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Decentralized Safe Path Following for Multiple Quadrotors on Intersecting Paths with Theoretical Guarantees

Hamza Tariq, Adeel Akhtar

arXiv:2610.00208v1eess.SYcs.ROcs.SYmath.OC

TL;DR

Multiple quadrotors on intersecting routes need to avoid collisions without leaving their assigned paths. The paper reformulates TFL as a QP that relaxes only along-path speed, and reports guarantees under stated assumptions alongside simulations in which conflicts are absorbed through speed rather than path or heading deviations.

  • Problem

    Existing approaches do not provide a decentralized guarantee of both collision avoidance and path invariance for multiple quadrotors on intersecting paths.

    Source: §I · p. 1

  • Method

    The method reformulates TFL as a QP with four objective constraints, relaxing only the speed constraint while retaining transverse and heading constraints and adding ECBF safety constraints.

    Source: §I · pp. 1–2

  • Results

    Under the stated assumptions and admissible initial conditions, the controller guarantees path following, collision avoidance, and avoidance of attitude singularities; Drake simulations show conflicts absorbed through speed while cascaded baselines leave paths and perturb heading.

    Source: D. Safe Path Following Guarantees · p. 7, §VI · p. 8

  • Takeaways & Limitations

    The simulations indicate that relaxing speed alone can resolve intersection conflicts while preserving the assigned paths and nominal heading dynamics.

    Source: §VI · p. 8

    Safety does not guarantee continued progress along a path, and symmetric encounters can in principle bring agents to a permanent stop.

    Source: D. Safe Path Following Guarantees · p. 7

Abstract

from arXiv · show

This paper studies decentralized safe path following for multiple quadrotors on intersecting paths, where safety requires both collision avoidance and strict adherence to pre-assigned routes. The proposed controller reformulates transverse feedback linearization as a constrained quadratic program with four equality constraints: two enforce convergence to and strict adherence to the path, and two prescribe the desired speed and heading. Safety is achieved by relaxing only the along-path speed constraint, while the path and heading constraints remain hard. Under the stated assumptions, and for admissible initial conditions, the controller guarantees that all agents converge to and thereafter follow their assigned paths, avoid collisions, and avoid attitude singularities. We evaluate the controller in the Drake physics engine on non-planar intersecting paths and compare it against two nominal-plus-safety-filter cascades. Code and additional results are available at https://gradslab.github.io/safe_multiquad_pf/.

I. INTRODUCTION

Safe path following must prevent collisions while keeping vehicles on assigned routes, since route deviations can create hazards and degrade sensing. The paper addresses the gap with a decentralized TFL-based QP that relaxes only speed, while ECBF constraints support safety.

I. INTRODUCTION

  • Route deviations can create collision hazards and worsen sensor measurements because recordings depend on where a vehicle flies.Path invariance is especially relevant in narrow corridors, tunnels, and close-range inspections requiring a prescribed camera heading.Source: §I · p. 1
  • No decentralized controller was known to guarantee both collision avoidance and path invariance for multiple quadrotors on intersecting paths.Prior approaches based on path design or timing coordination do not suit confined settings with shared intersections; safety filters generally do not preserve path invariance.Source: §I · p. 1
  • The controller reformulates TFL as a four-equality QP and relaxes only speed, preserving the transverse path and heading dynamics while adding ECBF safety constraints.Each equality corresponds to one control objective; keeping constraints hard preserves those objectives but shrinks the feasible set.Source: §I · pp. 1–2
  • The proposed decentralized controller guarantees path invariance, collision avoidance, and singularity avoidance under the stated assumptions.The paper reports this as the first such controller for multiple quadrotors on intersecting paths.Source: §I · p. 2

A. Exponential Control Barrier Functions

  • An ECBF supports safety constraints for systems with relative-degree greater than one; enforcing its condition renders the associated safe set forward invariant.The passage defines a safe set through h(x) ≥ 0 and states that satisfying the ECBF constraint preserves that set.Source: A. Exponential Control Barrier Functions · p. 2

B. Multi-Quadrotor System Model

The model considers homogeneous quadrotors whose agents compute controls locally while sharing state over a complete communication graph.

B. Multi-Quadrotor System Model

  • The system comprises N homogeneous quadrotors on a complete communication graph, with each agent computing its own input locally.Each agent can transmit its state to every other agent; this is the paper's sense of decentralization.Source: B. Multi-Quadrotor System Model · p. 2
  • The modeled vehicle state includes position and velocity, roll-pitch-yaw angles, and body rates, with mass, inertia, gravity, torques, and collective thrust in the dynamics.The passage specifies inertia as J = diag(Ix, Iy, Iz) and defines b3 as the vertical unit vector.Source: B. Multi-Quadrotor System Model · p. 2
  • The system's output is inertial position, represented as yi = xi_q.Source: B. Multi-Quadrotor System Model · p. 2

C. Dynamic Transverse Feedback Linearization (TFL)

Dynamic TFL turns path following into primary path-convergence and secondary speed-and-heading objectives. A local coordinate construction and dynamic extension support four decoupled control channels for those objectives.

