  We introduce a novel approach for safe control design based on the density
function. A control density function (CDF) is introduced to synthesize a safe
controller for a nonlinear dynamic system. The CDF can be viewed as a dual to
the control barrier function (CBF), a popular approach used for safe control
design. While the safety certificate using the barrier function is based on the
notion of invariance, the dual certificate involving the density function has a
physical interpretation of occupancy. This occupancy-based physical
interpretation is instrumental in providing an analytical construction of
density function used for safe control synthesis. The safe control design
problem is formulated using the density function as a quadratic programming
(QP) problem. In contrast to the QP proposed for control synthesis using CBF,
the proposed CDF-based QP can combine both the safety and convergence
conditions to target state into single constraints. Further, we consider
robustness against uncertainty in system dynamics and the initial condition and
provide theoretical results for robust navigation using the CDF. Finally, we
present simulation results for safe navigation with single integrator and
double-gyre fluid flow-field examples, followed by robust navigation using the
bicycle model and autonomous lane-keeping examples.
