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Bearing Rigidity and Almost Global Bearing-Only Formation Stabilization

Shiyu Zhao, Daniel Zelazo

arXiv:1408.6552v4eess.SY

TL;DR

The paper asks when bearings uniquely determine a formation and how formations can be stabilized using only bearing measurements. It develops arbitrary-dimensional bearing rigidity theory and distributed bearing-only controllers, proving unique-shape and almost-global stabilization results while identifying graph and collision-related scope boundaries.

  • Problem

    Existing bearing rigidity results largely focused on two-dimensional frameworks, while bearing-only control often required position information, estimated relative states, or specialized formation shapes.

  • Method

    The paper develops rank-based bearing rigidity theory in arbitrary dimensions and designs distributed nonlinear controllers for settings with and without a global reference frame.

  • Results

    Infinitesimally bearing rigid frameworks are uniquely determined up to translation and scaling, and the proposed controllers almost globally stabilize such formations.

  • Takeaways & Limitations

    Bearing rigidity provides a framework for analyzing and stabilizing arbitrary-dimensional formations using pure bearing measurements, including orientation synchronization without a global reference frame.

  • Takeaways & Limitations

    The analysis assumes undirected underlying graphs, although the bearing rigidity theory itself is independent of whether the graph is directed or undirected.

Abstract

from arXiv · show

A fundamental problem that the bearing rigidity theory studies is to determine when a framework can be uniquely determined up to a translation and a scaling factor by its inter-neighbor bearings. While many previous works focused on the bearing rigidity of two-dimensional frameworks, a first contribution of this paper is to extend these results to arbitrary dimensions. It is shown that a framework in an arbitrary dimension can be uniquely determined up to a translation and a scaling factor by the bearings if and only if the framework is infinitesimally bearing rigid. In this paper, the proposed bearing rigidity theory is further applied to the bearing-only formation stabilization problem where the target formation is defined by inter-neighbor bearings and the feedback control uses only bearing measurements. Nonlinear distributed bearing-only formation control laws are proposed for the cases with and without a global orientation. It is proved that the control laws can almost globally stabilize infinitesimally bearing rigid formations. Numerical simulations are provided to support the analysis.

I. INTRODUCTION

The paper extends bearing rigidity theory to arbitrary dimensions and applies it to distributed formation control using only bearing measurements. It establishes rank-based rigidity conditions and almost-global stabilization results for formations with and without a global reference frame.

  • Motivation: Bearing-only control avoids estimating relative positions or distances, reducing sensing-system complexity while directly using neighbors’ bearing measurements.Earlier approaches either required position measurements, estimated additional relative-state information, or targeted special formation shapes.
  • Bearing rigidity theory: The proposed bearing rigidity theory applies in arbitrary dimensions and characterizes when inter-neighbor bearings uniquely determine a framework up to translation and scaling.The theory connects bearing equivalence, bearing congruence, and framework shape through the bearing rigidity matrix.
  • Formation stabilization: Distributed nonlinear bearing-only control laws almost globally stabilize infinitesimally bearing rigid formations both with and without a global reference frame.In the no-reference-frame case, the control also synchronizes agents’ orientations; the global-frame analysis includes a sufficient collision-avoidance condition.
  • Bearing rigidity theory: Infinitesimal bearing rigidity is invariant to space dimensions, supporting the extension of the theory from lower-dimensional frameworks to higher-dimensional spaces.The corresponding bearing rigidity matrix ranks are equivalent across the dimension-embedding construction described in the paper.

A. Connections to Distance Rigidity Theory

The paper connects bearing rigidity with distance rigidity, proving equivalence in R2 while identifying important differences in higher dimensions. It also relates bearing-preserving motions to rotated distance-preserving motions in the plane.

  • Bearing rigidity and distance rigidity determine framework shape through inter-neighbor bearings and distances, respectively.
  • In R2, a framework is infinitesimally bearing rigid if and only if it is infinitesimally distance rigid.
  • The equivalence between bearing and distance rigidity does not extend to R3 or higher dimensions.Three-dimensional cubic and hexagonal pyramid frameworks are infinitesimally bearing rigid but not distance rigid.
  • For planar frameworks, rotating an infinitesimal bearing motion by π/2 produces an infinitesimal distance motion, and conversely.
  • Infinitesimal bearing rigidity remains dimension-invariant, whereas a rigid framework in a lower-dimensional space may become non-rigid in a higher-dimensional space.

III. BEARING-ONLY FORMATION CONTROL WITH A GLOBAL REFERENCE FRAME

This section formulates bearing-only formation control in arbitrary dimensions with a known global reference frame. The target is specified by feasible inter-agent bearings, and agents use only measured neighbor bearings to approach those constraints.

