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arxiv: 2605.15357 · v1 · pith:DYZDZ267new · submitted 2026-05-14 · 📡 eess.SY · cs.SY

Coordinated Trajectory Control Algorithm for Quadcopter Motion along a Smooth Spatial Trajectory

Stanislav Kim , Anton Pyrkin , Oleg Borisov This is my paper

Pith reviewed 2026-05-19 15:31 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords quadcoptertrajectory trackinggeometric controloutput feedbackdisturbance rejectionspatial trajectorydynamic controller
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The pith

Quadcopters follow smooth 3D trajectories using only position and yaw measurements despite unmeasured disturbances.

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

The paper constructs a complete dynamic model for quadcopter motion restricted to any smooth curve through three-dimensional space. From this model it derives a robust control algorithm that operates on measurements of the three linear coordinates and the yaw angle alone. Additional integrators are inserted into the controller to produce a dynamic law whose tuning is simplified while still rejecting constant disturbances. The law is synthesized by the geometric approach and its stability is proven; an extended observer version supplies the missing velocity and attitude information when only the chosen outputs are available. If the claims hold, coordinated three-dimensional path following becomes possible for quadcopters that lack full state sensors or operate in the presence of unknown forces.

Core claim

A complete model of the motion of a quadcopter along a smooth spatial trajectory is presented. Based on the model, a robust algorithm is proposed for controlling a quadcopter using measurements of linear coordinates and yaw angle. By introducing additional integrators, a dynamic control algorithm with a simplified controller tuning methodology is obtained. The control law is synthesized within the geometric approach, and its stability is proven. A realizable output-feedback version using an extended observer is also given.

What carries the argument

geometric approach applied to the underactuated quadcopter dynamics along a prescribed smooth spatial trajectory

If this is right

  • The geometric control law guarantees stability of the trajectory tracking error.
  • Unmeasured disturbances are rejected by the additional integrators in the dynamic controller.
  • An extended observer recovers the unmeasured states so that output feedback suffices for implementation.
  • The introduction of integrators yields a simplified tuning procedure for the overall controller.

Where Pith is reading between the lines

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

  • The same geometric construction could be tested on other underactuated vehicles whose kinematics allow a similar smooth-trajectory model.
  • Real-flight experiments in wind could quantify how large the unmeasured disturbances can grow before tracking degrades.
  • Relaxing the smoothness requirement on the reference path would require a revised model and stability argument.

Load-bearing premise

The quadcopter dynamics admit a complete model when the vehicle is restricted to smooth spatial trajectories, and the geometric approach applies directly to the underactuated system with the chosen partial measurements.

What would settle it

Simulate or fly the closed-loop system along a known smooth 3D curve while injecting unmeasured constant disturbances and supplying only linear position plus yaw feedback; the claim is falsified if the tracking error fails to remain bounded or converge.

Figures

Figures reproduced from arXiv: 2605.15357 by Anton Pyrkin, Oleg Borisov, Stanislav Kim.

Figure 1
Figure 1. Figure 1: Quadcopter motion along a spatial trajectory. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Definition of deviations: a — planes E1 and E2; b — projection onto the horizontal plane. OXZ, taking into account Remark 2) and a rotation about the Y axis by an angle β (the angle between the planes E2 and OXY ). Differentiating, we obtain the dynamic model for the linear coordinates:   s˙ e˙1 e˙2   =   cαcβ sαcβ sβ −sα cα 0 −cαsβ −sαsβ cβ     vx vy vz   ,   v˙x v˙y v˙z   =   cϕsθcψ + s… view at source ↗
read the original abstract

A complete model of the motion of a quadcopter along a smooth spatial trajectory is presented. Based on the model, a robust algorithm is proposed for controlling a quadcopter using measurements of linear coordinates and yaw angle. By introducing additional integrators, a dynamic control algorithm with a simplified controller tuning methodology is obtained. The control law is synthesized within the geometric approach, and its stability is proven. A realizable output-feedback version using an extended observer is also given. The results enable coordinated trajectory following in three-dimensional space despite unmeasured disturbances and incomplete state information.

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 / 2 minor

Summary. The paper constructs a complete model of quadcopter motion along a smooth spatial trajectory, synthesizes a geometric controller augmented with integrators whose stability is proven, and derives an extended-observer output-feedback realization. The control uses only position and yaw measurements and is claimed to achieve robust coordinated 3D trajectory tracking in the presence of unmeasured disturbances.

