Quaternions approach to solve the kinematic equation of rotation, A/sub a/A/sub x/=A/sub x/A/sub 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/sub a/A/sub x/=A/sub x/A/sub b/, which has to be solved for the rotational transformation matrix A/sub x/ given the transformations A/sub a/ and A/sub 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.<
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