GPI observer-based active disturbance rejection control for a morphing quadrotor

1. INTRODUCTION

Unmanned aerial vehicles (UAVs) are widely used in various fields such as agriculture, transport, and rescue ^[1]. As UAVs, quadrotors have attracted wide attention due to their advantage of vertical take-off and landing. However, quadrotors with fixed structures cannot cope with complex environments and tasks. For example, fixed-structure quadrotors usually need complex planning trajectories to cross narrow spaces ^[2,3]. In addition, the planning methods may fail to find an available solution when the space is too narrow. In recent years, morphing-structure quadrotors have begun to attract attention for their ability to adapt to different tasks by changing their morphology ^[4]. For example, Bucki et al. design a passive morphing quadrotor, which uses passive joints to enable morphing ^[5]; passive joints allow the arms of quadrotors to be folded down; meanwhile, a trajectory planning method is proposed to cross gaps. Zhao et al. design a foldable quadrotor by using a scissor-like foldable mechanism ^[6]; aerodynamic effects are investigated to ensure the morphological change. In addition, Falanga et al. devise a foldable quadrotor with rotating arms around its airframe ^[7]; a linear quadratic regulator controller is proposed to perform a stable flight; they also investigate a geometry-aware compensation scheme to guarantee higher flight accuracy ^[8]. Furthermore, Kim et al. design a morphing quadrotor, which can change its morphology according to payload mass ^[9]; an optimization method is proposed to determine the optimal morphology, and an adaptive control method is developed to realize stable flight; With the continuous research on morphing aircraft, Sakaguchi et al. use parallel link modules to design a novel quadrotor that can morph around the 3-axis and design a control method without thrust loss at any deformed state ^[10]. For the near term, Wu et al. develop a connected retractable ring-shaped quadrotor that can grasp and drop objects through the expansion and contraction of the ring-shaped structure, and a nonlinear model predictive control strategy and a time-variant physical parameter mode are applied to achieve stable flight ^[11]. Based on the current research situation above, it can be concluded that the current research on morphing quadrotors mainly focuses on mechanical structure design and control allocation design. Relatively speaking, a few studies have considered the anti-disturbance ability of morphing quadrotors despite the fact that they are more sensitive to disturbances, not to speak of estimating the internal and external disturbances accurately and quickly.

To increase the robustness of the morphing quadrotor system, Zhang et al. propose an active disturbance rejection control (ADRC) method ^[12,13] with an extended state observer (ESO) to eliminate the lumped disturbances ^[14]. The generalized proportional integral (GPI) observer as a high-order ESO is successfully applied in a series of robotic systems in recent years^[15–19] for estimating the disturbances more accurately. Despite all this, few studies have considered its application to morphing quadrotors.

Quadrotors usually suffer from airflow, sensor noises, ground effects, and unmodeled internal disturbances ^[20]. In addition, morphological changes make morphing quadrotor control more difficult. The challenges can be listed as follows: 1) Deviation of the center of gravity (CoG) and changing of the moment of inertia (MoI) of quadrotors; 2) High system sensitivity to disturbances; 3) The time-varying torque generated by motors around axes.

To address the above issues, in this paper, we design a morphing quadrotor whose arms can be folded horizontally. Meanwhile, considering the variation of CoG and MoI, a dynamic model and control allocation strategy for this morphing quadrotor are established. On this basis, the main contributions can be summarized as follows:

(1) By analyzing the morphing process of the designed quadrotor, the changing CoG and MoI are calculated in real time with the assistance of the physical formulas and professional modeling software. The calculated CoG and MoI will be combined with quadrotor dynamics and a mathematical model which can be applied to all potential quadrotor morphologies.

(2) A dynamic control allocation strategy is established to make the outputs of actuators more precise by analyzing the distance from the motors to the CoG during the process of morphing and setting the stage for a stable flight of the morphing quadrotor.

(3) A time-varying physical parameter model is combined with a new GPI observer-based ADRC method to effectively improve the estimation speed of disturbances and the robustness of morphing quadrotor system for the first time.

The remainder of this paper is organized as follows: In section 2, a dynamic model of the morphing quadrotor is established. An ADRC method for the morphing quadrotor based on the GPI observer will be designed in section 3. In section 4, simulations are carried out to verify the effectiveness of the proposed method. In section 5, the paper is summarized.

