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AFD-CAMLs: Agile Force-Distribution-Aware Planning and Control for Cable-Suspended Aerial Multi-Lifting Systems

Antreas Kourris, Sihao Sun

arXiv:2610.01185v1cs.RO

TL;DR

Redundant or ill-conditioned tension allocation can produce uneven cable forces even when payload tracking is accurate, especially with geometric mismatch and low-level errors. The paper combines null-space force planning, force-reference-aware NMPC, and tension-based admittance corrections; simulations and experiments report reduced imbalance while preserving agile tracking.

Abstract

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Multiple UAVs can cooperatively transport heavy payloads while controlling their position and orientation. Trajectory-based methods offer high agility while satisfying system constraints, but can produce uneven force distributions when the tension-to-wrench allocation is redundant or ill-conditioned, particularly under geometric mismatch and low-level tracking errors. We propose a hybrid planning-and-control framework to address this problem. A global planner generates payload trajectories and cable-force references by exploring the allocation null space under a prescribed internal-force setting. These references augment the cost of a centralized local planner, promoting feasible force distributions while generating trajectories for all UAVs. An admittance filter then compares the planned forces with onboard cable-tension estimates and adjusts the kinematic references to improve force tracking in degenerate or near-degenerate configurations. Simulations and experiments involving four to ten UAVs demonstrate more balanced tension distributions during both hovering and demanding agile maneuvers, without compromising agility or payload-tracking performance.

I. INTRODUCTION

Cooperative lifting increases payload capacity, but redundant cable-force allocation can leave individual UAVs overloaded or cables near slack even when payload tracking is accurate. The paper extends trajectory-based control with null-space force planning, force-aware NMPC, and tension-feedback corrections.

I. INTRODUCTION

  • Redundant force allocation can leave some UAVs overloaded and others near cable slackness despite accurate payload-pose tracking.Internal-force changes can alter individual cable tensions without changing the payload’s net wrench.Source: §I · p. 1
  • Trajectory-based control plans dynamically feasible trajectories for all UAVs, but geometric mismatch and tracking errors can shift realized tensions in redundant configurations.The issue is most pronounced when the tension-to-wrench allocation is degenerate or nearly degenerate.Source: §I · pp. 1–2
  • The proposed extension combines null-space force planning, force-reference-aware NMPC, and an admittance filter that adjusts kinematic references using onboard tension estimates.The global planner supplies feasible force and direction references, and the filter compares planned tensions with onboard estimates.Source: §I · p. 2
  • Simulations with four to ten UAVs and four-UAV experiments demonstrate improved force distribution under geometric and inertial-model mismatch while preserving agile payload tracking.Source: §I · p. 2

A. Notation

  • The notation distinguishes inertial, payload-fixed, and UAV body-fixed frames, and marks global-planner references with a superscript g.Source: A. Notation · p. 2

B. Multi-lifting system model

The model considers more than three unidirectional-thrust UAVs carrying a rigid payload through massless, inextensible cables. Cable geometry determines UAV positions and cable forces, which act oppositely on the payload and UAVs.

B. Multi-lifting system model

C. Redundancy analysis

With cable directions fixed, tension magnitudes must reproduce the payload wrench, but redundancy can make that allocation non-unique or highly sensitive. Symmetric configurations with four or more UAVs provide a concrete degenerate case.

C. Redundancy analysis

  • Cable forces must reproduce the required payload wrench while satisfying tension, actuation, collision, cable-interference, and kinematic constraints.Source: C. Redundancy analysis · p. 2
  • With cable directions fixed, the tension solution is unique when the allocation matrix A has full column rank.The analysis treats only cable magnitudes as unknown in the force-and-moment balance.Source: C. Redundancy analysis · p. 2
  • The inverse condition number k(A) measures allocation-matrix conditioning: values near one indicate good conditioning, while values approaching zero indicate near-dependence and sensitivity to small errors.It is zero when rank(A) < n.Source: C. Redundancy analysis · pp. 2–3
  • For symmetric attachment points and equally inclined cables, rank(A) ≤ 3, so configurations with n ≥ 4 have a nontrivial tension null space and k(A) = 0.Small departures from symmetry generally yield a small positive k(A), leaving force distribution highly sensitive.Source: C. Redundancy analysis · p. 3
  • In degenerate or near-degenerate configurations, model mismatch and tracking errors can redistribute tensions while preserving the payload wrench, so control must select and realize feasible tensions without sacrificing agility.Source: C. Redundancy analysis · p. 3

III. METHODOLOGY

The framework augments trajectory-based control with force-distribution planning and local tension feedback. Its global planner explores wrench-preserving allocations, and a centralized NMPC reconciles the resulting references with system dynamics and constraints.

