Cyclic Nullspace Coordination: Perpetual Flight of Aerial Carriers for Static Suspension

Antonio Franchi; Chiara Gabellieri; Martina Paolucci; Yaolei Shen

arxiv: 2503.03481 · v2 · submitted 2025-03-05 · 💻 cs.RO · cs.SY· eess.SY

Cyclic Nullspace Coordination: Perpetual Flight of Aerial Carriers for Static Suspension

Chiara Gabellieri , Yaolei Shen , Martina Paolucci , Antonio Franchi This is my paper

Pith reviewed 2026-05-23 01:20 UTC · model grok-4.3

classification 💻 cs.RO cs.SYeess.SY

keywords aerial roboticscable-suspended loadnullspace coordinationperpetual flightmulti-carrier systemstrajectory generationstatic suspensionHamiltonian cycle

0 comments

The pith

Three or more aerial carriers can fly continuously while keeping a cable-suspended load in a fixed pose.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that nonstop flights by three or more carriers are compatible with a cable-suspended load remaining completely still in position and orientation. It supplies an algorithm that generates the carriers' coordinated trajectories to achieve this balance. A reader would care because the result removes any requirement for the carriers to stop or land periodically during sustained transport or positioning tasks. The construction relies on selecting force directions from a Hamiltonian cycle on the attachment graph and building elliptical paths that keep each cable force moving on its constraint sphere.

Core claim

Non-stop flights of three or more carriers are compatible with holding a constant pose of a cable-suspended load. This is realized by choosing n linearly independent internal-force directions as the edges of a Hamiltonian cycle on the graph of cable attachment points, then mapping those edges to periodic coordinates via graph coloring so that no adjacent coordinates have simultaneous zero derivatives; the resulting elliptical force trajectories in distinct 2-D affine subspaces project onto the cable spheres with nonzero tangential velocity, satisfying load statics for any n greater than or equal to 3.

What carries the argument

Mapping each edge of the Hamiltonian cycle to a periodic coordinate through graph coloring, which guarantees that no adjacent coordinates have simultaneous zero derivatives and thereby produces force trajectories with nonzero tangential velocity on each cable constraint sphere.

If this is right

The construction supplies explicit tuning guidelines that scale to any number of carriers n greater than or equal to 3.
A fixed-wing-compatible planner is obtained that preserves load statics under speed, bank-angle, and flight-path constraints.
Quantitative bounds are given on sensitivity to parameter variation and on the effect of single-carrier failure.
The theoretical trajectories are confirmed by both numerical simulation and hardware experiments with quadrotor UAVs.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same cycle-based nullspace selection could be tested on systems with more than one suspended object or with non-rigid cables.
Extending the planner to include wind disturbances would reveal whether the perpetual-motion property survives realistic outdoor conditions.
The method suggests a general template for other multi-agent problems in which continuous motion must coexist with an equilibrium constraint at a central object.

Load-bearing premise

The graph-coloring assignment of periodic coordinates to Hamiltonian-cycle edges ensures that no two adjacent coordinates reach zero derivative at the same instant.

What would settle it

An experiment in which the carriers follow the generated trajectories yet the measured pose of the load begins to drift after a finite number of cycles.

Figures

Figures reproduced from arXiv: 2503.03481 by Antonio Franchi, Chiara Gabellieri, Martina Paolucci, Yaolei Shen.

**Figure 1.** Figure 1: A team of n ≥ 3 non-stop flying carriers maintains a cable-suspended load in a fixed position to enable a potential construction scenario, while all carriers continue on their respective non-zero speed flight paths parameters. In Sec. , we show numerical results supporting the theoretical findings. Eventually, conclusions are drawn in Sec. together with an outline of future work. Problem Statement Kinemati… view at source ↗

**Figure 2.** Figure 2: Geometric visualization of the force of the i-th carrier and its components. Remark 2. Notice the fact that with this choice the n forces belong to n different 2D affine subspaces despite the fact that the attachment points are in general distributed in 3D. In this section, we will explore how this result is instrumental in drawing important consequences for relating the time derivative of the force to the… view at source ↗

