Abstract
With the continuous development of science and technology, robotics is widely used in various fields. In recent years, more and more research studies have been done on the control of autonomous robotic manipulators. How to quickly, accurately, and smoothly grasp objects has always been a difficult point of research. As the robot’s executive mechanism, the robot arm plays an important role in whether the robot can complete a specific task. Therefore, the research on the robot arm is also the main topic in the development of robot technology. The control theory, kinematics, and human-computer interaction of robotic arms are the focus of the research in the field of robotic arms. Based on the above background, the research content of this paper is the research on the modeling method of autonomous robotic manipulator based on D-H algorithm. This paper uses D-H modeling method to model a four-degree-of-freedom robotic arm and gives the forward kinematics equation of the robotic arm. The inverse solution of the manipulator was given by the method and the geometric method, and the joint variable values were calculated. Finally, through experimental simulation, the experimental results show that the inverse solution of the end position of the machine by the geometric method is in the range of 2∼4 mm, and the inverse solution of the end position of the machine by the algebraic method is in the range of 6∼14 mm. It is more accurate to find the inverse solution of the geometrical method of the manipulator than the algebraic method.
1. Introduction
The birth of robots and the establishment and development of robotics are one of the most significant achievements of human science and technology in the 20th century. As a high-tech product, robots involve the cross-fusion of multiple disciplines such as mechanical design and automatic control. Integration: in daily production, robots are often used to perform different tasks instead of humans. For example, in environments where the implementation of the job is difficult and the risk factor is high, using robots will help reduce personnel injuries. Reducing personnel burden also improves work efficiency. Therefore, it can be foreseen that, with the continuous improvement of the robot’s intelligence, it can not only improve efficiency and save costs, but also avoid risks for personnel and ensure personal safety.
Manipulator is an important branch of robotics [1]. In the design and research of the robotic arm, the design of the control system is often inseparably related to the overall dynamic performance, especially for the control of the robotic arm joint, which has been one of the core contents of the research of the robotic arm [2, 3]. Autonomous robots can replace humans to complete some service work, thereby freeing humans from labor to do more meaningful things [4, 5]. The robotic arm, as the execution mechanism of the autonomous robot, also has the characteristics of high operational accuracy and continuous work, which can reduce people’s labor intensity and improve labor productivity [6, 7]. The continuous advancement of science and technology and the improvement of people’s living standards promote the continuous development and improvement of autonomous robot technology [8, 9]. The rapid development and widespread application of robot technology have also promoted the improvement of people’s quality of life and promoted the improvement of productivity and the progress of the whole society [10, 11]. Robots are ubiquitous in real life, play an important role in people’s daily life, and begin to slowly integrate into people’s lives [12].
Choosing the right robot from existing conventional configurations is one way to use a robot. However, with the increasing application of robot-assisted technology, the idea of custom design for designated working positions in specific task-oriented environments has drawn the attention of many researchers. One of the challenges in this regard is to achieve specific designs, including unconventional values for robot parameters. Singh discusses an adaptive module based on D-H parameters, which can be adjusted based on a given robot parameter value. There are three similar types of modules. Singh discussed in detail the design of basic length units, adaptive warping units, and extended length units. The focus of Singh’s work is to verify the adaptability of the proposed architecture through modeling, simulation, and development of standard configurations [13]. Zheng proposed a new joint attention intervention system for children’s ASD, which overcomes some existing limitations. In this area, such as the use of body wear sensors, nonautonomous robot operation requires human participation and lack of formal models robot-mediated joint attention interaction. Zheng proposed a completely autonomous robotic system, called a noncontact response robot-mediated intervention system, which can be inferred through a distributed noncontact gaze inference mechanism and embedded minimum-to-maximum (LTM) robot-mediated interaction models with attention to address current limitations. The system was tested in a multistage user study of 14 young children with autism. The results show that the participants’ joint attention skills have improved significantly, and the interest in robots has remained consistent throughout the experiment. The LTM interaction model has a significant effect on improving children’s academic performance [14, 15]. Jaradat proposed a complete highly mobile autonomous robot design for demining services. The method proposed by Jaradat provides an interesting compromise between the design requirements of a demining robot application and an overdriven autonomous robot. The robot body is mainly divided into two parts: the first part provides the required motion for the robot and is composed of a four-wheel drive/steering subsystem (4WD), a four-steering mechanism (4SW), and a passive suspension system. The second part is a three-degree-of-freedom manipulator