The precise transportation of suspended payloads using aerial robots, particularly quadrotor drones, represents a critical capability with extensive applications in fields such as search and rescue, construction, and logistics. The quadrotor drone platform offers unique advantages like vertical take-off and landing (VTOL) and agile maneuverability. However, controlling a quadrotor drone coupled with a cable-suspended payload introduces significant challenges. The system becomes a highly nonlinear, underactuated, and strongly coupled dynamic entity. The swinging motion of the payload acts as a persistent disturbance, complicating trajectory tracking. Furthermore, operational safety and environmental constraints often necessitate that the payload’s position remains within strict, predefined bounds throughout the mission. Traditional control strategies, while effective for stabilization, often guarantee only asymptotic convergence, where settling time can grow unbounded with initial conditions, or they may not explicitly enforce output constraints. This paper addresses these combined challenges by proposing a novel fixed-time convergent control scheme that ensures rapid trajectory tracking for the payload while rigorously adhering to prescribed positional constraints, even in the presence of external disturbances.
The dynamics of a quadrotor drone with a suspended payload are considerably more complex than those of an isolated drone. The system can be decomposed into three interconnected subsystems: the translational dynamics of the payload, the swing dynamics of the payload on the cable, and the rotational dynamics of the quadrotor drone itself. The coupling arises from the tension force in the cable, which links the acceleration of the drone to the swing angles of the payload. The control objective is to make the payload position $\mathbf{p}_l$ track a desired trajectory $\mathbf{p}_{ld}$, which implies indirect control of the quadrotor drone’s attitude and position. A common and practical modeling approach derives the error dynamics directly for the payload, treating the quadrotor drone’s states as intermediate control variables. Let $\mathbf{e}_p = \mathbf{p}_l – \mathbf{p}_{ld}$ and $\mathbf{e}_v = \dot{\mathbf{p}}_l – \dot{\mathbf{p}}_{ld}$ denote the position and velocity tracking errors of the payload in the inertial frame. A simplified form of the payload translational dynamics can be expressed as:
$$
\begin{align}
\dot{\mathbf{e}}_p &= \mathbf{e}_v \\
\dot{\mathbf{e}}_v &= \mathbf{f}(\mathbf{x}) + \mathbf{g}(\mathbf{x})\mathbf{u}_p + \boldsymbol{\Delta}_v(t)
\end{align}
$$
Here, $\mathbf{x}$ represents the full state vector (including quadrotor attitude, payload swing angles, etc.), $\mathbf{f}(\cdot)$ and $\mathbf{g}(\cdot)$ are nonlinear functions derived from system kinematics and dynamics, $\mathbf{u}_p$ is a virtual control input related to the quadrotor’s thrust vector, and $\boldsymbol{\Delta}_v(t)$ aggregates external disturbances (like wind) and unmodeled dynamics. The swing dynamics and quadrotor rotational dynamics follow similar structures but with their own states and control inputs (e.g., quadrotor torques $\mathbf{u}_\tau$). The underactuation is evident as the number of independent control inputs (4: total thrust and 3 torques) is less than the number of degrees of freedom one might wish to control independently (payload position in 3D and swing suppression).
