Trajectory control of a suspended load with non-stopping flying carriers

Technical Summary: Trajectory Control of the Suspended Load Pose using Non-stopping Flying Carriers

Problem Statement
The paper addresses the challenge of cooperative aerial transportation of cable-suspended payloads using Uncrewed Aerial Vehicles (UAVs). While existing research has extensively covered multirotor systems capable of hovering (stopping) to stabilize loads, these platforms suffer from limited endurance and range, making them unsuitable for long-distance logistics. Conversely, fixed-wing UAVs offer superior efficiency and endurance but possess a fundamental constraint: they cannot hover and must maintain a strictly positive forward velocity to remain airborne.

The core problem is to design a control framework for a system of $n \geq 3$ "non-stopping" flying carriers (e.g., fixed-wing UAVs) that can cooperatively transport a rigid payload along a desired trajectory. The trajectory may include dynamic phases (movement) and static phases (rest or slow movement). The control system must achieve two simultaneous, often conflicting, objectives:

1. Load Tracking: Regulate the 6-degree-of-freedom (6D) pose of the suspended load to follow a desired trajectory with bounded errors.
2. Non-Stopping Constraint: Ensure that every carrier maintains a velocity norm strictly greater than a minimum threshold ($\varepsilon > 0$), preventing them from stopping or hovering, even when the load itself is stationary or moving slowly.

Methodology
The authors propose a closed-loop control architecture that separates load tracking from carrier trajectory feasibility through three main components:

1. Outer-Loop Load-Wrench Controller:
   A feedback controller computes the desired wrench (force and moment) required at the load's center of mass to track the desired position and orientation. This module uses PID control laws based on pose and velocity errors to generate the target wrench $w_d(t)$, independent of the carrier constraints.

2. Cable Force Distribution and Grasp Matrix Formulation:
   The system utilizes the grasp matrix $G(R_L)$ to map cable forces to the load wrench. Since the number of cables ($3n$) exceeds the degrees of freedom of the load (6), the system possesses internal force redundancy. The cable forces are decomposed into:

   • A particular solution ($f_g$) that generates the required load wrench.
   • A nullspace component ($f_\lambda$) representing internal forces that redistribute tension without affecting the net load wrench.

3. Optimization Layer (Internal Force Modulation):
   This is the novel core of the proposed method. Instead of using fixed open-loop internal forces (as in prior work for static loads), the authors introduce an online optimization layer.

   • Parameterization: The internal force parameters $\lambda(t)$ are modeled as sinusoidal functions with time-varying amplitude $A(t)$ and frequency $\xi(t)$.
   • Objective: The optimizer dynamically adjusts $A(t)$ and $\xi(t)$ to ensure the carriers' velocities satisfy the non-stopping constraint ($\|\dot{p}_{Ri}\| \geq \varepsilon$).
   • Cost Function: The optimization minimizes the variation of the amplitude and frequency relative to the previous time step, as well as the variation of the internal forces and their derivatives. This discourages abrupt changes in cable tension and carrier trajectories.
   • Constraint: The optimization explicitly enforces that the origin is excluded from the set of possible carrier velocities, mathematically ensuring the velocity norm never drops below $\varepsilon$.

4. Carrier-Trajectory Generator:
   Using the desired wrench and the optimized internal forces, the system computes the required cable tensions and the corresponding carrier trajectories. These trajectories are then tracked by low-level carrier controllers.

Key Contributions
The paper claims the following primary contributions:

• First Closed-Loop Framework: It presents the first control architecture for cooperative payload transportation using non-stopping flying carriers that enables the tracking of dynamic load trajectories (including rest phases) while strictly enforcing non-stopping constraints.
• Feedback Wrench Control: The design of a feedback controller that actively regulates the load's pose by computing the necessary wrench.
• Online Internal Force Optimization: The formulation of an optimization layer that dynamically shapes internal force parameters (amplitude and frequency of sinusoidal functions) to guarantee persistent carrier motion without altering the desired load wrench. This allows carriers to "loiter" or move in elliptical paths even when the load is static.

Results and Validation
The method was validated through numerical simulations involving a system of four carriers transporting a 1.0 kg rigid load. The simulations accounted for full payload and cable dynamics, including non-idealities like cable elasticity (modeled as massless spring-dampers).

• Scenario: The load was tasked with moving from a static start, translating linearly, and returning to a static final position.
• Without Optimization: In a baseline scenario without the optimization layer, carriers attempting to follow the load kinematics performed sharp turns that caused their velocity norms to drop to zero, violating the non-stopping constraint.
• With Optimization: The proposed controller successfully maintained carrier velocities above the threshold ($\varepsilon = 0.2$ m/s) throughout the mission. When the load was static, the carriers continued moving in elliptical trajectories (loitering) to maintain aerodynamic feasibility.
• Tracking Performance: The load tracking errors remained bounded and converged to zero during static phases. The authors noted a slight increase in position error in the optimized case compared to the non-optimized case (mean error 0.015 m vs. 0.013 m), attributing this to the higher carrier velocities exciting the visco-elastic cables. However, the trade-off was deemed acceptable to satisfy the non-stopping constraint.

Significance and Claims
The authors position this work as a foundational step toward scalable, long-range aerial logistics using fixed-wing or convertible UAVs. By decoupling the load's desired motion from the carriers' kinematic constraints via internal force optimization, the method enables the use of efficient, non-hovering platforms for tasks that traditionally required hovering multirotors.

The paper modestly claims to provide the first solution for dynamic load tracking under non-stopping constraints, moving beyond previous open-loop strategies limited to static loads. It emphasizes that the proposed framework successfully prevents carrier stagnation while achieving accurate load trajectory tracking, thereby bridging the gap between the aerodynamic efficiency of fixed-wing vehicles and the dexterity required for cooperative manipulation. Future work is identified as refining the model to include full vehicle dynamics, handling potential optimization failures, and moving toward experimental validation on multirotor UAVs emulating fixed-wing constraints and eventually on actual convertible UAVs.