designed based on two parallelogram mechanisms. The proposed design guarantees many advantages of existing designs, including stability, operability, simplicity of autonomous navigation, and control effort constraints. In order to evaluate the robot’s motion in different environments, rough terrain, and slopes, Jaradat performed simulation analysis on the robot model and its stability. In addition, to help robots make accurate driving decisions in navigation and control, prototypes of the robots were developed and manufactured, and different types of sensors were used. In order to verify the proposed design, Jaradat tested the prototype in different scenarios and environments. Test results show that the robot has good performance in terms of autonomous navigation and target positioning [16, 17]. The robot competition has become a research platform for multiagent robot systems with its unique dynamics and real time. Zhang uses humanoid robots as the hardware platform to design a semiautonomous robot competition strategy including a communication subsystem, a vision subsystem, and a decision subsystem. Based on the motion characteristics of the humanoid robot, the system proposes offensive and defensive strategy models including robotic strategies such as situation assessment and path planning and verifies the stability of the countermeasure decision system through specific experiments [18, 19]. Industrial robots are in line with the development of the times and the industry has a bright future. However, the contradiction between the imbalance of supply and demand for talent in this field is becoming increasingly prominent. On the one hand, robot manufacturers, system integrators, and automotive processing industries are thirsty for their talents; on the other hand, the shortage of talents makes it difficult to meet the needs of enterprises.
This thesis mainly focuses on the design and implementation process of the manipulator used by the autonomous robot, including hardware system design, software system design, and kinematic analysis research. This article introduces the kinematics analysis of the robotic arm, introduces some basic knowledge of kinematics and the method of describing the coordinate space of the robotic arm, uses D-H parameter model method to establish the mathematical model of the robotic arm, analyzes the kinematics analysis method of the robotic arm, and obtains inverse solution of the robot arm. We derive the dynamic equation of the robot arm system. The dual-link manipulator was selected as the research object, and the system was modeled in the secant coordinate system through the Lagrange equation and hypothetical modal method, and its dynamic equation was deduced, which laid the foundation for the design of the subsequent controller basis. Finally, we used MATLAB’s Robotics Toolbox tool for simulation and verification.
2. Proposed Method
2.1. Kinematics
The kinematics analysis of the robotic arm is mainly to analyze and study the motion of the robotic arm relative to a fixed reference frame as a function of time, without considering the force and moment of the robotic arm when it moves; that is, the spatial displacement of the robotic arm is expressed as a function of time; in particular, the relationship between each joint variable and the position and attitude of the end effector of the robot arm should be studied [20, 21]. There are usually two issues of forward and inverse kinematics to be solved, as shown in Figure 1.
For a given robotic arm, given the link parameters and joint variables, find the pose of the end effector (grip) relative to a fixed reference coordinate system [22, 23]. This fixed reference coordinate system is usually a Cartesian coordinate system fixed on the ground or other objects [24]. This problem is called positive kinematics problem [25].
The known geometric parameters of the manipulator link determine the position of the end effector (hand grip) relative to the fixed reference coordinate system to solve the size of each joint variable [26, 27]. This problem is called the inverse kinematics problem. Each joint variable of the robot arm is an independent variable, and the pose of the end effector (hand grip) is usually expressed in the reference coordinate system [28, 29]. To determine the corresponding joint variables according to the position and posture of the hand in the reference coordinate system, the inverse kinematics problem needs to be solved [30]. Therefore, the solution of the inverse kinematics problem of the manipulator is necessary to control the manipulator.
2.2. D-H Algorithm
The robotic arm is composed of a plurality of rods connected by rotating joints. Each joint constitutes a degree of freedom. Starting from the base of the fixed robotic arm, it is the lumbar joint, shoulder joint, elbow joint, and wrist joint in turn. The final wrist joint is connected to the end manipulator to form a complete robotic arm. To analyze the kinematics of the four-degree-of-freedom robot arm, firstly, a link coordinate system must be established, and a homogeneous coordinate transformation matrix between two adjacent link coordinate systems must be determined. The commonly accepted and widely used representation method is the D-H parameter model method. The D-H parameter model method is to establish a homogeneous transformation matrix for the member coordinate system of each joint. Use this transformation matrix to represent its relationship with the previous member coordinate system, and then through successive transformations, we can get the end transformation matrix from actuator (hand grip) to base coordinate of robot arm. This method is called the D-H parameter model method. The rules for selecting the coordinate system: the axis is the direction of the joint axis; the axis is parallel to the common normals of and . If the common normal is not unique ( and are parallel), the d parameter is a free parameter; the axis, axis, and axis follow the right-hand rule (right-hand coordinate system).