2. MODELING

In this section, the dynamic model of the morphing quadrotor will be introduced. In detail, first, the dynamic equations will be established. The CoG and MoI for the morphing quadrotor will be acquired subsequently. In addition, a dynamic control allocation matrix will be introduced to obtain accurate revolutions per minute (RPM) of each motor.

2.2. Center of gravity

As shown in Figure 1, a morphing quadrotor consists of three parts: a body, arms, and motors. Four arms are connected with the body through four servo motors, and the servo motors can drive the arms to rotate horizontally for achieving morphological change. As the arm rotates, the CoG of quadrotors in the airframe coordinate system will change, which can be obtained as follows:

Figure 1. Morphing quadrotor system diagram.

(3) $$ \begin{eqnarray} \left\{ \begin{aligned} {\boldsymbol{r_{motor_1}}}= \begin{bmatrix} {x_{motor_1}^{ini}} \\ {y_{motor_1}^{ini}}\\ {z_{motor_1}^{ini}} \\ \end{bmatrix}+ \begin{bmatrix} {l_1(\rm{cos}{\theta_1}-1)} \\ {-l_1\rm{sin}{\theta_1}} \\ 0\\ \end{bmatrix} \\ {\boldsymbol{r_{motor_2}}}= \begin{bmatrix} {x_{motor_2}^{ini}} \\ {y_{motor_2}^{ini}}\\ {z_{motor_2}^{ini}} \\ \end{bmatrix}+ \begin{bmatrix} {l_1\rm{sin}{\theta_2}} \\ {l_1(\rm{cos}{\theta_2}-1)} \\ 0\\ \end{bmatrix} \\ {\boldsymbol{r_{motor_3}}}= \begin{bmatrix} {x_{motor_3}^{ini}} \\ {y_{motor_3}^{ini}} \\ {z_{motor_3}^{ini}} \\ \end{bmatrix}+ \begin{bmatrix} {l_1(1-\rm{cos}{\theta_3})} \\ {l_1\rm{sin}{\theta_3}} \\ 0\\ \end{bmatrix} \\ {\boldsymbol{r_{motor_4}}}= \begin{bmatrix} {x_{motor_4}^{ini}} \\ {y_{motor_4}^{ini}} \\ {z_{motor_4}^{ini}} \\ \end{bmatrix}+ \begin{bmatrix} {-l_1\rm{sin}{\theta_4}} \\ {l_1(1-\rm{cos}{\theta_4})} \\ 0\\ \end{bmatrix} \end{aligned} \right. \end{eqnarray} $$

where $$ {\boldsymbol{r_{motor_i}}} $$ is the CoG vector of the $$ i $$-th motor in the airframe coordinate system, $$ i $$=$$ {1, 2, 3, 4} $$, $$ {\boldsymbol{r_{motor_i}^{ini}}} = [{x_{motor_i}^{ini}} \; {y_{motor_i}^{ini}}\; {z_{motor_i}^{ini}}]^\top $$ is the initial position vector of the $$ i $$-th motor, $$ l $$ denotes the arm length, and $$ {\theta_{i}} $$ represents the $$ i $$-th arm rotation angle.

In addition, when the arm rotates, the CoG of arms can be obtained as follows:

(4) $$ \begin{eqnarray} \left\{ \begin{aligned} {\boldsymbol{r_{arm_1}}}= \begin{bmatrix} {x_{arm_1}^{ini}} \\ {y_{arm_1}^{ini}}\\ {z_{arm_1}^{ini}} \\ \end{bmatrix}+ \begin{bmatrix} {l_1(\rm{cos}{\theta_1}-1)} \\ {-l_1\rm{sin}{\theta_1}} \\ 0\\ \end{bmatrix} \\ {\boldsymbol{r_{arm_2}}}= \begin{bmatrix} {x_{arm_2}^{ini}} \\ {y_{arm_2}^{ini}}\\ {z_{arm_2}^{ini}} \\ \end{bmatrix}+ \begin{bmatrix} {l_1\rm{sin}{\theta_2}} \\ {l_1(\rm{cos}{\theta_2}-1)} \\ 0\\ \end{bmatrix} \\ {\boldsymbol{r_{arm_3}}}= \begin{bmatrix} {x_{arm_3}^{ini}} \\ {y_{arm_3}^{ini}} \\ {z_{arm_3}^{ini}} \\ \end{bmatrix}+ \begin{bmatrix} {l_1(1-\rm{cos}{\theta_3})} \\ {l_1\rm{sin}{\theta_3}} \\ 0\\ \end{bmatrix} \\ {\boldsymbol{r_{arm_4}}}= \begin{bmatrix} {x_{arm_4}^{ini}} \\ {y_{arm_4}^{ini}} \\ {z_{arm_4}^{ini}} \\ \end{bmatrix}+ \begin{bmatrix} {-l_1\rm{sin}{\theta_4}} \\ {l_1(1-\rm{cos}{\theta_4})} \\ 0\\ \end{bmatrix} \end{aligned} \right. \end{eqnarray} $$

