The coordinated transportation of payloads using a fleet of unmanned aerial vehicles (UAVs), specifically quadrotors, presents a promising paradigm for extending operational capabilities beyond the limits of a single agent. This multi-unmanned drone system offers significant advantages in terms of payload capacity, operational redundancy, and adaptability to complex or bulky cargo. However, the control problem is inherently challenging due to the strongly coupled, nonlinear dynamics involving multiple unmanned drones and a slung-load, compounded by practical constraints like actuator faults, input saturation, and persistent external disturbances such as wind gusts and aerodynamic downwash.
This paper addresses the formation control problem for a cooperative unmanned drone slung-load system. We propose an integrated control framework that combines an Improved Prescribed Performance Control (IPPC) strategy, an Integral Backstepping Sliding Mode Controller (IBSMC), and an Extended Disturbance Observer (EDO). The core objective is to ensure precise trajectory tracking and rigid formation-keeping for the unmanned drone fleet while strictly maintaining safe inter-agent distances, despite the presence of the aforementioned internal and external challenges. The IPPC guarantees that all tracking errors evolve within designer-specified performance bounds. The IBSMC provides robustness, and the EDO actively estimates and compensates for aggregated disturbances, leading to a cooperative control solution with enhanced stability and disturbance rejection.
1. Dynamic Modeling of the Multi-Unmanned Drone Slung-Load System
We consider a system comprising \(n\) quadrotor unmanned drones collaboratively transporting a point-mass payload via massless, rigid cables of constant length \(L_i\). The configuration is depicted in the conceptual figure, where each unmanned drone is connected to the shared load.
1.1 Quadrotor Unmanned Drone Model
A standard “X” configuration quadrotor unmanned drone is modeled as a rigid body. Let the inertial frame be \(\{O_g, x_g, y_g, z_g\}\) and the body-fixed frame be \(\{O_b, x_b, y_b, z_b\}\). The translational dynamics in the inertial frame and the rotational dynamics using Euler angles \(\boldsymbol{\Theta}_i = [\phi_i, \theta_i, \psi_i]^T\) are given by:
$$
\begin{aligned}
\ddot{x}_i &= \frac{1}{m_i}(u_{xi}\cos\psi_i\sin\theta_i\cos\phi_i + u_{xi}\sin\psi_i\sin\phi_i) \\
\ddot{y}_i &= \frac{1}{m_i}(u_{yi}\sin\psi_i\sin\theta_i\cos\phi_i – u_{yi}\cos\psi_i\sin\phi_i) \\
\ddot{z}_i &= \frac{1}{m_i}(U_{1i}\cos\theta_i\cos\phi_i) – g \\
\ddot{\phi}_i &= \frac{J_{yi}-J_{zi}}{J_{xi}}\dot{\theta}_i\dot{\psi}_i + \frac{l}{J_{xi}}U_{2i} \\
\ddot{\theta}_i &= \frac{J_{zi}-J_{xi}}{J_{yi}}\dot{\phi}_i\dot{\psi}_i + \frac{l}{J_{yi}}U_{3i} \\
\ddot{\psi}_i &= \frac{J_{xi}-J_{yi}}{J_{zi}}\dot{\phi}_i\dot{\theta}_i + \frac{1}{J_{zi}}U_{4i}
\end{aligned}
$$
where \(m_i\) is the mass, \(J_{xi}, J_{yi}, J_{zi}\) are moments of inertia, \(l\) is the arm length, \(g\) is gravity, and \(\mathbf{U}_i = [u_{xi}, u_{yi}, U_{1i}, U_{2i}, U_{3i}, U_{4i}]^T\) are the control inputs derived from rotor thrusts.
