Linear time-varying control law for stabilization of hopping robot during flight phase

The well-known Brocket's theorem revealed that nonholonomic systems, hopping robots, for example, can not be stabilized by smooth time-invariant state feedback controllers. In this manuscript, we propose a linear time-varying state feedback controller for stabilizing a nonholonomic hopping robot during flight mode in finite time. The current approach is novel in the sense that we modify the Pontryagin's minimum principle to formulate the linear state feedback control law. The existence of such a control law and its necessary conditions are presented in detail. The theoretical results are also validated through computer simulations.