We consider the problem of optimal navigation control design for navigation
on off-road terrain. We use traversability measure to characterize the degree
of difficulty of navigation on the off-road terrain. The traversability measure
captures the property of terrain essential for navigation, such as elevation
map, terrain roughness, slope, and terrain texture. The terrain with the
presence or absence of obstacles becomes a particular case of the proposed
traversability measure. We provide a convex formulation to the off-road
navigation problem by lifting the problem to the density space using the linear
Perron-Frobenius (P-F) operator. The convex formulation leads to an
infinite-dimensional optimal navigation problem for control synthesis. The
finite-dimensional approximation of the infinite-dimensional convex problem is
constructed using data. We use a computational framework involving the Koopman
operator and the duality between the Koopman and P-F operator for the
data-driven approximation. This makes our proposed approach data-driven and can
be applied in cases where an explicit system model is unavailable. Finally, we
demonstrate the application of the developed framework for the navigation of
vehicle dynamics with Dubin's car model