The D-H method is used for modeling, and the coordinate system of the manipulator needs to be established again. Because the D-H method is used for modeling, it is impossible to solve the situation that the parallel axis of the moving pair is parallel. Therefore, an error model is first established from the static platform center of the static platform to the dynamic system center of the moving platform, and then the tool vector homogeneous transformation matrix is added to complete the transition from the point of the complete mapping of the static center. First, the angle relationship of the ball pair is specified, and the direction of the axis of the rotating pair on the i-th branch chain is taken as the direction of the first angle of the ball pair; the direction of the rotation axis along the axis of the motion system is the direction of the second angle of the ball pair; the ball pair’s direction of the third rotation axis is perpendicular to the rotation of the first two axes. A basic coordinate system is established on the static platform, which is the same as the static coordinate system in the inverse kinematics analysis. The motion coordinate system is established on the moving platform, which is the same as that in the inverse kinematics analysis. A tool point coordinate system is established at the tool point , and its coordinate axis direction is the same as that of the moving system. After that, a coordinate system is established on the moving pair of a branch chain (O-Ai-Bi-P) of the robot arm in turn: the coordinate system based on the D-H method established above requires further analysis of the transmission relationship of the coordinate system. All the coordinate system will be split according to the transfer mode of moving first and then rotating.
The i-th link coordinate system is represented in the link coordinate system. The homogeneous transformation moments of the i-coordinate system and the coordinate system are established, and the i-th joint coordinate system and the i-th 1 joint coordinate system coincide.(1)Turn the coordinate system around the axis around the angle so that the axis is parallel to the axis and in the same direction(2)Translate the coordinate system along the axis by a distance di so that the axis coincides with the axis of (3)Translate the coordinate system around the axis by to make the origins of the two coordinate systems coincide(4)Turn the coordinate system around the axis by angle to make the two coordinate systems completely coincide
Through the above transformation, the homogeneous transformation matrix of the link i-coordinate system and the link coordinate system can be obtained. Then is called the D-H parameter model transformation matrix of the adjacent link coordinate systems i and .
The coordinate system is established on each kinematic pair of the mechanism, and the spatial relationship of the coordinate system is described by using the homogeneous transformation matrix. The spatial position relationship between the end and the basic coordinate system can also be established through sequential homogeneous transformation. If kinematic joint errors are added to the homogeneous transformation matrix, the geometric errors of all kinematic pairs can be brought into the kinematic model, and the geometric error model can be established. The geometric error model established by the D-H method can include the error terms of all the motion pairs in the mechanism, which is convenient to calculate the influence of the errors of all motion pairs on the ends. Applying the D-H method, for the i-th branch chain, the homogeneous transformation matrix between the motion coordinate system and the motion coordinate system iswhere h is the distance between two adjacent joints (link rod length), is the relative twist angle between adjacent joints (link twist angle), is the relative angular displacement between adjacent links, and d is the offset of the adjacent link on the motion pair. is the rotation attitude transformation matrix of adjacent coordinate systems, and is the position transformation matrix between adjacent coordinate systems. Because the working space of the robotic arm is known to be a space similar to a flat triangular prism, for this type of working space, and its rotation angle is more complicated, the measurement method needs to be optimized so that the measured value meets the error identification and solution conditions. And the operation is simple and easy to install.
2.3. Manipulator Modeling and Intelligent Control
In terms of system modeling methods, the more commonly used are finite element method, hypothetical mode method, etc. In both methods, the system is discretized before the dynamic equation is established. The finite element method divides the model into finite elements and approximates the deformation of each element to change linearly. Its characteristic is that each coordinate quantity has a certain physical meaning, but the natural frequency of vibration cannot be directly obtained; it is assumed that the modal method is one of the common methods for deriving the dynamic equation of the mechanical arm. The effect is countless independent resonance modes, different mode functions are selected according to different boundary conditions, and the truncation method is used to select the first few modes that have a dominant effect on the system to form the elastic deformation of the entire system. The natural frequency value obtained by the hypothetical modal method agrees well with the experimentally measured data. Therefore, this article will also use a hypothetical modal method to model the manipulator system. First, the dynamic equation of the system is given by the Lagrange equation:where q is a generalized coordinate, Q is a generalized force, and L is a Lagrange function, and its expression iswhere T is the total kinetic energy of the system, V is the total potential energy of the system, and U is the total strain energy of the system. It can be seen that if we want to derive the dynamic equation of the system, we must first derive the total kinetic energy, total potential energy, and total strain energy of the system.
