On Geometric Algorithms for Real-Time Grasping Force Optimization

Grasping force optimization with nonlinear friction constraints is a fundamental problem in dextrous manipulation with multifingered robotic hands. Over the last few years, by transforming the problem into convex optimization problems on Riemannian manifolds of symmetric and positive definite matrices, significant advances have been achieved in this area. Five promising algorithms: two gradient algorithms, two Newton algorithms, and one interior point algorithm have been proposed for real-time solutions of the problem. In this paper, we present in a unified geometric framework, the derivation of these five algorithms and the selection of step sizes for each algorithm. Using the geometric structure of the affine-scaling vector fields associated with the optimization problem, we prove that some of these algorithms have quadratic convergence properties, and their continuous versions are exponentially convergent. We evaluate the performance of these algorithms through simulation and experimental studies with the Hong Kong University of Science and Technology (HKUST) three-fingered hand. This study will facilitate selection and implementation of grasping force optimization algorithms for similar applications.