                  
                    -
                    
                      sin
                      ⁡
                      
                        (
                        
                          -
                          
                            α
                            T
                          
                        
                        )
                      
                    
                  
                
              
              
                
                  0
                
                
                  1
                
                
                  0
                
              
              
                
                  
                    sin
                    ⁡
                    
                      (
                      
                        -
                        
                          α
                          T
                        
                      
                      )
                    
                  
                
                
                  0
                
                
                  
                    cos
                    ⁡
                    
                      (
                      
                        -
                        
                          α
                          T
                        
                      
                      )
                    
                  
                
              
            
            )
          
        
        ,
      
    
    
      
        (
        12
        )
      
    
  




where M represents the transformation from the intermediate coordinate system 480 to the target horizontal coordinate system 520 (i.e., Remote Horizontal). This applies to the vectors whose coordinates are presented in the intermediate coordinate system 480. So, basis transition vectors (ê1R ê2R ê3R for the target radial aligned coordinates are formed through the calculations by eqn. (13):





  
    
      
        
          
            (
            
              
                
                  
                    
                      e
                      ^
                    
                    
                      1
                      ⁢
                      R
                    
                  
                
                
                  
                    
                      e
                      ^
                    
                    
                      2
                      ⁢
                      R
                    
                  
                
                
                  
                    
                      e
                      ^
                    
                    
                      3
                      ⁢
                      R
                    
                  
                
              
            
            )
          
          =
          
            
              (
              
                
                  
                    
                      cos
                      ⁡
                      
                        (
                        
                          -
                          
                            α
                            T
                          
                        
                        )
                      
                    
                  
                  
                    0
                  
                  
                    
                      -
                      
                        sin
                        ⁡
                        
                          (
                          
                            -
                            
                              α
                              T
                            
                          
                          )
                        
                      
                    
                  
                
                
                  
                    0
                  
                  
                    1
                  
                  
                    0
                  
                
                
                  
                    
                      sin
                      ⁡
                      
                        (
                        
                          -
                          
                            α
                            T
                          
                        
                        )
                      
                    
                  
                  
                    0
                  
                  
                    
                      cos
                      ⁡
                      
                        (
                        
                          -
                          
                            α
                            T
                          
                        
                        )
                      
                    
                  
                
              
              )
            
            ⁢
            
              (
              
                
                  
                    
                      
                        e
                        ^
                      
                      
                        1
                        ⁢
                        L
                      
                    
                  
                  
                    
                      
                        e
                        ^
                      
                      
                        2
                        ⁢
                        L
                      
                    
                  
                  
                    
                      
                        e
                        ^
                      
                      
                        3
                        ⁢
                        L
                      
                    
                  
                
              
              )
            
          
        
        ,
      
    
    
      
        (
        13
        )
      
    
  




where transition angle αT is defined in eqn. (8).

As an aside, one end of the vector {right arrow over (r)}H as line 440 is coincident with the origin (0, 0, 0) of the local horizontal plane at point 405. The other end lies at (rx, ry, 0) in the local horizontal coordinate system 510. For the remote horizontal coordinate system 520, the other end of the position vector {right arrow over (r)}H lies along the remote vertical axis (direction of ê3R) that runs in the remote horizontal coordinate system 520 through either the origin (0, 0, 0) or else the coordinates (0, 0, alt).
From this, the coordinates for the remote radially-aligned target velocity coordinate system 520 are formed by the calculation in eqn. (14):

{right arrow over (v)}T=(ê1R ê2R ê3R)T{right arrow over (v)}I,  (14)

which may alternatively be written as eqn. (15):





  
    
      
        
          
            v
            →
          
          T
        
        =
        
          
            (
            
              
                
                  
                    v
                    xT
                  
                
              
              
                
                  
                    v
                    yT
                  
                
              
              
                
                  
                    v
                    zT
                  
                
              
            
            )
          
          =
          
            
              (
              
                
                  
                    
                      