1.2 Constrained System Modeling via Udwadia-Kalaba (UK) Equation
The key to modeling the cooperative unmanned drone system is capturing the geometric constraints imposed by the cables. Let \(\boldsymbol{\zeta}_i = [x_i, y_i, z_i]^T\) and \(\boldsymbol{\zeta}_l = [x_l, y_l, z_l]^T\) be the positions of the \(i\)-th unmanned drone and the load, respectively. The vector from the load to the \(i\)-th unmanned drone is \(\mathbf{L}^e_i = \boldsymbol{\zeta}_i – \boldsymbol{\zeta}_l\). The rigid cable constraint is:
$$
g_{ci} = \mathbf{L}^{eT}_i \mathbf{L}^e_i – L_i^2 = 0.
$$
The Udwadia-Kalaba (UK) formulation provides an elegant framework for incorporating such constraints directly into the equations of motion without Lagrange multipliers. Defining the unconstrained acceleration vector \(\mathbf{q}_u\) and the constrained acceleration vector \(\mathbf{q}\), the constrained dynamics are given by:
$$
\mathbf{q} = \mathbf{q}_u + \mathbf{M}^{-1/2}\mathbf{V}^T(\mathbf{V}\mathbf{M}^{-1}\mathbf{V}^T)^{\dagger}(\mathbf{W} – \mathbf{V}\mathbf{q}_u)
$$
where \(\mathbf{M}\) is the mass matrix, \(\mathbf{V}\) and \(\mathbf{W}\) are derived from the constraint equations. To ensure robust constraint satisfaction (e.g., preventing slack cables from inducing instability), a stabilized constraint form is used: \(\ddot{g}_{ci} + \zeta_{1} \dot{g}_{ci} + \zeta_{2} g_{ci} = 0\), with \(\zeta_{1}, \zeta_{2} > 0\). The final compact model for the \(i\)-th unmanned drone and the load, accounting for disturbances \(\mathbf{d}_i\), can be expressed as:
$$
\ddot{\boldsymbol{\zeta}}_i = \mathbf{f}_i(\boldsymbol{\zeta}_i, \boldsymbol{\zeta}_l) + \mathbf{g}_i(\boldsymbol{\zeta}_i, \boldsymbol{\zeta}_l)\mathbf{U}_i + \mathbf{d}_i(t)
$$
where \(\mathbf{f}_i(\cdot)\) and \(\mathbf{g}_i(\cdot)\) encapsulate the nonlinear and input coupling terms derived from the UK formulation, and \(\mathbf{d}_i(t)\) represents aggregated unknown disturbances (wind, model uncertainties, etc.).
2. Control Framework Design
The overall control architecture is designed in a hierarchical manner. A cooperative formation algorithm generates desired trajectories for each unmanned drone based on the load’s reference path. An Improved Prescribed Performance Control (IPPC) module transforms the tracking errors to guarantee transient and steady-state performance. An Integral Backstepping Sliding Mode Controller (IBSMC) provides the primary control action, augmented by an Extended Disturbance Observer (EDO) for online disturbance estimation and compensation.
2.1 Distributed Cooperative Formation Algorithm
The communication network among the \(n\) unmanned drones is described by a directed graph \(\mathcal{G}=(\mathcal{V},\mathcal{E})\). The load’s desired trajectory \(\boldsymbol{\zeta}_{l}^d(t)\) acts as a virtual leader. The desired position for the \(i\)-th unmanned drone, \(\boldsymbol{\zeta}_{i}^d(t)\), is generated by the following distributed consensus-based protocol:
$$
\boldsymbol{\zeta}_{i}^d = \boldsymbol{\zeta}_{l}^d + \boldsymbol{\delta}_i – c_w(\boldsymbol{\zeta}_{i}^d – \boldsymbol{\zeta}_{l}^d – \boldsymbol{\delta}_i) – \sum_{j \in \mathcal{N}_i} a_{ij}[(\boldsymbol{\zeta}_{i}^d – \boldsymbol{\delta}_i) – (\boldsymbol{\zeta}_{j}^d – \boldsymbol{\delta}_j)]
$$
where \(\boldsymbol{\delta}_i\) defines the desired relative offset of the unmanned drone from the load in the formation, \(c_w>0\) is the leader-follower coupling weight, and \(a_{ij}\) are elements of the adjacency matrix. This algorithm ensures that the unmanned drone team maintains the predefined geometric formation \(\boldsymbol{\delta}_i\) while collectively tracking the load’s trajectory.
