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Adaptive stabilization of a leaderless bearing-constrained formation with disturbances

Minh Hoang Trinh, Chuong Van Nguyen, Quoc Van Tran, Tuynh Van Pham

arXiv:2609.06355v1cs.MAmath.OC

TL;DR

The paper addresses leaderless bearing-constrained formation control under unknown bounded disturbances, which can make the target formation time-varying. It proposes adaptive sliding-mode and smooth adaptive controllers; the former stabilize the desired formation and reject matched disturbances, while the latter mitigate chattering and guarantee boundedness.

  • Problem

    Leaderless bearing-constrained formations must be regulated under disturbances whose upper bound is unavailable, while prior robust approaches had assumptions including known bounds and/or leaders.

    Source: §I · pp. 1–2

  • Method

    Adaptive sliding-mode controllers adjust gains to address unknown disturbance bounds, and smooth adaptive laws mitigate chattering.

    Source: §I · p. 2, C. A smooth adaptive bearing-only… · p. 8, §VI · p. 10

  • Results

    Adaptive sliding-mode controllers uniformly stabilize the desired formation and completely reject matched disturbances, while smooth adaptive laws ensure uniform ultimate boundedness and eliminate chattering.

    Source: §VI · p. 10

  • Takeaways & Limitations

    The paper states that its adaptive control design strategies are applicable to other categories of leaderless formation control schemes.

    Source: §VI · p. 10

    The bearing-only adaptive gain update uses displacements, and the bearing-measurement-only modification is supported by simulations but lacks a proof.

    Source: C. Stability analysis · p. 5

Abstract

from arXiv · show

In this paper, we consider the problem of regulating and maintaining a target formation characterized by a set of bidirectional bearing constraints under disturbances. The agents in the formation are modeled by single integrators with bounded continuous disturbances of which the upper bound is unavailable for the control design. Due to the time-varying disturbances, the target formation is time-varying. We propose adaptive sliding mode control laws to uniformly globally asymptotically stabilizes the moving target formation and reject the matched disturbances. In addition, to alleviate chattering phenomena from sliding mode control, smooth adaptive control laws are then designed to guarantee uniform global boundedness of the desired formation. Finally, simulation results are given to support the analysis.

I. Introduction

The introduction motivates bearing-based formation control and identifies limits in existing disturbance-robust approaches. It introduces adaptive control for leaderless formations and reviews the graph and rigidity concepts used to specify bearing constraints.

I. Introduction

  • Bearing vectors can be obtained from onboard cameras, reducing per-agent sensor requirements and dependence on a global reference frame.The introduction also notes their suitability where signal emission must be limited because cameras are passive sensors.Source: §I · p. 1
  • Existing robust bearing-only methods assumed a known disturbance bound and/or leaders, while leaderless methods left bearing-error bounds state-dependent and uncomputable.The paper also notes that leaderless control with mismatched disturbances had not been considered in the cited works.Source: §I · pp. 1–2
  • Adaptive sliding-mode strategies address unknown bounded disturbances in leaderless formations, with smooth laws to mitigate chattering and keep the moving target uniformly ultimately bounded.The paper considers both relative-position and bearing-vector sensing and establishes uniform asymptotic stability for the moving target.Source: §I · p. 2

A. Bearing rigidity theory

  • A formation is represented by a graph whose edges identify neighboring agents, and each edge's bearing is the normalized relative displacement.The bearing rigidity matrix characterizes infinitesimal bearing rigidity; its rank is dn − d − 1.Source: A. Bearing rigidity theory · p. 2

B. Problem formulation

The problem formulation specifies single-integrator agents with unknown bounded disturbances and desired bearing constraints, then defines two sensing-based control problems. The section introduces a bearing-based control law and describes its adaptive gains and discontinuous solution interpretation.

B. Problem formulation

  • Agent motion follows a single-integrator model, with position and disturbance expressed in a global coordinate system.The agents' local coordinate axes are aligned, although their origins differ.Source: B. Problem formulation · p. 3
  • Disturbances are bounded, time-varying, and uniformly continuous, but their upper bound is unknown to each agent.The bound is denoted β in the formulation.Source: B. Problem formulation · p. 3
  • The desired formation is infinitesimally bearing rigid and defines the target bearing constraints, with the target set comprising formations bearing-congruent to it.The target formation has no collocated points.Source: B. Problem formulation · p. 3
  • The paper formulates separate objectives for displacement sensing and bearing-vector sensing: converge to formations bearing-congruent to the desired formation.Displacements and bearing vectors are defined relative to neighboring agents.Source: B. Problem formulation · p. 3
  • The section outlines control laws for the displacement-sensing problem, disturbance effects on formation acquisition, moving-target convergence, and smooth adaptive control for uniform ultimate boundedness.The moving target is described using desired bearing constraints and mismatched disturbances.Source: B. Problem formulation · p. 3

