  In order for automated mobile vehicles to navigate in the real world with
minimal collision risks, it is necessary for their planning algorithms to
consider uncertainties from measurements and environmental disturbances. In
this paper, we consider analytical solutions for a conservative approximation
of the mutual probability of collision between two robotic vehicles in the
presence of such uncertainties. Therein, we present two methods, which we call
unitary scaling and principal axes rotation, for decoupling the bivariate
integral required for efficient approximation of the probability of collision
between two vehicles including orientation effects. We compare the conservatism
of these methods analytically and numerically. By closing a control loop
through a model predictive guidance scheme, we observe through Monte-Carlo
simulations that directly implementing collision avoidance constraints from the
conservative approximations remains infeasible for real-time planning. We then
propose and implement a convexification approach based on the tightened
collision constraints that significantly improves the computational efficiency
and robustness of the predictive guidance scheme.
