Quaternions approach to solve the kinematic equation of rotation,A_aA_x=A_xA_b, of a sensor-mounted robotic manipulator

摘要

The problem of finding the relative orientation between thereference frames of a line-mounted sensor and the link is formulated asa kinematic equation of the form A_aA_x=A_xA_b, which has to besolved for the rotational transformation matrix A_xgiven the transformations A_a and A_b. This equation can be transformed to its equivalent form in termsof quaternion and then simplified to a well-structured linear system ofequations of the form Bx=0. Since B is rank-deficient,the solution is not unique. The generalized-inverse method usingsingular-value decomposition (SVD) is applied. Although the solution isreached using the analysis of SVD, the SVD is derived symbolically;therefore, the actual implementation of SVD is not required. A methodfor obtaining a unique solution is proposed where a system of nonlinearequations is solved using Newton-Raphson iteration. The iteration issimplified by a dimension-reduction technique that provides a set ofclosed-form formulas for solving the resulting linear system ofequations