Lyapunov Analysis of Robot Motion

The practice of automatic control has its origins in antiquity. It is only recently – within the middle decades of this century – that a body of scientific theory has been developed inform and improve that practice. Control theorists tend to divide their history into two periods. A “classical” period, prior to the sixties witnessed the systematization of feedback techniques based upon frequency domain analysis dominated by applications to electronics and telephony. A “modern” period in the sixties and seventies was characterized by a growing concern with formal analytical techniques pursued within the time domain motivated by the more stringent constraints posed by space applications and the enhanced processing capability of digital technology. The hallmark of control theory has been, by and large, a systematic exploitation of the properties of linear dynamical systems whether in the frequency or time domain. The dynamical behavior of mechanical systems appears to depart dynamically from the familiar linear case. The intent of this article is to show that a systematic application of Lyapunov Theory affords qualitative understanding of certain aspects of the input/output properties of a broad class of nonlinear systems (which includes all robots) analogous to that available for linear time invariant systems. For concreteness, this discussion is limited to robot arms – open kinematic chains with rigid links.