Constraint-Aware Payload Layer Fusion Control for Dual-Quadrotor Cooperative Slung-Load Transportation

Abstract

Low altitude logistics and aerial transport increasingly rely on multirotor unmanned aerial vehicles (UAVs) carrying slung payloads, where cable flexibility and load swing can degrade safety and delivery accuracy. This paper studies payload trajectory tracking for a dual-quadrotor cooperative slung-load system, targeting accurate tracking with swing suppression under thrust, attitude, and cable-tension limits. First, a payload-layer dynamic model is derived from d’Alembert’s principle with geometric cable constraints, and explicit tension reconstruction formulas are provided to enable direct enforcement of tension bounds. Building on this model, a payload-layer DEA nominal tracking controller is designed by applying dynamic extension to the tension-scalar channels and enforcing output-level linear error dynamics. To ensure real-time feasibility, a convex quadratic-programming (QP) projection layer minimally corrects the nominal command to satisfy thrust saturation, attitude-cone constraints, and cable-tension bounds. Moreover, an adaptive tuning control layer updates the DEA feedback gain and the projection weighting matrix within preset constraint limits based on energy residual and constraint-activation information, improving robustness and reducing manual tuning. Input-to-state stability is established under bounded disturbances and constraint-activation switching via a composite Lyapunov analysis. ROS–PX4–Gazebo simulations show low tracking error, suppressed swing, and sustained tension-limit compliance, validating the fusion controller.

1. Introduction

With the rapid development of the low altitude economy, the scope of unmanned aerial vehicle (UAV) operations has expanded to diverse cooperative tasks, ranging from area coverage and search and rescue to bird-driving strategies [1]. Among these applications, the use of UAVs for transportation tasks, such as urban logistics, last-mile delivery, and urban air mobility, has witnessed particularly rapid growth. In these scenarios, payload capacity and transportation safety are becoming key bottlenecks for overall system performance. How to accomplish safe and efficient payload transportation in complex low altitude environments has therefore emerged as an important research topic for multirotor systems [2,3,4].

From the viewpoint of payload handling, existing approaches can be broadly divided into two categories. One fixes the payload to the vehicle through a rigid gripper or cargo bay, while the other suspends the payload using cables, forming a flexible “UAV–cable–payload” structure. Compared to rigid robotic arms, cable slung configurations offer lower weight, simpler mechanics, and higher passive safety, which are attractive features for cooperative transportation systems such as dual-quadrotor slung load platforms. A recent survey has provided a comprehensive review of modeling, guidance, and control methods for quadrotor slung load systems and highlighted that cable flexibility and payload swing are critical factors limiting performance and safety [4]. Because rigid bodies and flexible cables coexist, these systems exhibit strong under actuation and nonlinear couplings: the payload motion is strongly coupled with the attitude and thrust of each UAV, which poses significant challenges for accurate modeling and control design.

For single UAV slung load systems, extensive work has focused on dynamic modeling and swing suppression. Energy-based nonlinear adaptive control has been used to achieve asymptotic tracking of both UAV and payload states with rigorous stability guarantees [5]. Adaptive disturbance rejection schemes have been developed to handle unknown payload mass and downwash effects and to improve trajectory tracking accuracy and suspension stability [6]. Dynamic feedback linearization/exact state feedback linearization with DEA-based controller derivation and disturbance robustness [7]. In addition, nonlinear robust controllers—such as (fractional-order) sliding-mode designs for quadrotors with slung loads—have been investigated to mitigate external disturbances and improve transient tracking performance under uncertainties [8]. Overall, single UAV slung load transportation has reached a relatively mature level in terms of system identification and swing control. However, the payload capacity, fault tolerance, and task redundancy of a single platform remain limited, which restricts its applicability to heavy load, long range, or safety critical missions.

To overcome these limitations, multi-quadrotor cooperative slung load systems have become a major research direction. Systematic studies have addressed analysis, planning, and control for tethered multirotor cooperative transportation, proposing unified frameworks that combine modeling, trajectory generation, and controller design [9]. For multi-UAV configurations, passivity-based cooperative controllers have been proposed to guarantee energy dissipative stability under external disturbances [10], and energy efficient strategies exploiting torque and inertia estimation have been shown to reduce energy consumption during transportation [11]. Geometric and robust control approaches have been developed to maintain controllability in the presence of contact or impact events [12]. Improved swing state estimation methods have also been introduced to provide more accurate feedback for the control of multirotor slung load system [13]. In the specific case of dual-quadrotor setups, nonlinear coupled models and two layer swing suppression controllers have been proposed to ensure rapid convergence of payload swing angles [14]. Robust cooperative controllers have been developed for dual-quadrotor systems transporting cable-suspended payloads (e.g., Y-shaped configurations), improving resilience to disturbances and parameter variations [15]. Along a complementary line, consistency-theory-based coordinated control has been proposed for quadrotor suspension systems to synchronize multiple vehicles while regulating load motion and suppressing oscillations [16].

These results demonstrate the potential of dual-quadrotor platforms for lifting heavier payloads, improving maneuverability, and increasing mission redundancy.

For more complex missions and tighter safety requirements, optimization and predictive control methods that explicitly handle constraints have received growing attention. Nonlinear model predictive control (NMPC) methods for cooperative transportation explicitly incorporate cable tension and attitude constraints into the optimization problem, yielding dynamically feasible trajectory planning and tracking solutions [17]. Adaptive NMPC can learn and compensate model uncertainties online (e.g., wind, unknown payloads), improving tracking without gain retuning [18]. Quadratic programming-based force allocation schemes have been proposed to compute near optimal thrust distributions in real time while satisfying actuator limits [19]. These studies show that optimizationbased control with explicit modeling of thrust, attitude, and tension constraints is an effective way to improve performance and safety in cooperative slung load transportation. However, in most cases the optimization layer is still formulated at the vehicle level, and the coupling among the UAVs, cables, and payload is not fully exploited at the payload layer.

Moreover, horizon-based constrained optimal control methods (e.g., receding-horizon MPC/NMPC) generally involve solving an online receding-horizon optimization, and the computational cost may increase notably with model fidelity and coupling complexity in multi-body slung-load systems. This motivates decoupled architectures that preserve explicit constraint enforcement through an instantaneous convex projection while producing the nominal tracking command via an analytical baseline controller.

At the same time, there has been increasing interest in nonlinear cooperative control and energy-consistency analysis under complex dynamics. Energy-dissipative control laws provide a unified perspective for ensuring dynamical consistency and regulating energy in slung-load systems [5,10]. Dynamic extension has also been applied to multi-UAV slung-load control; for instance, Al Lawati et al. reported a DEA-based nonlinear framework and validated its capability in addressing coupled multi-body dynamics [20]. Nevertheless, the joint integration of energy-consistent control, explicit constraint enforcement, and online gain/weight adjustment at the payload layer remains relatively less investigated.

In summary, the control of multi-UAV slung load systems is evolving from purely energy-consistent designs toward integrated schemes that also emphasize optimality and constraint feasibility. Yet, based on the above analysis, existing cooperative slung load control methods for multi-UAV systems still face several limitations in dual-quadrotor transportation: payload layer models that are both coupling-aware and optimization-friendly are scarce; thrust, attitude, and tension constraints are often treated separately; and controller/optimization parameters are usually fixed, which weakens robustness under varying payload and disturbance conditions. To address these issues, this paper proposes a payload-layer fusion cooperative control framework for dual-quadrotor slung load transportation, which combines a DEA-based nominal controller with a QP-based constraint projection and a slow-timescale adaptive tuning module. The main contributions are summarized as follows:

• Payload-Layer Dynamic Formulation: Compared with conventional vehicle-centric cascaded control strategies that treat cable tension as an external disturbance, our formulation derives an explicit payload-layer forward dynamic model via d’Alembert’s principle. This modeling perspective exposes the rigid-body coupling structure directly at the load level and enables analytical tension reconstruction, providing a more transparent foundation for constraint enforcement.
• Feasibility-Preserving Constrained Tracking: While many existing methods handle actuator or tension limits through penalty shaping or post hoc saturation, our framework integrates a Dynamic Extension Algorithm (DEA) with a convex QP-based projection layer. This structure separates nominal high-order tracking design from physical feasibility enforcement, enabling strict constraint satisfaction even during aggressive tracking transients.
• Constraint-Aware Adaptive Gain Regulation: In contrast to fixed-gain robust designs that require conservative tuning, the proposed adaptive module dynamically adjusts tracking aggressiveness based on constraint activation conditions. This reduces manual tuning effort and improves performance consistency under varying payload parameters.

