Global and Almost-Global Controllability of Underactuated Mechanical Systems by Using Time-Reversal Symmetry

Document Type

Conference Proceeding

Publication Date



Department of Electrical and Computer Engineering


This paper uses time-reversal symmetry (T-symmetry), which is inherent in many mechanical systems, to establish global controllability results for a class of underactuated mechanical systems, i.e., the systems with fewer actuators than the number of degrees of freedom (DoF). The idea is to find a control law guaranteeing a globally asymptotically stabilizable equilibrium (GAS) state, at which small-time local controllability (STLC) is also granted. Then, such equilibrium state can be used as a connection to design a global controller by using the time-reversal symmetry, i.e., the system can be driven from any initial state to any target state within finite time via the connection. By using the same line of reasoning, an UMS with an almost GAS equilibrium state proves to be almost-globally controllable. It shows that underactuated pendula with one degree of unactuation are almost-globally controllable (except a set of Lebesgue measure zero), for which the connection state is when all the links are at the downmost positions.

Publication Title

Proceedings of the American Control Conference