Stability of Second-order Asymmetric Linear Mechanical Systems with Application to Robot Grasping

This technical correspondence presents a surprisingly simple analytical criterion for the stability of general second-order asymmetric linear systems. The criterion is based on the fact that if a symmetric system is stable, adding a small amount of asymmetry would not cause instability. We compute analytically an upper bound on the allowed asymmetry such that the overall linear system is stable. This stability criterion is then applied to robot grasping arrangements which, due to physical effects at the contacts, are asymmetric mechanical systems. We present an application of the stability criterion to a 2D grasp arrangement. INTRODUCTION This technical correspondence is concerned with the stability of second-order linear systems that have an asymmetric stiffness matrix. Our goal is to provide an analytical criterion for the stability of systems of the form: Mp̈+Kd ṗ+Kp p = 0; (1) where M 2 IRn n and Kd 2 IRn n are symmetric positive definite, and Kp 2 IRn n is asymmetric. Such systems arise in the linearized dynamics of robot grasping arrangements [9], and in other applications such as feedback control. See, for instance, [7] and [5, p. 36]. Researchers have taken the following approach to the investigation of general asymmetric systems, where M, Kd , and Kp are asymmetric. Their approach is based on transforming the asymmetric system into a symmetric one. The subclass of asymmetric systems that can be transformed into symmetric systems is called symmetrizable systems. Inman has introduced necessary and sufficient conditions for a subclass of such systems to be symmetrizable via similarity transformation [4]. Ahmadian and Chou have developed a systematic technique for computing the coordinate system in which the symmetrizable system is symmetric [2]. Coghey and Ma have given a condition for transforming the system into a decoupled diagonal system [3]. Utilizing equivalence transformation rather than similarity transformation enables the subclass of symmetrizable systems to be enlarged [1, 8]. All these results are exact and give conditions for the stability of the original asymmetric system. However, only subclasses of asymmetric systems can be treated in these ways, and the application of stability criteria based on transformation to symmetric systems is cumbersome. In this technical correspondence we develop a simple criterion for the stability of asymmetric systems of the form (1). In the context of robot grasping applications, this stability criterion leads to a synthesis rule that indicates which contact points and what preloading profile guarantee stable grasp. We make the following two assumptions, which are motivated by consideration of the grasping application. First, as in many mechanical systems, we assume that the inertia and damping matrices, M and Kd , are symmetric positive definite matrices. Second, we