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Leader-Follower Formation Control with Prescribed Convergence Rates under Bearing Persistence of Excitation

Tarek Bouazza, Zhiqi Tang, Soulaimane Berkane, Tarek Hamel

arXiv:2609.19106v1eess.SYcs.SY

TL;DR

Fixed-gain bearing-based formation controllers can have convergence rates tied to the desired formation’s PE properties. This paper uses an adaptive time-varying matrix gain to prescribe rates independently of those properties. The design is extended to double-integrator dynamics, with uniform exponential stability established under the stated assumptions.

  • Problem

    Fixed-gain leader-follower bearing controllers have guaranteed convergence rates that depend on the desired formation’s PE properties and can become arbitrarily small.

    Source: §I · p. 1

  • Method

    A time-varying matrix gain adapts to accumulated bearing information to decouple the convergence rate from bearing-PE properties, and the design is extended to double-integrator dynamics.

    Source: §I · p. 1, §VII · pp. 7–8

  • Results

    Under Assumptions 1–3, the double-integrator closed-loop error equilibrium is uniformly exponentially stable, with convergence rates set by parameters λ_i.

    Source: §V · p. 5

  • Takeaways & Limitations

    For bearing-PE formations, the proposed gain is well-posed and uniformly bounded, while the exponential convergence rate is determined by its design parameter.

    Source: §VII · p. 7

Abstract

from arXiv · show

This paper addresses leader-follower formation control using only relative bearing and velocity measurements. Bearing-based leader-follower control strategies commonly use fixed control gains, for which the guaranteed convergence rates explicitly depend on the persistence of excitation (PE) properties of the desired formations. We propose a time-varying matrix gain that evolves according to the bearing information available to each follower and decouples the convergence rate from the PE bound. We establish the well-posedness and uniform boundedness of the proposed gain for bearing-persistently exciting formations, and characterize the exponential convergence rate through a tunable design parameter. The proposed control design is further extended to leader-follower formations with double-integrator dynamics. Simulations illustrate the resulting convergence properties compared with fixed gain designs.

I. INTRODUCTION

Bearing-based control uses lightweight relative-bearing measurements, but fixed-gain designs can have arbitrarily small guaranteed rates even for bearing-PE formations. This paper introduces an adaptive time-varying gain to prescribe convergence rates independently of the formation’s PE properties.

I. INTRODUCTION

  • Bearing-based methods require relative bearings rather than inter-agent distances or global positions, and can use lightweight sensors such as monocular cameras.Formation-control approaches are also classified as position-, displacement-, and distance-based –,,.Source: §I · p. 1
  • Bearing rigidity determines formations up to translation and scale, while directed graphs require additional constraint-consistency conditions captured by bearing persistence,.Bearing rigidity was studied in the plane, and generalized to arbitrary dimensions [19].Source: §I · p. 1
  • Tang et al. showed that PE of desired bearings can replace classical bearing-rigidity and constraint-consistency requirements while guaranteeing uniform exponential stabilization of shape and scale,.A single distance measurement is one approach to fixing scale.Source: §I · p. 1
  • Fixed-gain controllers for minimal-topology bearing-PE formations have guaranteed rates tied to the formation’s PE properties, which can become arbitrarily small.Minimal topologies include star graphs, where each follower is connected to a single leader.Source: §I · p. 1
  • The proposed time-varying matrix gain adapts to accumulated bearing information, decoupling exponential convergence rates from bearing-PE properties and allowing rates to be prescribed by design parameters.The framework uses bearing and relative-velocity measurements and is extended to double-integrator dynamics.Source: §I · p. 1

A. Notation

  • For vectors in R3, x× denotes the skew-symmetric matrix associated with the cross product.Source: A. Notation · p. 2
  • The notation defines the unit sphere and its orthogonal projection, along with matrix dimensions, positive-semidefinite ordering, spectral norm, and block-diagonal matrices.The projection π_y is onto the plane orthogonal to the unit vector y.Source: A. Notation · p. 2

B. Bearing persistence of excitation

The section states persistence-of-excitation definitions for positive-semidefinite matrix signals and time-varying directions. A direction is PE when its associated projection satisfies the matrix PE condition.