The remainder of this paper is organized as follows. Section 2 presents system modeling and relevant constraints. Section 3 details the proposed controller design. Section 4 provides stability analysis. Section 5 reports simulation/experimental results. Section 6 concludes the paper and discusses future work.

2. Dynamics of the Dual-Quadrotor Slung Load System

2.1. System Description and Coordinate Definitions

The dual-quadrotor cooperative slung load transportation system examined in this paper is illustrated schematically in Figure 1 [9]. The setup consists of two quadrotor UAVs (denoted by ${\mathrm{UAV}}_1$ and ${\mathrm{UAV}}_2$) that cooperatively transport a single rigid payload. The two vehicles are connected to the payload by two lightweight cables of fixed length, enabling cooperative control of the payload’s translation and attitude.

The mass of the $i$-th quadrotor is denoted by $m_i$, the corresponding cable length by $L_i$, the cable tension by $T_i$, and the payload mass by $m_L$. The gravitational acceleration is g. To describe the motion of the dual-quadrotor slung load system, an inertial frame $I=\left\{O_I;x_I,y_I,z_I\right\}$ and body-fixed frames $B_i=\{O_{B_i}; x_{B_i},y_{B_i},z_{B_i}\}, i=1,2$, are used. The position and attitude of the $i$-th quadrotor are given by

$$P_i={\left[x_i,y_i,z_i\right]}^T,\hspace{1em}{\eta }_i={\left[{\varphi }_i,{\theta }_i,{\psi }_i\right]}^T$$

(1)

where ${\varphi }_i$, ${\theta }_i$, and ${\psi }_i$ denote roll, pitch, and yaw, respectively. The position of the payload center of mass is denoted by $P_L={[x_L,y_L,z_L]}^T$. The cable attached to the $i$-th quadrotor has fixed length $L_i$ and its direction is represented by the unit vector

$$q_i={[sin{\theta }_{L_i}cos{\varphi }_{L_i}, sin{\varphi }_{L_i},-cos{\theta }_{L_i}cos{\varphi }_{L_i}]}^T$$

(2)

where ${\theta }_{L_i}$ and ${\varphi }_{L_i}$ denote the pitch and roll of the $i$-th cable, respectively. Throughout this paper, $q_i$ is defined as the unit vector pointing from the payload to the $i$-th UAV, and thus satisfies the geometric constraints

$$P_L=P_i+L_i{q}_i,\hspace{1em}{q}_i^T{q}_i=1$$

(3)

The cables are assumed to be rigid and always taut, i.e., $L_i$ is constant and $T_i>0$.

While the massless and inextensible cable assumption is widely adopted and highly effective for the considered low-to-medium speed operational envelope, it inherently omits “cable whip” and aeroelastic effects that become prominent in high-speed or highly aggressive maneuvers. As highlighted in [4], the taut-cable model remains the most common and utilized representation because it effectively captures the essential complex dynamics while maintaining mathematical tractability. Future work may explore the integration of finite-element cable models or learning-based residual dynamic compensation to address these extreme conditions.

The rotation matrix $R_i\in SO\left(3\right)$ from the body frame to the inertial frame is parameterized by Z–Y–X Euler angles $\left({\varphi }_i,{\theta }_i,{\psi }_i\right)$ as [21]

$$R_i=\left[\begin{array}{ccc}{c}_{{\mathsf{\theta }}_i}{c}_{{\mathsf{\psi }}_i}& c_{{\mathsf{\theta }}_i}{s}_{{\mathsf{\psi }}_i}& -s_{{\mathsf{\theta }}_i}\\ s_{{\varphi }_i}{s}_{{\mathsf{\theta }}_i}{c}_{{\mathsf{\psi }}_i}-c_{{\varphi }_i}{s}_{{\mathsf{\psi }}_i}& s_{{\varphi }_i}{s}_{{\mathsf{\theta }}_i}{s}_{{\mathsf{\psi }}_i}+c_{{\varphi }_i}{c}_{{\mathsf{\psi }}_i}& s_{{\varphi }_i}{c}_{{\mathsf{\theta }}_i}\\ c_{{\varphi }_i}{s}_{{\mathsf{\theta }}_i}{c}_{{\mathsf{\psi }}_i}+s_{{\varphi }_i}{s}_{{\mathsf{\psi }}_i}& c_{{\varphi }_i}{s}_{{\mathsf{\theta }}_i}{s}_{{\mathsf{\psi }}_i}-s_{{\varphi }_i}{c}_{{\mathsf{\psi }}_i}& c_{{\varphi }_i}{c}_{{\mathsf{\theta }}_i}\end{array}\right]$$

(4)

for brevity, $c_{·}=\cos \left(·\right),s_{·}=\sin \left(·\right)$.

In the following sections, the control objective is formulated at the load layer: given a desired payload trajectory and desired cable directions, we aim to design a load-focused QP–DEA fusion controller such that the payload position and selected cable attitudes accurately track their references with small tracking errors, while the quadrotor thrusts, attitudes, and cable tensions satisfy prescribed physical constraints.

2.2. Payload Layer Dynamics and Tension Reconstruction

Building on d’Alembert’s principle and the classical modeling of multirotor slung load systems, this work adopts a geometric constraint elimination strategy. By combining the UAV states $\left(P_i,R_i,f_i\right)$ with cable geometry, a forward dynamics model at the load layer and explicit formulas for reconstructing the cable tensions are derived. The complete derivation is presented in Appendix A; this section summarizes the key intermediate equations and the final expressions used in the subsequent QP–DEA controller design.

Given the payload acceleration ${\ddot{ P}}_L,$ cable direction $q_i,$ relative velocity $v_{rel,i},$ and the thrust vector of the $i$-th UAV $u_i=f_i{R}_i{e}_3$, which are assumed to be known or estimable, the tension in the $i$-th cable can be reconstructed as

$$T_i=m_i\left(q_i^{T}g{e}_3+{\displaystyle \frac{1}{m_i}}{q}_i^T\left(f_i{R}_i{e}_3+d_i\left(t\right)\right)+{\displaystyle \frac{\Vert v_{rel,i}{\Vert }^2}{L_i}}-q_i^T\ddot{P_L}\right),\hspace{1em}i=1,2$$

(5)

Here, $e_3={[0,0,1]}^{\mathrm{T}}$ is the unit vector along the inertial $z_I$-axis, $g$ is the gravitational acceleration, and $d_i$ denotes lumped disturbances acting on ${\mathrm{UAV}}_{\mathrm{i}}$. Equation (5) provides a direct mapping from the load layer state and UAV thrust inputs to the cable tensions, and these tensions serve as the measurable (or estimable) quantities for imposing tension constraints of the form $0\le T_i\le T_{i,max}$ in the QP layer.

Remark 1.

