A complete analytical characterization of optimal collision-free motion for two robots

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.

To enable an efficient interaction of non-communicating mobile robots in collision avoidance scenarios, we present a novel combined trajectory planning and prediction algorithm. Inverse optimal control is used to estimate unknown goal states of all robots based on observed past trajectories. Each robot also takes the perspective of other robots in considering self-prediction and solves a joint prediction problem using the estimated goal states. The resulting predictions are then considered for planning. Simulation results of scenarios with 2-8 robots show that the median of the durations until all vehicles reach their goals is 9.8 % faster compared to planning with constant acceleration based estimated goal states. Moreover, the proposed approach never leads to the solver being unable to find a solution to the planning or prediction problem.

Generating time-optimal, collision-free trajectories for autonomous mobile robots involves a fundamental trade-off between guaranteeing safety and managing computational complexity. State-of-the-art approaches formulate spline-based motion planning as a single Optimal Control Problem (OCP) but often suffer from high computational cost because they include separating hyperplane parameters as decision variables to enforce continuous collision avoidance. This paper presents a novel method that alleviates this bottleneck by decoupling the determination of separating hyperplanes from the OCP. By treating the separation theorem as an independent classification problem solvable via a linear system or quadratic program, the proposed method eliminates hyperplane parameters from the optimisation variables, effectively transforming non-convex constraints into linear ones. Experimental validation demonstrates that this decoupled approach reduces trajectory computation times up to almost 60% compared to fully coupled methods in obstacle-rich environments, while maintaining rigorous continuous safety guarantees.