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From Ellipsoids to Midair Control of Dynamic Hitches

Jiawei Xu, Subhrajit Bhattacharya, David Saldaña

TL;DR

This paper addresses dynamic hitch control formed by two interlacing taut cables actuated by four aerial vehicles. It introduces an ellipsoid-based pin-and-string kinematic model that yields a control-affine system and develops a CLF-HOCBF-QP controller to steer the hitch configuration to a reference while enforcing safety constraints like cable tautness. A cascade-error design ensures the Lyapunov function has relative degree one with respect to the input, enabling convergence guarantees and safe, high-speed tracking. Numerical simulations demonstrate stable performance under static, noisy, and dynamic references, highlighting real-time feasibility and robustness, with future hardware experiments planned to tackle uncertain hitch resistance.

Abstract

The ability to dynamically manipulate interaction between cables, carried by pairs of aerial vehicles attached to the ends of each cable, can greatly improve the versatility and agility of cable-assisted aerial manipulation. Such interlacing cables create hitches by winding two or more cables around each other, which can enclose payloads or can further develop into knots. Dynamic modeling and control of such hitches is key to mastering the inter-cable manipulation in context of cable-suspended aerial manipulation. This paper introduces an ellipsoid-based kinematic model to connect the geometric nature of a hitch created by two cables and the dynamics of the hitch driven by four aerial vehicles, which reveals the control-affine form of the system. As the constraint for maintaining tension of a cable is also control-affine, we design a quadratic programming-based controller that combines Control Lyapunov and High-Order Control Barrier Functions (CLF-HOCBF-QP) to precisely track a desired hitch position and system shape while enforcing safety constraints like cable tautness. We convert desired geometric reference configurations into target robot positions and introduce a composite error into the Lyapunov function to ensure a relative degree of one to the input. Numerical simulations validate our approach, demonstrating stable, high-speed tracking of dynamic references.

From Ellipsoids to Midair Control of Dynamic Hitches

TL;DR

This paper addresses dynamic hitch control formed by two interlacing taut cables actuated by four aerial vehicles. It introduces an ellipsoid-based pin-and-string kinematic model that yields a control-affine system and develops a CLF-HOCBF-QP controller to steer the hitch configuration to a reference while enforcing safety constraints like cable tautness. A cascade-error design ensures the Lyapunov function has relative degree one with respect to the input, enabling convergence guarantees and safe, high-speed tracking. Numerical simulations demonstrate stable performance under static, noisy, and dynamic references, highlighting real-time feasibility and robustness, with future hardware experiments planned to tackle uncertain hitch resistance.

Abstract

The ability to dynamically manipulate interaction between cables, carried by pairs of aerial vehicles attached to the ends of each cable, can greatly improve the versatility and agility of cable-assisted aerial manipulation. Such interlacing cables create hitches by winding two or more cables around each other, which can enclose payloads or can further develop into knots. Dynamic modeling and control of such hitches is key to mastering the inter-cable manipulation in context of cable-suspended aerial manipulation. This paper introduces an ellipsoid-based kinematic model to connect the geometric nature of a hitch created by two cables and the dynamics of the hitch driven by four aerial vehicles, which reveals the control-affine form of the system. As the constraint for maintaining tension of a cable is also control-affine, we design a quadratic programming-based controller that combines Control Lyapunov and High-Order Control Barrier Functions (CLF-HOCBF-QP) to precisely track a desired hitch position and system shape while enforcing safety constraints like cable tautness. We convert desired geometric reference configurations into target robot positions and introduce a composite error into the Lyapunov function to ensure a relative degree of one to the input. Numerical simulations validate our approach, demonstrating stable, high-speed tracking of dynamic references.
Paper Structure (16 sections, 1 theorem, 28 equations, 8 figures)

This paper contains 16 sections, 1 theorem, 28 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Hitch Dynamics
  3. Dynamics
  4. Control
  5. Equilibrium and nominal input
  6. Feasibility
  7. Configuration control through CLF-HOCBF-QP
  8. Error design and convergence
  9. Evaluation
  10. Experiment 1 - Static reference
  11. Experiment 2 - Noisy hitch
  12. Experiment 3 - Dynamic reference
  13. Conclusion
  14. Partial Derivatives
  15. Example Statistics in reference following
  16. ...and 1 more sections

Key Result

Proposition 1

Convergence of the Lyapunov function $V(\boldsymbol{x})\rightarrow0$ guarantees the convergence of the robot positions $\boldsymbol{p}_i\rightarrow\boldsymbol{p}_i^{ref}$ for $i=1,\dots 4$.

Figures (8)

  • Figure 1: Illustration of the cable-suspended system with the constraint ellipsoids overlay. Line segments of the same color denote a single cable connecting two robots. The cables interlace/wind around each other at the hitch point.
  • Figure 2: Two cables interlacing at the hitch $\boldsymbol{p}$, each connecting two robots. Each robots apply an input force vector $\boldsymbol{u}_i$, illustrated in 2D.
  • Figure 3: The feasibility test of the nonzero part of the input matrix $\boldsymbol{B}_{nonzero}$ with different configurations of the normal vectors $\boldsymbol{n}_{12}$ and $\boldsymbol{n}_{34}$.
  • Figure 4: Experiment 1: the cable-suspended aerial system in 3D following a static reference configuration from a random initialization, with first-order damping $c_d = 0.2$ at the hitch unknown to the controller.
  • Figure 5: The results of the system in 3D following a dynamic reference configuration of which the hitch moves at $0.1$m/s from a random initial state, with $c_d = 0.2$ and a Gaussian noise with a standard deviation of $5$N at the hitch both unknown to the controller.
  • ...and 3 more figures

Theorems & Definitions (2)

  • Proposition 1
  • proof