# Design and Control of a Deformable Trees-Pruning Aerial Robot

Authors: Changliang Xu; Zhong Yang; Zhao Zhang; Hao Xu; Jiying Wu; Dongsheng Zhou; Luwei Liao; Qiuyan Zhang
Journal: Complexity
 (2020)

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## Abstract

Tree branches near the electric power transmission lines are of great threat to the electricity supply. Nowadays, the tasks of clearing threatening tree branches are still mostly operated by hand and simple tools. Traditional structures of the multirotor aerial robot have the problem of fixed structure and limited performance, which affects the stability and efficiency of pruning operation. In this article, in order to obtain better environmental adaptability, an active deformable trees-pruning aerial robot is presented. The deformation of the aerial robot is implemented through two ways, arm telescopic and folding. In order to suppress the influence of internal and external disturbances on the system, Active Disturbance Rejection Control (ADRC) technology is adopted to build the flight controller. Firstly, active deformation aerial robot structure is given, followed by system dynamic model establishment under wind disturbance using the Newton–Euler method. Also, the analysis of the gusts influence on the system is considered. Then, the active deformation aerial robot system is decoupled into a combination of six SISO systems, so that a disturbance rejection controller is designed. Finally, the expanded state observer and the nonlinear state error feedback law are used to inspect and compensate the disturbance. Simulation results of attitude and position tracking as well as the antidisturbance capability show that the active deformation aerial robot with the ADRC flight controller designed in this paper has excellent attitude control capabilities during flight and trees-pruning operation.


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## Body

## 1. Introduction

With the feature of complexity, intelligent systems have significant research values and huge application potential [1, 2]. In the giant and complex power grid, it has been always a tough issue that trees near the high-voltage transmission line channel often exceed the safe distance, causing the line to discharge through the trees to the ground, causing safety accidents such as power outages and fires, bringing huge inconvenience to social production and life. Therefore, the power sector consumes a lot of manpower and material resources every year for tree barrier cleaning. At present, most of its main methods use manpower, supplemented by simple tools such as hand saws. Not only is this kind of cleaning method limited in efficiency, but also human safety cannot be guaranteed. Casualties caused by such tasks in our country occur from time to time. Therefore, it is particularly necessary to develop and promote a set of efficient and safe tree barrier cleaning aerial robots that can replace manual cleaning methods.The multirotor aerial robot that can take off and land vertically and hover in the air has the advantages of simple structure and good stability. The application research of using it as a work platform has attracted widespread attention [3–5].However, the traditional structure of the multirotor aerial robot has the problem of fixed structure and limited performance, which affects the stability and efficiency of pruning operation.In response to the above-mentioned problems of conventional multirotor aircraft, researchers have studied improving the controllability of the aircraft by optimizing the relative direction of the rotors or using tiltable rotors. Paper [6] puts forward a new type of rotor installation plan in which eight rotors are arranged in a cube structure in different directions through the analysis of the force and moment of ordinary motors. Paper [7] combines the advantages of conventional multirotor systems and omnidirectional control aircraft and proposes a new type of flying platform. Papers [8, 9] proposed a design scheme for a tiltable quadrotor and verified the effectiveness of the tiltable multirotor aircraft through simulation and actual flight experiments. Although these methods help achieve complex trajectories and operational tasks, they do not significantly change the shape of the multirotor aircraft frame.On the contrary, only by directly changing the shape and structure of the multirotor aircraft during flight can this problem be effectively solved. Paper [10] focused on kinematics design and studied a new type of deformable quadrotor based on a scissor-like foldable structure, which can dynamically adjust the volume of the quadrotor to adapt to obstacles of various sizes and narrower space. Paper [11] proposed a deformable aircraft with four rotors connected in series, established its dynamic model, and designed an attitude LQI controller and a position PID controller to achieve the purpose of encircling the object and grasping it without additional gripping device. Paper [12] improved the diversity of aircraft deformation, divided the standard quadrotor structure into multiconnected platforms, and proposed a new control method that decoupled thrust control and rotor gimbal control and realized 3D folding. Paper [13] optimized the structure of the quadrotor and designed the automatic telescopic structure in the air and the manual folding structure after landing. The rational selection of materials and the strength check ensure the rationality of the airframe and can achieve the expected functions. Paper [14] designed a miniature deformable quadrotor, which achieved the purpose of deformation by using flexible materials to manufacture aircraft brackets and designing torsion spring and gear deformation mechanism. PID was selected as its flight control method.The active deformation quadrotor is an aircraft that can actively change its shape during flight. Its shape changes have a strong influence on the mechanical properties, especially the expansion and folding of the arms, which not only directly change the position of the center of gravity and the inertia tensor of the aircraft but also change the mapping relationship between the lift generated by a single motor and the force and moment of the aircraft body [15]. This makes the active deformable quadrotor more uncertain and more coupled than the conventional quadrotor. In addition, the disturbance of the external environment during flight is inevitable, which makes the design of its controller more difficult. Researcher Han Jingqing inherited the essence of classic PID controllers and proposed Active Disturbance Rejection Control (ADRC) technology that requires low model accuracy and is easy to implement. ADRC has the advantages of fast control response speed and no overshoot, and transition process can be arranged. Also, the parameter adaptation range is large and has the ability to observe and compensate for disturbances inside and outside the system [16–18]. Because ADRC has obvious advantages in solving the control problems of nonlinear models with uncertainties and strong disturbances [19], researchers have tried ADRC on conventional quadrotor and verified the antidisturbance capability and robustness of the controller [20, 21]. Active Disturbance Rejection Control (ADRC) technology is a new type of control theory that does not require accurate modelling of the controlled object. A major feature of this theory is that it could effectively suppress the internal and external disturbances of the system, because it is able to estimate and compensate for the internal disturbances of the system and the unknown disturbances of the external environment in real time. During the operation of the tree barrier cleaning robot, the external force caused by the contact operation is a huge disturbance to the body, which brings about great challenges to the attitude stability control. In addition, ADRC has the characteristic of small amount of calculation, which makes it easy to realize real-time control of the robot's pose on the MCU (Microcontroller Unit).With the purpose of pruning trees near the power transmission line, this paper proposes an active deformable aerial robot to improve environmental adaptability. The deformation forms are divided into arm telescopic and folding. In order to better suppress the internal and external disturbances of the system, this paper designs a flight control method based on Active Disturbance Rejection Control technology. First, the structural design is carried out, followed by the establishment of the dynamic model. Then analysis and calculation of the changes in the center of gravity and inertia matrix caused by the deformation are presented. Second, the system is decoupled and the attitude ADRC controller is designed. Then, the control distribution matrix form is given. Finally, through simulation, the deformation ability of the designed active deformation aerial robot and the effectiveness, anti-interference performance, and robustness of the ADRC controller are verified.


## 2. Structural Design of Active Deformation Aerial Robot

The active deformation aerial robot designed in this paper is mainly composed of the following four parts: (1) the centered fuselage, (2) the deformable arm, (3) the motor and the rotor, and (4) deformation of the arm to achieve the aerial robot deforming various steering gears. Assume that the deformable aerial robot is a rigid body, as shown in Figure1. The earth fixed coordinate system OExEyEzE is defined to be fixed to the ground, and the origin of the body coordinate system OBxByBzB is fixed at the center of the robot fuselage. At the same time, the telescopic rudder of the active deformable active aerial robot is defined. The engine is installed at the midpoint of the arm, and the rotary servo is installed on the fuselage. For the convenience of writing, this article uses ith to represent the ith, i∈1,2,3,4. Define li∈15cm,25cm as the length of ith retractable arm. Define δi as the rotation angle of the ith arm around the zB axis of the machine system. When the ith arm rotates clockwise, δi is positive. Only δ3 is shown in Figure 1. The numbers of the arms and motors have been marked in the figure. Motors nos. 1 and 3 rotate counterclockwise, and motors nos. 2 and 4 rotate clockwise.Figure 1
Frames of actively deformable aerial robot.The active deformation aerial robot has two deformation functions. (1) The arm is stretched and deformed by the telescopic steering gear; that is, the length of the arm is changed. (2) The arm is folded and deformed around thezB axis of the machine system by rotating the steering gear changing δi. Its four arms can be deformed individually or in combination. In the following of the paper, expansion and folding are used to represent these two kinds of deformation.


