Controllability of kinematic control systems on stratified configuration spaces

This paper considers nonlinear kinematic controllability of a class of systems called stratified. Roughly speaking, such stratified systems have a configuration space which can be decomposed into submanifolds upon which the system has different sets of equations of motion. For such systems, considering controllability is difficult because of the discontinuous form of the equations of motion. The main result in this paper is a controllability test, analogous to Chow's theorem, is based upon a construction involving distributions, and the extension thereof to robotic gaits

Quasi-static legged locomotors as nonholonomic systems

We show how motion planning and control ideas for smooth nonholonomic systems can be extended to legged quasi-static locomotion via the notion of "stratified" configuration spaces and "stratified" control theory. We particularly consider "minimalist" legged systems, which are not well handled by conventional theories based on foot placement. We briefly discuss controllability issues, and then present a motion planning algorithm for stratified systems. The method does not depend upon the number of legs, nor is it based on foot placement concepts

Compositional boundedness of solutions for symmetric nonautonomous control systems

Abstract—This paper presents boundedness results for symmetric nonautonomous control systems. The results are Lyapunov-based and exploit the symmetric structure present in many compositional or distributed control system. The work extends some of our prior work which defined an equivalence class of symmetric control systems by determining conditions for boundedness of solutions for such systems. The extension is along two lines. First, the prior work was focused on stability for autonomous symmetric systems, and this work extends it to the nonautonomous case. Second, the prior work required exact symmetry in the system whereas the results in this paper allow for symmetry-breaking in the nonautonomous terms, which significantly broadens the class of systems to which these results will apply. These results will be useful for robotics and control engineers dealing with large-scale and distributed systems which are composed of many similar components because it will enable closed-form analysis on a very small system, and then guarantee system properties for much larger equivalent systems. The application of the theoretical results is illustrated with a nonlinear consensus example. Other potential application areas would be, for example, the design of control algorithms for fleets of autonomous robotic vehicles acting in a coordinated manner. I

Compositional stability of approximately symmetric systems: Initial results

Abstract—This paper considers nonlinear control systems that are approximately symmetric, and extends some prior work of the author related to stability of symmetric systems to the case where the system is not exactly symmetric. Many engineering systems are composed of components that are nominally identical, but due inherent variability in physical systems, can not be exactly symmetric. By exploiting the baseline symmetric structure of the system and constraining the deviations from exact symmetry, stability results are derived that are independent of the number of components in the system. This paper specifically focuses on the application of LaSalle’s Invariance Principle to approximately symmetric systems, which has broad applicability. The main utility of the stability result is one of scalability or compositionality because the main result shows that if the system is stable for a given number of components, under appropriate conditions, stability is then guaranteed for a larger system composed of the same type of components which are interconnected in a manner consistent with the smaller system. Index Terms—symmetric systems, multiagent coordination, nonlinear systems, compositionalit

˙x ˙x 0 0 ˙y ⎟ ⎜ ˙y ⎟ ⎜

The first part of these notes is a motivating example which justifies, apart from pure intellectual curiosity, our desire to delve into the details of Sussmann’s local controllability results presented in [2]. The following sections then summarize various results in the paper. Possibly the hardest part of reading Sussmann’s paper is keeping track of all the notation. The Appendix presents a list which summarizes most of the notation used in the paper. This actually lists the notation from the “easier ” paper [1], so there is not an exact correlation, but much of the notation is the same in both papers. The symbols are listed roughly in the order that they appear in [1], which is fortunately pretty well correlated with the order that they appear in [2].