New Graphical Approaches To Multi-Agent Formation Control

Distributed cooperative control of multiple autonomous vehicles has a lot of potential applications in practice.Compared to vehicles performing solo missions,the coordination of a team of autonomous vehicles can achieve greater efficiency and operational capability.Recently,the distributed formation control problem,which commonly arises in cooperative control,has been widely investigated,yet several problems still remain open.In this dissertation,we develop two extended graph theoretic tools and apply them to decentralized formation controller synthesis.Compared to the literature,the formation strategies proposed by us possess significant advantages,and provide new insights in improving formation performance.The details of our work can be summarized in the following outline.1.We develop “weak rigidity theory” to answer whether the shape of a geometric graph can be uniquely determined by pairwise inner products of inter-agent displacements.Compared to distance rigidity and bearing rigidity,weak rigidity requires fewer constrained edges in the graph to determine a geometric shape in an arbitrarily dimensional space.In the literature,rigidity is often determined by examining rank of the rigidity matrix.As a result,a formation of large scale will be costly to treat.In our work,we derive a necessary and sufficient graphical condition for infinitesimal weak rigidity of planar frameworks,which is simple and easy to check.As an application of the proposed weak rigidity theory,a gradient control law and a non-gradient control law are designed for a group of single-integrator modeled agents to stabilize a desired formation shape,respectively.Under the gradient control law,we prove that an infinitesimally weakly rigid formation is locally exponentially stable.In particular,if the number of agents is one greater than the dimension of the space,a minimally infinitesimally weakly rigid formation is almost globally asymptotically stable.Moreover,collisions between agents can be avoided for almost all initial positions.In the literature of rigid formation,the sensing graph is always required to be rigid.Under the non-gradient control law based on weak rigidity theory,the sensing graph is unnecessary to be rigid for local exponential stability of the formation.2.We develop “angle rigidity theory” to study whether the shape of a planar graph can be determined by angles between segments uniquely up to translations,rotations,scalings and reflections.We prove that a planar framework is infinitesimally angle rigid if and only if it is infinitesimally bearing rigid.Moreover,a strongly nondegenerate triangulated framework can always be uniquely determined by subtended angles.The proposed angle rigidity theory is applied to the formation stabilization problem,where multiple single-integrator modeled agents cooperatively achieve an angle-constrained formation.During the formation process,the global coordinate system is unknown for each agent and wireless communications between agents are not required.Compared to distance-constrained formations and bearing-constrained formations,an angle-constrained formation has higher degrees of freedom.By utilizing this advantage,we propose a distributed control law for agents to stabilize a target formation shape with desired orientation and scale.The angle-based formation control laws proposed by us can locally exponentially stabilize the desired equilibria corresponding to the target formation.3.Similar to the distance-based formation approach,when implementing the gradient control law induced by the angle-based potential function,undesired equilibria of the formation system often exist.To avoid convergence of undesired equilibria,we propose a novel decentralized controller for single-integrator agents based on an artificial potential function,which can(i)steer agents to meet some angle constraints;(ii)preserve formation rigidity during agents' motion;(iii)be implemented in the absence of the global coordinate system.Due to the rigidity preservation property,the proposed controller ensures almost global convergence to a formation satisfying predefined angle constraints.Moreover,we give a definition of “sign” for strongly nondegenerate triangulated frameworks.By implementing the proposed control strategy,the target formation shape is guaranteed to be stable provided its sign is identical to that of the initial formation.Finally,the scenario where each agent has double-integrator dynamics is also discussed.We propose a decentralized control law,under which the agents almost globally asymptotically achieve a target formation shape while moving with a common velocity.