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arxiv: 2606.23218 · v1 · pith:626VOWVJnew · submitted 2026-06-22 · 🧮 math.OC · cs.SY· eess.SY

Path-following Control of a Quadrotor using Quasi-Static Transverse Feedback Linearization

Mohamed Al Lawati , Adeel Akhtar This is my paper

Pith reviewed 2026-06-26 07:45 UTC · model grok-4.3

classification 🧮 math.OC cs.SYeess.SY
keywords quadrotorpath followingfeedback linearizationtransverse feedbackmanifold invarianceunderactuated controlgeometric pathUAV
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The pith

Quasi-static transverse feedback linearization renders a quadrotor's path-following manifold invariant and locally exponentially stable without extra controller states.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a controller that lets a quadrotor follow a geometric path by making the set of states on that path invariant under the closed-loop dynamics. It computes thrust algebraically from the current state rather than through derivatives or integration, and it regulates speed along the path and yaw angle at the same time. A coordinate transformation turns the transverse error dynamics into a linear form whose origin is shown to be locally exponentially stable. The design inverts only a 3-by-3 matrix, which is smaller than the 4-by-4 matrix required by earlier dynamic-feedback versions of the same idea. Numerical examples illustrate that trajectories started on the path stay on it.

Core claim

The QSTFL controller renders the path-following manifold invariant, ensuring trajectories initialized on the path remain on it for all future time, while regulating tangential velocity and yaw; a diffeomorphic coordinate transformation is established and local exponential stability of the path-following manifold is proved, with closed-form expressions for thrust and torque inputs requiring inversion of only a 3x3 decoupling matrix.

What carries the argument

Quasi-static transverse feedback linearization of the transverse dynamics to the path-following manifold, which produces an invariant manifold without dynamic extension.

If this is right

  • Trajectories that begin exactly on the path stay on it for all future time.
  • Tangential speed and yaw converge to their commanded values while the vehicle remains on the path.
  • Thrust is obtained by direct algebraic evaluation of the state, removing any need to measure or estimate its derivative.
  • The control law inverts a 3x3 matrix instead of a 4x4 matrix, reducing the number of arithmetic operations at each step.
  • Local exponential stability of the manifold holds inside some neighborhood whose size is determined by the linearization.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The algebraic thrust formula may reduce the sensor suite needed compared with methods that differentiate thrust.
  • The same quasi-static construction could be tried on other underactuated rigid-body systems whose output dimension matches the number of controls after the transverse linearization.
  • Because the manifold is invariant, initial errors transverse to the path may be driven to zero without separate capture logic.
  • The method's reliance on an exact diffeomorphism suggests that model mismatch will primarily affect the size of the region of attraction rather than destroy invariance outright.

Load-bearing premise

The quadrotor model together with the chosen path admits a diffeomorphic coordinate change that lets the transverse dynamics be linearized without adding controller states.

What would settle it

A closed-loop simulation started exactly on the path in which the distance to the path grows away from zero would falsify the invariance claim.

Figures

Figures reproduced from arXiv: 2606.23218 by Adeel Akhtar, Mohamed Al Lawati.

Figure 2
Figure 2. Figure 2: Snapshots, and input/output time signals for exampl [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 1
Figure 1. Figure 1: Controlled-invariance results under the QSTFL contr [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
read the original abstract

We propose a quasi-static transverse feedback linearization (QSTFL) controller for a quadrotor to follow a prescribed geometric path, rather than a time-parameterized trajectory. In contrast to existing dynamic-feedback approaches, the controller does not introduce additional controller states. The thrust input is computed algebraically from the current state, eliminating the need for thrust-derivative measurements and numerical integration. The proposed design renders the path-following manifold invariant, ensuring that trajectories initialized on the path remain on it for all future time, while simultaneously regulating tangential velocity and yaw. We establish a diffeomorphic coordinate transformation and prove local exponential stability of the path-following manifold. In addition, closed-form expressions are derived for the thrust and torque inputs. Compared with dynamic-feedback constructions, the controller requires inversion of only a $3\times 3$ decoupling matrix rather than a $4\times 4$ one, leading to a simpler control law and reduced computational complexity. Numerical simulations demonstrate the effectiveness of the proposed method. Code and animations are publicly available at \footnotesize{\texttt{\href{https://gitlab.com/a5akhtar/quasistatic-tfl-uav/}{https://gitlab.com/a5akhtar/quasistatic-tfl-uav/}}}.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper proposes a quasi-static transverse feedback linearization (QSTFL) controller for quadrotor geometric path-following. It asserts that a diffeomorphic coordinate transformation exists allowing the path-following manifold to be rendered invariant without dynamic extension or additional states, that local exponential stability of the manifold can be proved while regulating tangential velocity and yaw, and that closed-form thrust and torque inputs are obtained via algebraic computation requiring inversion of only a 3x3 decoupling matrix (versus 4x4 in dynamic-feedback methods). Effectiveness is illustrated via numerical simulations with publicly available code.

