  Multi-robot systems offer enhanced capability over their monolithic
counterparts, but they come at a cost of increased complexity in coordination.
To reduce complexity and to make the problem tractable, multi-robot motion
planning (MRMP) methods in the literature adopt de-coupled approaches that
sacrifice either optimality or dynamic feasibility. In this paper, we present a
convexification method, namely "parabolic relaxation", to generate optimal and
dynamically feasible trajectories for MRMP in the coupled joint-space of all
robots. We leverage upon the proposed relaxation to tackle the problem
complexity and to attain computational tractability for planning over one
hundred robots in extremely clustered environments. We take a multi-stage
optimization approach that consists of i) mathematically formulating MRMP as a
non-convex optimization, ii) lifting the problem into a higher dimensional
space, iii) convexifying the problem through the proposed computationally
efficient parabolic relaxation, and iv) penalizing with iterative search to
ensure feasibility and recovery of feasible and near-optimal solutions to the
original problem. Our numerical experiments demonstrate that the proposed
approach is capable of generating optimal and dynamically feasible trajectories
for challenging motion planning problems with higher success rate than the
state-of-the-art, yet remain computationally tractable for over one hundred
robots in a highly dense environment.
