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Translational and Scaling Formation Maneuver Control via a Bearing-Based Approach

Shiyu Zhao, Daniel Zelazo

arXiv:1506.05636v2eess.SY

TL;DR

The paper addresses translational and scaling maneuver control when target formations are specified by inter-neighbor bearings rather than relative positions or distances. It proposes linear bearing-based controllers for double-integrator agents and analyzes their stability under practical conditions. The results provide globally stable maneuver control in arbitrary dimensions, including settings with disturbances, acceleration saturation, and collision-avoidance requirements.

  • Problem

    Existing bearing-based studies mainly consider static target formations, while time-varying translation and scale and uniqueness of bearing-constrained target formations in arbitrary dimensions remain unresolved.

  • Method

    The paper uses inter-neighbor bearings and the bearing Laplacian to design linear double-integrator control laws, with extensions for disturbances, acceleration saturation, and collision avoidance.

  • Results

    The proposed laws achieve globally stable formation tracking, including global exponential convergence under constant input disturbances and global asymptotic convergence under acceleration saturation.

  • Takeaways & Limitations

    Bearing invariance provides a simple way to control formation translation and scale while preserving the desired geometric pattern in arbitrary dimensions.

Abstract

from arXiv · show

This paper studies distributed maneuver control of multi-agent formations in arbitrary dimensions. The objective is to control the translation and scale of the formation while maintaining the desired formation pattern. Unlike conventional approaches where the target formation is defined by relative positions or distances, we propose a novel bearing-based approach where the target formation is defined by inter-neighbor bearings. Since the bearings are invariant to the translation and scale of the formation, the bearing-based approach provides a simple solution to the problem of translational and scaling formation maneuver control. Linear formation control laws for double-integrator dynamics are proposed and the global formation stability is analyzed. This paper also studies bearing-based formation control in the presence of practical problems including input disturbances, acceleration saturation, and collision avoidance. The theoretical results are illustrated with numerical simulations.

I. INTRODUCTION

The paper formulates translational and scaling maneuver control using inter-neighbor bearings, whose invariance makes the target formation compatible with these maneuvers. It develops linear double-integrator controllers, analyzes stability, and addresses practical control issues in arbitrary dimensions.

  • Motivation: Bearing-based control defines the formation pattern through inter-neighbor bearings, which are invariant to translation and scale.This supports translational maneuvers at a common velocity and scaling maneuvers that preserve the geometric pattern.
  • Motivation: Existing relative-position and distance-based approaches complicate scale control, while the prior complex-Laplacian approach is limited to planar formations.The paper identifies time-varying translation and scale, arbitrary-dimensional applicability, and target-formation uniqueness as unresolved issues.
  • Contributions: The paper analyzes target-formation uniqueness using the bearing Laplacian, which captures both interconnection topology and inter-neighbor bearings.The bearing Laplacian is introduced as the matrix underlying the uniqueness analysis.
  • Contributions: Two linear bearing-based control laws for double-integrator agents track constant or time-varying leader velocities without requiring followers to estimate the desired maneuver.Global formation stability is analyzed for both control laws.
  • Practical issues: Additional control laws address constant input disturbances and acceleration saturation, while sufficient initial conditions ensure collision avoidance between any pair of agents.The collision-free guarantee applies even to agent pairs that are not neighbors.
  • Problem formulation: The formulation models leaders with prescribed motion and followers as double-integrators using relative-position and relative-velocity information over a fixed information-flow graph.The graph defines which agent information is accessible to each agent.
  • Problem formulation: The objective is to design follower acceleration inputs so real inter-neighbor bearings converge to desired bearings and follower position and velocity errors converge to zero.The target formation is jointly constrained by constant bearings and possibly time-varying leader positions.
  • Target formations: A target formation may fail to exist uniquely; nonuniqueness can prevent convergence to the desired geometric pattern despite satisfying the bearing constraints and leader positions.The paper therefore treats existence and uniqueness as a fundamental part of the maneuver-control problem.