A primary challenge in controlling this quadrotor drone system is the inherent uncertainty represented by $\boldsymbol{\Delta}(t)$. To address this, a Fixed-Time Disturbance Observer (FTDO) is designed. Unlike asymptotic or finite-time observers, an FTDO guarantees that the estimation error converges to zero within a time $T_{obs}$ that is bounded regardless of the initial estimation error. This property is crucial for providing rapid and reliable disturbance compensation. For a generic disturbance channel $\boldsymbol{\Delta}$, the observer structure is:
$$
\begin{align}
\dot{\hat{\boldsymbol{\eta}}} &= -\beta_1 \text{sig}^{\alpha_1}(\hat{\boldsymbol{\eta}} – \boldsymbol{\zeta}) -\beta_2 \text{sig}^{\alpha_2}(\hat{\boldsymbol{\eta}} – \boldsymbol{\zeta}) + \mathbf{G} \\
\hat{\boldsymbol{\Delta}} &= \hat{\boldsymbol{\eta}} + \gamma (\hat{\boldsymbol{\eta}} – \boldsymbol{\zeta})
\end{align}
$$
where $\hat{\boldsymbol{\Delta}}$ is the estimate of $\boldsymbol{\Delta}$, $\boldsymbol{\zeta}$ is an auxiliary variable from the system dynamics, $\mathbf{G}$ is a known nonlinear term, and $\beta_1, \beta_2, \gamma > 0$, $0 < \alpha_1 < 1$, $\alpha_2 > 1$. The function $\text{sig}^a(\mathbf{z}) = [|z_1|^a \text{sign}(z_1), …, |z_n|^a \text{sign}(z_n)]^\top$. With $\alpha_1 = 1 – 1/(1+\mu)$ and $\alpha_2 = 1 + 1/(1+\mu)$ for $\mu>0$, the convergence time $T_{obs}$ is upper-bounded by a constant dependent only on $\beta_1, \beta_2, \mu$ and not on initial conditions. This observer is applied separately to estimate disturbances $\boldsymbol{\Delta}_v$, $\boldsymbol{\Delta}_\omega$ (swing), and $\boldsymbol{\Delta}_\Omega$ (quadrotor rotation) acting on the respective subsystems of the quadrotor drone.
The second major challenge is enforcing output constraints on the payload position. We require $-\underline{\boldsymbol{\kappa}}(t) < \mathbf{e}_p(t) < \overline{\boldsymbol{\kappa}}(t)$ for all $t > 0$, where $\underline{\boldsymbol{\kappa}}(t)$ and $\overline{\boldsymbol{\kappa}}(t)$ are prescribed, smooth, positive, decaying performance functions. For example, $\overline{\kappa}_i(t) = (\kappa_{i0} – \kappa_{i\infty})e^{-\ell_i t} + \kappa_{i\infty}$, ensuring the error ultimately stays within the steady-state bound $\kappa_{i\infty}$. To incorporate these constraints into the controller design, a Barrier Lyapunov Function (BLF) is employed. A typical BLF candidate for a scalar constrained error $e$ with bounds $-\underline{\kappa}(t) < e < \overline{\kappa}(t)$ is:
$$
V_B = \frac{1}{2} \log \frac{\overline{\kappa}^2(t)}{\overline{\kappa}^2(t) – e^2} + \frac{1}{2} \log \frac{\underline{\kappa}^2(t)}{\underline{\kappa}^2(t) – e^2}
$$
This function approaches infinity as $e$ approaches either boundary $-\underline{\kappa}(t)$ or $\overline{\kappa}(t)$. By ensuring the time derivative of $V_B$ is negative definite, the error is guaranteed to remain within the prescribed funnel. For the vector case of the quadrotor drone’s payload error $\mathbf{e}_p$, a composite BLF is constructed as $V_{B,p} = \sum_{i=1}^3 V_{B,i}(e_{p,i})$.
The core of the proposed control strategy is a novel Fixed-Time Non-Singular Terminal Sliding Mode Control (FTNTSMC) law. This approach combines the strengths of several methods: the robustness of Sliding Mode Control (SMC), the fast finite-time convergence of Terminal SMC (TSMC), the elimination of the singularity problem associated with traditional TSMC, and the extension to fixed-time convergence independent of initial states. First, a non-singular terminal sliding manifold is defined for the payload subsystem. Using the transformed error variable $\boldsymbol{\xi} = \mathbf{F}(\mathbf{e}_p)$ derived from the BLF structure to incorporate constraints, the sliding surface is designed as:
$$
\mathbf{s}_p = \dot{\boldsymbol{\xi}} + \mathbf{H}(\boldsymbol{\xi}) = \mathbf{0}
$$
where $\mathbf{H}(\boldsymbol{\xi}) = [h(\xi_1), h(\xi_2), h(\xi_3)]^\top$ is a carefully designed function to ensure fixed-time attraction and avoid singularities. A common choice is:
$$
h(\xi_i) = \lambda_1 \phi(\xi_i) + \lambda_2 \text{sig}^{\frac{q}{p}}(\xi_i)
$$
with $\lambda_1, \lambda_2 > 0$, $1 < q/p < 2$, and $\phi(\xi_i)$ is a saturation-like function to bound the term when $\xi_i$ is large, preventing excessively high control signals. Once the sliding manifold $\mathbf{s}_p = \mathbf{0}$ is reached, the dynamics become $\dot{\boldsymbol{\xi}} = -\mathbf{H}(\boldsymbol{\xi})$. This differential equation guarantees that $\boldsymbol{\xi}$ converges to zero in fixed time, which in turn implies $\mathbf{e}_p$ converges to zero while respecting the constraints $\underline{\boldsymbol{\kappa}}(t), \overline{\boldsymbol{\kappa}}(t)$.
