Reference : Towards coordination algorithms on compact Lie groups
Dissertations and theses : Master of advanced studies dissertation
Engineering, computing & technology : Multidisciplinary, general & others
Towards coordination algorithms on compact Lie groups
Sarlette, Alain mailto [Université de Liège - ULiège > Dép. d'électric., électron. et informat. (Inst.Montefiore) > Systèmes et modélisation >]
Université de Liège
DEA en Sciences Appliquées, spécialité systèmes & contrôle
Sepulchre, Rodolphe mailto
Wehenkel, Louis mailto
Rochus, Pierre mailto
Lecomte, Pierre mailto
Absil, Pierre-Antoine mailto
[en] The present work considers the design of control algorithms to coordinate a swarm of identical, autonomous, cooperating agents that evolve on compact Lie groups. The objective is that the agents reach a so-called consensus state without using any external reference. In the same line of thought, a leader-follower approach where ’follower’ agents would track one ’leader’ agent is excluded, in favor of a fully cooperative strategy. Moreover, the presence of communication links between agents is explicitly restricted, leading to undirected, directed and/or time-varying communication structures.
Two levels of complexity are considered for the models of the agents. First, they are modeled as simple integrators on Lie groups. This setting is meaningful in a trajectory-planning context for swarms of mechanical vehicles, or to solve algorithmic problems involving multiple agent coordination. In a second step, the model of Newtonian mechanics is used for Lie group solids, which correspond to the abstraction of the Euler laws for the rotation of a rigid body to general Lie groups. This setting is relevant for the actual control of mechanical vehicles through torques and forces.
As a common starting point, the consensus problem is formulated in terms of the extrema of a cost function. This cost function is linked to a specific centroid definition on manifolds, which is referred to in this work as the induced arithmetic mean, that is easily computable in closed form and hence may be of independent interest. Using the integrator model, this naturally leads to efficient gradient algorithms to synchronize (i.e. maximizing the consensus) or balance (i.e. minimizing the consensus) the agents; the latter however can only implement the corresponding control laws if the communication graph is fixed and undirected. For directed and/or varying communication graphs, a convenient adaptation of the gradient algorithms is obtained using auxiliary estimator variables that evolve in an embedding vector space. An extension of these results to homogeneous manifolds is briefly discussed. For the mechanical model, the coordination objective is specialized to coordinated motion (i.e. moving such that the relative positions of the agents are conserved) and synchronization (i.e. having all the agents at the same position on the Lie group). Control laws are derived using two classical approaches of nonlinear control - tracking and energy shaping. They are both based on the ideas developed in the first part.
For the sake of easier understanding and given its practical importance as representing orientations of rigid bodies in 3-dimensional space, the group SO(3) (or more generally SO(n)) is used as a running example throughout this report. Other examples are the circle SO(2) and, for the extension to homogeneous manifolds, the Grassmann manifolds Grass(p, n).
As this report is written in the middle of research activities, it closes with several future research directions that can be explored in the continuity of the present work.

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