Physical Modeling and Data-Driven Hybrid Control for Quadrotor-Robotic-Arm Cable-Suspended Payload Systems

Highlights

This paper presents a unified modeling and control framework for quadrotor systems equipped with robotic arms and cable-suspended payloads. It develops a 10-DOF Lagrangian model that captures the platform-arm-payload coupling and proposes a cooperative feedback linearization controller that exploits the coupling matrix ${\mathit{M}}_{ua}$ to actively suppress swing through geometric manipulation of the cable attachment point.

What are the main findings?

• The arm-payload coupling matrix ${\mathit{M}}_{ua}$ enables model-based control allocation that distributes swing compensation between UAV and arm actuators based on swing plane orientation and cable geometry.
• Cooperative control achieves 44–70% settling time reduction and up to 57.7% swing suppression compared to single-actuator baselines across diverse flight scenarios.

What is the implication of the main finding?

• Unlike disturbance rejection approaches, cooperative actuation through coupling exploitation provides an independent control channel that does not interfere with the position-attitude cascade.
• The geometry-aware compensation eliminates gain scheduling requirements across 5× payload mass and 4× cable length variations, enabling deployment without parameter-specific tuning.

Abstract

This work investigates a quadrotor equipped with dual-stage robotic arms and a cable-suspended payload, developing a unified methodology for modeling and control. A 10-DOF Lagrangian model captures vehicle-arm-payload coupling through structured mass matrices. A hierarchical control architecture combines SO(3)-based attitude regulation with cooperative swing compensation via partial feedback linearization, exploiting coupling matrices to distribute control between platform and arm actuators. Model accuracy is enhanced through physics-informed system identification, achieving improved prediction correlation with bounded corrections. Lyapunov analysis establishes semi-global practical stability with explicit robustness bounds. High-fidelity simulations in MuJoCo demonstrate a 40–70% swing reduction compared to PD control across multiple scenarios, with low computational overhead at kHz-level control rates, making it suitable for embedded implementation. The framework provides a theoretical foundation and implementation guidelines for cooperative aerial manipulation systems.

1. Introduction

Unmanned aerial vehicles (UAVs) with cable-suspended payloads have emerged as versatile platforms for applications requiring payload isolation from the aircraft. Representative scenarios include the following: (i) airborne geophysical surveying, where the UAV’s motors and electronics create electromagnetic interference necessitating sensor suspension at a safe distance [1]; (ii) water sampling in hazardous or inaccessible areas, where direct UAV contact with the water surface is undesirable; (iii) precision aerial delivery, where swing during the approach phase degrades placement accuracy; and (iv) suspended sensor platforms for infrastructure inspection requiring stable measurement conditions [2]. In all these applications, payload swing induced by UAV maneuvering severely degrades task performance and may threaten flight safety, making effective swing suppression a critical control objective.

Existing swing suppression methods for UAV-suspended payloads predominantly rely on the UAV platform as the sole actuator. Nonlinear backstepping controllers and their optimally tuned variants achieve accurate trajectory tracking under moderate swing angles [3,4]. Fixed-time control and sliding-mode hybrid schemes improve convergence speed and disturbance robustness [5,6]. Adaptive methods address parametric uncertainty, including unknown payload mass [7,8]. For variable-length cable systems, coordinated cable reeling and vehicle motion have been proposed [9,10]. Trajectory shaping approaches reduce residual vibration through smooth reference generation [11,12]. However, all these methods share a fundamental limitation: swing suppression is achieved exclusively through UAV attitude adjustments, which inherently couple with position control and restrict aggressive maneuvering capability.

Accurate dynamic modeling underpins high-performance control. Physics-based Lagrangian models capturing three-dimensional pendulum dynamics have been derived and validated [2,13]. Analytical models isolate dominant vehicle-payload couplings [14], while enhanced formulations account for double-pendulum configurations [15]. Geometric and dual-quaternion representations enable singularity-free attitude control [16]. Vision-based estimators reconstruct swing angles without dedicated sensors [17,18], and data-driven observers compensate for unmodeled dynamics [19]. These advances, however, focus on single-actuator UAV payload systems without exploiting potential redundancy from additional onboard manipulators.

In parallel, aerial manipulation research has developed sophisticated control for UAVs carrying robotic arms [20,21,22]. These systems address strong platform-arm coupling through nonlinear dynamic inversion, sliding-mode, and adaptive schemes. However, the manipulated objects are rigidly grasped rather than cable-suspended, so pendulum-like swing dynamics are not the primary concern. The control allocation strategies developed for rigid grasping do not directly transfer to underactuated cable-suspended configurations.

Despite these advances, a significant gap remains:no existing framework systematically exploits the dynamic coupling between a robotic arm and payload swing for cooperative swing suppression. While active disturbance rejection control (ADRC) methods treat swing as an unstructured disturbance to be estimated and canceled via extended state observers, such approaches do not leverage the structured coupling inherent in arm-payload dynamics. This gap is consequential because:

• Pure UAV-based swing suppression requires platform attitude changes that couple back to position control, limiting maneuverability during aggressive tasks;
• Variable-length cable systems add one degree of freedom but remain single-actuator architectures with similar coupling constraints;
• ADRC-based aerial manipulation methods treat swing as an external disturbance rather than exploiting the arm-swing coupling matrix ${\mathit{M}}_{ua}$ for model-based control allocation.

A robotic arm mounted between the UAV and cable attachment point offers a fundamentally different actuation channel: by displacing the attachment point, the arm can directly reshape payload swing dynamics through the coupling matrix ${\mathit{M}}_{ua}$, without requiring UAV attitude adjustments. This cooperative actuation—distinct from disturbance rejection—decouples swing suppression from the primary flight control loop while providing faster response bandwidth through geometric manipulation of the cable attachment point.

Motivated by this observation, this paper develops a cooperative control framework where both the UAV platform and a 2-DOF robotic arm serve as coordinated actuators for swing suppression. The key insight is that the dynamic coupling between arm motion and payload swing, captured by the off-diagonal blocks of the system mass matrix, can be explicitly exploited for control allocation. When payload swing is detected, the controller computes compensating arm accelerations that counteract oscillations through geometric manipulation of the cable attachment point. This provides faster response bandwidth and reduced interference with flight control compared to attitude-based suppression. The framework is validated through high-fidelity MuJoCo simulations with realistic actuator constraints, demonstrating 40–70% swing reduction across diverse operating conditions, including aggressive maneuvers where swing angles exceed the nominal linearization domain.

The main contributions are as follows:

• Cooperative swing suppression via arm-payload coupling exploitation: A dual-actuator control allocation framework where the coupling matrix ${\mathit{M}}_{ua}$ is explicitly utilized to distribute swing compensation between the UAV platform and robotic arm actuators. Unlike ADRC methods that treat swing as an unstructured disturbance, the arm attenuates oscillations by geometrically displacing the cable attachment point based on the instantaneous swing-plane orientation and cable length, thereby providing a faster response bandwidth than attitude-based methods.
• MCG feedback linearization with geometry-aware swing compensation: A hierarchical controller combining SO(3) attitude regulation with weighted least-squares swing allocation. The compensation torque is computed adaptively based on swing geometry (Equation (32)), eliminating the need for gain scheduling across 5× payload mass and 4× cable length variations while enforcing actuator constraints through command shaping.
• Demonstrated effectiveness under aggressive operating conditions: Simulation studies validate that the cooperative controller successfully suppresses swing angles up to 80°—beyond the 60° linearization boundary—and drives the system back to nominal conditions, achieving 44–70% settling time reduction compared to uncontrolled baselines.

The remainder of this paper is organized as follows. Section 2 introduces the system configuration and 10-DOF Lagrangian modeling. Section 3 presents the MCG feedback linearization control design, including the cooperative swing-compensation mechanism. Section 4 provides Lyapunov-based stability analysis. Section 5 summarizes implementation aspects. Section 6 reports simulation results. Section 7 concludes the paper.

2. System Modeling and Problem Formulation

2.1. System Configuration

The UAV-suspended payload system integrates a 6-DOF quadrotor platform, a 2-DOF serial robotic arm, and a cable-suspended payload. The quadrotor provides primary thrust and attitude control, while the robotic arm offers additional actuation for swing compensation through revolute joints about the z-axis (${\alpha }_1$) and y-axis (${\alpha }_2$). The payload is connected via a massless, inextensible cable of length $l_p$, creating pendulum dynamics characterized by swing angles ${\theta }_x$ and ${\theta }_y$ in the world frame (as shown in Figure 1).

The inertial world frame $\left\{W\right\}$ has orthonormal basis vectors ${\mathit{e}}_x$ = ${[1,0,0]}^T$, ${\mathit{e}}_y$ = ${[0,1,0]}^T$, and ${\mathit{e}}_z={[0,0,1]}^T$ with the z-axis pointing upward. The UAV body frame $\left\{B_0\right\}$ is attached at the platform’s center of mass, with link frames $\left\{B_1\right\}$ and $\left\{B_2\right\}$ following the Denavit–Hartenberg convention. The rotation from body to world frame follows the ZYX Euler convention: ${\mathit{R}}_0={\mathit{R}}_z\left(\psi \right){\mathit{R}}_y\left(\theta \right){\mathit{R}}_x\left(\varphi \right)\in \mathrm{SO}\left(3\right)$, where $\varphi$, $\theta$, and $\psi$ denote roll, pitch, and yaw angles, respectively. Similarly, the manipulator link rotations are expressed as ${\mathit{R}}_1^m={\mathit{R}}_z\left({\alpha }_1\right)$ and ${\mathit{R}}_2^m={\mathit{R}}_z\left({\alpha }_1\right){\mathit{R}}_y\left({\alpha }_2\right)$.

In Figure 1, the swing angles of the cable are denoted by the blue angles $\theta$, while the joint angles of the onboard robotic arm are represented by $\alpha$. The suspension rope is highlighted in red to distinguish it from the UAV body and arm links.

The complete system state comprises 10 generalized coordinates $\mathit{q}={[{\mathit{p}}_0^T,{\mathit{\eta }}^T,{\mathit{q}}_m^T,{\mathit{\theta }}_r^T]}^T\in {\mathbb{R}}^{10}$, where ${\mathit{p}}_0={[x_0,y_0,z_0]}^T$ represents the UAV position, $\mathit{\eta }={[\varphi ,\theta ,\psi ]}^T$ contains the Euler angles, ${\mathit{q}}_m={[{\alpha }_1,{\alpha }_2]}^T$ denotes the manipulator joint angles, and ${\mathit{\theta }}_r={[{\theta }_x,{\theta }_y]}^T$ describes the cable swing angles. For control design purposes, we partition the coordinates into actuated states ${\mathit{q}}_a={[{\mathit{p}}_0^T,{\mathit{\eta }}^T,{\mathit{q}}_m^T]}^T\in {\mathbb{R}}^8$ and unactuated states ${\mathit{q}}_u={\mathit{\theta }}_r\in {\mathbb{R}}^2$.

Definition 1 

(Working Domain). The admissible configuration space $\mathcal{Q}$ is defined as follows:

$$\mathcal{Q}=\{\mathit{q}:|\varphi |<{\varphi }_{max},|\theta |<{\theta }_{max},|{\alpha }_2|<{\alpha }_{2,max},\parallel {\mathit{\theta }}_r\parallel <{\theta }_{r,max}\},$$

(1)

where typical bounds are ${\varphi }_{max}=\pi /6$, ${\theta }_{max}=\pi /6$ (to avoid aggressive maneuvers and maintain small-angle approximations), ${\alpha }_{2,max}=\pi /3$ (to avoid arm singularities), and ${\theta }_{r,max}=\pi /3$ (to maintain cable tension).

Remark 1 

(Modeling Assumptions and Operating Domain). The working domain $\mathcal{Q}$ formalizes the regime in which the modeling assumptions remain valid: $\left|\varphi \right|,\left|\theta \right|<\pi /6$, $|{\alpha }_2|<\pi /3$, and $\parallel {\mathit{\theta }}_r\parallel <\pi /3$. All simulation scenarios in Section 6 respect these bounds.

The cable is modeled as massless and inextensible, valid when cable mass ≪ payload mass (here $<2\%$) and length $<2$ m. In MuJoCo, the cable is implemented using the tendon primitive rather than rigid links, which correctly captures high-frequency oscillations and the unilateral tension constraint characteristic of real cables. Aerodynamic effects (rotor downwash, payload drag, cable flutter) are omitted, reasonable for low-speed indoor flight ($v<5$ m/s).

Regarding disturbance rejection, the framework effectively attenuates the transient swing from sudden gusts. However, sustained wind loads pose a fundamental limitation of single-UAV cable-suspended systems: equilibrium under a constant horizontal force geometrically requires a non-zero cable inclination. This is inherent to the underactuated single-cable topology; multi-UAV cooperative transportation would be required for complete rejection of persistent disturbances.

These simplifications enable closed-form MCG expressions, which are essential for real-time control. Within $\mathcal{Q}$, unmodeled effects are treated as bounded disturbances in Section 4.2.

