Tactical Ballistic Missile Tracking using the Interacting Multiple Model Algorithm

Transcription

1 Tactical Ballistic Missile Tracking using the Interacting Multiple Model Algorithm Robert L Cooperman Raytheon Co C 3 S Division St Petersburg, FL Robert_L_Cooperman@raytheoncom Abstract The problem of tracking a tactical ballistic missile is complicated by the varying target dynamics in the boost, eo-atmospheric and endo-atmospheric phases of flight A single Kalman filter tuned for constant velocity or constant acceleration is not based upon the correct underlying physical model of these dynamics By including models with all of the correct dynamics, an Interacting Multiple Model (IMM) simultaneously weights all of them and adapts to the one most closely matching the data based upon measurement residuals A multiple-sensor application of this algorithm requires either a single IMM driven by measurements from all sensors (measurement fusion) or an IMM for each sensor driven by its own measurements, followed by fusion across sensors (track fusion) This paper develops a tactical ballistic missile tracker within an IMM framework and gives an eample of the measurement fusion approach for a simulated trajectory and simulated sensor geometry Keywords: Tracking, filtering, Interacting Multiple Model, Tactical Ballistic Missile, Kalman filter 1 Introduction The Interacting Multiple Model (IMM) approach to target tracking has been in use for over a decade, mainly in the area of air defense, in which the goal is to reduce the lags that develop while tracking highly maneuvering manned aircraft These lags develop when the underlying motion model for the target is constant velocity (CV) and the motion deviates substantially from this model, as in a maneuver The simplest IMM for this application is a bank of tracking filters (usually Kalman or Etended Kalman filters (EKFs)), in which each model is tuned to a different acceleration, by means of the Kalman filter process noise A large value of process noise is used for a large acceleration and a small value for a small acceleration The track outputs of the multiple models included in the IMM are combined linearly, with weights that depend upon the likelihood that a measurement fits the assumption of each of the models The number of models to use in the IMM is largely a matter of eperiment, but most implementations use two, or at most three For the tactical ballistic missile (TBM) application in this paper, there are three models, corresponding to the three regions of a TBM trajectory: boost, eo-atmospheric (ballistic) and endo-atmospheric (re-entry) The boost model is a 9-state EKF, in Cartesian coordinates centered on the sensor declared to be local, for the purpose of composite tracking The state elements are position, velocity and acceleration The ballistic model is a 6-state EKF with gravity and Coriolis terms The state elements are position and velocity State propagation, however, includes gravity and Coriolis forces, even though the state does not contain acceleration The re-entry model is a 7-state EKF, identical to the ballistic state but has a 7 th element, the (inverse) ballistic coefficient In this paper, the issue of contact-to-track association is not addressed That is, all contacts from all sensors are assumed to be correctly associated to a track Also, the practical issue of track initiation is not addressed It is assumed that the first two contacts arriving in chronological order (from either one or two different sensors) initiate the track and all further contacts are used for track update Simulated truth trajectories and radar contact data were generated by numerically solving the nd order differential equations of motion for a unitary missile with the Runge-Kutta method and adding simulated measurement noise The code provided in [1] was used to compute the truth trajectories A multiple boost missile was simulated by allowing multiple burn periods, with given weight-thrust characteristics and eperiencing atmospheric drag The missile simulated for this paper was not intended to be representative of any real TBM, but serves for illustrative purposes [1] The single IMM in this study fused contacts from multiple sensors by injecting a chronological stream of associated measurement reports (AMRs); ie, measurements which have been assumed to be associated to the same track The study was limited to two sensors, one defined as local and one remote The local sensor is the sensor in which the track state is defined Prediction and filtering in each IMM EKF involved both the local and 84

