An extended Kalman filter (EKF) for systems with configuration given by matrix Lie groups is presented. The error dynamics are given by the logarithm of the Lie group and are based on the kinematic differential equation of the logarithm, which is given in terms of the Jacobian of the Lie group. The probability distribution is also described in terms of the logarithm as a concentrated Gaussian distribution that is a tightly focused distribution around the identity of the Lie group. The filter is applied to estimation on SO(3) a case where a stereo camera setup tracks a crane wire with a payload. The wire, which is under tension and forms a line is monitored by two 2D-cameras, and a line detector is used to obtain a description of how the wire is projected onto each image plane. A model of a spherical pendulum is applied and the estimator is validated by applying it on simulated data, as well as experimental data.
This paper presents results on kinematic controllers for the stabilization of rigid body displacements using dual quaternions. The paper shows how certain results for quaternion stabilization of rotation can be extended to dual quaternions stabilization of displacements. The paper presents a relevant background material on screw motion and the screw description of lines and twists. Moreover, results are presented on the computation of the exponential functions for dual quaternions for use in numerical integration. The paper presents and analyzes different controllers based on feedback from dual quaternions, where some of the controllers are known from the literature, and some are new. In particular, it is shown which controllers give screw motion, and it is discussed how this will affect the performance of the controlled system compared to other controllers that are not based on screw motion. This analysis is supported by Lyapunov analysis. Also, certain passivity properties for dual quaternions are presented as an extension to previously published results on passivity for quaternions.
An unscented Kalman filter for matrix Lie groups is proposed where the time propagation of the state is formulated on the Lie algebra. This is done with the kinematic differential equation of the logarithm, where the inverse of the right Jacobian is used. The sigma points can then be expressed as logarithms in vector form, and time propagation of the sigma points and the computation of the mean and the covariance can be done on the Lie algebra. The resulting formulation is to a large extent based on logarithms in vector form, and is therefore closer to the UKF for systems in R n . This gives an elegant and well-structured formulation which provides additional insight into the problem, and which is computationally efficient. The proposed method is in particular formulated and investigated on the matrix Lie group SE(3). A discussion on right and left Jacobians is included, and a novel closed form solution for the inverse of the right Jacobian on SE(3) is derived, which gives a compact representation involving fewer matrix operations.The proposed method is validated in simulations.
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