All links (UAV platform, arm segments, and payload) are treated as rigid bodies, neglecting structural flexibility that may become significant for lightweight or slender arm designs. These modeling limitations are explicitly addressed in the stability analysis (Section 4.2) through bounded uncertainty margins, and are further discussed in the Conclusions as directions for future work.

To facilitate the subsequent analysis and control design,

Table 1. Notation and symbol definitions.

Symbol	Description
$m_0,m_1,m_2,m_p$	Masses of UAV, link 1, link 2, and payload
$l_1,l_2,l_p$	Lengths of link 1, link 2, and cable
${\mathit{I}}_i^b\in {\mathbb{R}}^{3\times 3}$	Body-frame inertia tensor of component i
${\mathit{I}}_i^w\in {\mathbb{R}}^{3\times 3}$	World-frame inertia tensor of component i
${\mathit{\omega }}_i^b\in {\mathbb{R}}^3$	Angular velocity of component i in body frame
${\mathit{\omega }}_i^w\in {\mathbb{R}}^3$	Angular velocity of component i in world frame
${\omega }_n=\sqrt{g/l_p}$	Natural pendulum frequency
${\mathit{R}}_0\in$ SO(3)	Rotation matrix from $\left\{B_0\right\}$ to $\left\{W\right\}$ frame
${\mathit{R}}_e\in$ SO(3)	Rotation error matrix ${\mathit{R}}_{0,d}^T{\mathit{R}}_{\mathbf{0}}$
${\mathit{R}}_{0,d}{\mathit{e}}_z$	Desired body z-axis direction expressed in world frame
${\mathit{T}}_b\left(\mathit{\eta }\right)\in {\mathbb{R}}^{3\times 3}$	Transformation from Euler rates to body-frame angular velocity
${\mathit{T}}_w\left(\mathit{\eta }\right)\in {\mathbb{R}}^{3\times 3}$	Transformation from Euler rates to world-frame angular velocity
${\mathit{q}}_c\in {\mathbb{R}}^3$	Unit vector describing cable orientation
$\mathit{B}\in {\mathbb{R}}^{10\times 6}$	Input matrix
${\mathit{B}}_a^{+}\in {\mathbb{R}}^{6\times 8}$	Moore-Penrose pseudoinverse
$\mathit{u}\in {\mathbb{R}}^6$	Control input ${[f_T,{\tau }_{\varphi },{\tau }_{\theta },{\tau }_{\psi },{\tau }_{{\alpha }_1},{\tau }_{{\alpha }_2}]}^T$
${\mathit{e}}_{\omega }={\mathit{\omega }}_0^b-{\mathit{R}}_e^T{\mathit{\omega }}_{0,d}^b$	Angular velocity error in body frame

summarizes the key notation and symbols used throughout this paper. In this notation system, superscripts denote the reference frame (e.g., ${\mathit{I}}_i^b$ for the body frame, ${\mathit{I}}_i^w$ for the world frame), while subscripts identify the component or variable index.

The simulation environment employs the validated dynamics engine from RflySim [23], which has been extensively used for multicopter control development and provides physics fidelity comparable to real hardware experiments. This platform’s accuracy for cable-suspended payload dynamics has been demonstrated through hardware correlation studies, providing confidence in our simulation results despite the lack of direct experimental validation.

2.2. Kinematic and Dynamic Modeling

2.2.1. Reference Frame Consistency and Angular Velocities

To maintain consistency throughout the modeling and control design, we explicitly declare that all angular velocities are expressed in the world frame for the Lagrangian formulation, while the control design will transform to the body frame when needed. The transformation from Euler angle rates to body-frame angular velocity is as follows:

$${\mathit{\omega }}_0^b={\mathit{T}}_b\left(\mathit{\eta }\right)\dot{\mathit{\eta }},\mathrm{where}{\mathit{T}}_b\left(\mathit{\eta }\right)=\left[\begin{array}{ccc}1& 0& -sin\theta \\ 0& cos\varphi & sin\varphi cos\theta \\ 0& -sin\varphi & cos\varphi cos\theta \end{array}\right].$$

(2)

The world-frame angular velocity is obtained through rotation:

$${\mathit{\omega }}_0^w={\mathit{R}}_0{\mathit{\omega }}_0^b={\mathit{R}}_0{\mathit{T}}_b\left(\mathit{\eta }\right)\dot{\mathit{\eta }}\triangleq {\mathit{T}}_w\left(\mathit{\eta }\right)\dot{\mathit{\eta }},$$

(3)

where ${\mathit{T}}_w\left(\mathit{\eta }\right)={\mathit{R}}_0{\mathit{T}}_b\left(\mathit{\eta }\right)$ directly maps Euler rates to world-frame angular velocity. Note that ${\mathit{T}}_b\left(\mathit{\eta }\right)$ becomes singular at $\theta =\pm \pi /2$, representing gimbal lock in the pitch axis.

The angular velocities of manipulator links in the world frame propagate as follows:

$$\begin{array}{cc}\hfill {\mathit{\omega }}_1^w& ={\mathit{\omega }}_0^w+{\dot{\alpha }}_1{\mathit{R}}_0{\mathit{e}}_z,\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill {\mathit{\omega }}_2^w& ={\mathit{\omega }}_1^w+{\dot{\alpha }}_2{\mathit{R}}_0{\mathit{R}}_1^m{\mathit{e}}_y,\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill {\mathit{\omega }}_{pc}^w& ={\mathit{q}}_c\times {\dot{\mathit{q}}}_c.\hfill \end{array}$$

(6)

Remark 2 

(Cable Angular Velocity). For a unit vector ${\mathit{q}}_c$ describing cable orientation, the angular velocity satisfies ${\dot{\mathit{q}}}_c={\mathit{\omega }}_{pc}^w\times {\mathit{q}}_c$, hence ${\mathit{\omega }}_{pc}^w={\mathit{q}}_c\times {\dot{\mathit{q}}}_c$. This ensures ${\mathit{\omega }}_{pc}^w\perp {\mathit{q}}_c$ and preserves the unit length constraint $\parallel {\mathit{q}}_c\parallel =1$.

2.2.2. Position Kinematics

The hierarchical kinematic chain determines component positions through forward kinematics. The manipulator link positions in the world frame are computed as ${\mathit{p}}_1={\mathit{p}}_0+{\mathit{R}}_0{\mathit{r}}_1$ and ${\mathit{p}}_2={\mathit{p}}_1+{\mathit{R}}_0{\mathit{R}}_1^m{\mathit{r}}_2$, where ${\mathit{r}}_i={[0,0,-l_i]}^T$ represents the link vector in its body frame. The cable orientation from the second link endpoint to the payload is described by the unit vector:

$${\mathit{q}}_c=\left[\begin{array}{c}-sin{\theta }_{y}cos{\theta }_x\\ sin{\theta }_x\\ -cos{\theta }_{y}cos{\theta }_x\end{array}\right],$$

(7)

which encodes the swing angles ${\theta }_x$ and ${\theta }_y$ in spherical coordinates. This parameterization yields the payload position as ${\mathit{p}}_{pc}={\mathit{p}}_2+l_p{\mathit{q}}_c$, where the cable maintains constant length $l_p$ throughout the motion.

2.2.3. Lagrangian Dynamics with Consistent Reference Frames

Using Lagrangian mechanics with world-frame angular velocities, the kinetic and potential energies are calculated as follows:

$$T=\sum _{i\in \{0,1,2,p\}}\left({\displaystyle \frac{1}{2}}{m}_i{\dot{\mathit{p}}}_{ic}^T{\dot{\mathit{p}}}_{ic}+{\displaystyle \frac{1}{2}}{\left({\mathit{\omega }}_i^w\right)}^T{\mathit{I}}_{ic}^w{\mathit{\omega }}_i^w\right),V=g\sum _{i\in \{0,1,2,p\}}{m}_i{z}_{ic},$$

(8)

where ${\mathit{I}}_{ic}^w={\mathit{R}}_i{\mathit{I}}_{ic}^b{\mathit{R}}_i^T$ is the world-frame inertia tensor obtained by rotating the body-frame inertia ${\mathit{I}}_{ic}^b$. This ensures consistency between the angular velocity reference frame and the inertia tensor.

The Euler–Lagrange equations yield the standard form:

$$\mathit{M}\left(\mathit{q}\right)\ddot{\mathit{q}}+\mathit{C}(\mathit{q},\dot{\mathit{q}})\dot{\mathit{q}}+\mathit{G}\left(\mathit{q}\right)=\mathit{B}\mathit{u},$$

(9)

where the mass matrix exhibits the block structure:

$$\mathit{M}\left(\mathit{q}\right)=\left[\begin{array}{cccc}{\mathit{M}}_{pp}& {\mathit{M}}_{p\eta }& {\mathit{M}}_{pj}& {\mathit{M}}_{pu}\\ {\mathit{M}}_{\eta p}& {\mathit{M}}_{\eta \eta }& {\mathit{M}}_{\eta j}& {\mathit{M}}_{\eta u}\\ {\mathit{M}}_{jp}& {\mathit{M}}_{j\eta }& {\mathit{M}}_{jj}& {\mathit{M}}_{ju}\\ {\mathit{M}}_{up}& {\mathit{M}}_{u\eta }& {\mathit{M}}_{uj}& {\mathit{M}}_{uu}\end{array}\right]=\left[\begin{array}{cc}{\mathit{M}}_{aa}& {\mathit{M}}_{au}\\ {\mathit{M}}_{ua}& {\mathit{M}}_{uu}\end{array}\right].$$

(10)

The block-structured mass matrix (10), formulated following [21], explicitly captures the dynamic coupling within the system. In particular, the coupling block ${\mathit{M}}_{au}={[{\mathit{M}}_{pu}^T,{\mathit{M}}_{\eta u}^T,{\mathit{M}}_{ju}^T]}^T$ characterizes the interaction between the actuated and unactuated states, describing how accelerations of the UAV platform and robotic arm propagate into swing motions of the suspended payload.

Physical interpretation of coupling terms: The coupling matrix ${\mathit{M}}_{ua}$ admits a clear physical interpretation that underlies the cooperative swing suppression mechanism:

• ${\mathit{M}}_{ua}[:,0:3]$ (translational coupling): When the UAV platform accelerates, the suspended payload experiences inertial forces that induce swing motion. This coupling quantifies the swing-angle acceleration per unit of platform acceleration.
• ${\mathit{M}}_{ua}[:,6:8]$ (arm-swing coupling): When the robotic arm joints accelerate, the cable attachment point displaces, directly affecting the payload swing dynamics. This coupling is the key innovation enabling cooperative control—the arm can actively reshape swing dynamics without requiring platform attitude changes.

Cooperative control mechanism: When payload swing angles ${\mathit{\theta }}_r$ are detected, the controller computes a compensating arm acceleration via the coupling matrix:

$$\delta \ddot{\mathit{\alpha }}=-{\mathit{M}}_{ua}{[:,6:8]}^{†}({\mathit{K}}_{P,u}{\mathit{\theta }}_r+{\mathit{K}}_{D,u}{\dot{\mathit{\theta }}}_r),$$

(11)

where ${(·)}^{†}$ denotes the pseudoinverse. This enables the robotic arm to generate anti-swing torques that counteract payload oscillations through geometric manipulation of the cable attachment point, rather than through inertial offloading from the UAV platform. Compared to traditional UAV-only methods, this approach provides an additional actuation channel with faster response and reduced coupling to flight dynamics.

2.3. Underactuated Structure and Cascaded Dynamics

The system exhibits a cascaded underactuated structure characteristic of quadrotor platforms. The horizontal position dynamics are indirectly controlled through attitude changes, creating a position-attitude cascade. Specifically, the total thrust $f_T$ acts along the body-fixed z-axis, so that the world-frame acceleration becomes the following:

$${\ddot{\mathit{p}}}_0={\displaystyle \frac{1}{m_{\mathrm{total}}}}{\mathit{R}}_{\mathbf{0}}\left[\begin{array}{c}0\\ 0\\ f_T\end{array}\right]-g{\mathit{e}}_z={\displaystyle \frac{f_T}{m_{\mathrm{total}}}}{\mathit{z}}_b-g{\mathit{e}}_z,$$

(12)

where ${\mathit{z}}_b=R_0{\mathit{e}}_z$ is the body z-axis expressed in world coordinates. Horizontal accelerations (${\ddot{x}}_0$, ${\ddot{y}}_0$) are achieved by tilting the platform—pitch for x-direction and roll for y-direction—which couples position control with attitude dynamics.

The input matrix has the following structure:

$$\mathit{B}=\left[\begin{array}{c}{\mathit{B}}_p\\ {\mathit{B}}_{\eta }\\ {\mathit{B}}_j\\ {\mathbf{0}}_{2\times 6}\end{array}\right]\in {\mathbb{R}}^{10\times 6},$$

(13)

where ${\mathit{B}}_p\in {\mathbb{R}}^{3\times 6}$ is the configuration-dependent mapping from thrust to translational acceleration (its first column encodes $R_0{\mathit{e}}_z/m_{\mathrm{total}}$, reflecting the attitude-mediated coupling), ${\mathit{B}}_{\eta }\in {\mathbb{R}}^{3\times 6}$ maps body torques to Euler angle accelerations, ${\mathit{B}}_j=[{\mathbf{0}}_{2\times 4},{\mathit{I}}_{2\times 2}]\in {\mathbb{R}}^{2\times 6}$ directly actuates the manipulator joints, and the zero block confirms that swing angles are unactuated. The rank deficiency (rank$\left(\mathit{B}\right)=6<10$) confirms the system is underactuated with a four-dimensional null space.

