  This paper presents a new approach for guaranteed safety subject to input
constraints (e.g., actuator limits) using a composition of multiple control
barrier functions (CBFs). First, we present a method for constructing a single
CBF from multiple CBFs, which can have different relative degrees. This
construction relies on a soft minimum function and yields a CBF whose
$0$-superlevel set is a subset of the union of the $0$-superlevel sets of all
the CBFs used in the construction. Next, we extend the approach to systems with
input constraints. Specifically, we introduce control dynamics that allow us to
express the input constraints as CBFs in the closed-loop state (i.e., the state
of the system and the controller). The CBFs constructed from input constraints
do not have the same relative degree as the safety constraints. Thus, the
composite soft-minimum CBF construction is used to combine the input-constraint
CBFs with the safety-constraint CBFs. Finally, we present a feasible
real-time-optimization control that guarantees that the state remains in the
$0$-superlevel set of the composite soft-minimum CBF. We demonstrate these
approaches on a nonholonomic ground robot example.
