A Davenport dual angles approach for minimal parameterization of the rigid body displacement and motion

Abstract

In this paper, the problem of the decomposition of a rigid body displacement by a succession of three rigid body displacements with specified screw-axis is completely solved. Using the isomorphism between the Lie group of the rigid body displacements and the Lie group of the orthogonal dual tensors, the necessary and sufficient conditions for which the decomposition is possible are given. It will be determined either the closed form the dual angles of the screw displacements (translation and rotation). The results are coordinate-free, and they are obtained using only algebraic elements of tensor calculus. In the specific case of a screw axis is perpendicular to the other two, the decomposition is possible for any rigid body displacement. So, a result that generalizes the Davenport decomposition in case of rotation is obtained. A Davenport dual angles decomposition is a new minimal parameterization of the rigid body displacement.For rigid body motion, the kinematic equations that link the instantaneous dual angular velocity to the time variation of Davenport dual angles are deduced. The cases of singularity of the decomposition are identified and a physical interpretation of these cases is given. The results interest the inverse kinematic of robotics, control theory and astrodynamics (full body relative orbital motion problem).