(Sensitivity of Tension Reconstruction). Although the algebraic reconstruction in (5) is exact at the modeling level, it is practically sensitive to errors in the estimated kinematic states, especially the acceleration term ${\ddot{P}}_L$ and the relative-velocity term $\Vert v_{rel,i}{\Vert }^2$. Nevertheless, the mapping in (5) is locally Lipschitz with respect to its measured/estimated arguments; thus, bounded estimation errors and parameter biases (e.g., mass and cable-length uncertainties) induce only bounded perturbations in the reconstructed tension $T_i$. Since $T_i$ is used only to define the instantaneous feasible set of the QP projection layer, such bounded perturbations result in bounded variations in the constraint set, and the robustness claim based on bounded disturbances in Section 4 remains applicable. In implementation, ${\ddot{P}}_L$ and $q_i$ are obtained from onboard state estimation rather than raw numerical differentiation, and the reconstructed $T_i$ can be low-pass filtered before constraint evaluation to mitigate noise-driven constraint chattering.

Substituting (5) into the payload dynamics and eliminating $T_i$ yields a load layer forward dynamics model expressed solely in terms of the payload state, cable directions, and thrust vectors (see Appendix A for details):

$$\ddot{P_L}=g{e}_3+S^{-1}\left({\displaystyle \sum }_{i=1}^2{q}_i{q}_i^T\left(f_i{R}_i{e}_3+d_i\left(t\right)\right)+{\displaystyle \sum }_{i=1}^2{m}_i{q}_i{\displaystyle \frac{\Vert v_{rel,i}{\Vert }^2}{L_i}}+d_L\left(t\right)\right)$$

(6)

where $S≔m_L{I}_3+m_1{q}_1{q}_1^T+m_2{q}_2{q}_2^T$. Here, $I_3$ is the $3\times 3$ identity matrix, and $d_L$ represents lumped disturbances acting on the payload. Since $m_L>0$ and each term $m_i{q}_i{q}_i^T$ is positive semidefinite, the matrix S is positive definite and hence invertible.

From a physical viewpoint, (6) reduces the original multibody system consisting of two UAVs, cables, and a payload to a nonlinear model at the load layer: the inputs are the UAV thrust vectors (including terms induced by the relative velocity), and the output is the payload acceleration ${\ddot{P}}_L$. This compact representation provides a unified dynamical basis for the DEA-based load layer trajectory tracking design and the QP-based constraint projection developed in Section 3.

2.3. Unified State Space Model and Control Input Decomposition

Based on the above modeling, the generalized state vector of the dual-quadcopter slung load system is defined as

$$x={\left[\begin{array}{ccccc}{P}_1^T& P_2^T& P_L^T& q_1^T& q_2^T\end{array}\right]}^T$$

(7)

The control input is chosen as the thrust vectors of the two UAVs expressed in the inertial frame

$$u={\left[\begin{array}{cc}{u}_1^T& u_2^T\end{array}\right]}^T={\left[\begin{array}{cc}{f}_1{R}_1{e}_3^T& f_2{R}_2{e}_3^T\end{array}\right]}^T\in {ℝ}^6$$

(8)

By combining the load layer dynamics (6), the translational dynamics of the UAVs (5), and the geometric constraints (3), the overall system can be written in the standard nonlinear state space form

$$\dot{x}=f\left(x\right)+g\left(x\right)u+w\left(x,t\right)$$

(9)

where $f\left(x\right)$ collects the nonlinear drift terms arising from gravity, cable geometry, and velocity-dependent effects; $g\left(x\right)$ denotes the input gain matrix; and $w\left(x,t\right)$ represents bounded uncertainties and external disturbances, including $d_i$, $d_L$, and aerodynamic modeling errors.

To implement the proposed QP–DEA fusion control architecture, the control input is decomposed as

$$u=u_{DEA}+\Delta u_{QP}$$

(10)

where $u_{DEA}$ is the baseline thrust vector designed at the load layer via the DEA scheme to ensure the primary performance objectives such as trajectory tracking and swing suppression.

The term $u_{QP}$ is a correction computed by the QP layer under cable tension constraints, attitude cone constraints, and thrust saturation limits, to guarantee physical feasibility and safety while keeping the actual input as close as possible to $u_{DEA}$. Substituting (10) into (9) yields:

$$\dot{x}=f\left(x\right)+g\left(x\right)\left(u_{DEA}+\Delta u_{QP}\right)+w\left(x,t\right)$$

(11)

This unified system model serves as the basis for the QP–DEA fusion control design and the stability analysis developed in Section 3.

3. Controller Design

To achieve accurate payload trajectory tracking and swing suppression for the dual-quadrotor slung-load system under thrust, attitude, and cable-tension limits, we develop a fusion control framework based on the payload-layer dynamics in Section 2, where a nominal DEA-based law is combined with a QP constraint-projection layer for feasibility enforcement.

The proposed controller adopts a three-loop architecture that combines a payload-layer nominal controller, a QP constraint projection layer, and a low-level attitude and thrust inner loop. An adaptive tuning module runs on a slower timescale and updates $K_e\left(t\right),{\alpha }_f\left(t\right)$, and $H\left(t\right)$ online using the DEA error-energy indicator and the constraint-activation information ${[A_{\mathrm{ineq}}{u}^{*}\left(t\right)-b_{\mathrm{ineq}}]}_{+}.$

3.1. Control Architecture

Figure 2 illustrates the signal flow of the proposed controller. The reference generator first provides the desired payload trajectory $y_{ref}\left(t\right)$, while the Gazebo simulation returns the system state $x\left(t\right)$ and the measured output $y\left(t\right)$.

(1) DEA Nominal Layer.

Using $x\left(t\right)$ and $y\left(t\right)$, the DEA layer treats the payload position and selected cable attitude components as outputs and performs dynamic feedback linearization along the tension scalar channels. For given gains $K_e\left(t\right)$ and fusion factor ${\alpha }_f\left(t\right)$, it produces an unconstrained nominal thrust vector $u_{\mathrm{DEA}}\left(t\right)$, which defines the desired linear error dynamics at the payload layer in the absence of constraints.

(2) Constraint Projection (QP) Layer.

The QP layer receives $u_{DEA}\left(t\right)$ and, given the current state $x\left(t\right)$ and weight matrix H(t), computes the closest feasible control input $u^{*}\left(t\right)$ subject to thrust saturation, attitude cone bounds, and cable tension limits. This control input is then passed to the low-level attitude inner loop for execution.

(3) Adaptive Tuning Module.

On a slower timescale, the adaptive tuning module monitors the DEA error energy $V_{\mathrm{DEA}}\left(e\right)$ and the constraint activation residual ${[A_{\mathrm{ineq}}{u}^{*}\left(t\right)-b_{\mathrm{ineq}}]}_{+}$. Based on these quantities, it updates $H\left(t\right)$, $K_e\left(t\right)$, and ${\alpha }_f\left(t\right)$ to automatically balance DEA tracking performance and QP conservativeness, while always respecting the hard physical limits encoded in the constraints.

3.2. Nominal Control and Constraint Projection Design

3.2.1. DEA Layer: Dynamic Extension and Energy Consistent Control

To achieve accurate payload trajectory tracking with limited swing, we first construct a nominal payload-layer control law based on the port dynamics model derived in Section 2.2. A dynamic extension formulation is introduced to expose two tension scalar channels as the primary control directions of the dual-quadrotor system. The resulting payload-layer command is then minimally modified by the QP layer to satisfy thrust, attitude, and cable tension constraints.

(a)

Payload layer model and input decomposition

From (6), the payload layer dynamics can be written in compact form as

$$\ddot{P_L}=g{e}_3+S^{-1}\left(P_L,q_1,q_2\right)\left({\displaystyle \sum }_{i=1}^2{q}_i{q}_i^T{u}_i+h\left(P_L,q_i,v_{rel,i}\right)+d_L\left(t\right)\right)$$

(12)

where $u_i=f_i{R}_i{e}_3$ is the thrust vector of ${\mathrm{UAV}}_{\mathrm{i}}$,$v_{rel,i}={\dot{P}}_i-{\dot{P}}_L$ denotes the relative velocity, $h\left(\cdot \right)$ collects velocity-dependent nonlinear terms, and $S\left(\cdot \right)$ is the operating region matrix defined in (6).