2 remote measurement matri H, defined in the EKF formulation [] followed in this study specifying the acceleration terms which affect the transition step of the EKF IMM Formulation Figure 1 shows the architecture of the IMM algorithm used in this study At each time step k, a linear combination of the previous outputs (states and covariances) is input into each model Also, the current measurement is input into each model and residuals are computed, along with corresponding likelihood functions Normalized likelihood functions are used as weights in a linear combination of current model outputs to form the desired blended track state and covariance outputs, while each model outputs are stored for the net iteration In addition, model probabilities (µ) are updated for the net iteration µ BOOST (k-1), µ EXO (k-1), µ ENDO (k-1) X BOOST k-1 k-1 XEXO k-1 k-1 XENDO k-1 k-1 3 TBM Equations The TBM in this study is assumed to be a point target with three degrees of freedom, its position in space This is an obvious simplification of the real TBM problem, ignoring the additional three rotational degrees of freedom The acceleration in an Earth-Centered Earth-Fied (ECEF) is given by * 1 a = ρv vc ˆ D A g mg R * r rˆ * * * * * ( ω v) ω ( ω r) (1) M k Interaction (Miing) X BOOST k-1 k-1 X EXO k-1 k-1 X ENDO k-1 k-1 Boost Model Λ BOOST (k) Eo Model Endo Model Λ EXO (k) Λ ENDO (k) Model Probability Update State Estimate Combination where the first term is atmospheric drag, acting opposite in direction to velocity, the second term is gravity, the third term is Coriolis acceleration and the last term is centrifugal acceleration The missile mass is m, the gravitational acceleration is g (98 m/s ), the velocity is v, shown with the unit vector in its direction, the missile effective surface area is A and the (zero-lift) drag is C D It is further assumed that there is no acceleration due to lift The Earth s equatorial radius is R ( m) [4], the missile position is r and the Earth s rotational velocity is [ -5 rad/s) [4] Although Equation (1) is written for simplicity for a spherical Earth, the tracker developed in this study incorporated the WGS-84 Earth Model [4] The tracker was developed in the local East-North-Up (ENU) coordinate system In this coordinate system, the acceleration components (for a spherical Earth) are given by X BOOST k k X EXO k k X ENDO k k Figure 1 TBM IMM Architecture X k k µ BOOST (k), µ EXO (k), µ ENDO (k) In this figure, the state vectors (and corresponding covariance matri, not shown for simplicity) for all models are combined linearly with the model probabilities as coefficients The mathematical details of the IMM and the EKF can be found in many references [,3] and are not repeated here The mathematics related to the TBM problem involves ρg gr = R * β 3 n z u y * * R + R + ( ω γ ω γ ) u n 3 ( ω γ ω γ ) 3 1 ( ω ω ) ρg gr y y = R * y β 3 u e z * * R + R + e u ( ω ω ) () (3) 85

3 ρg z = R * z β gr ( R + z) * * 3 R + R + ( ω ω ) ( ω γ ω γ ) n 1 e y e where the ENU components of ω at geodetic latitude µ are given by ωe ωn = = ω cos µ n (4), (5) F () = f = y y (1) ωu = ω sin µ the components of vector γ are given by γ 1 = ωn γ = ωu ( R + z) ωu y ω ( R + z) e γ 3 = ωe y ωn the ballistic coefficient is given by, (6) β = mg (7) C D A and R * is the vector to the missile from the origin of the local ENU coordinate system The endo-atmospheric track state includes a drag term α which is defined as 1/β These acceleration components are used in the prediction step of the EKF for the 6-state eo-atmospheric and 7-state endo-atmospheric models The predicted state is given by 1 ˆ = ~ + f () ~ t + F()()( ~ f ~ t) (8) where f () ~ is the time derivative of the state vector, given by f () [ y y z z α ] T = (9) for the endo-atmospheric model and the same vector for the eo-atmospheric model without the drag term The function F () is a Jacobian matri of partial derivatives given by (for the endo-atmospheric model) (The α terms are dropped for the eo-atmospheric model) This matri is a very complicated combination of position (target and sensor), velocity and acceleration terms, which will not be presented here for the sake of brevity As mentioned above, a WGS-84 Earth model is included in these terms, allowing for a non-spherical Earth in the tracking algorithm [5] 4 Results This study developed a multi-sensor TBM tracker within an IMM framework and applied it to a simulated trajectory The results of the study measured the ability of a multiple-sensor TBM track to provide better track accuracy than a single-sensor TBM track for a hypothetical missile, sensors and sensor geometry For this purpose, one sensor was chosen to be close to the launch point of the TBM and one sensor was chosen to be close to the impact point of the TBM The sensors were chosen to have long enough detection range to follow the missile along all phases of its trajectory In reality, both sensors would not necessarily track the entire trajectory and further benefit would be derived from the viewpoint of track continuity, with one sensor filling in when the other sensor drops track However, this study only considered the track accuracy benefit from overlapping coverage Future work will evaluate the benefits derived from multiple geographically dispersed sensors Figure shows the modeled TBM trajectory in ENU coordinates with origin at Sensor 1 86

