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Vision-based Underwater Formation Control With Input Saturations via Barrier Lyapunov Functions

Nicola De Carli, João Zenário, Victor Nan Fernandez-Ayala, Dimos V. Dimarogonas

arXiv:2609.25917v1cs.ROeess.SY

TL;DR

Vision-based underwater formation control must account for limited sensing and preserve visibility of neighboring robots. This paper combines recentered barrier Lyapunov functions, command-filtered backstepping, and an actuator-aware quadratic program; Gazebo SITL simulations show adaptive relaxation while physical constraints remain satisfied.

  • Problem

    Vision-based formation control must account for limited field of view and sensing range to preserve visibility, while barrier Lyapunov functions become undefined after constraint violations.

    Source: §I · p. 1

  • Method

    A communication-free framework combines recentered barrier Lyapunov functions, command-filtered backstepping, and an actuator-aware quadratic program for underwater formation control.

    Source: §I · p. 1

  • Results

    In Gazebo SITL, adaptive distance-bound relaxation enlarged the conservative bound while the physical constraint remained satisfied.

    Source: §IV · p. 7

  • Takeaways & Limitations

    The approach was validated through realistic Software-in-the-Loop simulations with underwater vehicles.

    Source: §V · p. 8

    The framework assumes a common reference frame, which future work will seek to remove.

    Source: §V · p. 8

Abstract

from arXiv · show

In this work, we propose a communication-free framework for vision-based formation control of fully actuated underwater robots subject to sensing constraints, collision-avoidance requirements, and input saturations. Recentered barrier Lyapunov functions encode sensing and collision-avoidance constraints, while command-filtered backstepping extends the design to the second-order vehicle dynamics. The resulting control objective is enforced through a quadratic program that explicitly accounts for actuator limits. Conservative sensing domains provide margins from the physical limits and are adaptively relaxed when necessary, allowing temporary violation of the conservative bounds. The proposed approach is validated through realistic Software-in-the-Loop (SITL) simulations in Gazebo.

I. INTRODUCTION … A. System Model

The paper proposes communication-free formation control for fully actuated underwater vehicles while addressing sensing, collision-avoidance, and actuator constraints. Recentered barrier Lyapunov functions, command-filtered backstepping, and a saturation-aware quadratic program form the design, with adaptive relaxation of conservative constraint domains.

I. INTRODUCTION

  • Underwater formation control is challenged by uncertain hydrodynamics, unavailable GPS, and acoustic links limited in range, bandwidth, and reliability.These limitations make multi-robot coordination and deployment difficult.Source: §I · p. 1
  • Camera-based formation can avoid explicit communication, but limited field of view and sensing range require explicit visibility constraints.Barrier Lyapunov functions repel the system from constraint boundaries but become undefined after a constraint is violated, complicating temporary violations under conflicting objectives or actuator limits.Source: §I · p. 1
  • Related prescribed-performance approaches relax constraints temporarily, using either box constraints or smooth minimum operators with a common relaxation variable.The proposed method instead adapts each constraint independently according to its margin and is not restricted to box constraints.Source: §I · p. 1
  • The framework combines recentered barrier Lyapunov functions, command-filtered backstepping, a thruster-coordinate quadratic program, and adaptive, margin-based relaxation for independently handled constraints.A domain is enlarged only when its margin is small and the zero-relaxation CLF condition conflicts with actuator limits, then recovered toward its nominal value.Source: §I · p. 1

Notation

  • Notation defines vector, matrix, and set conventions; Euclidean and weighted norms; distance to a set; quaternion and rotation-group symbols; and skew, cross-product, and positive-part operators.The passages also define the identity and zero matrices, trace, vector concatenation, and the unit three-sphere.Source: Notation · pp. 1–2

A. System Model

  • Each vehicle’s pose comprises position and unit-quaternion orientation, while its body-fixed generalized velocity comprises linear and angular velocity.The team consists of fully actuated underwater vehicles with inertial, body-fixed, and camera frames.Source: A. System Model · p. 2
  • Vehicle dynamics include inertia, Coriolis and centripetal effects, hydrodynamic damping, and restoring forces; thruster forces generate the wrench subject to componentwise bounds.The allocation matrix has full row rank, and actuator limits are represented directly as box constraints.Source: A. System Model · p. 2

Camera Model:

The camera model uses a calibrated, rigidly attached forward-looking depth camera and defines visibility through front-facing, field-of-view, and range constraints. Visibility also depends on the observing robot’s translation and rotation.