Significance. If the diffeomorphism and stability results hold under the paper's conditions, the approach offers a simpler control law for underactuated UAV path-following by eliminating dynamic extension, thrust-derivative measurements, and higher-dimensional matrix inversions. Public release of code and animations is a positive contribution to reproducibility.

major comments (2)
  1. [Abstract / coordinate transformation derivation] Abstract and coordinate transformation section: the central claim that the quadrotor 6-DOF dynamics plus chosen path admit a diffeomorphism enabling QSTFL (no dynamic extension, 3x3 decoupling matrix always invertible) is load-bearing for manifold invariance and the local exponential stability proof. The manuscript must explicitly state and verify the relative-degree conditions, path curvature bounds, and rank conditions on the decoupling matrix; without these, the algebraic thrust expression and transverse linearization may fail for generic paths, as is typical for underactuated rigid-body systems.
  2. [Stability proof] Stability analysis: the proof of local exponential stability of the path-following manifold is asserted via the diffeomorphic change of coordinates, but the transverse error dynamics, the precise form of the feedback, and any assumptions on initial conditions or path regularity are not detailed enough in the provided material to confirm that the closed-loop transverse subsystem is indeed exponentially stable independently of the tangential dynamics.
minor comments (1)
  1. The public code repository link is helpful; consider adding a brief comparison table (e.g., matrix dimensions, number of controller states, required measurements) against a representative dynamic-feedback TFL method to quantify the claimed reduction in complexity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. We address each major comment below and will incorporate clarifications and expansions in a revised version to strengthen the presentation of the coordinate transformation and stability results.

read point-by-point responses

  1. Referee: [Abstract / coordinate transformation derivation] Abstract and coordinate transformation section: the central claim that the quadrotor 6-DOF dynamics plus chosen path admit a diffeomorphism enabling QSTFL (no dynamic extension, 3x3 decoupling matrix always invertible) is load-bearing for manifold invariance and the local exponential stability proof. The manuscript must explicitly state and verify the relative-degree conditions, path curvature bounds, and rank conditions on the decoupling matrix; without these, the algebraic thrust expression and transverse linearization may fail for generic paths, as is typical for underactuated rigid-body systems.

    Authors: We agree that the diffeomorphism and decoupling-matrix invertibility require explicit conditions. The current manuscript assumes sufficient path smoothness (C^4) and bounded curvature to ensure the transverse coordinates are well-defined and the 3x3 matrix remains nonsingular. In the revision we will add a dedicated subsection (or lemma) that (i) states the relative-degree conditions for the chosen outputs, (ii) gives an explicit curvature bound (e.g., |κ| < κ_max derived from vehicle parameters), and (iii) proves that the decoupling matrix has full rank 3 whenever these bounds hold. This will make clear that the algebraic thrust formula and QSTFL construction are valid on the stated class of paths rather than on completely arbitrary curves. revision: yes

  2. Referee: [Stability proof] Stability analysis: the proof of local exponential stability of the path-following manifold is asserted via the diffeomorphic change of coordinates, but the transverse error dynamics, the precise form of the feedback, and any assumptions on initial conditions or path regularity are not detailed enough in the provided material to confirm that the closed-loop transverse subsystem is indeed exponentially stable independently of the tangential dynamics.

    Authors: The stability argument proceeds by exhibiting a diffeomorphism that decouples the transverse and tangential subsystems; the transverse error dynamics are then rendered exponentially stable by a linear feedback whose gain is chosen from the controllability of the linearized transverse system. The tangential velocity and yaw are regulated separately on the manifold. To address the concern about insufficient detail, the revised manuscript will expand the stability section to include: (a) the explicit coordinate transformation and its inverse, (b) the closed-form expression for the transverse feedback, (c) the resulting linear transverse error system, and (d) the precise assumptions (initial conditions inside a neighborhood of the manifold and path regularity C^4). These additions will confirm that exponential stability of the transverse subsystem holds independently of the tangential dynamics. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained via stated proofs

full rationale

The paper's central claims rest on establishing a diffeomorphic coordinate transformation, proving local exponential stability of the path-following manifold, and deriving closed-form thrust/torque expressions via quasi-static transverse feedback linearization applied to the quadrotor model. These steps are presented as direct mathematical constructions from the system dynamics and path geometry, without any reduction of predictions to fitted parameters, self-definitional loops, or load-bearing self-citations. The abstract explicitly contrasts the approach with dynamic-feedback methods and notes the 3x3 decoupling matrix inversion as a simplification, but provides no evidence that the diffeomorphism or invariance results are assumed or imported circularly. The derivation chain is therefore independent and self-contained against external benchmarks such as the underlying rigid-body dynamics.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The approach rests on standard nonlinear control assumptions regarding the applicability of transverse feedback linearization and coordinate transformations to the quadrotor dynamics; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption The quadrotor dynamics and path admit a diffeomorphic coordinate transformation enabling quasi-static transverse feedback linearization
    Invoked to establish the coordinate change and prove local exponential stability of the manifold.

pith-pipeline@v0.9.1-grok · 5760 in / 1273 out tokens · 48835 ms · 2026-06-26T07:45:26.512585+00:00 · methodology

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Reference graph

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