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Aerial Non-Stop Trajectories Preserving the Equilibrium of Loads Suspended by Variable-Length Cables
Antonio Franchi, Alberto Landi, Chiara Gabellieri
TL;DR
The paper studies how aerial carriers can move without stopping while preserving a rigid suspended load's pose, including the challenge of level flight. It derives a cable-length law for shaping carrier positions and characterizes when non-stop motion is possible. Two carriers permit bounded unrestricted non-stop motion, but periodic level non-stop motion is obstructed for two and constructively achieved for three or more under stated conditions.
Problem
Fixed cable lengths constrain carrier motion to spheres and prevent independent altitude shaping, while level equilibrium-preserving non-stop trajectories with actuated cable lengths have not been enforced.
Source: §I · p. 1Method
The paper combines equilibrium-preserving internal-force motions with an explicit cable-length law for shaping carrier positions, including constant-altitude trajectories.
Source: §I · p. 1, A. Constant-Altitude Realization · p. 4Results
Two carriers admit bounded unrestricted non-stop motion, but periodic level non-stop motion is obstructed for two carriers and constructively achieved for three or more under stated conditions.
Source: §I · p. 1, B. Structural Limitation of… · p. 5, §VII · p. 6Takeaways & Limitations
Variable cable lengths provide radial freedom for shaping equilibrium-preserving carrier trajectories, with periodic level non-stop motion requiring at least three carriers in the stated construction.
Source: §VII · p. 6The analysis assumes taut cables, ideal cable-length and trajectory tracking, and a constant load pose, while neglecting winch and carrier constraints, slackness, aerodynamics, and collisions.
Source: §VII · p. 7
Abstract
from arXiv · showhide
This work studies equilibrium-preserving non-stop trajectories of aerial carriers connected to rigid loads by variable-length cables. Internal-force motions vary the cable directions without changing the load wrench, while cable-length actuation shapes the radial realization of the carrier trajectories. We derive an explicit cable-length law for realizing prescribed carrier-position profiles along a fixed direction, with constant altitude as a relevant specialization. Under this constraint, a carrier stops exactly when its cable direction stops changing. This characterization reveals a distinction between unrestricted and level motion: variable cable lengths enable bounded non-stop motion with two carriers, whereas periodic level non-stop motion is obstructed for two carriers and constructively achieved for three or more under explicit geometric and feasibility conditions. Numerical simulations illustrate the constant-altitude construction.
I. INTRODUCTION AND RELATED WORK
The paper addresses equilibrium-preserving trajectories for aerial carriers connected to a rigid load, focusing on non-stop motion and constant-altitude shaping. It uses variable cable lengths to shape carrier positions while internal-force motions preserve the load wrench.
I. INTRODUCTION AND RELATED WORK
- Fixed-length systems constrain carriers to spheres and do not permit independent altitude shaping, while level flight is especially relevant for wing-borne carriers.Prior persistent non-stop methods use fixed cable lengths and require at least three carriers.Source: §I · p. 1
- The work generates carrier and variable-cable-length trajectories compatible with a constant load pose.Its contributions include radial shaping, two-carrier unrestricted motion, a two-carrier periodic-level obstruction, and a level construction for three or more carriers.Source: §I · p. 1
II. MODELING
- With variable-length cables, carriers can remain in motion when their tangential velocity component vanishes, provided cable length is changing.Carrier velocity separates into tangential motion from cable-direction change and radial motion from winch actuation.Source: §II · p. 2
- Internal-force motions vary cable directions while preserving the load wrench, whereas cable-length actuation shapes the radial realization of those directions.The paper's model assumes straight, taut cables connecting each rigid-load attachment point to a carrier.Source: §II · p. 2
III. EQUILIBRIUM-PRESERVING FORCE MOTIONS
The paper characterizes force distributions that produce a prescribed equilibrium wrench. Internal forces provide the freedom to vary individual cable forces without changing that wrench.
III. EQUILIBRIUM-PRESERVING FORCE MOTIONS
- The equilibrium model expresses the cable wrench as the wrench required to balance gravity about the load center of mass.The prescribed static equilibrium is assumed feasible.Source: §III · p. 2
- Every tensile force distribution producing the prescribed equilibrium wrench is represented as a particular solution plus an internal-force variation.The admissible distributions are restricted to those with positive cable tensions.Source: §III · p. 3
- Internal forces can change individual cable forces and directions while preserving the load wrench.They are force variations in the kernel of the grasp map, which represents resultant force and moment about the load center of mass.Source: §III · pp. 2–3
A. Cycle-Induced Internal-Force Motions
A Hamiltonian cycle provides a structured family of internal-force motions. Its cycle-induced forces self-equilibrate, and each carrier's force block has a local two-direction representation.