Camera Model:

  • Each robot has a rigidly attached forward-looking depth camera whose optical axis is the camera-frame x-axis; its known, calibrated extrinsic parameters are assumed constant.The camera-frame y- and z-axes define the horizontal and vertical image directions, respectively.Source: Camera Model · p. 2
  • A target is detectable only in front of the camera, within its horizontal and vertical field of view, and within the maximum sensing range dmax.The range limit depends on hardware and operating conditions; underwater turbidity can reduce visibility, and dmax is defined using inter-robot distance rather than camera-center-to-target distance.Source: Camera Model · p. 2
  • Because image coordinates depend on relative position and the observing robot’s orientation, preserving visibility is coupled to the robot’s translational and rotational motion.The field-of-view constraints are expressed in normalized form.Source: Camera Model · p. 2

B. Interaction Graph

The interaction graph models camera-based sensing rather than communication: each robot uses its own state and relative measurements of neighbors. The assumed topology is a directed spanning tree rooted at the leader.

B. Interaction Graph

  • Robots exchange no information; each uses only its own state and camera-based relative measurements of neighboring vehicles.The architecture avoids inter-robot communication because underwater links typically have limited bandwidth, significant latency, and reduced reliability.Source: B. Interaction Graph · pp. 2–3
  • A directed edge (i, j) means robot i observes robot j, and field-of-view limits mean sensing need not be reciprocal.Robot i’s out-neighbors are the vehicles it observes through its onboard camera.Source: B. Interaction Graph · p. 3
  • The leader–follower topology assumes each follower observes a unique parent and the sensing graph is a directed spanning tree rooted at the leader [17, Section 3.3].Source: B. Interaction Graph · p. 3

C. Problem Statement

The problem is to design decentralized formation controllers that satisfy actuator limits and preserve physical sensing and collision constraints as control authority allows, while maintaining bounded tracking and formation errors. Under constant references and nominal conditions, the errors should converge to zero.

  • Robot 1 tracks an available desired trajectory, while each follower is assigned a desired relative displacement from its unique parent.Source: C. Problem Statement · p. 3
  • The nominal controller uses conservative domains strictly inside the physical admissible domains, with desired relative poses inside the conservative domains and initial poses inside the physical domains.Source: C. Problem Statement · p. 3
  • Every sensing edge must begin with a relative pose inside its physical admissible domain.Source: C. Problem Statement · p. 3
  • Controllers must satisfy actuator limits and preserve physical constraints when control authority permits, while adaptively enlarging conservative barrier domains only within physical domains and recovering them when possible.Source: C. Problem Statement · p. 3
  • For time-varying references, tracking and formation errors must remain bounded; under constant references and nominal conditions, both errors must converge to zero.Source: C. Problem Statement · p. 3

III. CONTROL DESIGN

The controller combines command-filtered backstepping with a BLF-derived desired velocity to address formation, sensing, and collision avoidance under actuator limits, while adapting conservative BLF domains toward physical limits.

III. CONTROL DESIGN

  • Command filtering avoids differentiating the BLF-derived desired velocity, making the CLF derivative affine in thruster forces for actuator-bounded QP enforcement with relaxed dissipation.Auxiliary dynamics adapt the BLF domains between their conservative and physical limits.Source: §III · p. 3

A. Recentered Barrier Design

The design combines formation objectives with sensing and collision-avoidance constraints in a composite barrier Lyapunov function. Recentered logarithmic barriers vanish, with zero gradient, at desired poses and diverge at constraint boundaries.