To drive the sliding variable $\mathbf{s}_p$ to zero in fixed time, the control law is synthesized using the disturbance estimate $\hat{\boldsymbol{\Delta}}_v$ and a fixed-time reaching law. The virtual control input $\mathbf{u}_p$ for the payload dynamics is designed as:
$$
\begin{aligned}
\mathbf{u}_p &= \mathbf{g}^{-1}(\mathbf{x}) \Big[ -\mathbf{f}(\mathbf{x}) – \hat{\boldsymbol{\Delta}}_v – \dot{\mathbf{H}}(\boldsymbol{\xi}) \\
&\quad – k_1 \text{sig}^{\frac{1}{2}}(\mathbf{s}_p) – k_2 \text{sig}^{\frac{3}{2}}(\mathbf{s}_p) – k_3 \mathbf{s}_p \Big]
\end{aligned}
$$
The terms $-k_1 \text{sig}^{1/2}(\mathbf{s}_p) – k_2 \text{sig}^{3/2}(\mathbf{s}_p)$ constitute a fixed-time reaching law. Lyapunov analysis using $V_s = \frac{1}{2}\mathbf{s}_p^\top \mathbf{s}_p$ shows that $\dot{V}_s \leq -\eta_1 V_s^{3/4} – \eta_2 V_s^{5/4}$, which, according to fixed-time stability theory, guarantees $\mathbf{s}_p \to \mathbf{0}$ within a settling time $T_{reach} \leq \frac{4}{\eta_1} + \frac{4}{\eta_2}$, independent of $\mathbf{s}_p(0)$. Similar sliding manifolds and control laws are derived for the swing angle subsystem (ensuring swing damping) and the quadrotor drone’s attitude subsystem. The virtual control $\mathbf{u}_p$ is then mapped to the actual quadrotor drone controls: total thrust $F$ and body-frame torques $\boldsymbol{\tau}$.
The overall stability of the complete closed-loop quadrotor drone system is proven using Lyapunov theory. A composite Lyapunov function candidate is constructed as the sum of the BLF for the payload, similar functions for other states, and terms for sliding surfaces and observer errors:
$$
V_{total} = V_{B,p} + V_{B,q} + V_{B,R} + \frac{1}{2}\mathbf{s}_p^\top\mathbf{s}_p + \frac{1}{2}\mathbf{s}_q^\top\mathbf{s}_q + \frac{1}{2}\mathbf{s}_R^\top\mathbf{s}_R + \frac{1}{2}\tilde{\boldsymbol{\Delta}}_v^\top\tilde{\boldsymbol{\Delta}}_v + …
$$
where $\tilde{\boldsymbol{\Delta}} = \boldsymbol{\Delta} – \hat{\boldsymbol{\Delta}}$ is the disturbance estimation error. The time derivative $\dot{V}_{total}$ is shown to be negative definite and upper-bounded by an expression of the form $\dot{V}_{total} \leq -\varpi_1 V_{total}^{c_1} – \varpi_2 V_{total}^{c_2}$ with $0 < c_1 < 1$, $c_2 > 1$. This proves that the entire system–payload tracking error, swing angles, quadrotor drone attitude, and observer errors–converges to a small neighborhood of zero within a fixed time $T_{total}$, establishing the core result of fixed-time stable, constraint-satisfying trajectory tracking for the quadrotor drone with a suspended payload.
