pith. sign in

arxiv: 2604.27705 · v1 · submitted 2026-04-30 · 📡 eess.SY · cs.SY· math.OC

Robust Geometric Control of Catenary Robots under Unstructured Force Uncertainties

Alexandre Anahory Simoes , Leonardo Colombo This is my paper

Pith reviewed 2026-05-07 06:58 UTC · model grok-4.3

classification 📡 eess.SY cs.SYmath.OC
keywords catenary robotgeometric controlinput-to-state stabilityquadrotor UAVSE(3) manifoldcable constraintsrobust trackingmulti-agent coordination
0
0 comments X

The pith

A geometric controller for two quadrotors connected by an inextensible cable renders tracking errors locally input-to-state stable under bounded catenary force perturbations.

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

The paper models a catenary robot as two quadrotors whose relative configuration on SE(3) induces the cable shape and the resulting forces that couple their translational dynamics. Instead of treating the cable as an independent dynamic element, the approach uses the inextensible cable purely as a geometric constraint whose forces act as exogenous inputs. A geometric tracking controller is designed specifically for the relative pose of the two vehicles. The analysis shows that this controller makes the closed-loop tracking errors locally input-to-state stable: they converge asymptotically to zero when the catenary forces are known exactly, and they remain ultimately bounded by an explicit function of the perturbation size when the forces contain bounded unstructured uncertainties. This result matters because many tethered multi-UAV tasks, such as cooperative transport or aerial manipulation, operate in environments where cable forces are hard to model precisely but can be treated as bounded disturbances.

Core claim

The paper establishes that the proposed geometric tracking controller for the relative configuration of the two quadrotors on SE(3) renders the closed-loop error dynamics locally input-to-state stable with respect to the catenary-induced forces treated as unstructured but bounded exogenous inputs. When the perturbations are identically zero, the tracking errors converge asymptotically to the origin. When the perturbations are bounded, the errors are ultimately bounded by an explicit constant that depends on the size of the bound and on the controller gains.

What carries the argument

Geometric tracking controller on SE(3) for the relative configuration of the two quadrotors, with the cable shape and forces treated as a configuration-dependent geometric constraint and the force uncertainties as bounded exogenous inputs.

If this is right

  • Tracking errors converge asymptotically to zero in the absence of catenary-force perturbations.
  • Tracking errors remain ultimately bounded by an explicit value that scales with the size of the bounded force perturbations.
  • The same controller structure applies to any desired relative configuration on SE(3) provided the initial conditions lie inside the local stability region.
  • The geometric modeling choice removes the need to augment the state with independent cable dynamics while still accounting for the coupling forces.

Where Pith is reading between the lines

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

  • The local character of the stability result implies that global convergence may require additional mechanisms such as a hybrid switching strategy or a different choice of error functions.
  • Because the cable is reduced to a pure geometric constraint, the approach may extend naturally to other inextensible or length-constrained multi-agent systems where full cable dynamics would be prohibitively complex.
  • If the bound on force perturbations can be estimated online from sensor data, the explicit ultimate-bound formula could be used to certify safe operating envelopes for real-world catenary-robot tasks.

Load-bearing premise

The cable is modeled as an inextensible geometric constraint whose instantaneous shape is completely determined by the two quadrotor configurations, and the force uncertainties are treated as unstructured bounded exogenous inputs rather than state-dependent or unbounded effects.

What would settle it

A numerical simulation or physical experiment in which the tracking errors grow without bound or exceed the predicted ultimate bound while the applied catenary-force perturbations remain strictly within the assumed bounded range, or in which the errors fail to converge to zero when the perturbations are set exactly to zero.

Figures

Figures reproduced from arXiv: 2604.27705 by Alexandre Anahory Simoes, Leonardo Colombo.

Figure 1
Figure 1. Figure 1: Configuration of the catenary robot. The body angular velocity Ωi ∈ R 3 and translational velocity vi ∈ R 3 satisfy the kinematics R˙ i = RiΩˆ i , p˙i = vi , (1) where (·) ∧ : R 3 → so(3) denotes the hat map (i.e., and isomorphism between vectors on R 3 with skew-symmetric matrices, see e.g., [12]). Each UAV is an underactuated aerial robot of quadrotor type, with control inputs ui = (fi , τi) ∈ R × R 3 , … view at source ↗
Figure 3
Figure 3. Figure 3: illustrates Corollary 1. The left panel shows that the ultimate tracking error grows approximately linearly with ∥∆d∥∞, comparing low damping (Kv = 4I3, kΩ = 3.5) and high damping (Kv = 10I3, kΩ = 8). The right panel fixes ∆ = 0 ¯ .40 and shows that increasing the translational damping gain Kv = kvI3 reduces the ultimate tracking error. 0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 0.0 0.2 0.4 0.6 |e(t)| Nominal, d … view at source ↗
Figure 2
Figure 2. Figure 2: The nominal error converges to zero, while bounded view at source ↗
read the original abstract

This paper considers the robust control of a catenary robot composed of two quadrotors connected by an inextensible cable. The system is modeled on \(SE(3)\), with the cable treated as a geometric subsystem induced by the UAV configuration rather than as an independent dynamical element. The catenary shape determines configuration-dependent forces that couple the translational dynamics of the vehicles. We propose a geometric tracking controller for the relative configuration of the agents and analyze its robustness with respect to unstructured uncertainties in the catenary-induced forces. The main theoretical result establishes local input-to-state stability of the closed-loop tracking errors. In particular, we obtain asymptotic convergence in the nominal case and an explicit ultimate bound for the tracking errors under bounded catenary-force perturbations.