A. Recentered Barrier Design

  • Each constraint uses a continuously differentiable function that is positive on admissible poses and zero at the conservative boundary, paired with a recentered barrier.The barrier has zero value and gradient at the desired pose and diverges at the boundary.Source: A. Recentered Barrier Design · pp. 3–4
  • With initially positive conservative constraints, d_ij > d_min, and |α_h,ij|, |α_v,ij| < 1, the conservative collision-avoidance and sensing constraints remain satisfied for all t ≥ 0.Under these conditions, p^C_x,ij(t) remains positive for all t ≥ 0.Source: A. Recentered Barrier Design · p. 3
  • Bounded relaxation states s_ℓ,i ∈ [0, 1] shift each conservative boundary toward its physical limit: s_ℓ,i = 0 is nominal, while s_ℓ,i = 1 reaches that limit.The relaxation-state dynamics are deferred to Sec. III-D.Source: A. Recentered Barrier Design · pp. 3–4
  • The leader tracks an absolute position reference, while each follower's objective regulates relative position error and includes an image-centering potential.The follower objective is defined relative to its unique parent.Source: A. Recentered Barrier Design · p. 4
  • For a follower with α^d_v,ij = 0, the desired set S_ij uniquely minimizes its potential, and every critical point outside S_ij is a strict saddle.The condition α^d_v,ij = 0 corresponds to centering the parent vertically in the image.Source: A. Recentered Barrier Design · p. 4

B. Command-Filtered Backstepping

The section uses generalized velocity as a virtual control to connect pose-level barrier potentials to acceleration-level wrench dynamics, then command-filters the desired velocity to avoid unavailable derivatives.

  • Because barrier potentials depend on pose while the wrench enters at generalized acceleration, backstepping first treats generalized velocity as the virtual control for the kinematic subsystem.Source: B. Command-Filtered Backstepping · p. 5
  • Followers use camera-measured parent-relative position, so unavailable parent velocity prevents cancellation of χ_i, which remains an interconnection term without inter-robot communication.Source: B. Command-Filtered Backstepping · p. 5
  • The virtual generalized-velocity command adds negative-definite barrier-gradient feedback, while the interconnection and relaxation terms remain disturbances.Source: B. Command-Filtered Backstepping · p. 5
  • A first-order command filter avoids differentiating desired velocity, whose follower derivatives would require unmeasured parent velocity, and provides its derivative directly.Source: B. Command-Filtered Backstepping · p. 5
  • The filtered command and actual velocity define separate command-filter and tracking errors, which enter the decomposition of actual velocity around feedforward and barrier feedback.Source: B. Command-Filtered Backstepping · p. 5
  • The backstepped velocity-error Lyapunov derivative retains barrier-gradient, filter-error, and interconnection terms, while an affine thruster-force term supports dissipation constraints in the following subsection.Source: B. Command-Filtered Backstepping · pp. 5–6

C. CLF-QP-Based Control Design

The design enforces desired Lyapunov dissipation through a CLF-QP that accounts for actuator limits. Under nominal conditions, the desired formation set has asymptotic stability and almost-global attractivity guarantees, with local robustness to disturbances.

C. CLF-QP-Based Control Design

  • The CLF-QP imposes desired dissipation while respecting actuator limits, centers actuation on a nominal trim allocation, and penalizes CLF relaxation.The trim allocation balances the restoring wrench; for overactuated vehicles it may be the minimum-norm admissible solution.Source: C. CLF-QP-Based Control Design · p. 6
  • With zero CLF relaxation and inactive adaptive-domain dynamics, the desired set Ai is asymptotically stable and almost globally attractive relative to the barrier domain when disturbances vanish.This holds under Proposition 2’s hypotheses; almost-global attractivity excludes initial conditions in stable manifolds of undesired saddle sets.Source: C. CLF-QP-Based Control Design · p. 6
  • Near Ai, the closed-loop subsystem is input-to-state stable with respect to di = col (˜νi, vj).The local robustness claim is established through a local ISS-Lyapunov inequality.Source: C. CLF-QP-Based Control Design · p. 6
  • Without thruster bounds, the CLF constraint is always feasible with zero relaxation.When eν,i ≠ 0 the affine input term can enforce dissipation; when eν,i = 0, ai = bi = 0.Source: C. CLF-QP-Based Control Design · p. 6

D. Adaptive Barrier-Domain Enlargement

The section defines when conservative sensing and collision-avoidance domains enlarge and establishes conditions under which adaptive boundaries remain outside the state. It also specifies when enlargement stops and physical-constraint preservation depends on control authority and parent motion.