A. Cycle-Induced Internal-Force Motions
- Each column of the cycle-induced internal-force matrix has zero resultant force and zero resultant moment about the load center of mass.Each edge applies equal and opposite forces along parallel directions.Source: A. Cycle-Induced Internal-Force Motions · p. 3
- The cycle-induced matrix parametrizes an internal-force subspace, allowing the constructions to avoid requiring a full basis of the grasp kernel.Its columns belong to the kernel of the grasp map.Source: A. Cycle-Induced Internal-Force Motions · p. 3
- At each cycle vertex, the associated internal-force block is represented by the signed direction arrays for its incoming and outgoing edges.Every vertex has exactly one incoming and one outgoing edge.Source: A. Cycle-Induced Internal-Force Motions · p. 3
IV. RADIAL SHAPING OF CARRIER TRAJECTORIES
Variable cable lengths allow the radial realization of cable directions to be shaped independently, enabling prescribed carrier-position profiles along a fixed direction. Fixed-length cables instead impose a compatibility constraint on those profiles.
IV. RADIAL SHAPING OF CARRIER TRAJECTORIES
- Cable lengths can be selected independently to shape the radial realization of cable directions, subject to their admissible bounds.With fixed lengths, the directional profile must instead satisfy the compatibility condition in Corollary 2.Source: §IV · p. 3
- For a fixed unit direction, variable cable lengths realize a continuously differentiable desired carrier-position profile when the cable direction's projection on that axis stays bounded away from zero.The resulting lengths must also remain within admissible positive bounds.Source: §IV · p. 3
- The carrier position along the chosen direction is obtained by projecting the carrier-position relation onto that direction and solving for cable length.The lower bound on the direction projection prevents singular length commands.Source: §IV · p. 3
A. Constant-Altitude Realization
The constant-altitude specialization uses cable-length actuation to realize carrier motion while maintaining altitude. Under its length law, a carrier stops exactly when its cable direction stops changing.
A. Constant-Altitude Realization
- The vertical-direction specialization realizes a desired constant carrier altitude when the induced cable length satisfies the admissibility condition (25).The altitude is defined relative to the load-side attachment point.Source: A. Constant-Altitude Realization · p. 4
- With fixed cable length, constant altitude requires q_i,z(t) to remain constant; variable length allows ℓ_i(t) and q_i,z(t) to vary while their product stays constant.The fixed-length direction is restricted to a constant-latitude circle on S2.Source: A. Constant-Altitude Realization · p. 4
- Under the constant-altitude length law, a carrier stops exactly when its cable-direction rate is zero, so periodic level motion is non-stop iff that rate never vanishes.The continuous periodic speed then has a strictly positive minimum over one period.Source: A. Constant-Altitude Realization · p. 4
V. STRUCTURAL EXISTENCE RESULTS
Admissible non-stop motion requires feasible tensile cable forces, bounded positive cable lengths, and a common positive lower bound on every carrier’s speed. Level motion additionally fixes each carrier’s altitude.
V. STRUCTURAL EXISTENCE RESULTS
- Admissibility requires cable forces to remain in F+ and every cable length to stay within positive finite bounds.The bounds apply to all carriers throughout the motion.Source: §V · p. 4
- Every carrier must maintain speed at least a common constant v > 0 for all time.This is the uniform non-stop requirement in the definition.Source: §V · p. 4
- A motion is level when each carrier remains at a constant altitude h_i.Source: §V · p. 4
A. Unrestricted Non-stop Motion
A single carrier cannot sustain admissible non-stop motion indefinitely while its cable length remains bounded. Cable-length actuation can move it temporarily, but does not remove that bounded-motion obstruction.
A. Unrestricted Non-stop Motion
- No admissible non-stop motion exists for one carrier.Cable-length actuation permits temporary motion while the load remains still, but that motion cannot remain bounded and non-stop for all time.Source: A. Unrestricted Non-stop Motion · p. 4
- With one cable and a constant load pose, translational equilibrium uniquely fixes the cable force and therefore the cable direction.Source: A. Unrestricted Non-stop Motion · p. 4
- Non-stop motion would require the cable length to keep changing in one direction at a rate bounded away from zero, eventually violating its bounds.The cable-length derivative is continuous and never vanishes in this argument.Source: A. Unrestricted Non-stop Motion · p. 4
1) Two carriers:
Under condition (37), two carriers admit bounded unrestricted non-stop motion. A sinusoidal internal-force parameter and cable lengths provide a construction whose carrier speeds have a common positive lower bound.
