This paper presents a framework for aerial manipulation of an extensible cable that combines a high-fidelity model based on partial differential equations (PDEs) with a reduced-order representation suitable for real-time control. The PDEs are discretised using a finite-difference method, and proper orthogonal decomposition is employed to extract a reduced-order model (ROM) that retains the dominant deformation modes while significantly reducing computational complexity. Based on this ROM, a nonlinear model predictive control scheme is formulated, capable of stabilizing cable oscillations and handling hybrid transitions such as payload attachment and detachment. Simulation results confirm the stability, efficiency, and robustness of the ROM, as well as the effectiveness of the controller in regulating cable dynamics under a range of operating conditions. Additional simulations illustrate the application of the ROM for trajectory planning in constrained environments, demonstrating the versatility of the proposed approach. Overall, the framework enables real-time, dynamics-aware control of unmanned aerial vehicles (UAVs) carrying suspended flexible cables.
Flexible cables offer a versatile alternative to rigid appendages in aerial manipulation. By keeping the vehicle at a distance, they enhance safety when interacting with hazardous or delicate objects, and their compliant nature makes them well-suited for operation in confined or cluttered environments. In addition, as illustrated in Fig. 1, the passive dynamics of the cable may be leveraged to facilitate certain maneuvers, potentially reducing actuation effort.
These advantages, however, come with challenges: cables are continuum systems governed by nonlinear dynamics, and when interacting with payloads, they exhibit hybrid behaviors due to switching boundary conditions at the tip. of fidelity in the cable model.
Rigid pendulum models. These represent the payload either as a single rigid body or as a point mass connected by a link. Such formulations enable efficient swing attenuation and are also used in cooperative transport [1]- [4]. Their main limitation is that they neglect slackness and distributed deformation, restricting applicability to scenarios where the cable remains taut and aligned beneath the vehicle.
Hybrid slack/taut models. In these approaches, the payload is modeled as switching between free-fall and tensioned regimes [5]- [7]. This allows explicit handling of events such as re-tensioning, enabling controllers to maintain stability and performance through mode transitions. However, the underlying dynamics typically remain simplified.
Reduced-order continuum models. The authors of [8] pioneered the usage of proper orthogonal decomposition (POD) to reduce the state dimension of soft robot models, allowing fast, but still meaningful, reduced-order models (ROMs). In fact, [9] applied POD to approximate the cable manipulated by a UAV by extracting its principal modes, enabling accurate capture of the deformation in a way suitable for model predictive control (MPC); However, [9] neglected any payload interactions.
Each modeling choice embodies a trade-off between fidelity and tractability. Importantly, none of the existing approaches simultaneously combine accurate continuum modeling with hybrid payload dynamics in a form suitable for online control.
This paper proposes a hybrid control framework for UAVs hosting flexible, extensible cables and executing dynamic payload manipulation. The main contributions are: and explicitly captures payload attachment and detachment events.
• A POD-based reduced-order model (ROM) specifically adapted to preserve the dominant deformation modes and ensure modal coherence across hybrid transitions. • A nonlinear MPC scheme that exploits the ROM to achieve real-time trajectory tracking while actively suppressing undesired cable dynamics, ensuring both accuracy and robustness during manipulation tasks. Crucially, this framework is not just a refinement of existing approaches. Its novelty lies in the first integration of continuum tether modeling, model reduction, and hybrid predictive control into a single pipeline that is both theoretically sound and practically deployable. This combination provides tangible advantages in accuracy, robustness, and real-time feasibility, significantly extending the range of scenarios that can be addressed compared to prior methods.
Consider a perfectly flexible string of length L, parametrized by the curvilinear coordinate s ∈ [0, L] and time t. In the world frame F w = {O, E x , E y , E z }, its configuration is given by r(s, t) ∈ R 3 (see Fig. 2). Using Lagrange’s notation, r t := ∂r ∂t and r tt := ∂ 2 r ∂t 2 denote velocity and acceleration, and r s := ∂r ∂s the tangent vector. For compactness, the explicit time dependence (t) is omitted throughout this paper unless required for clarity.
The internal contact force at s, i.e. n(s), represents the force transmitted through the cross-section of the string, exerted by the portion (s, L) on the portion (0, s). Because of the assumption of perfect flexibility, the constitutive equation links the string deformation to the contact force through
with ∥ • ∥ denoting the L 2 -norm, E being the Young modulus of the string, and A its cross-sectional area. Take a string material segment (s 0 , s 1 ), with 0 < s 0 < s 1 < L, subject to the contact forces -n(s 0 ) and n(s 1 ) and to the body force per unit length β(s). Below, the integral equation expresses the conservation of linear momentum for the material segment:
where ρ c is the density of the string. Differentiation returns the classical form:
where n s (s) := ∂n ∂s (s). At the endpoints, boundary conditions model the interaction of the string with any attached bodies.
A finite-difference method (FDM) is used to solve the string PDEs [10]. The idea is to discretize the domain [0, L] in N intervals of length h s := L N , approximating the continuum as a chain of N + 1 nodes. For any field quantity, for instance r(s), its value at node i is denoted r i := r(i h s ). Nodal velocities, accelerations and contact forces are writte
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