where $$ {\boldsymbol{r_{arm_i}}} $$ is the CoG vector of the $$ i $$-th arm in the airframe coordinate system, $$ {\boldsymbol{r_{arm_i}^{ini}}} $$ = [$$ {x_{arm_i}^{ini}} $$$$ {y_{arm_i}^{ini}} $$$$ {z_{arm_i}^{ini}} $$]$$ ^\top $$ is the initial position vector of the $$ i $$-th arm, and $$ l_1 $$ denotes the length from the CoG of arms to a body.

Meanwhile, the CoG of the body will not change, which can be obtained as follows:

(5) $$ \begin{eqnarray} {\boldsymbol{r_{body}}}= {\boldsymbol{r_{body}^{ini}}}= \begin{bmatrix} {x_{body}^{ini}} \\ {y_{body}^{ini}}\\ {z_{body}^{ini}} \\ \end{bmatrix} \end{eqnarray} $$

Finally, the CoG of the morphing quadrotor can be derived as follows:

(6) $$ \begin{eqnarray} {\boldsymbol{{r}_{cog}}}=\frac{{m_{body}}\boldsymbol{r_{body}}+\sum\limits_{i=1}^{4}({m_{arm_i}}{\boldsymbol{r_{arm_i}}}+{m_{motor_i}}{\boldsymbol{r_{motor_i}}})}{{m_{body}}+\sum\limits_{i=1}^{4}({m_{arm_i}}+{m_{motor_i}})} \end{eqnarray} $$

where $$ {\boldsymbol{{r}_{cog}}} $$ is the CoG vector of the morphing quadrotor in the airframe coordinate system, $$ {m_{body}} $$, $$ {m_{arm_i}} $$, and $$ {m_{motor_i}} $$ denote body mass, mass of each arm, and mass of each motor, respectively.

All the initial position vectors mentioned previously can be obtained through SolidWorks.

3. CONTROLLER DEVELOPMENT

In this section, an ADRC method for the morphing quadrotor based on a GPI observer will be introduced. Firstly, a GPI observer will be proposed for accurate state estimation. Furthermore, a controller consisting of a GPI observer and a PD controller is proposed. Theoretical analysis demonstrates that the tracking error $$ {e} $$ will converge to a small neighborhood near the origin.

4. SIMULATION RESULTS

In this section, comparative simulations for disturbance estimation of the four-order GPI observer and ESO will be carried out. Meanwhile, the proposed PD+GPI observer controller is compared with a PD controller to demonstrate the effectiveness of the anti-disturbance capability.

The parameters of the four-order GPI observer are selected as $$ \lambda_0 $$ = 6$$ \omega_0 $$, $$ \lambda_1 $$ = 15$$ \omega_0^2 $$, $$ \lambda_2 $$ = 13$$ \omega_0^3 $$, $$ \lambda_3 $$ = 15$$ \omega_0 $$, $$ \lambda_4 $$ = 6$$ \omega_0 $$, and $$ \lambda_5 $$ = $$ \omega_0 $$, where $$ \omega_0 $$ = 50.

The controller parameters are selected as $$ K_p $$ = 100, $$ K_d $$ = 20, $$ b $$ = $$ \frac{1}{J_{yy}} $$, where $$ J_{yy} $$ can be obtained from $$ J $$, and it is changing in real-time with respect to the rotation of arms.

