Integrating rotation from angular velocity

Abstract\ud The integration of the rotation from a given angular velocity is often required in practice. The present paper explores how the choice of the parametrization of rotation, when employed in conjuction with different numerical time-integration schemes, effects the accuracy and the computational efficiency. Three rotation parametrizations – the rotational vector, the Argyris tangential vector and the rotational quaternion – are combined with three different numerical time-integration schemes, including classical explicit Runge–Kutta method and the novel midpoint rule proposed here. The key result of the study is the assessment of the integration errors of various parametrization–integration method combinations. In order to assess the errors, we choose a time-dependent function corresponding to a rotational vector, and derive the related exact time-dependent angular velocity. This is then employed in the numerical solution as the data. The resulting numerically integrated approximate rotations are compared with the analytical solution. A novel global solution error norm for discrete solutions given by a set of values at chosen time-points is employed. Several characteristic angular velocity functions, resulting in small, finite and fast oscillating rotations are studied

Unscented Orientation Estimation Based on the Bingham Distribution

Orientation estimation for 3D objects is a common problem that is usually tackled with traditional nonlinear filtering techniques such as the extended Kalman filter (EKF) or the unscented Kalman filter (UKF). Most of these techniques assume Gaussian distributions to account for system noise and uncertain measurements. This distributional assumption does not consider the periodic nature of pose and orientation uncertainty. We propose a filter that considers the periodicity of the orientation estimation problem in its distributional assumption. This is achieved by making use of the Bingham distribution, which is defined on the hypersphere and thus inherently more suitable to periodic problems. Furthermore, handling of non-trivial system functions is done using deterministic sampling in an efficient way. A deterministic sampling scheme reminiscent of the UKF is proposed for the nonlinear manifold of orientations. It is the first deterministic sampling scheme that truly reflects the nonlinear manifold of the orientation

On least-cost path for realistic simulation of human motion

We are interested in "human-like" automatic motion simulation with applications in ergonomics. The apparent redundancy of the humanoid wrt its explicit tasks leads to the problem of choosing a plausible movement in the framework of redundant kinematics. Some results have been obtained in the human motion literature for reach motion that involves the position of the hands. We discuss these results and a motion generation scheme associated. When orientation is also explicitly required, very few works are available and even the methods for analysis are not defined. We discuss the choice for metrics adapted to the orientation, and also the problems encountered in defining a proper metric in both position and orientation. Motion capture and simulations are provided in both cases. The main goals of this paper are: to provide a survey on human motion features at task level for both position and orientation, to propose a kinematic control scheme based on these features, to define properly the error between motion capture and automatic motion simulation

A Unifying Approach to Quaternion Adaptive Filtering: Addressing the Gradient and Convergence

A novel framework for a unifying treatment of quaternion valued adaptive filtering algorithms is introduced. This is achieved based on a rigorous account of quaternion differentiability, the proposed I-gradient, and the use of augmented quaternion statistics to account for real world data with noncircular probability distributions. We first provide an elegant solution for the calculation of the gradient of real functions of quaternion variables (typical cost function), an issue that has so far prevented systematic development of quaternion adaptive filters. This makes it possible to unify the class of existing and proposed quaternion least mean square (QLMS) algorithms, and to illuminate their structural similarity. Next, in order to cater for both circular and noncircular data, the class of widely linear QLMS (WL-QLMS) algorithms is introduced and the subsequent convergence analysis unifies the treatment of strictly linear and widely linear filters, for both proper and improper sources. It is also shown that the proposed class of HR gradients allows us to resolve the uncertainty owing to the noncommutativity of quaternion products, while the involution gradient (I-gradient) provides generic extensions of the corresponding real- and complex-valued adaptive algorithms, at a reduced computational cost. Simulations in both the strictly linear and widely linear setting support the approach

Three-dimensional graphics

Three-dimensional graphics is the area of computer graphics that deals with producing two-dimensional representations, or images, of three-dimensional synthetic scenes, as seen from a given viewing configuration. The level of sophistication of these images may vary from simple wire-frame representations, where objects are depicted as a set of segment lines, with no data on surfaces and volumes, to photorealistic rendering, where illumination effects are computed using the physical laws of light propagation. All the different approaches are based on the metaphor of a virtual camera positioned in 3D space and looking at the scene. Hence, independently from the rendering algorithm used, producing an image of the scene always requires the resolution of the following problems: 1. Modeling geometric relationships among scene objects, and in particular efficiently representing the situation in 3D space of objects and virtual cameras; 2. Culling and clipping, i.e. efficiently determining which objects are visible from the virtual camera; 3. Projecting visible objects on the film plane of the virtual camera in order to render them. This chapter provides an introduction to the field by presenting the standard approaches for solving the aforementioned problems.168-17