                        
                          e
                          ^
                        
                        
                          1
                          ⁢
                          R
                        
                      
                      ·
                      
                        
                          v
                          →
                        
                        I
                      
                    
                  
                
                
                  
                    
                      
                        
                          e
                          ^
                        
                        
                          2
                          ⁢
                          R
                        
                      
                      ·
                      
                        
                          v
                          →
                        
                        I
                      
                    
                  
                
                
                  
                    
                      
                        
                          e
                          ^
                        
                        
                          3
                          ⁢
                          R
                        
                      
                      ·
                      
                        
                          v
                          →
                        
                        I
                      
                    
                  
                
              
              )
            
            .
          
        
      
    
    
      
        (
        15
        )
      
    
  




In this manner, the target velocity vector {right arrow over (v)}T may be expressed as a transformation from the intermediate velocity vector {right arrow over (v)}I using unit transformation coordinates.

Upon establishing the coordinates, the calculation is straightforward. For any coordinate system, obtaining the flight path angle θ or FPA with respect to the horizontal plane involves determining the components of the velocity vector {right arrow over (v)} in the horizontal plane for that coordinate system, and subsequently performing the calculation by eqn. (16):




  
    
      
        θ
        =
        
          
            arccos
            (
            
              
                
                
                  
                    v
                    →
                  
                  H
                
                
              
              
                
                
                  v
                  →
                
                
              
            
            )
          
          .
        
      
    
    
      
        (
        16
        )
      
    
  




The flight path angle θ sought can be determined by eqn. (17):




  
    
      
        θ
        =
        
          
            arccos
            (
            
              
                
                  
                    v
                    xT
                    2
                  
                  +
                  
                    v
                    yT
                    2
                  
                
              
              
                
                  
                    v
                    xT
                    2
                  
                  +
                  
                    v
                    yT
                    2
                  
                  +
                  
                    v
                    xT
                    2
                  
                  +
                  
                    v
                    zT
                    2
                  
                
              
            
            )
          
          .
        
      
    
    
      
        (
        17
        )
      
    
  




Consequently, the flight path angle θ>0 if the vertical velocity component of the target horizontal coordinate transition is above zero vzT>0. Further, the flight path angle is below zero θ<0 if that same component is below zero vzT<0.

FIGS. 6A and 6B list an exemplary text 600 that provides coordinate transition equations in MATLAB code. The transition portion 610 in FIG. 6A lists full equations and intermediate variables used to transform from local horizontal coordinates to the remote horizontal coordinates employed for calculating flight path and subsequent angles. The results portion 620 in FIG. 6B lists the corresponding step-by-step algorithmic process executed to generate the transformation coordinates.
FIGS. 7A through 7E list an exemplary text 700 that provides exemplary annotated calculations for the above processes. FIG. 7A provides a first section 710 that includes definitions 715. FIG. 7B provides a second section 720 for a first method with calculations for unit vectors 722, velocity 724 and rotation 726. FIG. 7C continues with a third section 730 for the first method with calculations for coordinate translation 732, projector 734, velocity magnitude 736 and magnitude of the horizontal component of velocity 738.
FIG. 7D completes the first method with a fourth section 740 with time and angle calculations 742, followed by calculations 744 for a second but conventional method for comparison purposes with simplified assumptions compared to the first method. This is followed by calculations 746 for a third method, also being a less calculation-intensive version of the first method. Thus, both second and third methods employ simplified assumptions compared to the first method. FIG. 7E provides a fifth section 750 with time calculations 755 and printer formats.
An exemplary problem can be demonstrated in association with signal-to-noise. For a vector of




  
    (
    
      
        
          x
        
      
      
        
          y
        
      
      
        
          z
        
      
    
    )
  
   




variables with standard deviations





  
    (
    
      
        
          
            σ
            x
          
        
      
      
        
          
            σ
            y
          
        
      
      
        
          
            σ
            z
          
        
      
    
    )
  
  ,




the expected standard deviations can be summarized for velocity vector error statistics σ|{right arrow over (v)}| and velocity vector angular error σ∠ based on:





