An overbounding ellipse approach for set based tracking with range-bearing measurements Fiona Fletcher M. Sanjeev Arulampalam Maritime Operations Division, Defence Science and Technology Organisation PO Box 1500, Edinburgh, SA 5111 AUSTRALIA Fiona.Fletcher@defence.gov.au Sanjeev.Arulampalam@defence.gov.au Abstract – In many tracking applications, the target state is es- timated using range-bearing measurements. For statistical track- ers, this problem may be dealt with using a converted measure- ment Kalman Filter. In this paper, measurement conversion proce- dures are proposed for using range-bearing measurements in a set based tracker. These procedures involve overbounding the range- bearing error region with an ellipse. The measurement conver- sions have been implemented in a set based tracker and numerical results are presented demonstrating their relative performance. Keywords: Set based estimation, tracking, range-bearing mea- surements 1 Introduction It is common in target tracking applications for measure- ments from a target to be non-linearly related to the tar- get state. A particularly prevalent example of this is where range-bearing measurements are obtained from a target whose state is modelled in Cartesian coordinates. This is often the case for active sonar or radar [1]. There are two main approaches for dealing with range- bearing measurements in statistical tracking. An Extended Kalman Filter (EKF) may be used, where the non-linear measurement equation is approximated by a linear equa- tion using Taylor series expansion [2]. An alternative ap- proach is that of the Converted Measurement Kalman Filter (CMKF), where the measurements are converted to Carte- sian position and covariance and are then used in a lin- ear Kalman Filter. Examples of these measurement con- versions are the standard measurement conversion [2], the debiased measurement conversion [3] and the unbiased measurement conversion [4]. These trackers are all based around the Kalman filter, which calculates a Gaussian den- sity for the target state based on assumptions of Gaussianity for the noises and initial states. The estimate provided by such a tracker is simply the mean of the Gaussian density for target state. Set based estimation provides an alternative philosophy for tracking. Rather than a point estimate, the estimate ob- tained using a set based tracker is a set of possible target states. This set is guaranteed to contain the true target state provided the system has been accurately modelled. An- other advantage of set based tracking is that it only requires bounds on the process and measurement noises. It makes no assumptions on how these noises are distributed, except that they must satisfy the specified bounds. This can pro- vide a more robust estimate of target state than Kalman fil- ter based trackers in the particular case where noises are non-Gaussian. The aim of a set based tracker is to maintain a guaranteed bound on the target state at all times. This means that the EKF approach of approximating the non-linear measure- ment equation by its Taylor series expansion may not be used as it does not maintain a guaranteed bound. The mea- surement conversions used in the CMKF do not apply to set based estimation, as a set based estimator requires a bound on the measurement error rather than a Gaussian distribu- tion approximation. The principles behind the CMKF do apply to set based tracking, and in this paper, we propose a measurement conversion procedure specific to set based tracking. The premise of maintaining a guaranteed bound on target state continues throughout the paper. In Section 2 the problem is formulated, including a brief description of set based estimation and discussion of ellip- soidal overbounding. A measurement conversion procedure specific to set based tracking is described in Section 3, and included in this is an improvement to the conversion for cer- tain sets of parameters. Section 4 includes results of monte carlo simulations and discussion. Conclusions are drawn in Section 5. 2 Problem Formulation 2.1 System Equations The state of a target is described by Cartesian position and velocity, x k =[x k , ˙ x k ,y k , ˙ y k ] ′ . We assume that the target obeys a nearly constant velocity model, x k+1 = F k x k + v k (1) where F k =  F 0 0 F  is the transition matrix and v k is the process noise. Here F =  1 T 0 1  where T is the time increment between updates. Rather than requiring sta- tistical properties of the noise v k , the set based framework only requires that it be bounded, with the bound known, v k ∈ Ω Q (k). (2)