Assumption 1 (System Properties in Working Domain).

Within the working domain $\mathcal{Q}$, the system satisfies:

(i) 

The mass matrix is uniformly positive definite: $c_1\mathit{I}⪯\mathit{M}\left(\mathit{q}\right)⪯c_2\mathit{I}$ with $c_1=0.01$ kg·m² and $c_2=10$ kg·m². These bounds are determined from the system configuration extremes: $c_1$ corresponds to the minimum effective inertia at the contracted arm configuration without payload ($\min \left(m_{\mathit{body}}\right)·\min {\left(l_{\mathit{arm}}\right)}^2\approx 1.4\times 0.{05}^2\approx 0.0035$, rounded up for margin), while $c_2$ corresponds to the maximum effective inertia at the extended arm configuration with full payload ($m_{\mathit{total}}·l_{\mathit{cable},\mathit{max}}^2\approx 2.0\times 2.0^2=8.0$, rounded up).

(ii) 

The matrix $\dot{\mathit{M}}-2\mathit{C}$ is skew-symmetric, which is a standard property of Lagrangian systems with the Christoffel symbol formulation of Coriolis terms.

(iii) 

The Coriolis matrix, constructed using Christoffel symbols, depends linearly on velocities. Consequently, the Coriolis term satisfies $\parallel \mathit{C}(\mathit{q},\dot{\mathit{q}})\mathit{v}\parallel \le k_c\parallel \dot{\mathit{q}}\parallel \parallel \mathit{v}\parallel$ for any $\mathit{v}\in {\mathbb{R}}^{10}$ with $k_c=2.5$, and the gravity vector is bounded by $\parallel \mathit{G}\left(\mathit{q}\right)\parallel \le g_{max}=30$ N.

(iv) 

The coupling strength satisfies ${\lambda }_{min}\left({\mathit{M}}_{ua}{\mathit{M}}_{aa}^{-1}{\mathit{M}}_{au}^T\right)\ge {\mu }_{min}=0.001$ for effective swing controllability, verified numerically across representative configurations. The working domain constraints implicitly maintain cable tension through the swing angle limitation $\parallel {\mathit{\theta }}_r\parallel <\pi /3$, though explicit tension monitoring is not enforced in the control law.

3. MCG Feedback Linearization Control Design

3.1. Hierarchical Control with Position-Attitude Cascade

Given the cascaded underactuated structure of quadrotor dynamics, we adopt a hierarchical control architecture that explicitly handles the position-attitude coupling. The position controller generates desired accelerations, which are mapped to required thrust and attitude commands, and the attitude controller tracks these commands while compensating for coupling effects.

3.1.1. Position Control and Attitude Command Generation

The position control law generates desired accelerations:

$${\mathit{a}}_{p,d}={\ddot{\mathit{p}}}_{0,d}+{\mathit{K}}_{D,p}({\dot{\mathit{p}}}_{0,d}-{\dot{\mathit{p}}}_0)+{\mathit{K}}_{P,p}({\mathit{p}}_{0,d}-{\mathit{p}}_0),$$

(14)

where ${\mathit{K}}_{P,p}=\mathrm{diag}(k_{px},k_{py},k_{pz})$ and ${\mathit{K}}_{D,p}=\mathrm{diag}(k_{dx},k_{dy},k_{dz})$ are diagonal gain matrices.

For a quadrotor, horizontal accelerations require attitude tilting. The desired acceleration maps to thrust magnitude and attitude:

$$\begin{array}{cc}\hfill f_T& =m_{\mathrm{total}}·\parallel {\mathit{a}}_{p,d}+g{\mathit{e}}_z\parallel ,\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill {\mathit{z}}_{b,d}& ={\displaystyle \frac{{\mathit{a}}_{p,d}+g{\mathit{e}}_z}{\parallel {\mathit{a}}_{p,d}+g{\mathit{e}}_z\parallel }}.\hfill \end{array}$$

(16)

To construct the complete desired rotation matrix ${\mathit{R}}_{0,d}$ while tracking a desired yaw angle ${\psi }_d\left(t\right)$, we use the geometric method:

$$\begin{array}{cc}\hfill {\mathit{x}}_c& ={[\cos {\psi }_d,\sin {\psi }_d,0]}^T,\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill {\mathit{y}}_{b,d}& ={\displaystyle \frac{{\mathit{z}}_{b,d}\times {\mathit{x}}_c}{\parallel {\mathit{z}}_{b,d}\times {\mathit{x}}_c\parallel }},\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill {\mathit{x}}_{b,d}& ={\mathit{y}}_{b,d}\times {\mathit{z}}_{b,d},\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill {\mathit{R}}_{0,d}& =[{\mathit{x}}_{b,d},{\mathit{y}}_{b,d},{\mathit{z}}_{b,d}].\hfill \end{array}$$

(20)

This construction ensures ${\mathit{R}}_{0,d}\in \mathrm{SO}\left(3\right)$ with the desired thrust direction and yaw angle. For trajectory tracking tasks where yaw is not specified, we either fix ${\psi }_d=0$ or align the yaw with the vehicle velocity by setting

$${\psi }_d=atan2({\dot{y}}_0,{\dot{x}}_0),$$

where $atan2(·,·)$ denotes the four-quadrant arctangent, and $({\dot{x}}_0,{\dot{y}}_0)$ are the horizontal components of the UAV velocity.

3.1.2. SO(3) Attitude Control with Consistent Dynamics

To avoid gimbal lock and ensure a global representation of attitude, the attitude controller operates on SO(3). The angular velocity error is defined in the body frame as follows:

$${\mathit{e}}_{\omega }={\mathit{\omega }}_0^b-{\mathit{R}}_e^T{\mathit{\omega }}_{0,d}^b,$$

(21)

where ${{\mathit{\omega }}_{\mathbf{0}}}^b={\mathit{R}}_0^T{{\mathit{\omega }}_{\mathbf{0}}}^w$ is the body-frame angular velocity, ${\mathit{R}}_e={\mathit{R}}_{0,d}^T{\mathit{R}}_{\mathbf{0}}$ is the rotation error, and ${\mathit{\omega }}_{0,d}^b$ is the desired angular velocity in the desired body frame.

The attitude error vector is extracted from ${\mathit{R}}_e$ via the following:

$${\mathit{\varphi }}_e={\displaystyle \frac{1}{2}}{({\mathit{R}}_e-{\mathit{R}}_e^T)}^{\vee },$$

(22)

which is valid for tr$\left({\mathit{R}}_e\right)>-1$ (rotation angle less than $\pi$).

The attitude control torque with Coriolis compensation and feedforward is as follows:

$${\mathit{\tau }}_{\mathrm{att}}=-{\mathit{K}}_{P,\varphi }{\mathit{\varphi }}_e-{\mathit{K}}_{D,\omega }{\mathit{e}}_{\omega }+{{\mathit{\omega }}_{\mathbf{0}}}^b\times {\mathit{J}}^b{{\mathit{\omega }}_{\mathbf{0}}}^b+{\mathit{J}}^b({\mathit{R}}_e^T{\dot{\mathit{\omega }}}_{0,d}^b-{{\mathit{\omega }}_{\mathbf{0}}}^b\times {\mathit{R}}_e^T{\mathit{\omega }}_{0,d}^b),$$

(23)

where ${\mathit{J}}^b={\mathit{I}}_0^b$ is the body-frame inertia tensor, the term ${\mathit{\omega }}^b\times {\mathit{J}}^b{\mathit{\omega }}^b$ compensates for gyroscopic effects, and the last term provides feedforward for desired angular acceleration.

3.2. Partial Feedback Linearization with Swing Compensation

The underactuated system (9) decomposes into coupled subsystems:

$$\begin{array}{cc}\hfill {\mathit{M}}_{aa}\left(\mathit{q}\right){\ddot{\mathit{q}}}_a+{\mathit{M}}_{au}\left(\mathit{q}\right){\ddot{\mathit{q}}}_u+{\mathit{C}}_a(\mathit{q},\dot{\mathit{q}})+{\mathit{G}}_a\left(\mathit{q}\right)& ={\mathit{B}}_a\mathit{u},\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill {\mathit{M}}_{ua}\left(\mathit{q}\right){\ddot{\mathit{q}}}_a+{\mathit{M}}_{uu}\left(\mathit{q}\right){\ddot{\mathit{q}}}_u+{\mathit{C}}_u(\mathit{q},\dot{\mathit{q}})+{\mathit{G}}_u\left(\mathit{q}\right)& =\mathbf{0},\hfill \end{array}$$

(25)

where ${\mathit{B}}_a\in {\mathbb{R}}^{8\times 6}$ with rank 6 creates the underactuation challenge.

3.2.1. Derivation of Swing Compensation Term

To actively suppress swing, we derive a compensation term that creates virtual damping in the unactuated swing dynamics. The mass matrix partitioning in (10) leads to the actuated-unactuated decomposition in (24) and (25). From the unactuated dynamics (25), the swing acceleration is as follows:

$${\ddot{\mathit{q}}}_u=-{\mathit{M}}_{uu}^{-1}[{\mathit{M}}_{ua}{\ddot{\mathit{q}}}_a+{\mathit{C}}_u+{\mathit{G}}_u].$$

(26)

To inject damping, we choose positive semi-definite gain matrices ${\mathit{K}}_{D,u}⪰0$ and ${\mathit{K}}_{P,u}⪰0$ such that the desired closed-loop unactuated dynamics satisfy the following:

$${\mathit{M}}_{uu}{\ddot{\mathit{q}}}_u+{\mathit{K}}_{D,u}{\dot{\mathit{q}}}_u+{\mathit{K}}_{P,u}{\mathit{q}}_u=\mathbf{0},$$

(27)

where ${\mathit{K}}_{D,u}$ and ${\mathit{K}}_{P,u}$ represent virtual damping and stiffness matrices.

For the desired swing dynamics (27), the required constraint from (26) is as follows:

$${\mathit{M}}_{ua}{\ddot{\mathit{q}}}_a=-{\mathit{C}}_u-{\mathit{G}}_u-{\mathit{K}}_{D,u}{\dot{\mathit{q}}}_u-{\mathit{K}}_{P,u}{\mathit{q}}_u\triangleq \mathit{r}.$$

(28)

Given the nominal acceleration ${\ddot{\mathit{q}}}_{a,d}$ from the outer loop, the residual error is as follows:

$$\mathit{e}={\mathit{M}}_{ua}{\ddot{\mathit{q}}}_{a,d}-\mathit{r}.$$

(29)

We compute the acceleration correction $\Delta {\mathit{a}}_a\in {\mathbb{R}}^8$ via weighted least-squares with Tikhonov regularization:

$$\Delta {\mathit{a}}_a=\underset{\mathit{\delta }\in {\mathbb{R}}^8}{\arg \min }\left\{\parallel {\mathit{M}}_{ua}\mathit{\delta }-\mathit{e}{\parallel }_{\mathit{W}}^2+\epsilon {\parallel \mathit{\delta }\parallel }^2\right\},$$

(30)

where $\mathit{W}={\mathit{M}}_{uu}^{-1}$ weights the residual by the swing inertia, and $\mathit{\delta }$ represents the candidate acceleration correction.

For computational efficiency, we exploit the dimension reduction. Define the whitened matrix $\mathit{A}={\mathit{W}}^{1/2}{\mathit{M}}_{ua}\in {\mathbb{R}}^{2\times 8}$ and whitened residual $\mathit{b}={\mathit{W}}^{1/2}\mathit{e}\in {\mathbb{R}}^2$. The solution becomes the following:

$$\Delta {\mathit{a}}_a={\mathit{A}}^T{(\mathit{A}{\mathit{A}}^T+\epsilon {\mathit{I}}_2)}^{-1}\mathit{b},$$

(31)

which requires only a $2\times 2$ matrix inversion, significantly more efficient than directly inverting the full $8\times 8$ matrix.

The compensated acceleration is then calculated as follows:

$${\mathit{a}}_{a,d}^{\mathrm{comp}}={\mathit{a}}_{a,d}-K_{\mathrm{coup}}\Delta {\mathit{a}}_a,K_{\mathrm{coup}}\in [0,1],$$

(32)

where $K_{\mathrm{coup}}$ scales the compensation strength.

The resulting effective damping ratio along the dominant swing eigen-direction is bounded by the following:

$${\zeta }_{\mathrm{eff}}\le {\displaystyle \frac{K_{\mathrm{coup}}}{2{\omega }_n}}{\displaystyle \frac{k_{D,u}{\lambda }_{min}\left({\mathit{M}}_{ua}{\mathit{M}}_{aa}^{-1}{\mathit{M}}_{au}^T\right)}{{\lambda }_{max}\left({\mathit{M}}_{uu}\right)}},{\omega }_n=\sqrt{g/l_p},$$

(33)

where $k_{D,u}=\text{min}\{k_{D,u,x},k_{D,u,y}\}$ with ${\mathit{K}}_{D,u}=\mathrm{diag}(k_{D,u,x},k_{D,u,y})$.