### 2.1. Kinematics and Dynamics

This article usesR⋅⋅∈SO3 to represent the rotation matrix between coordinate systems, and RBE represents the rotation matrix from the machine system to the earth’s fixed coordinate system (in the text, s⋅ means sin⋅, c⋅ means cos⋅, and t⋅ means tan⋅):(1)RBE=cθcψsϕsθcψ−cϕsψcϕsθcψ+sϕsψcθsψsϕsθsψ+cϕcψcϕsθsψ−sϕcψ−sθsϕcθcϕcθ.The rigid body kinematics model of the actively deformed aerial robot is(2)P˙E=VE,Θ˙=WΩB,where PE=x;y;z and VE=u;v;w, respectively, represent the airframe position and triaxial velocity of the robot in the earth's fixed coordinate system, Θ=ϕ;θ;ψ represents the Euler angle of the robot, and ΩB=p;q;r represents the triaxial angular velocity of the robot in the airframe coordinate system.(3)W=1tθsϕtθcϕ0cϕ−sϕ0sϕcθcϕcθ.The force and moment changes caused by the deformation of the active deforming aerial robot are more complicated than the conventional aerial robots. The dynamic model of the active deforming aerial robot is analyzed and established below.The fuselage is regarded as a mass point during the deformation process. As a six-degree-of-freedom rigid body, the dynamic model of the actively deformed aerial robot is obtained by Newton–Euler equations [22]. In this paper, the system translation equation is defined in the earth’s fixed coordinate system, and the rotation equation is defined in the body coordinate system:(4)mV˙EIBΩ˙B+0ΩB×IBΩB=FEMB,where m represents the mass of the machine body and IB is the inertial matrix. Its value will change due to the deformation of the body, which will be analyzed in detail later.The force analysis of the active deformation aerial robot system is carried out, and the following is obtained:(5)FE=GE+FtE+FwindE+DE,FtE=RBE∑i=14Ti,Ti=00−kfni2T.where GE represents gravity. FtE represents the lift generated by the rotor. FwindE represents the wind disturbance. DE represents the air resistance. kf is the lift coefficient of the rotor, and ni is the speed of the ith rotor.Before analyzing the torque experienced by the active deformable aerial robot, the position of the center of mass and the inertia matrix are first analyzed. Compared with conventional quadrotors, the position of the center of mass and the inertia matrix of the actively deformed aerial robots will change due to the expansion and folding of the arms. Therefore, when the system is deformed, the position of the center of mass and the inertia matrix of the system need to be recalculated. The offset between the center of mass of the active deformed aerial robot and the origin of the system coordinates [23] rCoG∈R3 is expressed as follows:(6)rCoG=mbodyrbody+Ambody+∑i=14marm+mmot+mrot+melo,A=∑i=14marmrarm,i+mmotrmot,i+mrotrrot,i+melorelo,i,where m⋅ represents the mass of ⋅ and r⋅ represents the vector diameter from the origin of the machine system coordinates to ⋅. The meaning of each subscript English letter is shown in Table 1.Table 1
Meaning of subscript letter.
SubscriptsDescriptionsbodyRobot bodyarm,iith robot propeller armmot,iith motorrot,iith rotationelo,iith telescopic steering gearFor the body inertia matrixIB of an actively deformed aerial robot, the parallel axis theorem can be used for calculation. Here are some assumptions: the fuselage containing the rotating steering gear is regarded as a rectangular parallelepiped with length and width of lbody and height of hbody. The cuboid length, width, and height are larm,i, warm, and harm, and the rotor and the motor are regarded as cylinders, in which the rotor radius is rmot, the height is hmot, the motor radius is rrot, and the height is hrot; the telescopic steering gear is regarded as the cuboid with length, width, and height, respectively, of lelo, welo, and helo. The calculation formula for the moment of inertia of each part [24] is as follows:(7)Ibody=mbody12diaghbody2+lbody2,hbody2+lbody2,lbody2+lbody2,I˜arm,i=marm12diagwarm2+harm2,harm2+larm,i2,warm2+larm,i2,Imot=mmot12diag3rmot2+hmot2,3rmot2+hmot2,6rmot2,Irot=mrot12diag3rrot2+hrot2,3rrot2+hrot2,6rrot2,I˜elo,i=melo12diagwelo2+helo2,helo2+lelo2,welo2+lelo2,where I⋅ represents the inertia value of ⋅.Considering that the motor, rotor, arm, and telescopic steering gear are actively deforming the aerial robot during folding and deformation, inertia must also be recalculated [25]. Since the inertia of the cylinder does not change when it is folded and deformed, this change can be ignored for the inertia of the motor and the rotor. Since the fuselage structure is fixedly connected to the fuselage coordinate system, the inertia of the fuselage does not need to be changed. Therefore, the rotation matrix is introduced to rerepresent them:(8)Iarm,i=RzδiI˜arm,iRzδiTIelo,i=RzδiI˜elo,iRzδiT,i∈1,2,3,4.The rotation matrix is expressed as follows:(9)Rzδi=cδisδi0−sδicδi0001.So, the inertia calculation formula can be expressed as(10)IB=Ibody−mbodyrbody−rCoG2+∑i=14Iarm,i−marmrarm,i−rCoG2+Imot−mmotrmot,i−rCoG2+Irot−mrotrrot,i−rCoG2+Ielo,i−melorelo,i−rCoG2.Analyzing the moment of the active deforming aerial robot, we get(11)MB=MtB+MantiB+MgyroB+MwindB+MGB,where MtB is the torque generated by the lift of the rotor, MantiB is the rotation reaction torque of the rotor, MgyroB is the system gyro effect item, and MwindB is the wind disturbance moment. Different from the conventional quadrotor, there is an additional gravitational moment MGB.The torque generated by the rotor thrust is(12)MtB=∑i=14Li×Ti,where Lirepresents the coordinate vector of the lift application point of the ithrotor in the robot system. The expression is as follows:(13)Li=li−cαi+i−1/2πsαi+i−1/2π0T,i∈1,3,li−sαi+i−2/2π−cαi+i−2/2π0T,i∈2,4,where αi=π/4−δi is converted to angle system, namely, αi∈0∘,90∘.The antitorque expression is(14)MantiB=00∑i=14QiT,Qi=kmni2i∈1,3−kmni2i∈2,4,km>0.The system gyro effect is mainly caused by the rotation of the body:(15)MgyroB=JP∑i=14ΩB×−1fi00ni,fi=1,i∈1,3,0,i∈2,4.The expression of gravitational moment is(16)MGB=rCoG×REBGE.