To highlight the dominant channel along each cable, we decompose $u_i$ into the cable direction and its orthogonal plane:

$$u_i=u_{i1}{q}_i+u_{i\perp },u_{i1}=q_i^T{u}_i,\hspace{1em}{u}_{i\perp }=\left(I_3-q_i{q}_i^T\right)u_i$$

(13)

It follows that

$${\displaystyle \sum }_{i=1}^2{q}_i{q}_i^T{u}_i={\displaystyle \sum }_{i=1}^2{q}_i{q}_i^T\left(u_{i1}{q}_i\right)={\displaystyle \sum }_{i=1}^2{u}_{i1}{q}_i$$

(14)

Thus, the scalars $u_{11}$ and $u_{21}$ directly drive the payload translation and are regarded as the tension scalar channels for DEA design, whereas the vectors $u_{i\perp }$ mainly affect cable swing and vehicle attitudes and will be shaped in the QP layer.

Reordering the inputs, we define the six-dimensional thrust input as

$$u\triangleq {\left[\begin{array}{cccc}{u}_{11}& u_{21}& u_{1\perp }^{\top }& u_{2\perp }^{\top }\end{array}\right]}^{\top }\in {ℝ}^6$$

(15)

Combining the payload and UAV translational dynamics, the overall system can be written in affine form

$$\dot{x}=f\left(x\right)+{\displaystyle \sum }_j{g}_j\left(x\right) u_j+w\left(x,t\right)$$

(16)

where $x$ is the generalized state vector containing $P_1,P_2,P_L,q_1,q_2,$ …, and $w\left(x,t\right)$ denotes bounded uncertainties and disturbances.

Throughout this section, $u$ in (15) is also the decision variable of the QP layer in Section 3.2.2.

(b)

Dynamic extension and flat output selection

The core idea of DEA is to perform dynamic extension on the key input channels so that the chosen output has full relative degree with respect to the extended system, enabling exact feedback linearization. In this work, dynamic extension is applied only to the two tension scalars $u_{11}$ and $u_{21}$.

We introduce an extension state $\xi$ and virtual inputs ${\nu }_{11}$,${\nu }_{21}$, and construct the extended state and input vectors as

$$x^{\langle r\rangle }={\left[\begin{array}{cc}{x}^T& {\xi }^T\end{array}\right]}^T, u^{\langle r\rangle }\triangleq {\left[\begin{array}{cccc}{\nu }_{11}& {\nu }_{21}& u_{1\perp }^T& u_{2\perp }^T\end{array}\right]}^T$$

(17)

Following standard dynamic-extension calculations, the dynamics can be rewritten as

$$\dot{x^{\langle r\rangle }}=f^{\langle r\rangle }\left(x^{\langle r\rangle }\right)+g^{\langle r\rangle }\left(x^{\langle r\rangle }\right) u^{\langle r\rangle }$$

(18)

where the superscript $\langle r\rangle$ indicates quantities associated with the system extended by DEA. For the extended system, the payload position and two components of the second cable direction are chosen as outputs:

$$y={\left[\begin{array}{ccc}{P}_L^{\top }& q_{21}& q_{22}\end{array}\right]}^{\top }={\left[\begin{array}{ccccc}{y}_1& y_2& y_3& y_4& y_5\end{array}\right]}^{\top }\in {ℝ}^5$$

(19)

Given a desired output trajectory

$$y_{ref}\left(t\right)={\left[\begin{array}{ccc}{P}_{L,ref}^{\top }\left(t\right)& q_{21,ref}\left(t\right)& q_{22,ref}\left(t\right)\end{array}\right]}^{\top }$$

(20)

the tracking error is defined as

$$e=y_{ref}\left(t\right)-y$$

(21)

Let $r_k$ denote the relative degree of the $k$-th output component $y_k$, and collect them into the relative degree vector $\mathrm{r}={\left[4 4 4 2 2\right]}^{\mathrm{T}}$.

By repeatedly differentiating each $y_k$ along the vector fields in (18), and using the unit-length constraints $\Vert q_i\Vert =1$ together with the associated geometric relations, the extended system is shown to admit a full relative degree with respect to $y$ in the considered operating region [7].

Define the stacked vectors of the highest order derivatives for the actual and reference outputs as

$$y^r={\left[\begin{array}{ccc}{y}_1^{\left(r_1\right)}& \cdots & y_5^{\left(r_5\right)}\end{array}\right]}^{\top },y_{ref}^r\left(t\right)={\left[\begin{array}{ccc}{y}_{1,ref}^{\left(r_1\right)}\left(t\right)& \dots & y_{5,ref}^{\left(r_5\right)}\left(t\right)\end{array}\right]}^{\top }$$

(22)

where ${\left(\cdot \right)}^{\left(r_k\right)}$ denotes the $r_k$-th time derivative and the superscript $\langle r\rangle$ indicates stacking according to the relative-degree vector $r={\left[r_1,\dots ,r_5\right]}^T$.

Then, there exist a decoupling matrix $A^{\langle r\rangle }\left(x^{\langle r\rangle }\right)\in {ℝ}^{5\times 6}$ and a drift term $b^{\langle r\rangle }\left(x^{\langle r\rangle }\right)$ such that

$$y^{\langle r\rangle }=A^{\langle r\rangle }\left(x^{\langle r\rangle }\right) u^{\langle r\rangle }+b^{\langle r\rangle }\left(x^{\langle r\rangle }\right)$$

(23)

The decoupling matrix $A^{\langle r\rangle }(x^{\langle r\rangle })$ maintains full row rank provided the system avoids singular configurations, which primarily occur when the cables become perfectly horizontal or when the tension vectors become completely collinear. In our practical operational envelope, the perfectly horizontal singularity is physically unreachable due to gravity compensation. Furthermore, to strictly guarantee that $A^{\langle r\rangle }$ remains well-conditioned for inversion, the trajectory planner and constraint projection layer explicitly restrict the cable elevation angles ${\mathsf{\theta }}_{rope,i}$ to operate within $\left[{30}^{\circ },{60}^{\circ }\right]$. This geometric bound rigorously confines the system state to a non-singular submanifold, which enables exact feedback linearization at the output level in the subsequent DEA design.

(c)

Linear error dynamics and nominal DEA control law

Let $r_k$ denote the relative degree of the $k$-th output component $y_k$, and collect them into the vector $r={[r_1,\dots ,r_5]}^{\top }$. For each output channel, we construct an error chain

$$z_{k,\mathcal{l}}=e_k^{\left(\mathcal{l}-1\right)},\hspace{1em}\mathcal{l}=1,\dots ,r_k, k=1,\dots ,5$$

(24)

where $e_k$ is the tracking error of $y_k$ and $e_k^{\left(\mathcal{l}-1\right)}$ denotes its $(\mathcal{l}-1$)-th time derivative. The error states are stacked into

$$z={[z_{1,1},\dots ,z_{1,r_1},\dots ,z_{5,1},\dots ,z_{5,r_5}]}^{\top }$$

(25)

Neglecting disturbances, the DEA-extended system yields a standard Brunovský (chain-of-integrators) error dynamics

$$\dot{z}=A_{c}z+B_c\nu$$

(26)

where $\nu \in {ℝ}^5$ is the virtual input after the DEA transformation, and $A_c$,$B_c$ are constant matrices determined by the relative degrees $r_k$. They have a block diagonal structure composed of five integrator chains. When disturbances are taken into account, a bounded term $d_e\left(t\right)$ is added to the right-hand side of (26).