Remark 1: The parameter $$ \omega_0 $$ determines the response speed of the observer, and when it decreases, the estimated speed of the observer will respond slower. The controller parameters $$ K_p $$ and $$ K_d $$ are the PD controller gains that determine the response speed and overshoot of states, and $$ K_p $$ and $$ K_d $$ can be adjusted by following the tuning process of PID controllers.

As illustrated in Figure 2 and Figure 3, the quadrotor arms are rotating between 7 s to 8 s. In detail, the 1-th arm rotates from 45 deg to 0 deg, the 2-th arm rotates from 45 deg to 90 deg, the 3-th arm rotates from 45 deg to 0 deg, and the 4-th arm rotates from 45 deg to 90 deg.

Figure 2. Morphing quadrotor arms rotating diagram. On the right is the initial morphology, and on the left is the morphology after rotating.

Figure 3. Rotation of the morphing quadrotor arms.

As illustrated in Figure 4 and Figure 5, it can be seen the response time and estimation of the GPI observer are faster and more accurate than those of ESO. The GPI observer can track rapidly even if there exists a step part in the external disturbances, and for internal disturbances, the estimated delay will also not exceed 0.2 s. A potential reason is that the GPI observer has more augmented states to ensure the accuracy of estimation.

Figure 4. Simulation results of the internal dynamic change estimation. The actual internal dynamic change (black solid lines), the internal dynamic change estimated by a GPI observer (red solid lines), and the internal dynamic change estimated by an ESO (blue dashed lines).

Figure 5. Simulation results of external disturbance estimation. The actual external disturbance (black solid lines), the external disturbance estimated by a GPI observer (red solid lines), and the external disturbance estimated by an ESO (blue dashed lines).

As shown in Figure 5, a square wave, as the external disturbances, is given to the $$ \phi $$ channel, which will affect the morphing quadrotor position of $$ y $$. Simulation results are shown in Figure 6. The quadrotor will oscillate near the desired position only with a PD controller. Compared with the PD controller, it can be seen that the PD+GPI observer controller can effectively suppress disturbances and achieve precise position control without oscillation within 3 s, even the quadrotor suffers from external disturbances and morphological changes at the same time between 7 s to 8 s.

Figure 6. Simulation results of position control (Case 1). The desired position (black dashed lines), the position with PD+GPI observer controller (red solid lines), and the position with PD controller (blue dashed lines).

A sine wave signal $$ 8 $$$$ {\sin(2t)} $$ is given to $$ \theta $$ channel as external disturbances in Figure 7, which will affect the morphing quadrotor position of $$ x $$. From Figure 6 and Figure 7, it can be concluded that regardless of which channel is affected by disturbances, the proposed method can achieve satisfactory anti-disturbance and stable flight performances within 3 s, although the disturbance forms and desired operating points are different.

Figure 7. Simulation results of position control (Case 2). The desired position (black dashed lines), the position with PD+GPI observer controller (red solid lines), and the position with PD controller (blue dashed lines).

To further validate the effectiveness of the proposed method, a sine wave signal $$ 8 $$$$ {\sin(2t)} $$ is given to the $$ \phi $$ channel and $$ \theta $$ channel at the same time as external disturbances in Figure 8, which will affect the morphing quadrotor positions of $$ x $$ and $$ y $$ and the desired operating positions are also changed. It is clearly shown in Figure 8 that the proposed method can quickly suppress disturbances and maintain system stability in the $$ \phi $$ channel and $$ \theta $$ channel simultaneously, which demonstrates that the proposed method has a wider application range.

Figure 8. Simulation results of position control (Case 3). The desired position (black dashed lines), the position with PD+GPI observer controller (red solid lines), and the position with PD controller (blue dashed lines).

5. CONCLUSIONS

This paper presents a morphing quadrotor, which can cross through obstacles and narrow spaces by rotating arms. Meanwhile, a dynamic model is established by considering real-time changes of the CoG and MoI. According to arm rotation angles, a dynamic control allocation matrix is obtained. In addition, a four-order GPI observer is proposed to estimate the lumped disturbances. On this basis, an ADRC method is proposed for the morphing quadrotor position control. Finally, simulation results verify the good performance of a four-order GPI observer and the anti-disturbance ability of a proposed control method. In the future, the proposed method and other intelligent control methods will be applied to the actual system.

DECLARATIONS