Optimality, reduction and collective motion

Abstract

The planar self-steering particle model of agents in a collective gives rise to dynamics on the N-fold direct product of SE(2), the rigid motion group in the plane. Assuming a connected, undirected graph of interaction between agents, we pose a family of symmetric optimal control problems with a coupling parameter capturing the strength of interactions. The Hamiltonian system associated with the necessary conditions for optimality is reducible to a Lie-Poisson dynamical system possessing interesting structure. In particular, the strong coupling limit reveals additional (hidden) symmetry, beyond the manifest one used in reduction: this enables explicit integration of the dynamics, and demonstrates the presence of a 'master clock' that governs all agents to steer identically. For finite coupling strength, we show that special solutions exist with steering controls proportional across the collective. These results suggest that optimality principles may provide a framework for understanding imitative behaviours observed in certain animal aggregations.

Symmetry Reduction in Optimal Control of Multiagent Systems on Lie Groups

“…Along the proof for reduced optimality conditions we will employ Pontryagin's maximum principle for left invariantcontrol systems (see Theorem 2.1 in [15] and [14]).…”

“…Recent studies include dynamic programming [22], interconnected systems [11] and soft robotics [6]. While most of the applications of symmetry reduction provided in the literature focus on the single agent situation, only a few works studied the relation between multi-agent systems and symmetry reduction (see for instance the early work on the topic [15]), in this work we introduce a new application in optimal control, by extending the reduction technique to multi-agent systems modeled by left-invariant control systems with a decentralized communication topology determined by an undirected graph, i.e., the information between the agents is only shared between nearest neighbors. Such a method is proposed from a Lagrangian and a Hamiltonian point of view.…”

Energy-Time Optimal Control of Wheeled Mobile Robots

“…Wheeled mobile robots have been extensively studied motivated by their potential applications in floor cleaning [6], personal transportation [7], lawn mowing [8], and have been used as rudimentary models for cars and trailers [9]. They have also served as particle models for formation control of multi-agent systems [10]. Klancar et al [11] provide a concise introduction to wheeled mobile robots including aspects of modeling, estimation, path planning and control.…”

Optimal Control of Left-Invariant Multi-Agent Systems with Asymmetric Formation Constraints

“…As an application we study a minimum-energy problem for three unicycles and characterize the exact solution for the control law of one of the agents. To the best of authors' knowledge, this is one of the first attempts where a formation constraint for a coordination motion of unicycles is expressed in absolute configurations on the Lie group SE(2) (a different approach for relative configurations has been studied in [17] and [18]), allowing to explore more the Lie group framework in formation problems with non-compact configuration spaces. This approach can be seen as a complement to the related literature for formation problems on Lie groups is the case of agents evolving on the Lie group of rotations SO (3) where the constraint is written as the geodesic distance between two points (since SO(3) is compact and therefore a complete manifold such a distance is well defined) and the use of Rodrigues' formula allows the use of trackable formation constraints.…”

“…This approach can be seen as a complement to the related literature for formation problems on Lie groups is the case of agents evolving on the Lie group of rotations SO (3) where the constraint is written as the geodesic distance between two points (since SO(3) is compact and therefore a complete manifold such a distance is well defined) and the use of Rodrigues' formula allows the use of trackable formation constraints. Moreover, the optimization problem considered in [17] and [18] is based on minimizing the strength of interactions by using a coupling parameter among particles while our optimization problem is a minimum-energy problem avoiding collisions among agents by using artificial potentials created to simulate a fictitious repulsion among them in the configuration space inspired by the approach given in [19] for robotic manipulators.…”