The virtual input is chosen as

$$\nu =-K_e z+y_{ref}^{\langle r\rangle }\left(t\right)$$

(27)

where $K_e$ is a feedback gain matrix selected by pole placement, and $y_{ref}^{\langle r\rangle }\left(t\right)$ collects the reference output and its derivatives up to order $r_k$. Substituting (27) into (26) gives the closed-loop error dynamics

$$\dot{z}=\left(A_c-B_c{K}_e\right)z+d_e\left(t\right)$$

(28)

If the matrix $A_c-B_c{K}_e$ is Hurwitz, then there exist symmetric positive definite matrices $P,Q>0$ such that

$${(A_c-B_c{K}_e)}^{\top }P+P\left(A_c-B_c{K}_e\right)=-Q$$

(29)

For the Lyapunov function

$V_{DEA}={\displaystyle \frac{1}{2}}{z}^{T}Pz$, its time derivative satisfies

$$\dot{V_{DEA}}\le -z^{\top }Qz+z^{\top }P d_e\left(t\right)$$

(30)

Hence, in the absence of disturbances, or when $d_e\left(t\right)$ is bounded and sufficiently small, the tracking error $e\left(t\right)$ converges exponentially, and the transient performance is fully determined by the linear gain $K_e$.

To implement feedback linearization at the output layer, the virtual input relation (27) is combined with the decoupled output Equation (23) by enforcing $y^{\langle r\rangle }=\nu$, which yields the algebraic equation

$$A^{\langle r\rangle }\left(x^{\langle r\rangle }\right) u^{\langle r\rangle }+b^{\langle r\rangle }\left(x^{\langle r\rangle }\right)=-K_{e}z+y_{ref}^{\langle r\rangle }\left(t\right)$$

(31)

Under the full-rank assumption on $A^{\langle r\rangle }$, the extended input is obtained as

$$u^{\langle r\rangle }=A^{\langle r\rangle }{(x^{\langle r\rangle })}^{+}\left(-K_{e}z+y_{ref}^{\langle r\rangle }\left(t\right)-b^{\langle r\rangle }\left(x^{\langle r\rangle }\right)\right)$$

(32)

where ${(\cdot )}^{+}$ denotes the generalized inverse. Applying the inverse DEA transformation and the geometric mapping $T_u^{-1}$, the extended input $u^{\langle r\rangle }$ is mapped back to the physical thrust vectors $u_i$, leading to the nominal DEA control law in the thrust input domain

$$u_{DEA}=\left[\begin{array}{c}{u}_1^{DEA}\\ u_2^{DEA}\end{array}\right]=K\left(x^{\langle r\rangle },r\left(t\right),r^{\left(1\right)}\left(t\right),\dots ,r^{\left(r\right)}\left(t\right)\right)$$

(33)

where the nonlinear feedback mapping $K\left(\cdot \right)$ is determined jointly by (18) and the inverse geometric transformation. This nominal thrust command serves as the reference input for the minimum deviation projection in the QP layer.

Remark 2.

*In real-world sensing applications, calculating the higher-order derivatives required by DEA via direct numerical differentiation of raw sensor data is practically infeasible due to severe noise amplification. To fundamentally mitigate state estimation noise propagation, our control architecture strictly avoids pure numerical differentiation. The reference derivatives are generated analytically, while the state feedback terms in (33) are computed through model-based algebraic recursion. By substituting the explicit system dynamics, higher-order terms are expressed entirely as functions of lower-order states. These lower-order states can be smoothly and robustly obtained using established multi-sensor fusion frameworks, such as visual-inertial odometry (VIO) [22] and cooperative slung-load estimators [23].

3.2.2. Constraint Projection (QP) Layer: Constrained Optimal Correction

The DEA layer provides a nominal thrust command $u_{\mathrm{DEA}}\in {ℝ}^6$. However, under certain attitudes or payload configurations, this command may violate physical limits on thrust, attitude, or cable tension and therefore cannot be applied directly. To address this issue, a convex QP is formulated on top of the DEA output, which computes a minimum deviation correction consistent with thrust, attitude, tension, and cooperation constraints and yields an executable control input $u_{\mathrm{QP}}^{*}$.

(a)

Constrained optimization problem

Given the nominal control $u_{\mathrm{DEA}}\in {ℝ}^6$, the QP layer solves

$$u^{*}=\underset{u\in {ℝ}^6}{argmin} {\displaystyle \frac{1}{2}}{(u-u_{\mathrm{DEA}})}^{T}H\left(u-u_{\mathrm{DEA}}\right)\hspace{1em}\left\{\begin{array}{l}{A}_{\mathrm{eq}}\left(x\right)u=b_{eq}\left(x\right)\hfill \\ A_{\mathrm{ineq}}\left(x\right)u\le b_{\mathrm{ineq}}\left(x\right)\hfill \end{array}\right.$$

(34)

where $H>0$ is a positive definite weighting matrix. The matrices $A_{\mathrm{eq}}\left(x\right)$ and $A_{\mathrm{ineq}}\left(x\right)$ encode thrust saturation, attitude angle bounds, cable tension limits, and simple load coordination constraints via linear or piecewise affine inequalities, summarized as

$$F_{i,\min }\le \Vert F_i\Vert \le F_{i,\max },\hspace{1em}\left|{\varphi }_i\right|,\left|{\theta }_i\right|\le {\varphi }_{\max },\hspace{1em}{T}_{\min }\le T_i\left(x,u\right)\le T_{\max }$$

(35)

together with the cooperation relations

$$A_{\mathrm{eq}}\left(x\right)u=b_{\mathrm{eq}}\left(x\right)$$

(36)

These constraints define the feasible set $U\left(x\right)$, and (34) performs a minimum deviation projection of $u_{\mathrm{DEA}}$ onto $U\left(x\right)$ in the H weighted norm.

When the system approaches critical conditions and hard feasibility may be lost, soft constraints with slack variables $s\ge 0$ are introduced, leading to the following QP

$$\begin{gathered} \left(u^{*},s^{*}\right)=arg\underset{u,s}{\min }{\displaystyle \frac{1}{2}} {(u-u_{\mathrm{DEA}})}^{\top }H \left(u-u_{\mathrm{DEA}}\right)+{\displaystyle \frac{\rho }{2}} \Vert s{\Vert }_2^2 \\ s.t.\hspace{1em}{A}_{\mathrm{eq}}\left(x\right) u=b_{\mathrm{eq}}\left(x\right),\hspace{1em}{A}_{\mathrm{ineq}}\left(x\right) u\le b_{\mathrm{ineq}}\left(x\right)+s,\hspace{1em}s\ge 0 \end{gathered}$$

(37)

where $\rho \gg 1$ ensures that the slack variables are activated only when hard feasibility is about to be lost. This implements the strategy “guarantee feasibility first, then stay as close as possible to the DEA command within the feasible region”.

Remark 3.

(Physical meaning of slack variable). The slack variable $s\ge 0$ is introduced only to relax the conservative margin $T_{min}>0$ for avoiding occasional QP infeasibility. The unilateral physical constraint $T_i\ge 0$ is always enforced as a strict hard constraint and is never softened. Hence, $s>0$ does not imply cable slackness; it merely relaxes the safety margin with a large penalty to keep the cable taut.

(b)

Properties and implementation

Since the cost function is strictly convex and the constraints are linear, the optimization problem in (34)–(36) guarantees a unique global optimum provided the (soft) feasible set is nonempty. Geometrically, $u_{\mathrm{QP}}^{*}$ is the H weighted projection of $u_{\mathrm{DEA}}$ onto the constraint set, where the active constraints reorient the deviation vector via their Lagrange multipliers.

In practice, the QP dimension is small (six thrust components plus a few slack variables). With warm start and simple hysteresis/filtering on the active set, each QP can be solved reliably within one inner-loop period, thereby providing a real time constraint projection layer for the stability analysis in Section 4 and the simulations in Section 5.

