  In this paper we propose a novel method for estimating rigid body motion by
modeling the object state directly in the space of the rigid body motion group
SE(2). It has been recently observed that a noisy manoeuvring object in SE(2)
exhibits banana-shaped probability density contours in its pose. For this
reason, we propose and investigate two state space models for moving object
tracking: (i) a direct product SE(2)xR3 and (ii) a direct product of the two
rigid body motion groups SE(2)xSE(2). The first term within these two state
space constructions describes the current pose of the rigid body, while the
second one employs its second order dynamics, i.e., the velocities. By this, we
gain the flexibility of tracking omnidirectional motion in the vein of a
constant velocity model, but also accounting for the dynamics in the rotation
component. Since the SE(2) group is a matrix Lie group, we solve this problem
by using the extended Kalman filter on matrix Lie groups and provide a detailed
derivation of the proposed filters. We analyze the performance of the filters
on a large number of synthetic trajectories and compare them with (i) the
extended Kalman filter based constant velocity and turn rate model and (ii) the
linear Kalman filter based constant velocity model. The results show that the
proposed filters outperform the other two filters on a wide spectrum of types
of motion.
