Passive Stiffness Shaping in Cable-Suspended Aerial Manipulation via Movable Compliant Anchors

Antonio Franchi; Amr Afifi

cs.ROSep 30, 2026

Passive Stiffness Shaping in Cable-Suspended Aerial Manipulation via Movable Compliant Anchors

Authors: Antonio Franchi, Amr Afifi

Organizations: Robotics and Mechatronics Lab, Faculty of Electrical Engineering, Mathematics & Computer Science, University of Twente, Enschede, The Netherlands · Department of Computer, Control and Management Engineering, Sapienza University of Rome, 00185 Rome, Italy · Department of Information and Computing Sciences, Utrecht University, Utrecht, The Netherlands

Abstract

Cable-suspended aerial manipulation offers a lightweight architecture for cooperative transportation and physical interaction, yet the passive mechanical response perceived at the load remains insufficiently understood and systematically exploited. This work interprets aerial vehicles as movable compliant anchors and develops a gravity-aware quasi-static theory for predicting and shaping the passive Cartesian stiffness of a suspended load. The formulation applies to an arbitrary number of aerial vehicles connected to a point load by taut, straight, inextensible cables. At a selected gravity-loaded equilibrium, aerial-anchor compliance and transverse cable geometric compliance combine in series within each leg, while the leg stiffnesses act in parallel on the load. For isotropic aerial-anchor behavior, each leg is exactly equivalent to a virtual unilateral elastic cable, revealing an axial--transverse stiffness decomposition governed by the equilibrium tension. These results define a nonlinear map from commanded-anchor configuration to passive load stiffness, whose differential enables local constraint-preserving shaping through anchor repositioning. A dynamic rigid-body validation framework with nonlinear vehicle control, elastic-damped tendons, and environmental contact is defined to assess when and to what extent the derived stiffness remains predictive beyond the assumptions of the analytical model.

Figures & tables

Panel of Figure 1 — Fig. 1 : System architecture and passive-stiffness-shaping principle. Each taut cable connects the load point $P$ to an actual aerial-anchor point $P_{a,i}$ , while the finite closed-loop stiffness of aerial vehicle $i$ relates this point to its commanded anchor $P_{d,i}$ . Gravity determines the equilibrium cable directions and tensions. Repositioning the commanded anchors changes these operating-point quantities and thereby reshapes the passive Cartesian stiffness ellipsoid at the load. The three-vehicle drawing represents the general $n$ -vehicle system.

Panel of Figure 2 — Fig. 2 : Physical and virtual representations of one cable–anchor leg. The physical leg combines aerial-anchor compliance and the transverse geometric compliance of the taut inextensible cable in series. Under isotropic aerial-anchor stiffness, the leg is quasi-statically equivalent to a virtual unilateral elastic cable connecting the load directly to the commanded anchor point.

Panel of Figure 3 — Fig. 3 : Axial–transverse eigenstructure of an isotropic cable–anchor leg. The Euclidean-identified stiffness operator has the simple eigenvalue $k_{i}$ along the cable direction and the eigenvalue $\gamma_{i}$ , with algebraic and geometric multiplicity two, on the transverse plane. Increasing the operating-point tension changes the leg contribution from nearly rank one toward isotropic.

Panel of Figure 4 — Fig. 4 : Gravity-aware stiffness pipeline and parallel intrinsic and coordinate descriptions. The aerial-anchor setpoints determine a selected gravity-loaded equilibrium branch and thereby the equilibrium cable directions and tensions. These quantities determine the leg and total stiffness maps. Their coordinate representations yield the stiffness-coordinate vector and its Jacobian for sensitivity analysis and control. Gravity affects stiffness through the selected equilibrium rather than through an additive stiffness term.

Panel of Figure 5 — Fig. 5 : Constraint-preserving local stiffness-regulation loop.

Parameter	Value
System: number of quadrotors and cables	$4$
Quadrotor: mass	$1.28\,\mathrm{kg}$
Quadrotor: principal inertia	$\operatorname{diag}(0.015,\,0.015,\,0.007)\,\mathrm{kg\,m^{2}}$
Payload: mass	$2.00\,\mathrm{kg}$
Payload: principal inertia	$\operatorname{diag}(0.05,\,0.05,\,0.05)\,\mathrm{kg\,m^{2}}$
Translational control: position gain $\bm{K}_{p}$	$\operatorname{diag}(12,\,12,\,14)\,\mathrm{N/m}$

TABLE I : MuJoCo model and controller parameters.