  
    
      
        
          
          
            v
            →
          
          
        
        =
        
          
            
              x
              2
            
            +
            
              y
              2
            
            +
            
              z
              2
            
          
        
      
    
    
      
        (
        18
        )
      
    
  
  
    
      
        
          ∠
          =
          
            arccos
            (
            
              
                
                  
                    x
                    2
                  
                  +
                  
                    y
                    2
                  
                
              
              
                
                  
                    x
                    2
                  
                  +
                  
                    y
                    2
                  
                  +
                  
                    z
                    2
                  
                
              
            
            )
          
        
        ,
      
    
    
      
        (
        19
        )
      
    
  




such that v=√{square root over (x2+y2+z2)} represents a Maxwell-Boltzmann distribution for orthogonal independent variables x, y and z and q=√{square root over (x2+y2)} represents a Rayleigh distribution. From this, the square of directional coefficient a for each orthogonal direction x, y and z can be determined by:





  
    
      
        
          a
          x
          2
        
        =
        
          
            ∫
            
              -
              ∞
            
            ∞
          
          ⁢
          
            
              x
              2
            
            ⁢
            
              p
              ⁡
              
                (
                x
                )
              
            
            ⁢
            
              d
              x
            
          
        
      
    
    
      
        (
        20
        )
      
    
  
  
    
      
        
          a
          y
          2
        
        =
        
          
            ∫
            
              -
              ∞
            
            ∞
          
          ⁢
          
            
              y
              2
            
            ⁢
            
              p
              ⁡
              
                (
                y
                )
              
            
            ⁢
            
              d
              y
            
          
        
      
    
    
      
        (
        21
        )
      
    
  
  
    
      
        
          a
          z
          2
        
        =
        
          
            ∫
            
              -
              ∞
            
            ∞
          
          ⁢
          
            
              z
              2
            
            ⁢
            
              p
              ⁡
              
                (
                z
                )
              
            
            ⁢
            
              d
              z
            
          
        
      
    
    
      
        (
        22
        )
      
    
  




where p(•) is the probability density function of the independent variable.

From Bar-Shalom, the expected value of a function of a random variable may be provided for expected value E of a function g for random variable x may be provided by integration:




  
    
      
        
          
            E
            ⁡
            
              [
              
                g
                ⁡
                
                  (
                  x
                  )
                
              
              ]
            
          
          =
          
            
              ∫
              
                -
                ∞
              
              ∞
            
            ⁢
            
              
                g
                ⁡
                
                  (
                  x
                  )
                
              
              ⁢
              
                p
                ⁡
                
                  (
                  x
                  )
                
              
              ⁢
              
                d
                x
              
            
          
        
        ,
      
    
    
      
        (
        23
        )
      
    
  




where g(x) represents a deviation in the x variable.

Both Maxwell-Boltzman and Rayleigh distributions can be assumed to have the same standard deviation. The distribution of a function of two well-known distributions can follow a normal (i.e., Gaussian) distribution and have the same standard deviation. The function g defined is the vector angle calculation defined in eqn. (17).
As an exemplary test, a true transient axial displacement vector {right arrow over (x)}(t) can be generated as a first step:




  
    
      
        
          
            
              x
              →
            
            ⁢
            
              (
              t
              )
            
          
          =
          
            
              (
              
                
                  
                    
                      x
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                  
                
                
                  
                    
                      y
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                  
                
                
                  
                    
                      z
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                  
                
              
              )
            
            =
            
              (
              
                
                  
                    
                      
                        
                          m
                          x
                        
                        ⁢
                        t
                      
                      +
                      
                        b
                        x
                      
                    
                  
                
                
                  
                    
                      
                        
                          m
                          y
                        
                        ⁢
                        t
                      
                      +
                      
                        b
                        y
                      
                    
                  
                
                
                  
                    
                      
                        
                          m
                          z
                        
                        ⁢
                        t
                      
                      +
                      
                        b
                        z
                      
                    
                  
                
              
              )
            
          
        
        ,
      
    
    
      
        (
        24
        )
      
    
  




where t is time, m is representative of velocity (linear variation slope) and b is representative of initial position (ordinate intercept, respectively for variable subscripts x, y and z.

