Artificial Spacetimes for Reactive Control of Resource-Limited Robots

TL;DR

Addresses how to control microrobots with minimal onboard computation by introducing artificial spacetimes, a geometry-based framework that maps reactive trajectories to light-like geodesics in a Lorentzian metric. The main method uses a two-stage, conformal composition to embed boundary information into a virtual space and then map it back to physical space, yielding composite control fields that realize behaviors such as rallying, confinement, and sorting under static fields. The paper provides analytic results for a radial control metric, demonstrates exponential convergence to targets, and validates the theory with simulations and experiments on silicon microrobots whose paths agree with geodesic predictions. This approach promises low-overhead, reusable control primitives for scalable microrobotic applications, with potential impact in drug delivery and environmental remediation.

Abstract

Field-based reactive control provides a minimalist, decentralized route to guiding robots that lack onboard computation. Such schemes are well suited to resource-limited machines like microrobots, yet implementation artifacts, limited behaviors, and the frequent lack of formal guarantees blunt adoption. Here, we address these challenges with a new geometric approach called artificial spacetimes. We show that reactive robots navigating control fields obey the same dynamics as light rays in general relativity. This surprising connection allows us to adopt techniques from relativity and optics for constructing and analyzing control fields. When implemented, artificial spacetimes guide robots around structured environments, simultaneously avoiding boundaries and executing tasks like rallying or sorting, even when the field itself is static. We augment these capabilities with formal tools for analyzing what robots will do and provide experimental validation with silicon-based microrobots. Combined, this work provides a new framework for generating composed robot behaviors with minimal overhead.

Figures (4)

• Figure 1: Motion in gradient fields. a) On top, the kinematics of a basic robot with two motors that move at speeds $V_1$ and $V_2$. The overall body speed $V_B$ is given by the average of the motor velocities, and the curvature $\kappa$ is proportional to the normalized difference in motor velocity. The robot's width is given by $2\delta$. Below, a plot of normalized intensity illustrates the control field in three dimensions. b) Plotted with the third dimension of time, robots travel along trajectories connecting cones of accessible positions. The width of the cone $w_1$ or $w_2$ depends on the velocity of the robot. c) Trajectories with an initial angle in the range $+\pi/2 \rightarrow -\pi/2$ are guaranteed to reach the target location, and trajectories outside of this range escape. d) Convergence time for a variety of initial orientations. The black lines are the predicted results for the derived exponential relationship $r(t) = r_0exp(-t\cos(\alpha_0))$.
• Figure 2: Two-stage process to building artificial spacetimes. Coordinate transformations enable metrics for behaviors to be composed with metrics for boundary avoidance. First, we map a physical space with a complex boundary to a simple virtual space by a conformal transformation. A control metrics is then implemented in virtual space to dictate the robot's behavior. For example, here robots all travel to the target location. When mapped back to physical space, we produce a composite metric that both encodes the behavior and boundary information in a single scalar field.
• Figure 3: Simulating robot trajectories. a) A maze traversal demonstration with an embedded proportional control metric that directs robots to the same target location. For visualization, the control field magnitude is plotted logarithmically. b) A turning metric acts as a sorter that directs robots to different sides of the space based on the initial trajectory. Note both robots use the same static field.
• Figure 4: Experimental demonstrations of the general relativity correspondence. a) A silicon-based microrobot with two electrokinetic motors and differential drive kinematics and an optical setup that creates spatially varying intensity fields. b) Experimental demonstrations of various control metrics implemented as intensity fields (top) compared to simulation results (bottom). In the 90$^\circ$ and double $90^\circ$ turning metric experiments, the simulated traces in red most closely match the initial pose of the experimental demonstration above.