D. Adaptive Barrier-Domain Enlargement

  • The conservative domain enlarges only near its adaptive boundary when actuator limits make the zero-relaxation CLF condition infeasible; if initially unrelaxed and feasible, nominal Proposition 3 is recovered.Source: D. Adaptive Barrier-Domain Enlargement · pp. 6–7
  • When the constraint margin is small, recovery is disabled and the adaptive domain cannot shrink; when recovery is active, the relaxation state decays exponentially toward zero.Source: D. Adaptive Barrier-Domain Enlargement · p. 7
  • Under the stated finite-time conditions, each adaptive state remains in [0, 1] and its constraint boundary stays strictly outside the state.The proposition assumes an initially positive constraint function, bounded d_i, and δ_i ∈ L1([0, T)).Source: D. Adaptive Barrier-Domain Enlargement · p. 7
  • At full enlargement, the adaptive and physical boundaries coincide, so no further enlargement is possible and preservation of the physical constraint depends on control authority and, for relative constraints, parent motion.Source: D. Adaptive Barrier-Domain Enlargement · p. 7

IV. SIMULATION RESULTS

The simulations evaluate five-robot formation control in Gazebo/PX4 SITL and a two-robot stress test of adaptive-domain relaxation under limited control authority. The reported results show the adaptive bound enlarging while the physical distance constraint remains satisfied.

IV. SIMULATION RESULTS

  • The nominal SITL evaluation used piecewise-constant formation references, a leader trajectory, specified controller and barrier gains, and QP parameters.The passage gives Kη,i = diag(0.55, 0.55, 0.55, 0.80, 0.80, 0.80), Ke,i = 0.8 Mi, Λi = diag(3, 3, 3, 4, 4, 4), barrier weights µδ,i = µ∆,i = 0.18 and µh,i = µv,i = 0.25, and QP parameters Hi = 3.77×10−4I8, p1,i = 100, and p2,i = 5 × 103.Source: §IV · p. 7
  • Five-robot SITL simulations in an experimental water-tank model included vehicle-parameter mismatch and unmodeled thruster dynamics; Fig. 4 shows sensing constraints and formation errors.The sensing-graph topology and water-tank model are shown in Fig. 3.Source: §IV · p. 7
  • When follower thrust limits were reduced to 20% of nominal, the leader moved away at 0.45 m/s for 5 s from t = 60 s, inducing saturation and distance relaxation; the adaptive bound enlarged while the physical constraint remained satisfied.The reported maximum was 2.40 m.Source: §IV · p. 7

V. CONCLUSIONS

The paper presents a decentralized underwater formation-control framework for limited sensing and bounded actuation. It combines adaptive sensing constraints with dynamics- and actuator-aware control, validates the approach in SITL Gazebo simulations, and identifies two directions for future work.

V. CONCLUSIONS

  • The framework addresses underwater multi-robot formation control under limited sensing and bounded actuation.Constraint satisfaction is encoded through barrier Lyapunov functions.Source: §V · p. 7
  • Conservative sensing domains are adaptively enlarged when necessary, while command-filtered backstepping and a thruster-level CLF-QP account for vehicle dynamics and actuator limits.The framework uses barrier Lyapunov functions to encode the constraints.Source: §V · pp. 7–8
  • The approach is validated through realistic SITL Gazebo simulations with underwater vehicles.The reported SITL results involve five robots.Source: §V · p. 8, Figure 4 · p. 8
  • Future work will lift the common-reference-frame assumption and deploy the algorithm fully onboard the robots.Source: §V · p. 8
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