2.2 Improved Prescribed Performance Control (IPPC)
The core idea of Prescribed Performance Control (PPC) is to enforce the tracking error \(e_{i,k}(t)\) (for the \(k\)-th state of the \(i\)-th unmanned drone) to evolve within a predefined performance envelope:
$$
-\delta_{i,k} \rho_{i,k}(t) < e_{i,k}(t) < \rho_{i,k}(t), \quad \text{if } e_{i,k}(0) > 0
$$
$$
-\rho_{i,k}(t) < e_{i,k}(t) < \delta_{i,k} \rho_{i,k}(t), \quad \text{if } e_{i,k}(0) < 0
$$
where \(0 < \delta_{i,k} \le 1\) and \(\rho_{i,k}(t)\) is a smooth, bounded, and decreasing performance function. The traditional choice \(\rho(t) = (\rho_0 – \rho_{\infty})e^{-\alpha t} + \rho_{\infty}\) has limitations: a tight bound may cause input saturation for large initial errors, while a loose bound compromises steady-state accuracy.
We propose an Improved Prescribed Performance Function (IPPF):
$$
\rho_{i,k}(t) = (\rho_{0_{i,k}} – \rho_{\infty_{i,k}}) \left[ \frac{1}{1+\gamma_{1_{i,k}}\exp(-\gamma_{2_{i,k}}|e_{i,k}|^{\gamma_{3_{i,k}}})} \right] e^{-\alpha_{i,k} t} + \rho_{\infty_{i,k}}
$$
The key innovation is the term \(\frac{1}{1+\gamma_{1}\exp(-\gamma_{2}|e|^{\gamma_{3}})}\), which makes the performance boundary \(\rho(t)\) adaptive to the instantaneous error magnitude. When the error \(|e|\) is large (e.g., during initial transients or sudden disturbances), this term increases, effectively widening the performance envelope to prevent control saturation and instability. As the error converges, the term diminishes, allowing the envelope to tighten for high steady-state precision. This adaptive mechanism overcomes the rigidity of conventional PPC, enhancing both safety and performance for the unmanned drone team.
To incorporate this constraint into controller design, a smooth, strictly increasing transformation function \(T(z)\) is used:
$$
e_{i,k}(t) = \rho_{i,k}(t) T(\xi_{i,k}(t))
$$
where \(\xi_{i,k}\) is a transformed, unconstrained error. The derivative introduces a term \(E_{i,k} = \frac{1}{\rho_{i,k}} \frac{\partial T^{-1}}{\partial (e_{i,k}/\rho_{i,k})} > 0\), which is crucial for the stability proof.
2.3 Controller Design with Disturbance Observer
The controller is designed using an Integral Backstepping Sliding Mode Control (IBSMC) approach, integrated with the IPPC transformation and an EDO.
2.3.1 Error Transformation and Virtual Control
For the translational subsystem (underactuated dynamics in \(x, y\)), define the transformed error \(\xi_{1,i}\) from the original tracking error \(e_{1,i} = \zeta_{1,i}^d – \zeta_{1,i}\) using the IPPC relation. An integral term is added: \(\xi_{0,i} = \int_0^t \xi_{1,i}(\tau)d\tau\). A Lyapunov function \(V_{1,i} = \frac{1}{2}K_0 \xi_{0,i}^2 + \frac{1}{2}K_1 \xi_{1,i}^2\) is chosen. Its derivative leads to the design of a virtual control law \(\boldsymbol{\beta}_{1,i}\):
$$
\boldsymbol{\beta}_{1,i} = \dot{\boldsymbol{\zeta}}_{1,i}^d + \frac{1}{E_{1,i}} \left( K_0 \xi_{0,i} + K_1 \xi_{1,i} – \frac{\dot{\rho}_{1,i}}{\rho_{1,i}} e_{1,i} \right)
$$
Define the virtual control error \(e_{2,i} = \boldsymbol{\beta}_{1,i} – \dot{\boldsymbol{\zeta}}_{1,i}\).
