  Traditional task assignment approaches for multi-agent motion control do not
take the possibility of collisions into account. This can lead to challenging
requirements for path planning. We derive an assignment method that not only
minimises the largest distance between an agent and its assigned destination
but also provides local constraints for guaranteed collision avoidance. To this
end, we introduce a sequential bottleneck optimisation problem and define a
notion of robustness of an optimising assignment to changes of individual
assignment costs. Conditioned on a sufficient level of robustness in relation
to the size of the agents, we construct time-varying position bounds for every
individual agent. These local constraints are a direct byproduct of the
assignment procedure and only depend on the initial agent positions, the
destinations that are to be visited, and a timing parameter. We prove that no
agent that is assigned to move to one of the target locations collides with any
other agent if all agents satisfy their local position constraints. We
demonstrate the method in a illustrative case study.
