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Dynamics Modeling of a Multi-UAV Slung Load System Using a Discrete-Link Cable Approach

Harvey Merton, Ian W. Hunter

arXiv:2610.07456v1cs.RO

TL;DR

MUSLS simulation needs cable models that work with standard rigid-body toolchains and remain stable across tension transitions, while common approaches use massless, always-taut rods. This paper models cables as discrete rigid links and tunes the model using real-world flights; the selected configuration reproduces real-world behavior at real-time speed with orientation errors below 11.4° and translational errors of approximately 5.5% of cable length.

  • Problem

    MUSLS simulation needs stable, near-real-time toolchain integration through tension transitions, while common cable models assume massless, always-taut rods.

    Source: §I · p. 1

  • Method

    The paper models cables as discrete rigid links with link inertias and tunable joint properties, selecting parameters through sensitivity analysis.

    Source: A. Organization and contributions · p. 2, A. Parameter sensitivity study · p. 5

  • Results

    With 15 links per cable, orientation errors are below 11.4° and translational errors are approximately 5.5% of cable length, with real-time-speed simulation.

    Source: §V · p. 8

  • Takeaways & Limitations

    The selected configuration balances fidelity and efficiency for real-time simulation of the modeled system.

    Source: §V · p. 8

    The setup does not use load-pose feedback, so its results are not directly comparable to prior closed-loop approaches.

    Source: B. Performance evaluation · p. 7

Abstract

from arXiv · show

A common assumption to simplify the problem of controlling a multi-UAV slung load system (MUSLS) is that the flexible cables can be modeled as massless rigid rods. In this work, we propose an alternative Euler-Newton derived dynamical model which uses a series of rigid links to model the flexible cables. The model is specifically designed to allow efficient simulation using Featherstone's articulated body algorithm. We perform real-world validation of this model on gentle, aggressive, and tension-engagement maneuvers and run a parameter sweep to determine the number of links, joint damping, and joint friction to achieve the greatest model fidelity. The model closely matches real-world flight data with mean load translation errors below 132 mm (5.5% of the cable length) and orientation errors below 11.4 degrees. We make the real-world flight data publicly available for the development of future cable models.

I. INTRODUCTION

MUSLS simulation needs models that integrate with standard rigid-body toolchains, run near real time, and remain stable through cable-tension transitions. Existing cable approaches either simplify cable behavior or complicate integration with general-purpose simulators.

I. INTRODUCTION

  • Multiple UAVs can carry heavier payloads and, in principle, regulate the load’s full six-degree-of-freedom pose.The system suspends a shared load from flexible cables.Source: §I · p. 1
  • MUSLS models for off-the-shelf toolchains must run near real time and remain stable through cable-tension loss and re-engagement.These requirements motivate a model suited to standard rigid-body simulators.Source: §I · p. 1
  • Flexible cables have nonlinear, high-dimensional dynamics and abrupt transitions when tension is lost or reestablished.Cable models include discrete mass–spring, rigid-link, and finite-element approaches, as well as continuous elastic-rod and spline-based formulations.Source: §I · p. 1
  • The common massless, always-taut rod assumption reduces simulation fidelity, particularly when cables go slack.Hybrid tension models handle slack transitions but still use custom simulators that are difficult to integrate with general-purpose robotics toolchains.Source: §I · p. 1
  • Prior discrete-link MUSLS models concentrate mass at joints, increasing the elements needed for convergence and preventing links from being treated as standard rigid bodies.This obstructs integration with simulators that use tree-structured rigid bodies and Featherstone’s articulated-body algorithm.Source: §I · p. 2

A. Organization and contributions

The paper presents a multi-link cable dynamics model for MUSLS and evaluates its performance. Its contributions include tunable link and joint properties, systematic parameter selection, trajectory testing, and publicly available flight data.