In a coordinate system, there is a point, and the position of this point is represented by a vector. For a rectangular coordinate system , there is a point , and the position of the point is AP. It is a column vector, that is,
In order for the robot to move according to our instructions, it needs to give the specific position of the object in space and its direction. The relevant knowledge of the coordinate system is needed here to describe the position and orientation of the object in detail. In order to determine the orientation of a rigid body B in space, it is necessary to connect the rigid body B with the rectangular coordinate system and select the selected coordinate system as a reference. The orientation of the rigid body B relative to it is equivalent to the coordinate system in the reference, and the cosine of the coordinate system relative to it is represented by a matrix in the formwhere is called a rotation matrix, where the coordinate system and the coordinate system are represented by superscript A and subscript B, respectively. Consider the following conditions:
The rotation transformation matrices corresponding to the rotation angle of x, y, and z are
Among them, s means sin and c means cos.
3. Experiments
3.1. Experimental Environment and Data
There are many simulation software packages for the robotic arm. According to different requirements, different methods can be used, such as directly using existing software for simulation or VC++ for self-programming and simulation. Robot simulation software includes Robotics Toolbox and Sim Mechanics Toolbox from MATLAB, Pro/e, ADAMS, and OPENGL. Robotics Toolbox is a robot simulation tool maintained by Peter Corke, which is widely used. Therefore, this system uses the Robotics Toolbox for MATLAB toolbox for simulation. Compared with the traditional methods in the past, this method has faster operation speed, more convenient implementation, and better visual effects. Using Robotics Toolbox for MATLAB to analyze and simulate the workspace of the robot requires first obtaining the values of the parameters in the D-H parameter model. Then use the function programming provided by the toolbox for simulation.
In this paper, before positioning and grabbing the target, when the target is outside the working range of the robot arm, the robot needs to first determine the direction and time of movement according to the distance between the target and the robot and then autonomously move closer to the target for the robot arm to grab. The main moving directions are horizontal and vertical directions. The D-H parameter table is shown in Table 1.
Set the horizontal movement speed to 210 mm/s and the vertical movement speed to 225 mm/s. After using the ultrasonic sensor to measure the moving distance of the mobile platform relative to the wall, the data was fitted by MATLAB. Using Robotics Toolbox for MATLAB to analyze and simulate the workspace of the robot requires first obtaining the values of the parameters in the D-H parameter model. Then use the function programming provided by the toolbox for simulation. Connect the whole system, turn on the power, and perform linkage experiments through the control interface of the host computer to verify whether the design of the robot arm is reasonable and whether the movement is stable and flexible. Because the precision of mechanical processing is not very high and the gap between the joint and the motor reduces, each joint has different degrees of jitter. This paper uses a higher processing technology to connect rods and a better motor to achieve precise control. The grip experiments of the robotic arm were carried out to verify whether the structural design of the robotic arm’s flexibility and hand grip was reasonable. Because the torque and reduction ratio of the shoulder joint motor are not very large, in order to avoid damage to the motor in the experiment, an empty mineral water bottle was used as the object to be caught.
3.2. Experimental Evaluation Indicators
The mechanical arm designed in this paper is shown in Figure 2.
The hardware structure design of the manipulator in this paper mainly includes FPGA core board, motor driver, DC servo motor, incremental photoelectric encoder, Kinect camera, host PC, power management module, and so on. The embedded processor chooses FPGA. Compared with other processors, FPGA has a simple design process, lower volume and cost, shorter design time, reduced PCB area, and increased system reliability. Do not need to invest a lot of time and effort, reducing the risk of design. This system uses a DC servomotor. The motor is selected from Shanghai Aiqi’s 36SYK71 series hollow cup motor. Because the hollow cup motor has outstanding energy-saving characteristics, sensitive and convenient control characteristics, and stable running characteristics, as a high-efficiency energy conversion device, it represents the development direction of the motor.