3.3. Adaptive Tuning Module

The closed-loop performance of the proposed fusion controller is governed not only by the DEA feedback gain $K_e$ that shapes the nominal convergence rate and damping, but also by the trade off in the constraint-projection layer between (i) tracking/swing-suppression objectives and (ii) constraint accommodation. In practice, fixed parameters often perform well only within a limited range of operating conditions: overly aggressive $K_e$ or overly large projection penalties may lead to overly conservative responses or even oscillations, whereas small gains and penalties typically slow down convergence and cause more frequent constraint activation.

To enhance robustness and reduce manual tuning, we introduce an adaptive tuning layer that updates the DEA feedback gain $K_e$ and the projection weighting matrix H on a slow timescale. It is important to emphasize that physical bounds including thrust saturation, attitude limits, (if enforced) swing-angle limits, and cable-tension bounds are implemented as fixed and non-relaxable constraint limits in the projection layer. The adaptation acts only within these preset limits, reshaping the penalty metric and the nominal convergence rate rather than relaxing the physical constraints.

(a)

Energy residual and constraint-activation measures

Let $V_{\mathrm{DEA}}\left(e\right)$ denote the Lyapunov function associated with the DEA error dynamics in Section 3.2.1. We define an energy-residual index

$$E_t\triangleq V_{\mathrm{DEA}}\left(e\left(t\right)\right)+\alpha \Vert e\left(t\right){\Vert }_2^2, \alpha >0$$

(38)

which summarizes the remaining error “energy”. Persistently large $E_t$ indicates insufficient damping or slow decay of the DEA error, suggesting that the nominal feedback authority should be moderately strengthened.

To quantify how close the QP solution is to the inequality boundaries, we define a constraint-activation index

$${\sigma }_t\triangleq \Vert {[A_{\mathrm{ineq}}{u}^{*}\left(t\right)-b_{\mathrm{ineq}}]}_{+}{\Vert }_2^2$$

(39)

where $\left[\cdot {]}_{+}\triangleq \max (\cdot ,0\right)$ is applied element-wise. Thus, ${\sigma }_t\approx 0$ indicates that the solution stays strictly inside the feasible set, while larger ${\sigma }_t$ implies that some inequality constraints are repeatedly driven close to activation.

To emphasize which directions in the input space are frequently constrained, we construct a direction-weighting matrix

$${\mathsf{{\rm Y}}}_t\triangleq A_{\mathrm{ineq}}^{\top } \left({\widehat{\mu }}_t\right) A_{\mathrm{ineq}}\ge 0$$

(40)

where ${\widehat{\mu }}_t\ge 0$ is a non-negative weight vector. In implementation, ${\widehat{\mu }}_t$ can be chosen as either (i) the non-negative inequality dual variables returned by the QP solver, or (ii) a residual-based surrogate such as ${\widehat{\mu }}_t={[A_{\mathrm{ineq}}{u}_{\mathrm{QP}}^{*}\left(t\right)-b_{\mathrm{ineq}}]}_{+}.$

(b)

Slow-timescale update laws with projection

Let $u\in {ℝ}^6$ be the QP decision variable, hence $H\in {\mathbb{S}}_{++}^6$. We update H using a projected discrete-time law

$$H_{k+1}={\mathsf{\Pi }}_{{\mathbb{S}}_{++}^6}\left[H_k+{\rho }_H\left(E_t{I}_6+\gamma {\mathsf{{\rm Y}}}_t-{\sigma }_H{H}_k\right)\right],\hspace{1em}{\rho }_H,\gamma ,{\sigma }_H>0$$

(41)

where ${\mathsf{\Pi }}_{{\mathbb{S}}_{++}^6}\left(\cdot \right)$ denotes projection onto the cone of $6\times 6$ symmetric positive definite matrices. The term $E_t{I}_6$ increases the overall penalty when the error energy is large; $\gamma {\mathsf{{\rm Y}}}_t$ selectively reinforces constraint-active directions; and $-{\sigma }_H{H}_k$ is a leakage term that prevents unbounded growth.

For the DEA feedback gain in (27), $\nu =-K_{e}z+y_{\mathrm{ref}}^{\langle r\rangle }$, we use a similar projected update:

$$K_{e,k+1}={\mathsf{\Pi }}_{\mathcal{H}}\left[K_{e,k}+{\rho }_K\left(E_{t}G-{\sigma }_K{K}_{e,k}\right)\right],\hspace{1em}{\rho }_K,{\sigma }_K>0$$

(42)

where G is a shaping matrix with the same dimension as $K_e$ (e.g., a fixed stabilizing template $K_{e,0}$), and ${\mathsf{\Pi }}_{\mathcal{H}}\left(\cdot \right)$ projects $K_e$ onto a predefined admissible set \mathcal{H} that preserves a desired stability margin (i.e., ensuring $A_c-B_c{K}_e$ remains Hurwitz).

The updates (41)–(42) are executed on a timescale much slower than the DEA/QP inner loops, so the parameter variations can be treated as small perturbations of the frozen-gain analysis. In this way, the controller automatically coordinates (i) convergence rate, (ii) constraint conservativeness, and (iii) swing suppression across different operating conditions, while the QP hard constraints continue to guarantee feasibility with respect to physical limits.

4. Stability and Optimality Analysis

This section analyzes the closed-loop properties of the QP–DEA integrated controller proposed in Section 3. The aim is twofold:

(i) to establish input-to-state stability (ISS) of the closed loop with respect to bounded disturbances;

(ii) to clarify the “minimum deviation” optimality and continuity properties of the QP layer.

The analysis assumes that the DEA layer is exponentially stable and that the QP problem is strictly convex.

4.1. Composite Lyapunov Function and Main Result

To measure both the DEA tracking error and the deviation introduced by the QP projection, we define the composite Lyapunov function:

$$V_{tot}=V_{DEA}\left(e\right)+{\displaystyle \frac{1}{2}} \delta u^{\top }H \delta u,\hspace{1em}\delta u\triangleq u_{QP}^{*}-u_{DEA}$$

(43)

where $V_{\mathrm{DEA}}\left(e\right)$ is the Lyapunov function of the DEA error system (Section 3.2.1), $\delta u$ is the difference between the QP output and the nominal DEA control, and $H>0$ is the QP weighting matrix.

On time intervals during which the active constraint set does not change $H$ and $K_e$ can be regarded as constant, the exponential stability of the DEA error system (26)–(28) and the KKT orthogonality of the QP solution lead to the estimate:

$$\dot{V_{tot}}\le - e^{\top }Qe-{\lambda }_{min}\left(H\right) \Vert \delta u{\Vert }^2+c_w \Vert w\left(t\right){\Vert }^2$$

(44)

where $Q>0$, $w\left(t\right)$ is the aggregate disturbance (including modeling errors and external disturbances), $c_w>0$ is a constant, and ${\lambda }_{\min }\left(H\right)$ is the smallest eigenvalue of $H$. Inequality (44) shows that the pair ($e,\delta u$) decays quadratically, up to an additive term determined by $w\left(t\right)$.

In Section 3.3, the matrices $H\left(t\right)$ and $K_e\left(t\right)$ are updated on a slow timescale by projection adaptive laws. Their rates of change are explicitly bounded by the parameters ${\rho }_H,{\rho }_K,{\sigma }_H,{\sigma }_K$, and their values are projected into bounded sets that preserve closed-loop stability. With suitable parameter choices, the variations of $H\left(t\right)$ and $K_e\left(t\right)$ can be treated as small perturbations of (44). The closed-loop boundedness under bounded disturbances is characterized using standard ISS Lyapunov arguments.

We now state the main result.

Theorem 1.

ISS and practical exponential stability.