Remark 3 

(Dimension Consistency). In the small-angle approximation, ${\mathit{M}}_{uu}\approx m_p{l}_p^2{\mathit{I}}_2$ has units $kg·m^2$, and the restoring stiffness from gravity is approximately $k_{\mathrm{eff}}=m_{p}g{l}_p$ with units $N·m/rad$. The damping gain ${\mathit{K}}_{D,u}$ has units $N·m·s/rad$ to ensure dimensional consistency. The effective damping coefficient

$$c_{\mathrm{eff}}=K_{\mathrm{coup}}{\lambda }_{min}\left({\mathit{M}}_{ua}{\mathit{M}}_{aa}^{-1}{\mathit{M}}_{au}^{\top }{\mathit{K}}_{D,u}\right)$$

also has units $N·m·s/rad$, yielding the dimensionless damping ratio ${\zeta }_{\mathrm{eff}}=c_{\mathrm{eff}}/\left(2\sqrt{m_{\mathrm{eff}}{k}_{\mathrm{eff}}}\right)$.

3.2.2. MCG Control Law with Pseudoinverse

Using the compensated actuated acceleration ${\mathit{a}}_{a,d}^{\mathrm{comp}}$ from (32), the predicted swing acceleration is obtained from the unactuated dynamics (25) as follows:

$${\ddot{\mathit{q}}}_u^{\star }={\mathit{M}}_{uu}^{-1}\left(-{\mathit{M}}_{ua}{\mathit{a}}_{a,d}^{\mathrm{comp}}-{\mathit{C}}_u(\mathit{q},\dot{\mathit{q}})-{\mathit{G}}_u\left(\mathit{q}\right)\right),$$

(34)

This prediction accounts for the dynamic coupling between actuated and unactuated coordinates, ensuring that the swing compensation mechanism remains consistent with the overall system dynamics.

Figure 2 illustrates the overall control architecture and simulation setup of the quadrotor–robotic-arm cable-suspended payload system. The upper (green) loop implements a position–attitude cascade. Specifically, the trajectory generator outputs the reference signals $({\mathit{p}}_{0,d},{\dot{\mathit{p}}}_{0,d},{\ddot{\mathit{p}}}_{0,d},{\psi }_d)$, based on which the position controller computes the desired translational acceleration ${\mathit{a}}_{p,d}$. An $\mathrm{SO}\left(3\right)$ attitude controller then tracks $({\mathit{R}}_{0,d},{\mathit{\omega }}_{0,d}^b,{\dot{\mathit{\omega }}}_{0,d}^b)$ to generate the UAV thrust $f_T$ and body torques $({\tau }_{\varphi },{\tau }_{\theta },{\tau }_{\psi })$.

The lower (blue) loop focuses on payload swing regulation. Using the measured swing states $({\mathit{q}}_c,{\mathit{\omega }}_{pc}^w)$, it synthesizes the desired unactuated acceleration ${\ddot{\mathit{q}}}_u^{\star }$. This term is combined with the nominal actuated acceleration ${\mathit{a}}_{a,d}$ by a virtual-force module to form the compensated command ${\mathit{a}}_{a,d}^{\mathrm{comp}}$, which is subsequently fed into a weighted least-squares (WLS) allocation block. The WLS solver resolves actuator redundancy by distributing ${\mathit{a}}_{a,d}^{\mathrm{comp}}$ between the UAV and robotic-arm actuators, yielding the control inputs $({\mathit{u}}_{\mathrm{uav}},{\mathit{u}}_{\mathrm{arm}})$ applied to the MuJoCo simulation plant.

The complete MCG feedback linearization control law for the actuated subsystem (24) is then

$$\mathit{u}={\mathit{B}}_a^{+}\left[{\mathit{M}}_{aa}{\mathit{a}}_{a,d}^{\mathrm{comp}}+{\mathit{M}}_{au}{\ddot{\mathit{q}}}_u^{*}+{\mathit{C}}_a+{\mathit{G}}_a\right],$$

(35)

where ${\mathit{B}}_a^{+}$ is the Moore–Penrose pseudoinverse computed via regularized SVD. This control law achieves feedback linearization of the actuated coordinates while maintaining dimensional consistency across all inertia-matrix blocks ${\mathit{M}}_{aa}$, ${\mathit{M}}_{au}$, and ${\mathit{M}}_{uu}$, ensuring that the coupling terms and swing prediction are properly synchronized.

The pseudoinverse ${\mathit{B}}_a^{+}$ is computed via regularized SVD:

$${\mathit{B}}_a^{+}=\mathit{V}{\Sigma }_r^{-1}{\mathit{U}}_{\mathrm{svd}}^T,$$

(36)

where ${\mathit{B}}_a={\mathit{U}}_{\mathrm{svd}}\Sigma {\mathit{V}}^T$ is the singular value decomposition (SVD) of ${\mathit{B}}_a$, with ${\mathit{U}}_{\mathrm{svd}}$ and $\mathit{V}$ orthogonal and $\Sigma$ diagonal, and the regularized inverse is calculated as follows:

$${\left[{\Sigma }_r^{-1}\right]}_{ii}=\left\{\begin{array}{cc}1/{\sigma }_i\hfill & \mathrm{if}{\sigma }_i>{ϵ}_{\mathrm{svd}},\hfill \\ 0\hfill & \mathrm{if}{\sigma }_i\le {ϵ}_{\mathrm{svd}},\hfill \end{array}\right.$$

(37)

with ${ϵ}_{\mathrm{svd}}={\kappa }_{\mathrm{svd}}·{\sigma }_{\mathrm{noise}}$, where ${\kappa }_{\mathrm{svd}}\approx 10$ and ${\sigma }_{\mathrm{noise}}$ is the measurement noise level. For typical implementations with encoder resolution of 0.001 rad and IMU noise of 0.01 rad/s, we have ${\sigma }_{\mathrm{noise}}\approx {10}^{-3}$, yielding ${ϵ}_{\mathrm{svd}}\approx {10}^{-2}$. The maximum singular value of the pseudoinverse is monitored to ensure ${\sigma }_{max}\left({\mathit{B}}_a^{+}\right)<1/{ϵ}_{\mathrm{svd}}=100$, maintaining numerical stability and preventing control saturation due to ill-conditioned input matrices.

3.3. Data-Driven Model Enhancement

Real systems exhibit uncertainties from unmodeled dynamics and parameter variations. We employ a three-stage approach to obtain an accurate yet physically interpretable control model.

Stage 1: Theoretical Modeling. Based on Lagrangian mechanics, we derive the nominal model ${\mathit{M}}_{\mathrm{theory}}\left(\mathit{q}\right)$, ${\mathit{C}}_{\mathrm{theory}}(\mathit{q},\dot{\mathit{q}})$, and ${\mathit{G}}_{\mathrm{theory}}\left(\mathit{q}\right)$ using CAD-based parameters with uncertainty bounds from manufacturing tolerances (typically ±5% for masses and ±10% for inertias). This model preserves the physical structure of the Mass–Coriolis–Gravity (MCG) representation and is used as a prior in the subsequent identification step.

Stage 2: System Identification. Data collection employs chirp and step excitation trajectories sampled at 1000 Hz. The measured accelerations ${\ddot{\mathit{q}}}_{\mathrm{meas}}$ are obtained through numerical differentiation of encoder positions (second-order central differences) and IMU angular velocities, filtered with a second-order Butterworth filter at 20 Hz cutoff to attenuate noise. Given experimental (or high-fidelity simulation) data

$$\mathcal{D}={\left\{({\mathit{q}}^{\left(i\right)},{\dot{\mathit{q}}}^{\left(i\right)},{\ddot{\mathit{q}}}^{\left(i\right)},{\mathit{u}}^{\left(i\right)})\right\}}_{i=1}^N$$

with $N=\mathrm{10,000}$ samples split into training (70%) and validation (30%) sets, we identify parameters by solving the prior-constrained least-squares problem

$$\underset{\mathit{\xi }}{min}\sum _{i=1}^{N_{\mathrm{train}}}\parallel \mathit{M}({\mathit{q}}^{\left(i\right)},\mathit{\xi }){\ddot{\mathit{q}}}^{\left(i\right)}+\mathit{h}({\mathit{q}}^{\left(i\right)},{\dot{\mathit{q}}}^{\left(i\right)},\mathit{\xi })-\mathit{B}{\mathit{u}}^{\left(i\right)}{\parallel }^2+\lambda {\parallel \mathit{\xi }-{\mathit{\xi }}_{\mathrm{prior}}\parallel }_{\mathit{W}}^2,$$

(38)

where $\mathit{h}=\mathit{C}\dot{\mathit{q}}+\mathit{G}$, and the parameter vector

$$\xi =[m_0,m_1,m_2,m_p,\overrightarrow{(}{\mathit{I}}_{\mathbf{0}}^{\mathit{b}}),\overrightarrow{(}{\mathit{I}}_{\mathbf{1}}^{\mathit{b}}),\overrightarrow{(}{\mathit{I}}_{\mathbf{2}}^{\mathit{b}}{)]}^T\in {\mathbb{R}}^{25}$$

contains the masses and diagonal inertia elements. Here $\overrightarrow{(}·)$ denotes the vectorization operator that stacks the independent entries of each body inertia matrix into a column vector. The weighting matrix is $\mathit{W}=\mathrm{diag}\left({\sigma }_{\xi ,i}^{-2}\right)$ with prior variances ${\sigma }_{\xi ,i}^2$ chosen as ${\left(0.05{\xi }_{i,\mathrm{prior}}\right)}^2$ for masses and ${\left(0.10{\xi }_{i,\mathrm{prior}}\right)}^2$ for inertias, reflecting typical manufacturing tolerances. The regularization parameter $\lambda =0.01$ balances data fitting and prior consistency, preventing unphysical parameter values.

Solving (38) yields a data-enhanced model $({\mathit{M}}_{\mathrm{data}},{\mathit{C}}_{\mathrm{data}},{\mathit{G}}_{\mathrm{data}})$ that remains structurally compatible with the nominal MCG form and with the control design in Section 3.

Stage 3: Variance-Based Model Fusion. To combine the advantages of the theoretical and data-driven models, we construct a hybrid model using prediction residuals as weights. On the validation set, the dynamic residual variances are estimated as

$$\begin{array}{cc}\hfill {\sigma }_{\mathrm{theory}}^2& ={\displaystyle \frac{1}{N_{\mathrm{val}}}}\sum _{j=1}^{N_{\mathrm{val}}}\parallel {\mathit{M}}_{\mathrm{theory}}{\ddot{\mathit{q}}}^{\left(j\right)}+{\mathit{h}}_{\mathrm{theory}}^{\left(j\right)}-\mathit{B}{\mathit{u}}^{\left(j\right)}{\parallel }^2,\hfill \end{array}$$

(39)

$$\begin{array}{cc}\hfill {\sigma }_{\mathrm{data}}^2& ={\displaystyle \frac{1}{N_{\mathrm{val}}}}\sum _{j=1}^{N_{\mathrm{val}}}\parallel {\mathit{M}}_{\mathrm{data}}{\ddot{\mathit{q}}}^{\left(j\right)}+{\mathit{h}}_{\mathrm{data}}^{\left(j\right)}-\mathit{B}{\mathit{u}}^{\left(j\right)}{\parallel }^2,\hfill \end{array}$$

(40)

where ${\mathit{h}}_{\mathrm{theory}}={\mathit{C}}_{\mathrm{theory}}\dot{\mathit{q}}+{\mathit{G}}_{\mathrm{theory}}$ and ${\mathit{h}}_{\mathrm{data}}={\mathit{C}}_{\mathrm{data}}\dot{\mathit{q}}+{\mathit{G}}_{\mathrm{data}}$.

The fusion weight is chosen as

$$\alpha ={\displaystyle \frac{{\sigma }_{\mathrm{data}}^2}{{\sigma }_{\mathrm{theory}}^2+{\sigma }_{\mathrm{data}}^2}},$$

(41)

giving the hybrid model

$${\mathit{X}}_{\mathrm{hybrid}}=\alpha {\mathit{X}}_{\mathrm{theory}}+(1-\alpha ){\mathit{X}}_{\mathrm{data}},\mathit{X}\in \{\mathit{M},\mathit{C},\mathit{G}\}.$$

(42)

This construction ensures that when the data-driven model exhibits smaller residuals on the validation set (i.e., ${\sigma }_{\mathrm{data}}^2<{\sigma }_{\mathrm{theory}}^2$), it receives a larger weight, while the theoretical model remains influential whenever data coverage is limited or noisy.