Significance. If the stability proof and observer convergence hold under the stated assumptions, the work supplies a geometrically motivated, tuning-simplified method for underactuated quadcopters that explicitly handles incomplete state information and disturbances. This would be a useful contribution to the geometric-control literature for UAVs, particularly if the derivations are machine-checkable or accompanied by reproducible simulation code.

major comments (2)
  1. [model construction and geometric controller synthesis] Model and geometric-synthesis sections: the central claim that the chosen position-plus-yaw outputs remain differentially flat for arbitrary smooth 3D trajectories rests on an implicit assumption that the underactuated rotational coupling (thrust direction constraint) does not destroy the required invertibility. No explicit verification or additional regularity condition is supplied to confirm that the flat-output map and its derivatives remain well-defined and invertible when the translational acceleration is forced to align with the body z-axis. This assumption is load-bearing for both the geometric inversion and the subsequent observer-error dynamics.
  2. [extended observer and output-feedback realization] Observer design and stability proof: the extended observer is asserted to recover full state and disturbance estimates from position+yaw measurements alone, yet the error dynamics are not shown to remain asymptotically stable when the control input is constrained by the underactuation (only four independent torques/thrust). A concrete bound or gain-selection criterion that accounts for this coupling is missing and directly affects the output-feedback claim.
minor comments (2)
  1. Notation for the flat outputs and their time derivatives should be introduced once and used consistently; several symbols appear to be redefined without cross-reference.
  2. The stability theorem statement would benefit from an explicit list of the standing assumptions (smoothness class of the reference trajectory, boundedness of disturbances, etc.).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments and for recognizing the potential utility of the geometric approach with dynamic extension and extended observer. We address the two major comments point by point below.

read point-by-point responses

  1. Referee: [model construction and geometric controller synthesis] Model and geometric-synthesis sections: the central claim that the chosen position-plus-yaw outputs remain differentially flat for arbitrary smooth 3D trajectories rests on an implicit assumption that the underactuated rotational coupling (thrust direction constraint) does not destroy the required invertibility. No explicit verification or additional regularity condition is supplied to confirm that the flat-output map and its derivatives remain well-defined and invertible when the translational acceleration is forced to align with the body z-axis. This assumption is load-bearing for both the geometric inversion and the subsequent observer-error dynamics.

    Authors: The manuscript derives the flat outputs explicitly in the model-construction section by expressing the required thrust vector, body-frame orientation, and angular-velocity commands as functions of the second- and higher-order derivatives of the position and yaw. The map remains invertible provided the commanded acceleration (after gravity compensation) is nonzero and the trajectory is C^4-smooth, which is already stated as an assumption for the reference trajectory. We agree that an explicit regularity remark would remove any ambiguity. In the revised manuscript we will insert a short paragraph immediately after the flat-output definition that recalls the standard non-vanishing-acceleration condition and confirms that the underactuated thrust-alignment constraint does not introduce additional singularities for admissible trajectories. revision: yes

  2. Referee: [extended observer and output-feedback realization] Observer design and stability proof: the extended observer is asserted to recover full state and disturbance estimates from position+yaw measurements alone, yet the error dynamics are not shown to remain asymptotically stable when the control input is constrained by the underactuation (only four independent torques/thrust). A concrete bound or gain-selection criterion that accounts for this coupling is missing and directly affects the output-feedback claim.

    Authors: The observer-error analysis is carried out on the closed-loop system that already incorporates the geometric controller; the four control inputs (thrust and three torques) are used to realize the desired thrust direction and yaw rate that the flat-output inversion prescribes. High-gain observer theory is applied to the resulting cascaded dynamics, yielding asymptotic convergence when the observer gains exceed a threshold determined by the Lipschitz constants of the vector fields and the disturbance bounds. We acknowledge that the manuscript does not spell out an explicit numerical bound that isolates the effect of underactuation. In the revision we will add a corollary stating a sufficient gain-selection inequality that explicitly incorporates the maximum thrust-to-mass ratio and the attitude-tracking error bound supplied by the geometric controller. revision: yes

Circularity Check

0 steps flagged

No significant circularity; model, geometric synthesis, and observer are independent constructions

full rationale

The paper first presents a complete model of quadcopter motion along a smooth spatial trajectory, then synthesizes a geometric controller (with added integrators for simplified tuning) whose stability is proven, and finally provides an extended-observer output-feedback realization. None of these steps reduce by construction to fitted parameters, self-definitions, or self-citation chains; the geometric approach and observer error dynamics are built as separate, verifiable steps from the model without the target performance being presupposed in the inputs. This matches the reader's assessment that the abstract and description show no evident reduction of claimed performance to self-referential elements. The underactuation concern is an assumption-validity issue rather than a circularity reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on abstract only; the complete model and geometric synthesis likely rest on standard domain assumptions for rigid-body quadcopter dynamics and bounded disturbances, with no explicit free parameters or invented entities identifiable.

axioms (1)
  • domain assumption A complete model of quadcopter motion along a smooth spatial trajectory exists and is suitable for control design.
    Directly stated in the abstract as the foundation for the proposed algorithm.

pith-pipeline@v0.9.0 · 5620 in / 1229 out tokens · 48207 ms · 2026-05-19T15:31:12.087124+00:00 · methodology

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Reference graph

Works this paper leans on

8 extracted references · 8 canonical work pages

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