In the evolving landscape of aerial robotics, the quadrotor drone has emerged as a versatile platform for numerous industrial applications. Its compact size, agility, and capability for vertical take-off and landing make it an ideal candidate for carrying specialized payloads and performing tasks in challenging environments. One such innovative application involves augmenting a quadrotor drone with a twin propeller propulsion system suspended beneath it. This system acts as a dynamic work platform, enabling precise positional control in three-dimensional space to reach specific work points, thereby potentially replacing human labor in hazardous conditions. However, the effective operation of this suspended system is fraught with challenges, primarily due to strong coupling effects and susceptibility to external disturbance forces. These disturbances can destabilize the platform, compromising its ability to maintain a stable working posture. Therefore, developing a robust control strategy that ensures accurate position tracking, rapid response, and resilience against perturbations is paramount. In this work, I address these challenges by proposing an integrated control framework that combines an optimal Linear Quadratic Regulator (LQR) with a reduced-order disturbance observer. The core objective is to enhance the position response accuracy and anti-interference capability of the twin propeller propulsion system attached to a quadrotor drone.
The foundation of any control system design lies in an accurate mathematical model of the physical system. For the twin propeller propulsion system suspended from a quadrotor drone, we can simplify the dynamics by considering its motion in a single plane, such as the pitch direction, after decoupling. The system essentially behaves like an inverted pendulum, where the control inputs are the thrust forces generated by the two propellers. Let us denote the combined thrust from both motors as \( F \), the mass of the entire platform as \( m \), the distance from the suspension point to the system’s center of mass as \( l \), the moment of inertia about the suspension point as \( I_o \), and the pitch angle from the vertical as \( \theta \). Assuming small angle approximations and neglecting secondary frictional forces, the linearized equation of motion derived from Newton’s laws is:
$$ I_o \ddot{\theta} + m g l \theta = F l $$
Here, \( g \) is the acceleration due to gravity. The transfer function relating the pitch angle \( \theta(s) \) to the thrust input \( F(s) \) in the Laplace domain is:
$$ G(s) = \frac{\theta(s)}{F(s)} = \frac{\frac{l}{I_o}}{s^2 + \frac{m g l}{I_o}} $$
For the specific system parameters used in this study, the values are summarized in the table below. These parameters are typical for a medium-sized quadrotor drone payload system and form the basis for controller design and simulation.
| Parameter | Symbol | Value | Unit |
|---|---|---|---|
| Moment of Inertia | \( I_o \) | 0.36 | kg·m² |
| Distance to Center of Mass | \( l \) | 0.60 | m |
| System Mass | \( m \) | 1.0 | kg |
| Gravitational Acceleration | \( g \) | 9.81 | m/s² |
The primary control challenge for this quadrotor drone-mounted system is twofold: achieving precise tracking of a desired pitch angle \( \theta_0 \) and robustly rejecting external disturbance forces, such as sudden wind gusts or payload interactions. A common external disturbance is a lateral force \( F_d \) acting on the platform. Incorporating this disturbance, the dynamic equation becomes:
$$ I_o \ddot{\theta} + m g l \theta = F l – F_d $$
To address this, my proposed control strategy employs a two-tier approach. The outer loop utilizes an optimal LQR controller to minimize tracking error and control effort, ensuring efficient operation—a critical consideration for battery-powered quadrotor drones. The inner loop features a reduced-order disturbance observer that estimates the unknown disturbance force \( F_d \) in real-time and feeds it forward for compensation. This combined structure aims to deliver high performance while maintaining stability against uncertainties.
The design of the optimal LQR controller begins by formulating the system in state-space form. For the pitch dynamics, I define the state vector as \( \mathbf{x} = [\theta, \dot{\theta}]^T \), the control input as \( u = F \), and the output as \( y = \theta \). From the linearized equation, the state-space matrices are:
$$ \dot{\mathbf{x}} = A \mathbf{x} + B u $$
$$ y = C \mathbf{x} $$
where
$$ A = \begin{bmatrix} 0 & 1 \\ -\frac{m g l}{I_o} & 0 \end{bmatrix}, \quad B = \begin{bmatrix} 0 \\ \frac{l}{I_o} \end{bmatrix}, \quad C = \begin{bmatrix} 1 & 0 \end{bmatrix} $$
For the quadrotor drone’s propulsion system, energy efficiency is vital to extend flight time. The LQR framework optimally balances state regulation and control effort. I define a performance index \( J \) that penalizes both the tracking error and the control input:
$$ J = \int_0^\infty \left[ \rho \, z(t)^2 + u(t)^2 \right] dt $$
Here, \( z(t) \) is a weighted output representing the tracking performance. I choose \( z(t) = H_1 \mathbf{x} \), with \( H_1 = [2, 1] \) to emphasize both position and velocity errors. The scalar \( \rho > 0 \) is a tuning parameter that adjusts the trade-off between error minimization and control energy consumption. After extensive simulation trials for this quadrotor drone application, I selected \( \rho = 12 \). The optimal control law that minimizes \( J \) is given by \( u = -K \mathbf{x} \), where the gain matrix \( K \) is obtained by solving the Algebraic Riccati Equation. Using computational tools, the LQR gain for our system is:
$$ K = \begin{bmatrix} 2.1306 & 3.7844 \end{bmatrix} $$
Thus, the controller transfer function from error to input becomes:
$$ K(s) = K (sI – A)^{-1} B = \frac{6.32 s + 3.56}{s^2 + 16.35} $$
This controller ensures that the closed-loop system for the quadrotor drone’s propulsion unit has desirable transient response and zero steady-state error for step inputs, provided disturbances are absent.
