Collective Periodic Motion Coordination
This chapter introduces a collective periodic motion coordination problem. Coordinated periodic motions play an important role in applications involving multi-agent networks with repetitive movements such as cooperative patrol, mapping, sampling, or surveillance. We introduce two types of algorithms. For the first type, we introduce Cartesian coordinate coupling to existing distributed consensus algorithms for respectively, single-integrator dynamics and double-integrator dynamics, to generate collective motions, namely, rendezvous, circular patterns, and logarithmic spiral patterns in the three-dimensional space. It is shown that the interaction graph and the value of the Euler angle in the case of single-integrator dynamics and the interaction graph, the damping gain, and the value of the Euler angle in the case of double-integrator dynamics affect the resulting collective motions. We show that when the nonsymmetric Laplacian matrix has certain properties, the damping gain is above a certain bound in the case of double-integrator dynamics, and the Euler angle is below, equal, or above a critical value, the agents will eventually rendezvous, move on circular orbits, or follow logarithmic spiral curves lying on a plane normal to the Euler axis. For the second type, we introduce coupled second-order linear harmonic oscillators with local interaction to generate synchronized oscillatory motions. We analyze convergence conditions under, respectively, directed fixed and switching interaction graphs. It is shown that the coupled harmonic oscillators can be synchronized under mild network connectivity conditions. The theoretical result is also applied to synchronized motion coordination in multi-agent systems as a proof of concept.
KeywordsCircular Orbit Euler Angle Negative Real Part Nonzero Eigenvalue Interaction Graph
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