Angle-only Filtering in 3D using Modified Spherical and Log Spherical Coordinates Mahendra Mallick Propagation Research Associates, Inc. Marietta, GA 30066, USA mahendra.mallick@gmail.com Sanjeev Arulampalam Maritime Operations Division, DSTO Edinburgh SA 5111, Australia Sanjeev.Arulampalam@dsto.defence.gov.au Lyudmila Mihaylova School of Computing and Communications Lancaster University, LA1 4WA, UK mila.mihaylova@lancaster.ac.uk Yanjun Yan ARCON Corporation Waltham, MA, 02451, USA yanjun@arcon.com Abstract— This paper considers the angle-only filtering prob- lem in 3D using bearing and elevation angle measurements from a single maneuvering sensor. We develop continuous-discrete extended Kalman filter (EKF) based algorithms using modified spherical coordinates (MSC) and log spherical coordinates (LSC), where the dynamic and measurement models are described in continuous and discrete times, respectively. The predicted state estimate and covariance are calculated numerically by integrating a joint vector differential equation. Numerical results show that, while the discrete-time Cartesian EKF outperforms the proposed algorithms for highly accurate measurements, the new algorithms show superior performance as the measurement accuracy decreases. EKF-MSC and EKF-LSC have comparable performance for the examples examined. Keywords: Angle-only filtering in 3D, Nonlinear stochastic dif- ferential equations, Modified spherical coordinates (MSC), Log spherical coordinates (LSC), Continuous-discrete filtering. I. I NTRODUCTION The angle-only filtering problem in 3D using bearing and elevation angles from a single maneuvering sensor is the counterpart of the bearing-only filtering problem in 2D [1], [5], [17]. This problem arises in passive ranging using an infrared search and track (IRST) sensor, passive sonar, and passive radar in the presence of jamming [6]. A great deal of research has been conducted for the bearing-only filtering problem in 2D - see for e.g. [1], [5], [17], and the references therein. However, the number of publications for the angle- only filtering problem in 3D is relatively small [19], [18], [15], [10], [16]. The angle-only filtering problem in 3D using modified spherical coordinates (MSC) [19] or log spherical coordinates (LSC) is much harder than the bearing-only filtering problem in 2D using modified polar coordinates (MPC) [1], [17] or log polar coordinates (LPC) [8]. This is due to the fact that discretization of the dynamic model for the 3D case using MSC or LSC is not as straightforward as for the 2D case using MPC [1] or LPC [8]. Therefore, we use the continuous- discrete extended Kalman filter (EKF) using MSC and LSC. Stallard [19] first extended the method from [12] and [14] to three dimensions and proposed the MSC. There are three important differences between our approach and Stallard’s approach, as follows: Firstly, we use standard conventions [17] in defining the coordinate frames for the tracker and sensor, and coordinates for MSC and LSC. In [19], the coordinate frames, bearing and elevation angle are defined in a non-standard way. Secondly, we clearly show the equivalence of the nearly constant velocity model (NCVM) using Cartesian state vector and stochastic differential equations using the MSC or LSC. This is missing in [19]. We illustrate this by first presenting the NCVM in Cartesian coordinates with a continuous white noise acceleration process [3], together with its power spectral density matrix, expressed in the tracker coordinate frame (T frame). Then we derive the first order nonlinear stochastic differential equations for MSC and LSC corresponding to the NCVM in Cartesian coordinates. Thirdly, and most importantly, we present a new derivation of the differential equation for the predicted covariance fol- lowing the Brownian motion process [13], when the nonlin- ear time evolution function depends on the target state and continuous-time process noise. In most common nonlinear stochastic differential equations, the continuous-time process noise appears as an additive term as in chapter 6 of [9]. The nonlinear time evolution function for MSC or LSC depends on the target state, continuous-time process noise and ownship acceleration in the sensor frame (S frame). The predicted state estimate and covariance are integrated numerically and jointly to provide better numerical accuracy. This is a key contribution of our paper. In [19], the state transition matrix, discrete-time process noise covariance matrix, and discrete time predicted covariance matrix are derived using the approximation that the relative geometry between the target and ownship changes relatively slowly. The paper is organized as follows. Section II introduces notations and the coordinate frames with respect to the target and ownship. Section III presents the dynamic models of the target and ownship, and measurement model with respect to Cartesian relative state. Sections IV and V present the MSC