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

Abstract

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.

A new methodology to compute stabilizing control laws for continuous‐time LTV systems

“…A large family of practical systems can be modeled using the LTV framework, as for example, the aerodynamics in high‐speed aircrafts, space stations carrying moving vehicles, and diffusion coefficients in chemical processes . Moreover, the stability analysis and the control synthesis for nonlinear systems through linearization procedures yield more reliable results when performed around a set of operation points, or in terms of a time‐varying operation point, instead of linearizing the system around a static operation point . Therefore, the development of techniques suited to deal with LTV systems is an important research subject in automatic control theory.…”