  The safety-critical control of robotic systems often must account for
multiple, potentially conflicting, safety constraints. This paper proposes
novel relaxation techniques to address safety-critical control problems in the
presence of conflicting safety conditions. In particular, Control Barrier
Function (CBFs) provide a means to encode safety as constraints in a Quadratic
Program (QP), wherein multiple safety conditions yield multiple constraints.
However, the QP problem becomes infeasible when the safety conditions cannot be
simultaneously satisfied. To resolve this potential infeasibility, we introduce
a hierarchy between the safety conditions and employ an additional variable to
relax the less important safety conditions (Relaxed-CBF-QP), and formulate a
cascaded structure to achieve smaller violations of lower-priority safety
conditions (Hierarchical-CBF-QP). The proposed approach, therefore, ensures the
existence of at least one solution to the QP problem with the CBFs while
dynamically balancing enforcement of additional safety constraints.
Importantly, this paper evaluates the impact of different weighting factors in
the Hierarchical-CBF-QP and, due to the sensitivity of these weightings in the
observed behavior, proposes a method to determine the weighting factors via a
sampling-based technique. The validity of the proposed approach is demonstrated
through simulations and experiments on a quadrupedal robot navigating to a goal
through regions with different levels of danger.
