Properties of the Grasp Stiffness Matrix and Conservative Control Strategies

In this paper, we present fundamental properties of stiffness matrices as applied in analysis of grasping and dextrous manipulation in configuration spaces and linear Euclidean R^3×3 space without rotational components. A conservative-stiffness matrix in such spaces needs to satisfy both symmetric and exact differential criteria. Two types of stiffness matrices are discussed: constant and configuration-dependent matrices. The symmetric part of a constant-stiffness matrix can be derived from a conservative quadratic potential function in the Hermitian form; while the skew-symmetric part is a function of the nonconservative curl vector field of the grasp. A configuration-dependent stiffness matrix needs to be symmetric and must simultaneously satisfy the exact differential condition to be conservative. The theory is most relevant to the Cartesian stiffness control, where the stiffness of the end effector is usually constant, such as that in RCC wrists. Conservative control strategies are proposed for a configuration-dependent stiffness matrix. One of the most important results of this paper is the nonconservative congruence mapping of stiffness between the joint and Cartesian spaces. In general, the congruence transformation (or its inverse transformation), K_θ = J^T _θK_pJ_θ, is a nonconservative mapping over finite paths for a configuration-dependent Jacobian. Thus, to obtain a conservative system with respect to the Cartesian space, one has to either find the corresponding K_θ at every configuration due to the constant and symmetric Cartesian stiffness matrix, or determine the symmetric yet configuration-varying K_θ which makes the resulting configuration-dependent K_p conservative. In addition, the stiffness matrix also must be positive definite to maintain stability.