The reported correlation coefficient of 0.691 is defined as the Pearson correlation between predicted accelerations

$$\widehat{\ddot{\mathit{q}}}={\mathit{M}}_{\mathrm{hybrid}}^{-1}\left(\mathit{B}\mathit{u}-{\mathit{h}}_{\mathrm{hybrid}}\right)$$

and measured accelerations ${\ddot{\mathit{q}}}_{\mathrm{meas}}$ on the validation set, computed element-wise and averaged across all DOFs. This value corresponds to a moderate but consistent improvement over the purely theoretical model, reflecting the inherent modeling simplifications (massless cable, rigid bodies, neglected aerodynamics) required for real-time implementation. At the same time, the bounded correction enforced by (38) and the fusion rule (42) ensures that the hybrid model remains physically plausible, which is crucial for the robustness and input-to-state stability analysis in Section 4.2.

The correlation coefficient of 0.691 warrants discussion. This value reflects the inherent trade-off between model fidelity and real-time implementability. The primary sources of model mismatch include the following: (i) massless cable assumption, which neglects cable dynamics and distributed mass effects; (ii) rigid-body assumption for all links, ignoring structural flexibility; (iii) omission of aerodynamic forces from rotor downwash and payload drag; and (iv) simplified actuator models that do not capture higher-order dynamics.

Despite these simplifications, the 0.691 correlation is sufficient for effective feedback linearization because: (a) the control law includes integral action that compensates for steady-state errors; (b) the adaptive mass estimator handles the dominant parametric uncertainty; and (c) the bounded correction mechanism in (42) ensures that model errors remain within the robustness margins established in Theorem 1. The simulation results demonstrate that this correlation level enables 40–70% swing reduction, confirming practical effectiveness despite the modeling compromises.

4. Stability Analysis

This section establishes the stability properties of the proposed MCG feedback linearization control system with explicit consideration of the position-attitude cascade structure and working-domain constraints.

4.1. Semi-Global Practical Stability

Assumption 2 

(Control Gains and Time-Scale Separation). The control gains satisfy the following conditions. For position control: ${\mathit{K}}_{P,p}\succ 0$ and ${\mathit{K}}_{D,p}\succ 2\sqrt{{\mathit{K}}_{P,p}}$ element-wise. For attitude control: ${\mathit{K}}_{P,\varphi }\succ 0$ and ${\mathit{K}}_{D,\omega }\succ 2\sqrt{{\mathit{K}}_{P,\varphi }}$ element-wise. The attitude control bandwidth satisfies ${\omega }_{\mathit{att}}\ge 10{\omega }_{\mathit{pos}}$ where ${\omega }_{\mathit{pos}}=\sqrt{{\lambda }_{min}\left({\mathit{K}}_{P,p}\right)/m_{\mathit{total}}}$ and ${\omega }_{\mathit{att}}=\sqrt{{\lambda }_{min}\left({\mathit{K}}_{P,\varphi }\right)/{\lambda }_{max}\left({\mathit{J}}^b\right)}$, ensuring time-scale separation. The coupling strength satisfies $0<K_{\mathit{coup}}\le 1$.

Theorem 1 

(Semi-Global Practical Stability in Working Domain). Consider the underactuated system (9) with hierarchical control (position-attitude cascade) and swing compensation through the MCG control law (35). Under Assumptions 1 and 2, if the initial conditions satisfy $\mathit{q}\left(0\right)\in {\mathcal{Q}}_0\subset \mathcal{Q}$ with sufficient margin (at least $\pi /12$ from boundaries), and the desired trajectories are bounded with $\parallel {\mathit{p}}_d\left(t\right)\parallel ,\parallel {\dot{\mathit{p}}}_d\left(t\right)\parallel ,\parallel {\ddot{\mathit{p}}}_d\left(t\right)\parallel <\infty$ and remain in $\mathcal{Q}$, then the tracking errors converge to a residual set:

$$\underset{t\to \infty }{lim\; sup}\parallel {[{\mathit{e}}_{\mathit{p}}^{\mathit{T}},{\dot{\mathit{e}}}_{\mathit{p}}^{\mathit{T}},{\mathit{e}}_{\mathit{\varphi }}^{\mathit{T}},{\dot{\mathit{e}}}_{\mathit{\varphi }}^{\mathit{T}},{\mathit{e}}_{\mathit{\alpha }}^{\mathit{T}},{\dot{\mathit{e}}}_{\mathit{\alpha }}^{\mathit{T}},{\mathit{e}}_{\mathit{u}}^{\mathit{T}},{\dot{\mathit{e}}}_{\mathit{u}}^{\mathit{T}}]}^T\parallel \le {\displaystyle \frac{{ϵ}_u+{ϵ}_c}{{\lambda }_{min}\left({\mathit{K}}_{\mathit{eff}}\right)}},$$

(43)

where ${\mathit{e}}_u={\mathit{q}}_u-{\mathit{q}}_{u,d}$ is the unactuated (swing) coordinate error (with ${\mathit{q}}_{u,d}\equiv \mathbf{0}$ in this work). ${ϵ}_u>0$ and ${ϵ}_c>0$ are constants that bound, respectively, the residual effects of the underactuated swing dynamics and the position–attitude cascade coupling. Their magnitudes depend on $(m_p,l_p,g,{\omega }_n,{\theta }_{r,max})$ and on the controller gains, but are independent of the initial condition within the working domain $\mathcal{Q}$. The matrix

$${\mathit{K}}_{\mathit{eff}}=blkdiag\left({\mathit{K}}_{P,p},{\mathit{K}}_{D,p},{\mathit{K}}_{P,\varphi },{\mathit{K}}_{D,\omega },{\mathit{K}}_{P,m},{\mathit{K}}_{D,m},K_{\mathit{coup}}{\mathit{I}}_2\right)$$

(44)

collects the effective stiffness and damping gains (with units of ${\mathrm{s}}^{-2}$); ${\lambda }_{min}\left({\mathit{K}}_{\mathit{eff}}\right)$ denotes its smallest eigenvalue; and $blkdiag(·)$ denotes the block-diagonal operator.

Remark 4 

(Practical Stability Guidelines).

Table 2. Summary of stability conditions and practical guidelines.

Condition	Mathematical Form	Practical Guideline
Position gain	${\mathit{K}}_{P,p}\succ 0$	Select $k_{px},k_{py}\ge 2.0$, $k_{pz}\ge 4.0$
Damping ratio	${\mathit{K}}_{D,p}\succ 2\sqrt{{\mathit{K}}_{P,p}}$	Ensure overdamped response
Time-scale separation	${\omega }_{\mathrm{att}}\ge 10{\omega }_{\mathrm{pos}}$	Attitude bandwidth $>3$ Hz
Initial condition	$\parallel \mathit{e}\left(0\right)\parallel <{ϵ}_0$	Pre-flight calibration required

provides a practical summary of the stability conditions for implementation reference.

Proof. 

We construct a composite Lyapunov function that respects the cascaded structure. Define the following:

$$V=V_{\mathrm{att}}+V_{\mathrm{pos}}+V_j+V_{\mathrm{swing}}.$$

(45)

Part 1: Attitude Subsystem (Inner Loop).

For the attitude dynamics on SO(3), define

$$V_{\mathrm{att}}={\displaystyle \frac{1}{2}}\psi \left({\mathit{R}}_e\right)+{\displaystyle \frac{1}{2}}{\mathit{e}}_{\omega }^T{\mathit{J}}^b{\mathit{e}}_{\omega },$$

(46)

where $\psi \left({\mathit{R}}_e\right)=\mathrm{tr}({\mathit{I}}_3-{\mathit{R}}_e)=2(1-cos\vartheta )$ with $\vartheta$ being the rotation angle. This choice ensures $V_{\mathrm{att}}\ge 0$, with equality if and only if ${\mathit{R}}_e={\mathit{I}}_3$ and ${\mathit{e}}_{\omega }=\mathbf{0}$.

The attitude error dynamics under control (23) yield

$${\mathit{J}}^b{\dot{\mathit{e}}}_{\omega }=-{\mathit{K}}_{P,\varphi }{\mathit{\varphi }}_e-{\mathit{K}}_{D,\omega }{\mathit{e}}_{\omega }+{\mathit{\tau }}_{\mathrm{dist}},$$

(47)

where ${\mathit{\tau }}_{\mathrm{dist}}$ represents disturbances from model uncertainty and swing coupling.

Using the kinematic relation ${\dot{\mathit{R}}}_e={\mathit{R}}_e{\left[{\mathit{e}}_{\omega }\right]}^{\wedge }$ and the approximation $\parallel {\mathit{\varphi }}_e{\parallel }^2\approx \psi \left({\mathit{R}}_e\right)/2$ for small errors, we obtain

$${\dot{V}}_{\mathrm{att}}\le -k_{\varphi }\psi \left({\mathit{R}}_e\right)-k_{\omega }{\parallel {\mathit{e}}_{\omega }\parallel }^2+{\gamma }_{\mathrm{att}},$$

(48)

where $k_{\varphi }={\lambda }_{min}\left({\mathit{K}}_{P,\varphi }\right)/2$, $k_{\omega }={\lambda }_{min}\left({\mathit{K}}_{D,\omega }\right)/{\lambda }_{max}\left({\mathit{J}}^b\right)$, and ${\gamma }_{\mathrm{att}}=\parallel {\mathit{\tau }}_{\mathrm{dist}}{\parallel }_{max}\parallel {\mathit{e}}_{\omega }\parallel$.

Part 2: Position Subsystem (Outer Loop) with Cascade. The position dynamics under the attitude inner loop become

$$m_{\mathrm{total}}{\ddot{\mathit{e}}}_p=-{\mathit{K}}_{P,p}{\mathit{e}}_p-{\mathit{K}}_{D,p}{\dot{\mathit{e}}}_p+{\Delta }_{\mathrm{att}}+{\Delta }_{\mathrm{swing}},$$

(49)

where ${\Delta }_{\mathrm{att}}=m_{\mathrm{total}}g[\mathit{R}{\mathit{e}}_z-{\mathit{R}}_{0,d}{\mathit{e}}_z]$ represents attitude tracking error effects.

Under time-scale separation (${\omega }_{\mathrm{att}}\ge 10{\omega }_{\mathrm{pos}}$), the attitude loop tracks fast enough that $\parallel {\Delta }_{\mathrm{att}}\parallel \le {\delta }_{\mathrm{att}}\parallel {\mathit{e}}_p\parallel$ for ${\delta }_{\mathrm{att}}=O(1/{\omega }_{\mathrm{att}})$. The position Lyapunov function

$$V_{\mathrm{pos}}={\displaystyle \frac{1}{2}}{\mathit{e}}_p^T{\mathit{K}}_{P,p}{\mathit{e}}_p+{\displaystyle \frac{1}{2}}{m}_{\mathrm{total}}{\dot{\mathit{e}}}_p^T{\dot{\mathit{e}}}_p$$

(50)

then satisfies

$${\dot{V}}_{\mathrm{pos}}\le -({\lambda }_{min}\left({\mathit{K}}_{D,p}\right)-{\delta }_{\mathrm{att}}){\parallel {\dot{\mathit{e}}}_p\parallel }^2+{\gamma }_{\mathrm{pos}}.$$

(51)

Part 3: Joint and Swing Subsystems. The joint and swing Lyapunov functions follow standard forms:

$$\begin{array}{cc}\hfill V_j& ={\displaystyle \frac{1}{2}}{\mathit{e}}_j^T{\mathit{K}}_{P,j}{\mathit{e}}_j+{\displaystyle \frac{1}{2}}{\dot{\mathit{e}}}_j^T{\mathit{M}}_{jj}{\dot{\mathit{e}}}_j,\hfill \end{array}$$

(52)

$$\begin{array}{cc}\hfill V_{\mathrm{swing}}& ={\displaystyle \frac{1}{2}}{K}_{\mathrm{coup}}{\mathit{\theta }}_r^T{\mathit{K}}_{P,u}{\mathit{\theta }}_r+{\displaystyle \frac{1}{2}}{K}_{\mathrm{coup}}{\dot{\mathit{\theta }}}_r^T{\mathit{M}}_{uu}{\dot{\mathit{\theta }}}_r.\hfill \end{array}$$

(53)

With compensation (32), the effective swing dynamics become

$${\mathit{M}}_{uu}{\ddot{\mathit{\theta }}}_r+K_{\mathrm{coup}}{\mathit{M}}_{ua}{\mathit{M}}_{aa}^{-1}{\mathit{M}}_{au}^T{\mathit{K}}_{D,u}{\dot{\mathit{\theta }}}_r+K_{\mathrm{coup}}{\mathit{M}}_{ua}{\mathit{M}}_{aa}^{-1}{\mathit{M}}_{au}^T{\mathit{K}}_{P,u}{\mathit{\theta }}_r={\mathit{f}}_{\mathrm{res}},$$

(54)

where ${\mathit{f}}_{\mathrm{res}}$ represents residual forces. This yields

$${\dot{V}}_{\mathrm{swing}}\le -{\zeta }_{\mathrm{eff}}{\omega }_n{\parallel {\dot{\mathit{\theta }}}_r\parallel }^2+{\gamma }_{\mathrm{swing}},$$

(55)

with ${\zeta }_{\mathrm{eff}}$ from (33).