III. METHODOLOGY

  • A centralized NMPC converts payload and cable references into dynamically feasible, constraint-satisfying trajectories for the complete multi-lifting system.It retains the system model, constraints, solver, and trajectory decoder of, while augmenting the objective with global-planner cable-force and direction references.Source: §III · p. 3

A. Internal-force-aware global planner

  • The global planner generates payload and cable references from the current and target payload poses and a user-specified internal-force level.Payload position uses a minimum-snap polynomial, orientation uses SLERP, and their derivatives and the nominal model determine the required wrench.Source: A. Internal-force-aware global planner · p. 3
  • Wrench-preserving force allocations are parameterized using a null-space basis, with the coefficient chosen to maintain cable tautness and impose a user-specified outward preload.Source: A. Internal-force-aware global planner · p. 3
  • The global planner optimizes the force allocation using penalties that include minimum cable tension and minimum outward force, then converts optimized forces into tension references.Source: A. Internal-force-aware global planner · p. 3

B. Force-reference-aware NMPC local planner

The NMPC starts from the estimated load–cable state and plans dynamically feasible trajectories under system constraints. Its objective adds cable-force and direction tracking so the local plan accounts for the desired allocation.

B. Force-reference-aware NMPC local planner

  • The estimated load–cable state initializes finite-horizon NMPC, whose coupled model and path constraints cover cable tautness, actuation limits, and collision avoidance.The optimized load–cable trajectory is decoded into kinematic and feedforward references for each quadrotor.Source: B. Force-reference-aware NMPC local… · pp. 3–4
  • The centralized planner maps payload and cable references into system trajectories, which are decoded into quadrotor references for downstream adjustment.The framework flow places the NMPC between the global planner and the admittance filter.Source: Figure 2 · p. 4, B. Force-reference-aware NMPC local… · p. 4
  • The stage cost adds cable-direction and cable-force tracking to payload tracking and smooth-control costs, reconciling the desired allocation with coupled dynamics and constraints.The global planner selects a desired force allocation; the NMPC reconciles it over the prediction horizon.Source: B. Force-reference-aware NMPC local… · p. 4
  • Force-tracking costs improve the planned distribution, but cannot alone reject differences between nominal allocation and realized tensions because forces are realized through vehicle motion.This motivates a force-feedback layer beyond the NMPC.Source: B. Force-reference-aware NMPC local… · p. 4

C. Admittance filter for hybrid cable-tension and trajectory tracking

The admittance filter uses observed tension errors to adjust individual quadrotor references, reducing discrepancies between planned and realized cable tensions. It leaves the global payload command and the NMPC’s coordinated motion and constraint role intact.

C. Admittance filter for hybrid cable-tension and trajectory tracking

  • As the tension-allocation matrix becomes degenerate or nearly degenerate, small geometry, cable-length, or tracking errors can cause large tension redistributions with little change in payload wrench.The filter uses observed tension error to correct each quadrotor’s kinematic reference.Source: C. Admittance filter for hybrid… · p. 4
  • The local planner supplies commanded tensions, cable directions, and kinematic trajectories, while the onboard wrench observer estimates external force for tension feedback.The observed tension error is converted to a cable-aligned correction force that drives a virtual mass–spring–damper system.Source: C. Admittance filter for hybrid… · p. 4
  • The admittance corrections modify NMPC-generated kinematic references, and the filter is evaluated for degenerate and near-degenerate allocation configurations.These correspond to k(A) = 0 and k(A) ≈ 0, respectively.Source: C. Admittance filter for hybrid… · pp. 4–5
  • The filter adds local compliant motion to individual quadrotor references, reducing the discrepancy between predicted and realized cable tension without changing the global payload command.The NMPC remains responsible for coordinated whole-body motion and constraint satisfaction.Source: C. Admittance filter for hybrid… · p. 4

D. Onboard trajectory tracking

The work retains the flatness-based PD and INDI onboard controller from to execute the admittance-corrected trajectories.

D. Onboard trajectory tracking

  • The unchanged flatness-based PD and INDI controller executes admittance-corrected local trajectories and generates rotor commands.Flatness mapping supplies attitude and feedforward quantities; INDI uses onboard inertial measurements to reject unmodeled force and torque disturbances,.Source: D. Onboard trajectory tracking · p. 5
  • Flatness mapping generates the attitude and feedforward quantities needed for trajectory execution.Source: D. Onboard trajectory tracking · p. 5
  • INDI rejects unmodeled force and torque disturbances using onboard inertial measurements,.Source: D. Onboard trajectory tracking · p. 5

IV. RESULTS

The evaluation includes simulation and real-world experiments, with a four-quadrotor hover used to validate null-space tension allocation. Enforcing tension constraints substantially reduces tracking error by giving the local planner feasible references.