**Figure 3.** Figure 3: Simulation for the 3 Hamiltonian cycles of a 4-carrier system. The cycles are in the first plot on the left; the carriers’ velocities for each of the 3 cycles are reported, in the same order, in the other plots. The minimum of the carriers’ velocities in the first cycle is indicated by a dashed line in the three plots. As expected, it is greater than the minimum carrier’s velocity when the other two cycles… view at source ↗

**Figure 4.** Figure 4: Four carriers (top row) 6 carriers (bottom row) manipulating a load. In the first column, a snapshot of an animation, where the carriers follow the colored planned paths, the cross indicates the load CoM, and dotted lines indicate the edges of the Hamiltonian cycle (connecting the cable anchoring point to the carrier on the corresponding graph vertex). In the second column, |p˙Ri(t)| > 0. In the third and … view at source ↗

**Figure 5.** Figure 5: Results of 5-carrier system. Each column corresponds to one of the 12 Hamiltonian cycles, displayed in the first row, where the cable anchoring points of the load are displayed. The second row shows the values of epL and the third of eRL. run, one per each of the (5 − 1)!/2 = 12 Hamiltonian cycles. For the sake of space, being the outcomes of the three tests comparable, in [PITH_FULL_IMAGE:figures/full_fi… view at source ↗

read the original abstract

This work demonstrates that the non-stop flights of three or more carriers are compatible with holding a constant pose of a cable-suspended load. It also presents an algorithm for generating the carriers' coordinated non-stop trajectories. The proposed method builds upon two pillars: (1) the choice of n special linearly independent directions of internal forces within the 3n-6-dimensional nullspace of the grasp matrix of the load, chosen as the edges of a Hamiltonian cycle on the graph that connects the cable attachment points on the load. Adjacent pairs of directions are used to generate n forces evolving on distinct 2D affine subspaces, despite the attachment points being generically in 3D; (2) the construction of elliptical trajectories within these subspaces by mapping, through appropriate graph coloring, each edge of the Hamiltonian cycle to a periodic coordinate while ensuring that no adjacent coordinates exhibit simultaneous zero derivatives. Combined with conditions for load statics and attachment point positions, these choices ensure that each of the n force trajectories projects onto the corresponding cable constraint sphere with non-zero tangential velocity, enabling perpetual motion of the carriers while the load is still. The work provides a scalable constructive design for any n greater than or equal to 3 with tuning guidelines, quantifies sensitivity and single-carrier failures, and provides a fixed-wing-compatible planner that preserves load statics under speed/bank/flight-path constraints. The theoretical findings are validated through simulations and laboratory experiments with quadrotor UAVs.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a graph-theoretic construction for perpetual non-stop carrier trajectories that keep a cable-suspended load at fixed pose, using Hamiltonian-cycle nullspace directions plus coloring to avoid zero-velocity instants.

read the letter

The core contribution is a scalable way to pick n linearly independent nullspace directions from the edges of a Hamiltonian cycle on the attachment graph, then build elliptical force trajectories in 2D subspaces so the carriers can keep moving while the load stays still. The second step maps cycle edges to periodic coordinates via graph coloring to guarantee that no two adjacent coordinates hit zero derivative at the same time, which keeps the projected velocities non-zero on the cable spheres. That construction, plus the statics conditions, is what lets them claim perpetual motion for any n >= 3. They also add a fixed-wing planner variant and some sensitivity checks for single-carrier loss. The simulations and quadrotor experiments appear to back the claim in practice. The approach is constructive and stays inside linear algebra and graph theory rather than fitting parameters, which is a plus. The potential soft spot is whether the coloring step plus the chosen periodic functions truly rules out simultaneous zero derivatives for every attachment geometry and every n; the abstract asserts it does, but if that link has edge cases the experiments might not have hit, the perpetual-motion guarantee could be narrower than stated. Overall the math looks internally consistent and the validation is concrete enough to be useful. This is for researchers working on multi-UAV cable manipulation or formation control who need explicit trajectory generators rather than optimization-based ones. It deserves a serious referee because the method is new, the validation is there, and the claim is falsifiable. I would send it to review.

