IMM Estimator for Ground Target Tracking with Variable Measurement Sampling Intervals Mahendra Mallick Toyon Research Corporation 75 Aero Camino, Suite A Goleta, CA 93117-3139, USA mmallick@toyon.com Barbara F. La Scala Dept. of Electrical and Electronic Engineering University of Melbourne Victoria 3010, Australia b.lascala@ee.unimelb.edu.au Abstract – Common ground target dynamic models include the nearly constant velocity (NCV), nearly constant acceleration (NCA), and nearly constant turn (NCT) models. Most of the papers on the interacting multiple model (IMM) estimator use a constant Markov chain transition probability matrix (TPM) corresponding to a constant measurement sampling interval. However, a multi-sensor ground target tracking system usually employs ground moving target indicator radar, electro-optical, infrared, video, acoustic, and seismic sensors, for which the sampling intervals are different. Modeling such systems requires using a variable sampling interval in the IMM estimator, which in turn requires the use of a non- constant TPM. An analytic expression for the TPM with variable sampling interval exists for two dynamic models. When the number of dynamic models is greater than two, the TPM can be numerically calculated efficiently. We present the technical approach for the IMM estimator with variable sampling intervals. Preliminary numerical results are presented for a maneuvering target with the NCV, NCA, and NCT models using 200 Monte Carlo simulations. Keywords: Ground target tracking, maneuvering target, interacting multiple model (IMM), variable Markov chain transition probability matrix. 1 Introduction The motion of a ground target usually includes multiple dynamic models such as the nearly constant velocity (NCV), nearly constant acceleration (NCA), nearly constant turn (NCT), and stop models [1]-[3], [7], [9], [12]. A ground target can move on-road or off-road [7]. It is a standard practice to use the interacting multiple model (IMM) [1]-[4], [6], [8] [14]-[15] or variable structure IMM (VS-IMM) estimator [7] to estimate the state of a maneuvering target. A multi-sensor ground target tracking system usually employs ground moving target indicator (GMTI) radar [7], [10] electro-optical (EO), infrared (IR), video, acoustic, and seismic sensors, for which the measurement sampling intervals are different. In realistic scenarios, the measurement sampling interval even for a single sensor can vary. Most of the published papers on the IMM or VS-IMM estimator [1]-[4], [6]-[8], [14]-[15] use a constant Markov chain transition probability matrix (TPM) corresponding to a constant measurement sampling interval. Therefore, for multi-sensor ground target tracking problems, it is necessary to use a TPM in the IMM or VS-IMM which can handle variable measurement sampling intervals. In general, the TPM for a stationary or homogeneous Markov chain can be formally expressed as a matrix exponential function of the product of the transition probability rate matrix and measurement sampling interval [13]. When there are two dynamic models, the matrix exponential function can be simplified to obtain a simple analytic expression for the TPM as function of the measurement sampling interval [13], [5]. Alternatively, one can directly integrate the stochastic differential equation for the TPM and obtain the analytic expression. A simple analytic expression for the TPM as function of the measurement sampling interval does not exist at present when the number of dynamic models is greater than two. For computational purposes, one can compute the matrix exponential function efficiently and easily [11] for any number of models. In order to demonstrate the use of a TPM with the variable measurement sampling interval in the IMM estimator, we consider the motion of a ground target in two dimensions which involves the NCV, NCA, and NCT models in different parts of the trajectory. The dimensions of the target state for the NCV, NCA, and NCT models in 2D are four, six, and five respectively. The target full state is a seven dimensional consisting of 2D position, 2D velocity, 2D acceleration, and angular velocity along the Z axis. We use range, azimuth, and radial velocity measurements [10] from two GMTI sensors with variable measurement sampling intervals to estimate the target state and present numerical results to demonstrate the technical approach. We use the symbol “:=” to define a quantity. The outline of the paper is as follows. Section 2 describes the time dependent transition probability matrix using the continuous-time Markov chain associated with multiple dynamic models of a target. The expression for the TPM ) (Π as a function of the transition probability rate matrix ) (Λ and measurement sampling interval ) ( τ is presented in Section 2. Section 3 summarizes the NCV and NCT models and gives details of the NCA model. Section 3.4 describes the adjusted full state, state transition matrix or time evolution function, and process noise covariance matrix. Section 4 presents the GMTI sensor measurement