### 2.2. Wind Disturbance Modelling

The influence of the wind field environment on the multirotor aircraft is partly caused by the aerodynamic effect of the rotor and is also partly caused by the air resistance of the frame of the multirotor aircraft [26]. The active deformation aerial robot will inevitably be affected by the external wind field when it is flying, especially when it is deformed. So, wind gusts will have a certain impact on it. The wind disturbance model can calculate the forces and moments acting on the actively deformed aerial robot in a given wind field, so modelling and analysis of wind disturbance are essential [27].The schematic diagram of the aerodynamic effect of the rotor under the action of the wind field is shown in Figure2. In the figure, Vwind represents the wind speed, Vd represents the induced speed, and Vtotal represents the vector sum of the wind speed and the induced speed.Figure 2
Aerodynamics of rotor under wind gust.According to the rotor slip flow theory [28], the calculation formula of induced speed is as follows:(17)Vt=kfn22πρr2,where ρ represents the local air density and r represents the radius of the rotor blade. When there is a wind field, the total aerodynamic force experienced by a single rotor [29] is(18)Ti+Fwi=2πρr2VtVwind+Vt,i∈1,2,3,4.So, the wind disturbance force and wind disturbance moment of theith rotor are as follows:(19)Fwi=kfni2−2πρr200kfn22πρr2+REBVwindE,(20)Mwi=−kmkfFwi,i∈1,3,kmkfFwi,i∈2,4.Thereby,(21)FwindE=RBE∑i=4400Fwi,(22)MwindB=∑i=4400Mwi.The air resistance calculation formula is(23)DE=12cρSairVair2,where c represents the coefficient of air resistance, Sair represents the windward area, and Vair represents the relative speed of the robot and the air; that is, Vair=VwE−VE. When calculating the windward area of a deformable robot, the robot is simplified as a cylinder, and the average windward area is taken as S¯=σ2πl¯h+1−σπl¯2, where σ∈0,1 is the crosswind coefficient, h is the height of the robot body, and l¯ is the average length of the arms, so(24)DE=12cρS¯airVairVair.


### 2.3. Control Distribution

Like the conventional quadrotor, the active deformation aerial robot system control force and moment are mainly the thrust and reaction torque generated by the rotor. Combining equations (4)–(24), UFE and UMB are used to represent the force and torque generated by the rotor. The dynamic model can be reexpressed:(25)mV˙EIBΩ˙B+ΩB×IBΩB=GE+FwindE+DE+UFEMgyroB+MwindB+MGB+UMB.The relationship betweenUFE and UMB and rotor speed nis(26)UFEUMB=Cn12n22n32n42,(27)C=kfkfkfkf−kfl1cα1−kfl2sα2kfl3cα3kfl4sα4kfl1sα1−kfl2cα2−kfl3sα3kfl4cα4km−kmkm−km.MatrixC is the control efficiency matrix. It can be seen that matrix C is a time-varying matrix. The following verification matrix C is singular only at the singular point α1,α2,α3,α4∈0∘,90∘,0∘,90∘,90∘,0∘,90∘,0∘, and the other matrices C are all nonsingular.Proof.
Using the method of proof by contradiction and assuming that the square matrixC is a singular matrix, its columns must not be full of rank. Because of kf,km≠0, only column 1 and column 3 or column 2 and column 4 are proportional. Let Cij be the element in row i and column j for discussion:(1)
Assuming that the sine and cosines of all angles are not 0, column 1 and column 3 are proportional.C11/C13=C41/C43=1, so −kfl1cα1/kfl3cα3=1, that is, cα1/cα3=−l3/l1, and sα1/sα3=−l3/l1 derives cα1/cα3=sα1/sα3, which is further written as cα1sα3−sα1cα3=0, that is, sα1−α3=0, which is also equivalent to α1−α3=kπ,k∈Z. At this time, only the situation where α1=α3=0∘ exists, but it is contrary to the assumption, so there is no solution.(2)
Assuming that the sine and cosines of all angles are not 0, column 2 and column 4 are proportional: with the same principle, there is no solution.(3)
Suppose that there are an angular sine and cosine of 0: whencα=0 or sα=0, αis 0∘ or 90∘, and the empirical calculation is that only in the case of α1,α2,α3,α4∈0∘,90∘,0∘,90∘,90∘,0∘,90∘,0∘ is matrix C not full of rank, which is a singular matrix.
Proof ends.
Since the deformation process of the actively deformed aerial robot in this paper does not involve the location of singular points, it can be considered that matrixC is fully reversible. So, the form of control distribution is as follows:(28)n12n22n32n42=C−1UFEUMB.
It can be seen from equations (25)–(28) that the active deformation aerial robot model has strong nonlinearity and coupling. The deformation causes the length and angle of the arm to change, which leads to the change of model parameters. Compared with the conventional aerial robot, the internal disturbance is greater. At the same time, external disturbances such as the wind field experienced by the actively deformed aerial robot cannot be ignored.