A. Proposed control law

  • The proposed bearing-based control law uses edge-associated adaptive gains initialized above zero and a positive gain-update constant.The gains are non-decreasing; the control law is discontinuous and its solutions are interpreted in the Filippov sense.Source: A. Proposed control law · p. 3
  • The multi-agent dynamics under the control law are written using stacked control inputs and adaptive gains, with the gain matrix formed diagonally from edge gains.The gain matrix is extended using the identity matrix for the agent-coordinate dimensions.Source: A. Proposed control law · p. 3

B. Effects of the matched and mismatched disturbances

The analysis separates disturbances into matched control directions, whole-formation translation, and scaling, then studies stability under an additional restriction on the scaling disturbance. It establishes convergence to the desired formation and conditions for collision avoidance.

B. Effects of the matched and mismatched disturbances

  • The disturbance decomposes into a matched component handled by control, a translation component, and a scaling component about the formation centroid.The scaling component lies in the uncontrollable space alongside translation, where the sliding-mode law has no effect.Source: B. Effects of the matched and… · p. 3
  • The scaling disturbance is restricted so its cumulative effect remains below the initial scale, preventing the agents from shrinking to a point.The restriction is expressed as an integral bound on the scaling disturbance.Source: B. Effects of the matched and… · p. 4

C. Stability analysis

  • If every adaptive gain exceeds the disturbance bound, the formation error is globally finite-time stable; otherwise, it converges asymptotically to zero.In both cases, the adaptive gains converge to a constant vector.Source: C. Stability analysis · p. 4
  • The formation converges to the desired set, and the stated collision-avoidance corollary guarantees no collisions for all time under its assumptions.The proof passage says pairwise agent distances remain positive.Source: C. Stability analysis · p. 5
  • The adaptive mechanism uses displacement measurements, although simulations suggest a bearing-only modification works; the authors report no proof for that modification.When disturbance magnitude is known, the underlying control law can be implemented using bearing measurements alone.Source: C. Stability analysis · p. 5

D. A smooth bearing-based control law

The section addresses chattering from the signum discontinuity by proposing a smooth adaptive bearing-based control law. Under the stated disturbance assumption, its ultimate bound is adjustable through the control parameters.

D. A smooth bearing-based control law

  • The signum discontinuity in sliding-mode control causes chattering, motivating a smooth adaptive bearing-based control law.The proposed compromise relaxes perfect disturbance rejection to avoid chattering.Source: D. A smooth bearing-based control law · p. 5
  • Under Assumption 3.1, the smooth control law preserves the earlier lemma's claims and permits defining the time-varying target formation.The section identifies the control law as (17).Source: D. A smooth bearing-based control law · p. 5
  • Under control law (36), the formation error and adaptive gains are globally uniformly ultimately bounded, with an adjustable ultimate bound.The bound depends on control parameters kp, kγ, ε, and α.Source: D. A smooth bearing-based control law · p. 5

IV. Adaptive bearing-only formation control with disturbance rejection

The section considers bearing-only formation control under disturbances, where agents must satisfy desired bearing constraints and suppress disturbances using measured bearing vectors. It outlines adaptive sliding-mode and smooth adaptive approaches, with the latter guaranteeing uniform ultimate boundedness.

IV. Adaptive bearing-only formation control with disturbance rejection

  • The bearing-only task requires agents to satisfy desired bearing constraints while suppressing disturbances using only measured bearing vectors.The section describes this as substantially more complex than the preceding problem.Source: §IV · p. 6
  • The section proposes an adaptive sliding-mode bearing-only control law and provides a corresponding stability analysis.Source: §IV · p. 6
  • A smooth adaptive bearing-only control law is analyzed to guarantee uniform ultimate boundedness of the desired formation.Source: §IV · p. 6

A. Proposed control law

The section analyzes bearing-only control law (19) under stated disturbance assumptions and derives convergence and collision-avoidance guarantees, with a qualification for scaling disturbances.

