Real-time SVD for the control of redundant robotic manipulators

Abstract

By taking advantage of the well-behaved nature of singular values and vectors, the computational expense of determining the SVD of the Jacobian matrix can be reduced to such an extent that on-line calculation becomes feasible. This permits more intelligent use of the extra degrees of freedom present with redundant manipulators, particularly with respect to the optimization of various secondary criteria, including obstacle avoidance, under the constraint of a specified end effector trajectory.However, the control of robotic systems is not based on the solution of arbitrary matrix equations but quite frequently involves the solution of equations based on the Jacobian matrix. The current Jacobian for a system can be regarded as a perturbation of a previously known matrix for which perturbation bounds on the singular values and singular vectors can be established. Knowledge of the previous state can be exploited during the current calculation of the SVD in order to reduce the overall computational burden. This results in a computational scheme capable of calculating the SVD of the Jacobian for use in real-time control of manipulators. u,v; (4) ;=1 II. Perturbation Bounds on the SVD,=k+lThe perturbation bounds on the rotation of subspaces defined by singular vectors are not as widely known but are also well-behaved [11]. In particular, consider a partitioning of the Jacobian into two singular subspaces of the form All SVD algorithms possess an iterative component designed to orthogonalize the columns (or rows) of the matrix being decomposed. Clearly, the more orthogonal these columns are to begin with, the fewer the number of calculations are required to reach convergence. Thus, if one considers the current manipulator Jacobian, denoted by J(t + At), to be a perturbation of the previous Jacobian (2)(1) J(t) = U(t)D(t)VT(t)the SVD of which is known and given by then the matrix J(t + At)V(t) will have nearly orthogonal columns provided the perturbation AJ(t) is small relative to J(t). The foundation of the above lies in the fundamentally well-behaved nature of the SVD of a matrix. The perturbation bounds on singular values, denoted by ai, are very well-known and it is easy to show that lai(J(t + At)) -a,(J(t))1~IILlJ(t)ll.(3)

An on-line task modification method for singularity avoidance of robot manipulators

“…The determinant of the manipulator's Jacobian matrix gives illegible information about the absolute proximity to singularities, since the minimum singular value is the only reliable measure of the absolute proximity to singularities. 21,22 Decomposition of the Jacobian matrix has the form…”