⚡ electrical engineering

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

This paper presents a scalable algorithm and theoretical framework enabling three or more aerial carriers to perform perpetual, coordinated non-stop flights while maintaining a static cable-suspended load by constructing elliptical trajectories within specific 2D affine subspaces derived from a Hamiltonian cycle on the load's attachment points.

Original authors: Chiara Gabellieri, Yaolei Shen, Martina Paolucci, Antonio Franchi

Published 2026-05-12

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CC BY 4.0

Original authors: Chiara Gabellieri, Yaolei Shen, Martina Paolucci, Antonio Franchi

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a group of drones trying to hold a heavy object, like a construction beam or a package, perfectly still in mid-air. Usually, to keep something hanging from a rope perfectly still, the people holding the ropes have to stand perfectly still too. If they start walking around, the object would swing and spin.

This paper asks a tricky question: Can a team of drones keep a heavy object perfectly still while all of them are flying in circles (or loops) without ever stopping or landing?

The answer, according to this research, is yes, but only if you have at least three drones, and they fly in a very specific, mathematically choreographed dance.

Here is how they did it, explained simply:

1. The Problem: The "Stop-and-Go" Dilemma

If you have one drone holding a load, it must hover in one spot. If it moves, the load swings.
If you have two drones, they can't hold a load steady while moving in a circle because the physics just doesn't work out; the load would start spinning or falling.
But with three or more drones, there is enough "wiggle room" in the physics to let them move while the load stays put.

2. The Secret Sauce: The "Hamiltonian Cycle" (The Magic Loop)

The researchers came up with a clever way to plan the flight paths. They used a concept from math called a Hamiltonian Cycle.

Think of the drones as points on a map. A Hamiltonian Cycle is a path that visits every single drone exactly once and then returns to the start, forming a giant loop (like a necklace of drones).

The Trick: They didn't just pick any loop. They picked a loop and assigned specific "internal forces" to the cables connecting the drones to the load.
The Analogy: Imagine the load is a spinning plate on a stick. The drones are the people spinning the stick. Usually, if they move their hands, the plate wobbles. But this team figured out a way for the drones to move their hands in a specific, synchronized pattern so that the "wobbles" cancel each other out perfectly. The load feels like it's being held by a ghost that never moves, even though the drones are zooming around.

3. The "Edge Coloring" Dance

To make sure the drones never stop moving, the researchers had to solve a timing puzzle.

They assigned each drone a specific "rhythm" (a wave-like motion) to follow.
They used a math trick called edge coloring (like coloring a map so no two touching countries have the same color) to make sure that when one drone slows down to turn a corner, its neighbor is speeding up.
The Result: At no single moment does the whole team stop. As long as the load is being pulled, at least one drone is always pulling hard enough to keep the tension, and the others are moving in a way that balances it out. It's like a relay race where the baton is never dropped, but everyone is running in a circle at the same time.

4. The "Elliptical" Path

The drones don't fly in perfect circles. They fly in ellipses (ovals).

The researchers proved that if the drones fly in these specific oval shapes, the load stays perfectly still.
They even showed that for certain setups (like a flat load with cables attached evenly), the drones can fly at a constant speed, which is great for saving battery and making the flight smooth.

5. What They Actually Tested

The paper didn't just do math on a computer; they proved it works in the real world.

Simulations: They ran thousands of computer simulations to see what happens if a drone fails or if the wind blows. They found the system is quite robust.
Real Experiments: They used four quadcopter drones (the kind with four propellers) in a lab. They tied them to a small cross-shaped object with strings.
- With their method: The drones flew in loops, and the object hung there, barely moving (like a statue).
- Without their method: They deliberately messed up the timing of one drone. Suddenly, the object swung wildly and spun around.

Summary

This paper proves that you can have a team of three or more drones fly non-stop in loops while holding a heavy object perfectly still. They did this by using a specific mathematical map (the Hamiltonian cycle) to choreograph the drones' movements so that their individual motions cancel out any shaking, leaving the load in a state of "perpetual suspension."

They validated this with computer models and real-life drone experiments, showing that the object stays still while the drones keep flying.

Technical Summary: Cyclic Nullspace Coordination for Perpetual Flight of Aerial Carriers

Problem Statement
This work addresses the challenge of maintaining a cable-suspended rigid-body load in a constant static pose (fixed position and orientation) while the $n$ aerial carriers ( $n \geq 3$ ) executing the manipulation perform non-stop flights with non-zero forward speeds. While previous research established the theoretical feasibility for $n=3$ and demonstrated that $n < 3$ is impossible, a general constructive method for arbitrary $n \geq 3$ was lacking. The core difficulty lies in reconciling the static equilibrium constraints of the load (requiring specific force balances) with the dynamic constraints of the carriers (requiring continuous motion to avoid hovering, which is essential for fixed-wing or convertible UAVs).