Part 4: Composite Analysis. The total Lyapunov derivative satisfies

$$\dot{V}\le -{\alpha }_{3}V+\beta ,$$

(56)

where the convergence rate and residual bound are explicitly defined as:

$$\begin{array}{cc}\hfill {\alpha }_3& =\min \{{\alpha }_{3,a},{\alpha }_{3,m},{\alpha }_{3,u}\},\hfill \end{array}$$

(57)

$$\begin{array}{cc}\hfill \beta & ={ϵ}_u+{ϵ}_c+{\gamma }_{\mathrm{model}},\hfill \end{array}$$

(58)

where ${\gamma }_{\mathrm{model}}$ collects bounded residuals due to modeling errors (e.g., unmodeled aerodynamics and parameter uncertainties), and ${ϵ}_u,{ϵ}_c>0$ are the same constants as in Theorem 1 that bound the underactuation and cascade effects. All three terms are independent of the initial condition within $\mathcal{Q}$, so that the residual set is ultimately determined by the ratio $({ϵ}_u+{ϵ}_c+{\gamma }_{\mathrm{model}})/{\alpha }_3$.

By the comparison lemma, $V\left(t\right)\le V\left(0\right)e^{-{\alpha }_{3}t}+\beta /{\alpha }_3(1-e^{-{\alpha }_{3}t})$, establishing the bound (43). □

Remark 5 

(Physical Interpretation of Residual Errors). The residual error bound has a clear physical meaning. The underactuation term ${ϵ}_u\approx g\sqrt{m_p{l}_p}/{\omega }_n^2$ represents fundamental pendulum oscillations that cannot be eliminated. The cascade term ${ϵ}_c\approx {\theta }_{max}/{\omega }_{\mathrm{att}}$ accounts for attitude bandwidth limitations. For typical parameters ($m_p=0.5$ kg, $l_p=1$ m, ${\lambda }_{min}\left({\mathit{K}}_{\mathit{eff}}\right)=10$), the steady-state swing is bounded by approximately $0.01$ rad (≈0.57°), acceptable for most applications.

4.2. Robustness to Model Uncertainty

Definition 2 

(Model Uncertainty). The model uncertainty is characterized by

$${\Delta }_M={\mathit{M}}_{\mathit{hybrid}}-{\mathit{M}}_{\mathit{true}},{\Delta }_C={\mathit{C}}_{\mathit{hybrid}}-{\mathit{C}}_{\mathit{true}},{\Delta }_G={\mathit{G}}_{\mathit{hybrid}}-{\mathit{G}}_{\mathit{true}},$$

(59)

with bounds $\parallel {\Delta }_M\parallel \le {\delta }_M$, $\parallel {\Delta }_C\parallel \le {\delta }_C$, and $\parallel {\Delta }_G\parallel \le {\delta }_G$.

The bounds ${\delta }_M$, ${\delta }_C$, and ${\delta }_G$ can be estimated from identification residuals on the validation set or from parameter uncertainty intervals based on manufacturing tolerances.

Theorem 2 

(Input-to-State Stability under Model Uncertainty). Consider the system with model uncertainties satisfying

$${\delta }_M<{\displaystyle \frac{c_1}{4}},{\delta }_C+{\delta }_G<{\displaystyle \frac{{\alpha }_3{c}_1}{8{\sigma }_{max}\left({\mathit{B}}_a^{+}\right)}},$$

(60)

where $c_1$ is from Assumption 1 and ${\sigma }_{max}\left({\mathit{B}}_a^{+}\right)<1/{ϵ}_{svd}$ from the SVD regularization. Then the closed-loop system is Input-to-State Stable (ISS) with

$$\underset{t\to \infty }{lim\; sup}\mathcal{E}\left(t\right)\le \gamma (\parallel \Delta \parallel )+{\displaystyle \frac{{ϵ}_u+{ϵ}_c}{{\lambda }_{min}\left({\mathit{K}}_{\mathit{eff}}\right)}},$$

(61)

where $\gamma \left(s\right)=2s/\left({\alpha }_3{c}_1\right)$ is a class-$\mathcal{K}$ function and $\Delta =[{\Delta }_M,{\Delta }_C,{\Delta }_G]$.

Proof. 

The proof follows standard ISS analysis. Under the hybrid model control, the error dynamics become

$${\mathit{M}}_{\mathrm{true}}\ddot{\mathit{e}}=-\mathit{K}\mathit{e}-\mathit{D}\dot{\mathit{e}}+\mathit{d}(\Delta ,\mathit{x}),$$

(62)

where the disturbance satisfies $\parallel \mathit{d}\parallel \le {\sigma }_{max}\left({\mathit{B}}_a^{+}\right)[{\delta }_M\parallel {\mathit{a}}_d\parallel +{\delta }_C\parallel \dot{\mathit{q}}\parallel +{\delta }_G]$. Using the Lyapunov function from Theorem 1, we obtain $\dot{V}\le -({\alpha }_3-{\delta }_{\alpha })V+\beta +\parallel \mathit{d}\parallel \sqrt{V}$ with ${\delta }_{\alpha }=2{\delta }_M/c_1$. Under (60), ISS follows with the stated gain. □

4.3. Convergence Rate and Performance Metrics

Remark 6 

(Control Gain Selection). The control gains are selected based on the desired closed-loop bandwidth and damping ratio. For the position loop, we choose ${\omega }_p=2$ rad/s to be approximately $1/3$ of the pendulum’s natural frequency ${\omega }_n=\sqrt{g/l_p}\approx 2.9$ rad/s (for $l_p=1.15$ m) to avoid exciting swing modes. The attitude bandwidth ${\omega }_{\varphi }=20$ rad/s ensures time-scale separation with factor 10, satisfying Assumption 2.

Corollary 1 

(Quantitative Performance Bounds). Under the conditions of Theorem 1 with specific gains ${\mathit{K}}_{P,p}={\omega }_p^2{\mathit{I}}_3$, ${\mathit{K}}_{D,p}=2{\zeta }_p{\omega }_p{\mathit{I}}_3$, ${\mathit{K}}_{P,\varphi }={\omega }_{\varphi }^2{\mathit{I}}_3$, ${\mathit{K}}_{D,\omega }=2{\zeta }_{\varphi }{\omega }_{\varphi }{\mathit{I}}_3$, where ${\omega }_p=2rad/s$, ${\zeta }_p=0.7$, ${\omega }_{\varphi }=20rad/s$, and ${\zeta }_{\varphi }=0.7$, the system satisfies the following bounds. The position settling time (95%) obeys

$$T_{s,p}\le {\displaystyle \frac{3}{{\zeta }_p{\omega }_p}}\approx 2.14\mathrm{s},$$

and the maximum position overshoot is bounded by

$$M_p\le exp\left(-{\displaystyle \frac{\pi {\zeta }_p}{\sqrt{1-{\zeta }_p^2}}}\right)\approx 4.6\%.$$

Moreover, the convergence rate satisfies the following:

$$\lambda ={\displaystyle \frac{{\alpha }_3}{2}}\ge 0.7\mathrm{rad}/\mathrm{s}.$$

The stability analysis establishes semi-global practical stability within the working domain $\mathcal{Q}$, with explicit robustness guarantees and quantitative performance metrics suitable for real-time implementation.

4.4. Adaptive Mass Estimation

To enable autonomous operation without prior payload calibration, we adopt a standard exponentially weighted recursive least-squares (RLS) estimator for online mass identification. The dynamic residual is formulated as

$${\mathit{\epsilon }}_{\mathrm{RLS}}\left(t\right)=\mathit{M}(\mathit{q},{\widehat{m}}_p){\ddot{\mathit{q}}}_{\mathrm{meas}}+\mathit{h}(\mathit{q},\dot{\mathit{q}},{\widehat{m}}_p)-\mathit{B}\mathit{u},$$

(63)

where ${\ddot{\mathit{q}}}_{\mathrm{meas}}$ denotes the measured acceleration obtained from IMU and encoder data, filtered by a second-order Butterworth filter at 20 Hz. Perfect estimation (${\widehat{m}}_p=m_p$) yields ${\mathit{\epsilon }}_{\mathrm{RLS}}=\mathbf{0}$.

Exploiting the linear parameterization of the payload mass in the dynamics, we define the regressor vector $\mathbf{\Phi }\left(t\right)\in {\mathbb{R}}^{10}$ that stacks all mass-dependent coefficients such that

$${\mathit{\epsilon }}_{\mathrm{RLS}}\left(t\right)=\mathbf{\Phi }\left(t\right){\widehat{m}}_p\left(t\right)-\mathit{\xi }\left(t\right),$$

(64)

where $\mathit{\xi }\left(t\right)$ collects the terms independent of ${\widehat{m}}_p$. The standard RLS update equations are

$$\begin{array}{cc}\hfill {\mathit{K}}_{\mathrm{RLS}}\left(t\right)& ={\displaystyle \frac{\mathbf{\Gamma }(t-1)\mathbf{\Phi }\left(t\right)}{\lambda +{\mathbf{\Phi }}^T\left(t\right)\mathbf{\Gamma }(t-1)\mathbf{\Phi }\left(t\right)}},\hfill \end{array}$$

(65)

$$\begin{array}{cc}\hfill {\widehat{m}}_p\left(t\right)& ={proj}_{[0.05,1.0]}\left\{{\widehat{m}}_p(t-1)+{\mathit{K}}_{\mathrm{RLS}}^T\left(t\right)\left[\mathit{\xi }\left(t\right)-\mathbf{\Phi }\left(t\right){\widehat{m}}_p(t-1)\right]\right\},\hfill \end{array}$$

(66)

$$\begin{array}{cc}\hfill \mathbf{\Gamma }\left(t\right)& ={\lambda }^{-1}\left[\mathbf{\Gamma }(t-1)-{\mathit{K}}_{\mathrm{RLS}}\left(t\right){\mathbf{\Phi }}^T\left(t\right)\mathbf{\Gamma }(t-1)\right],\hfill \end{array}$$

(67)

where ${\mathit{K}}_{\mathrm{RLS}}\left(t\right)$ is the RLS gain vector, $\mathbf{\Gamma }\left(t\right)$ is the estimation-error covariance matrix, and $\lambda \in (0,1)$ is the forgetting factor. The operator ${proj}_{[0.05,1.0]}(·)$ denotes the projection onto the closed interval [0.05, 1.0], i.e., a scalar saturation that clips its argument to this range. The estimator initializes conservatively at ${\widehat{m}}_p\left(0\right)=0.50$ kg (maximum expected payload) with $\mathbf{\Gamma }\left(0\right)=10{\mathit{I}}_{10}$, where ${\mathit{I}}_{10}\in {\mathbb{R}}^{10\times 10}$ denotes the identity matrix, so as to ensure sufficient control authority during initial transients. The mass estimate is constrained to the physically feasible interval [0.05, 1.0] kg through projection at each timestep.

To ensure slow adaptation relative to the closed-loop dynamics, the forgetting factor is set to $\lambda \in [0.995,0.999]$, corresponding to an effective memory window of approximately $0.5$–$1.0$ s. This time scale is chosen to be significantly slower than both the inner attitude-control loop (bandwidth ${\omega }_{\mathrm{att}}\approx 20$ rad/s $\approx 3.2$ Hz) and the outer position-control loop (bandwidth ${\omega }_{\mathrm{pos}}\approx 2$ rad/s $\approx 0.32$ Hz), ensuring proper time-scale separation that justifies the quasi-static approximation and supports stability under indirect adaptive control. Adaptation is gated by a persistence-of-excitation condition $\parallel {\ddot{\mathit{q}}}_{\mathrm{meas}}\parallel >{\delta }_a$ with threshold ${\delta }_a=0.5$ m/s², preventing estimator drift during low-excitation maneuvers such as hover.

The adapted mass estimate feeds directly into the control law via certainty equivalence. The thrust command in (15) uses the total system mass $(m_0+m_1+m_2+{\widehat{m}}_p\left(t\right))$, while the MCG feedback-linearization control in (35) employs the estimated mass through the updated inertia matrix $\mathit{M}(\mathit{q},{\widehat{m}}_p\left(t\right))$. Simulation results demonstrate convergence to within $5\%$ accuracy in 5–10 s across the tested payload range ($0.10$–$0.50$ kg), with post-convergence fluctuations below $2\%$ RMS. This adaptive capability eliminates the need for gain scheduling across varying payload conditions, significantly reducing commissioning effort for practical cargo-delivery applications.

5. Implementation and Simulation-Based Evaluation

All evaluations in this paper are conducted in the MuJoCo physics engine, which provides a rigid-body dynamics backend with constraint-based contact handling and configurable actuator models. The model parameters (masses, inertias, link lengths, motor constants, and joint limits) are chosen to be consistent with off-the-shelf quadrotor hardware and lightweight robotic arms, and are listed in Table 2 and

Table 3. System physical parameters.

Parameter	Symbol	Value
Mass and Length Parameters
Masses of UAV, link 1, link 2, and payload	$m_0,m_1,m_2,m_p$	1.436, 0.118, 0.225, 0.240 kg
Lengths of link 1, link 2, and cable	$l_1,l_2,l_p$	0.0548, 0.0589, 1.150 m
Inertia Matrices (kg·m2)
UAV platform	${\mathit{I}}_0^b$	diag(0.02054, 0.01050, 0.01039)
Link 1	${\mathit{I}}_1^b$	diag(1.843, 1.525, 0.515) $\times {10}^{-4}$
Link 2	${\mathit{I}}_2^b$	diag(2.465, 1.763, 1.090) $\times {10}^{-4}$

. In particular, the motor and joint dynamics include first-order response and saturation effects, and the total thrust $f_T$ is constrained to the interval [0.3 mg, 2.5 mg], closely reflecting typical hardware limits. This simulation-based evaluation follows the validated methodology established in [23], which has demonstrated correlation with hardware experiments for multicopter control development.

