Formation Matrix and Energy-based Control of Multi-Agent Systems

Formation Matrix and Energy-based Control of Multi-Agent Systems is a paper by Martín Crespo, Sergio Junco, and Matías Nacusse, submitted to arXiv on September 3, 2026 (arXiv:2609.04158). The work introduces an Interconnected Physical Structure (IPS) that models multi-agent interactions using virtual spring-damper modules (Elementary Coupling Blocks) to achieve formation control and collision avoidance.

The controller’s novelty lies in the Formation Matrix, a graph-theoretic tool that maps agent velocities to distance changes, enabling a Port-Hamiltonian Systems (pHS) framework. This approach utilizes Bond Graphs to derive a Control-by-Interconnection (CbI) scheme, which guarantees stability and allows for a static state-feedback implementation without integrating velocity states.

Key features of the proposed method include: Collision Avoidance: Spring forces are designed to approach infinity as inter-agent distances near zero, naturally repelling agents to prevent collisions. Physical Consistency: The system is modeled as a network of energy-storing and dissipating elements, with the Formation Matrix modulating power exchange between agents and couplings. Stability: The paper demonstrates Lyapunov stability for both point-mass and rigid-body agent models, leveraging Casimir invariants derived from the bond graph structure. Strategies: Two control strategies are presented: Leader Agent Control (treating the formation as a semi-rigid body) and Position Control (using a Virtual Rigid Structure).

The paper develops an energy-based control framework for multi-agent formation control, with a particular focus on coordinating agents into a desired geometric configuration while avoiding collisions. A central element is the use of a formation matrix to encode the desired relative positions or offsets among agents, allowing the formation error to be represented compactly and used directly in the control law. The controller is built around a spring–damper analogy: agents behave as inertial bodies, inter-agent formation constraints are enforced through spring-like restorative forces, and damping is introduced to dissipate energy and drive the system toward the desired formation. Collision avoidance is incorporated as an additional energy-based interaction, so that repulsive or safety-related effects are not appended as an afterthought but are integrated into the same energetic structure.

The key contribution is the enhanced bond-graph model that encapsulates the agents, formation springs, damping, and collision-avoidance interactions in a single power-flow representation. This gives the approach a strong physical and structural interpretation: energy storage, dissipation, and exchange between agents and their environment can be tracked explicitly. Such a model is valuable because it supports energy-based stability reasoning, clarifies where forces and constraints enter the system, and makes the controller more modular and composable. In effect, the bond-graph formulation turns a potentially ad hoc combination of formation tracking and avoidance logic into a unified, physically grounded architecture.

The work matters because it addresses a common practical challenge in multi-agent systems: maintaining coordinated geometry while remaining safe in the presence of nearby agents. By treating formation control and collision avoidance through a shared energy-based lens, the paper offers a principled alternative to designs that rely on loosely coupled potential fields or separate control layers. This makes the approach especially relevant to safety-critical coordination problems, such as robotic swarms, aerial teams, and mobile manipulators, where stability, interpretability, and extensibility to more complex interactions are important.