As a second step, noise can be added to produce observed values as:




  
    
      
        
          
            
              y
              →
            
            ⁡
            
              (
              
                t
                k
              
              )
            
          
          =
          
            
              (
              
                
                  
                    
                      u
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                  
                
                
                  
                    
                      v
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                  
                
                
                  
                    
                      w
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                  
                
              
              )
            
            =
            
              (
              
                
                  
                    
                      
                        
                          m
                          x
                        
                        ⁢
                        
                          t
                          k
                        
                      
                      +
                      
                        b
                        x
                      
                      +
                      
                        
                          N
                          
                            t
                            kx
                          
                        
                        ⁡
                        
                          (
                          
                            0
                            ,
                            
                              σ
                              2
                            
                          
                          )
                        
                      
                    
                  
                
                
                  
                    
                      
                        
                          m
                          y
                        
                        ⁢
                        
                          t
                          k
                        
                      
                      +
                      
                        b
                        y
                      
                      +
                      
                        
                          N
                          
                            t
                            ky
                          
                        
                        ⁡
                        
                          (
                          
                            0
                            ,
                            
                              σ
                              2
                            
                          
                          )
                        
                      
                    
                  
                
                
                  
                    
                      
                        
                          m
                          z
                        
                        ⁢
                        
                          t
                          k
                        
                      
                      +
                      
                        b
                        z
                      
                      +
                      
                        
                          N
                          
                            t
                            kz
                          
                        
                        ⁡
                        
                          (
                          
                            0
                            ,
                            
                              σ
                              2
                            
                          
                          )
                        
                      
                    
                  
                
              
              )
            
          
        
        ,
      
    
    
      
        (
        25
        )
      
    
  




where Nu(0,σ2) is representative of normal Gaussian distribution about zero with standard deviation σ, respectively for variable subscripts x, y and z.

As a third step, true angle ∠({right arrow over (x)}(t)) can be computed for the true displacement as:




  
    
      
        
          
            ∠
            ⁡
            
              (
              
                
                  x
                  →
                
                ⁡
                
                  (
                  t
                  )
                
              
              )
            
          
          =
          
            arccos
            (
            
              
                
                  
                    x
                    2
                  
                  +
                  
                    y
                    2
                  
                
              
              
                
                  
                    x
                    2
                  
                  +
                  
                    y
                    2
                  
                  +
                  
                    z
                    2
                  
                
              
            
            )
          
        
        ,
      
    
    
      
        (
        26
        )
      
    
  




where x, y and z are true horizontal displacements.

As a fourth step, measured angle ∠({right arrow over (y)}(t)) can be computed with the added noise as:




  
    
      
        
          
            ∠
            ⁡
            
              (
              
                
                  y
                  →
                
                ⁡
                
                  (
                  t
                  )
                
              
              )
            
          
          =
          
            arccos
            (
            
              
                
                  
                    u
                    2
                  
                  +
                  
                    v
                    2
                  
                
              
              
                
                  
                    u
                    2
                  
                  +
                  
                    v
                    2
                  
                  +
                  
                    w
                    2
                  
                
              
            
            )
          
        
        ,
      
    
    
      
        (
        27
        )
      
    
  




where u, v and w are the horizontal displacements with added noise, for these relations.

As a fifth step, a difference function Δ(t) can be computed as:

Δ(t)=|∠({right arrow over (x)}(t))−∠({right arrow over (y)}(t))|,  (28)

with the difference function representing the absolute value of the differences between true angle ∠({right arrow over (x)}(t)) and noisy measured angle ∠({right arrow over (y)}(t)).