Phenomenon	Analytical representation	MuJoCo representation	Remaining simulation-to-reality gap
Payload	Point load with translational displacement	Free rigid body with mass, inertia, attitude, and four spatially separated attachment sites	Structural flexibility, uncertain inertial parameters, and unmodeled payload aerodynamics
Aerial vehicles	Movable anchors with prescribed linear Cartesian stiffness	Rigid quadrotors with nonlinear $SE(3)$ tracking and finite translational gains	Motor dynamics, thrust uncertainty, battery effects, and aerodynamic interaction
Anchor compliance	Symmetric positive-definite linear stiffness map	Controller-induced displacement under cable loading	Gain variation, saturation, delays, estimation errors, and off-equilibrium nonlinearities
Cables	Taut, straight, massless, and inextensible	Elastic-damped spatial tendons with finite axial stiffness and rest length	Distributed mass, sagging, bending, drag, transverse vibration, and cable contact
Dynamics	Quasi-static evolution along a selected equilibrium branch	Coupled vehicle–payload transients with inertia and dissipation	Unmodeled high-frequency dynamics and hardware-dependent settling behavior
Contact	Excluded from the stiffness derivation	Optimization-based contact between the payload and environment	Surface compliance, friction uncertainty, impact dynamics, and geometry imperfections

TABLE II : Relationship among the analytical model, the MuJoCo implementation, and the corresponding physical system.

Panel of Figure 8 — Fig. 6 : Parity-plot comparison across the selected stiffness and deformation quantities.

Panel of Figure 9 — Fig. 7 : Stiffness tracking plots for 2 of the 4 cases used in zigzag sliding

Panel of Figure 10 — Fig. 8 : Stiffness tracking plots for 2 of the 4 cases used in zigzag sliding

Panel of Figure 11 — Fig. 9 : Cable tensions during stiffness regulation.

Panel of Figure 12 — Fig. 10 : Commanded-anchor velocities during stiffness regulation.

Parameter	Value	Role
Payload regulation gain $\lambda_{p}$	1.0	Weight on payload-position error
Stiffness regulation gain $\lambda_{K}$	1.0	Weight on stiffness-tracking error
Payload objective weight $w_{p}$	1.0	Relative weight of payload term in the QP cost
Stiffness objective weight $w_{K}$	25.0	Relative weight of stiffness term in the QP cost
Velocity regularization $\varepsilon_{v}$	$10^{-4}$	Quadratic penalty on anchor-velocity magnitude
Admissibility gain $\eta_{T}$	1.0	Gain in the minimum-tension inequality

TABLE III : Default parameters of the implemented QP anchor-setpoint controller.

Fig. 11 : Schematic of the zigzag guided-sliding task, including the longitudinal direction of progression and the lateral contact direction.

Panel of Figure 15 — Fig. 12 : Longitudinal and lateral tracking errors during guided sliding.

Panel of Figure 16 — Fig. 13 : Commanded-anchor formations used to generate the four passive-stiffness profiles for the guided-sliding comparison.

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Ana Maria Nascimento, Augusto Sales, Antonio Marcus Lima, Tiago Nascimento

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Open paper AskPDF arXiv

This work proposes a novel control and estimation approach for aerial manipulation of a cable-suspended load using Unmanned Aerial Vehicles (UAVs). Common approaches in the state of the art have practical limitations, relying on direct load measurements and Lagrangian methods for dynamic modeling. The lack of a straightforward dynamic model of the system led us to propose adopting the Udwadia-Kalaba method to explicitly incorporate the cable's geometric constraints. This formulation allowed for the consistent derivation of the tension force and its direct integration into the NMPC prediction model. Additionally, we propose a sensorless load state estimation based on the same geometric constraints. Results from real-robot experiments demonstrated that the explicit inclusion of load dynamics in the optimization problem significantly reduces trajectory-tracking errors and yields better overall performance compared to strategies based on incomplete models.

Abstract

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