C. Dynamic Transverse Feedback Linearization (TFL)

  • The primary objective is convergence to and continued residence on the prescribed path, while the secondary objective sets along-path speed and yaw.The path-coordinate rate is the along-path speed, and the desired speed and yaw may be specified as constants.Source: C. Dynamic Transverse Feedback… · pp. 2, 4
  • Dynamic extension appends two integrators to thrust, after which the extended model is written in control-affine form and transformed into four decoupled integrator chains.The chains are the ξ-, ζ-, η-, and μ-subsystems.Source: C. Dynamic Transverse Feedback… · pp. 2–3
  • Two transverse coordinates define the path-following manifold, and stabilizing its maximal controlled-invariant subset achieves path convergence and invariance.The lifted path set Γ_i is defined by two transverse coordinates; the path-following manifold is its maximal controlled-invariant subset.Source: C. Dynamic Transverse Feedback… · p. 3
  • The construction uses a path coordinate and yaw as virtual outputs, with the path coordinate allowed to use a nearest-point map or another smooth function meeting stated conditions.For closed paths, the path coordinate is interpreted in a local chart; the controller is unchanged by an additive change of chart lift.Source: C. Dynamic Transverse Feedback… · p. 3
  • The operating set requires vector relative degree {4, 4, 4, 2}, a local-diffeomorphism coordinate map, and a nonsingular decoupling matrix.For the path representations used in the simulations, the relative-degree conditions are confirmed by direct computation.Source: C. Dynamic Transverse Feedback… · p. 3
  • The TFL controller stabilizes the transverse and tangential subsystems to achieve path following and the prescribed speed and yaw, provided the decoupling matrix stays invertible.The controller is undefined at roll or pitch angles of ±π/2, which the paper identifies as attitude singularities.Source: C. Dynamic Transverse Feedback… · p. 4

III. PROBLEM FORMULATION

The problem is to design decentralized quadrotor controllers that achieve path following while avoiding collisions and attitude singularities. A nominal controller cascaded with a safety filter may perturb both the route and heading when paths intersect.

III. PROBLEM FORMULATION

  • On intersecting paths, a safety filter can preserve collision avoidance yet perturb the nominal controller enough to break path invariance and turn body-fixed sensors away from their targets.The filter chooses the smallest control change that enforces the collision constraint.Source: §III · p. 4
  • The design goal is decentralized collision avoidance while achieving the primary path-following objective and avoiding attitude singularities for all time.The TFL controller is well-defined only away from those singularities.Source: §III · p. 4
  • Assumption 1 requires each quadrotor’s collective thrust to remain positive, excluding unusual conditions such as free fall and aggressive acrobatics.The condition can also be enforced with an additional ECBF constraint.Source: §III · p. 4
  • For admissible initial conditions near the path-following manifold, the controller must make that manifold locally exponentially attractive, maintain an attitude-singularity margin, and guarantee pairwise separation of at least d_s.These requirements apply under Assumption 1 to the extended quadrotor model.Source: §III · p. 4

IV. PROPOSED CONTROLLER

The nominal TFL controller is reformulated as a quadratic program so safety constraints can be added while selectively relaxing only its speed component. The path and heading requirements remain hard constraints.

IV. PROPOSED CONTROLLER

  • Reformulating TFL as a parametric QP allows safety constraints to be added while relaxing only the controller’s speed component.The TFL input relation is affine in the control because its terms depend only on the state.Source: §IV · p. 4
  • On U_i, invertibility of D_i makes the equality-constrained TFL-QP’s unique optimizer equal to the nominal TFL controller.Source: §IV · p. 4
  • The two transversal constraints remain hard, preserving the nominal transverse dynamics exactly whenever the QP is feasible.Source: §IV · p. 4
  • The heading constraint also remains hard to preserve the pointing direction of body-fixed sensors.Source: §IV · p. 4

A. Roll and Pitch Constraints

Roll and pitch bounds are represented by upper- and lower-side barriers with a prescribed margin from attitude singularities. ECBF constraints enforce those barriers.

A. Roll and Pitch Constraints

B. Pairwise Collision Constraints

Pairwise collision avoidance is encoded by a safe set and an ECBF constraint for each agent pair. Responsibility weights split the centralized constraint into decentralized local constraints.

