⚡ electrical engineering

Robust Geometric Control of Catenary Robots under Unstructured Force Uncertainties

This paper proposes a geometric tracking controller for a two-quadrotor catenary robot system modeled on SE(3) and proves its local input-to-state stability against unstructured uncertainties in the cable-induced forces, ensuring asymptotic convergence in nominal conditions and bounded tracking errors under perturbations.

Original authors: Alexandre Anahory Simoes, Leonardo Colombo

Published 2026-05-01

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CC BY 4.0

Original authors: Alexandre Anahory Simoes, Leonardo Colombo

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine two drones flying in the sky, connected by a single, unbreakable rope. This isn't just a loose string; it's a heavy rope that sags under its own weight, forming a specific curve known as a catenary (think of the shape a power line makes between two poles, or a chain hanging from your hand).

This paper is about teaching these two drones to fly together in a coordinated dance, even when the physics of that hanging rope gets messy or unpredictable.

Here is the breakdown of their approach, using simple analogies:

1. The Problem: The "Lazy" Rope

Usually, when engineers control robots, they treat the connection between them as a rigid stick or a simple distance rule. But a real rope is different.

The Shape Matters: Because the rope has weight, it doesn't stay straight. It hangs in a curve.
The Pull: This curve creates a specific pull (tension) on each drone. If the drones move closer, the rope tightens and the pull changes. If they move apart, the rope sags more, and the pull changes again.
The Messy Reality: In the real world, the rope might wiggle, catch the wind, or stretch slightly. The paper calls these "unstructured uncertainties." It's like trying to walk a tightrope while someone is occasionally blowing on the rope or shaking it.

2. The Solution: Seeing the Rope as a "Shadow"

Instead of trying to model the rope as a separate, complicated machine with its own moving parts, the authors treat the rope as a shadow cast by the drones' positions.

The Analogy: Imagine the drones are two people holding a heavy blanket between them. The shape of the blanket is entirely determined by where the people are standing. You don't need to calculate the physics of every thread in the blanket; you just need to know where the people are, and the blanket's shape (and the pull on their hands) is automatically decided.
The Math: They built a mathematical model where the rope's shape and the forces it exerts are calculated instantly based on the drones' relative positions. This simplifies the problem significantly.

3. The Controller: The "Smart Pilot"

The authors designed a "geometric controller." Think of this as a super-smart autopilot for each drone.

The Plan: The autopilot knows exactly where the drones should be relative to each other to create a specific rope shape.
The Compensation: It calculates the exact pull the rope should be exerting and adds that to its flight plan. It's like a cyclist who knows there's a headwind and pedals harder to compensate, so they don't slow down.
The Goal: The drones try to maintain a specific distance and angle, which in turn forces the rope to hold a specific shape.

4. The Safety Net: Robustness

This is the most important part of the paper. What happens if the wind blows the rope, or the rope isn't perfectly heavy as the math predicted?

The Claim: The authors proved mathematically that their controller is robust.
The Analogy: Imagine you are walking on a moving walkway that sometimes speeds up or slows down unexpectedly. A "robust" walker doesn't fall over; they just adjust their steps slightly and stay on the path.
The Result:
- Perfect World: If the rope behaves exactly as predicted, the drones fly perfectly to their target, and the error (how far off they are) goes to zero.
- Messy World: If the rope is wobbly or the wind is strong, the drones won't be perfect, but they won't crash either. They will stay within a small, predictable "bubble" of error.
- The Tuning Knob: The paper shows that if you make the drones' "damping" (their ability to resist shaking) stronger, this error bubble gets smaller. It's like tightening the suspension on a car to make the ride smoother over bumps.

Summary

The paper presents a new way to control two drones connected by a heavy, sagging rope. Instead of fighting the complexity of the rope, they treat the rope's shape as a direct result of where the drones are. They proved that their control system works perfectly in ideal conditions and stays safe and stable even when the rope behaves unpredictably, keeping the drones close to their target path.

1. Problem Statement

The paper addresses the control of a catenary robot, a system consisting of two quadrotor UAVs connected by a single, inextensible cable of fixed length $l$ .

The Challenge: Unlike rigid-link systems or simple point-mass cable models, a heavy cable under gravity assumes a catenary shape. This shape is not static; it depends entirely on the relative configuration (position and orientation) of the two UAVs. Consequently, the cable induces configuration-dependent forces (tensions) that couple the translational dynamics of the two vehicles.
The Gap: Existing approaches often rely on simplified planar models or treat the cable as a kinematic constraint or a nominal force element. These simplifications fail to capture the full geometric coupling in 3D space and make robustness analysis difficult when the ideal quasi-static catenary model is perturbed by unmodeled effects (e.g., aerodynamic drag, cable elasticity, or inertia).
Objective: The goal is to design a geometric tracking controller that regulates the relative configuration of the two UAVs to a desired trajectory while ensuring robustness against unstructured uncertainties in the catenary-induced forces.