## 2.1. Kinematics and Dynamics

This article usesR⋅⋅∈SO3 to represent the rotation matrix between coordinate systems, and RBE represents the rotation matrix from the machine system to the earth’s fixed coordinate system (in the text, s⋅ means sin⋅, c⋅ means cos⋅, and t⋅ means tan⋅):(1)RBE=cθcψsϕsθcψ−cϕsψcϕsθcψ+sϕsψcθsψsϕsθsψ+cϕcψcϕsθsψ−sϕcψ−sθsϕcθcϕcθ.The rigid body kinematics model of the actively deformed aerial robot is(2)P˙E=VE,Θ˙=WΩB,where PE=x;y;z and VE=u;v;w, respectively, represent the airframe position and triaxial velocity of the robot in the earth's fixed coordinate system, Θ=ϕ;θ;ψ represents the Euler angle of the robot, and ΩB=p;q;r represents the triaxial angular velocity of the robot in the airframe coordinate system.(3)W=1tθsϕtθcϕ0cϕ−sϕ0sϕcθcϕcθ.The force and moment changes caused by the deformation of the active deforming aerial robot are more complicated than the conventional aerial robots. The dynamic model of the active deforming aerial robot is analyzed and established below.The fuselage is regarded as a mass point during the deformation process. As a six-degree-of-freedom rigid body, the dynamic model of the actively deformed aerial robot is obtained by Newton–Euler equations [22]. In this paper, the system translation equation is defined in the earth’s fixed coordinate system, and the rotation equation is defined in the body coordinate system:(4)mV˙EIBΩ˙B+0ΩB×IBΩB=FEMB,where m represents the mass of the machine body and IB is the inertial matrix. Its value will change due to the deformation of the body, which will be analyzed in detail later.The force analysis of the active deformation aerial robot system is carried out, and the following is obtained:(5)FE=GE+FtE+FwindE+DE,FtE=RBE∑i=14Ti,Ti=00−kfni2T.where GE represents gravity. FtE represents the lift generated by the rotor. FwindE represents the wind disturbance. DE represents the air resistance. kf is the lift coefficient of the rotor, and ni is the speed of the ith rotor.Before analyzing the torque experienced by the active deformable aerial robot, the position of the center of mass and the inertia matrix are first analyzed. Compared with conventional quadrotors, the position of the center of mass and the inertia matrix of the actively deformed aerial robots will change due to the expansion and folding of the arms. Therefore, when the system is deformed, the position of the center of mass and the inertia matrix of the system need to be recalculated. The offset between the center of mass of the active deformed aerial robot and the origin of the system coordinates [23] rCoG∈R3 is expressed as follows:(6)rCoG=mbodyrbody+Ambody+∑i=14marm+mmot+mrot+melo,A=∑i=14marmrarm,i+mmotrmot,i+mrotrrot,i+melorelo,i,where m⋅ represents the mass of ⋅ and r⋅ represents the vector diameter from the origin of the machine system coordinates to ⋅. The meaning of each subscript English letter is shown in Table 1.Table 1
Meaning of subscript letter.
SubscriptsDescriptionsbodyRobot bodyarm,iith robot propeller armmot,iith motorrot,iith rotationelo,iith telescopic steering gearFor the body inertia matrixIB of an actively deformed aerial robot, the parallel axis theorem can be used for calculation. Here are some assumptions: the fuselage containing the rotating steering gear is regarded as a rectangular parallelepiped with length and width of lbody and height of hbody. The cuboid length, width, and height are larm,i, warm, and harm, and the rotor and the motor are regarded as cylinders, in which the rotor radius is rmot, the height is hmot, the motor radius is rrot, and the height is hrot; the telescopic steering gear is regarded as the cuboid with length, width, and height, respectively, of lelo, welo, and helo. The calculation formula for the moment of inertia of each part [24] is as follows:(7)Ibody=mbody12diaghbody2+lbody2,hbody2+lbody2,lbody2+lbody2,I˜arm,i=marm12diagwarm2+harm2,harm2+larm,i2,warm2+larm,i2,Imot=mmot12diag3rmot2+hmot2,3rmot2+hmot2,6rmot2,Irot=mrot12diag3rrot2+hrot2,3rrot2+hrot2,6rrot2,I˜elo,i=melo12diagwelo2+helo2,helo2+lelo2,welo2+lelo2,where I⋅ represents the inertia value of ⋅.Considering that the motor, rotor, arm, and telescopic steering gear are actively deforming the aerial robot during folding and deformation, inertia must also be recalculated [25]. Since the inertia of the cylinder does not change when it is folded and deformed, this change can be ignored for the inertia of the motor and the rotor. Since the fuselage structure is fixedly connected to the fuselage coordinate system, the inertia of the fuselage does not need to be changed. Therefore, the rotation matrix is introduced to rerepresent them:(8)Iarm,i=RzδiI˜arm,iRzδiTIelo,i=RzδiI˜elo,iRzδiT,i∈1,2,3,4.The rotation matrix is expressed as follows:(9)Rzδi=cδisδi0−sδicδi0001.So, the inertia calculation formula can be expressed as(10)IB=Ibody−mbodyrbody−rCoG2+∑i=14Iarm,i−marmrarm,i−rCoG2+Imot−mmotrmot,i−rCoG2+Irot−mrotrrot,i−rCoG2+Ielo,i−melorelo,i−rCoG2.Analyzing the moment of the active deforming aerial robot, we get(11)MB=MtB+MantiB+MgyroB+MwindB+MGB,where MtB is the torque generated by the lift of the rotor, MantiB is the rotation reaction torque of the rotor, MgyroB is the system gyro effect item, and MwindB is the wind disturbance moment. Different from the conventional quadrotor, there is an additional gravitational moment MGB.The torque generated by the rotor thrust is(12)MtB=∑i=14Li×Ti,where Lirepresents the coordinate vector of the lift application point of the ithrotor in the robot system. The expression is as follows:(13)Li=li−cαi+i−1/2πsαi+i−1/2π0T,i∈1,3,li−sαi+i−2/2π−cαi+i−2/2π0T,i∈2,4,where αi=π/4−δi is converted to angle system, namely, αi∈0∘,90∘.The antitorque expression is(14)MantiB=00∑i=14QiT,Qi=kmni2i∈1,3−kmni2i∈2,4,km>0.The system gyro effect is mainly caused by the rotation of the body:(15)MgyroB=JP∑i=14ΩB×−1fi00ni,fi=1,i∈1,3,0,i∈2,4.The expression of gravitational moment is(16)MGB=rCoG×REBGE.


## 2.2. Wind Disturbance Modelling

The influence of the wind field environment on the multirotor aircraft is partly caused by the aerodynamic effect of the rotor and is also partly caused by the air resistance of the frame of the multirotor aircraft [26]. The active deformation aerial robot will inevitably be affected by the external wind field when it is flying, especially when it is deformed. So, wind gusts will have a certain impact on it. The wind disturbance model can calculate the forces and moments acting on the actively deformed aerial robot in a given wind field, so modelling and analysis of wind disturbance are essential [27].The schematic diagram of the aerodynamic effect of the rotor under the action of the wind field is shown in Figure2. In the figure, Vwind represents the wind speed, Vd represents the induced speed, and Vtotal represents the vector sum of the wind speed and the induced speed.Figure 2
Aerodynamics of rotor under wind gust.According to the rotor slip flow theory [28], the calculation formula of induced speed is as follows:(17)Vt=kfn22πρr2,where ρ represents the local air density and r represents the radius of the rotor blade. When there is a wind field, the total aerodynamic force experienced by a single rotor [29] is(18)Ti+Fwi=2πρr2VtVwind+Vt,i∈1,2,3,4.So, the wind disturbance force and wind disturbance moment of theith rotor are as follows:(19)Fwi=kfni2−2πρr200kfn22πρr2+REBVwindE,(20)Mwi=−kmkfFwi,i∈1,3,kmkfFwi,i∈2,4.Thereby,(21)FwindE=RBE∑i=4400Fwi,(22)MwindB=∑i=4400Mwi.The air resistance calculation formula is(23)DE=12cρSairVair2,where c represents the coefficient of air resistance, Sair represents the windward area, and Vair represents the relative speed of the robot and the air; that is, Vair=VwE−VE. When calculating the windward area of a deformable robot, the robot is simplified as a cylinder, and the average windward area is taken as S¯=σ2πl¯h+1−σπl¯2, where σ∈0,1 is the crosswind coefficient, h is the height of the robot body, and l¯ is the average length of the arms, so(24)DE=12cρS¯airVairVair.


## 2.3. Control Distribution

Like the conventional quadrotor, the active deformation aerial robot system control force and moment are mainly the thrust and reaction torque generated by the rotor. Combining equations (4)–(24), UFE and UMB are used to represent the force and torque generated by the rotor. The dynamic model can be reexpressed:(25)mV˙EIBΩ˙B+ΩB×IBΩB=GE+FwindE+DE+UFEMgyroB+MwindB+MGB+UMB.The relationship betweenUFE and UMB and rotor speed nis(26)UFEUMB=Cn12n22n32n42,(27)C=kfkfkfkf−kfl1cα1−kfl2sα2kfl3cα3kfl4sα4kfl1sα1−kfl2cα2−kfl3sα3kfl4cα4km−kmkm−km.MatrixC is the control efficiency matrix. It can be seen that matrix C is a time-varying matrix. The following verification matrix C is singular only at the singular point α1,α2,α3,α4∈0∘,90∘,0∘,90∘,90∘,0∘,90∘,0∘, and the other matrices C are all nonsingular.Proof.
Using the method of proof by contradiction and assuming that the square matrixC is a singular matrix, its columns must not be full of rank. Because of kf,km≠0, only column 1 and column 3 or column 2 and column 4 are proportional. Let Cij be the element in row i and column j for discussion:(1)
Assuming that the sine and cosines of all angles are not 0, column 1 and column 3 are proportional.C11/C13=C41/C43=1, so −kfl1cα1/kfl3cα3=1, that is, cα1/cα3=−l3/l1, and sα1/sα3=−l3/l1 derives cα1/cα3=sα1/sα3, which is further written as cα1sα3−sα1cα3=0, that is, sα1−α3=0, which is also equivalent to α1−α3=kπ,k∈Z. At this time, only the situation where α1=α3=0∘ exists, but it is contrary to the assumption, so there is no solution.(2)
Assuming that the sine and cosines of all angles are not 0, column 2 and column 4 are proportional: with the same principle, there is no solution.(3)
Suppose that there are an angular sine and cosine of 0: whencα=0 or sα=0, αis 0∘ or 90∘, and the empirical calculation is that only in the case of α1,α2,α3,α4∈0∘,90∘,0∘,90∘,90∘,0∘,90∘,0∘ is matrix C not full of rank, which is a singular matrix.
Proof ends.
Since the deformation process of the actively deformed aerial robot in this paper does not involve the location of singular points, it can be considered that matrixC is fully reversible. So, the form of control distribution is as follows:(28)n12n22n32n42=C−1UFEUMB.
It can be seen from equations (25)–(28) that the active deformation aerial robot model has strong nonlinearity and coupling. The deformation causes the length and angle of the arm to change, which leads to the change of model parameters. Compared with the conventional aerial robot, the internal disturbance is greater. At the same time, external disturbances such as the wind field experienced by the actively deformed aerial robot cannot be ignored.