Methodology
The proposed methodology, termed "Cyclic Nullspace Coordination," generates coordinated, periodic trajectories for the carriers by manipulating the internal forces within the system's nullspace. The approach relies on two primary pillars:

Hamiltonian Cycle-Based Nullspace Construction:
The method selects $n$ linearly independent directions of internal forces within the $(3n-6)$ -dimensional nullspace of the grasp matrix $G$ . These directions are derived from the edges of a Hamiltonian cycle $H$ on the complete graph connecting the $n$ cable attachment points on the load. By mapping the cycle edges to the nullspace matrix $N$ , the internal force of each cable is decomposed into two distinct 2D affine subspaces. This construction ensures that the forces evolve in a way that satisfies the static equilibrium equation $G(\bar{R}_L)f(t) = w_0$ while allowing the forces to vary over time.
Graph Coloring for Periodic Trajectories:
To ensure the carriers never stop (i.e., their velocity norm $\|\dot{p}_{R_i}(t)\|$ remains strictly positive), the time-varying coefficients $\lambda(t)$ of the nullspace forces are chosen as periodic functions. The paper employs a graph coloring strategy where each edge of the Hamiltonian cycle is assigned a specific periodic function (e.g., sinusoids with specific phase shifts). The coloring ensures that for any carrier $i$ , the derivatives of the two adjacent edge functions ( $\dot{\lambda}_i$ and $\dot{\lambda}_{i+1}$ ) are never simultaneously zero. This guarantees that the time derivative of the cable force, $\dot{f}_i(t)$ , is non-zero and non-collinear with the force itself, which, via the kinematic relationship $\dot{p}_{R_i} \propto \dot{f}^\perp_i$ , ensures the carrier maintains a non-zero velocity.

The algorithm computes these trajectories by:

Selecting a valid Hamiltonian cycle that satisfies geometric non-degeneracy conditions (attachment points not aligned, force directions not collinear with the equilibrium force).
Assigning periodic functions to cycle edges via edge coloring (using 2 functions for even $n$ , 3 for odd $n$ ).
Calculating the resulting cable forces and deriving the carrier positions and velocities through kinematic inversion.

Key Contributions

General Existence Proof: The paper provides the first proof of the existence of non-stop flight trajectories for holding a static load for any number of carriers $n \geq 3$ , extending previous minimal findings limited to $n=3$ .
Constructive Algorithm: It presents a complete, scalable algorithm for generating these trajectories, including guidelines for tuning parameters (frequency, amplitude, edge coloring) and selecting Hamiltonian cycles to optimize performance.
Fixed-Wing Compatibility: The method is adapted to include a planning layer that enforces constraints typical of fixed-wing UAVs (speed, bank angle, flight-path angle bounds), demonstrating that non-stop manipulation is compatible with fixed-wing dynamics.
Robustness Analysis: The work analyzes the system's sensitivity to parameter uncertainties (mass, cable length, attachment points) and investigates the impact of single-carrier failures, showing that load pose errors remain bounded even if one carrier detaches.

Results

Simulations: Extensive numerical simulations in MATLAB-Simulink validated the theoretical findings. Results confirmed that carriers can maintain a static load pose while executing periodic, non-zero-speed trajectories. The simulations demonstrated that:
- Load pose errors remain low (position error < few cm, attitude error < 2 degrees) despite sensor noise and tracking errors.
- The choice of Hamiltonian cycle affects the minimum carrier velocity; cycles with orthogonal internal force directions yield constant-speed profiles, while non-orthogonal directions increase speed variability.
- The system is robust to single-carrier failures, with load pose errors remaining bounded.
- A fixed-wing compatible planner successfully generated trajectories satisfying speed and angle constraints.
Experiments: Laboratory experiments were conducted using four Crazyflie 2.1 quadrotors tethered to a cross-shaped payload.
- With the proposed method: The load remained essentially static (average position error ~5.6 cm, attitude errors ~1-3 degrees) while the quadrotors moved in continuous loops.
- Without the method (control case): When one quadrotor's trajectory was desynchronized, the load exhibited significant linear and angular velocities, confirming that the cyclic coordination is necessary for static suspension.
- The experiments validated the "cable abstraction," showing that the theoretical model holds in practice despite unmodeled cable elasticity and lack of load-state feedback.

Significance
The paper claims to constitute a "substantial leap forward" by solving the general case for $n \geq 3$ and providing the first constructive design for non-stop aerial manipulation. Its significance lies in enabling energy-efficient aerial manipulation by integrating long-distance transport (non-stop flight) with precise load control. This capability is particularly relevant for fixed-wing and convertible UAVs, which cannot hover, thereby expanding the operational envelope of cooperative aerial manipulation to include continuous flight scenarios while maintaining load stability. The work bridges the gap between theoretical nullspace control and practical, non-stop flight constraints.