  We consider stochastic model predictive control of a multi-agent systems with
constraints on the probabilities of inter-agent collisions. We first study a
sample-based approximation of the collision probabilities and use this
approximation to formulate constraints for the stochastic control problem. This
approximation will converge as the number of samples goes to infinity, however,
the complexity of the resulting control problem is so high that this approach
proves unsuitable for control under real-time requirements. To alleviate the
computational burden we propose a second approach that uses probabilistic
bounds to determine regions with increased probability of presence for each
agent and formulate constraints for the control problem that guarantee that
these regions will not overlap. We prove that the resulting problem is
conservative for the original problem with probabilistic constraints, ie. every
control strategy that is feasible under our new constraints will automatically
be feasible for the original problem. Furthermore we show in simulations in a
UAV path planning scenario that our proposed approach grants significantly
better run-time performance compared to a controller with the sample-based
approximation with only a small degree of sub-optimality resulting from the
conservativeness of our new approach.
