Geometric approach to feedback stabilization of a hopping robot in the flight phase

Using a model of a hopping robot it is shown that a previously introduced novel approach for the synthesis of time-varying stabilizing feedback control for drift free systems, which is based on the trajectory intersection idea and primarily applies to systems whose controllability Lie algebra is finite dimensional, is also applicable to systems whose controllability algebra is infinite dimensional. The original model of the hopping robot is first approximated to yield a simplified model whose controllability Lie algebra is finite dimensional. A time varying stabilizing feedback law is then constructed for the simplified model. The latter can be viewed as a composition of a standard stabilizing feedback control for a Lie algebraic extension of the system and a periodic continuation of a parametrized solution to a certain open-loop, finite horizon trajectory interception problem which is stated and solved in logarithmic coordinates of flows. An adequately large stability robustness margin for the extended controlled system can always be insured and is shown to guarantee that the constructed feedback control is also stabilizing for the original model