Swarm and Multi-Agent Robotics: Graph Laplacian Consensus, Formation Control, and Decentralized Learning
Abstract
This educational module presents a comprehensive academic introduction to Swarm and Multi-Agent Robotics within the Prep4Uni Robotics and AI curriculum. Core Theoretical and Engineering Foundations: Network Representation & Algebraic Graph Theory: Modeling multi-agent systems via time-varying graphs G = (V, E), topological neighborhoods, degree matrices, and adjacency matrices. Dynamic Consensus Protocols: Discrete and continuous consensus update laws, row-stochastic and doubly stochastic coupling weights, and asymptotic convergence to average consensus states. Graph Laplacian Spectra & Network Stability: Formulation of the Laplacian matrix L = D - A, stability analysis of continuous dynamics x_dot = -L x, and the physical role of the Fiedler eigenvalue (algebraic connectivity lambda_2) in governing convergence rates and link failure resilience. Formation Control & Leader-Follower Architectures: Relative displacement consensus laws, pinning control gains for trajectory-following swarms, and artificial potential field repulsive vectors for decentralized collision avoidance. Distributed Convex Optimization: Collaborative parameter estimation via the Distributed Subgradient Method (DSM), minimizing aggregate network cost functions without central coordinators. Contemporary 2026 AI-Era Computational Paradigms: Multi-Agent Reinforcement Learning (MARL) via Centralized Training with Decentralized Execution (CTDE / MAPPO / QMIX), and Decentralized Physics-Informed Neural Networks (Dec-PINNs) with consensus regularization for mapping continuous environmental spatiotemporal fields. Pedagogical components include 10 conceptual review questions, 12 thought-provoking analytical problems, and 12 fully worked step-by-step numerical engineering derivations.