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

1 major / 2 minor

Summary. The manuscript develops a geometric tracking controller for a catenary robot formed by two quadrotors linked by an inextensible cable, modeled on the special Euclidean group SE(3). The cable is treated as a geometric constraint inducing configuration-dependent forces rather than as an independent dynamic subsystem. The authors propose a controller for the relative configuration and establish local input-to-state stability (ISS) of the closed-loop tracking errors with respect to bounded unstructured uncertainties in the catenary-induced forces. In the nominal case without perturbations, asymptotic convergence is shown, and an explicit ultimate bound on the tracking errors is derived under bounded perturbations.

Significance. If the local ISS result holds with the domain of validity properly ensured, the work contributes a rigorous robustness analysis for geometric controllers applied to multi-agent systems with configuration-dependent forces. The explicit ultimate bound is a positive feature, as is the use of standard Lyapunov arguments on Lie groups for the nominal case. This could be useful for applications in aerial transportation and manipulation where cable forces have uncertainties. The geometric setup on SE(3) is standard and the nominal asymptotic convergence follows from the usual Lyapunov argument on Lie groups.

major comments (1)
  1. [Main stability theorem (stability analysis section)] The central local ISS result (main theorem in the stability analysis section) asserts an explicit ultimate bound on tracking errors that grows with the perturbation magnitude, but provides no explicit threshold on the perturbation bound to ensure this ball lies strictly inside the neighborhood where the attitude error function is positive definite (i.e., excluding antipodal rotations on SO(3)). Without such a condition or a proof of forward invariance of the local domain under the ISS inequality, trajectories may escape the region of validity, invalidating the bound. This is load-bearing for the local ISS claim.
minor comments (2)
  1. [Modeling section] The abstract states that forces are 'configuration-dependent' yet treated as 'unstructured exogenous inputs'; the modeling section should explicitly state the assumption under which configuration-induced forces can be bounded independently of the state for the ISS analysis.
  2. [Controller design section] Notation for the relative configuration error on SE(3) and the associated error functions should be defined with a reference to the standard definitions (e.g., the attitude error function) to improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the thorough review and constructive comments. The primary concern regarding the domain of validity for the local ISS result has been addressed through revisions that provide an explicit perturbation threshold and a forward-invariance argument.

read point-by-point responses

  1. Referee: The central local ISS result (main theorem in the stability analysis section) asserts an explicit ultimate bound on tracking errors that grows with the perturbation magnitude, but provides no explicit threshold on the perturbation bound to ensure this ball lies strictly inside the neighborhood where the attitude error function is positive definite (i.e., excluding antipodal rotations on SO(3)). Without such a condition or a proof of forward invariance of the local domain under the ISS inequality, trajectories may escape the region of validity, invalidating the bound. This is load-bearing for the local ISS claim.

    Authors: We agree that the local ISS claim requires an explicit condition ensuring the ultimate bound remains strictly inside the region where the attitude error function is positive definite. In the revised manuscript, we have added a corollary to the main stability theorem that derives an explicit upper bound on the perturbation magnitude (expressed in terms of the initial error, the desired ultimate bound, and the system parameters including the minimum eigenvalue of the inertia matrices and the cable length). This bound guarantees that the ISS ball lies within the open set where the Lyapunov function is positive definite and the attitude error is bounded away from antipodal rotations. We have also inserted a supporting lemma proving forward invariance of this local domain: on the boundary of the domain, the time derivative of the Lyapunov function is shown to be strictly negative when the perturbation is below the derived threshold, using the standard properties of the attitude error function on SO(3) and the ISS inequality. These additions preserve the original nominal asymptotic stability result and the explicit ultimate bound while making the local ISS statement fully rigorous. revision: yes

Circularity Check

0 steps flagged

No circularity: local ISS derived from closed-loop SE(3) dynamics via standard Lyapunov analysis

full rationale

The central result (local ISS of tracking errors with explicit ultimate bound under bounded perturbations) follows from the closed-loop vector field on SE(3) and standard ISS definitions applied to geometric error functions. The catenary is modeled as an inextensible geometric constraint inducing configuration-dependent forces treated as exogenous bounded inputs; the controller is a geometric tracking law whose stability is analyzed directly without fitting parameters or renaming known results. No load-bearing step reduces by construction to a self-citation, fitted input, or self-definitional loop. The derivation remains self-contained against external benchmarks (standard ISS theorems and SE(3) attitude error functions).