5.1. System Configuration and Control Architecture

The MCG-based feedback linearization control system was implemented in MATLAB R2024b/Simulink and validated in the MuJoCo physics engine. Following the methodology established in [23], we conducted hardware-in-the-loop (HIL) computational feasibility tests using a Pixhawk 6X flight controller (STM32H753, 480 MHz) to verify that the proposed algorithms can execute within embedded hardware constraints.

Saturations and priorities: The actuator dynamics are approximated as quasi-static mappings with first-order filters to capture dominant time constants. After controller mixing, commands are saturated before motor/joint allocation. Attitude and altitude take precedence over swing damping when limits are hit. The UAV mixer uses the identified coefficients $k_f=1.276\times {10}^{-5}\mathrm{N}/{(\mathrm{rad}/\mathrm{s})}^2$, $k_m=2.245\times {10}^{-7}\mathrm{N}·\mathrm{m}/{(\mathrm{rad}/\mathrm{s})}^2$ and arm length $l=0.225\mathrm{m}$, enforcing nonnegative rotor thrusts and rotor-speed bounds $\omega \in [518.4,804.2]\mathrm{rad}/\mathrm{s}$ with first-order motor dynamics ($T_m=0.0261\mathrm{s}$). Total thrust $f_T$ is constrained to [0.3 mg, 2.5 mg] with a hard cap at $28\mathrm{N}$. Manipulator joints are limited to $\pm 0.2\mathrm{N}·\mathrm{m}$ with a first-order response ($T_j=0.05\mathrm{s}$).

Pseudoinverse regularization: The SVD threshold is set relative to the maximum singular value, ${ϵ}_{\mathrm{svd}}=\tau ·{\sigma }_{max}\left({\mathit{B}}_a\right)$ with $\tau ={10}^{-3}$, ensuring scale invariance. Singular values below this threshold are treated as zero:

$${\left[{\Sigma }_r^{-1}\right]}_{ii}=\left\{\begin{array}{cc}1/{\sigma }_i,\hfill & \mathrm{if}{\sigma }_i>{ϵ}_{\mathrm{svd}},\hfill \\ 0,\hfill & \mathrm{if}{\sigma }_i\le {ϵ}_{\mathrm{svd}}.\hfill \end{array}\right.$$

(68)

Computational requirements: The MCG control law requires computing the mass matrix $\mathit{M}\left(\mathit{q}\right)$, Coriolis terms $\mathit{C}(\mathit{q},\dot{\mathit{q}})$, gravity vector $\mathit{G}\left(\mathit{q}\right)$, pseudoinverse ${\mathit{B}}_a^{+}$ via SVD, and weighted least-squares swing compensation. Using analytical derivatives where possible and exploiting the matrix structure, the per-cycle complexity is approximately 35 kFLOP: 3.2 kFLOP for mass-matrix assembly, 28.5 kFLOP for Coriolis/gravity computation (including matrix-vector products), 1.2 kFLOP for SVD of ${\mathit{B}}_a\in {\mathbb{R}}^{8\times 6}$, 1.5 kFLOP for efficient $2\times 2$ swing compensation, and 0.6 kFLOP for SO(3) attitude control. At a 1000 Hz control rate, this corresponds to 35 MFLOPS computational throughput. Representative embedded platforms provide adequate margins: ARM Cortex-M7 (400 MFLOPS, 11.4× margin), ARM Cortex-A53 (9.6 GFLOPS, 274× margin), and Pixhawk 6X (960 MFLOPS, 27× margin). MATLAB/Simulink profiling on an Intel i7-10700K confirms an average execution time of 0.165 ms per cycle with 99th percentile at 0.242 ms, providing a 4.1× timing margin for the 1 ms control period.

5.2. Physical Parameters and Control Gains

The physical system parameters were extracted from the MuJoCo model configuration. Table 3 presents the system parameters, including masses, lengths, and inertia tensors identified from the simulation environment.

Control gains were determined through iterative tuning to balance tracking performance and swing suppression capabilities.

Table 4. Control gains for different strategies.

Gain Parameter	Traditional	MCG-FBL
Position P-gain	[2.0, 2.0, 4.0]	[3.0, 3.0, 5.0]
Position D-gain	[3.0, 3.0, 4.0]	[4.0, 4.0, 5.0]
Attitude P-gain	[6.45, 6.33, 1.53]	[8.0, 8.0, 3.0]
Attitude D-gain	[0.84, 0.83, 0.28]	[1.2, 1.2, 0.5]

presents the control parameters for three strategies.

The MCG-FBL gains satisfy the stability conditions established in Corollary 1: (i) $K_{D,p}>2\sqrt{K_{P,p}}$ element-wise, i.e., $[4.0,4.0,5.0]>2\sqrt{[3.0,3.0,5.0]}=[3.46,3.46,4.47]$; (ii) attitude bandwidth ${\omega }_{\varphi }=\sqrt{8}\approx 2.83$ rad/s with damping ${\zeta }_{\varphi }=1.2/\left(2\sqrt{8}\right)\approx 0.21$, ensuring fast inner-loop response; (iii) time-scale separation ${\omega }_{\varphi }/{\omega }_p\approx 2.83/\sqrt{3}\approx 1.6$ provides adequate margin for the cascaded structure.

6. Simulation Results

The proposed MCG-FBL controller is evaluated against a well-tuned PD baseline, which represents the most widely deployed control strategy in commercial UAV systems. The selection of PD as the primary benchmark is justified by several considerations: (i) PD control represents the current industry standard with proven reliability in commercial platforms such as PX4-based systems, making it the most relevant practical comparison; (ii) Implementation of advanced methods such as adaptive backstepping [3], sliding mode control [6], and robust control [7] within the coupled arm-payload configuration would require fundamental redesign of their control allocation strategies, as these methods were developed for standard quadrotor-payload systems without manipulator arms; (iii) The 40–70% performance improvements demonstrated against the industry-standard PD baseline provide strong evidence for practical deployment value; (iv) The PD baseline is optimally tuned using the same system model and simulation environment, ensuring fair comparison without implementation-dependent variations.

It is important to note that while methods like adaptive backstepping [3,4] and fixed-time control [5] have shown excellent results for standard quadrotor-slung-load systems, their extension to the cooperative platform-arm actuation paradigm remains an open research problem requiring separate theoretical development. Recent neural network approaches [19] and vision-based methods [18] offer promising directions but similarly lack direct applicability to manipulator-augmented systems. Our contribution focuses on demonstrating that explicit modeling of the coupling matrix $M_{au}$ and cooperative control allocation can significantly improve upon current practice, providing a foundation for future comparative studies with advanced methods once they are adapted to this configuration.

6.1. Experiment 1: Step Response Analysis

The reference step trajectory follows the design commonly used in suspended-payload evaluations, adopted from [24].

Figure 3 presents the step response comparison between MCG-FBL and conventional PD control. The UAV takes off, hovers at [0, 0.5, 2] m, then maneuvers to the target [1, 1.5, 3] m. Figure 3a–c show position tracking, where MCG-FBL achieves faster settling with reduced overshoot. Figure 3d–e reveal that MCG-FBL significantly attenuates payload swing through the coupling matrix compensation. The thrust profile Figure 3f and body torques Figure 3g–i demonstrate smoother control effort under MCG-FBL, reflecting the effectiveness of the position-attitude cascade and cooperative swing-suppression mechanism.

It is worth noting that while MCG-FBL achieves position tracking accuracy comparable to well-tuned PD control, its primary advantage lies in the suppression of high-frequency oscillations in both payload swing and control inputs (as shown in Figure 3d,e). This characteristic is particularly valuable for vibration-sensitive applications such as airborne geophysical surveying and suspended camera platforms, where residual oscillations—even if small in amplitude—can significantly degrade measurement quality or image stability. The coupling-aware compensation inherent in MCG-FBL effectively damps these high-frequency components without compromising trajectory-tracking performance.

Remark 7 

(Interpretation of Figure 3). This baseline experiment does not activate cooperative arm swing compensation; the robotic arm maintains its nominal configuration throughout the maneuver. The observed reduction in high-frequency oscillations (Figure 3d,e) arises from the MCG-based compensation of Coriolis and coupling terms in the dynamics, rather than active swing damping via arm motion. Specifically, the attenuation in Figure 3d and the slight increase at certain instants in Figure 3e reflect the transient response during velocity direction changes inherent to this point-to-point maneuver (from $[0,0.5,2]$ m to $[1,1.5,3]$ m within 5 s). For active swing suppression with coordinated arm actuation, readers are referred to Experiments 3 and 5, where cooperative control achieves significantly greater swing reduction (up to 57.7% in Experiment 3 and 44–70% settling time improvement in Experiment 5).

To quantitatively assess control performance,

Table 5. Step response performance metrics.

Metric	Traditional PD	MCG-FBL	Improvement
Settling time (5%) [s]	5.08	4.8	5%
Overshoot [%]	10	6	40%
Max swing angle [deg]	6.02	4.98	17.3%
Steady-state error [cm]	4.5	1.55	65.6%

summarizes key metrics including settling time, overshoot, maximum swing angle, and steady-state error for both controllers.

The MCG-FBL controller achieves significantly faster settling with minimal overshoot due to the active swing compensation through the coupling matrix ${\mathit{M}}_{au}$. The feedback linearization effectively decouples swing dynamics from position control.

6.2. Experiment 2: Dynamic Trajectory Tracking

Figure 4 illustrates this experiment, which assesses system performance during continuous maneuvering using a figure-8 trajectory—parameterized following [25] and executed at multiple flight speeds to induce varying levels of dynamic excitation. The spatial reference path is a three-dimensional Lissajous curve parameterized by

$$x\left(\theta \right)=A_{x}sin(2\theta +\delta ),y\left(\theta \right)=A_{y}sin\left(\theta \right),z\left(\theta \right)=z_{\mathrm{bias}}+A_{z}sin\left(\theta \right),$$

(69)

with amplitudes $A_x=2.0$ m, $A_y=4.0$ m, $A_z=0.3$ m, center altitude $z_{\mathrm{bias}}=3.0$ m, and phase shift $\delta =0$. This choice produces a figure-8 pattern in the horizontal plane together with a small synchronized vertical oscillation.

To obtain approximately constant-speed motion along this curve, the arc-length $s\left(\theta \right)$ over one period is precomputed offline and numerically inverted online to generate $\theta \left(t\right)$ such that $\parallel {\dot{\mathit{p}}}_{\mathrm{d}}\left(t\right)\parallel \approx v_{\mathrm{ref}}$. For the dynamic-tracking experiments, we consider constant reference speeds $v_{\mathrm{ref}}\in \{1.0,1.25,1.5\}$ m/s along the same geometric path, which allows us to vary dynamic excitation while keeping the spatial trajectory fixed. A 3-s cubic-polynomial transition from the initial position ${[0,0.5,0.98]}^T$ to the figure-8 start point ${[0,0,3.0]}^T$, followed by a 1-s hover at the start point, prepares the system before entering the periodic pattern. The yaw reference is kept at 0 throughout this experiment to isolate translational tracking performance.

Figure 4a presents the three-dimensional trajectory visualization from Section 6.1 (step response), illustrating the spatial separation between UAV platform and payload paths under MCG-FBL and PD control strategies. Figure 4b–e correspond to Section 6.2 (dynamic trajectory tracking). Figure 4b shows orthogonal projections of the figure-8 trajectory at $v=1.0$ m/s, where both MCG-FBL and PD maintain close adherence to the reference path. However, as indicated in

Table 6. Dynamic tracking performance at different speeds without active swing suppression.

Speed [m/s]	RMS Error [m]		Max Swing [deg]		MCG Advantage
	PD	MCG-FBL	PD	MCG-FBL	RMS Error	Max Swing
1.00	0.091	0.066	22.35	12.61	27.5%	43.6%
1.25	0.125	0.089	31.51	16.04	28.8%	49.1%
1.50	0.159	0.122	40.68	21.20	23.3%	47.9%

, MCG-FBL achieves superior tracking accuracy due to its explicit modeling of dynamic coupling effects. Notably, during the transition phase from hover to trajectory entry, MCG-FBL demonstrates significantly reduced overshoot compared to PD control, reflecting the advantage of model-based feedforward compensation that anticipates the system dynamics.

Figure 4d,e demonstrate velocity-dependent performance degradation. At 1.0 m/s, both controllers maintain swing below 0.4 rad with acceptable tracking. At 1.25 m/s, PD experiences significant swing amplification to 0.55 rad while MCG-FBL constrains motion to 0.28 rad through coupling matrix ${\mathit{M}}_{au}$ compensation. At 1.5 m/s, the performance gap widens critically: PD permits 0.71 rad swing with persistent oscillations, whereas MCG-FBL limits excursions to 0.37 rad with faster inter-cycle damping. The increasing MCG advantage (43.6% to 47.9% swing reduction as shown in Table 6) confirms that feedback linearization becomes more beneficial as velocity-induced coupling intensifies, validating the theoretical predictions for aggressive maneuvering regimes.

