This paper establishes novel results regarding the global convergence properties of a large class of consensus protocols for multi-agent systems that evolve on the $n$-dimensional unit sphere or $n$-sphere. For any connected, undirected graph and all $n\epsilon\mathbb{IN}\{{1\}}$, each protocol in said class is shown to yield almost global consensus. The feedback laws are intrinsic gradients of a Lyapunov function and include the canonical gradient descent protocol as a special case. This convergence result sheds new light on the general problem of consensus on Riemannian manifolds; the case of the n-sphere for $n\epsilon\mathbb{IN}\{{1\}}$ differs from those of the circle and SO(3) where the corresponding protocols fail to achieve almost global consensus. Finally, we derive a novel consensus protocol on SO(3) by combining two almost globally convergent protocols on the $n$sphere for $n\epsilon\mathbb{IN}\{{1,2\}}$. Theoretical and simulation results suggest that the combined protocol yields almost global consensus on SO(3).

From the flashing of fireflies to autonomous robot swarms, synchronization phenomena are ubiquitous in nature and technology. They are commonly described by the Kuramoto model that, in this paper, we generalise to networks over n-dimensional spheres. We show that, for almost all initial conditions, the sphere model converges to a set with small diameter if the model parameters satisfy a given bound. Moreover, for even n, a special case of the generalized model can achieve phase synchronization with nonidentical frequency parameters. These results contrast with the standard n = 1 Kuramoto model, which is multistable (i.e., has multiple equilibria), and converges to phase synchronization only if the frequency parameters are identical. Hence, this paper shows that the generalized network Kuramoto models for n ≥ 2 displays more coherent and predictable behavior than the standard n = 1 model, a desirable property both in flocks of animals and for robot control.

This note concerns synchronization of multi-agent systems on networks over Riemannian manifolds. The motion of each agent is governed by the gradient descent flow of a disagreement function that is a sum of (squared) distances between pairs of communicating agents. Two metrics are considered: geodesic distances and chordal distances for manifolds that are embedded in an ambient Euclidean space. We show that networks which, roughly speaking, are dominated by a large cycle yield a multistable systems if the manifold is multiply connected or contains a closed geodesic that is of locally minimum length in a space of closed curves. This result summarizes previous results on the stability of splay or twist state equilibria of the Kuramoto model on the circle and its generalization, the quantum sync model on SO(n). It also extends them to the Lohe model on U(n).

In the multi-agent systems setting, this paper addresses continuous-time distributed synchronization of columns of rotation matrices. More precisely, k specific columns shall be synchronized and only the corresponding k columns of the relative rotations between the agents are assumed to be available for the control design. When one specific column is considered, the problem is equivalent to synchronization on the (d - 1)-dimensional unit sphere and when all the columns are considered, the problem is equivalent to synchronization on SO(d). We design dynamic control laws for these synchronization problems. The control laws are based on the introduction of auxiliary variables in combination with a QR-factorization approach. The benefit of this QR-factorization approach is that we can decouple the dynamics for the k columns from the remaining d k ones. Under the control scheme, the closed loop system achieves almost global convergence to synchronization for quasi-strong interaction graph topologies. (C) 2018 Elsevier Ltd. All rights reserved.

This paper investigates the reduced attitude formation control problem for a group of rigid-body agents using feedback based on relative attitude information. Under both undirected and directed cycle graph topologies, it is shown that reversing the sign of a classic consensus protocol yields asymptotical convergence to formations whose shape depends on the parity of the group size. Specifically, in the case of even parity the reduced attitudes converge asymptotically to a pair of antipodal points and distribute equidistantly on a great circle in the case of odd parity. Moreover, when the inter-agent graph is an undirected ring, the desired formation is shown to be achieved from almost all initial states.

This paper proposes a local information based control law in order to solve the planar manipulation problem of rotating a grasped rigid object to a desired orientation using multiple mobile manipulators. We adopt a multi-agent systems theory approach and assume that: (i) the manipulators (agents) are capable of sensing the relative position to their neighbors at discrete time instances, (ii) neighboring agents may exchange information at discrete time instances, and (iii) the communication topology is connected. Control of the manipulators is carried out at a kinematic level in continuous time and utilizes inverse kinematics. The mobile platforms are assigned trajectory tracking tasks that adjust the positions of the manipulator bases in order to avoid singular arm configurations. Our main result concerns the stability of the proposed control law.