## 3. Design of Active Disturbance Rejection Flight Control Law

One of the significant features of Active Disturbance Rejection Control technology is that it has low dependence on the accuracy of the system modelling. It could estimate and dynamically compensate for the internal and external disturbances of the system as well, which contributes to the system's disturbance rejection and robustness ability. In this paper, Active Disturbance Rejection Control algorithms are implemented using the following three parts: tracking differentiator (TD), extended state observer (ESO), and nonlinear state error feedback (NLSEF).


### 3.1. Preliminary of the ARDC Law Design

There is a coupling between the position and attitude of the actively deformed aerial robot studied in this paper, which is an underactuated system. Only six-degree-of-freedom motion can be achieved in the incomplete sense, and the roll and pitch angles need to be changed while doing horizontal motion [30]. ADRC provides a more convenient way for the decoupling control of the multivariable coupling system, in which 6 states in the dynamic model of the active deformation aerial robot system are regarded as 6 channels, and ADRC can cause the coupling and deformation between the channels of the system. The parameter perturbation is treated as internal disturbance, and ESO is used to estimate and compensate for the internal and external disturbances of the system to realize the decoupling of the state of each channel. The virtual control quantity is introduced to realize the decoupling of system control, thereby converting the system description form from MIMO to six SISO systems combination. Therefore, the system's six-degree-of-freedom dynamic model equation (25) is rewritten in the following form:(29)x¨=s1x,x˙,ωout1,ωdef1+b1u1,y¨=s2y,y˙,ωout2,ωdef2+b2u2,z¨=s3z,z˙,ωout3,ωdef3+b3u3,ϕ¨=s4p,q,r,ϕ,ϕ˙,θ,θ˙,ψ,ψ˙,ωout4,ωdef4+b4u4,θ¨=s5p,q,r,ϕ,ϕ˙,θ,θ˙,ψ,ψ˙,ωout5,ωdef5+b5u5,ψ¨=s6p,q,r,ϕ,ϕ˙,θ,θ˙,ψ,ψ˙,ωout6,ωdef6+b6u6,where si⋅ is the uncertainty item and ωouti and ωdefi are the disturbances caused by the external disturbance and deformation of the system, respectively. b1,b2,b3 is an adjustable parameter with a size near 1/m. b4,b5,b6 is an adjustable parameter with a size near 1/Ixx,1/Iyy,1/Izz, and u1,u2,u3,u4,u5,u6 is the introduced virtual control quantity.Based on this, the control law adopts an inner and outer loop strategy, where the inner loop is attitude control and the outer loop is position control. The introduction of the control quantityU1,U2,U3,U4, respectively, represents the expected value of the total lift and the expected value of the torque around the three axes of the machine system. The actual position value x,y,z and the position expected value xd,yd,zd are used as the input of the outer loop position ADRC control law, and the expected values of roll and pitch angles ϕd,θd and U1 are output. The actual attitude value ϕ,θ,ψ and the expected attitude value ϕd,θd,ψd are input to the ADRC control law of the inner loop attitude. The output is U2,U3,U4. Finally, U1,U2,U3,U4 obtains the desired motor speed n1d,n2d,n3d,n4d through control distribution, where the conversion relations between the virtual control variables u1,u2,u3,u4,u5,u6 and U1,ϕd,θd are(30)U1=mu12+mu22+mu3−mg2,ϕd=−arcsinmu1sψ−mu2cψU1,θd=−arcsinmu1cψ+mu2sψU1cϕd,U2U3U4=IBW−1u4u5u6.


### 3.2. Design of Attitude Active Disturbance Rejection Control Law

Take the pitch angleθ channel as an example to illustrate the design process of the attitude ADRC control law:(1)
Design a tracking differentiator (TD) to arrange the transition process with the given signalθd as a reference input:(31)e=v1−θd,fh=fhane,v2,r,h,v1=v1+hv2,v2=v2+hfh,wherev1 is the transition value of TD from the initial value to θd. v2 is the derivative value of v1, and the parameters r,h are the fast factor and the filter factor, respectively.(2)
Design an extended state observer (ESO) to use the system outputθ and control input u5 to observe the system state and disturbances in real time:(32)e=z1−θ,fe=fale,0.5,δ,fe1=fale,0.25,δ,z1=z1+hz2−β01e,z2=z2+hz3−β02fe+b5u5,z3=z3+h−β03fe1,wherez1,z2 tracks θ,θ˙. z3 estimates the total disturbance s5, and β01,β02,β03 is a set of adjustable parameters.(3)
Design the nonlinear state error feedback (NLSEF) law, calculateu0, and combine it with disturbance compensation to calculate the control variable u5:(33)e1=v1−z1,e2=v2−z2,u0=−fhane1,ce2,r,h,u5=u0−z3b5.The parameterr,h,c is an adjustable parameter.The expressions of the fastest tracking control integrated functionfhanx1,x2,r,h and nonlinear function fale,α,δ are as follows:(34)fale,α,δ=signeeα,e≤δ,δ1−αe,e>δ,d=rh,d0=hd,y=x1+hx2,a0=d2+8ry,a=x2+0.5a0−dsigny,y>d0,x2+yh,y≤d0,fhan=−rsigna,a>d,rad,a≤d,where δ>0. The design of the ϕ and ψ channel control law is similar to that of the θchannel and will not be described in detail here.


### 3.3. Position Active Disturbance Rejection Control Law Design

During the design of the position ADRC control law, the design of the position’s three-channel TD, ESO, and NLSEF is similar to the design of theθ channel in the previous section. The only difference is that the z channel NLSEF adopts the following form:(35)e1=v1−z1,e2=v2−z2,u0=k1e1+k2e2,u3=u0−z3b3,where k1,k2 is an adjustable parameter.


