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Computational kinematics and dynamics of robot manipulators are dealt with. A recursive method based on the vector form of Rodrigues' equation is presented for the computation of the associated coordinate transformations. The method allows for forward, backward, and two-way recursions and is applied to the computations of the Jacobian matrices and dynamic equations for mechanical manipulators. The computational complexities of the resulting equations are also evaluated and compared to some of the existing methods in each case. It is shown that the algorithms presented have certain computational advantages over most of the existing methods.