Coordinated Path Following: A Nested Invariant Sets Approach
In many situations team work helps people achieve their goals more efficiently than working alone; there are tasks that can only be completed through collaboration. Similarly, in the field of control systems, the use of cooperative control systems is unavoidable in many occasions and a large amount of research has concentrated on cooperative control systems. A cooperative control system is a system in which a group of autonomous control systems work together to achieve a common goal. An important cooperative control problem is coordinated path following. Coordinated path following involves designing feedback controllers that make each agent's output approach and traverse a pre-assigned path while simultaneously coordinating its motion with the other agents. Coordinated motion along paths includes tasks like maintaining formations, traversing paths at a common speed and more general tasks like making the positions of some agents obey functional constraints that depend on the states of other agents.
We view the coordinated path following problem as a nested set stabilization problem. In the nested set stabilization approach, stabilization of the larger set corresponds to driving the agents to their assigned paths. This set, under suitable assumptions, is an embedded, controlled invariant, product submanifold and is called the multi-agent path following manifold. Stabilization of the nested set, contained in the multi-agent path following manifold, corresponds to meeting the coordination specification. Under appropriate assumptions, this set is also an embedded controlled invariant submanifold which we call the coordination set.
Our approach to locally solving nested set stabilization problems is based on feedback equivalence of control systems. We propose and solve two local feedback equivalence problems for nested invariant sets. The first, less restrictive, solution gives necessary and sufficient conditions for the dynamics of a system restricted to the larger submanifold and transversal to the smaller submanifold to be linear and controllable. This normal form facilitates designing controllers that locally stabilize the coordination set relative to the multi-agent path following manifold. The second, more restrictive, result additionally imposes that the transversal dynamics to the larger submanifold be linear and controllable. This result can simplify designing controllers to locally stabilize the multi-agent path following manifold. We propose sufficient conditions under which these normal forms can be used to locally solve the nested set stabilization problem.
To illustrate these ideas we consider a coordinated path following problem for a multi-agent system of dynamic unicycles. The multi-agent path following manifold is characterized for arbitrary paths. We show that each unicycle is feedback equivalent, in a neighbourhood of its assigned path, to a system whose transversal and tangential dynamics to the path following manifold are both double integrators. We provide sufficient conditions under which the coordination set is nonempty. The effectiveness of the proposed approach is demonstrated experimentally on two robots.