⚡ electrical engineering

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

This paper proposes an energy-based control framework for multi-agent formation systems that utilizes a bond graph model and a novel formation matrix to map the system's dynamics to the Control-by-Interconnection scheme within the port-Hamiltonian framework, thereby achieving stable formation maintenance, collision avoidance, and leader-following capabilities validated through numerical simulations.

Original authors: Martín Crespo, Sergio Junco, Matías Nacusse

Published 2026-09-04

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CC BY 4.0

Original authors: Martín Crespo, Sergio Junco, Matías Nacusse

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a swarm of small robots moving across a flat floor, each one needing to stay a specific distance from its neighbors while avoiding collisions and following a shared path. This is the challenge of multi-agent formation control, a field that sits at the intersection of robotics, physics, and mathematics. For these machines to work together effectively, they cannot simply act as independent individuals; they must behave as a single, coordinated unit. The difficulty lies in designing a control system that tells every robot exactly how to move to maintain a precise shape, whether that shape is a line, a circle, or a complex geometric pattern, all while the group is in motion. If the robots get too close, they crash; if they drift apart, the formation breaks. Solving this requires a deep understanding of how energy flows between moving objects and how to use that energy to guide them into a desired arrangement.

In a recent study, researchers from the National University of Rosario in Argentina tackled this problem by looking at the swarm not just as a collection of machines, but as a physical structure held together by invisible forces. Instead of programming each robot with a complex set of rules to calculate its position relative to every other robot, the team designed a system that mimics a network of springs and dampers connecting the agents. In this virtual setup, every pair of robots that needs to stay at a fixed distance is linked by a "spring" that pulls them together if they drift too far apart and pushes them apart if they get too close. To prevent crashes, these virtual springs are designed with a special property: as two robots get dangerously close, the force pushing them apart grows infinitely strong, acting as an impenetrable barrier. This approach transforms the abstract problem of formation control into a tangible physical phenomenon, where the robots naturally settle into their correct positions as if they were real objects connected by elastic bands.

The core innovation of this work is a new mathematical tool the authors call the "formation matrix." While previous methods often relied on measuring angles or simple distances, this matrix captures both the distance between robots and how fast they are moving relative to one another. It acts as a translator, converting the complex movements of the entire group into a clear set of instructions. By using this matrix, the researchers could prove that the virtual spring network creates a stable system where the robots are mathematically guaranteed to reach and hold their desired formation. They demonstrated that this structure is not just a simulation trick but a fundamental geometric property of the group, meaning the control system can be simplified into a direct, static feedback loop rather than a complex, dynamic calculation. This means the robots can react instantly to changes without needing to solve difficult equations in real-time.

To test their ideas, the team ran detailed computer simulations with a group of six mobile robots modeled as rigid bodies, similar to small, omnidirectional vehicles that can move in any direction. They tested two different strategies. In the first, they designated one robot as a "leader" and used the virtual springs to guide the entire group to a specific location and orientation. The simulations showed that the group successfully formed the desired shape and moved together, with the leader pulling the rest of the swarm into place. In the second strategy, they focused on position control, where the entire group was guided along a specific path while maintaining its shape. They found that while the virtual springs alone could prevent collisions and keep the robots somewhat organized, an additional control signal was necessary to ensure the group converged perfectly to the intended path and shape. When this extra signal was added, the robots not only avoided crashing but also settled into the exact formation with zero error.

The results of these simulations confirm that treating a robot swarm as a physical system of interconnected springs and dampers is a powerful way to solve formation control problems. The researchers showed that by using their new formation matrix, they could create a controller that is both physically intuitive and mathematically robust. The system successfully handled the dual challenges of maintaining a specific shape and avoiding collisions, even when the robots were modeled with realistic physical properties like mass and rotation. The study concludes that this energy-based approach offers a reliable framework for coordinating groups of robots, providing a clear path for future applications where swarms need to move together in complex environments. The work suggests that by understanding the energy exchange between agents, engineers can design simpler, more effective control systems that allow robots to work together as naturally as a flock of birds or a school of fish.

Technical Summary: Formation Matrix and Energy-based Control of Multi-Agent Systems

Problem Statement
The paper addresses the formation control problem for a multi-agent system (MAS) consisting of $n$ mobile omnidirectional robots operating on a plane. The primary objectives are to achieve and maintain a specific geometric formation, avoid collisions between agents, and, in certain configurations, control the global position and orientation of the formation. The authors seek a control design that is physically intuitive yet theoretically rigorous, leveraging energy-based methods to handle the complex interplay of storage, flow, and dissipation inherent in multi-agent interactions.

Methodology
The proposed approach integrates graph theory, bond graph (BG) modeling, and port-Hamiltonian systems (pHS) theory within the Control by Interconnection (CbI) framework.

