Multi-agent robotic systems have attracted growing interest in recent years, with applications ranging from coordinated drone shows to autonomous operations on land, in the air, and on water. At the planning level, robot motion is often modeled using double-integrator dynamics (with low-level controllers handling the physical dynamics during execution) with a minimum-energy objective subject to collision-avoidance constraints and limits on position, velocity, and acceleration. The complexity of the problem resides in the constraints, which couple the trajectories and make the optimization problem nonconvex. Existing numerical approaches typically discretize and optimize entire trajectories, which becomes expensive for large fleets. We exploit the structure of optimal trajectories by organizing optimization around contacts, at which robots reach the minimum allowed separation. We formulate a coupled algebraic system in contact variables and reconstruct trajectories analytically between contacts. Our algorithm, sequential contact insertion (SCI), starts from unconstrained trajectories, inserts contacts at separation violations, and jointly adjusts their times and states. Numerical experiments demonstrate planning for one million robots in minutes and thousands of robots in fractions of a second. On problems feasible for a fixed-grid sequential convex programming baseline, SCI achieves median speedups of several orders of magnitude at matching solution accuracy.

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