Tracking the Orientation and Axes Lengths of an Elliptical Extended Object

Abstract

Extended object tracking considers the simultaneous estimation of the kinematic state and the shape parameters of a moving object based on a varying number of noisy detections. A main challenge in extended object tracking is the nonlinearity and high-dimensionality of the estimation problem. This work presents compact closed-form expressions for a recursive Kalman filter that explicitly estimates the orientation and axes lengths of an extended object based on detections that are scattered over the object surface. Existing approaches are either based on Monte Carlo approximations or do not allow for explicitly maintaining all ellipse parameters. The performance of the novel approach is demonstrated with respect to the state-of-the-art by means of simulations.

Fusion of Elliptical Extended Object Estimates Parameterized with Orientation and Axes Lengths

“…This representation explicitly allows for capturing uncertainties for the different dimensions, providing important information which can be utilized for the fusion. Its usefulness has been demonstrated in, e.g., [7]. We also add a kinematic state r, which can be realized as, e.g., a Cartesian velocityṁ, a polar velocity consisting of velocity v and orientation ψ, or even only v while using α as the orientation, in either case resulting in a state vector…”

“…For simplicity, we assume only a single target which is tracked by multiple sensors. Each sensor can observe the entire target's shape, i.e., we assume that the sensors observe a point cloud distributed across the entire target's surface [2], [6], [7], [9], [11], [24], [48], [49]. Due to aspects such as different sensor quality or sensor to target geometry, the sensor estimates have different uncertainties, represented by their covariances.…”

“…(4n + k) ← w + . (4n + k) • w.n end end x + , C + , w + ←reduce_mixture(x + , C + , w + ) returnx + , C + , w + Algorithm 1: The RED fusion algorithm relies on the following functions: K() switches according to (7) with the last input as k when given state means and switches covariances accordingly as well, len() provides the length of the input vector, kalman() is the Kalman filter correction (restricting the innovation in α between −π and π) providing updated mean, covariance, and likelihood, and reduce mixture() is a mixture reduction function also providing normalized weights.…”

“…Elliptic targets can be tracked using Random Matrices, pioneered by Koch et al [6], [24]. However, it was demonstrated in [7] that under the assumption that the semi-axes don't change, but the orientation does, tracking the orientation and semi-axes as individual parameters can improve the estimation, especially in turns. Also, if the same target is detected by different sensors, the estimates may have different uncertainties depending on the sensor's qualities and position relative to the target (see Figure 1).…”

“…To counter this, the shape orientation could be aligned with the velocity vector. However, in tracking elliptic targets, velocity and shape orientation are often decoupled, e.g., [6], [7]. The reason is that the ellipse orientation is not necessarily equal to the orientation of the velocity vector, e.g., due to drifting of the vessels in case of ships [22], [25].…”