1) Two carriers:
- Under condition (37), an admissible non-stop motion exists with two carriers.Together with the one-carrier impossibility, the paper concludes that unrestricted admissible non-stop motion exists iff n ≥ 2 under the stated feasibility and nondegeneracy conditions.Source: 1) Two carriers · pp. 4–5
- The construction uses λ(t) = λ0 + a cos(ξt) and ℓ_i(t) = ℓ_i0 + b_i sin(ξt), with constants chosen to preserve force and length feasibility.The force distribution remains in F+ and cable lengths stay strictly within their admissible bounds.Source: 1) Two carriers · pp. 4–5
- The λ and cable-length derivatives never vanish simultaneously, so neither cable direction stops changing and both periodic carrier speeds have a common positive lower bound.For n ≥ 3, the paper also notes an admissible fixed-length construction from.Source: 1) Two carriers · p. 5
B. Structural Limitation of Two-Carrier Level Motion
With two carriers, periodic level motion cannot remain non-stop under the stated assumptions. Periodicity forces a moment when both carriers stop, because exact constant altitude prevents cable length from independently compensating for a stationary cable direction.
B. Structural Limitation of Two-Carrier Level Motion
- For two carriers, no periodic level carrier motion is non-stop for all time when the internal-force parameter is continuously differentiable and stays in a compact admissible interval.The obstruction also assumes condition (37).Source: B. Structural Limitation of… · p. 5
- A continuously differentiable periodic scalar internal-force parameter has a stationary point each period, when both cable directions and carriers stop.Under level motion, a carrier stops exactly when its cable direction stops changing.Source: B. Structural Limitation of… · p. 5
- Unlike unrestricted motion, exact constant altitude fixes the cable-length rate, so it cannot compensate when the cable direction stops changing.In the unrestricted construction, cable length is deliberately varied when cable-direction motion vanishes.Source: B. Structural Limitation of… · p. 5
C. Level Non-stop Motion with n ≥3
For three or more carriers, the paper constructs equilibrium-preserving, non-stop motion at constant altitude under geometric and feasibility conditions. The construction combines cycle-based internal-force variation with a cable-length law, and illustrates reduced altitude oscillations when cable lengths vary.
C. Level Non-stop Motion with n ≥3
- For the cycle-based force family with n ≥3, force motion induces cable-direction motion when the stated force conditions hold.The force derivative retains a nonzero component orthogonal to the cable direction.Source: C. Level Non-stop Motion with n ≥3 · p. 5
- With cable-length variation, carrier speeds remain positive while altitude oscillations fall from several decimetres without variation to around 0.001 m.Figure 1 compares the level-flight results with and without cable-length variation.Source: Figure 1 · p. 5
- Under the theorem’s geometric and feasibility conditions, the resulting trajectories preserve load equilibrium, remain at constant altitudes, and are non-stop for n ≥3.The conditions include force-trajectory admissibility, nonzero vertical cable-direction components, and admissible parameter bounds.Source: C. Level Non-stop Motion with n ≥3 · p. 5
- A feasible phase choice sets φ_j = (j − 1)π/n, making every consecutive cycle-edge phase difference, including across the cycle boundary, not an integer multiple of π.The construction uses cycle parameters λ_j(t) = λ_j0 + a sin(ξt + φ_j).Source: C. Level Non-stop Motion with n ≥3 · pp. 5–6
- Within the periodic internal-force constructions considered, admissible unrestricted non-stop motion requires at least two carriers, while admissible level non-stop motion requires at least three.The two-carrier level obstruction and the n ≥3 construction establish this threshold.Source: C. Level Non-stop Motion with n ≥3 · p. 6
VI. NUMERICAL ILLUSTRATION
The numerical illustration uses four aerial carriers to demonstrate the constant-altitude construction, with an unperturbed load pose and positive minimum cable tension. The conclusion notes that the analysis assumes ideal tracking and taut cables while omitting several practical constraints and effects.
VI. NUMERICAL ILLUSTRATION
- With four carriers, the simulation leaves the load pose unperturbed and maintains a minimum cable tension of 0.71 N in both cases.The simulated load has mass 0.3 kg.Source: §VI · p. 6
VII. CONCLUSIONS AND LIMITATIONS
- Variable cable lengths shape equilibrium-preserving trajectories; two carriers suffice for unrestricted non-stop motion, but periodic level motion requires n ≥3 under the stated feasibility conditions.Source: §VII · p. 6
- The analysis assumes taut cables, ideal cable-length and trajectory tracking, and a constant load pose, while neglecting winch and carrier constraints, slackness, aerodynamics, and collisions.Source: §VII · p. 7