However, in real-world operations, a quadrotor drone often encounters unpredictable external forces. To enhance robustness, I design a reduced-order disturbance observer to estimate and compensate for such forces. The key advantage of a reduced-order observer is that it only estimates the unmeasured states, leveraging the directly measured output—here, the pitch angle \( \theta \). I augment the state vector to include the disturbance force \( F_d \) as an additional state. Assuming the disturbance varies slowly, its derivative is approximately zero. The augmented state-space model is:
$$ \begin{bmatrix} \dot{\theta} \\ \ddot{\theta} \\ \dot{F}_d \end{bmatrix} = \begin{bmatrix} 0 & 1 & 0 \\ -\frac{m g l}{I_o} & 0 & -\frac{1}{I_o} \\ 0 & 0 & 0 \end{bmatrix} \begin{bmatrix} \theta \\ \dot{\theta} \\ F_d \end{bmatrix} + \begin{bmatrix} 0 \\ \frac{l}{I_o} \\ 0 \end{bmatrix} F $$
I partition the states into the measured component \( x_1 = \theta \) and the components to be estimated \( \mathbf{x}_2 = [\dot{\theta}, F_d]^T \). The system matrices are correspondingly partitioned as:
$$ \begin{aligned}
A_{11} &= 0, \quad A_{12} = \begin{bmatrix} 1 & 0 \end{bmatrix}, \quad B_1 = 0 \\
A_{21} &= \begin{bmatrix} -\frac{m g l}{I_o} \\ 0 \end{bmatrix}, \quad A_{22} = \begin{bmatrix} 0 & -\frac{1}{I_o} \\ 0 & 0 \end{bmatrix}, \quad B_2 = \begin{bmatrix} \frac{l}{I_o} \\ 0 \end{bmatrix}
\end{aligned} $$
The reduced-order observer for \( \mathbf{x}_2 \) has the form:
$$ \dot{\mathbf{w}} = (A_{22} – L A_{12}) \hat{\mathbf{x}}_2 + (A_{21} – L A_{11}) y + (B_2 – L B_1) u $$
where \( \mathbf{w} = \hat{\mathbf{x}}_2 – L y \) is an auxiliary vector to avoid direct differentiation of the output, \( L = [l_1, l_2]^T \) is the observer gain matrix, and \( \hat{\mathbf{x}}_2 \) is the estimate of \( \mathbf{x}_2 \). The estimate is recovered as \( \hat{\mathbf{x}}_2 = \mathbf{w} + L y \). Substituting the matrices yields the observer dynamics:
$$ \begin{aligned}
\dot{w}_1 – l_1 \dot{y} &= -l_1 \hat{\dot{\theta}} – \frac{1}{I_o} \hat{F}_d – \frac{m g l}{I_o} y – \frac{l}{I_o} u \\
\dot{w}_2 – l_2 \dot{y} &= -l_2 \hat{\dot{\theta}}
\end{aligned} $$
To ensure rapid and accurate estimation, the observer poles must be placed appropriately. A rule of thumb is to set the observer poles 2 to 6 times faster than the dominant closed-loop poles of the LQR-controlled system. The LQR controller yielded closed-loop poles with real parts around -6.4. Therefore, I choose observer poles at \( \lambda_1 = -6.4 + 6.3j \) and \( \lambda_2 = -6.4 – 6.3j \). The desired characteristic polynomial is:
$$ \alpha(s) = (s – \lambda_1)(s – \lambda_2) = s^2 + 12.8s + 80.65 $$
Matching this with the observer characteristic polynomial \( \det(sI – (A_{22} – L A_{12})) = s^2 + l_1 s – \frac{l_2}{I_o} \), we solve for the gains:
$$ l_1 = -(\lambda_1 + \lambda_2) = 12.8 $$
$$ l_2 = -\lambda_1 \lambda_2 I_o = -80.65 \times 0.36 = -29.034 $$
The observer gain matrix is thus \( L = [12.8, -29.034]^T \). With these parameters, the disturbance observer can quickly estimate both the angular velocity \( \dot{\theta} \) and the disturbance force \( F_d \). The estimated disturbance \( \hat{F}_d \) is then used in a feedforward compensation path, modifying the control input to \( u = -K \mathbf{x} + \frac{\hat{F}_d}{l} \). This effectively cancels the disturbance effect before it significantly impacts the quadrotor drone’s platform position.
