A report discusses the inversion of the Jacobian matrix for a class of six-degree - of-freedom arms with spherical wrist, i.e., with the last three joints intersecti ng. The authors show that by taking advantage of the simple geometry of such arms, the closed-form solution of , which represents a linear transformation from task space to joint space, can be obtained efficiently. The inverse transformation, represented as , is essential for the implementation of the advanced task-space control schemes since the actuation; i e. the application of forces/torques is ultimately performed at the joints level. Howeve r, while the forward transformation given as , presents no major problem, the computation of the inverse transformation is more difficult, and previous authors have studied ways to perform it efficiently. The main motivation in the development of the previous methods has been to avoid the prohibitive computational cost which arises from the symbolic or numerical computation of the Jacobian and its numerical inversion. The theoretical development in this paper involves the derivation of the Jacobian vectorial matrix, which is a general form of the Jacobian from which specific Jacobians can be derived by a set of appropriate transformation, i.e., velocity reference point transformation as well as projection onto the applicable coordinates. The authors develop a relatively simple method in which the Jacobian vectorial matrix and corresponding transformations are derived by relating the velocity of two points. It is shown that any specific Jacobian can b e described in terms of another Jacobian with arbitrary choice of velocity referenc e point and coordinate frame, which can be chosen to yield the Jacobian in its simplest symbolical form leading to its symbolic inversion and the closed-form solution of . The report presents solutions for a PUMA arm, a JPL/Stanford arm, and a six- revolute-joint coplanar arm along with all singular points. The solution for each joint variable is found as an explicit function of the singular points, thus prov iding better insight into the effect of the singular points on the motion of, and force/torque exerted by, each joint. The main contribution of this paper is to show that the simple geometry of this type of arms can be exploited in performing the inverse transformation without any need to compute the Jacobian or its inverse explicitly. The cost of computing the inverse transformation becomes of the same order as that of the cost of computing forward kinematic position transformation, and it is significantly reduced if this transformation is already performed. The implication of this computational efficiency is that advanced task-space control schemes for spherical-wrist arms can be implemented more efficiently.
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Last updated: May 10, 1996