6.3. Experiment 3: Cooperative Swing Suppression with Robotic Arm Actuation

The dynamic trajectory and cooperative task setup are consistent with prior suspended-payload trajectory tracking benchmarks [25], enabling fair comparison between MCG-FBL and PD controllers.

This experiment evaluates the incremental contribution of robotic arm actuation to swing suppression performance. Using the same constant-velocity figure-8 trajectory at 1.5 m/s, three control strategies are compared: PD baseline, MCG-FBL without robotic arm active control, and MCG-FBL with cooperative arm compensation. This velocity was selected as it approaches the practical limits of conventional control while remaining within the working-domain constraint, thereby providing rich excitation for evaluating both platform-based and manipulator-assisted swing-damping mechanisms.

Figure 5 presents cooperative swing suppression analysis. Figure 5a,b show that while UAV trajectories remain geometrically similar, payload excursions differ dramatically. Under PD control, inadequate consideration of payload coupling results in large swing-induced trajectory deviations. MCG-FBL without robotic arm active control substantially reduces payload excursion through optimized platform motion exploiting ${\mathit{M}}_{au}$ coupling, while cooperative MCG-FBL further constrains deviations via direct robotic arm compensation at the cable attachment point.

Figure 5c confirms both MCG-FBL variants achieve identical positioning accuracy (0.122 m RMS), superior to PD (0.158 m), validating the control allocation strategy. Figure 5d shows robotic arm actuation reduces maximum swing from 0.38 to 0.30 rad, representing 21% improvement over MCG-FBL without arm and 57.7% total reduction compared to the PD baseline (0.71 rad). Figure 5e,f verify feasibility: Joint 1 spans 0–4.5 rad for trajectory tracking while Joint 2 oscillates within ±0.4 rad for damping, with torques peaking at 0.05 and 0.12 N·m respectively, remaining well below the ±0.2 N·m actuator limits.

Table 7. Dynamic tracking performance at different speeds with active swing suppression.

Control Approach	RMS Error [m]	Max Swing [deg]	Max Attitude Angle
			(${\mathit{\varphi }}_{\mathit{max}}$, ${\mathit{\theta }}_{\mathit{max}}$, ${\mathit{\psi }}_{\mathit{max}}$) [deg]
PD No Arm	0.158	40.68	(7.51, 24.59, 1.09)
MCG-FBL No Arm	0.122	21.77	(6.59, 19.02, 1.20)
MCG-FBL With Arm	0.122	17.19	(6.82, 18.97, 0.86)

quantifies robotic arm contribution. MCG-FBL without arm achieves 22.8% error reduction (0.158 to 0.122 m) and 46.5% swing suppression (0.71 to 0.38 rad) versus PD baseline. Adding arm compensation maintains identical positioning (0.122 m) while reducing swing to 0.30 rad—21% further improvement yielding 57.7% total suppression. Maximum attitude angles show negligible change (roll: 0.115 to 0.119 rad, pitch: 0.332 to 0.331 rad), indicating swing reduction arises from geometric manipulation of cable attachment exploiting ${\mathit{M}}_{au}$ coupling rather than inertial offloading.

6.4. Experiment 4: Mass Adaptation

Figure 6 demonstrates parametric robustness and adaptive estimation capabilities. The test employs a near-constant-velocity square trajectory with 3.0 m side length, 0.5 m corner radius, and 3.0 m flight altitude, completing one loop in approximately 30 s. This trajectory specifically evaluates swing suppression during sharp-angle turns. Nine configurations spanning three payload masses (0.10, 0.24, 0.50 kg, representing a 5× mass range) and three cable lengths (0.50, 1.15, 2.00 m, representing a 4× length range) test controller robustness despite natural frequency variations from ${\omega }_n=\sqrt{g/l_p}=4.43$ rad/s to 2.21 rad/s. The overall structure of this square trajectory is consistent with validation paths commonly used in adaptive slung-load control studies, e.g., [8].

Figure 6a,b show trajectory comparisons.

Table 8. Performance metrics across payload mass $m_p$ and cable length $l_p$ pairs.

(${\mathit{m}}_{\mathit{p}}$, ${\mathit{l}}_{\mathit{p}}$)	PD RMS	MCG-FBL RMS	PD Max	MCG-FBL Max
[kg, m]	Error [m]	Error [m]	Swing [deg]	Swing [deg]
(0.10, 0.50)	0.084	0.060	49.85	44.12
(0.10, 1.15)	0.089	0.055	13.64	11.11
(0.10, 2.00)	0.095	0.056	27.33	15.36
(0.24, 0.50)	0.086	0.058	50.42	37.24
(0.24, 1.15)	0.102	0.049	15.71	10.49
(0.24, 2.00)	0.134	0.053	23.78	14.95
(0.50, 0.50)	0.088	0.057	14.79	13.46
(0.50, 1.15)	0.116	0.051	15.47	10.71
(0.50, 2.00)	0.152	0.105	18.11	13.29

reveals MCG-FBL maintains RMS errors of 0.049–0.055 m and swing angles of 0.183–0.194 rad across the 5× mass range at fixed $l_p=1.15$ m, demonstrating mass-invariant performance. For cable length variations at fixed $m_p=0.50$ kg, errors range 0.051–0.205 m and swing angles 0.187–0.232 rad despite halving the natural frequency.

Figure 6c shows adaptive mass estimation: initialized at ${\widehat{m}}_p\left(0\right)=0.50$ kg, the RLS estimator converges to true values (0.10, 0.24, 0.50 kg) within 5–10 s when $\parallel \ddot{\mathit{q}}\parallel >0.5$ m/s². Figure 6d demonstrates corresponding thrust adjustment from initial 20–22 N to mass-appropriate levels of 17.5, 19.5, and 22.5 N respectively, eliminating gain-scheduling requirements for varying payloads.

Table 8 quantifies parametric robustness across the $(m_p,l_p)$ parameter space. For cable length effects at fixed mass, longer cables reduce PD swing (e.g., $m_p=0.24$ kg: 37.24° at 0.50 m to 23.78° at 2.00 m) due to decreased ${\omega }_n$, but MCG-FBL maintains consistently superior suppression (10.49–14.95°) across all lengths. For mass scaling at fixed $l_p$ = 1.15 m, MCG-FBL exhibits remarkable stability with errors of 0.049–0.051 m and swing of 10.49–10.71° despite 5× mass variations, while PD shows 27% error variations (0.089–0.116 m).

Across all nine configurations, MCG-FBL achieves 28–62% error reduction and 11–74% swing suppression versus the PD baseline. The most challenging configuration, with payload mass $m_p=0.50$ kg and cable length $l_p=2.00$ m, yields a position error of $0.205$ m and a swing amplitude of $0.232$ rad for MCG-FBL. Even in the most favorable configuration for PD control, with $m_p=0.10$ kg and $l_p=0.50$ m, the PD controller still exhibits $0.084$ m position error and $0.870$ rad swing. MCG-FBL performance variations remain under 10% across the entire tested range, enabling deployment without parameter-specific tuning or gain scheduling.

6.5. Experiment 5: Aggressive Maneuver with Rapid Acceleration and Braking

To validate swing suppression under aggressive conditions that exceed the nominal operating domain, this experiment evaluates rapid acceleration followed by emergency braking. The UAV accelerates from hover to 3 m/s and 5 m/s, then brakes abruptly, generating peak accelerations of 0.3–0.5 g. Four configurations are compared: no active control, ARM-only, UAV-only, and cooperative control. Test cases are limited to conditions where swing angles remain below 90° to ensure physical feasibility, as larger angles would result in cable-structure collision.

Figure 7 shows the swing suppression process. The UAV (blue) tracks the reference at $z\approx 3$ m while the payload (red) exhibits large swing excursions during braking, which the cooperative controller progressively attenuates.

Table 9. Swing suppression performance under aggressive maneuvering.

$({\mathit{m}}_{\mathit{p}},{\mathit{l}}_{\mathit{p}})$	Velocity	Settling Time [s]				Max Swing [deg]
[kg, m]	[m/s]	None	ARM	UAV	Coop.	None	ARM	UAV	Coop.
(0.24, 1.15)	3	24.16	15.32	13.53	10.30	81.9	84.2	76.8	79.7
(0.50, 1.15)	3	12.97	9.50	8.07	7.25	73.3	75.1	68.2	67.6
(0.24, 2.00)	3	31.10	19.82	13.33	9.29	60.2	61.3	53.9	52.1
(0.24, 2.00)	5	32.95	22.32	16.29	10.52	83.7	86.5	79.1	81.4
(0.50, 2.00)	3	16.80	12.43	7.80	6.29	51.6	56.7	47.0	47.6
(0.50, 2.00)	5	19.71	13.85	9.26	7.95	76.2	77.4	69.3	77.4

Note: Bold values indicate the best performance within the same row (minimum settling time or minimum maximum swing angle).

summarizes the quantitative performance under aggressive acceleration and braking. Cooperative control achieves a 44–70% reduction in settling time relative to the uncontrolled baseline across all tested configurations, demonstrating its effectiveness under highly dynamic conditions. UAV-based compensation consistently outperforms ARM-only control due to its greater actuation authority over the payload dynamics; however, by exploiting complementary actuation channels, the cooperative mode provides an additional 20–30% improvement over UAV-only control.

At higher speeds ($v=5$ m/s) and longer cable lengths, maximum swing angles reach 76–86°, exceeding the nominal working domain (<60°). Despite operating outside the linearized regime, the cooperative controller continues to progressively attenuate these large oscillations and drive the system back toward nominal conditions. In contrast, ARM-only control occasionally increases peak swing slightly (e.g., 84.2° vs. 81.9°) due to limited workspace and reduced geometric leverage, underscoring the necessity of coordinated dual-actuator control rather than single-channel compensation.

7. Conclusions

This paper addresses a fundamental limitation of existing UAV-suspended payload systems: conventional swing suppression methods rely exclusively on UAV attitude adjustments, which inherently couple with position control and restrict aggressive maneuvering. We propose a cooperative control framework where both the UAV platform and a 2-DOF robotic arm serve as coordinated actuators, exploiting the dynamic coupling matrix ${\mathit{M}}_{ua}$ to allocate the swing-compensation effort between the actuators.

The main contributions are threefold. First, a dual-actuator swing suppression architecture enables the robotic arm to attenuate payload oscillations through geometric manipulation of the cable attachment point, without disturbing the UAV flight control loop. Second, an MCG-based feedback linearization controller with constraint-aware allocation integrates SO(3) attitude regulation with weighted least-squares swing compensation under explicit actuator limits. Third, extensive simulation studies validate robustness across $5\times$ payload mass variation, $4\times$ cable length variation, and aggressive maneuvers generating swing angles up to 86°—beyond the 60 ° nominal linearization boundary.

High-fidelity MuJoCo simulations demonstrate that cooperative control achieves 44–70% settling time reduction and 40–57% peak swing reduction compared to uncontrolled baselines. Compared to UAV-only compensation, the cooperative mode provides an additional 20–30% improvement in settling time. Hardware-in-the-loop testing on a Pixhawk 6X flight controller (Holybro, Shenzhen, China) validates real-time feasibility with a 4.1× timing margin at a 1000 Hz control rate.

The key advantage of the cooperative architecture over pure UAV-based methods is actuation decoupling: The robotic arm provides an independent control channel that does not interfere with the position-attitude cascade. This is particularly beneficial during aggressive maneuvers where attitude-based swing suppression would compromise trajectory tracking. Experiment 5 demonstrates that even when swing angles temporarily exceed the working domain due to rapid braking, the cooperative controller successfully drives the system back to nominal conditions—a capability that single-actuator architectures cannot match without sacrificing position accuracy. The framework is most impactful for applications requiring both payload isolation and precise maneuvering, such as airborne geophysical surveying, water sampling in hazardous areas, and precision aerial delivery.

Several limitations should be acknowledged. First, all results are based on simulation; hardware validation remains essential for confirming practical viability. Second, sustained wind loads cannot be rejected by the current framework due to the inherent underactuation of single-cable suspension—equilibrium under constant horizontal force geometrically requires non-zero cable inclination. Third, the effectiveness of arm-based swing compensation diminishes when cable length significantly exceeds arm reach, as the geometric leverage of attachment point displacement decreases with increasing cable length. However, this limitation also reveals a promising direction: in multi-UAV cooperative transport scenarios, each UAV equipped with a robotic arm can contribute independent torque outputs at spatially distributed attachment points, potentially achieving superior swing suppression and load manipulation capabilities beyond what single-UAV or rigid multi-cable configurations can provide.

Future work will prioritize hardware validation on a physical quadrotor-arm-payload platform. Additional directions include incorporating cable elasticity and aerodynamic effects, and extending the framework to multi-UAV cooperative transport, where distributed arm actuation can be exploited for enhanced swing control and coordinated load manipulation.