B. Bearing persistence of excitation

C. Preliminaries on graph theory

The preliminaries model agent interactions as a directed sensing graph, where edges specify which agents can access information about others. Leader-follower graphs are acyclic and rooted at the leader.

C. Preliminaries on graph theory

  • The sensing graph has vertices for agents and directed edges for sensed-neighbor relations; N_i collects agent i’s neighbors.A directed path is a sequence of distinct vertices connected by directed edges.Source: C. Preliminaries on graph theory · p. 2
  • A sensing digraph edge (i, j) means agent i senses information about agent j, with the leader shown as agent 1 and followers as agents i ≥ 2.Source: Figure 1 · p. 2, C. Preliminaries on graph theory · p. 2
  • A leader-follower graph is acyclic with a directed spanning tree rooted at agent 1; a minimal structure gives each follower exactly one neighbor.Source: C. Preliminaries on graph theory · p. 2
  • The paper’s leader-follower structure generalizes the leader-first-follower structure in, in which followers generally have two neighbors and agent 2 is connected only to the leader.Source: C. Preliminaries on graph theory · p. 2

III. PROBLEM DESCRIPTION

The problem is to control a three-dimensional leader-follower formation toward a desired, time-varying geometry using bearing-based control. The desired bearings are PE, and the formulation includes relative position and velocity measurements.

III. PROBLEM DESCRIPTION

  • The problem is posed for multi-agent systems in three-dimensional space, with agents interconnected through a leader-follower topology.Source: §III · p. 2
  • The setup defines relative position from a follower to a neighbor and relative velocity as the neighbor’s velocity minus the follower’s.Source: §III · p. 2
  • The formation is required to be bearing-PE, with each follower’s bearing information satisfying the PE condition.Source: §III · p. 2
  • The controller must drive each neighbor-relative position toward its desired value in a leader-follower system of at least two agents.Source: §III · pp. 2–3

IV. FIRST-ORDER BEARING LEADER-FOLLOWER FORMATION CONTROL DESIGN

The section introduces a time-varying matrix gain for bearing-based leader–follower control, addressing fixed-gain rates that can become arbitrarily small for bearing-PE formations. Under the bearing-PE condition, the proposed control yields uniform exponential convergence, with the rate prescribed by a design parameter rather than the PE parameters.

IV. FIRST-ORDER BEARING LEADER-FOLLOWER FORMATION CONTROL DESIGN

  • The design assumes a bearing-PE desired formation and well-defined bearing measurements, with collisions and occlusions excluded throughout the formation’s evolution.The controller is presented for single-integrator agents with velocity as the control input.Source: §IV · p. 3
  • Fixed-gain rates depend on the PE parameters and can approach zero when μ_i is arbitrarily small, including in quasi-static minimal-topology formations.The section motivates a gain design that decouples the rate from those parameters.Source: §IV · p. 3
  • The time-varying gain is defined by a Riccati-type equation that accounts for accumulated bearing excitation; bearing-PE ensures it is well-posed, uniformly bounded, and uniformly positive definite.Its inverse is a low-pass filtered local bearing Gramian, with forgetting rate λ_i.Source: §IV · p. 3
  • The stability proof proceeds through the directed formation: each follower’s error dynamics are treated as a cascade driven by upstream errors that converge exponentially.For the first follower, the proof uses a storage function whose decay is tied to λ_i; later followers extend the argument by induction.Source: §IV · pp. 4–5
  • The proposed control makes follower errors uniformly exponentially stable, with each agent’s convergence rate determined by λ_i rather than the PE parameters δ_i and μ_i.The PE parameters instead affect the gain bound.Source: §IV · pp. 4–5

V. SECOND-ORDER DOUBLE INTEGRATOR DYNAMICS

The design extends the bearing-based controller to agents with double-integrator dynamics, where the input acts at the acceleration level. Under the stated assumptions, the proposed gain is uniformly bounded and positive definite, and the error equilibrium is uniformly exponentially stable with rates set by λ_i.