B. Properties of the Target Formation

This section introduces the bearing Laplacian and its null-space properties, then uses them to characterize bearing-preserving motions and target-formation uniqueness.

  • B. Properties of the Target Formation: The section develops properties of target formations used throughout the paper.
  • 1) Bearing Laplacian Matrix:: The bearing Laplacian is defined as a block matrix whose edge blocks are orthogonal projection matrices associated with desired bearings.
  • 1) Bearing Laplacian Matrix:: The bearing Laplacian is a matrix-weighted graph Laplacian that characterizes both formation topology and inter-neighbor bearings.
  • 1) Bearing Laplacian Matrix:: Its null space contains translational and scaling motions because these motions preserve all formation bearings.Other bearing-preserving motions may also belong to the null space.
  • 1) Bearing Laplacian Matrix:: For undirected graphs, the null space equals the span of translation and scaling motions exactly when the formation is infinitesimally bearing rigid.
  • 1) Bearing Laplacian Matrix:: Partitioning the bearing Laplacian isolates Bff, the follower-follower submatrix that plays an important role in the subsequent analysis.The partition includes leader-leader, leader-follower, follower-leader, and follower-follower blocks.
  • 1) Bearing Laplacian Matrix:: Under bidirectional information flow among followers, Bff is symmetric and positive semi-definite.The proof decomposes Bff into a follower-subgraph bearing Laplacian and a positive semi-definite block-diagonal matrix.

2) Uniqueness of the Target Formation:

The paper characterizes when bearings and leader positions uniquely determine a target formation, then specifies leader motions that achieve prescribed centroid translation and scale dynamics.

  • 2) Uniqueness of the Target Formation:: The bearing Laplacian analyzes whether feasible bearing constraints and leader positions admit a unique target formation.
  • 2) Uniqueness of the Target Formation:: The target formation is unique if and only if the follower block Bff is nonsingular.
  • 2) Uniqueness of the Target Formation:: A unique target formation requires at least two leaders, while infinitesimal bearing rigidity with two leaders provides a sufficient condition.
  • 2) Uniqueness of the Target Formation:: The analysis assumes the target formation remains unique for all t ≥0, equivalently that Bff stays nonsingular.
  • 3) Target Formation Maneuvering:: The target formation is described through its centroid and scale, whose desired dynamics combine common translation with scale variation.
  • 3) Target Formation Maneuvering:: The scale expands when α(t) > 0 and contracts when α(t) < 0.
  • 3) Target Formation Maneuvering:: Theorem 2 gives leader velocities that achieve the desired centroid and scale dynamics.
  • 3) Target Formation Maneuvering:: Each leader combines the common translational velocity, scale-induced velocity, and the target formation centroid.

III. BEARING-BASED FORMATION CONTROL LAWS

The paper proposes distributed linear bearing-based laws for double-integrator agents, proves convergence under constant leader velocity, and demonstrates maneuvering while preserving the formation pattern.

  • III. BEARING-BASED FORMATION CONTROL LAWS: Two distributed control laws steer followers toward maneuvering target formations.
  • III. BEARING-BASED FORMATION CONTROL LAWS: The first law uses relative position and velocity feedback for constant leader velocities, while the second adds acceleration feedback for time-varying velocities.
  • A. Formation Maneuvering with Constant Leader Velocity: The control term uses positive semidefinite orthogonal projection matrices rather than scalar edge weights, steering neighbor directions toward desired bearings.
  • A. Formation Maneuvering with Constant Leader Velocity: Under control law (7), tracking errors globally and exponentially converge to zero when leader velocity is constant.
  • A. Formation Maneuvering with Constant Leader Velocity: The constant-velocity analysis derives follower error dynamics and a characteristic equation governed by the follower block Bff.
  • A. Formation Maneuvering with Constant Leader Velocity: Because all relevant eigenvalues have negative real parts for positive gains and positive Bff eigenvalues, the error system is Hurwitz.
  • A. Formation Maneuvering with Constant Leader Velocity: When leader velocity varies, the constant-velocity law may not eliminate tracking errors; acceleration feedback is required for perfect tracking.
  • A. Formation Maneuvering with Constant Leader Velocity: With two leaders and two followers, simulation shows continuously varying translation and scale while maintaining the desired square pattern.

