Lie Algebraic Unscented Kalman Filter for Pose Estimation

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 $\mathbb{R}^n$. 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

An EKF for Lie Groups with Application to Crane Load Dynamics

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

Kinematic Feedback Control Using Dual Quaternions

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

Lie Algebraic Unscented Kalman Filter for Pose Estimation

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 Rn. 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

Kinematic Feedback Control Using Dual Quaternions

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.acceptedVersion© 2018 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works

Estimation of ship-deck motion using lidar, gyroscopes and cameras

This paper presents a system for the estimation of ship deck motion using camera, lidar and gyroscopes. A camera is used in a vision system that is based on the detection of lines as input to a vanishing point detector. This is done under a Manhattan assumption for man-made structures where the majority of lines are along 3 orthogonal axes. Two sets of parallel orthogonal lines are detected for the ship deck, and this is used to estimate the attitude using a complementary filter with input from lidar and gyroscopes. Since the vision algorithm depends on lines rather than points, the system is more resistant to occlusions like vision algorithms based on point tracking. In addition, a lidar is used to measure the distance between the sensor frame and the plane, and gyroscopes are used to improved the accuracy of the estimates. The system is validated in real time in lab experiments on a model of a ship

Extending educational effectiveness: A critical review of research approaches in International Effectiveness Research, and proposals to improve them

A review is given of internationally orientated educational effectiveness research over time, looking at the waves of studies from 1960 to 2001, the beginning of the PISA studies. It is concluded that whilst PISA has been a major methodological improvement upon earlier work, there are issues to do with the ‘culture fairness’ of test items, OECD misrepresenting of its own work, sampling adequacy and manipulation of samples by some governments. It is argued that educational effectiveness and improvement research can provide useful perspectives for international effectiveness research in terms of its focus upon pedagogy/teaching, its value-added approach, its often longitudinal design, its use of ‘supply side’ as well as ‘demand side’ policy variables and its use of ‘efficiency’ as well as ‘effectiveness’ measures. It is also argued that the cultural and contextual factors/features of societies would additionally require study.It is finally argued that high quality international comparative work may help in the generation of more useful educational effectiveness research, particularly in the generation of more valid, sensitive context specific formulations