2.3.2 Sliding Surface and Final Control Law
A sliding surface is defined to incorporate the integral action and the virtual control error:
$$
\mathbf{s}_{1,i} = K_2 \xi_{1,i} + e_{2,i}
$$
Its derivative is:
$$
\dot{\mathbf{s}}_{1,i} = K_2 \dot{\xi}_{1,i} + \dot{\boldsymbol{\beta}}_{1,i} – \ddot{\boldsymbol{\zeta}}_{1,i}
$$
$$
= K_2 \dot{\xi}_{1,i} + \dot{\boldsymbol{\beta}}_{1,i} – \mathbf{f}_{1,i} – \mathbf{g}_{1,i}\mathbf{U}_{xy,i} – \mathbf{d}_{1,i}
$$
where \(\mathbf{U}_{xy,i}=[u_{xi}, u_{yi}]^T\).
2.3.3 Extended Disturbance Observer (EDO) Design
An EDO is designed to estimate the lumped disturbance \(\mathbf{d}_{1,i}\):
$$
\begin{aligned}
\dot{\mathbf{z}}_{0,i} &= -\mathbf{l}_{0,i}(\mathbf{z}_{0,i} – \dot{\boldsymbol{\zeta}}_{1,i}) + \mathbf{f}_{1,i} + \mathbf{g}_{1,i}\mathbf{U}_{xy,i} + \hat{\mathbf{d}}_{1,i} \\
\hat{\mathbf{d}}_{1,i} &= \mathbf{z}_{1,i} + \mathbf{l}_{1,i}(\mathbf{z}_{0,i} – \dot{\boldsymbol{\zeta}}_{1,i}) \\
\dot{\mathbf{z}}_{1,i} &= -\mathbf{l}_{1,i}(\dot{\mathbf{z}}_{0,i} – \ddot{\boldsymbol{\zeta}}_{1,i}) + \hat{\dot{\mathbf{d}}}_{1,i} \\
\hat{\dot{\mathbf{d}}}_{1,i} &= \mathbf{z}_{2,i} + \mathbf{l}_{2,i}(\mathbf{z}_{0,i} – \dot{\boldsymbol{\zeta}}_{1,i}) \\
\dot{\mathbf{z}}_{2,i} &= -\mathbf{l}_{2,i}(\dot{\mathbf{z}}_{0,i} – \ddot{\boldsymbol{\zeta}}_{1,i}) + \boldsymbol{\varphi}_i(\hat{\mathbf{d}}_{1,i}, \hat{\dot{\mathbf{d}}}_{1,i})
\end{aligned}
$$
where \(\mathbf{l}_{0,i}, \mathbf{l}_{1,i}, \mathbf{l}_{2,i} > 0\) are observer gains, and \(\boldsymbol{\varphi}_i(\cdot)\) is a bounded function (e.g., involving hyperbolic tangent) to ensure observer stability. The estimates \(\hat{\mathbf{d}}_{1,i}\) and \(\hat{\dot{\mathbf{d}}}_{1,i}\) are used for compensation.
2.3.4 Final Control Law for Underactuated Subsystem
The actual control law for the translational subsystem is designed as:
$$
\mathbf{U}_{xy,i} = \mathbf{g}_{1,i}^{-1} \left[ -K_2 \dot{\xi}_{1,i} – \dot{\boldsymbol{\beta}}_{1,i} + \mathbf{f}_{1,i} – K_3 \mathbf{s}_{1,i} – K_4 \tanh\left(\frac{\mathbf{s}_{1,i}}{\mu}\right) + \hat{\mathbf{d}}_{1,i} \right]
$$
where \(K_3, K_4, \mu > 0\). The \(\tanh(\cdot)\) function approximates the signum function to reduce chattering.
