Consensus and Synchronization: From the Euclidean Space Circle to Lie Groups

Connecting natural phenomena with advanced engineering, this work redefines coordination and synchronization by moving beyond basic Euclidean models into the realm of Lie groups with bi-invariant metrics. It transforms the harmony of flocking birds and schooling fish into insights for robotics, autonomous vehicles, and spacecraft. Coordination, consensus, and synchronization are found in diverse natural phenomena and engineering applications. Examples are flocking birds, illuminating fireflies, a school of fish, and distributed control and sensing. The simplest of such problems are set in the Euclidean spaces and the circle. Consensus and Synchronization: From the Euclidean Space and the Circle to Lie Groups moves beyond this domain to the more sophisticated setting of Lie groups with bi-invariant metrics and extends the mathematical theories of consensus and synchronization for generic scenarios. This is relevant to applications such as robotics, autonomous vehicles, and spacecraft.