4 This was done in order to compare results with the single-sensor case The first comparison provided the same total amount of data, while the second comparison provided twice the amount of data, because each sensor in the network was assumed to have the same capability as if it acted alone Both sensors were assumed perfect, ecept for measurement noise, with no false alarms or missed detections and no bias (either sensor or gridlock) The 1 σ measurement noise was assumed to be 1 m in range and 5 mrad in bearing and elevation It is also noted that the sensors are assumed to measure position only, but not velocity Figure Simulated TBM Trajectory The missile was launched at t= from a range of approimately 4 km from Sensor 1, placed near the impact point Sensor was located near the launch point Figure 3 shows the speed and velocity components of the missile in Sensor 1 coordinates The speed and z- component clearly show the two boost stages, one lasting from t= to t=3 seconds and the net from t=3 to t=138 seconds This is followed by the ballistic motion with z- component decreasing to zero at apogee and then by negative z-component as the missile falls to Earth Atmospheric drag acts to slow it down as it continues, so that all velocity components approach zero Figure 4 shows the corresponding acceleration and components The two boost stages appear as two impulses, followed by constant ballistic acceleration of approimately 1g in the negative z-directon (It is actually slightly less than 1g because the acceleration due to gravity decreases with altitude) The decceleration due to drag appears as a positive acceleration because the negative speed is getting less negative, so that the speed profile (Figure 3) has a positive slope, equal to the acceleration AMRs were processed by the multiple EKFs in the TBM IMM filter in chronological order received The approach taken was to quantify the tracking performance using Sensor 1 alone and to quantify the benefit of an additional sensor Three cases were considered : Case 1 had Sensor 1 alone, with an update interval of 1 second Case had both sensors, but each had an update interval of seconds, staggered by 1 second, giving a network update interval of 1 second ; ie, the same total amount of data as for Case 1, but from geometrically diverse sources Case 3 had both sensors, but each was allowed to have an update interval of 1 second (staggered by 5 second), as in Case 1 The TBM IMM algorithm was applied to this set of data For brevity, the plots below compare Case 1 and Case, while the error plots compare all three cases Figures 5 and 6 show the track and truth along the trajectory for Cases 1 and, respectively Ecept for initial fluctuations in the boost phase of Case 1, the track appears reasonably close to the truth Figure 6 shows that measurement fusion improves the track, especially in boost phase Figure 3 TBM Velocity Figure 4 TBM Acceleration 87

5 Figure 5 TBM Truth and Single Sensor IMM Track (Case 1) Figure 7 IMM Model Probabilities transitions occur at the proper times and are very sharp The only noisy region in these curves is in the endo-atmospheric motion, where the algorithm is slightly confused in selecting between the eo- and endo- models, which differ only in the presence of a small non-zero element in the 7th component of the track state in the endo-atmospheric model Figures 8 and 9 show the IMM-derived velocity components and speed corresponding to the boost stage truth data in Figure 3, for Case 1 and, respectively The z- component and the speed are positive, while the - and y- components are negative Figure 6 TBM Truth and Two-Sensor IMM Track (Case ) The eo-atmospheric two-sensor track is smoother than the track with Sensor 1 updates alone, but appears not to quite reach apogee The endo-atmospheric two-sensor track appears approimately the same as for Sensor 1 alone Figure 7 shows the IMM model probabilities for the three models vs time At the bottom is shown the target acceleration (magnitude curve from Figure 4) for comparison As seen from this acceleration plot, the model Figure 8 IMM and True Boost Stage Velocity (Case 1) 88

