Asymptotically Optimal Kinodynamic Motion Planning for Self-reconfigurable Robots

Abstract

Self-reconfigurable robots are composed of many individual modules that can autonomously move to transform the shape and structure of the robot. In this paper we present a kinodynamically optimal algorithm for the following “x-axis to y-axis” reconfiguration problem: given a horizontal row of n modules, reconfigure that collection into a vertical column of n modules. The goal is to determine the sequence of movements of the modules that minimizes the movement time needed to achieve the desired reconfiguration of the modules. Prior work on self-reconfigurable (SR) robots assumed a constant velocity bound on module movement and so required time linear in n to solve this problem.

In this paper we define an abstract model that assumes unit bounds on various physical properties of modules such as shape, aspect ratio, mass, and the maximum magnitude of force that an individual module can exert. We also define concrete instances of our abstract model similar to those found in the prior literature on reconfigurable robots, including various examples where the modules are cubes that are attached and can apply forces to neighboring cubes. In one of these concrete models, the cube’s sides can contract and expand with controllable force, and in another the cubes can apply rotational torque to their neighbors. Our main result is a proof of tight Θ($\sqrt{n}$) upper and lower bounds on the movement time for the above reconguration problem for concrete instances of our abstract model.

This paper’s analysis characterizes optimal reconfiguration movements in terms of basic laws of physics relating force, mass, acceleration, distance traveled, and movement time. A key property resulting from this is that through the simultaneous application of constant-bounded forces by a system of modules, certain modules in the system can achieve velocities exceeding any constant bounds. This delays modules with the least distance to travel when reconfiguring in order to accelerate modules that have the farthest to travel. We utilize this tradeoff in our algorithm for the x-axis to y-axis problem to achieve an O($\sqrt n$) movement time.