FPGA, as the core device of manipulator control, plays a decisive role in the whole manipulator system. It is mainly responsible for generating motor drive signals, measurement of rotation angle, communication with the host computer, and implementation of control algorithm PID.
Figure 3 shows the basic connection block diagram of FPGA design. It can be seen from the connection block diagram that the FPGA is connected to the PC through the RS232 communication interface and is used to receive the control instructions issued by the PC and at the same time feed back the control status and angle information to the PC, so that the PC can download the information according to the different feedback information. Send different control instructions. The four DC servomotors are connected to the FPGA through a motor driver, and the encoder rotates while rotating, thereby transmitting the encoded signal to the FPGA. After the FPGA collects the encoded signal, it is converted into angle information after filtering, quadruple frequency, and discrimination, used to control the rotation angle of the joint. The end manipulator (hand grip) is connected to the FPGA. The FPGA determines the grip of the grip according to the instructions issued by the PC. At the same time, in order to prevent the grip from exceeding the limit position during the grip, a limit switch is used to avoid it.
4. Discussion
4.1. Motion System Analysis
The positioning results of the robot in this paper are shown in Figure 4.
As shown in Figure 4, the steady-state speed of the lateral distance robot is about 186.9 mm/s, the startup acceleration time is about 3-4 s, and the motion error is about 6.67%. The actual steady-state speed of the longitudinal distance robot is about 193.9 mm/s, the startup acceleration time is about 3-4 s, and the error is about 3.4%. Before capturing the target, we need to know the 3D coordinates of the target. This coordinate is based on the left camera and is limited by the measurement method. In this paper, the Z value of the 3D coordinate, that is, the distance, is used as the evaluation index to determine the accuracy of the target positioning. The experiment measured 10 different Z values, as shown in Table 2.
As can be seen from Table 2, the error within 2000 mm does not exceed 1.6%, and the error range within 1000 mm is only about 0.4%; in particular, the error within 500 mm is less than 0.1%. The distance is only 400 mm, so the accuracy fully meets the target grasping requirements.
4.2. Analysis of Adaptive Sliding Mode Controller
In order to verify the feasibility of the designed adaptive sliding mode controller, this paper uses MATLAB software to simulate the flexible manipulator system. In this section, based on the flexible manipulator system model and the adaptive sliding mode variable structure controller designed in the previous section, the MATLAB/Simulink software is used to control and simulate the autonomous robot manipulator system.
In the adaptive sliding mode control rate, the initial value of the adaptive parameters of , , and is . The simulation results of the parameters of the robot arm, the optimal control parameters of the fast-changing subsystem, and the expected trajectory are shown in Figure 5.
As can be seen from the comparison chart, within 0∼2 s, there is a large error between the position response curve of the joint 1 and the expected curve. After 2 s, the given curve is tracked, and the tracking error approaches zero. Given the curve, it shows that, under adaptive sliding mode control, the system can quickly track the desired trajectory; the input curve of adaptive sliding mode control is relatively smooth, which greatly reduces the chattering phenomenon. When only sliding mode variable structure control is used, although the stable state can be reached in a short period of time, the control curve is not smooth and there is obvious chattering phenomenon; when using an improved adaptive sliding mode controller for control, chattering phenomenon obviously weakened, the system has strong stability, and the control effect is more ideal.
In order to reduce chattering and improve the stability and response speed of the system, this paper improves the sliding mode controller of the slow-changing subsystem of the manipulator. Combining adaptive control with sliding mode control, an adaptive sliding mode variable structure controller is designed, and compared with previous simulations, it is verified that the improved system has faster response speed and stronger stability and effectively weakens chattering phenomenon, which has more ideal control effect.
4.3. Experimental Verification of Inverse Kinematics
In this paper, a verification kinematic inverse kinematics experiment is designed to compare the advantages and disadvantages of the algebraic method and the geometric method to find the inverse kinematics solution. The experiment needs to preset a set of data that a robot arm needs to reach a three-dimensional position, input the given data to the controller, and measure the actual position to which the end of the arm moves. The inverse kinematics experimental verification data are shown in Table 3.
As shown in Table 3, when using the autonomous robot developed by the laboratory to conduct experiments, the data of the expected position and the reached position of the 8 end claws are randomly selected from a large amount of experimental data. The geometric method to find the inverse solution of the end position error of the machine is shown in Figure 6.