For a numerical example, true vector {right arrow over (x)} values for true displacements (x, y, z) can be (3, 4, 5) in scalar form. These values would yield true angle value of 45°. For the corresponding measured vector {right arrow over (y)} values to be extremely noisy, this may produce measured displacements (u, v, w)=(5.5, 6.5, 7.5). Under this circumstance, the measured angle would be 41.4°.
Alternatively, for corresponding measured values being nominally noisy with measured displacements may be (u, v, w)=(3.25, 4.25, 5.25), the measured angle would be 44.5°. For corresponding measured values being only slightly noisy, measured displacements correspond to (u, v, w)=(3.025, 4.025, 5.025). For that example, the measured angle would be 44.9°. Consequently, the difference function Δ(t) can be plotted in a histogram.
FIG. 8 shows a windows view 800 of angle deviations with ten example histograms, each with 100,000 samples showing maximum counts varying as deviation in degrees. Plots 810, 815, 820, 825 and 830 show counts ranging up to two-hundred, with apparently little output standard deviation σout ̃39 despite variation between these plots of input standard deviation from σin=10 to σin=100,000. By contrast, plots 835, 840, 845, 850 and 855 show maximum counts from about two-thousand and ten-thousand, despite the similarly wide range in input standard, but with output standard deviation constrained between less than one-tenth to slightly over ten.
Plots 810 and 835 demonstrate the difference in response to increasing the original magnitude of the input quantities. At all the left-side plots, their respective fixed points (x, y, z) are set to zero. This is followed by performing the angle calculation with noisy values, resulting in an arbitrary angle calculated from noisy zeroes. On the right-side, the point values (x, y, z) have been increased while maintaining a fixed measurement noise.
This causes (somewhat unintentionally) the original magnitude of the vector (x, y, z) to increase with respect to the noise. This was realized to be similar to a signal-to-noise type problem. Typically, the greater the vector length, the less likely that its direction is be affected by comparatively small noise. The high speed of missiles constitutes a benefit to this aspect of the problem.
For the graphical views 800, the left-side histogram plots 810, 815, 820, 825 and 830 have steady-state conditions as x(t)=y(t)=z(t)=0. The right-side histogram plots 835, 840, 845, 850 and 855 are transient, with x(t)=y(t)=25.0×t and z(t)=0.1×t. For both sides, true x, y and z values can be computed to calculate true angle. Also, noisy x, y and z as u, v and w values can be computed for a noisy angle ∠({right arrow over (y)}(t)). The difference Δ(t) can be calculated between true and noisy angles, and can be plotted as histograms.
FIGS. 9A and 9B show expected standard deviation plot views 900 of deviation as a function of signal-to-noise ratio based on empirical comparisons of simulated results. For plot 910 in FIG. 9A, x(t)=y(t)=0 and z(t)=a constant. This was selected for simplicity, because the accompanying assumption of uniform noise for each coordinate implies the problem can typically be transformed into the problem considered.
For plot 920 in FIG. 9B, the expected value integral provides the analytical predicted maximum 970 based on an integral restriction:




  
    
      
        
          
            E
            ⁡
            
              [
              
                ∠
                2
              
              ]
            
          
          ≤
          
            
              ∫
              0
              
                π
                2
              
            
            ⁢
            
              
                
                  
                    e
                    
                      
                        
                          -
                          
                            k
                            2
                          
                        
                        2
                      
                      ⁢
                      
                        cos
                        2
                      
                      ⁢
                      φ
                    
                  
                  ⁡
                  
                    (
                    1
                    )
                  
                
                ·
                
                  φ
                  2
                
              
              ⁢
              cos
              ⁢
              
                
              
              ⁢
              
                φ
                ·
                
                  
                
                ⁢
                
                  d
                  φ
                
              
            
          
        
        ,
      
    
    
      
        (
        29
        )
      
    
  




where k represents the ratio between the original constant z and the noise standard deviation value σz (i.e., signal-to-noise ratio in the corresponding direction), and φ represents the deviation angle. For both plots, signal-to-noise ratio represents the abscissa 930 and standard deviation represents the ordinate 940.