## 3.1. Preliminary of the ARDC Law Design

There is a coupling between the position and attitude of the actively deformed aerial robot studied in this paper, which is an underactuated system. Only six-degree-of-freedom motion can be achieved in the incomplete sense, and the roll and pitch angles need to be changed while doing horizontal motion [30]. ADRC provides a more convenient way for the decoupling control of the multivariable coupling system, in which 6 states in the dynamic model of the active deformation aerial robot system are regarded as 6 channels, and ADRC can cause the coupling and deformation between the channels of the system. The parameter perturbation is treated as internal disturbance, and ESO is used to estimate and compensate for the internal and external disturbances of the system to realize the decoupling of the state of each channel. The virtual control quantity is introduced to realize the decoupling of system control, thereby converting the system description form from MIMO to six SISO systems combination. Therefore, the system's six-degree-of-freedom dynamic model equation (25) is rewritten in the following form:(29)x¨=s1x,x˙,ωout1,ωdef1+b1u1,y¨=s2y,y˙,ωout2,ωdef2+b2u2,z¨=s3z,z˙,ωout3,ωdef3+b3u3,ϕ¨=s4p,q,r,ϕ,ϕ˙,θ,θ˙,ψ,ψ˙,ωout4,ωdef4+b4u4,θ¨=s5p,q,r,ϕ,ϕ˙,θ,θ˙,ψ,ψ˙,ωout5,ωdef5+b5u5,ψ¨=s6p,q,r,ϕ,ϕ˙,θ,θ˙,ψ,ψ˙,ωout6,ωdef6+b6u6,where si⋅ is the uncertainty item and ωouti and ωdefi are the disturbances caused by the external disturbance and deformation of the system, respectively. b1,b2,b3 is an adjustable parameter with a size near 1/m. b4,b5,b6 is an adjustable parameter with a size near 1/Ixx,1/Iyy,1/Izz, and u1,u2,u3,u4,u5,u6 is the introduced virtual control quantity.Based on this, the control law adopts an inner and outer loop strategy, where the inner loop is attitude control and the outer loop is position control. The introduction of the control quantityU1,U2,U3,U4, respectively, represents the expected value of the total lift and the expected value of the torque around the three axes of the machine system. The actual position value x,y,z and the position expected value xd,yd,zd are used as the input of the outer loop position ADRC control law, and the expected values of roll and pitch angles ϕd,θd and U1 are output. The actual attitude value ϕ,θ,ψ and the expected attitude value ϕd,θd,ψd are input to the ADRC control law of the inner loop attitude. The output is U2,U3,U4. Finally, U1,U2,U3,U4 obtains the desired motor speed n1d,n2d,n3d,n4d through control distribution, where the conversion relations between the virtual control variables u1,u2,u3,u4,u5,u6 and U1,ϕd,θd are(30)U1=mu12+mu22+mu3−mg2,ϕd=−arcsinmu1sψ−mu2cψU1,θd=−arcsinmu1cψ+mu2sψU1cϕd,U2U3U4=IBW−1u4u5u6.


## 3.2. Design of Attitude Active Disturbance Rejection Control Law

Take the pitch angleθ channel as an example to illustrate the design process of the attitude ADRC control law:(1)
Design a tracking differentiator (TD) to arrange the transition process with the given signalθd as a reference input:(31)e=v1−θd,fh=fhane,v2,r,h,v1=v1+hv2,v2=v2+hfh,wherev1 is the transition value of TD from the initial value to θd. v2 is the derivative value of v1, and the parameters r,h are the fast factor and the filter factor, respectively.(2)
Design an extended state observer (ESO) to use the system outputθ and control input u5 to observe the system state and disturbances in real time:(32)e=z1−θ,fe=fale,0.5,δ,fe1=fale,0.25,δ,z1=z1+hz2−β01e,z2=z2+hz3−β02fe+b5u5,z3=z3+h−β03fe1,wherez1,z2 tracks θ,θ˙. z3 estimates the total disturbance s5, and β01,β02,β03 is a set of adjustable parameters.(3)
Design the nonlinear state error feedback (NLSEF) law, calculateu0, and combine it with disturbance compensation to calculate the control variable u5:(33)e1=v1−z1,e2=v2−z2,u0=−fhane1,ce2,r,h,u5=u0−z3b5.The parameterr,h,c is an adjustable parameter.The expressions of the fastest tracking control integrated functionfhanx1,x2,r,h and nonlinear function fale,α,δ are as follows:(34)fale,α,δ=signeeα,e≤δ,δ1−αe,e>δ,d=rh,d0=hd,y=x1+hx2,a0=d2+8ry,a=x2+0.5a0−dsigny,y>d0,x2+yh,y≤d0,fhan=−rsigna,a>d,rad,a≤d,where δ>0. The design of the ϕ and ψ channel control law is similar to that of the θchannel and will not be described in detail here.


## 3.3. Position Active Disturbance Rejection Control Law Design

During the design of the position ADRC control law, the design of the position’s three-channel TD, ESO, and NLSEF is similar to the design of theθ channel in the previous section. The only difference is that the z channel NLSEF adopts the following form:(35)e1=v1−z1,e2=v2−z2,u0=k1e1+k2e2,u3=u0−z3b3,where k1,k2 is an adjustable parameter.


## 4. Simulation Verification

In order to verify the rationality of the designed active deformable aerial robot and the effectiveness of its control law, simulations are arranged for verification in this section. First, we set up attitude response and position tracking experiments in order to verify the dynamic performance of the ADRC control algorithm. Then, in order to verify the control performance of the deformation function, experiments were arranged for the extension and folding of a single arm and two arms under the control of hovering, respectively. The folding experiments of the four arms are also arranged. Finally, the deformable aerial robot needs to perform cleaning operations, so the anti-interference performance is also one of the key points to be verified in this article. Therefore, an attitude control experiment of the robot under the action of wind disturbance and trees-pruning contact force is arranged. Based on MATLAB/Simulink platform, the deformable aerial robot simulation built with simulation step is set to 0.001s, and the system simulation parameters are shown in Table2.Table 2
Simulation parameters of the system.
SymbolsDescriptionsValuesmbodyBody mass31.58kgmarmArm mass0.43kgmrotMotor mass0.63kgmeloMass of telescopic steering gear0.31kggAcceleration of gravity9.76m/s2Ixx/Iyyx/y-axis moment of inertia16.38kg⋅m2Izzz-axis moment of inertia30.65kg⋅m2kfLift coefficient6.33×10−5N/rad2kmAntitorque coefficient6.1×10−7N⋅m/rad2JPGyro moment coefficient5.1×10−5N⋅m/rad2ρAir density1.293kg/m3S¯Average windward area0.096m2cCoefficient of air resistance2.92lbody/hbodyBody length/width/height282/73/41cmlarm/warm/harmArm length/width/height55/13/4cmrmot/hmotPropeller radius/height64/0.33cmrrot/hrotMotor radius/height15/3cmlelo/welo/heloTelescopic servo length/width/height6/9.2/3.1cmThe parameters of the active disturbance rejection flight controller are shown in Table3. In this paragraph, we have presented some method and experience for the tuning of ADRC controller to obtain a satisfying control performance. There are two adjustable parameters of TD, which are r and h. There are tricks that the larger h is, the better the filtering effect of TD will be, and the greater phase delay will be. The larger r is, the faster TD can track the input signal. Among the adjustable parameters of ESO, β01,β02,β03 affects the convergence speed of ESO, which is generally set according to the empirical formula. The tuning of b refers to equation (29). The meaning of r,h in NLSEF is the same as that in TD. c is the damping factor, and the meaning is the same as the damping coefficient of the second-order system. Increasing c would make the system overshoot decrease but the response speed becomes slower. k1,k2 functions, respectively, equivalent to the proportional module and the differential module in classical PID control. Increasing k1 increases the system’s response speed but is prone to overshoot or even oscillation, while increasing k2 will make the system’s dynamic performance better, but too much of which will lead to longer adjustment time.Table 3
Simulation parameters of active disturbance rejection ﬂight controller.
Parametersx/yzϕ/θψTD−r0.98215.810.42TD−h0.0010.0010.0015.1TD−δ0.0020.0040.0040.002ESO−β0110010001000100ESO−β021201738017380350ESO−β03400133820133820710ESO−b1.40.981.840.98NLSEF−h1—11NLSEF−c1.128—77NLSEF−r2.11—44NLSEF−k1—45.9——NLSEF−k2—25——


### 4.1. Deformation Control

#### 4.1.1. Attitude Control Law Experiment

Figure3 shows the attitude step response of the actively deformed aerial robot. It can be seen that the attitude 3-channel controller performs well.Figure 3
Curve of attitude response.


