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

*Intell Robot*2023;3(3):274-87.

## Abstract

Quadrotors are widely used in transportation, aerial photography, agricultural protection, and other important fields. Nevertheless, quadrotors with a fixed structure will face great challenges when crossing through or entering narrow spaces for operations. To improve quadrotor crossing ability in different environments, a morphing quadrotor is designed in this paper, and four servo motors are added to independently change four arm rotation angles. Meanwhile, the dynamic model and dynamic control allocation matrix are established. In addition, considering that the internal dynamic variation caused by morphologic changes and external disturbances may compromise system stability, a control method based on the generalized proportional integral (GPI) observer is proposed to increase the system robustness, and the corresponding stability analysis is provided. Finally, simulation results demonstrate the effectiveness of the proposed GPI observer-based active disturbance rejection control method.

## Keywords

*,*dynamic control allocation

*,*generalized proportional integral observer

*,*active disturbance rejection control

## 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.

### 2.1. Dynamic model

First, morphing quadrotor position state equations are obtained as follows:

where

In addition, attitude models can be obtained as follows:

where

### 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:

where

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

where

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

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

where

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

### 2.3. Moment of inertia

Morphology change not only affects the CoG but also changes the MoI of the morphing quadrotor. The MoI of a morphing quadrotor can be obtained through CoGs calculated in the previous subsection.

In addition, it can be assured that the MoIs of the body and motors are hardly affected, which can be obtained as

where

The MoI of arms can be obtained as follows:

where

The MoI of a morphing quadrotor can be obtained through the relevant physical equations

where

All the initial MoI matrices mentioned above can be obtained through SolidWorks.

### 2.4. Control allocation

The control allocation matrix will change when the morphology changes. It is necessary to obtain the control allocation matrix in real time to convert control inputs into accurate RPM generated by actuators.

As shown in Figure 1, the lift force

where

In addition, the control allocation equations can be derived as follows:

where

Due to the arms rotating horizontally,

where

Define the following equations

The RPM of a motor can be derived as follows:

where

## 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

### 3.1. The GPI observer

According to the dynamic model of the morphing quadrotor, each state can be rewritten; for example, a rolling channel can be taken as

where

On this basis, the GPI observer can be designed as follows^{[15,16]}:

where

Defining

According to Equation (18), Equation (19) can be derived as follow:

It can be assumed that ^{[19]}.

### 3.2. Controller

Each channel of the morphing quadrotor dynamic model can be written as follow:

The controller is designed as

where

Defining

Furthermore, Equation (22) can be rewritten as

It can be known that ^{[19]}.

## 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

The controller parameters are selected as

**Remark 1:** The parameter

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.

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

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

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

## 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

### Authors' contributions

Made substantial contributions to the conception and design of the study and performed simulation and interpretation: He C, Sun H, and Su Y;

Provided administrative, technical, and material support: Sun N, Wu Q

### Availability of data and materials

Not applicable

### Financial support and sponsorship

This work was supported by the National Natural Science Foundation of China under Grant 52205019, the China Postdoctoral Science Foundation under Grant 2021M701779, and the State Key Laboratory of Robotics and Systems (HIT) under Grant SKLRS-2023-KF-13.

### Conflicts of interest

All authors declared that there are no conflicts of interest.

### Ethical approval and consent to participate

Not applicable.

### Consent for publication

Not applicable.

### Copyright

© The Author(s) 2023.

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## Cite This Article

## How to Cite

He, C.; Sun H.; Wu Q.; Su Y.; Sun N. GPI observer-based active disturbance rejection control for a morphing quadrotor. *Intell. Robot.* **2023**, *3*, 274-87. http://dx.doi.org/10.20517/ir.2023.18

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