Differential geometry measures of nonlinearity for the bearing-only tracking problem

Abstract

The bearing-only tracking problem arises in many radar and sonar tracking applications. Since the bearing measurement model is a nonlinear function of the target state, the filtering problem is nonlinear in nature. A great deal of attention has been focused on this problem due to the difficulty posed by the so-called high degree of nonlinearity (DoN) in the problem. However, a quantitative measure of the DoN is not calculated in previous works. It has been observed that the extended Kalman filter (EKF) in which the state vector consists of the Cartesian components of position and velocity is unstable and diverges in some cases. The range parametrized EKF (RPEKF) and particle filter (PF) have been shown to produce improved estimates for the bearing-only tracking problem. In this paper, we calculate two measures of nonlinearity, (1) the parameter-effects curvature and (2) intrinsic curvature for the bearing-only tracking problem using the differential geometry measures of nonlinearity. We present numerical results using simulated data for the constant velocity motion of a target in 2D with bearing-only measurements where the sensor platform uses a higher order motion than the target to achieve observability. We analyze the DoN by varying the distance between the target and sensor.