Consensus Algorithm for Multiagents Evolving on Smooth Riemannian Manifolds

In this paper, consensus algorithms are proposed for second-order dynamics for multiagents having the structure of a simple mechanical control systems evolving on a smooth connected Riemannian manifold without boundary which is geodesically complete. The error between the configurations of two neighboring agents is measured using a tracking error function, defined on a general smooth Riemannian manifold. The velocities of two agents are compared using a transport map, which converts the tangent velocity of an agent to the tangent space of its neighboring agent. The compatibility condition between a tracking error function and a transport map is discussed. A compatible tracking error and transport map based on the geodesic distance and parallel transport is defined along with logarithm function. A generalized version of Lyapunov theory applicable for systems evolving on a smooth manifold is used for proving the stability of the consensus equilibrium. The position and velocity consensus algorithms are proposed for undirected and directed communication graphs. The Lyapunov proof is provided for both connected tree and connected communication graph topology. The presented position and velocity consensus is extended so that all the agents track a predefined reference trajectory. An application of the proposed consensus algorithm is discussed, where agents evolving on [Formula: see text], a two dimensional sphere defined as [Formula: see text] is considered. The tracking error function and a transport map is discussed for this case and the consensus algorithm is validated by numerical simulations.