Referee Report

2 major / 2 minor

Summary. The paper claims that non-stop flights of three or more aerial carriers are compatible with holding a constant pose of a cable-suspended load. It presents a scalable constructive algorithm based on two pillars: (1) selecting n linearly independent internal-force directions in the 3n-6-dimensional nullspace of the grasp matrix, taken as the edges of a Hamiltonian cycle on the attachment-point graph; (2) generating elliptical trajectories in the resulting 2D affine subspaces by mapping cycle edges to periodic coordinates via graph coloring such that no adjacent coordinates have simultaneous zero derivatives. Combined with load-statics and attachment conditions, this ensures each force trajectory projects onto its cable sphere with non-zero tangential velocity at all times. The work includes tuning guidelines, sensitivity analysis, a fixed-wing-compatible planner, and validation via simulations and quadrotor experiments.

Significance. If the construction is verified, the result supplies a parameter-light, graph-theoretic method for perpetual multi-carrier suspension that scales with n and extends to fixed-wing platforms. The explicit use of the grasp-matrix nullspace and Hamiltonian-cycle structure, together with hardware validation, constitutes a concrete contribution to aerial manipulation.

major comments (2)

[Pillar (2) construction] Pillar (2) construction (as described in the abstract and the section outlining the second pillar): the claim that the graph-coloring mapping of Hamiltonian-cycle edges to periodic coordinates guarantees that no adjacent coordinates exhibit simultaneous zero derivatives is load-bearing for the non-zero tangential-velocity condition. The manuscript asserts this property but supplies neither a general proof for arbitrary n nor an exhaustive check for small n and generic attachment geometries; without this, the perpetual-motion guarantee remains unsecured.
[Validation section] Validation section: the abstract states that simulations and laboratory experiments with quadrotors confirm the findings, yet no quantitative metrics (e.g., maximum load-pose deviation, minimum carrier speed over multi-period runs, or force-projection error) are referenced. This weakens assessment of how closely the statics and non-stop conditions are realized in practice.

minor comments (2)

[Introduction / Preliminaries] Early in the manuscript, restate the precise dimension of the grasp-matrix nullspace (3n-6) and the definition of the grasp matrix itself to aid readers outside the cable-suspension literature.
[Figures] In trajectory figures, explicitly annotate the periodic coordinates and overlay the tangential-velocity magnitude to make the non-zero condition visually verifiable.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback. We address each major comment below and will revise the manuscript accordingly to strengthen the presentation.

read point-by-point responses

Referee: [Pillar (2) construction] Pillar (2) construction (as described in the abstract and the section outlining the second pillar): the claim that the graph-coloring mapping of Hamiltonian-cycle edges to periodic coordinates guarantees that no adjacent coordinates exhibit simultaneous zero derivatives is load-bearing for the non-zero tangential-velocity condition. The manuscript asserts this property but supplies neither a general proof for arbitrary n nor an exhaustive check for small n and generic attachment geometries; without this, the perpetual-motion guarantee remains unsecured.

Authors: We agree that explicitly establishing the property strengthens the perpetual-motion guarantee. In the revised manuscript we will add a general proof for arbitrary n that relies on the cycle structure and the proper 2-coloring of the cycle edges (ensuring that at most one coordinate per adjacent pair can have zero derivative at any instant). We will also include an exhaustive numerical verification for n = 3 to 6 over generic attachment geometries. revision: yes
Referee: [Validation section] Validation section: the abstract states that simulations and laboratory experiments with quadrotors confirm the findings, yet no quantitative metrics (e.g., maximum load-pose deviation, minimum carrier speed over multi-period runs, or force-projection error) are referenced. This weakens assessment of how closely the statics and non-stop conditions are realized in practice.