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard Lie-group geometry and nonlinear stability theory plus two domain assumptions about the cable. No new physical constants or fitted parameters are introduced in the abstract; the controller gains are design parameters left free.

free parameters (1)
  • controller gains
    Positive definite gain matrices appearing in the geometric tracking law; their specific values are not fixed by the paper but must be chosen to satisfy the local stability conditions.
axioms (2)
  • domain assumption The cable is inextensible and its shape is instantaneously determined by the relative configuration of the two UAVs on SE(3).
    Invoked in the modeling section to replace independent cable dynamics with a configuration-dependent force map.
  • domain assumption Force uncertainties are unstructured but bounded exogenous inputs.
    Used to apply the ISS framework; stated explicitly in the robustness analysis.

pith-pipeline@v0.9.0 · 5426 in / 1707 out tokens · 64863 ms · 2026-05-07T06:58:16.218759+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

  1. [1]

    Dynamics and control of a quadrotor with a payload suspended through an elastic cable,

    P. Kotaru, G. Wu, and K. Sreenath, “Dynamics and control of a quadrotor with a payload suspended through an elastic cable,” inProceedings of the 2017 American Control Conference, 2017, pp. 3906–3913

    work page 2017

  2. [2]

    Geometric control of quadrotor uavs transporting a cable- suspended rigid body,

    T. Lee, “Geometric control of quadrotor uavs transporting a cable- suspended rigid body,”IEEE Transactions on Control Systems Technol- ogy, vol. 26, no. 1, pp. 255–264, 2018

    work page 2018

  3. [3]

    Geometric control of cooperating multiple quadrotor uavs with a suspended load,

    T. Lee, K. Sreenath, and V . Kumar, “Geometric control of cooperating multiple quadrotor uavs with a suspended load,” inProceedings of the IEEE Conference on Decision and Control, 2013, pp. 5510–5515

    work page 2013

  4. [4]

    Geometric control for load transportation with quadrotor uavs by elastic cables,

    J. R. Goodman, T. Beckers, and L. J. Colombo, “Geometric control for load transportation with quadrotor uavs by elastic cables,”IEEE Transactions on Control Systems Technology, vol. 31, no. 6, pp. 2848– 2862, 2023

    work page 2023

  5. [5]

    Geometric control of two quadrotors carrying a rigid rod with elastic cables,

    J. R. Goodman and L. Colombo, “Geometric control of two quadrotors carrying a rigid rod with elastic cables,”Journal of Nonlinear Science, vol. 32, p. 65, 2022

    work page 2022

  6. [6]

    Tension-based reconfigurable multi-agent formation for aerial load transportation,

    V . P. Bacheti, D. Villa, R. Lozano, M. Sarcinelli-Filho, and P. Castillo, “Tension-based reconfigurable multi-agent formation for aerial load transportation,”IEEE Access, vol. 13, pp. 114 098–114 116, 2025

    work page 2025

  7. [7]

    A leader- follower control system for a package-delivery drone,

    E. D. S. Cardoso, V . P. Bacheti, and M. Sarcinelli-Filho, “A leader- follower control system for a package-delivery drone,”IEEE Access, vol. 13, pp. 55 313–55 323, 2025

    work page 2025

  8. [8]

    The catenary robot: Design and control of a cable propelled by two quadrotors,

    D. S. D’antonio, G. A. Cardona, and D. Saldana, “The catenary robot: Design and control of a cable propelled by two quadrotors,”IEEE Robotics and Automation Letters, vol. 6, no. 2, pp. 3857–3863, 2021

    work page 2021

  9. [9]

    Adaptive control for cooperative aerial transportation using catenary robots,

    G. A. Cardona, D. S. D’Antonio, R. Fierro, and D. Salda ˜na, “Adaptive control for cooperative aerial transportation using catenary robots,” in IEEE International Conference on Robotics and Automation (ICRA), 2021

    work page 2021

  10. [10]

    Scalable cooperative transport of cable-suspended loads with uavs using distributed trajectory optimization,

    B. E. Jackson, T. A. Howell, K. Shah, M. Schwager, and Z. Manchester, “Scalable cooperative transport of cable-suspended loads with uavs using distributed trajectory optimization,”IEEE Robotics and Automation Letters, vol. 5, no. 2, pp. 3368–3374, 2020

    work page 2020

  11. [11]

    Aerial transporta- tion control of suspended payloads with multiple agents,

    F. Oliva-Palomo, D. Mercado-Ravell, and P. Castillo, “Aerial transporta- tion control of suspended payloads with multiple agents,”Journal of the Franklin Institute, vol. 361, no. 7, p. 106787, 2024

    work page 2024

  12. [12]

    T. Lee, M. Leok, and N. H. McClamroch,Global formulations of Lagrangian and Hamiltonian dynamics on manifolds. Springer

    work page

  13. [13]

    Geometric tracking control of a quadrotor uav on se (3),

    ——, “Geometric tracking control of a quadrotor uav on se (3),” in49th IEEE conference on decision and control (CDC). IEEE, 2010, pp. 5420–5425

    work page 2010