  We propose a hybrid feedback control law that guarantees both safety and
asymptotic stability for a class of Lagrangian systems in environments with
obstacles. Rather than performing trajectory planning and implementing a
trajectory-tracking feedback control law, our approach requires a sequence of
locations in the environment (a path plan) and an abstraction of the
obstacle-free space. The problem of following a path plan is then interpreted
as a sequence of reach-avoid problems: the system is required to consecutively
reach each location of the path plan while staying within safe regions.
Obstacle-free ellipsoids are used as a way of defining such safe regions, each
of which encloses two consecutive locations. Feasible Control Barrier Functions
(CBFs) are created directly from geometric constraints, the ellipsoids,
ensuring forward-invariance, and therefore safety. Reachability to each
location is guaranteed by asymptotically stabilizing Control Lyapunov Functions
(CLFs). Both CBFs and CLFs are then encoded into quadratic programs (QPs)
without the need of relaxation variables. Furthermore, we also propose a
switching mechanism that guarantees the control law is correct and well-defined
even when transitioning between QPs. Simulations show the effectiveness of the
proposed approach in two complex scenarios.
