Computing collision-free trajectories is of prime importance for safe navigation. We present an approach for computing the collision probability under Gaussian distributed motion and sensing uncertainty with the robot and static obstacle shapes approximated as ellipsoids. The collision condition is formulated as the distance between ellipsoids and unlike previous approaches we provide a method for computing the exact collision probability. Furthermore, we provide a tight upper bound that can be computed much faster during online planning. Comparison to other state-of-the-art methods is also provided. The proposed method is evaluated in simulation under varying configuration and number of obstacles.
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