2. Methodology

A. Geometric Modeling on SE(3)

The authors model the system on the Special Euclidean group $SE(3)$, representing the full 3D position and attitude of each UAV.

Cable as a Geometric Subsystem: Instead of treating the cable as an independent dynamical element with its own states, the cable is modeled as a geometric subsystem induced by the UAV configuration.
Configuration Parameters: The relative position of the UAVs uniquely determines the catenary shape via three parameters:
1. Orientation ( $R_C$ ): The vertical plane containing the cable.
2. Half-span ( $s$ ): Half the horizontal distance between UAVs.
3. Shape Parameter ( $a$ ): Determined by the cable length $l$ and span $s$ via the intrinsic relation $l/2 = a \sinh(s/a)$ .
Force Derivation: The endpoint tensions ( $T_1, T_2$ ) are derived analytically based on the static equilibrium of the catenary under gravity. These forces are smooth functions of the UAV positions and act as coupling terms in the translational dynamics equations.

B. Control Design

The control strategy employs a geometric tracking controller that operates directly on the manifold $SE(3)$ without linearization.

Nominal Control Law:
- Translational Dynamics: A feedback controller is designed for the position and velocity errors. Crucially, the nominal catenary-induced forces are computed in real-time based on the current configuration and added as feedforward terms to cancel the coupling effects.
- Attitude Dynamics: A standard geometric controller on $SO(3)$ is used to align the UAV thrust vectors with the desired total force direction (which includes gravity, tracking errors, and the nominal cable tension).
Robustness Formulation:
- The authors model deviations from the ideal quasi-static catenary (due to drag, elasticity, etc.) as additive unstructured uncertainties ( $d_i$ ) in the force terms.
- The control objective is modified to ensure that the tracking errors remain bounded (Input-to-State Stability) despite these bounded disturbances.

C. Stability Analysis

The theoretical core of the paper is the proof of Local Input-to-State Stability (ISS).

Lyapunov Function: A composite Lyapunov function is constructed, combining terms for relative translational errors ( $e_p, e_v$ ) and attitude errors ( $e_R, e_\Omega$ ) for both agents.
Derivation: The authors derive the time derivative of the Lyapunov function, showing that the closed-loop error dynamics satisfy an inequality of the form:
$\dot{V} \leq -c_1 \|e\|^2 + c_2 \|\Delta d\|^2$
where $\Delta d$ represents the effective disturbance on the relative dynamics.
Result: This inequality proves that the system is ISS with respect to the force uncertainties.

3. Key Contributions

Geometric Modeling Framework: The paper provides a rigorous $SE(3)$ formulation where the cable is treated as a configuration-induced geometric subsystem, avoiding the need for independent cable states while capturing the full 3D coupling forces.
Robust Geometric Controller: A novel controller is proposed that explicitly compensates for the nominal catenary forces via feedforward while maintaining geometric tracking properties.
ISS Stability Proof: The authors establish local Input-to-State Stability for the closed-loop system. This is a significant theoretical advancement, proving that bounded uncertainties in the cable forces lead to bounded tracking errors.
Explicit Ultimate Bound: The analysis provides an explicit formula for the ultimate bound of the tracking errors. This bound is shown to be inversely proportional to the damping gains, offering a clear design guideline for tuning robustness.

4. Results

Theoretical Results:
- Nominal Case: In the absence of uncertainties, the tracking errors converge asymptotically to zero.
- Perturbed Case: Under bounded force perturbations, the tracking errors converge to a small neighborhood of the origin (Uniformly Ultimately Bounded - UUB). The size of this neighborhood decreases as the control gains (specifically damping gains) increase.
Numerical Simulations:
- Simulations were conducted with a constant desired relative position and added sinusoidal disturbances to the force terms.
- Figure 2: Demonstrates that while the nominal error converges to zero, the perturbed system settles into a bounded oscillation consistent with the ISS estimate.
- Figure 3: Validates the theoretical bound, showing that the ultimate error magnitude grows linearly with the disturbance amplitude and decreases significantly when higher damping gains are used.

5. Significance

This work bridges the gap between complex physical modeling and robust control for multi-agent aerial systems.

Practical Relevance: By treating the cable as a geometric subsystem, the approach is computationally efficient and suitable for real-time implementation on UAVs, avoiding the complexity of flexible cable dynamics.
Robustness: The explicit handling of unstructured uncertainties makes the controller viable for real-world applications where aerodynamic drag and cable elasticity cannot be perfectly modeled.
Theoretical Rigor: The use of geometric control on $SE(3)$ avoids singularities associated with Euler angles and provides a global perspective on attitude dynamics, while the ISS analysis offers rigorous guarantees for safety-critical cooperative tasks.

Future Directions: The authors suggest extending this framework to dynamic and elastic cables, contact-aware tasks, and experimental validation on physical multi-UAV platforms.