The first expected plot 910 features results from a fixed length z against a variable input noise, calculating output statistics of the distance function in a simulation curve 950. The second expected plot 920 includes a legend 960 identifying an analytical predicted curve 970 of eqn. (29) and a simulated actual curve 980, which corresponds to the same curve 950 of plot 910.
The deviation angle φ in eqn. (29) represents a variable of integration developed based on transformation from Cartesian coordinates to spherical coordinates when attempting to evaluate the three-dimensional version of eqn. (21), which is not readily amenable to closed-form solutions. Thus, the solutions presented employ a Monte Carlo type approach for generating an expectation plot 910 in FIG. 9A, and providing an adequate approximation plot 920 in FIG. 9B.
The approach described herein has resulted from an investigation into resolving considerations for operating an existing conventional system. The conventional system included ability to indicate circumstances in which an observed track exhibits ballistic behavior. Many analytical paths have been investigated to distinguish ballistic and flight trajectories. The difference method provides the most preferred results, and from approximating the Expectation integral of eqn. (21) due to difficulty in developing closed solutions.
FIG. 10 shows a flowchart diagram 1000 of a bogey identification process for designation as a ballistic missile target. A noisy signal 1005, such as radar return energy from the bogey, reaches a sensor 1010 that detects the signal 1005. The sensor 1010 submits that information to a preprocessing filter 1020 for estimating the observed bogey state, including its position and velocity. The altitude of the bogey can also be determined from its relative position in relation to earth's radius.
A coordinate query 1030 can be performed on the resulting estimated state as to whether that state is provided in the target horizontal coordinate system (THCS). If FALSE (as generally expected), the process diverts to a conversion operation 1040 using the coordinate transformations described above. Then, or otherwise if the query is TRUE, the process proceeds to a signal quality query 1050 as to whether the filtered and transformed state has an adequate signal-to-noise (SNR) ratio. If not, the process determines at operation 1060 that the bogey's state is indeterminate, and that further information may be required. Noisy signals compromise the ability to determine acceleration, unless change in velocity exceeds some SNR threshold, such as for a ballistic missile.
If SNR is adequate, the process diverts to a calculation operation 1070 to determine flight path and acceleration path angles (FPA and APA) respectively from eqns. (1) and (5). A set of trajectory decision queries 1080 uses the calculated path angles to categorize phase states 1090 based on the signal 1005. In particular, the first query 1080a inquires whether absolute acceleration exceeds twice gravity (|a|>2 g), and both acceleration path and flight path angles greatly exceed zero (APA>>0°, FPA>>0°), which if all satisfied determine that the bogie can be identified as a ballistic missile with the BOOST phase identifier 1090a. Otherwise, the second query 1080b inquires whether acceleration path angle points vertically downward (APA ̃−90°), flight path angle is significantly greater than zero (FPA>>0°), and altitude is above commercial flights, which if all satisfied determine that the bogie can be identified as a missile with the ASCENT phase identifier 1090b. 
Otherwise, the third query 1080c inquires whether absolute acceleration approximates gravity (|a| ̃g), acceleration path angle points vertically downward (APA ̃−90°, and absolute flight path angle approximates zero (|FPA| ̃0°), which if all satisfied determine that the bogie can be identified as a missile with the APOGEE state identifier 1090c. Otherwise, the fourth query 1080d inquires whether absolute acceleration approximates gravity (|a| ̃g), acceleration path angle points vertically downward (APA ̃−90°), and flight path angle is approximately down (FPA ̃<0°), which if all satisfied determine that the bogie can be identified as a missile with the DESCENT identifier 1090d. For conditions in which none of these criteria are satisfied, the process concludes at determination 1095, such that the bogey constitutes a possible aircraft or other aerial object, that identification is indeterminate, beyond not being a ballistic missile. One should note that application Ser. No. 12/930,168 includes the second inquiry 1080b and the corresponding ASCENT categorization 1090b. 
These operations, and portions thereof, can be performed as procedural steps being performable by machine operation—e.g., a programmable code for a general purpose computer, or else an ASIC. The process results in an automated determination, based on noisy input data and optional transformations, with an identity determination that a bogey constitutes a ballistic threat.
The considerations are indicated in letter from Naval Surface Warfare Center, Dahlgren Division to Aegis Ballistic Missile Defense Program Office, reference 9000 Ser W05/058, “Certification of Aegis Ballistic Missile Defense (BMD) Aegis Weapon System (AWS) Baseline 3.6.1.1 Computer Programs”, dated Mar. 20, 2009.
In summary, conventional systems rely on information other than velocity and are sensitive either to relative position information or to delays in reporting the alternative information. Testing indicated the referenced system would perform poorly against certain types of targets, which led to an examination of techniques in use by that system, and subsequent development of the exemplary process described above.
While certain features of the embodiments of the invention have been illustrated as described herein, many modifications, substitutions, changes and equivalents will now occur to those skilled in the art. It is, therefore, to be understood that the appended claims are intended to cover all such modifications and changes as fall within the true spirit of the embodiments.

