A research team has described a way to plan collision-free motion for fleets as large as one million robots. The method, called sequential contact insertion (SCI), focuses on moments when two robots reach their minimum allowed separation, rather than optimizing every robot’s full route at once. In numerical experiments, the researchers report planning for one million robots in minutes, and for thousands in fractions of a second.
The work appeared on arXiv on November 2, 2026. It addresses a planning problem where each robot’s movement is linked to the others: their paths must avoid collisions while also respecting limits on position, speed and acceleration. The paper models motion using double-integrator dynamics, a simplified description of how acceleration changes velocity and position. It also assumes a minimum-energy objective; low-level controllers are left to handle the robots’ physical dynamics during execution.
How does the method work?
Many existing numerical methods divide the whole journey into time steps and optimize the resulting trajectories. That can become expensive as the fleet grows. SCI instead begins with unconstrained trajectories, identifies where robots violate the required separation, and inserts a contact at each violation. It then adjusts the contacts’ times and states together. Between contacts, the trajectories can be reconstructed analytically, rather than optimized as a long sequence of time steps.
The paper compares SCI with a fixed-grid sequential convex programming method, a baseline that also tackles the motion constraints through repeated optimization. On problems that the baseline could solve, SCI achieved median speedups of several orders of magnitude while matching its solution accuracy, according to the paper. That comparison is limited to those shared, solvable problems; the abstract does not give a single speedup figure or say that the method was tested with physical robots.
What the results show
The results suggest that organizing calculations around separation events can make very large motion-planning problems much faster to solve. The examples connect to settings such as coordinated drone shows and autonomous operations on land, in the air and on water. But the reported million-robot result is a numerical planning result, not evidence that a million-robot fleet has been deployed or controlled in the real world. The paper describes an approach to the planning stage, while execution still depends on low-level controllers.
Why it matters
For teams planning routes for large robot fleets, faster calculations could make it practical to evaluate more complex coordinated movements. The paper reports numerical planning results, but does not establish how the method performs on physical fleets or what hardware and operating conditions would be needed.