V. SECOND-ORDER DOUBLE INTEGRATOR DYNAMICS

  • The extension addresses platforms whose control input appears at the acceleration level, rather than directly controlling velocity.Here u_i is the acceleration control input.Source: §V · p. 5
  • For double-integrator dynamics, the cited fixed-gain baseline guarantees uniform exponential stability when its positive gains satisfy stated bounds.The result is adapted from [16, Theorem 2].Source: §V · p. 5
  • Unlike the fixed-gain double-integrator result, the proposed controller sets convergence rates through λ_i rather than the desired formation’s bearing-PE properties.The fixed-gain rates remain dependent on those PE properties.Source: §V · p. 5
  • The proposed gain yields uniform exponential stability of the position- and velocity-error equilibrium for double-integrator agents under Assumptions 1–3.The gain is uniformly bounded and uniformly positive definite.Source: §V · p. 5

VI. SIMULATION RESULTS

The section presents simulations to illustrate the effectiveness of the proposed control laws.

VI. SIMULATION RESULTS

  • The simulations illustrate the effectiveness of the proposed control laws (11) and (26).Source: §VI · p. 7

A. Formation control for single-integrator motion dynamics

In the rotating-pyramid simulation, both controllers reduce relative-position errors, but the proposed Riccati-based controller converges faster and more uniformly than the fixed-gain law.

A. Formation control for single-integrator motion dynamics

  • The comparison uses four agents in a chain leader-follower graph, with desired relative positions rotating about the z-axis to form a rotating pyramid.The desired bearings are well-excited.Source: A. Formation control for… · p. 7
  • Both controllers converge, but the proposed controller achieves faster, more uniform relative-position-error convergence than the fixed-gain controller, especially for p̃_43.The p̃_43 error is most affected by the chain topology; the comparison is consistent with the proposed rate being independent of the PE bound.Source: A. Formation control for… · p. 7, Figure 2 · p. 7

B. Formation control for double-integrator motion dynamics

In the star-formation simulation with small bearing variations, the proposed double-integrator controller achieves exponential convergence for all agents, while the fixed-gain controller converges significantly more slowly.

B. Formation control for double-integrator motion dynamics

  • The simulated five-follower star formation places followers at regular-pentagon vertices, with small bearing variations and a small PE lower bound.The desired positions evolve along small arcs.Source: B. Formation control for… · p. 7
  • The proposed controller achieves exponential convergence for all agents, whereas the fixed-gain controller converges significantly more slowly under the same bearing-PE conditions.Source: B. Formation control for… · p. 7, Figure 3 · p. 7

VII. CONCLUSIONS

The paper introduces a time-varying gain that uses accumulated desired-bearing information to decouple exponential convergence rate from bearing-PE parameters. It establishes boundedness and convergence under bearing-PE, extends the framework to double-integrator dynamics, and illustrates its performance in simulations.

VII. CONCLUSIONS

  • The time-varying gain uses accumulated bearing information to decouple the exponential convergence rate from parameters characterizing bearing persistence of excitation.The gain is defined through a Riccati-type equation whose inverse corresponds to a filtered bearing Gramian.Source: §VII · p. 7
  • Bearing-PE ensures the gain is well-posed and uniformly bounded, and convergence of formation error is established without bearing rigidity assumptions.The exponential convergence rate is determined by the gain design parameter.Source: §VII · p. 7
  • The framework extends to double-integrator dynamics, and simulations illustrate convergence performance and an advantage over classical designs.Collision avoidance and extensions to more general interaction topologies and time-varying formations are identified as future work.Source: §VII · p. 8

APPENDIX

The appendix states conditions for a projected-inverse result and gives supporting arguments involving restricted operators and non-PE bearing signals.

APPENDIX

  • For a matrix-valued bearing function that is not persistently exciting, Lemma 3 gives a sequence and unit vector, and states that y(t) converges to a constant vector.The passage does not specify the sequence's further properties in the supplied text.Source: APPENDIX · p. 8
  • Lemma 4 considers symmetric positive invertible Aε approaching an orthogonal projector P, with PAεP invertible on Im(P) for small ε.The passage introduces a projected-inverse result under these conditions but does not state its conclusion.Source: APPENDIX · p. 8
  • In the Lemma 2 argument, Bε=PAεP restricted to Im(P) approaches the identity and is invertible for small ε.The proof uses this restriction to establish the pseudo-inverse property on Im(P).Source: APPENDIX · p. 8
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