Assume that:

• The QP layer is always feasible or soft constraints of the form (37) are used so that a feasible solution exists at all times.
• The weighting matrix $H\left(t\right)$ is bounded and there exists a constant $h_0>0$ such that ${\lambda }_{\min }\left(H\left(t\right)\right)\ge h_0$ for all $t$
• The DEA gain matrix $K_e\left(t\right)$ always belongs to an admissible set $\mathcal{H}$ that guarantees the Hurwitz property of $A_c-B_c{K}_e$
• Switching of the active constraint set satisfies a minimum dwell time condition, or high frequency switching is avoided by hysteresis or filtering strategies.
• The aggregate disturbance $w\left(t\right)$ is bounded.

Then, the closed-loop system is ISS with respect to the disturbance input $w\left(t\right)$. In particular:

If $w\left(t\right)\equiv 0$ and the active constraint set becomes fixed after some finite time, the output error $e\left(t\right)$ and the control deviation $\delta u\left(t\right)=u_{\mathrm{QP}}^{*}\left(t\right)-u_{\mathrm{DEA}}\left(\mathrm{t}\right)$ converge to zero exponentially.

If constraint switching occurs with finite frequency, the system is practically exponentially stable, i.e., $e\left(t\right)$ and $\delta u\left(t\right)$ converge to a small ball around the origin, whose radius can be made arbitrarily small by appropriate design. The proof is outlined in Appendix B, based on the composite Lyapunov function (43), the derivative inequality (44), and standard Lyapunov and small-gain ISS arguments.

4.2. Minimal Deviation Optimality and Continuity of the QP

We now summarize the basic optimality and continuity properties of the QP layer.

Consider the QP problem in Section 3.2.2. For a given state $x$, weighting matrix $H>0$, and nominal control $u_{\mathrm{DEA}}$, the cost function is a strictly convex quadratic form and the constraints are linear equalities and inequalities. Let the feasible set be $U\left(x\right)$, and denote the optimal solution by $u_{\mathrm{QP}}^{*}=u_{\mathrm{QP}}^{*}\left(x,u_{\mathrm{DEA}},H\right)$.

Theorem 2.

Minimal deviation optimality.

If Slater’s condition holds (i.e., there exists a strictly feasible point), then for any given $\left(x,u_{\mathrm{DEA}},H\right)$, the optimal solution $u_{\mathrm{QP}}^{*}$ is unique and satisfies

$$u_{\mathrm{QP}}^{*}=arg\underset{u\in U\left(x\right)}{\min }\Vert u-u_{\mathrm{DEA}}{\Vert }_H^2,\Vert v{\Vert }_H^2\triangleq v^{\top }Hv$$

(45)

Thus, $u_{\mathrm{QP}}^{*}$ is the weighted closest point in the feasible set $U\left(x\right)$ to the nominal control $u_{\mathrm{DEA}}$. This directly realizes the design goal of “staying as close as possible to the DEA command while satisfying all constraints’’ [24].

Theorem 3.

Local continuity with respect to parameters.

Under Slater’s condition and for a fixed active constraint set, the mapping

$$\left(x,u_{\mathrm{DEA}},H\right)\mapsto u_{\mathrm{QP}}^{*}\left(x,u_{\mathrm{DEA}},H\right)$$

(46)

is locally Lipschitz continuous.

In other words, small perturbations in the state or in the nominal control lead only to small changes in the QP solution, so that large jumps in the control input do not occur [24].

Combined with the ISS result in Section 4.1, these properties imply that the QP layer not only enforces a minimal deviation from the DEA nominal control but also avoids introducing high frequency excitations that could compromise stability. The QP layer and the DEA layer therefore form a well-coupled closed-loop structure.

Remark 4.

(On QP switching and possible oscillations). When constraints are active, the closed loop becomes piecewise-smooth due to QP active-set switching, where the QP correction can be viewed as a bounded state-dependent perturbation $\Delta u\left(t\right)$. In this case, the stability result is in the sense of Uniform Ultimate Boundedness (UUB) rather than strict asymptotic convergence. The projection operator confines the adaptive gains to a compact set and bounds their update rate, mitigating chattering and limit-cycle excitation associated with active-set switching (see Appendix B.3).

5. Simulation Experiments

This chapter evaluates the trajectory-tracking and swing-suppression performance of a dual-quadrotor slung-load system on a ROS–PX4–Gazebo software-in-the-loop (SITL) simulation platform. Gazebo is used to simulate the dual-UAV–cable–payload dynamics and the cable-length geometric constraints, while PX4 SITL provides the attitude–thrust inner-loop control and motor allocation. The proposed fusion controller runs as the outer loop. Through the constraint-projection (QP) layer, it outputs feasible thrust commands that satisfy thrust, attitude, and tension constraints, which are then tracked by the PX4 inner loop. In simulation, the payload position, cable attitudes, and the poses of the two UAVs are directly obtained from Gazebo feedback. All simulations are conducted on a standard desktop computer (AMD Ryzen 5 3600 CPU (Advanced Micro Devices, Santa Clara, CA, USA), 32 GB RAM.).

5.1. Simulation Platform

As shown in Figure 3, the Gazebo scene consists of two quadrotors and one rigid payload connected to the UAVs by two inextensible cables of length 1 m. Gazebo handles gravity, inertia, and the cable-length geometric constraints.

To physically mitigate severe inter-vehicle downwash interference, the operational trajectories and cable lengths are configured to strictly maintain a horizontal clearance of at least 0.75 m (approximately three rotor diameters) between the UAVs. Furthermore, the residual aerodynamic drag and downwash interactions are subsumed into the bounded lumped disturbance terms $\left(d_L\left(t\right),d_i\left(t\right)\right)$ the system dynamics. As proven in Section 4, the ISS property of the closed-loop system inherently guarantees the safe rejection of these unmodeled aerodynamic forces. Key system parameters are summarized in

Table 1. Parameters of the suspend-payload UAV.

Parameters	Value
M	1.5 kg
g	9.81 m/s2
$L_i$	1 m
Ix	1.98 × 10−2 kgm2
Iy	1.98 × 10−2 kgm2
Iz	3.55 × 10−2 kgm2

.

PX4 SITL provides the attitude–thrust inner loop and motor allocation, while the fusion controller runs as the outer loop at 200 Hz. At each control step, the payload position, cable attitudes, and the poses of the two UAVs are read from Gazebo; the controller then generates feasible thrust commands and sends them to the PX4 inner loop for execution.

5.2. Payload Trajectory-Tracking Experiments

To cover two typical task types—step waypoint switching and continuous curve tracking—two reference trajectories are designed: a staircase-like step-rectangular waypoint sequence and a planar figure-eight trajectory. The staircase step-waypoint task is used for performance comparison against baseline controllers and for ablation studies of the adaptive module. The planar figure-eight task is used to validate the tracking capability of the proposed method under continuous maneuvering, and the corresponding swing-suppression performance and constraint-consistency results are also reported. The payload mass is set to 320 g, which exceeds the nominal payload capacity of a single quadrotor in our setup.

For each reference trajectory, the desired payload position $P_{L,\mathrm{ref}}\left(t\right)$ and the actual payload position $P_L\left(t\right)$ are recorded, and the following metrics are computed:

Axis-wise tracking errors: $e_x=x-x_{\mathrm{ref}}, \left|e_y=y-y_{\mathrm{ref}},\right| e_z=z-z_{\mathrm{ref}}$;

Position RMSE and the maximum absolute error ${\left|e\right|}_{max}$ able swing angles ${\mathsf{\theta }}_{\mathrm{rope},i}$ (deviations relative to the hovering equilibrium cable orientations); smoothness of cable tensions and control inputs (for validating constraint consistency).

5.2.1. Step-Rectangular Waypoint Sequence (Staircase Profile)

To evaluate the convergence behavior and swing suppression under multiple waypoint transitions, a step-rectangular waypoint sequence is constructed as the payload reference. Four vertices in the horizontal plane form a $1 \mathrm{m}\times 1 \mathrm{m}$ rectangle. This rectangle is duplicated at two altitudes, ${\mathrm{z}}_1=1.5 {\mathrm{m}, \mathrm{z}}_2=2 \mathrm{m}$, resulting in a staircase-like sequence of eight waypoints (“lower rectangle + upper rectangle”). The results are shown in

Table 2. Cycle tracking error metrics for the stair-step rectangular waypoint mission (RMSE and maximum absolute error).

