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

Martín Crespo, Sergio Junco, Matías Nacusse

arXiv:2609.04158v1eess.SYcs.ROcs.SY

TL;DR

The paper addresses formation control for mobile robot teams that must regulate inter-agent distances and avoid collisions. It combines an energy-based interconnected model with a Formation Matrix, then presents two control strategies whose equilibrium stability is analyzed.

  • Problem

    The paper addresses how to control formations of omnidirectional mobile robots while regulating inter-agent distances and avoiding collisions.

    Source: §I · p. 1

  • Method

    The approach combines virtual spring-damper couplings and a Formation Matrix with Bond Graph and Control by Interconnection modeling, then develops leader-agent and position-based strategies.

    Source: §I · pp. 1–2, §IX · p. 13

  • Results

    Above 0.9 meters, inter-agent distances remained throughout the position-control simulation, while both schemes’ equilibrium stability was demonstrated in the Lyapunov sense.

    Source: B. Scenario 2: Position Control · p. 12, §IX · p. 13

  • Takeaways & Limitations

    Leader Agent Control combines formation control and collision prevention in one controller with a physical interpretation.

    Source: C. Discussions · p. 12

Abstract

from arXiv · show

This paper presents an energy-based controller for a multiagent robotic system designed to achieve and maintain a specific formation while moving on a plane and avoiding collisions between agents. The controller emulates a network of elementary spring-damper modules connecting pairs of agents. This network, with its de-energized states representing the desired formation, determines the system's dynamics, which is fully encapsulated by a bond graph model. The modeling is further enhanced through the introduction of a formation matrix, using a graph-theoretic approach, that describes both the distances and relative velocities among the agents of the arrangement. This matrix mathematically represents the interconnection and energy-exchange structure of the bond graph, allowing us to put it in correspondence with the control-by-interconnection CbI-scheme of the IDA-PBC theory, facilitating the solution of the formation control problem within the port-Hamiltonian system framework. Furthermore, the paper presents leader-following and position-based formation control systems based on the CbI scheme, including a stability analysis of the corresponding closed-loop systems. The theoretical findings are validated through numerical simulations across various scenarios.

I. INTRODUCTION

The introduction frames formation control as a physically inspired multi-agent design problem and proposes an interconnected structure whose Formation Matrix captures both geometry and relative motion.

I. INTRODUCTION

  • The Formation Matrix represents inter-agent distances and relative velocities using edge-angle cosines and sines.Source: §I · p. 1
  • Virtual spring-damper couplings form an Interconnected Physical Structure for regulating inter-agent distances while avoiding collisions.The stated system comprises n mobile wheeled omnidirectional robots.Source: §I · p. 1
  • Bond Graphs connect the physical interconnection model to port-Hamiltonian representations and Control by Interconnection.The CbI controller connects to the plant through a power-preserving interconnection.Source: §I · pp. 1–2
  • Casimir invariant sets relate closed-loop controller states to plant states, and holonomic constraints permit a static feedback interpretation.The paper attributes the invariants to the interconnection geometry.Source: §I · p. 2

II. PRELIMINARIES

  • Section II introduces notation, port-Hamiltonian systems, graph theory, agent models, and the formal problem statement.Source: §II · p. 2

A. Notation

  • The notation defines identity and zero matrices, block-diagonal matrices, gradients, and the Kronecker product.Source: A. Notation · p. 2

B. Port-Hamiltonian systems

  • The port-Hamiltonian model characterizes stored energy, interconnection, dissipation, input, and power-related output.Its state is x, Hamiltonian H(x) is stored energy, and the interconnection matrix J(x) is skew-symmetric.Source: B. Port-Hamiltonian systems · p. 2

C. Graph Theory

  • Graph theory represents agents as nodes and interactions as edges; the incidence matrix and node positions describe neighboring-agent displacements and distances.Source: C. Graph Theory · pp. 2–3

D. Mobile Base Models

The paper models mobile bases both as point masses and as rigid bodies, using the latter to represent position, orientation, and locomotion.