Interconnected Physical Structure (IPS): The authors model the MAS as a network of "Elementary Coupling Blocks" (ECBs). Each ECB consists of virtual spring-damper modules connecting pairs of agents.
- Springs: These are designed with a barrier function potential energy. They exert repulsive forces (approaching infinity) as inter-agent distances approach zero (collision avoidance) and attractive forces when distances exceed a desired reference (formation maintenance).
- Dampers: These attenuate oscillations within the network.
The Formation Matrix ( $F(r)$ ): A central contribution of the work is the introduction of the Formation Matrix. Unlike previous approaches that rely solely on angle-based or distance-based representations, $F(r)$ mathematically encapsulates both the Euclidean distances and the relative velocities between agents. It is defined using the incidence matrix of the interaction graph and the sine/cosine of the edge angles. This matrix serves as the linear mapping between agent velocities and the rates of change of inter-agent distances.
Modeling:
- The agents are modeled first as point masses and later extended to rigid bodies with omnidirectional wheels.
- The IPS is modeled using Bond Graphs, which capture the energy exchange and causal relationships. This BG model is translated into a port-Hamiltonian system representation.
Control Design (CbI):
- The controller is designed using the Control by Interconnection (CbI) technique. The ECBs act as the controller, and the agents act as the plant.
- The interconnection generates Casimir functions, which are conserved quantities determined by the system's geometry (the interconnection structure). Specifically, the function $K(r, \tilde{z}) = F(r)r - \tilde{z}$ is identified as a Casimir function, relating the plant states (positions) to the controller states (spring deformations).
- This relationship allows the dynamic controller to be reduced to a static state-feedback law, eliminating the need to integrate velocity signals and simplifying the initial condition requirements.
Control Strategies:
- Leader Agent Control: A leader node and a specific edge are selected to define the global position and orientation of the formation. A PI controller regulates the leader's position and the edge's angle. Stability is proven via Lyapunov analysis, requiring the interaction graph to be minimally rigid to ensure the transformation matrix is full rank.
- Position-Based Control: A Virtual Reference Structure (VRS) is defined, and agents track corresponding Virtual Reference Points (VRPs). An additional control signal $\omega$ (based on a sliding mode-like term) is introduced to ensure asymptotic stability of the equilibrium point, overcoming limitations where the distance-based control alone might not guarantee convergence to the exact desired topology.

Key Contributions

Formation Matrix: The introduction of a matrix that simultaneously represents inter-agent distances and relative velocities, facilitating a unified energy-based control formulation.
Static Feedback via Casimir Functions: The demonstration that the specific geometry of the IPS and the holonomic constraints lead to Casimir invariants. This allows the dynamic CbI controller to be implemented as a static state-feedback law, avoiding the complexity of dynamic controllers and arbitrary initial conditions for the controller states.
Unified Framework: The integration of Bond Graphs with pHS and CbI provides a structured, physically consistent architecture for formation control, explicitly linking the physical interpretation (springs/dampers) with the mathematical control structure.
Rigid Body Extension: The methodology is shown to be applicable to rigid-body agents with omnidirectional wheels, not just point masses, by decoupling translational and rotational dynamics.

Results
Numerical simulations were conducted with six agents modeled as rigid bodies in two scenarios:

Leader Agent Control: Using a minimally rigid graph ( $m=9$ ), the system successfully stabilized the formation at a desired position and orientation. The agents converged to the desired distances, and the formation moved along a trajectory while maintaining its shape.
Position Control: Using a fully rigid graph ( $m=15$ ), the system demonstrated that without the auxiliary control signal $\omega$ , the agents might stabilize at a local equilibrium that does not match the desired global topology. With $\omega$ active, the agents achieved asymptotic stability, converging to the desired formation and trajectory while maintaining collision-free distances (greater than 0.9m).

Significance and Claims
The paper claims that its primary significance lies in providing a physically inspired yet theoretically supported approach to formation control. By leveraging the Formation Matrix and the CbI scheme, the authors achieve a control architecture where:

The energy-based design naturally handles collision avoidance and formation maintenance.
The geometric properties of the system (via the Formation Matrix) inherently generate the necessary invariants (Casimir functions) to simplify the control law to a static feedback form.
The approach offers a more comprehensive and physically consistent alternative to methods that decouple energy models from interaction geometry or rely solely on angle-based representations.

The authors emphasize that while the Leader Agent Control requires a minimally rigid graph to ensure stability and full rank of the transformation matrix, the Position Control strategy offers robustness in topology maintenance through the use of a Virtual Reference Structure and an additional stabilizing signal. The work concludes that the Bond Graph technique was instrumental in deriving these results, particularly in revealing the causal relationships that lead to the static feedback interpretation.

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