  • The control problem considers n agents in R^d with n ≥2 and d ≥2, a fixed undirected sensing graph, and a global reference frame.The framework is represented by agent positions and inter-agent bearings on the graph edges.
  • Each agent measures the relative bearings of its neighbors, while the target formation is specified by constant feasible bearing constraints.Feasibility means that some formation satisfies the prescribed target bearings.
  • Figures 4 examples illustrate target and initial formations using black and grey configurations, respectively.
  • The designed velocity input must use only each agent’s neighbor-bearing measurements and drive every measured bearing toward its corresponding target bearing.

A. A Bearing-Only Control Law

The proposed distributed control law uses projected bearing errors to adjust agent velocities without measuring distances. Its geometry preserves the formation centroid and scale while reducing bearing errors, with explicit trajectory bounds.

  • The control law is distributed and uses only neighbor-bearing measurements, while each control input remains bounded by the agent’s neighbor count.
  • The projected control term is perpendicular to the current bearing, so it reduces bearing error while preserving each affected inter-agent distance.
  • The control law is a modified gradient law that removes the distance factor required by the corresponding gradient control.
  • The centroid and scale remain invariant under the control law.The proof uses velocity orthogonality to the translation and scaling directions.
  • For all time, s ≤ max_i∈V ∥p_i(t)−p̄∥ ≤ s√(n−1), and ∥p_i(t)−p_j(t)∥ ≤ 2s√(n−1).

B. Formation Stability Analysis

Under the infinitesimal bearing-rigidity assumption, the target formation exists uniquely with the initial centroid and scale, and the control dynamics converge exponentially to it from almost all initial states. The only other isolated equilibrium is a point-reflected formation with opposite bearings.

  • Target formation: The target formation exists uniquely under the infinitesimal bearing-rigidity assumption and matches the initial formation's centroid and scale.It satisfies all target bearing constraints.
  • State geometry: The formation error evolves on a sphere because centroid preservation and scale invariance keep its centered norm constant.The state satisfies ∥δ + r∗∥ = ∥r∗∥.
  • Equilibria: The system has exactly two isolated equilibria: the desired δ = 0 and the undesired δ = −2r∗.The undesired equilibrium is the target formation's point reflection about the centroid, with opposite bearings but the same centroid, scale, and shape.
  • Stability analysis: The undesired equilibrium is unstable because its Jacobian is symmetric positive semidefinite with at least one positive eigenvalue.The desired equilibrium's Jacobian is symmetric negative semidefinite, so indirect linearization alone does not establish its stability.
  • Stability analysis: Theorem 11 establishes exponential convergence to δ = 0 from every point on the state sphere except δ(0) = −2r∗.Thus the formation is almost globally exponentially stable under the stated rigidity assumption.
  • Convergence rate: The smallest positive eigenvalue λd+2 of ˜R^T(p∗)˜R(p∗) affects convergence rate and measures the degree of infinitesimal bearing rigidity.λd+2 > 0 exactly when the target formation is infinitesimally bearing rigid.

C. Collision Avoidance

The collision-avoidance analysis supplies a sufficient initial-error condition for maintaining a minimum separation, while the control law alone cannot guarantee collision avoidance globally. The sufficient bound is conservative and shrinks with the number of agents.

  • Limitation: The control law cannot globally guarantee collision avoidance, so the stability result is valid only until a neighbor collision occurs.The paper suggests combining the controller with mechanisms such as artificial potentials in practice.
  • Scope of guarantee: The allowable upper bound on the initial error is inversely proportional to √n.The authors describe this condition as conservative, noting simulations avoided collisions even when it was not satisfied.

IV. BEARING-ONLY FORMATION CONTROL WITHOUT A GLOBAL REFERENCE FRAME

Without a global reference frame, agents use local bearing and relative-orientation measurements to control positions and orientations in three dimensions. The distributed law synchronizes orientations while stabilizing the target formation almost globally.

  • Problem setting: The setting considers agents in R3 whose global reference frame is unknown and whose measurements are expressed in local body frames.Each agent has position, linear velocity, angular velocity, and a body-to-global rotation matrix.
  • Control design: The distributed control law is designed to control both the position and orientation of every agent.The inputs include body-frame linear and angular velocities.
  • Measurements: Each agent measures neighboring bearings in its local frame and relative orientations through Q_i^T Q_j.The control problem is formulated using only these local measurements.
  • Synchronization and stability: The orientations synchronize to a common value, after which the synchronized local frames serve as a common frame for satisfying the bearing constraints.The final synchronized orientation itself is not the quantity of interest; the formation shape is.

A. A Bearing-Only Control Law

The paper proposes distributed bearing-only position and orientation control laws, including one implementable without a global frame. Under the control law, formation centroid and scale remain invariant, with bounded agent positions and pairwise distances.

  • A. A Bearing-Only Control Law: The proposed position and orientation control laws are distributed, and the no-global-frame version requires only local bearing measurements.The control law can be implemented without knowledge of the global frame.
  • A. A Bearing-Only Control Law: The closed-loop position dynamics preserve the formation centroid and scale under the proposed control law.The position derivative is orthogonal to the translation and scaling directions.
  • A. A Bearing-Only Control Law: The centroid and scale invariance results also apply to any position control law with the stated structural form.This extends the invariance property beyond the specific proposed law.
  • A. A Bearing-Only Control Law: The formation trajectory satisfies s ≤ maxi∈V ∥pi(t) −¯p∥≤s√n −1 for all t ≥0.The same result gives the pairwise-distance bound ∥pi(t) −pj(t)∥≤2s√n −1 for all agents and all t ≥0.

