  Guaranteeing safety for robotic and autonomous systems in real-world
environments is a challenging task that requires the mitigation of stochastic
uncertainties. Control barrier functions have, in recent years, been widely
used for enforcing safety related set-theoretic properties, such as forward
invariance and reachability, of nonlinear dynamical systems. In this paper, we
extend this rich framework to nonlinear discrete-time systems subject to
stochastic uncertainty and propose a framework for assuring risk-sensitive
safety in terms of coherent risk measures. To this end, we introduce risk
control barrier functions (RCBFs), which are compositions of barrier functions
and dynamic, coherent risk measures. We show that the existence of such barrier
functions implies invariance in a coherent risk sense. Furthermore, we
formulate conditions based on finite-time RCBFs to guarantee finite-time
reachability to a desired set in the coherent risk. Conditions for
risk-sensitive safety and finite-time reachability of sets composed of Boolean
compositions of multiple RCBF are also formulated. We show the efficacy of the
proposed method through its application to a cart-pole system in a
safety-critical scenario.