A. Organization and contributions

  • The MUSLS model uses discrete-link cables with link inertias and tunable joint properties.The formulation is designed for integration with rigid-body robotics simulators.Source: A. Organization and contributions · p. 2
  • A systematic parameter-selection procedure balances computational efficiency and predictive performance for multi-link cables.Source: A. Organization and contributions · p. 2
  • The model’s performance is quantified over varied trajectories, including tension engagement, and real-world flight data are made publicly available for future cable-model development.Source: A. Organization and contributions · p. 2

II. DYNAMICS MODELING

The MUSLS is modeled as a tree of rigid bodies, with each flexible cable represented by rigid links and the equations derived using the Euler–Newton approach. This structure supports solution with the articulated body algorithm while retaining cable inertial effects.

II. DYNAMICS MODELING

  • The model represents each cable as cylindrical rigid links with mass, length, and inertia, connected between the load and a drone.The cable links are part of a tree of rigid bodies solved using the articulated body algorithm.Source: §II · p. 2, Figure 2 · p. 2
  • The Euler–Newton equations describe load translation and rotation, drone dynamics, and the dynamics of each cable link.The model assumes negligible cable aerodynamic forces and inextensible cables with negligible bending stiffness.Source: §II · pp. 2–3
  • Joint moments combine stiffness, damping, and static Coulomb friction; equal stiffness and damping on the two active axes give isotropic properties.The joints can be universal joints because the moment around each cable joint’s X axis is zero, reducing each joint’s degrees of freedom by one.Source: §II · p. 3
  • Unlike a massless rigid-rod model, the multi-link formulation retains cable inertial effects that matter during aggressive maneuvers, tension re-engagement, or when cable mass is substantial.Prior point-mass-chain formulations simplify state representation and force resolution, but omit link inertias required by this articulated-body formulation.Source: §II · p. 3
  • The articulated body algorithm solves the tree-structured chain in O(n) time without explicitly computing internal cable forces.The link directionality makes the kinematic equations a tree-structured open chain rooted at the load.Source: §II · p. 3

III. EXPERIMENTAL PLATFORM

The experimental platform evaluates the dynamics model using a three-quadrotor slung-load system in simulation and hardware. Drone waypoints are tracked independently, while onboard control and state estimation close each vehicle’s loop.

III. EXPERIMENTAL PLATFORM

  • The evaluation uses a three-quadrotor MUSLS in simulation and in real-world experiments.The hardware experiments use Holybro x500 v2 drones, while simulations run with DART in Gazebo.Source: §III · pp. 3–4, Figure 3 · p. 3
  • Each drone independently tracks a waypoint without inter-agent coupling or formation feedback, keeping the evaluation focused on cable and load dynamics.Waypoints derive from a virtual cone whose base places drones evenly around the desired load pose.Source: §III · p. 4
  • An onboard cascaded PID pose controller regulates each drone’s angular velocity, attitude, linear velocity, and position through thrust and body torque commands.Drones use motion-capture measurements or simulator feedback fused with IMU data by an onboard EKF; controller gains are identical in hardware and simulation.Source: §III · p. 4
  • The drones’ dynamics are coupled through the load and cables, and the setup uses motion-capture data at 100 Hz in real-world experiments.The hardware system uses 20 Vicon cameras.Source: Figure 4 · p. 4, §III · p. 4

IV. RESULTS

The model is evaluated against real-world MUSLS trajectories for quasi-static, aggressive, and cable tension-engagement maneuvers. Agreement is assessed through translational and rotational errors averaged over each trajectory, with oscillations able to inflate those averages.

IV. RESULTS

  • The evaluation covers a quasi-static circle, simple harmonic motion along the Y-axis, and cable tension engagement.The circle uses a 1.5 m radius and 0.3 m/s tangential velocity; the SHM has 1 m amplitude and 2 rad/s angular frequency; engagement raises load height by 0.5 m and yaw by 1.7 rad.Source: §IV · p. 4
  • Trajectory agreement is measured by translational error and rotational geodesic distance, averaged across each trajectory’s timesteps.For oscillatory motion such as SHM, minor period mismatches can produce large instantaneous amplitude differences and overstate mean error.Source: §IV · pp. 4–5
  • Real-world and final simulation flights were repeated five times; mean trajectories were formed across runs, with 95% confidence intervals.Two real-world circles and one yaw-engagement cycle were rejected because communication dropout affected the desired drone arrangement; Section IV-A simulations used one full flight each.Source: §IV · p. 5

A. Parameter sensitivity study

The sensitivity study tunes joint damping, joint friction, and cable-element count to balance fidelity, stability, and simulation speed. It selects 15 elements, 2 × 10−3 N·s/rad damping, and zero friction, while highlighting instability and accuracy trade-offs.

