A Distributed Linear Quadratic Discrete-Time Game Approach to Formation Control with Collision Avoidance

TL;DR

This paper introduces a distributed formation control method with collision avoidance using a linear quadratic game framework, solving state-dependent Riccati equations and iteratively updating control inputs through simple matrix-vector operations.

Contribution

It extends distributed LQDTG methods to include collision avoidance by incorporating penalty terms and solving state-dependent Riccati equations in a distributed manner.

Findings

Distributed implementation for formation control with collision avoidance demonstrated.

Collision avoidance incorporated via penalty terms on network edges.

Numerical example validates the proposed receding horizon approach.

Abstract

Formation control problems can be expressed as linear quadratic discrete-time games (LQDTG) for which Nash equilibrium solutions are sought. However, solving such problems requires solving coupled Riccati equations, which cannot be done in a distributed manner. A recent study showed that a distributed implementation is possible for a consensus problem when fictitious agents are associated with edges in the network graph rather than nodes. This paper proposes an extension of this approach to formation control with collision avoidance, where collision is precluded by including appropriate penalty terms on the edges. To address the problem, a state-dependent Riccati equation needs to be solved since the collision avoidance term in the cost function leads to a state-dependent weight matrix. This solution provides relative control inputs associated with the edges of the network graph. These relative inputs then need to be mapped to the physical control inputs applied at the nodes; this can be done in a distributed manner by iterating over a gradient descent search between neighbors in each sampling interval. Unlike inter-sample iteration frequently used in distributed MPC, only a matrix-vector multiplication is needed for each iteration step here, instead of an optimization problem to be solved. This approach can be implemented in a receding horizon manner, this is demonstrated through a numerical example.