IV. RESULTS

  • The experiments assess tension-reference feasibility, force tracking and load sharing, payload tracking, and robustness to model mismatch.Source: §IV · p. 5

A. Experimental setup

  • The evaluation combines RotorS simulation and indoor Vicon experiments, usually using four quadrotors with 1 m cables and larger teams for scalability tests.The centralized planner ran at 10 Hz, onboard control at 300 Hz, and Vicon measurements at 100 Hz.Source: A. Experimental setup · p. 5
  • Figure 3 illustrates how changing radial preload magnitude affects the resulting quadrotor formation.Source: Figure 3 · p. 5, 1) Null-space optimization validation · p. 5

1) Null-space optimization validation:

  • Null-space optimization substantially reduces tension-tracking error in a four-quadrotor hover at 30° payload pitch by supplying references that satisfy the local planner’s tension constraints.Without optimization, the global planner generated references below the local planner’s 3 N minimum; Fig. 4 compares tension tracking with and without optimization.Source: 1) Null-space optimization validation · p. 5, Figure 4 · p. 5

2) Cable-tension redundancy:

Cable-tension tracking degrades when the allocation is redundant or poorly conditioned, while the admittance filter improves tension regulation across four- and ten-UAV configurations. Under tested payload-model mismatches, the NMPC configurations varied less than the geometric controller, and filtering did not noticeably reduce inertial-uncertainty robustness.

2) Cable-tension redundancy:

  • At k(A)=0.0012, filtering substantially reduced four-UAV level-hover tension error; at k(A)=0.1916, error remained low without filtering.The level configuration was poorly conditioned, whereas the inclined configuration was better conditioned.Source: 2) Cable-tension redundancy · p. 6
  • Ten-UAV tension error grew in both level and inclined configurations when filtering was disabled, because the allocation matrix lacked full column rank.The ten-UAV allocation admitted infinitely many tension solutions.Source: 2) Cable-tension redundancy · pp. 5–6

3) Effect of the admittance filter:

  • With nominal hover conditions, filtering brought measured tensions closer to their references, reduced cable-to-cable spread, and reduced payload position and orientation errors in four- and ten-quadrotor cases.The same decentralized correction law remained effective for ten quadrotors.Source: 3) Effect of the admittance filter · p. 6, Figure 6 · p. 6

4) Sensitivity to inertial model uncertainties:

5) Agile trajectory tracking:

During agile trajectory tracking, the admittance filter reduced payload error under a modeled cable-length mismatch, while tension corrections adapted the UAV trajectories. In real-world hovering tests, two UAVs initially carried more of the payload load than the others.

5) Agile trajectory tracking:

  • 0.221 m to 0.199 m: with a −1 cm modeled cable-1 length error, filtering reduced three-dimensional mean absolute payload-tracking error.Without mismatch, the two configurations achieved comparable tracking; allocation conditioning was generally larger during agile motion than during hovering.Source: Table II · p. 6, 5) Agile trajectory tracking · p. 6
  • With cable-length mismatch, filtering adapted individual quadrotor trajectories to tension errors, reducing the planned-to-observed force discrepancy and payload-tracking error.The corresponding errors and allocation conditioning are shown in Fig. 9.Source: 5) Agile trajectory tracking · p. 7

C. Real-world experiments

  • Before the admittance filter was activated during real-world hovering, UAVs 2 and 4 carried a larger share of the payload weight while the other cable tensions approached zero.The filter was activated after approximately 15 s.Source: C. Real-world experiments · p. 7

1) Hovering:

Hovering tests found that filtering moved measured cable tensions toward their references and improved balance, while experiments with an unknown load still showed tension-tracking gains. The paper concludes that its framework improves force distribution and robustness to geometric mismatch while maintaining accurate payload tracking.

1) Hovering:

  • After filter activation, measured tensions converged toward their references, producing a more balanced distribution.Fig. 10 shows cable tensions during hovering.Source: 1) Hovering · p. 7, Figure 10 · p. 7
  • A residual steady-state offset of approximately 2 N remained after activation, versus approximately 0.5 N in simulation; unmodeled aerodynamic effects such as rotor downwash may contribute.Direct tension measurements could help distinguish observer bias from aerodynamic loading.Source: 1) Hovering · p. 8
  • 3.51 N to 2.09 N: enabling the admittance filter reduced aggregate cable-tension tracking RMSE across all four cables in the unknown-load experiment.With the unknown load, the framework outperformed the baseline on cables 1, 3, and 4, while error increased on cable 2.Source: 1) Hovering · p. 8, Figure 11 · p. 8

V. CONCLUSIONS

  • The framework combines null-space force-distribution planning, centralized NMPC, and tension-estimate-based admittance corrections; simulations and four-UAV experiments showed improved force balance and geometric-mismatch robustness while retaining accurate payload tracking.The reported scope includes teams of up to ten UAVs in simulation and four UAVs in experiments.Source: §V · p. 8
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