6 Acceleration estimates from the IMM are shown in Figures 1 and 13, only for the boost stage The boost stage filter is the only filter with acceleration included in the state vector The smooth velocity behavior shown in Figures 9 through 11 for the other stages will suffice to demonstrate good performance for those stages, rather than to compute acceleration with a noisy divided difference approimation Figure 9 IMM and True Boost Stage Velocity (Case ) A comparison shows that the additional sensor smoothes the fluctuations The corresponding curves for the eo- and endo- atmospheric regions differ less from each other on this scale, compared to the boost region and so only the curves for Case are presented in Figures 1 and 11 Figure 1 IMM and True Boost Acceleration (Case 1) Figure 1 IMM and True Ballistic Velocity (Case ) Figure 13 IMM and True Boost Acceleration (Case ) An improvement in Sensor 1 s track in the boost phase due to the contribution of Sensor is seen by comparing these figures The peak acceleration in the first boost stage is detected and estimated with only a few seconds delay in Figure 13 compared to a delay of approimately 15 seconds in Figure 1 In addition, the acceleration estimate itself is closer to the truth for both boost stages Also, the fluctuations in Figure 1 are reduced in Figure 13 Figure 11 IMM and True Re-entry Velocity (Case ) Figure 14 shows the estimate of the ballistic coefficient during the endo-atmospheric phase Also plotted is the value of α corresponding to the value of β = 1/α built into the simulated truth trajectory (5 lb/ft ) It was computed 89

7 for both cases and the results were essentially the same A non-zero estimate begins to be computed as the endoatmospheric model increases in importance, as seen in Figure 7 Figure 14 Ballistic Coefficient Estimation To complete the study, attention is turned to Figures These figures show position and speed errors (root-sumsquared error between truth and track) over the entire trajectory for the three cases, as well as corresponding estimation error (square root of filtered covariance trace) Statistics were computed over time, for each of the three regions of target motion The covariance error is very Figure 16 Position and Speed Errors (Case ) Figure 17 Position and Speed Errors (Case 3) smooth and its close correspondence to the track error shows the near consistency of the IMM Figure 15 Position and Speed Errors (Case 1) Case 1 shows the performance of Sensor 1 alone, with a 1-second update interval Sensor 1 was near the impact point, far from the launch point The large error in its track during the boost phase resulted from the large magnification 83

8 of the angular error over the long range between Sensor 1 and the launch point This large relative boost-phase error also appeared in velocity Cases and 3 demonstrate the improvement in tracking error from measurement fusion in this eample The improvement was demonstrated in position and speed errors, as well A comparison of the composite -second update track (Case ) with the single-sensor track (Case 1) showed that boost and eo-atmospheric position errors decreased for the composite compared to Sensor 1 alone, with a slight increase in the endo-atmospheric region The 1-second composite track (Case 3) showed improvement over Case 1 for all three regions Speed errors decreased in all three regions for both Case and 3 [3] Bar Shalom, Y and KC Chang, Tracking a Maneuvering Target Using Input Estimation Versus the Interacting Multiple Model Algorithm, IEEE Transactions on Aerospace and Electronic Systems, AES-5, (March 1989), 96-3 [4] Ballistic Missile Defense Organization (BMDO), Ballistic Missile Defense (BMD) Navigation Standard, BMD-P-SD-9--A, June 3, 1993 [5] Ballistic Missile Defense Organization (BMDO) and Office of Naval Research (ONR), Functional Description of the Ballistic Missile Defense (BMD) Benchmark, Version 4, December 1,1 These results are summarized in Tables 1 and, below Table 1 Mean Position Error (m) Boost Eo Endo Case Case Case Table Mean Speed Error (m/s) Boost Eo Endo Case Case Case Conclusions A TBM tracking algorithm was developed within the framework of the IMM and applied to simulated data from multiple sensors A tracking accuracy improvement compared to a single sensor was observed when a composite track was computed for all three phases of TBM motion 6 References [1] Zarchan, P, Tactical and Strategic Missile Guidance,Vol 176, Progress in Astronautics and Aeronautics, AIAA, Washington, DC, 1997 [] Castella, FR, Multisensor, Multisite Tracking Filter, IEE Proc Radar,Sonar,Navig,Vol 141, No,April