Safety-critical and constrained geometric control synthesis using control Lyapunov and control Barrier functions for systems evolving on manifolds

Geometric controllers for mechanical system have been well-developed in past research work. Although geometric controllers possess the advantages of being coordinate-free and compact, these controllers don't take into account constraints such as safety-critical collision constraints or input saturations. In this paper, we utilize the concepts of control Lyapunov function(CLF) and control Barrier function(CBF) to incorporate various constraints in a state-dependent quadratic programming(QP). Through relaxation, a proper trade-off has been made between the constraints imposed by CLF and CBF. Then qualitative analysis of this design method is derived in detail, and we provide convincing simulation results on a point mass, a spherical pendulum, and a 3D pendulum.