2.3.5 Altitude and Attitude (Fully-Actuated) Subsystem
The control design for the altitude \(z\) and attitude \((\phi, \theta, \psi)\) subsystems follows a similar IBSMC+IPPC+EDO structure but must also handle actuator faults and saturation. An auxiliary system is introduced to compensate for the effect of input saturation/faults \(\Delta \mathbf{U}\):
$$
\dot{\mathbf{h}}_i = -K_h \mathbf{h}_i + \mathbf{g}_{2,i} \Delta \mathbf{U}_i
$$
The sliding surface is modified as \(\mathbf{s}_{2,i}’ = \mathbf{s}_{2,i} – \mathbf{h}_i\), and the control law \(\mathbf{U}_{1234,i} = [U_{1i}, U_{2i}, U_{3i}, U_{4i}]^T\) is derived accordingly, ensuring boundedness despite input constraints.
2.3.6 Attitude Reference Calculation
From the translational control outputs \(u_{xi}\) and \(u_{yi}\), the desired roll and pitch angles for each unmanned drone are computed under small-angle approximation:
$$
\phi_i^d = \arcsin\left( \frac{u_{xi}\sin\psi_i^d – u_{yi}\cos\psi_i^d}{U_{1i}} \right), \quad \theta_i^d = \arcsin\left( \frac{u_{xi}\cos\psi_i^d + u_{yi}\sin\psi_i^d}{U_{1i}\cos\phi_i^d} \right)
$$
The desired yaw angle \(\psi_i^d\) is typically specified independently by the mission.
2.4 Stability Analysis
The closed-loop stability of the entire cooperative unmanned drone system is rigorously proven using Lyapunov theory. A composite Lyapunov candidate function \(V\) is constructed, encompassing:
- The transformed tracking errors (\(\xi_0, \xi_1\)) from IPPC.
- The sliding surfaces (\(\mathbf{s}_1, \mathbf{s}_2’\)).
- The disturbance observer estimation errors (\(\tilde{\mathbf{d}}, \tilde{\dot{\mathbf{d}}}\)).
- The states of the auxiliary system (\(\mathbf{h}\)).
Using the properties of the IPPC transformation (\(E_{i,k}>0\)), the boundedness of disturbance estimates from the EDO, and the robust control terms, the derivative \(\dot{V}\) is shown to be negative definite outside a compact set. This proves that all signals in the cooperative unmanned drone system are Uniformly Ultimately Bounded (UUB). Furthermore, the prescribed performance bounds guarantee that the original tracking errors \(e_{i,k}(t)\) remain within the user-defined envelopes \(\pm \rho_{i,k}(t)\) for all time \(t \ge 0\), ensuring transient and steady-state performance as well as collision avoidance within the unmanned drone team.
3. Simulation Results and Performance Evaluation
To validate the proposed cooperative control framework, a simulation of four unmanned drones (\(n=4\)) transporting a load is conducted under severe conditions. Key simulation parameters and disturbances are summarized below.
| Parameter | Value |
|---|---|
| Unmanned Drone Mass (\(m_i\)) | 1.8 kg |
| Arm Length (\(l\)) | 0.2 m |
| Cable Length (\(L_i\)) | 3 m |
| Moments of Inertia (\([J_x, J_y, J_z]\)) | [0.03, 0.03, 0.04] kg·m² |
| Time-Varying Payload Mass (\(m_l\)) | 0.6 kg (0-30s), 0.2 kg (30-50s), 1.0 kg (>50s) |
| Actuator Fault (occurs at t=40s) | Effectiveness reduced to 80% with constant bias |
| Control Input Saturation | Modeled via \(\tanh(\cdot)\) function |
| Wind Gust Disturbance | Time-varying profile in all axes |
| Downwash on Load | Modeled as vertical disturbance proportional to unmanned drone proximity |
| State (\(k\)) | \(\rho_0\) | \(\rho_{\infty}\) | \(\alpha\) | \(\gamma_1\) | \(\gamma_2\) | \(\gamma_3\) | \(\delta\) |
|---|---|---|---|---|---|---|---|
| \(x, y\) | 0.5 m | 0.3 m | 0.5 | 5 | 2 | 2 | 0.8 |
| \(z\) | 1.0 m | 0.5 m | 0.5 | 5 | 2 | 2 | 0.8 |
| \(\phi, \theta\) | 1.0 rad | 0.5 rad | 0.8 | 10 | 2 | 2 | 0.8 |
| \(\psi\) | 0.2 rad | 0.1 rad | 0.8 | 10 | 2 | 2 | 0.8 |
| Gain Set | Values |
|---|---|
| IPPC/IBSMC (\([K_0, K_1, K_2, K_3, K_4]\)) | Translational: [0.2, 0.1, 0.1, 1, 2] Rotational: [20, 15, 2, 1, 2] (for \(\phi,\theta\)), [10, 5, 1, 1, 0.5] (for \(\psi\)) |
| EDO Gains (\([l_0, l_1, l_2]\)) | [5, 50, 100] (scaled per channel) |
| Auxiliary System Gain (\(K_h\)) | 5 |
The load is commanded to follow a 3D trajectory. The formation is defined by the relative offsets \(\boldsymbol{\delta}_i\). The proposed controller (IPPC+IBSMC+EDO) is compared against a standard Sliding Mode Control (SMC) and a traditional fixed-boundary PPC with SMC.
