A RANGE-ONLY MULTIPLE TARGET PARTICLE FILTER TRACKER Volkan Cevher, Rajbabu Velmurugan, and James H. McClellan Georgia Institute of Technology Atlanta, GA 30332-0250 ABSTRACT We propose a particle ﬁlter tracker to track multiple maneuvering targets using a batch of range measurements. The state update is for- mulated through a locally linear motion model and the observability of the state vector is proved using geometrical arguments. The data likelihood treats the range observations as an image using template models derived from the state update equation, and incorporates the possibility of missing data as well as spurious range observations. The particle ﬁlter handles multiple targets, using a partitioned state- vector approach. The ﬁlter proposal function uses a Gaussian ap- proximation of the full-posterior to cope with target maneuvers for improved efﬁciency. By treating the range measurements as images and using smoothness constraints, the particle ﬁlter is able to avoid the data association problems. Computer simulations demonstrate the performance of the tracking algorithm. 1. INTRODUCTION Radar range tracking problem is a challenging signal processing prob- lem that has attracted very little interest in the literature [1–3]. This tracking problem is usually formulated using state-space models, where the target’s motion is approximated as local linear (e.g., con- stant velocity) and the observations are temporal snapshots of radar range and range-rate estimates. Then, to estimate the state vector consisting of the target’s position and velocity using range-only mea- surements, a mobile platform must be used that executes known ma- neuvers for system observability [2]. Otherwise, multiple beacons must be used to track the state vector by virtue of triangulation [3]. In this paper, we present a particle ﬁlter algorithm to track a state vector that consists of the target direction-of-arrival (DOA) θ(t), the logarithm of the target range R(t), the target speed v(t), and the target heading φ(t), using a batch of range-only measurements, ob- tained at a stationary sensor. The angles are measured counterclock- wise with respect to the x-axis. We prove that the particle ﬁlter state vector is observable given at least three range measurements under rotational and planar-symmetrical ambiguities. Our proof makes use of the Stewart’s triangle theorem in geometry. The motivation for the state vector of the particle ﬁlter is the low power RF sensor, implemented at the University of Florida that transmits a microwave signal to determine the range, the velocity, and the size of the detected targets [4]. The sensor is capable of pro- viding range estimates at 32ms intervals with a range resolution of approximately 2m on a range-Doppler map. Up to 100m, the current system is capable of producing range estimates for multiple ground vehicles as well as human targets. The radar hardware is envisioned Prepared through collaborative participation in the Advanced Sensors Consortium sponsored by the U. S. Army Research Laboratory under the Col- laborative Technology Alliance Program, Cooperative Agreement DAAD19- 01-02-0008. to have a larger detection range with hemispherical coverage in the future. Note that the ﬁlter equations in this paper are developed using range-only measurements so that it is also applicable to amplitude tracking problems. Additional velocity measurements or range-rate measurements can be easily incorporated through the data likelihood equation via independence assumptions. The particle ﬁlter uses a batch of range measurements to deter- mine the state vector, based on an image template matching idea. The template matching idea is very effective when accurate models are available [5]. In our problem, a temporal range image is ﬁrst formed, when a batch of range measurements are received. Then, candidate image templates are formed by using the state update func- tion and the target state vectors. By determining the best matching image template, the target state vectors are determined. It is assumed that the range measurements are normally distributed around the true range measurements, with constant data miss-probability and clutter density. The presence of multiple targets increases the tracking complex- ity, because the received data must be sorted for each target. Since the particle ﬁlter treats its range-only measurements as an image, the data association and ordering problems are naturally alleviated. To handle multiple targets, the particle ﬁlter uses a partitioning ap- proach, where a particle consists of the concatenation of multiple tar- get state vectors. We use the probabilistic data association methods to estimate the states by summing over all the association hypothesis weighted by the likelihood probabilities [6]. The particle ﬁlter pro- posal function independently proposes particles for its partitions by using a Gaussian approximation of the full posterior density for efﬁ- ciency. Hence, the presented particle ﬁlter is robust against the curse of dimensionality problem [7], when the number of targets increase. To derive the proposal function, the multi target posterior den- sity is approximately factorized for each target. Then the Laplace’s method is used to approximate each partition posterior by a Gaussian around its mode [8]. We calculate the partition modes using a robust Newton-Raphson recursion with a backtracking step size selection that imposes smoothness constraints on the target motion [9]. This approach is similar to the one for an angle-only tracking ﬁlter [10]. 2. DATA MODELS 2.1. State Update Model The ﬁlter state vector xt = x T 1 (t),x T 2 (t), ··· ,x T K (t) T consists of the concatenation of the partition vectors x k (t) for each target, indexed by k, k =1,...,K. Each partition has the corresponding target motion parameters x k (t)  [θ k (t),R k (t),v k (t),φ k (t)] T , where θ k (t) is the DOA, R k (t) is the logarithm of the range, v k (t) is the speed, and φ k (t) is the heading direction. For notational conve- nience, the logarithm of the range is used in the state vector because the range errors are modeled multiplicative.