B. Formation Stability Analysis

The stability analysis treats the orientation and position dynamics as a cascade: orientations synchronize first, while the position subsystem receives a vanishing input. This yields almost-global convergence to the desired bearing formation, with one undesired opposite-bearing equilibrium.

  • B. Formation Stability Analysis: The closed-loop system is a cascade in which orientation dynamics are independent of position dynamics, but position dynamics depend on orientation.The analysis uses orientation synchronization and input-to-state stability of the position subsystem.
  • B. Formation Stability Analysis: Under Assumption 2 and a fixed strongly connected graph, the orientation control law guarantees asymptotic orientation synchronization to a common Q∗.The specific value of Q∗ is not needed for formation stability.
  • B. Formation Stability Analysis: The target formation exists and is unique under Assumptions 1 and 2.The target formation is defined through the desired inter-agent bearings.
  • B. Formation Stability Analysis: The closed-loop system has two equilibria: the desired δ = 0 and the undesired δ = −2r∗, with all orientations equal to Q∗.The undesired formation has the same centroid, scale, and shape as the desired one but opposite bearings.
  • B. Formation Stability Analysis: The trajectory δ(t) asymptotically converges to δ = 0 from any initial state on S except a set of measure zero.The proof combines almost-global ISS of the position subsystem with the asymptotic convergence of h(t) to zero.
  • B. Formation Stability Analysis: Consequently, the inter-agent bearings almost globally converge to Q∗g∗ij as t →∞.This establishes that the control law solves the bearing-only formation stabilization problem.

V. SIMULATION EXAMPLES

The simulations use randomly generated initial positions, and the reported examples illustrate convergence under bearing-only control with and without a global reference frame.

  • V. SIMULATION EXAMPLES: Figures 11 and 12 use random initial positions and orientations to illustrate the control law without a global reference frame.The reported simulations show that agent orientations finally synchronize.

VI. CONCLUSIONS AND FUTURE WORKS

The paper extends bearing rigidity theory to arbitrary dimensions and applies it to distributed bearing-only formation stabilization. It concludes with almost-global stability results and identifies directed interaction topologies as future work for control stability analysis.

  • VI. CONCLUSIONS AND FUTURE WORKS: The proposed bearing rigidity theory applies to arbitrary dimensions and characterizes unique determination by bearings through infinitesimal bearing rigidity.The characterization is up to translation and scaling, and infinitesimal bearing rigidity can be checked by a rank condition.
  • VI. CONCLUSIONS AND FUTURE WORKS: In R2, infinitesimal bearing rigidity is equivalent to infinitesimal distance rigidity.The paper also explores the connection between the two rigidity notions.
  • VI. CONCLUSIONS AND FUTURE WORKS: Two distributed bearing-only formation control laws are proposed for settings with and without global reference frames, and almost-global formation stability is proved.The conclusion summarizes the paper's application of bearing rigidity theory to formation stabilization.
  • VI. CONCLUSIONS AND FUTURE WORKS: Bearing rigidity results remain valid for directed graphs, but the stability analysis of the proposed control laws is valid only for undirected graphs.Bearing-only formation control with directed interaction topologies is identified as future work.

APPENDIX

The appendix connects bearing rigidity in R2 to distance rigidity through equality of their rigidity-matrix ranks, then uses this relationship to establish equivalent infinitesimal-rigidity conditions.

  • Distance rigidity: The distance rigidity matrix is the Jacobian of the framework’s edge-length distance function, and its null motions define infinitesimal distance rigidity.An infinitesimal distance motion preserves edge lengths; rigidity requires such motions to be only rigid-body rotations and translations.
  • Rank conditions: In R2 with n ≥ d, infinitesimal distance rigidity requires rank(RD(p)) = 2n −3, while in R3 it requires rank(RD(p)) = 3n −6.These rank conditions are the dimension-specific cases of the distance-rigidity rank criterion.
  • Numerical illustrations: The appendix includes simulations for cases without a global reference frame in R2 with n = 4, m = 5 and R3 with n = 8, m = 13.The figures identify the dimensions and framework sizes for the two illustrated cases.
  • Matrix relationship: For frameworks in R2, the bearing rigidity matrix and distance rigidity matrix always have the same rank.This equality is stated as Proposition 2 and derived by rewriting the matrices using a π/2 rotation, whose associated transformation is invertible.
  • Theorem 8: In R2, a framework is infinitesimally bearing rigid if and only if rank(R(p)) = 2n −3, which is equivalent to infinitesimal distance rigidity.The equivalence follows by combining the bearing-rigidity criterion with Proposition 2 and the distance-rigidity criterion.
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