D. Mobile Base Models

1) Point Mass (State-equation) Model:

2) Rigid Body (Euler-Lagrange) Model:

  • The rigid-body model adds orientation to position and represents locomotion with three omni-wheels symmetrically placed at 120°.Its Euler-Lagrange dynamics include Coriolis terms and map wheel torques to efforts at the base’s center of mass.Source: 2) Rigid Body (Euler-Lagrange) Model · p. 3

E. Problem Statement and Solution Approach

The proposed spring-damper network targets inter-agent distances while avoiding collisions. The Formation Matrix captures the setup’s distance constraints and maps agent velocities to changes in those distances.

E. Problem Statement and Solution Approach

  • Virtual spring-damper couplings address both distance control and collision avoidance: their rest lengths set desired distances, while forces repel agents near zero separation.When agents are farther apart than the desired distance, springs attract them; dampers attenuate mesh oscillations.Source: E. Problem Statement and Solution… · p. 3

III. FORMATION MATRIX

  • The Formation Matrix compactly represents how edge lengths depend on robot positions and how those lengths change with robot velocities.It expresses the setup’s kinematic and differential kinematic constraints.Source: §III · p. 3
  • F(r) maps agent velocities to differentiated edge distances; its range contains achievable distance rates, and its null space contains motions that leave distances unchanged.Source: §III · p. 4
  • The m kinematic constraints are holonomic because they are integrable and expressible as constraints on the distances.Source: §III · p. 5
  • For a non-degenerate configuration, a minimally rigid graph gives F(r) full rank m = 2n − 3.Source: §III · p. 5

IV. INTERCONNECTED PHYSICAL STRUCTURE

The ECB model represents a virtual spring whose de-energized length is the desired distance, with damping and distance-dependent forces shaping the interaction. Its Hamiltonian has a minimum at that reference length.

IV. INTERCONNECTED PHYSICAL STRUCTURE

1) pH ECB-Model:

  • The de-energized spring length represents the desired distance and is treated as constant.The model defines the spring extension as the difference between its current and de-energized lengths.Source: 1) pH ECB-Model · p. 5
  • The virtual spring repels agents as separation approaches zero and attracts them when separation exceeds the reference length.Its compressive force tends to infinity as the distance approaches zero.Source: 1) pH ECB-Model · p. 5
  • The spring Hamiltonian is chosen to have a minimum when its length equals the de-energized reference length.Source: 1) pH ECB-Model · p. 5
  • The ECB input is the spring-length change rate, its output is the total effort, and its damping coefficient is bk.Source: 1) pH ECB-Model · p. 5

2) BG ECB-Model:

The ECB and agent models are connected through a power-preserving interconnection, yielding a port-Hamiltonian model of the complete IPS. The ECB and agent subsystems are passive under the stated conditions.

2) BG ECB-Model:

  • The ECB Bond Graph uses R and C elements joined by a 1-junction to represent dissipation and potential-energy storage, with ζk as input and fk as output.Source: 2) BG ECB-Model · p. 5

1) pH MAS-Model:

  • The ECB effort is resolved into forces on the two endpoint agents, and the agent subsystem is passive.Source: 1) pH MAS-Model · p. 6
  • The interconnection links shared agent and spring velocities and efforts while preserving power.Source: 1) pH MAS-Model · p. 6
  • The ECB subsystem is passive when its damping matrix B is nonnegative.Source: 1) pH MAS-Model · p. 6
  • The complete IPS dynamics can be expressed in port-Hamiltonian form using the sum of agent and ECB Hamiltonians.Source: 1) pH MAS-Model · p. 6

2) BG MAS-Model:

The MAS Bond Graph exposes an interface whose input-output dependence yields the control law. Its three-part structure couples the ECB models and point-mass agents through a power-conserving transformation.