B. Formation Maneuvering with Time-Varying Leader Velocity

The paper establishes uniqueness conditions for bearing-defined target formations and proposes a linear control law that tracks time-varying leader velocities with global exponential convergence. Simulations show simultaneous translation and scaling while preserving the desired formation pattern.

  • Control law: The proposed control law uses acceleration feedback to handle time-varying leader velocities without allowing leader acceleration to affect tracking-error convergence.The acceleration-feedback term eliminates the component containing the leader acceleration from the error dynamics.
  • Formation uniqueness: The target formation is unique only when the relevant bearing-defined matrix K_i is nonsingular; singularity occurs when neighboring desired bearings are collinear.If K_i is singular, an agent can move along a desired bearing without changing bearings, so the target formation is not unique.
  • Stability: For any time-varying leader velocity, control law (10) makes the position and velocity tracking errors globally and exponentially converge to zero.The associated error dynamics have eigenvalues in the open left-half plane for positive gains k_p and k_v.
  • Simulation: A three-dimensional cube with two leaders and six followers converges from an initial configuration while continuously varying its translation and scale.The simulation reports that the desired formation pattern is maintained exactly despite time-varying leader velocities.
  • Simulation: The simulation also illustrates obstacle avoidance and passage through narrow passages, while detailed obstacle detection and path generation remain outside the paper’s scope.The paper identifies sophisticated mechanisms such as obstacle detection and path generation as practical requirements for collision avoidance.

IV. BEARING-BASED FORMATION CONTROL WITH PRACTICAL ISSUES

This section examines bearing-based formation control under practical implementation issues, including input disturbances, input saturation, and collision avoidance among agents.

  • Practical issues: The practical-issues analysis covers input disturbances, acceleration saturation, and collision avoidance among the agents.These issues are presented as implementation concerns for bearing-based formation control.

A. Constant Input Disturbance

The paper augments bearing-based control with integral action to address unknown constant follower input disturbances and constant leader acceleration. Under gain conditions, tracking errors converge globally and exponentially.

  • Disturbance model: Unknown constant follower input disturbances can represent constant sensor or actuator biases.The disturbance is modeled as an unknown constant signal for each follower.
  • Integral control: Integral control is added to eliminate constant input disturbances and handle constant nonzero leader acceleration.The same integral mechanism addresses both effects in the stated setting.
  • Stability condition: For control law (12), 0 < k_I < k_p k_v λ_min(B_ff) guarantees global and exponential convergence under constant disturbance and constant leader acceleration.The gain bound depends on the minimum eigenvalue of B_ff.
  • Stability analysis: The stability proof uses a state-space error analysis and the Routh-Hurwitz criterion to place the closed-loop eigenvalues in the open left-half plane.The resulting gain restriction is tied to the eigenvalues of B_ff.
  • Time-varying leader velocity: For control law (14), 0 < k_I < k_p k_v guarantees global and exponential convergence under constant disturbance and time-varying leader velocity.The convergence applies to both position and velocity tracking errors.

B. Acceleration Saturation

The paper studies acceleration saturation by replacing unbounded acceleration commands with bounded saturation functions and proves asymptotic formation stability using Lyapunov analysis. The resulting bounded-input laws remain effective for time-varying leader velocities.