Authors: We concur that quantitative metrics improve the assessment of practical performance. The revised validation section will report explicit values for maximum load-pose deviation, minimum carrier speed over multi-period runs, and force-projection error, drawn from both the simulation suite and the quadrotor experiments. revision: yes

Circularity Check

0 steps flagged

No significant circularity; constructive nullspace and graph-coloring design is self-contained

full rationale

The paper's core contribution is an explicit constructive algorithm: select n independent nullspace directions as Hamiltonian-cycle edges on the attachment graph, then map those edges to periodic coordinates via graph coloring so that adjacent coordinates avoid simultaneous zero derivatives. These choices are shown to ensure each force trajectory lies on its cable sphere with non-zero tangential velocity while preserving load statics. No parameter is fitted to data and then relabeled a prediction; no result is justified solely by prior self-citation; the conditions are enforced directly by the algebraic and combinatorial construction rather than by definition or renaming. The derivation therefore stands on independent linear-algebra and graph-theoretic steps.

Axiom & Free-Parameter Ledger

1 free parameters · 3 axioms · 0 invented entities

The central claim rests on standard linear-algebra properties of the grasp matrix nullspace and graph-theoretic existence of Hamiltonian cycles and proper colorings; no free parameters or invented entities are introduced beyond tuning guidelines for the elliptical trajectories.

free parameters (1)

trajectory tuning parameters
The paper mentions tuning guidelines for the elliptical trajectories and periods, implying adjustable scalars chosen to satisfy the non-zero velocity condition.

axioms (3)

domain assumption Existence of a Hamiltonian cycle on the graph whose vertices are the cable attachment points on the load
Invoked to select n linearly independent directions in the 3n-6 nullspace.
standard math The chosen directions remain linearly independent in the nullspace of the grasp matrix
Required for the internal forces to produce no net wrench on the load.
standard math A proper graph coloring of the cycle edges exists such that no adjacent edges share the same periodic coordinate
Ensures that adjacent force components do not have simultaneous zero derivatives.

pith-pipeline@v0.9.0 · 5807 in / 1474 out tokens · 51763 ms · 2026-05-23T01:20:49.864120+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

mapping, through appropriate graph coloring, each edge of the Hamiltonian cycle to a periodic coordinate while ensuring that no adjacent coordinates exhibit simultaneous zero derivatives
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

libraries c1(t)=A cos(ξt), c2(t)=A cos(ξt+π/3), c3(t)=A cos(ξt+2π/3) via edge coloring

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Geometric Inverse Flight Dynamics on SO(3) and Application to Tethered Fixed-Wing Aircraft
cs.RO 2026-02 unverdicted novelty 7.0

A coordinate-free formulation on SO(3) yields closed-form trajectory-to-input maps for fixed-wing inverse flight dynamics, with analytic bank-angle expressions for tethered flight on spherical paths.
Trajectory control of a suspended load with non-stopping flying carriers
eess.SY 2025-10 unverdicted novelty 7.0

A wrench-based feedback controller with dynamic internal-force optimization enables non-stopping carrier trajectories during cooperative suspended-load transport.

Reference graph

Works this paper leans on

27 extracted references · 27 canonical work pages · cited by 2 Pith papers

[1]

Lippiello, and A

Ruggiero, F., V . Lippiello, and A. Ollero. Aerial manipulation: A literature review. V ol. 3, No. 3, 2018, pp. 1957–1964

work page 2018
[2]

Tognon, A

Ollero, A., M. Tognon, A. Suarez, D. Lee, and A. Franchi. Past, present, and future of aerial robotic manipulators. V ol. 38, No. 1, 2021, pp. 626–645

work page 2021
[3]

Garate, J

Estevez, J., G. Garate, J. M. Lopez-Guede, and M. Larrea. Review of Aerial Transportation of Suspended-Cable Payloads with Quadrotors. Drones, V ol. 8, No. 2. doi:10.3390/ drones8020035

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Michael, and V

Sreenath, K., N. Michael, and V . Kumar. Trajectory generation and control of a quadrotor with a cable-suspended load-a differentially-flat hybrid system. In2013. 2013, pp. 4888–4895

work page 2013
[5]

Pereira, P. O., M. Herzog, and D. V . Dimarogonas. Slung load transportation with a single aerial vehicle and disturbance removal. In 2016 24th. 2016, pp. 671–676

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Bernard, M. and K. Kondak. Generic slung load transportation system using small size helicopters. In 2009. 2009, pp. 3258– 3264

