|This paper develops a motion planning algorithm which exploits symmetries
in distributed systems to reduce motion planning computation
complexity. Symmetries allow for algebraic manipulations that are
computationally costly, which normally must be carried out for each
component in a distributed system, to be related among various symmetric
components in a distributed system by a simple algebraic relationship.
This leads to a large reduction in the complexity of the overall
motion planning problem for a group of distributed mobile robotic
agents. In particular, due to the manner in which a symmetric system
is defined, the structure of the Chen–Fliess–Sussmann differential
equations has a simple relationship among various symmetric components
of a distributed system. Essentially, symmetries are defined in
a manner which preserves the Lie algebraic structure of each component.
In a system with distributed computational capability, the motion
planning computations may be distributed throughout formation
in such a way that the objectives of the formation are satisfied and
collision avoidance is guaranteed. The algorithm maintains a rigid
body formation at the beginning and end of the trajectory, as well as
possibly specified intermediate points. Due to the generally nonholonomic
nature of mobile robots, guaranteeing a rigid body formation
during the intermediate motion is impossible. However, it is possible
to bound the magnitude of the deviation from the rigid body formation
at any point along the trajectory. Simulation and experimental results
are provided to demonstrate the utility of the algorithm.