Quaternions approach to solve the kinematic equation of rotation, AaAx=AxA b, of a sensor-mounted robotic manipulator

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