#### 4.1.2. Simulation of Spot Hover during Deformation

For the convenience of observation, the simulation in this paper reverses the value sign of positionz; that is, the negative sign becomes a positive sign. Set the initial position of the active deformation aerial robot P0E=0;0;0 and the desired position PdE=2;4;6. As shown in Figure 4, the active deformed aerial robot using the ADRC controller designed in this paper can quickly reach and hover at the desired position stably. Deformation is carried out at 10 s of the simulation. Figures 5 and 6 are the simulation curves of the elongation and folding of a single arm, and Figures 7 and 8 are the simulation curves of the double-arm elongation and folding. Also Figure 9 is the simulation result of the four-arm folding.Figure 4
Curve of position response.Figure 5
Curve of single-arm elongation.
(a)(b)Figure 6
Curve of single-arm folding.
(a)(b)(c)Figure 7
Curve of double-arm elongation.
(a)(b)Figure 8
Curve of double-arm folding.
(a)(b)(c)Figure 9
Curve of four-arm folding.
(a)(b)(c)It can be seen from Figures5 and 6 that the active deformation aerial robot can still hover steadily when a single arm is deformed, with the position hardly changing. Its attitude is less affected by deformation and the error convergence quickly under the action of the ADRC controller. Conclusion could be made that the control effect is very satisfactory. According to Figures 7 and 8, when the two arms of a deformable robot are deformed, the effect is similar to the deformation with a single arm. It can be seen from Figure 9 that when the four arms of the actively deformed aerial robot are deformed with the same speed simultaneously, there is almost no change in the center of gravity and the inertia tensor, and also its position and posture remain unchanged before the deformation.


#### 4.1.3. Trajectory Tracking Deformation Simulation

The trajectory tracking route of the active deforming aerial robot is designed, and deform is set to happen at 10 s of tracking. As shown in Figure10, the deformation method is selected to extend and fold arm 1. Figure 11 shows the trajectory tracking curve of the actively deformed aerial robot, and Figure 12 shows the position change curve during the trajectory tracking of the robot.Figure 10
Change curve of arm length and angle.
(a)(b)Figure 11
Curve of trajectory tracking.Figure 12
Curve of changing position.
(a)(b)(c)It can be seen from Figures11 and 12 that the active deforming aerial robot is deformed during trajectory tracking and can still maintain a good position tracking effect, which verifies the good position control performance of the designed ADRC controller.


### 4.2. Simulation of Anti-Interference Performance

Any aircraft in actual flight will inevitably be disturbed by the external environment, especially the disturbance generated by the wind field. In this paper, the Dryden model [31] is used to simulate atmospheric turbulence by shaping and filtering standard Gaussian white noise to obtain white noise. The transfer function of the shaping filter [32] is as follows:(36)Gus=KuTus+1,Ku=σuLuπv,Tu=Luv,Gvs=KvTvs+1,Kv=σvLvπv,Tv=2Lv3v,Gws=KwTws+1,Kw=σwLwπv,Tw=2Lw3v,where Lu,Lv,Lw and σu,σv,σw represent turbulence intensity and turbulence scale, respectively. Rotor aircrafts mainly fly at low altitudes and the calculation formulas for turbulence intensity and turbulence scale under this flight condition are as follows:(37)Lu=h0.177+0.00082312,Lv=Lu2,Lw=h2,σu=σw0.177+0.0008230.4,σv=σu0.177+0.0008230.4,σw=0.1u20,where u20 represents the wind speed at a height of 6.096 m.The turbulent wind module is built in MATLAB/Simulink platform, and the wind field effect formed by superimposingVwE=−3;3;0 continuous wind is shown in Figure 13. This paper compares ADRC controller, PID controller, and LQR controller and tests the antidisturbance performance of the controller by hovering deformation simulation under 4 different wind field conditions. The deformation is added at 10 s, and the deformation is the same as the single-arm folding in the previous section. The PID parameters used for comparison are set by trial and error. The wind field conditions are as follows:(1)
No wind(2)
Turbulent wind and no continuous wind(3)
Turbulent wind and continuous windVwE=−2;2;0(4)
Turbulent wind and continuous windVwE=−3;3;0Figure 13
Turbulent ﬂow ﬁeld.Set the initial height toz0=0 and the expected height to zd=3. It can be seen from Figure 14 that when there is no wind, the control effects of ADRC controller, PID controller, and LQR controller are not significantly different. When there is turbulent wind, PID controller and LQR controller begin to fluctuate slightly; and when there are turbulent wind and continuous wind, the PID controller and LQR controller fluctuate more and the actual height is slightly higher than the expected height. The LQR controller is affected by the dual effects of deformation and wind disturbance from 10 s. Compared with the PID controller, the altitude fluctuation is greater. When the continuous wind increases, the flying altitude of the actively deformed aerial robot adjusted by the PID controller and the LQR controller is significantly higher than the expected altitude. Under the same wind field, the ADRC controller can stably control the active deformable aerial robot at the desired height, and the control effect is basically the same as that when there is no wind, which verifies the good antidisturbance performance of the ADRC controller.Figure 14
Curve ofz under wind and without wind.
(a)(b)(c)(d)(e)Since ESO can observe, control, and compensate for the internal and external disturbances of the system, compared with PID controllers and LQR controllers, ADRC controllers can perfectly suppress deformation disturbances and wind disturbances. It can be seen from equation (18) that the crosswind will cause the rotor to produce wind disturbance in the same direction as the original lift, which increases the total lift. Figure 15 shows the disturbance observed by ESO under wind field condition 94). It can be seen that ESO has good disturbance observation performance, and the observed disturbance direction is consistent with the wind disturbance force direction.Figure 15
Disturbance estimation ofz channel under wind condition.Lastly, to check the antidisturbance ability during the pruning operation, a disturbance of constant pruning force acting around the robot axisY is added during the attitude tracking control. Figure 16 shows the attitude tracking control with disturbances.Figure 16
Tracking control with pruning disturbances.In Figure16, the constant pruning contact force begins to act from time of 3.0 s to time of 3.6 s, and it is presented that the robot pitch angle has a slight tremor, but the pruning robot attitude is still in control.


## 4.1. Deformation Control

### 4.1.1. Attitude Control Law Experiment

Figure3 shows the attitude step response of the actively deformed aerial robot. It can be seen that the attitude 3-channel controller performs well.Figure 3
Curve of attitude response.


### 4.1.2. Simulation of Spot Hover during Deformation

For the convenience of observation, the simulation in this paper reverses the value sign of positionz; that is, the negative sign becomes a positive sign. Set the initial position of the active deformation aerial robot P0E=0;0;0 and the desired position PdE=2;4;6. As shown in Figure 4, the active deformed aerial robot using the ADRC controller designed in this paper can quickly reach and hover at the desired position stably. Deformation is carried out at 10 s of the simulation. Figures 5 and 6 are the simulation curves of the elongation and folding of a single arm, and Figures 7 and 8 are the simulation curves of the double-arm elongation and folding. Also Figure 9 is the simulation result of the four-arm folding.Figure 4
Curve of position response.Figure 5
Curve of single-arm elongation.
(a)(b)Figure 6
Curve of single-arm folding.
(a)(b)(c)Figure 7
Curve of double-arm elongation.
(a)(b)Figure 8
Curve of double-arm folding.
(a)(b)(c)Figure 9
Curve of four-arm folding.
(a)(b)(c)It can be seen from Figures5 and 6 that the active deformation aerial robot can still hover steadily when a single arm is deformed, with the position hardly changing. Its attitude is less affected by deformation and the error convergence quickly under the action of the ADRC controller. Conclusion could be made that the control effect is very satisfactory. According to Figures 7 and 8, when the two arms of a deformable robot are deformed, the effect is similar to the deformation with a single arm. It can be seen from Figure 9 that when the four arms of the actively deformed aerial robot are deformed with the same speed simultaneously, there is almost no change in the center of gravity and the inertia tensor, and also its position and posture remain unchanged before the deformation.


