  Many state-of-the-art methods for safety assessment and motion planning for
automated driving require estimation of the probability of collision (POC). To
estimate the POC, a shape approximation of the colliding actors and probability
density functions of the associated uncertain kinematic variables are required.
Even with such information available, the derivation of the POC is in general,
i.e., for any shape and density, only possible with Monte Carlo sampling (MCS).
Random sampling of the POC, however, is challenging as computational resources
are limited in real-world applications. We present expressions for the POC in
the presence of Gaussian uncertainties, based on multi-circular shape
approximations. In addition, we show that the proposed approach is
computationally more efficient than MCS. Lastly, we provide a method for upper
and lower bounding the estimation error for the POC induced by the used shape
approximations.
