  Safety requirements in dynamical systems are commonly enforced with set
invariance constraints over a safe region of the state space. Control barrier
functions, which are Lyapunov-like functions for guaranteeing set invariance,
are an effective tool to enforce such constraints and guarantee safety when the
system is represented as a point in the state space. In this paper, we
introduce extent-compatible control barrier functions as a tool to enforce
safety for the system including its volume (extent) in the physical world. In
order to implement the extent-compatible control barrier functions framework, a
sum-of-squares based optimization program is proposed. Since sum-of-squares
programs can be computationally prohibitive, we additionally introduce a
sampling based method in order to retain the computational advantage of a
traditional barrier function based quadratic program controller. We prove that
the proposed sampling based controller retains the guarantee for safety.
Simulation and robotic implementation results are also provided.
