Tactical Ballistic Missile

Tactical Ballistic Missile Tracking using the Interacting Multiple Model Algorithm
Robert L. Cooper” Raytheon Co. C3S Division
St. Petemburg, FL ’ Robert-L-Cooperman @raytheon.com
Abstract The problem of tracking a tactical ballistic missile is complicated by the varying target dynamics in the boost, exo-atmospheric and endo-amspheric phases offlight. A single K a h n f i l t e r tuned forconstant velociry 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 dnta based upon measurement residuals. A multiple-sensor application of this algorithm requires either a singleIMM driven by measurements from all sensors (measurement fusion) or an IMM far 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 example of the measurement fusion approach for a simulated trajectory and simulated sensor geometry.

For the tactical ballisticmissile (TBM) application in this paper, there are three models, corresponding to the three regions of a TBM trajectory: boost, exo-atmospheric (ballistic) and endo-atmospheric (reentry). 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. Theballistic model is a &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” element the (inverse) ballistic coefficient.
In this paper, the issue of contact-to-trackassociation 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 ppdate.

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 thetarget 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 Extended 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 asmall 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 experiment, but most implementations use two, or at most three.

Simulated huth trajectories and radarcontact data were generated by numerically solving the 2nd order differential equations of motlon for a unitary missile with the Runge-Kutta method and adding simulated measurement noise. The code provided in [I] was used to compute the m t h trajectories. A multiple boost missile was simulated by allowing multiple burn periods, with given weight-thrust characteristics and experiencing atmospheric...