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.
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