Component Position	PD		DEA		AT-DEA
	RMSE	$|\mathit{e}{|}_{\mathit{m}\mathit{a}\mathit{x}}$	RMSE	$|\mathit{e}{|}_{\mathit{m}\mathit{a}\mathit{x}}$	RMSE	$|\mathit{e}{|}_{\mathit{m}\mathit{a}\mathit{x}}$
x (m)	0.077	0.271	0.093	0.255	0.029	0.094
y (m)	0.042	0.177	0.074	0.281	0.021	0.071
z (m)	0.061	0.312	0.045	0.155	0.042	0.097

, Figure 4 and Figure 5. This aggressive multi-waypoint sequence effectively emulates the severe transient accelerations, rapid attitude adjustments, and tracking demands characteristic of emergency obstacle avoidance maneuvers.

As shown in Figure 5, for the stair-step rectangular waypoint sequence, AT-DEA exhibits the closest axis-wise position responses to the reference, with smaller error peaks at waypoint transitions and faster error decay. The error trajectories remain near the prescribed bounds and are consistently lower than those of the baseline methods. In contrast, DEA shows more pronounced oscillations and larger peak errors in the x and y directions, with more evident error accumulation over repeated transitions. PD can achieve basic tracking, but it produces larger overshoots and rebounds during switching; in particular, the z-axis error peak increases noticeably, indicating higher sensitivity to step commands and slower settling.

The quantitative results are summarized in Table 2: AT-DEA achieves substantially lower cycle RMSE than DEA and PD, while keeping the three-axis errors within the $\pm 0.1 \mathrm{m}$ bounds. Overall, AT-DEA delivers higher accuracy and smaller peak errors under step-switching conditions, demonstrating improved transient regulation and constraint-consistent stable tracking.

5.2.2. Figure-Eight Trajectory

The second test uses a planar figure-eight as the payload reference trajectory. The motion lies in a horizontal plane at fixed altitude $z = 1.5 \mathrm{m}$. The reference in the x–y plane is given by $x_{L,ref}\left(t\right)=1.5 \sin (0.5\pi t),y_{L,ref}\left(t\right)=0.7 \sin (\pi t)$.

As shown in Figure 6, under the planar figure-eight reference, the payload’s actual 3D trajectory nearly overlaps the reference, with no evident phase lag or accumulated drift. The axis-wise tracking errors remain small and stay within the prescribed ±0.1 m bounds. In particular, the x and y errors exhibit only mild periodic fluctuations, while the z-axis shows more pronounced periodic variations that are still bounded. This behavior is mainly attributed to payload–cable coupling during coordinated turns and feasibility adjustments introduced by the constraint-projection layer. Overall, the results indicate stable tracking with bounded errors for continuous maneuvering tasks.

As shown in Figure 7 and Figure 8, the thrust commands remain smooth and bounded without evident saturation or high-frequency chattering: the lateral components vary periodically to execute the maneuver, while the vertical components stay close to the hover level with only mild adjustments. Meanwhile, the two cable swing angles exhibit bounded periodic variations with no growth or divergence, indicating stable swing regulation and constraint-consistent transportation during continuous motion.

5.3. Real-Time Computational Performance

While standard payload-free UAV navigation typically operates at 50–100 Hz, the tightly coupled rigid-flexible dynamics of the dual-quadrotor slung-load system necessitate a higher bandwidth. A 200 Hz outer-loop frequency enables the controller to sample states and update thrust commands with minimal latency, effectively suppressing high-frequency transient swings.

To evaluate the real-time computational viability of the proposed framework subject to a $5 \mathrm{ms}$ ($5000 \mathsf{\mu }\mathrm{s}$) deadline, the execution time was rigorously recorded over 13,000 continuous control cycles. These data were extracted from the aggressive multi-waypoint transitions during the step-rectangular waypoint sequence task presented in Section 5.2.1.

The statistical results are detailed in

Table 3. Real-time computational performance analysis of the QP-based fusion controller (200 Hz/5 ms deadline) over 13,000 control cycles during the step-rectangular waypoint sequence.

Metric	Mean	Median	99th%	WCET	Violations
Controller Exec (μs)	808.46	696.31	2018.66	7954.05	0.02%
QP Solver (μs)	481.04	434.78	1062.99	7791.67	0.02%
QP Iterations	53.32	50	-	325	-
QP/Total Ratio	59.5%	-	-	-	-
200 Hz Deadline (μs)	5000	5000	5000	5000	-
Computational Margin	84%	84%	60%	−59%

. The table breaks down the execution metrics for both the overall fusion controller and the isolated QP solver, including the mean, Median, 99th percentile (99th%), Worst-Case Execution Time (WCET), and the percentage of deadline violations. As shown, the mean execution time of the entire controller is $808.46 \mathsf{\mu }\mathrm{s}$, utilizing only 16% of the 5 ms deadline and leaving an 84% computational margin. Notably, the QP solver accounts for approximately 59.5% of the total computational load, with an average of 53 iterations per cycle.

Figure 9 visually depicts the execution time distribution across the 13,000 control cycles. The scatter plot illustrates that the vast majority of execution times tightly cluster below 1 ms. The 99th percentile execution time is $2.02 \mathrm{ms}$ ($2018.66 \mathsf{\mu }\mathrm{s}$), demonstrating that the system leaves a massive 60% computational margin in 99% of the operational scenarios. Occasional peak execution times, such as the WCET of $7954.05 \mathsf{\mu }\mathrm{s}$ shown as isolated spikes crossing the 5 ms red deadline limit, occurred due to extreme QP iteration caps (up to 325 iterations during aggressive trajectory switching) and soft real-time OS scheduling jitter. Nevertheless, only two instances out of 13,000 samples violated the deadline. This 99.98% deadline success rate conclusively demonstrates the real-time practical feasibility of the proposed 200 Hz QP-DEA architecture.

6. Conclusions

This paper presented a hybrid-layer cooperative control framework for dual-quadrotor slung-load transportation. By formulating an explicit payload-layer dynamic model via d’Alembert’s principle, the rigid-body coupling structure was directly exposed at the load level. A Dynamic Extension Algorithm (DEA) was designed for nominal high-order trajectory tracking, while a convex Quadratic Programming (QP) projection layer ensured the strict enforcement of thrust, attitude, and cable tension constraints. Furthermore, a constraint-aware adaptive tuning module was introduced to dynamically balance tracking aggressiveness and feasibility preservation.

Simulation results in ROS-PX4-Gazebo demonstrated accurate trajectory tracking performance under aggressive multi-waypoint and figure-eight maneuvers. Quantitative comparisons confirmed improved steady-state accuracy and constraint compliance relative to baseline DEA and cascaded PD controllers. Computational analysis further verified the real-time feasibility of the proposed architecture under repeated constraint activation cycles. From a practical perspective, this framework provides a structured methodology for separating nominal tracking design from physical feasibility enforcement, which is particularly beneficial for cooperative aerial transportation tasks where actuator saturation and cable tension limits are critical safety constraints.

Despite these contributions, certain limitations remain. Although the closed-loop system satisfies an input-to-state stability (ISS) property with respect to bounded disturbances, sustained high-amplitude wind gusts may degrade tracking performance due to severe actuator saturation and frequent constraint activations. While theoretical stability is preserved, performance optimization under extreme aerodynamic disturbances requires further investigation. Therefore, future work will focus on enhancing wind-resilient optimal control strategies, integrating predictive disturbance compensation mechanisms, and extending the framework to experimental validation on physical dual-UAV platforms.