2) BG MAS-Model:

  • The IPS Bond Graph groups ECB models, agent storage elements for point masses, and an interconnection block.Source: 2) BG MAS-Model · p. 6, Figure 5 · p. 7
  • The interconnection uses a power-conserving transformation modulated by the transpose of the Formation Matrix to relate efforts and flows across agents and ECBs.Source: 2) BG MAS-Model · p. 6
  • Reading the power-bond interface between the MAS and IPS gives the control law, with ˙r as input and ˙p as output under the assigned causality.Source: 2) BG MAS-Model · p. 6

V. FORMATION CONTROL

The bond-graph model is recast as a control-by-interconnection scheme with agents and elementary coupling blocks modeled as port-Hamiltonian systems. The Formation Matrix connects the interconnection structure to invariant sets and supports a static state-feedback solution.

V. FORMATION CONTROL

  • The bond-graph model is transformed into a control-by-interconnection scheme with agents and elementary coupling blocks modeled as port-Hamiltonian systems.Associating an elementary coupling block with each edge relates the multi-agent interconnection structure to the Formation Matrix.Source: §V · p. 7, A. Control by Interconnection · p. 7

A. Control by Interconnection

  • The resulting Casimir functions relate plant and controller states through invariant sets, with the constant c representing the de-energized spring length when c = z∗.The function C(r) = F(r)r defines the invariant relationship.Source: A. Control by Interconnection · p. 7
  • The Casimir invariants restrict controller initialization: for a given initial position r(0), the controller state z(0) cannot be assigned arbitrarily.The text describes imposing this initial-condition restriction as unnatural.Source: A. Control by Interconnection · p. 7
  • The function C(r) = F(r)r is obtained naturally by this approach, whereas obtaining such functions generally involves solving partial differential equations.The supplied passages also state that the solution holds in this case, without specifying it further.Source: A. Control by Interconnection · pp. 7–8

B. State Feedback Formation Control Law

  • The proposed feedback obtains z directly from the Formation Matrix, avoiding velocity integration and the controller-initialization restriction.In the feedback representation, this also eliminates the virtual springs’ dynamic behavior.Source: B. State Feedback Formation Control Law · p. 8
  • The CbI approach yields a static state-feedback solution to the formation-control problem.The passage introduces its control law and corresponding block diagram.Source: B. State Feedback Formation Control Law · p. 8

C. Extension to Rigid Body Robots Models

For omnidirectional rigid-body bases, translation can use the point-mass control law while rotation is controlled separately. A leader-following extension regulates the formation’s position and orientation, with stability results under stated graph and detectability conditions.

C. Extension to Rigid Body Robots Models

  • Omnidirectional bases are split into translational and angular coordinates, allowing the point-mass law to control translation and a separate law to control rotation.The control inputs are translated into wheel-motor torques through the inverse of G(q), with low-level motor controllers managing them.Source: C. Extension to Rigid Body Robots Models · p. 8

VI. LEADER AGENT CONTROL

  • The leader-following extension regulates the interconnected formation’s position and orientation by selecting a leader and adding a control law.Fixing the leader’s position and a neighboring edge’s orientation determines the formation’s three planar degrees of freedom; the bases are assumed identical for simplicity.Source: §VI · pp. 8–9
  • For a minimally rigid graph, the combined matrix T(r) has full rank 2n.The passage derives this from the Formation Matrix’s rank m = 2n − 3 and three independent rows of the Leader matrix.Source: §VI · p. 9
  • With the specified proportional-integral leader input, the closed-loop equilibrium is stable and is asymptotically stable when the output is detectable.The analysis uses a Lyapunov candidate for stability and a detectability condition for asymptotic stability.Source: §VI · p. 9

VII. POSITION CONTROL

Position-based control matches real agents to virtual reference points moving along a predetermined trajectory while retaining the elementary coupling blocks for collision avoidance. The analysis establishes asymptotic stability of the position-error equilibrium with the additional control signal.

