Multirobot Mixing via Braid Groups

This paper presents a framework for multirobot motion planning that characterizes pairwise interactions between agents, e.g., crossing paths while en route to a destination. Mixing is identified as the number of pairwise crossings exhibited by the robot motion. Mixing patterns specified through elements of the braid group provide sufficient level of abstraction to describe interactions without concern for the geometry of the motion. Controllers are constructed explicitly reasoning about the spatial collocation of robots to execute mixing patterns, achieving rich motion in a shared space, e.g., to exchange inter-robot information. We do not focus on achieving a particular pattern, but rather on the problem of being able to execute a whole class of them (e.g., all patterns with at most $M$ pairwise interactions). The result is a hybrid system driven by symbolic inputs that are mapped onto paths, realizing desired mixing levels. Controllers derived from optimal control provide theoretical bounds on the achievable amount of mixing, satisfaction of spatio-temporal constraints, and collision-free trajectories. Designs are carried to implementation on real robot platforms.