We present a novel sample reduction scheme for random variables belonging to the SE(2) group by means of Dirac mixture approximation. For this, dual quaternions are employed to represent uncertain planar transformations. The Cramér-von Mises distance is modified as a smooth metric to measure the statistical distance between Dirac mixtures on the manifold of planar dual quaternions. Samples of reduced size are then obtained by minimizing the probability divergence via Riemannian optimization while interpreting the correlation between rotation and translation. We further deploy the proposed scheme for nonparametric modeling of estimates for nonlinear SE(2) estimation. Simulations show superior tracking performance of the sample reduction-based filter compared with Monte Carlo-based as well as parametric model-based planar dual quaternion filters.

Estimating orientations of objects in Euclidean space is an omnipresent challenge in robotics and autonomous systems. A useful representation of orientations involves unit quaternions. While the space of all unit quaternions forms a three-dimensional unit hypersphere, inverting the sign of a quaternion does not change the orientation described by it. Therefore, all possible orientations can be described by considering only a hemisphere of the unit hypersphere. In this paper, we propose a grid filter for arbitrary-dimensional unit hyperhemispheres and apply it to an orientation estimation task and another evaluation scenario. Our approach outperforms previous approaches that consider densities on the entire hypersphere.

In this paper, we present a novel nonlinear progressive filtering approach for estimatingSE(2) states represented by unit dual quaternions. Unlike previously published approaches, the measurement model no longer needs to be assumed as identity. Our solution utilizes deterministic sampling on a Bingham-like probability distribution, which has been adapted to simultaneously model orientation and translation. During the measurement update step, the estimate gets progressively updated. Our approach inherently incorporates the nonlinear structure ofSE(2) and enables a flexible measurement update step. We also give an evaluation for planar rigid body motion estimation with a case study that is close to real-world scenarios.

In this paper, we present a novel approach to perform simultaneous localization and mapping (SLAM) for planar motions based on stochastic filtering with dual quaternion particles using low-cost range and gyro sensor data. Here, SE(2) states are represented by unit dual quaternions and further get stochastically modeled by a distribution from directional statistics such that particles can be generated by random sampling. To build the full SLAM system, a novel dual quaternion particle filter based on Rao-Blackwellization is proposed for the tracking block, which is further integrated with an occupancy grid mapping block. Unlike previously proposed filtering approaches, our method can perform tracking in the presence of multi-modal noise in unknown environments while giving reasonable mapping results. The approach is further evaluated using a walking robot with on-board ultrasonic sensors and an IMU sensor navigating in an unknown environment in both simulated and real-world scenarios.