To validate the theoretical development, numerical simulations are performed. The parameters for the quadrotor drone and payload system used in the simulation are listed in the table below. The controller gains and observer parameters are tuned to achieve satisfactory performance.
| Symbol | Parameter | Value |
|---|---|---|
| $m_Q$ | Mass of Quadrotor Drone | 2.0 kg |
| $m_l$ | Mass of Payload | 0.6 kg |
| $l$ | Cable Length | 0.8 m |
| $\mathbf{J}$ | Quadrotor Inertia Matrix | diag(0.03, 0.03, 0.055) kg·m² |
| $g$ | Gravitational Acceleration | 9.81 m/s² |
| $\kappa_{0}, \kappa_{\infty}$ | Performance Function Parameters | 0.3 m, 0.02 m |
| $\ell$ | Performance Function Decay Rate | 0.8 |
The desired trajectory for the payload is a helical path: $p_{ld,x} = 0.5\cos(0.5t)$, $p_{ld,y} = 0.5\sin(0.5t)$, $p_{ld,z} = 0.1t + 1$. External disturbances $\boldsymbol{\Delta}_v = [0.2\sin(0.8t), 0.1\cos(0.6t), 0.15]^\top$ N are applied. The initial position error is set to $e_{p,x}(0)=0.25$ m, which is close to the initial constraint boundary $\kappa_x(0)=0.3$ m, testing the constraint-handling capability of the controller for the quadrotor drone. The simulation results demonstrate the key features of the proposed controller. The payload position tracking errors $\mathbf{e}_p$ converge to near zero rapidly while strictly remaining within the funnel defined by $\pm \boldsymbol{\kappa}(t)$. The sliding variables $\mathbf{s}_p, \mathbf{s}_q, \mathbf{s}_R$ reach and maintain a small neighborhood of zero in fixed time. The Fixed-Time Disturbance Observer accurately estimates the applied disturbances within a short transient period. The control inputs (thrust and torques) remain smooth and feasible throughout the maneuver.
For quantitative comparison, the performance of the proposed Fixed-Time controller (FTC) is compared against a conventional Backstepping Sliding Mode Control (BSMC) designed for the same quadrotor drone system. Two standard performance indices are used: the Integral of Squared Error (ISE) and the Integral of Absolute Error (IAE).
| Performance Index | Channel | Proposed FTC | BSMC |
|---|---|---|---|
| ISE ($\times 10^{-3}$) | $e_x$ | 12.5 | 45.7 |
| $e_y$ | 11.8 | 48.2 | |
| $e_z$ | 9.6 | 52.1 | |
| IAE ($\times 10^{-2}$) | $e_x$ | 5.2 | 18.9 |
| $e_y$ | 4.9 | 20.3 | |
| $e_z$ | 4.1 | 22.5 | |
| Max Constraint Violation | None | 0.08 m | |
| Settling Time (to 2% of final bound) | ~3.2 s | ~7.8 s | |
The comparison clearly shows the superiority of the proposed method for the quadrotor drone system. The FTC achieves significantly lower ISE and IAE values, indicating higher tracking accuracy. Crucially, it maintains the payload error within the prescribed constraints at all times, whereas the BSMC briefly violates them. Most notably, the settling time of the FTC is bounded and substantially shorter, demonstrating the fixed-time convergence property. This makes the proposed controller far more reliable for time-critical operations where the quadrotor drone must complete its transport task swiftly and predictably.
In conclusion, this paper has presented a comprehensive fixed-time sliding mode control framework for a quadrotor drone transporting a cable-suspended payload. The design successfully integrates a fixed-time disturbance observer to rapidly estimate and compensate for uncertainties, a barrier Lyapunov function-based transformation to enforce prescribed performance constraints on the payload output, and a non-singular terminal sliding mode control law to achieve fixed-time convergence of all tracking errors. The stability of the entire closed-loop quadrotor drone system is rigorously proven. Simulation results confirm the theoretical claims, showing that the controller enables fast, accurate, and safe trajectory tracking, outperforming conventional asymptotic methods. Future work may focus on extending this strategy to multiple cooperative quadrotor drones carrying a shared payload or adapting it to scenarios with unknown or variable cable lengths, further enhancing the versatility and robustness of quadrotor drone-based transportation systems.