### 4.1.3. Trajectory Tracking Deformation Simulation

The trajectory tracking route of the active deforming aerial robot is designed, and deform is set to happen at 10 s of tracking. As shown in Figure10, the deformation method is selected to extend and fold arm 1. Figure 11 shows the trajectory tracking curve of the actively deformed aerial robot, and Figure 12 shows the position change curve during the trajectory tracking of the robot.Figure 10
Change curve of arm length and angle.
(a)(b)Figure 11
Curve of trajectory tracking.Figure 12
Curve of changing position.
(a)(b)(c)It can be seen from Figures11 and 12 that the active deforming aerial robot is deformed during trajectory tracking and can still maintain a good position tracking effect, which verifies the good position control performance of the designed ADRC controller.


## 4.1.1. Attitude Control Law Experiment

Figure3 shows the attitude step response of the actively deformed aerial robot. It can be seen that the attitude 3-channel controller performs well.Figure 3
Curve of attitude response.


## 4.1.2. Simulation of Spot Hover during Deformation

For the convenience of observation, the simulation in this paper reverses the value sign of positionz; that is, the negative sign becomes a positive sign. Set the initial position of the active deformation aerial robot P0E=0;0;0 and the desired position PdE=2;4;6. As shown in Figure 4, the active deformed aerial robot using the ADRC controller designed in this paper can quickly reach and hover at the desired position stably. Deformation is carried out at 10 s of the simulation. Figures 5 and 6 are the simulation curves of the elongation and folding of a single arm, and Figures 7 and 8 are the simulation curves of the double-arm elongation and folding. Also Figure 9 is the simulation result of the four-arm folding.Figure 4
Curve of position response.Figure 5
Curve of single-arm elongation.
(a)(b)Figure 6
Curve of single-arm folding.
(a)(b)(c)Figure 7
Curve of double-arm elongation.
(a)(b)Figure 8
Curve of double-arm folding.
(a)(b)(c)Figure 9
Curve of four-arm folding.
(a)(b)(c)It can be seen from Figures5 and 6 that the active deformation aerial robot can still hover steadily when a single arm is deformed, with the position hardly changing. Its attitude is less affected by deformation and the error convergence quickly under the action of the ADRC controller. Conclusion could be made that the control effect is very satisfactory. According to Figures 7 and 8, when the two arms of a deformable robot are deformed, the effect is similar to the deformation with a single arm. It can be seen from Figure 9 that when the four arms of the actively deformed aerial robot are deformed with the same speed simultaneously, there is almost no change in the center of gravity and the inertia tensor, and also its position and posture remain unchanged before the deformation.


## 4.1.3. Trajectory Tracking Deformation Simulation

The trajectory tracking route of the active deforming aerial robot is designed, and deform is set to happen at 10 s of tracking. As shown in Figure10, the deformation method is selected to extend and fold arm 1. Figure 11 shows the trajectory tracking curve of the actively deformed aerial robot, and Figure 12 shows the position change curve during the trajectory tracking of the robot.Figure 10
Change curve of arm length and angle.
(a)(b)Figure 11
Curve of trajectory tracking.Figure 12
Curve of changing position.
(a)(b)(c)It can be seen from Figures11 and 12 that the active deforming aerial robot is deformed during trajectory tracking and can still maintain a good position tracking effect, which verifies the good position control performance of the designed ADRC controller.


## 4.2. Simulation of Anti-Interference Performance

Any aircraft in actual flight will inevitably be disturbed by the external environment, especially the disturbance generated by the wind field. In this paper, the Dryden model [31] is used to simulate atmospheric turbulence by shaping and filtering standard Gaussian white noise to obtain white noise. The transfer function of the shaping filter [32] is as follows:(36)Gus=KuTus+1,Ku=σuLuπv,Tu=Luv,Gvs=KvTvs+1,Kv=σvLvπv,Tv=2Lv3v,Gws=KwTws+1,Kw=σwLwπv,Tw=2Lw3v,where Lu,Lv,Lw and σu,σv,σw represent turbulence intensity and turbulence scale, respectively. Rotor aircrafts mainly fly at low altitudes and the calculation formulas for turbulence intensity and turbulence scale under this flight condition are as follows:(37)Lu=h0.177+0.00082312,Lv=Lu2,Lw=h2,σu=σw0.177+0.0008230.4,σv=σu0.177+0.0008230.4,σw=0.1u20,where u20 represents the wind speed at a height of 6.096 m.The turbulent wind module is built in MATLAB/Simulink platform, and the wind field effect formed by superimposingVwE=−3;3;0 continuous wind is shown in Figure 13. This paper compares ADRC controller, PID controller, and LQR controller and tests the antidisturbance performance of the controller by hovering deformation simulation under 4 different wind field conditions. The deformation is added at 10 s, and the deformation is the same as the single-arm folding in the previous section. The PID parameters used for comparison are set by trial and error. The wind field conditions are as follows:(1)
No wind(2)
Turbulent wind and no continuous wind(3)
Turbulent wind and continuous windVwE=−2;2;0(4)
Turbulent wind and continuous windVwE=−3;3;0Figure 13
Turbulent ﬂow ﬁeld.Set the initial height toz0=0 and the expected height to zd=3. It can be seen from Figure 14 that when there is no wind, the control effects of ADRC controller, PID controller, and LQR controller are not significantly different. When there is turbulent wind, PID controller and LQR controller begin to fluctuate slightly; and when there are turbulent wind and continuous wind, the PID controller and LQR controller fluctuate more and the actual height is slightly higher than the expected height. The LQR controller is affected by the dual effects of deformation and wind disturbance from 10 s. Compared with the PID controller, the altitude fluctuation is greater. When the continuous wind increases, the flying altitude of the actively deformed aerial robot adjusted by the PID controller and the LQR controller is significantly higher than the expected altitude. Under the same wind field, the ADRC controller can stably control the active deformable aerial robot at the desired height, and the control effect is basically the same as that when there is no wind, which verifies the good antidisturbance performance of the ADRC controller.Figure 14
Curve ofz under wind and without wind.
(a)(b)(c)(d)(e)Since ESO can observe, control, and compensate for the internal and external disturbances of the system, compared with PID controllers and LQR controllers, ADRC controllers can perfectly suppress deformation disturbances and wind disturbances. It can be seen from equation (18) that the crosswind will cause the rotor to produce wind disturbance in the same direction as the original lift, which increases the total lift. Figure 15 shows the disturbance observed by ESO under wind field condition 94). It can be seen that ESO has good disturbance observation performance, and the observed disturbance direction is consistent with the wind disturbance force direction.Figure 15
Disturbance estimation ofz channel under wind condition.Lastly, to check the antidisturbance ability during the pruning operation, a disturbance of constant pruning force acting around the robot axisY is added during the attitude tracking control. Figure 16 shows the attitude tracking control with disturbances.Figure 16
Tracking control with pruning disturbances.In Figure16, the constant pruning contact force begins to act from time of 3.0 s to time of 3.6 s, and it is presented that the robot pitch angle has a slight tremor, but the pruning robot attitude is still in control.


## 5. Conclusion

In order to improve the environmental adaptability of trees-pruning aerial robots, this paper proposes an active deformable aerial robot with telescopic and foldable arms and designs a control distribution scheme and ADRC controller for it. The simulation results show that the active deformation aerial robot based on the ADRC controller has excellent attitude control ability; not only can it deform well during flight but also it effectively estimates and compensates for the disturbance and turbulent wind disturbance generated during deformation with better performance.



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*Source: 6627339-2020-12-28.xml*