Cooperative shape control of particle formations

The shape of a robot formation is key to performance in cooperative missions. Existing approaches of shape control require the formation dynamics be expressed in a unified coordinate frame. We present a shape theoretical approach which allows each robot, modeled as a particle, freedom to choose a coordinate frame to describe observed formation shape dynamics. Without knowing such coordinate frames of other robots, each robot is able to perform cooperative control. This is made possible by a class of gauge covariant cooperative control laws to control position, orientation, and shape of the formation. The formation dynamics are modeled as controlled Lagrangian systems on Jacobi shape space, we use measurements of shape variables and angular velocity as feedback to control the shape and to stop the rotation of the entire formation. Using Lyapunov functions, we prove that controlled dynamics converge to an invariant set where desired shape is achieved. Our method allows the robots to use direct observations from their onboard sensors to stabilize the formation.