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Kondak, I

Bernard, M., K. Kondak, I. Maza, and A. Ollero. Autonomous transportation and deployment with aerial robots for search and rescue missions. V ol. 28, No. 6, 2011, pp. 914–931. Prepared using TRR.cls Gabellieri and Franchi 11

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[8]

Fink, and V

Michael, N., J. Fink, and V . Kumar. Cooperative manipulation and transportation with aerial robots. V ol. 30, 2011, pp. 73–86

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[9]

Michael, S

Fink, J., N. Michael, S. Kim, and V . Kumar. Planning and control for cooperative manipulation and transportation with aerial robots. In 14th. Springer, 2011, pp. 643–659

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Ge, and G

Li, G., R. Ge, and G. Loianno. Cooperative transportation of cable suspended payloads with mavs using monocular vision and inertial sensing. V ol. 6, No. 3, 2021, pp. 5316–5323

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Cieslewski, and D

Gassner, M., T. Cieslewski, and D. Scaramuzza. Dynamic collaboration without communication: Vision-based cable- suspended load transport with two quadrotors. In 2017. 2017, pp. 5196–5202

work page 2017
[12]

Wahba, K. and W. H ¨onig. Efficient Optimization-based Cable Force Allocation for Geometric Control of a Multirotor Team Transporting a Payload

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Goodman, J. and L. Colombo. Geometric control of two quadrotors carrying a rigid rod with elastic cables. V ol. 32, No. 5, 2022, p. 65

work page 2022
[14]

Tognon, D

Gabellieri, C., M. Tognon, D. Sanalitro, and A. Franchi. Equilibria, Stability, and Sensitivity for the Aerial Suspended Beam Robotic System Subject to Parameter Uncertainty

work page
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Pereira, P. O., P. Roque, and D. V . Dimarogonas. Asymmetric collaborative bar stabilization tethered to two heterogeneous aerial vehicles. In 2018. 2018, pp. 5247–5253

work page 2018
[16]

Chen, T. and J. Shan. Cooperative Transportation of Cable- suspended Slender Payload Using Two Quadrotors. In 2019 IEEE Int. Conf. on Unmanned Systems. 2019, pp. 432–437. doi: 10.1109/ICUS48101.2019.8995928

work page doi:10.1109/icus48101.2019.8995928 2019
[17]

Jiang, Q. and V . Kumar. The inverse kinematics of cooperative transport with multiple aerial robots. V ol. 29, No. 1, 2012, pp. 136–145

work page 2012
[18]

Sirouspour, and A

Mohammadi, K., S. Sirouspour, and A. Grivani. Passivity- Based Control of Multiple Quadrotors Carrying a Cable- Suspended Payload. V ol. 27, No. 4, 2022, pp. 2390–2400. doi: 10.1109/TMECH.2021.3102522

work page doi:10.1109/tmech.2021.3102522 2022
[19]

Masone, C., H. H. B ¨ulthoff, and P. Stegagno. Cooperative transportation of a payload using quadrotors: A reconfigurable cable-driven parallel robot. In 2016. 2016, pp. 1623–1630. doi: 10.1109/IROS.2016.7759262

work page doi:10.1109/iros.2016.7759262 2016
[20]

H ¨urzeler, A

Leutenegger, S., C. H ¨urzeler, A. K. Stowers, K. Alexis, M. W. Achtelik, D. Lentink, P. Y . Oh, and R. Siegwart. Flying robots. Springer Handbook of Robotics, 2016, pp. 623–670

work page 2016
[21]

Gabellieri, C. and A. Franchi. On the Existence of Static Equilibria of a Cable-Suspended Load with Non-stopping Flying Carriers. In 2024. Chania, Greece, 2024, pp. 638–644. doi:10.1109/ICUAS60882.2024.10556930

work page doi:10.1109/icuas60882.2024.10556930 2024
[22]

Sanalitro, D., H. J. Savino, M. Tognon, J. Cort ´es, and A. Franchi. Full-pose manipulation control of a cable- suspended load with multiple UA Vs under uncertainties. V ol. 5, No. 2, 2020, pp. 2185–2191

work page 2020
[23]

Virtual truss model for characterization of internal forces for multiple finger grasps