Trajectory Tracking: The unmanned drone team successfully tracks the load’s desired trajectory in 3D space. The load’s position tracking errors remain minimal despite mass changes and wind gusts, validating the cooperative algorithm and the overall control robustness.
Formation Keeping and Collision Avoidance: The inter-unmanned drone distances are calculated in real-time. Due to the IPPC constraints, these distances are strictly maintained within safe bounds derived from the performance functions \(\rho(t)\) for the position errors. No collision or dangerous proximity occurs, even during the actuator fault event at t=40s.
Disturbance Rejection and Estimation: The EDO accurately estimates the combined effect of wind, downwash, and model uncertainties. The estimation error converges rapidly and remains small, demonstrating the effectiveness of the disturbance compensation mechanism in the control law.
Quantitative Performance Comparison: The following table summarizes key performance indices, averaged across all unmanned drones, comparing the three methods.
| Performance Index | SMC [31] | PPC+SMC | Proposed (IPPC+IBSMC+EDO) |
|---|---|---|---|
| Mean Square Error (MSE) – Position (m²) | ~0.1 – 0.5 | ~1e-3 – 1e-2 | ~1e-4 – 1e-6 |
| Max Absolute Error (MAE) – Position (m) | ~0.9 – 1.7 | ~0.1 – 0.3 | ~0.06 – 0.3 |
| MSE – Attitude (rad²) | ~1e-1 | ~5e-3 – 1e-2 | ~4e-4 – 4e-5 |
| Integral of Squared Error (ISE) | High | Medium | Lowest |
| Control Input Chattering | Severe | Moderate | Minimal (smooth) |
The results clearly demonstrate the superiority of the proposed framework. The IPPC ensures smooth, bounded error transients and high steady-state accuracy. The IBSMC+EDO combination provides strong robustness against disturbances, faults, and saturation without inducing excessive chattering. The cooperative unmanned drone system maintains a stable formation and accurate load transportation under highly challenging conditions.
4. Conclusion
This paper presented a comprehensive solution for the cooperative control of a multi-unmanned drone slung-load system. The framework integrates a graph-theoretic formation algorithm, an Improved Prescribed Performance Control (IPPC) strategy, an Integral Backstepping Sliding Mode Controller (IBSMC), and an Extended Disturbance Observer (EDO). The IPPC, with its error-adaptive performance boundary, is a key contribution that guarantees predefined transient and steady-state performance while providing flexibility to handle large initial errors and disturbances, thus ensuring safe inter-unmanned drone spacing. The IBSMC offers a robust control backbone, and the EDO effectively compensates for complex, aggregated disturbances. The Udwadia-Kalaba formulation provided a concise and accurate model of the constrained dynamics. Lyapunov-based stability analysis proves the uniform ultimate boundedness of all closed-loop signals. Extensive simulations under realistic disturbances and faults (actuator failure, input saturation, wind, payload variation) confirm the effectiveness, robustness, and superior performance of the proposed approach compared to conventional methods. This work provides a reliable control framework for deploying cooperative unmanned drone teams in demanding aerial transportation tasks.