## Claims

1. A computer-implemented analysis method executed on an automated processor for determining whether a flight trajectory of a bogey relative to Earth's surface surface constitutes either a ballistic projectile or else a non-ballistic projectile, said method comprising:
obtaining first and second velocities of the bogey separated by a first time interval;
calculating a first difference between said first and second velocities divided by said first time interval to obtain an acceleration vector,
calculating an acceleration magnitude from said acceleration vector;
determining a flight path state of whether said second velocity vector satisfies a first condition;
determining an acceleration path state of whether said acceleration vector satisfies a second condition;
determining an acceleration magnitude state of whether said acceleration magnitude satisfies a third condition; and
reporting an identified threat that the bogey represents a phase for the ballistic projectile in response to satisfaction of said first, second and third conditions by respective flight path, acceleration path and acceleration magnitude states, wherein
said first condition constitutes one of said second velocity vector being non-horizontal beyond a velocity threshold as a first result, an absolute value of said second velocity vector being horizontal within said velocity threshold as a second result, and said second velocity vector being below horizontal beyond said velocity threshold as a third result,
said second condition constitutes one of said acceleration vector being above horizontal beyond an acceleration vector threshold as a fourth result, and said acceleration vector being negative and substantially vertical as a fifth result,
said third condition constitutes one of an absolute value of acceleration magnitude exceeding twice gravity as a sixth result, and said absolute magnitude of acceleration vector approximating gravity as a seventh result, and
said phase constitutes one of BOOST for said first, second and third conditions corresponding to said first, fourth and sixth results, APOGEE for said first, second and third conditions corresponding to said second, fifth and seventh results, and DESCENT for said first, second and third conditions corresponding to third, fifth and seventh results.

2. The method according to claim 1, further including:
measuring an altitude of the bogey at one of said first and second time intervals; and
determining whether said altitude satisfies a fourth condition of being above a commercial threshold that constitutes satisfaction of an eighth result, wherein said phase constitutes ASCENT for said first, second and fourth conditions corresponding to said first, fourth and eighth results.

3. The method according to claim 1, further including:
translating said first and second velocity vectors in a local observed coordinate system into transform velocity vectors in an alternate coordinate system, and replacing said velocity vectors with said transformed velocity vectors.

4. The method according to claim 1, wherein determining one of said velocity vectors by:




  
    
      FPA
      i
    
    =
    
      
        sign
        ⁡
        
          (
          
            v
            
              z
              ,
              i
            
          
          )
        
      
      ⁢
      
        arccos
        (
        
          
            
              
                v
                
                  x
                  ,
                  i
                
                2
              
              +
              
                v
                
                  y
                  ,
                  i
                
                2
              
            
          
          
            
              
                v
                
                  x
                  ,
                  i
                
                2
              
              +
              
                v
                
                  y
                  ,
                  i
                
                2
              
              +
              
                v
                
                  z
                  ,
                  i
                
                2
              
            
          
        
        )
      
    
  
