SQP-Based Cable-Tension Allocation for Multi-Drone Load Transport

Technical Summary: SQP-Based Cable-Tension Allocation for Multi-Drone Load Transport

Problem Statement
Multi-Agent Aerial Load Transport Systems (MAATS), which utilize multiple drones to carry a single payload, offer superior payload capacity and fault tolerance compared to single-drone solutions. However, these systems face a fundamental control challenge: the force-balance equations are underdetermined. With $n$ agents and 3 degrees of freedom for the load, there are infinitely many ways to distribute the required total tension. Naive allocation strategies often lead to uneven energy distribution (causing premature battery depletion in specific agents), cable slack, or unsafe configurations where cables converge, increasing the risk of inter-drone or cable collisions. Existing solutions often focus solely on minimizing energy surrogates or enforcing rigid geometric formations, failing to simultaneously balance force distribution and maintain safe inter-cable alignment without restricting the operational envelope.

Methodology
The paper proposes a real-time optimization layer integrated into a hierarchical control architecture. The system employs a four-layer cascade controller:

1. Load Controller: Generates a total virtual force ($u_L$) required for trajectory tracking.
2. Tension Allocation (The Core Contribution): Distributes $u_L$ among the $n$ UAVs.
3. Position Controller: Computes individual thrust commands for each UAV.
4. Attitude Controller: Generates final torque and thrust commands.

The central innovation is the Sequential Quadratic Programming (SQP) formulation used in the tension allocation layer. Given the virtual control input, the SQP solver determines the desired tension magnitudes ($T_{id}$) and unit direction vectors ($\alpha_{id}$) by solving the following optimization problem:

$$\min_{T_{id}, \alpha_{id}} J = \frac{1}{2} \sum_{i=1}^{n} T_{id}^2 + \mu \sum_{i=1}^{n-1} \sum_{j=i+1}^{n} (\alpha_{id}^\top \alpha_{jd})^2$$

Subject to:

• Force balance: $\sum T_{id} \alpha_{id} = -u_L$
• Physical constraints: $T_{id} \geq 0$ and $\|\alpha_{id}\| = 1$

Key Contributions

• Dual-Objective Optimization: Unlike prior works that minimize only energy, this formulation minimizes the sum of squared tensions (promoting balanced effort) while incorporating a cable-alignment penalty. The penalty term, weighted by $\mu$, discourages small angles between cables (where the dot product approaches 1), thereby preventing tether convergence and collision risks.
• Real-Time Feasibility: The problem is structured to utilize quadratic subproblems with closed-form solutions for the gradient. The authors implement a warm-start initialization strategy, using the solution from the previous time step to initialize the current iteration. This ensures continuity in cable configuration and drastically reduces the number of iterations needed for convergence.
• Scalability and Flexibility: The approach does not impose a fixed geometric formation. Instead, it allows the optimizer to find a safe, energy-efficient arrangement from any feasible initial state, adapting to obstacles or operational constraints that might prevent maintaining a predefined shape.

Results
The method was validated through numerical simulations in MATLAB/Simulink using a four-quadrotor system transporting a 0.225 kg payload.

• Tracking Performance: The system maintained centimeter-level trajectory tracking accuracy during an ascending spiral mission, with a Root Mean Square (RMS) position error of 2.97 cm and a maximum excursion under 8 cm. This confirms the optimization layer does not degrade control performance.
• Tension Balancing: Compared to a baseline geometric approach, the SQP allocator achieved a significantly more balanced force distribution, preventing individual motors from saturating and ensuring uniform battery discharge.
• Safety: With an alignment weight $\mu = 0.15$, the system maintained all pairwise cable angles above 44.4°, comfortably exceeding the 30° safety threshold. The baseline approach frequently dropped below this limit.
• Computational Efficiency: On standard hardware (Intel Core Ultra 7), a complete SQP cycle averaged 1.13 ms, with a worst-case time of 12.29 ms. This demonstrates clear feasibility for real-time embedded implementation.
• Sensitivity Analysis: The study showed that the alignment weight $\mu$ acts as a tunable parameter, allowing operators to trade off between safety margin (larger angles) and energy consumption (integrated tension cost) without measurable degradation in tracking performance.

Significance and Claims
The paper claims to provide a scalable path to safe and energy-balanced cooperative load transport. By explicitly addressing both tension balancing and cable-to-cable alignment within a lightweight SQP framework, the method fills a gap in current literature where controllers often prioritize one safety aspect over the other or rely on prohibitive cost functions. The authors assert that their approach enables controllable trade-offs between safety and energy efficiency while maintaining the flexibility to operate without rigid formation constraints.

Limitations
The authors modestly acknowledge that the current study assumes rigid, massless cables and relies on a centralized optimization layer, which introduces a single point of failure. Furthermore, the results are based solely on simulations; hardware validation is required to assess performance under real-world conditions such as wind disturbances, communication delays, and modeling errors. Future work is proposed to include hardware experiments and distributed optimization strategies.