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.