Safety-Critical Geometric Control for Systems on Manifolds Subject to Time-Varying Constraints

We address the safety-critical control problem for systems whose dynamics evolve on manifolds using the concepts of control Lyapunov functions (CLFs) and control Barrier functions (CBFs). We first extend the concepts of CLFs and CBFs from Cartesian spaces to manifolds, resulting in geometric CLFs and CBFs. We then formulate a state-dependent online quadratic program (QP) that imposes the constraints of the geometric CLF and geometric CBF to compute the control input. The resulting geometric CBF-CLF-QP controller trades-off between tracking stability and critical safety. We test the controller’s performance, with both time invariant and time-varying safety constraints, on simple mechanical systems such as, a 3D moving point mass, a spherical pendulum, and a 3D pendulum. Keywords—Geometric Control Lyapunov Function, Geometric Control Barrier Function, Time-varying Safety Critical Control.