A. Parameter sensitivity study

  • During the tension-engagement trajectory, reducing cable-joint damping improves the match between simulated and real-world load yaw oscillations, but excessively low damping values lead to unstable simulations.Source: A. Parameter sensitivity study · p. 5
  • Joint friction can destabilize tension engagement and produce irregular yaw oscillations; it also lowers run speed from about 99% to 40%–56% of real time, so friction is set to 0 N·m.The reported speed varies with cable-element count.Source: A. Parameter sensitivity study · p. 5, Table II · p. 5, Figure 6 · p. 6
  • 15 cable elements are selected: more elements improve SHM pose error up to about 25, but counts above 15 raise tension-engagement error variability, while fewer than five destabilize SHM.The selected count is intended to balance stability and accuracy.Source: A. Parameter sensitivity study · p. 5, Figure 7 · p. 6
  • Mean SHM translational error and tension-engagement geodesic error exceed those for the other trajectories, consistent with oscillations amplifying average errors.The comparison varies cable-element count with joint damping of 2 × 10−3 N·s/rad and zero joint friction.Source: A. Parameter sensitivity study · p. 6, Figure 7 · p. 6

B. Performance evaluation

The final tuned model mostly follows the real-world load and drone poses across the tested trajectories, with translation and orientation errors generally limited. Tension engagement remains a notable mismatch, and simulated pose variability is sensitive to small link-orientation changes.

B. Performance evaluation

  • The selected settings include joint stiffness of 0 N·m/rad and joint friction of 0 N·m, with each cable joint modeled as universal.Source: B. Performance evaluation · p. 6
  • The simulated load and drone 1 mostly follow their real-world poses across all trajectories.Fig. 8 compares their mean poses; yaw is clipped on the circular trajectory to highlight roll and pitch.Source: B. Performance evaluation · p. 6, Figure 8 · p. 7
  • Discrepancies include real-world drone 1 roll and pitch spikes after a communication loss, altered simulated load SHM amplitude and pitch, and differing tension-engagement oscillations.The simulated tension-engagement yaw oscillates for a shorter period with faster damping, while drone 1 has higher roll and pitch oscillations.Source: B. Performance evaluation · p. 6
  • The model does not capture cable tension engagement successfully; other discrepancies may reflect imperfect load or cable inertial properties and need further testing.Source: B. Performance evaluation · p. 6
  • Small changes in link orientation can alter the impulse transmitted to the load during tension engagement, producing broad simulated confidence intervals.Source: B. Performance evaluation · p. 6
  • 17.2° is the largest geodesic error during tension engagement, versus a maximum of 11.4° in other cases.The reported discrepancy reflects a period mismatch between simulated and real-world yaw oscillations.Source: B. Performance evaluation · pp. 6–7, Table III · p. 7
  • Translation errors stay below 132.4 mm (approximately 5.5 % of the cable length) for all load and most drone trajectories.Drone 3 in the circular trajectory is the exception, at 8.4 % of the cable length.Source: B. Performance evaluation · p. 7, Table III · p. 7

V. CONCLUSION

The paper presents a multi-rigid-link cable model for multi-UAV slung-load systems that is efficiently solved via Featherstone’s algorithm. With selected parameters, it reproduces real-world behavior with low translation and orientation errors, while tension engagement and aggressive maneuvers remain targets for future improvement.

V. CONCLUSION

  • The selected configuration uses 15 links per cable, joint stiffness of 0 N·m/rad, damping of 2 × 10−3 N·s/rad, and joint friction of 0 N·m.Source: §V · p. 8
  • The model reproduces real-world behavior at real-time speed, with orientation errors below 11.4° and translational errors of approximately 5.5 % of the cable length.Source: §V · p. 8
  • Future work will compare the model directly with the massless rigid-rod assumption and improve tension-engagement prediction and performance under aggressive maneuvers.Source: §V · p. 8
  • Real-world MUSLS flight data has been made available to support continued research.Source: §V · p. 8
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