  • Saturation model: Acceleration saturation bounds the input using either a sign-based limiter or β tanh(x), applied component-wise to vector inputs.In both cases, β > 0 is the constant bound on the input magnitude per scalar component.
  • Lyapunov analysis: The saturation-induced nonlinear dynamics are analyzed with Lyapunov functions built from the integral Φ(x) of the saturation function.Φ is nonnegative and vanishes only at zero, while its derivative is related to sat(x).
  • Constant leader velocity: Under control law (15) with constant leader velocity, position and velocity tracking errors globally and asymptotically converge to zero.The proof establishes a nonincreasing Lyapunov function whose invariant zero-derivative set is δp = δv = 0.
  • Time-varying leader velocity: Under control law (18), the acceleration bound is independent of initial formation position and velocity and depends on target rigidity and leader accelerations.The same control law yields global asymptotic convergence for any time-varying leader velocity.
  • Error convergence: For control law (18), convergence is shown by proving ε and ε̇ approach zero, which is equivalent to convergence of δp and δv.The error variables satisfy ε = B_ff δp and ε̇ = B_ff δv.

C. A Collision-Free Condition

The section establishes sufficient initial-condition criteria for preventing collisions under bearing-based control laws. These criteria guarantee pairwise separation, though the authors note they can be conservative in practice.

  • Collision-free guarantee: Theorem 9 guarantees that all agent pairs remain more than the minimum distance γ apart under control law (7) with constant leader velocity.The guarantee applies when the initial position and velocity errors satisfy the theorem’s sufficient condition.
  • Collision-free guarantee: The collision-free condition is obtained by lower-bounding each pairwise distance and requiring that bound to remain greater than γ.The analysis uses a Lyapunov function together with the error dynamics to establish the bound.
  • Condition interpretation: The sufficient conditions require the initial formation to be sufficiently close to the target formation.This is the stated intuition behind the condition in Theorem 9.
  • Condition interpretation: The sufficient conditions are likely conservative: in Figure 4, no collision occurs although inequality (19) fails, with 325.88 on the left and 14.53 on the right for γ = 0.The simulation therefore illustrates that violating the sufficient inequality does not necessarily produce a collision.

V. CONCLUSIONS

The paper proposes and analyzes bearing-based translational and scaling maneuver control for formations in arbitrary-dimensional spaces. It establishes global formation stability while identifying bidirectional information flow and double-integrator dynamics as scope assumptions for future extension.

  • Conclusion: The work proposes and analyzes bearing-based control for translational and scaling formation maneuvers in arbitrary-dimensional spaces.The conclusion frames this as the paper’s central problem and approach.
  • Conclusion: The paper proposes multiple bearing-based formation control laws and analyzes their global formation stability.These are the principal analytical outcomes stated in the conclusion.
  • Future research: The analysis assumes bidirectional information flow between every pair of followers.The directional case is identified as future work involving bearing persistence.
  • Future research: The dynamics are modeled with double integrators, while more complicated models such as nonholonomic systems remain for future study.The paper notes that double-integrator dynamics approximately model some practical physical systems.

APPENDIX

The appendix reviews bearing-rigidity concepts used to characterize formation uniqueness. It defines bearings and the bearing rigidity matrix, then states equivalent conditions for infinitesimal bearing rigidity and uniqueness up to translation and scaling.

  • Bearing-rigidity definitions: For each directed edge, the appendix defines the edge vector e_k and bearing g_k as e_k divided by its norm.The bearings are assembled into the bearing function F_B(p).
  • Bearing-rigidity definitions: The bearing rigidity matrix R_B(p) is defined as the Jacobian of the bearing function F_B(p).The matrix has dimensions R^dm×dn according to the appendix.
  • Bearing-rigidity definitions: Translation and scaling of the entire formation are the two trivial infinitesimal bearing motions.These motions belong to the null space of the bearing rigidity matrix.
  • Equivalent conditions: Theorem 11 equates infinitesimal bearing rigidity with unique determination up to translation and scaling, maximum rank, and the stated null-space characterization.The equivalent rank condition is rank(R_B) = dn − d − 1.
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