This paper studies the torque applied by a large number of particles on a long aspect-ratio rod. The particles are all pushed in the same direction by a global signal. We calculate the force and torque generated by three canonical position distributions of a swarm: uniform, triangular, and normal. The model shows that for a pivoted rod the uniform distribution produces the maximum torque for small swarm standard deviations, but the normal distribution maximizes torque for large standard deviations. In the simulation, we use these results to design proportional-derivative controllers to orient rigid objects. We conclude showing the experiments with up to 97 hardware robots to evaluate our theory in practice.

We motivate, visualize and demonstrate recent work for minimizing the total execution time of a coordinated, parallel motion plan for a swarm of N robots in the absence of obstacles. Under relatively mild assumptions on the separability of robots, the algorithm achieves constant stretch: If all robots want to move at most d units from their respective starting positions, then the total duration of the overall schedule (and hence the distance traveled by each robot) is O(d) steps; this implies constant-factor approximation for the optimization problem. Also mentioned is an NP-hardness result for finding an optimal schedule, even in the case in which robot positions are restricted to a regular grid. On the other hand, we show that for densely packed disks that cannot be well separated, a stretch factor Omega(N^{1/4}) is required in the worst case; we establish an achievable stretch factor of O(N^{1/2}) even in this case. We also sketch geometric difficulties of computing optimal trajectories, even for just two unit disks.

Microrobotics has the potential to revolutionize many applications including targeted material delivery, assembly, and surgery. The same properties that promise breakthrough solutions-small size and large populations-present unique challenges for controlling motion. Robotic manipulation usually assumes intelligent agents, not particle systems manipulated by a global signal. To identify the key parameters for particle manipulation, we used a collection of online games in which players steer swarms of up to 500 particles to complete manipulation challenges. We recorded statistics from more than 10 000 players. Inspired by techniques in which human operators performed well, we investigate controllers that use only the mean and variance of the swarm. We prove that mean position is controllable and provide conditions under which variance is controllable. We next derive automatic controllers for these and a hysteresis-based switching control to regulate the first two moments of the particle distribution. Finally, we employ these controllers as primitives for an object manipulation task and implement all controllers on 100 kilobots controlled by the direction of a global light source.