Yoshikawa, T. Virtual truss model for characterization of internal forces for multiple finger grasps. V ol. 15, No. 5, 1999, pp. 941–947

work page 1999
[24]

Gabellieri, L

Tognon, M., C. Gabellieri, L. Pallottino, and A. Franchi. Aerial co-manipulation with cables: The role of internal force for equilibria, stability, and passivity. V ol. 3, No. 3, 2018, pp. 2577– 2583

work page 2018
[25]

Franchi, H

Zelazo, D., A. Franchi, H. H. B ¨ulthoff, and P. R. Giordano. Decentralized Rigidity Maintenance Control with Range Measurements for Multi-Robot Systems. The International Journal of Robotics Research , V ol. 34, No. 1, 2015, pp. 105–

work page 2015
[26]

doi:10.1177/0278364914546173

work page doi:10.1177/0278364914546173
[27]

The Mathematical Coloring Book

Soifer, A. The Mathematical Coloring Book. Springer-Verlag, 2008. Acknowledgements This work was partially funded by the Horizon Europe research and innovation programs under agreement no. 101059875 (Flyflic) and agreement no. 101120732 (Autoassess). Prepared using TRR.cls

work page 2008

[1] [1]

Lippiello, and A

Ruggiero, F., V . Lippiello, and A. Ollero. Aerial manipulation: A literature review. V ol. 3, No. 3, 2018, pp. 1957–1964

work page 2018

[2] [2]

Tognon, A

Ollero, A., M. Tognon, A. Suarez, D. Lee, and A. Franchi. Past, present, and future of aerial robotic manipulators. V ol. 38, No. 1, 2021, pp. 626–645

work page 2021

[3] [3]

Garate, J

Estevez, J., G. Garate, J. M. Lopez-Guede, and M. Larrea. Review of Aerial Transportation of Suspended-Cable Payloads with Quadrotors. Drones, V ol. 8, No. 2. doi:10.3390/ drones8020035

work page

[4] [4]

Michael, and V

Sreenath, K., N. Michael, and V . Kumar. Trajectory generation and control of a quadrotor with a cable-suspended load-a differentially-flat hybrid system. In2013. 2013, pp. 4888–4895

work page 2013

[5] [5]

Pereira, P. O., M. Herzog, and D. V . Dimarogonas. Slung load transportation with a single aerial vehicle and disturbance removal. In 2016 24th. 2016, pp. 671–676

work page 2016

[6] [6]

Bernard, M. and K. Kondak. Generic slung load transportation system using small size helicopters. In 2009. 2009, pp. 3258– 3264

work page 2009

[7] [7]

Kondak, I

Bernard, M., K. Kondak, I. Maza, and A. Ollero. Autonomous transportation and deployment with aerial robots for search and rescue missions. V ol. 28, No. 6, 2011, pp. 914–931. Prepared using TRR.cls Gabellieri and Franchi 11

work page 2011

[8] [8]

Fink, and V

Michael, N., J. Fink, and V . Kumar. Cooperative manipulation and transportation with aerial robots. V ol. 30, 2011, pp. 73–86

work page 2011

[9] [9]

Michael, S

Fink, J., N. Michael, S. Kim, and V . Kumar. Planning and control for cooperative manipulation and transportation with aerial robots. In 14th. Springer, 2011, pp. 643–659

work page 2011

[10] [10]

Ge, and G

Li, G., R. Ge, and G. Loianno. Cooperative transportation of cable suspended payloads with mavs using monocular vision and inertial sensing. V ol. 6, No. 3, 2021, pp. 5316–5323

work page 2021

[11] [11]

Cieslewski, and D

Gassner, M., T. Cieslewski, and D. Scaramuzza. Dynamic collaboration without communication: Vision-based cable- suspended load transport with two quadrotors. In 2017. 2017, pp. 5196–5202

work page 2017

[12] [12]

Wahba, K. and W. H ¨onig. Efficient Optimization-based Cable Force Allocation for Geometric Control of a Multirotor Team Transporting a Payload

work page

[13] [13]