As shown in Figure 6, the inverse solution of the end position of the machine by the geometric method is in the range of 2∼4 mm, and the inverse solution of the end point of the machine by the algebraic method is in the range of 6∼14 mm. These errors are because there is an error in the mechanical mechanism itself. In addition, there is an error in the conversion of angle and radian when the program is running. As far as the experimental results are concerned, the geometric method for the researched manipulator designed in this paper is more accurate than the algebraic method. However, these errors are all acceptable ranges for the competition, and because the geometric method is simple, the calculation speed is fast, and the effect is good. In practice, this paper chooses the geometric method as the inverse solution method.
4.4. Optimization Identification Method
The ridge estimation method is used to optimize the identification matrix, and then the identification model obtained by the first combined measurement method and the second measurement method is optimized. After that, the end geometric error source identification value is obtained, and compared with the given value, the verification is performed. Identify the effectiveness of model optimization. Firstly, the first combination measurement method is optimized. After introducing random measurement noise value and adding the end error value, the L curve method can be used to obtain the ridge estimation parameter value of 0.0588. The obtained L curve chart is shown in Figure 7.
After the ridge estimates are obtained, they are brought into their identification models, and identification simulations of the compensable geometric error terms at the ends are performed. The identification results obtained by the identification are shown in Table 4.
As shown in Table 4, it can be seen that the identification value of the geometric error term obtained by the error identification model optimized by the ridge estimation optimization method is closer to the true value, which can prove that the ridge estimation algorithm measures the existence of the identification matrix based on the least square method. The optimization situation in the case of noise is obvious. Due to the small difference between the two measurement methods after the optimization, it is impossible to quantitatively reflect the advantages and disadvantages of the two. Here we introduce an evaluation coefficient based on the statistical concept. Its expression is
In the formula, represents the given value of the i-th error in the compensable error matrix Δε. Using this evaluation coefficient, it can be shown that the overall recognition result is approached. In the case of a fixed value, the smaller indicates that the overall identification result approaches the true value of the error. The evaluation coefficient values of the geometric error terms obtained under the two measurement methods are calculated, respectively. Because this data is small, it is magnified 1000 times to solve it. It can be obtained that the evaluation coefficient obtained by the first combination measurement method is = 5.9667, and the evaluation coefficient obtained by the fourth combination measurement method is = 6.0586. Thus, the improvement effect of the ridge estimation algorithm on the identification model is obvious. The first measurement method: the identification values obtained with the fourth measurement method are closer to the true value, and it can be seen that the items of the first combined measurement method are closer to the given value as a whole compared to the compensable geometric error terms identified by the fourth combined measurement method. It shows that the first combined measurement method is better. Combining the above simulations, it can be seen that the identification effect of the first combination detection method is better than that of the fourth combination detection method, and when performing error identification work, using the ridge estimation optimization method to optimize the identification model can improve the identification accuracy of the identification model.
5. Conclusions
This paper focuses on the design of autonomous robot manipulators and elaborates the design method and implementation process of self-help robot manipulators in detail. A robotic arm for the application requirements and design requirements of an autonomous robot was completed for the autonomous robot design, laying a good foundation for subsequent research and development.
According to the characteristics of poor man-machine interaction of the current robotic arm, this paper writes PC software using C++ language. This software can easily and flexibly implement various control and debugging of the robotic arm, and it has become an important means of humanized control of the robotic arm. Aiming at the characteristics of elastic deformation of the manipulator in actual work, this paper decomposes the system into a slow-changing subsystem representing a large-scale rigid motion and a fast-changing subsystem representing an elastic deformation. Sliding mode controllers are designed for slow-varying subsystems; optimal control methods are used for fast-varying subsystems to suppress vibration.
There are still many shortcomings in this paper. In the process of modeling a robotic arm, for the convenience of research, external factors such as friction are ignored, and the flexibility of the joint is ignored. Only the flexibility of the arm is considered. In fact, the joint shaft and transmission mechanism of the robotic arm will also undergo a certain degree of distortion during the working process. Although its joint flexibility is much smaller than that of the boom, it will also have a certain impact on the control accuracy. Therefore, in the further research, we will consider the coexistence of the flexibility of the boom and the flexibility of the joint to make a more accurate controller design.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the Zhejiang Provincial Key Research and Development Project (No. 2021C01149), Zhejiang Provincial Science Foundation (No. LGF19F020010), and Hangzhou Science and Technology Development Plan Project (No. 20200706B06).