Analytical computation of gradient and Hessian for three types of manipulability measures

Abstract

The gradient and Hessian of manipulability, a key measure for avoiding kinematic singularities, are often used in optimization-based motion planning. However, despite their importance, a comprehensive analytical derivation of these quantities has not been extensively studied. This paper presents an analytical framework for computing the gradient vector and Hessian matrix of three representative manipulability measures. While previous studies have addressed only the gradient of the first measure, this work, to the best of our knowledge, provides the analytical derivations of both the first- and second-order differentials for all three measures for the first time. The derivation leverages trace properties and the adjoint operator to systematically differentiate the vectorized Jacobian matrix with respect to joint variables. The derived expressions are shown to satisfy a zero-boundary condition, based on the equivalence between body and space Jacobians and their independence from the first and last joint variables. The proposed analytical method is validated through both simulation and experimental results, with a focus on the approximation errors associated with the Taylor expansion. Furthermore, its computational advantages are demonstrated through comparisons of matrix operations and execution times in simulation. These results establish a rigorous and efficient foundation for manipulability-based motion optimization.