A. Proposed control law

  • Because law (20) has a discontinuous right-hand side, its solutions are interpreted in the Filippov sense; γ_ij(t) stays positive.Source: A. Proposed control law · p. 6

B. Stability analysis

  • The analysis assumes disturbances are uniformly continuous, bounded, and perpendicular to p for all t ≥ 0.Source: B. Stability analysis · p. 6
  • If every initial γ_ij exceeds β5, the desired configuration is finite-time stable under control law (19).Source: B. Stability analysis · p. 6
  • Under adaptive bearing-only law (19), p(t) converges to p_d and gains converge to γ*; with the stated gain-margin condition, convergence is finite-time.The finite-time claim requires the condition on γ(T) specified in the theorem.Source: B. Stability analysis · p. 7
  • Corollary 4.1 guarantees no inter-agent collisions for all t ≥ 0 under law (19), subject to its stated condition.Source: B. Stability analysis · p. 8
  • Scaling disturbance can disrupt the monotone-angle argument; convergence is retained if f3 is sufficiently small, scale stays bounded away from zero, and α(0) is sufficiently large.Source: B. Stability analysis · p. 8

C. A smooth adaptive bearing-only control law

The smooth adaptive bearing-only law modifies sliding-mode control to mitigate chattering. Its theorem guarantees global uniform ultimate boundedness of the formation error and adaptive gain, with an adjustable ultimate bound.

C. A smooth adaptive bearing-only control law

  • The bearing-only law (19) is modified to mitigate chattering arising from the signum function's discontinuity.The modification uses small positive ε and α, a positive control gain kp, and adaptive rate kγ.Source: C. A smooth adaptive bearing-only… · p. 8
  • The matrix-form system (37) preserves the lemma's claims, and a similar collision-avoidance condition applies.The motion is rewritten in matrix form before the boundedness theorem; the passage also states that ṗ(t) is orthogonal to ker(Ṙb).Source: C. A smooth adaptive bearing-only… · pp. 8–9
  • Theorem 4.2 guarantees global uniform ultimate boundedness of δ and γ, with an ultimate bound adjustable through kp, kγ, ε, and α.The result assumes Problem 2.2 and uses control law (36).Source: C. A smooth adaptive bearing-only… · p. 9

V. Simulation results

The simulation setup uses six agents in two-dimensional space with an interaction graph and a desired bearing-constrained formation. Initial agent positions and adaptive gains are specified for the simulations.

V. Simulation results

  • The simulations use six agents in two-dimensional space and the interaction graph shown in Fig. 2.The desired formation is specified by 12 bearing vectors corresponding to a six-regular polygon configuration.Source: §V · p. 9
  • The desired formation uses 12 bearing vectors corresponding to a six-regular polygon configuration.Source: §V · p. 9
  • For every simulation, the initial values assigned to agents’ positions and adaptive gains were chosen as specified in Table I.Source: §V · p. 9

A. Simulation 1: Bearing-based control

The bearing-based simulations report that sliding-mode control attains the desired bearings while compensating matched disturbances. Under the smooth adaptive law, bearing error remains small despite mismatched disturbances that move and rescale the formation.

A. Simulation 1: Bearing-based control

  • Bearing error converges to 0 and adaptive gains approach constants after about 5 seconds under law (4).The simulation reports compensation of matched disturbances alongside attainment of the desired bearing vectors.Source: A. Simulation 1: Bearing-based control · p. 9, Figure 3 · p. 9
  • After 5 seconds, bearing error remains within [0, 5 · 10^-3] under smooth adaptive law (17).Adaptive gains vary within a positive bounded interval rather than converging to fixed values; mismatched disturbances translate and rescale the formation.Source: A. Simulation 1: Bearing-based control · p. 9

B. Simulation 2: Bearing-only control

In the bearing-only simulations, the adaptive sliding-mode law nearly achieves the desired bearings within 10 seconds, while the smooth adaptive law keeps bearing error bounded later in the run. The conclusion also states that the control strategies apply beyond bearing-constrained formations.

B. Simulation 2: Bearing-only control

VI. Conclusions

  • The conclusion states that the adaptive sliding-mode controllers stabilize desired formations and reject matched disturbances.The smooth adaptive laws are described as ensuring uniform ultimate boundedness and eliminating chattering.Source: §VI · p. 10

B. Simulation 2: Bearing-only control

VI. Conclusions

  • The control design strategies are stated to apply to other categories of leaderless formation control schemes.Source: §VI · p. 10

Appendix

The two proofs use Lyapunov bounds to establish ultimate boundedness: Theorem 3.2 bounds the equilibrium error, while Theorem 4.2 establishes ultimate boundedness of δ and γ.

Appendix

A. Proof of Theorem 3.2

  • Theorem 3.2 uses a Lyapunov function with β4 greater than the disturbances’ supremum norm to establish global uniform ultimate boundedness of the equilibrium error.The proof derives a decaying Lyapunov bound with an additive term and applies the comparison lemma.Source: A. Proof of Theorem 3.2 · pp. 11–12

B. Proof of Theorem 4.2

  • Theorem 4.2 uses a Lyapunov function with β7 greater than the disturbances’ supremum norm to show δ and γ are uniformly ultimately bounded.The proof states that this bound holds after a finite time T0.Source: B. Proof of Theorem 4.2 · p. 12
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