To validate the proposed control scheme for the quadrotor drone’s twin propeller system, I conducted comprehensive simulations using a computational environment. The simulation model incorporates the full nonlinear dynamics, the LQR controller, and the reduced-order disturbance observer. The test scenario involves commanding a step change in the desired pitch angle to \( 10^\circ \) while applying a sudden lateral disturbance force of 5 N between 0.3 s and 2.3 s. The performance is compared between two cases: the standalone LQR controller and the LQR controller augmented with the disturbance observer. Key performance metrics such as overshoot, settling time, and maximum deviation due to disturbance are recorded.
The simulation results clearly demonstrate the efficacy of the disturbance observer. The observer accurately estimates the applied disturbance force, as shown in the table below, which compares the actual and estimated disturbance values at key time instances.
| Time (s) | Actual Disturbance \( F_d \) (N) | Estimated Disturbance \( \hat{F}_d \) (N) | Estimation Error (N) |
|---|---|---|---|
| 0.5 | 5.00 | 4.98 | 0.02 |
| 1.0 | 5.00 | 5.01 | -0.01 |
| 2.0 | 5.00 | 4.99 | 0.01 |
| 2.4 | 0.00 | 0.05 | -0.05 |
The closed-loop response metrics are summarized in the following table, highlighting the improvement brought by the observer-based compensation for the quadrotor drone system.
| Performance Metric | LQR Controller Only | LQR with Disturbance Observer |
|---|---|---|
| Rise Time (to 90% of final value) | 0.45 s | 0.44 s |
| Overshoot | 0% | 0% |
| Settling Time (for step command) | 1.2 s | 1.2 s |
| Max Deviation due to 5N Disturbance | 0.5° | 0.1° |
| Recovery Time after Disturbance | 1.8 s | 0.3 s |
The numerical results underscore the significant enhancement in disturbance rejection. With the observer, the maximum pitch deviation caused by the disturbance is reduced by 80%, and the system recovers to its reference trajectory five times faster. This is crucial for a quadrotor drone operating in dynamic environments where stability is paramount. The control effort, measured as the total thrust variation, also remains within practical limits, ensuring efficient use of onboard battery power.
To further illustrate the theoretical underpinnings, let me present the key equations governing the combined controller-observer system. The overall control law is:
$$ u = -K \begin{bmatrix} \theta \\ \hat{\dot{\theta}} \end{bmatrix} + \frac{\hat{F}_d}{l} $$
where \( \hat{\dot{\theta}} \) and \( \hat{F}_d \) are provided by the reduced-order observer. The observer dynamics, after substituting gains, are:
$$ \begin{aligned}
\dot{\hat{\dot{\theta}}} &= 12.8 (\dot{\theta} – \hat{\dot{\theta}}) – \frac{1}{0.36} \hat{F}_d – \frac{1 \times 9.81 \times 0.6}{0.36} \theta – \frac{0.6}{0.36} u \\
\dot{\hat{F}}_d &= -29.034 (\dot{\theta} – \hat{\dot{\theta}})
\end{aligned} $$
In practice, for implementation on a quadrotor drone’s flight controller, these continuous-time equations would be discretized. The integration of this control module with the drone’s existing attitude and position control loops is a straightforward process, given the modular design.
The successful simulation outcomes lead to several important conclusions regarding the control of quadrotor drone-based propulsion platforms. First, the reduced-order disturbance observer design is both feasible and effective. By utilizing the directly measurable pitch angle, it avoids the complexity of a full-state observer and accurately estimates unmeasured states and external forces. Second, the feedforward compensation mechanism using the estimated disturbance force seamlessly integrates with the optimal LQR controller, creating a cohesive control strategy that does not compromise the stability guarantees of the LQR. Third, the combined approach markedly improves the system’s resistance to sudden disturbances, which are inevitable in outdoor operations of a quadrotor drone. The significant reduction in position deviation and recovery time translates to higher operational precision and safety.
Looking ahead, there are promising avenues for extending this work. Future research could explore adaptive versions of the observer to handle time-varying system parameters, such as changes in the suspended mass—a common scenario when a quadrotor drone delivers or picks up a payload. Additionally, extending the control framework to manage the full six degrees of freedom of the quadrotor drone and its suspended platform, including roll and yaw motions, would provide a comprehensive solution for complex manipulation tasks. Experimental validation on a physical quadrotor drone prototype is the logical next step to confirm the simulation findings under real-world noise and uncertainties.
In summary, this study has presented a robust optimal control strategy tailored for a twin propeller propulsion system carried by a quadrotor drone. The synergy between the LQR controller and the reduced-order disturbance observer addresses the dual needs of precise tracking and robust disturbance rejection. For engineers and researchers working on aerial robotic systems, this method offers a practical and theoretically sound approach to enhancing the performance of quadrotor drone-mounted manipulators and platforms, paving the way for more reliable and capable drones in industrial applications.