VII. POSITION CONTROL

  • Position-based control configures the external signal ν for formation control and adds a signal ω to ensure asymptotic stability.The design retains the elementary coupling blocks used for collision avoidance.Source: §VII · pp. 9–10
  • The Virtual Reference Structure consists of points moving along a predetermined trajectory, and formation control associates each real agent with its corresponding point.The real agents are intended to take on the topology of the virtual structure.Source: §VII · p. 9
  • The analysis establishes stability of the desired equilibrium, and the reduced position-error equilibrium is asymptotically stable with the designed ω signal.The text states that ω makes the Lyapunov derivative negative definite, implying ξ → 0.Source: §VII · p. 10
  • In this scheme, elementary coupling blocks prevent collisions rather than position agents into a specific formation, so the Formation Matrix rank does not affect stability.The passage notes that using as many coupling blocks as possible is advantageous.Source: §VII · p. 10

VIII. SIMULATIONS

The simulations use six rigid-body agents moving in a horizontal plane across two scenarios. The scenarios vary the interaction graph while sharing the desired trajectory and path.

VIII. SIMULATIONS

  • The simulations model six rigid-body agents moving in a horizontal plane to validate the presented control strategies.The passage describes two scenarios but does not report their outcomes here.Source: §VIII · p. 10
  • The scenarios use either a minimally rigid directed graph with m = 9 or a rigid graph with m = 15.The desired topology and orientation are shown in Fig. 8a, with the graph alternatives in Fig. 8b and Fig. 8c.Source: §VIII · p. 10, Figure 8 · p. 11
  • Both scenarios use the same desired trajectory and corresponding path, and the setup includes initial conditions for the mobile bases.The trajectory and path are shown in Fig. 9a and Fig. 9b; initial conditions are listed in Table II.Source: §VIII · p. 10, Figure 9 · p. 11, Table II · p. 11

A. Scenario 1: Leader Agent Control

With agent 1 leading on a constant reference, the formation reaches its desired distances and orientation while following the trajectory. The controller’s topology and orientation are constrained by graph rigidity.

A. Scenario 1: Leader Agent Control

  • In graph G1, agent 1 is selected as leader and edge 1 controls the angle; the first two reference-vector components’ evolution is shown in Fig. 9a and Fig. 9b, while the desired-angle component remains constant.Source: A. Scenario 1: Leader Agent Control · p. 11
  • The MAS reaches the stable equilibrium when the reference becomes constant: agents stop, desired distances are attained, and the leader error tends to zero.The plane snapshots show the agents following the trajectory with the desired topology and orientation.Source: A. Scenario 1: Leader Agent Control · p. 11, Figure 10 · p. 11, Figure 11 · p. 12
  • The Leader Agent Control graph must be minimally rigid for the Formation Matrix to have full rank; topology changes require new controlled distances to preserve rigidity.Distance constraints can still permit mirrored formations or multiple possible agent positions.Source: A. Scenario 1: Leader Agent Control · pp. 11–12

B. Scenario 2: Position Control

Position Control without ω does not reach the desired formation, while adding ω brings trajectory and distance errors to zero and maintains collision spacing. The discussion distinguishes the schemes’ interpretations and limitations, and the conclusion reports Lyapunov stability for both.

B. Scenario 2: Position Control

C. Discussions

  • Leader Agent Control combines formation and collision prevention in one physically interpreted controller; Position Control ensures desired orientation without inter-agent collisions.Leader control uses mechanically inspired interconnected subsystems, whereas Position Control derives its expressions from formation geometry.Source: C. Discussions · pp. 12–13
  • Leader Agent Control requires a minimally rigid graph with m = 9 edges, and cannot prevent collisions between agents whose distances are absent from G1.Position Control lacks a physical interpretation and focuses more on individual agents than on the formation.Source: C. Discussions · p. 13
  • The proposed strategy requires centralized implementation because computing the Formation Matrix needs global agent positions.The positions may be obtained by fusing camera measurements, IMU data, and odometry.Source: C. Discussions · p. 13
  • Bond Graph modeling represents storage, dissipation, interconnection, and power exchange; causal relationships and the Formation Matrix support a CbI reformulation with static state feedback.The authors report that this structure suggests a Casimir function and allows deriving a dynamic control law a priori.Source: C. Discussions · p. 13

IX. CONCLUSION

  • Both schemes’ equilibrium stability is demonstrated in the Lyapunov sense; future work includes differential mobile bases, distance-control limitations, and three-dimensional formations.For Position Control, ω is intended to ensure convergence to the desired equilibrium.Source: §IX · p. 13
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