Goodman, J. and L. Colombo. Geometric control of two quadrotors carrying a rigid rod with elastic cables. V ol. 32, No. 5, 2022, p. 65

work page 2022

[14] [14]

Tognon, D

Gabellieri, C., M. Tognon, D. Sanalitro, and A. Franchi. Equilibria, Stability, and Sensitivity for the Aerial Suspended Beam Robotic System Subject to Parameter Uncertainty

work page

[15] [15]

Pereira, P. O., P. Roque, and D. V . Dimarogonas. Asymmetric collaborative bar stabilization tethered to two heterogeneous aerial vehicles. In 2018. 2018, pp. 5247–5253

work page 2018

[16] [16]

Chen, T. and J. Shan. Cooperative Transportation of Cable- suspended Slender Payload Using Two Quadrotors. In 2019 IEEE Int. Conf. on Unmanned Systems. 2019, pp. 432–437. doi: 10.1109/ICUS48101.2019.8995928

work page doi:10.1109/icus48101.2019.8995928 2019

[17] [17]

Jiang, Q. and V . Kumar. The inverse kinematics of cooperative transport with multiple aerial robots. V ol. 29, No. 1, 2012, pp. 136–145

work page 2012

[18] [18]

Sirouspour, and A

Mohammadi, K., S. Sirouspour, and A. Grivani. Passivity- Based Control of Multiple Quadrotors Carrying a Cable- Suspended Payload. V ol. 27, No. 4, 2022, pp. 2390–2400. doi: 10.1109/TMECH.2021.3102522

work page doi:10.1109/tmech.2021.3102522 2022

[19] [19]

Masone, C., H. H. B ¨ulthoff, and P. Stegagno. Cooperative transportation of a payload using quadrotors: A reconfigurable cable-driven parallel robot. In 2016. 2016, pp. 1623–1630. doi: 10.1109/IROS.2016.7759262

work page doi:10.1109/iros.2016.7759262 2016

[20] [20]

H ¨urzeler, A

Leutenegger, S., C. H ¨urzeler, A. K. Stowers, K. Alexis, M. W. Achtelik, D. Lentink, P. Y . Oh, and R. Siegwart. Flying robots. Springer Handbook of Robotics, 2016, pp. 623–670

work page 2016

[21] [21]

Gabellieri, C. and A. Franchi. On the Existence of Static Equilibria of a Cable-Suspended Load with Non-stopping Flying Carriers. In 2024. Chania, Greece, 2024, pp. 638–644. doi:10.1109/ICUAS60882.2024.10556930

work page doi:10.1109/icuas60882.2024.10556930 2024

[22] [22]

Sanalitro, D., H. J. Savino, M. Tognon, J. Cort ´es, and A. Franchi. Full-pose manipulation control of a cable- suspended load with multiple UA Vs under uncertainties. V ol. 5, No. 2, 2020, pp. 2185–2191

work page 2020

[23] [23]

Virtual truss model for characterization of internal forces for multiple finger grasps

Yoshikawa, T. Virtual truss model for characterization of internal forces for multiple finger grasps. V ol. 15, No. 5, 1999, pp. 941–947

work page 1999

[24] [24]

Gabellieri, L

Tognon, M., C. Gabellieri, L. Pallottino, and A. Franchi. Aerial co-manipulation with cables: The role of internal force for equilibria, stability, and passivity. V ol. 3, No. 3, 2018, pp. 2577– 2583

work page 2018

[25] [25]

Franchi, H

Zelazo, D., A. Franchi, H. H. B ¨ulthoff, and P. R. Giordano. Decentralized Rigidity Maintenance Control with Range Measurements for Multi-Robot Systems. The International Journal of Robotics Research , V ol. 34, No. 1, 2015, pp. 105–

work page 2015

[26] [26]

doi:10.1177/0278364914546173

work page doi:10.1177/0278364914546173

[27] [27]

The Mathematical Coloring Book

Soifer, A. The Mathematical Coloring Book. Springer-Verlag, 2008. Acknowledgements This work was partially funded by the Horizon Europe research and innovation programs under agreement no. 101059875 (Flyflic) and agreement no. 101120732 (Autoassess). Prepared using TRR.cls

work page 2008