Consensus Control of Multi-Agent Rigid Body Systems using Rotation Matrices and Exponential Coordinates

Abstract

This dissertation addresses the 6-DOF consensus control problem of multi-agent rigid body systems. The consensus protocols are designed using two different attitude representations: rotation matrices and principal rotation vectors (exponential coordinates). The control objective is stabilizing the system of rigid bodies to a configuration where all the rigid bodies have a common attitude and prescribed relative positions with velocity synchronization. In this work, for the most part, a fixed and undirected communication topology is considered for the consensus control design and analysis. However, the stability of consensus in multi-agent systems with periodically switched communication topology is also studied using Floquet theory. In addition, the application of Floquet theory in analyzing cases such as switched systems with joint connectivity, unstable subsystems (antagonistic interactions), and nonlinear systems is also studied. As the first methodology for consensus control of multi-agent rigid body systems, the configurations of the rigid bodies are described in terms of the exponential coordinates associated with the Lie groups SO(3) and SE(3). Moreover, the stability of the consensus in multi-agent rigid body systems with periodically switched communication topology is studied using Floquet theory and linearizing the closed-loop systems. The second type of protocols for consensus control of a multi-agent system of $N$ heterogeneous rigid bodies are proposed in the framework of the tangent bundles TSO(3) and TSE(3) associated with Lie groups SO(3) and SE(3), respectively. The feedback control design uses the rotation matrix as opposed to various attitude parameterizations. Almost global asymptotic stability of the consensus subspace is demonstrated using an extension of the Morse-Lyapunov (M-L) approach. Also, the presence of unstable non-consensus equilibria in the closed-loop dynamics is discussed and shown in illustrative examples. A new strategy for full pose and velocity consensus control of multi-agent rigid body systems in the presence of communication delays is presented in this dissertation. Specifically, consensus protocols are proposed on the Banach manifold associated with the tangent bundle TSE(3)^N. The stability argument is strengthened from that used in prior studies by using an extension of Morse-Lyapunov-Krasovskii (M-L-K) approach, and sufficient conditions are derived to achieve almost global asymptotic stability of the consensus subspace. This work also investigates the finite-time pose consensus control of multi-agent rigid body systems using Morse-Lyapunov analysis in the framework of the tangent bundle TSE(3) associated with SE(3). Almost global finite-time stability of the consensus subspace in the nonlinear state space is demonstrated. As another finite-time consensus control problem, the prescribed-time consensus of multi-agent rigid body systems using exponential coordinates is also studied. Specifically, the control objective is to stabilize the relative pose configurations with velocity synchronization of a multi-agent rigid body system in a user-defined convergence time. In this dissertation, the consensus control of multi-agent rigid body spacecraft in orbital relative motion is explored using two approaches. In the first approach, a proportional-derivative (PD) consensus control method, an extension of the Morse-Lyapunov analysis in the framework of the tangent bundle TSE(3) associated with Lie group SE(3) is used. In the second approach, a proportional-integral-derivative (PID) consensus control protocol is introduced where the configurations of the rigid bodies are described in terms of the exponential coordinates associated with the Lie group SE(3). In general, the rigid-body attitude control problems are formulated in terms of full attitude configurations. However, in cases involving control objectives stated in terms of pointing the rigid body, reduced-attitude configurations defined in S^2 are exploited. In this dissertation, distributed control algorithms are proposed for asymptotically stable synchronization and balancing of a multi-agent rigid body reduced-attitude system using Lyapunov analysis. The control objective in the balancing problem is the maximization of the minimum relative angular distance between each pair of rigid body reduced attitudes.