  ,




where FPAi is flight path angle at interval i, vx,i, vy,i and vz,i represent velocity components in the axial, lateral and vertical directions for said interval, and




  
    
      APA
      i
    
    =
    
      
        sign
        ⁡
        
          (
          
            a
            
              z
              ,
              i
            
          
          )
        
      
      ⁢
      
        arccos
        (
        
          
            
              
                a
                
                  x
                  ,
                  i
                
                2
              
              +
              
                a
                
                  y
                  ,
                  i
                
                2
              
            
          
          
            
              
                a
                
                  x
                  ,
                  i
                
                2
              
              +
              
                a
                
                  y
                  ,
                  i
                
                2
              
              +
              
                a
                
                  z
                  ,
                  i
                
                2
              
            
          
        
        )
      
    
  
  ,




where APAi is flight path angle at said interval, ax,i, ay,i and az,i represent acceleration components in the axial, lateral and vertical directions for said interval.

5. The method according to claim 3, further including:
translating said acceleration vector into said alternate coordinate system.

6. The method according to claim 3, wherein translating said velocity vectors into said alternate coordinate system further includes:
translating said velocity vectors from a local observation coordinate system having a horizontal plane into an intermediate coordinate system in said horizontal plane as intermediate velocity vectors; and
translating said intermediate velocity vectors from said intermediate coordinate system into a remote coordinate system as target transformed velocity vectors.

7. The method according to claim 4, wherein said acceleration components are determined as




  
    
      a
      
        x
        ,
        i
      
    
    =
    
      
        
          v
          
            x
            ,
            i
          
        
        -
        
          v
          
            x
            ,
            
              i
              -
              1
            
          
        
      
      
        
          t
          i
        
        -
        
          t
          
            i
            -
            1
          
        
      
    
  
  ,
  
    
      a
      
        y
        ,
        i
      
    
    =
    
      
        
          v
          
            y
            ,
            i
          
        
        -
        
          v
          
            y
            ,
            
              i
              -
              1
            
          
        
      
      
        
          t
          i
        
        -
        
          t
          
            i
            -
            1
          
        
      
    
  
  ,
  
    
      and
      ⁢
      
        
      
      ⁢
      
        a
        
          z
          ,
          i
        
      
    
    =
    
      
        
          
            v
            
              z
              ,
              i
            
          
          -
          
            v
            
              z
              ,
              
                i
                -
                1
              
            
          
        
        
          
            t
            i
          
          -
          
            t
            
              i
              -
              1
            
          
        
      
      .

8. The method according to claim 6, wherein said acceleration components are determined as




  
    
      a
      
        x
        ,
        i
      
    
    =
    
      
        
          v
          
            x
            ,
            i
          
        
        -
        
          v
          
            x
            ,
            
              i
              -
              1
            
          
        
      
      
        
          t
          i
        
        -
        
          t
          
            i
            -
            1
          
        
      
    
  
  ,
  
    
      a
      
        y
        ,
        i
      
    
    =
    
      
        
          v
          
            y
            ,
            i
          
        
        -
        
          v
          
            y
            ,
            
              i
              -
              1
            
          
        
      
      
        
          t
          i
        
        -
        
          t
          
            i
            -
            1
          
        
      
    
  
  ,
  
    
      and
      ⁢
      
        
      
      ⁢
      
        a
        
          z
          ,
          i
        
      
    
    =
    
      
        
          
            v
            
              z
              ,
              i
            
          
          -
          
            v
            
              z
              ,
              
                i
                -
                1
              
            
          
        
        
          
            t
            i
          
          -
          
            t
            
              i
              -
              1
            
          
        
      
      .

9. The method according to claim 6, wherein translating said velocity vectors into said alternate coordinate system further includes:
translating said velocity vectors from a local observation coordinate system having a horizontal plane into an intermediate coordinate system in said horizontal plane as intermediate velocity vectors; and
translating said intermediate velocity vectors from said intermediate coordinate system into a remote coordinate system as target transformed velocity vectors.