Dextrous hand grasping force optimization

A key goal in dextrous robotic hand grasping is to balance external forces and at the same time achieve grasp stability and minimum grasping energy by choosing an appropriate set of internal grasping forces. Since it appears that there is no direct algebraic optimization approach, a recursive optimization, which is adaptive for application in a dynamic environment, is required. One key observation in this paper is that friction force limit constraints and force balancing constraints are equivalent to the positive definiteness of a certain matrix subject to linear constraints. Based on this observation, we formulate the task of grasping force optimization as an optimization problem on the smooth manifold of linearly constrained positive definite matrices for which there are known globally exponentially convergent solutions via gradient flows. There are a number of versions depending on the Riemannian metric chosen, each with its advantages, Schemes involving second derivative information for quadratic convergence are also studied. Several forms of constrained gradient flows are developed for point contact and soft-finger contact friction models. The physical meaning of the cost index used for the gradient flows is discussed in the context of grasping force optimization. A discretized version for real-time applicability is presented. Numerical examples demonstrate the simplicity, the good numerical properties, and optimality of the approach.