B. Pairwise Collision Constraints

  • The pairwise collision barrier defines a safe set requiring inter-agent separation of at least d_s.Source: B. Pairwise Collision Constraints · p. 5
  • The pairwise safe-set barrier has relative degree 4 under the combined dynamics on the stated domain.The passage identifies the rank-3 input matrix for the fourth derivative of position.Source: B. Pairwise Collision Constraints · p. 5
  • Responsibility weights split each centralized pairwise ECBF inequality into local constraints; summing the two agents’ inequalities recovers the centralized condition.For homogeneous quadrotors, the weights are equal.Source: B. Pairwise Collision Constraints · p. 5

C. Decentralized Multi-Quadrotor Safe Path Following

The decentralized controller incorporates the path-following and safety constraints into a per-agent QP, relaxing only the speed row while retaining other constraints as hard.

C. Decentralized Multi-Quadrotor Safe Path Following

  • The QP uses a scalar slack that perturbs only the speed row, leaving the other controller equalities hard.The safety inequality collects attitude-barrier rows and collision rows involving neighboring agents.Source: C. Decentralized Multi-Quadrotor Safe… · p. 5

D. Safe Path Following Guarantees

Under strict-feasibility assumptions, the controller has a unique, locally Lipschitz solution and preserves exponentially stable transverse and heading dynamics. For admissible initial conditions and forward-complete solutions, it guarantees path following, collision avoidance, and avoidance of attitude singularities, though it does not guarantee continued progress.

D. Safe Path Following Guarantees

  • Strict feasibility of the reduced QP yields a unique minimizer and a locally Lipschitz controller on Ω̄.Assumption 2 requires strict feasibility at every point of Ω̄.Source: D. Safe Path Following Guarantees · pp. 5–6
  • Transverse coordinates converge exponentially to the path-following manifold, and every agent’s heading error converges exponentially to zero.The claim applies to closed-loop trajectories contained in Ω̄.Source: D. Safe Path Following Guarantees · p. 6
  • The safety argument uses a sublevel set that keeps outputs inside their path tubes and attitude angles within barrier limits; forward completeness is also required.For non-compact paths, the relevant bounds must hold uniformly along the path.Source: D. Safe Path Following Guarantees · p. 6
  • For initial states in Ω0 and forward-complete solutions, all agents avoid attitude singularities with margin ϵ and remain at least ds apart.Together with Lemma 3, these guarantees solve the stated problem under Assumptions 1 and 2.Source: D. Safe Path Following Guarantees · pp. 6–7
  • Safety does not guarantee liveness: symmetric encounters can in principle bring agents to a permanent stop.Liveness is continued progress along the path.Source: D. Safe Path Following Guarantees · p. 7
  • After 15 s in the Drake circular-path scenario, the proposed controller stays within 0.5 mm of its path, while both cascades deviate to avoid collisions.The figure compares three-dimensional trajectories and top views.Source: Figure 2 · p. 7

V. SIMULATION RESULTS

In Drake simulations, the proposed controller stayed close to assigned paths and maintained the safety distance, while the cascades showed larger path deviations or heading perturbations. Conflicts were handled through along-path speed, with path and heading channels held at their references.

Evaluation setup

  • In Drake, the QP controller was evaluated against TFL + filter and SE(3) + filter baselines on identical scenarios, including four quadrotors on intersecting 1.5 m-radius circles with safety distance ds = 0.5 m.Source: §V · p. 7

Implementation

  • The 400 Hz implementation approximates the continuous-time controller; the authors claim no separate sampled-data invariance guarantee.The reduced QP ran without a numerical solver, using the clipping rule.Source: §V · p. 7

Circle scenario

  • For t ≥15 s, maximum path distance stayed below 0.5 mm with the proposed controller, versus up to 40 cm with TFL [17] + filter and 71 cm with SE(3) [27] + filter.Minimum pairwise distance never fell below the safety distance for the proposed controller or TFL [17] + filter; SE(3) [27] + filter dipped 7 mm inside it.Source: §V · p. 7
  • Below 10^-3 degrees, proposed heading error stayed; the cascades perturbed heading by up to 1.4° and 19.6°, while the proposed controller absorbed conflicts through along-path speed.The proposed controller held transverse, altitude, and heading channels at their references; both cascades perturbed every channel.Source: §V · p. 7, Figure 3 · p. 8

VI. CONCLUSION

The paper concludes that its decentralized QP controller handles intersecting paths by relaxing only speed while preserving path-following and heading dynamics. Drake simulations show conflicts resolved through speed alone; actuator constraints and fallback control remain future work.

Conclusion

  • The proposed QP preserves transversal path-following and nominal heading-error dynamics while incorporating ECBF safety constraints by relaxing only the speed constraint.Drake simulations showed the controller resolving intersection conflicts through speed alone, unlike safety-filtered baselines that left their paths and perturbed heading.Source: §VI · p. 8
  • The paper presents a decentralized safe path-following controller for multiple quadrotors on intersecting paths.Its stated guarantees include path invariance, collision avoidance, and singularity avoidance under the stated assumptions.Source: §VI · p. 8

Future work

  • Future work will incorporate actuator constraints and design a fallback controller for instances where Assumption 2 fails.Source: §VI · p. 8

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