Control system of trajectory tracking of discretely-actuated manipulator based on computed torque method

Abstract

In the past, in the process of tracking and controlling the discretely-actuated manipulator, the tracking result has a large error. To this end, a control system of trajectory tracking of discretely-actuated manipulator based on computed torque method was designed. The overall framework of the system was analyzed and designed, and the hardware of the system was designed in an integrated DSP information processing environment. The design of the monocular stereo sensor, drive controller, servo drive, etc. was described in detail. Then the control algorithm, the computed torque method, was analyzed. Finally, with the help of the control platform of discretely-actuated manipulator, the control system of trajectory tracking of discretely-actuated manipulator based on computed torque method was verified. After the different control algorithms were added, the trajectory tracking accuracy of the discretely-actuated manipulator was compared. The experimental results show that the maximum errors of each joint in the system are 0.0568 rad, 0.0347 rad and 0.0044 rad, which are lower than the traditional system, it is shows that the reliability of the system is verified.

Keywords

Monocular stereo sensor drive controller servo driver calculation moment method

1 Introduction

With the continuous development of the discretely-actuated manipulator field, the discretely-actuated manipulator technology has made great progress. The discretely-actuated manipulator technology is a combination of traditional engineering and is a relatively new technology field. Discretely-actuated manipulator technology and its applications require a variety of modern information technology and mathematical knowledge such as automatic control, mechanical, electronic, sensing, communication, computer and artificial intelligence. In recent years, the discretely-actuated manipulator technology has been applied to aerospace, defense military, rehabilitation and other fields, and has developed extremely rapidly. It has also been used as a measure of a country’s technological innovation and high-end manufacturing level [5]. The discretely-actuated manipulator industry is also regarded as an important strategic goal of the country or region by the world’s major economies such as the United States, the European Union, and Japan. The industry discretely-actuated manipulator is a multi-joints discretely-actuated manipulator or multi-degree-of-freedom machine for industrial applications that can perform tasks based on human-planned trajectories to accomplish human-defined tasks. Therefore, the research on the trajectory planning and trajectory tracking control of discretely-actuated manipulator has important practical significance.

At present, the traditional methods applied to the control system of trajectory tracking of discretely-actuated manipulator mainly include PD control, variable structure control, and adaptive control. PD control is the most classical control method in automatic control, and its control algorithm is simple and easy to implement. As long as the error amount of the control system is input, the control amount of the controlled system can be obtained by the set control law, thereby realizing the control of the controlled object discretely-actuated manipulator. However, in the complex industrial application environment, the operation of the discrete drive manipulator will be affected by the nonlinear factors, joint coupling, friction and load disturbance of the joint flexible components, so it is difficult to accurately model the system. Variable structure control is a kind of nonlinear control method, which belongs to robust control. Its main idea is to find a suitable sliding surface, and use the control law to make the control track finally stay on the sliding surface to achieve the purpose of trajectory control. However, the experimental results show that the trajectory tracking control accuracy of the system is low. Adaptive Control: When the uncertain part of the discretely-actuated manipulator can be represented by the mathematical model with unknown parameters, it can be identified and adjusted online by adaptive control to achieve tracking control, this method also has the problem of low control accuracy [1, 16].

In order to solve the problem of large tracking error in the process of track tracking control of discrete drive manipulator by traditional system, this paper takes the industrial discrete drive manipulator as the research object, aiming at improving the track tracking accuracy of discrete drive manipulator, and studies the track tracking control system of discrete drive manipulator based on the calculation moment method. The results show that the control system based on the method of calculating moment is used to track the trajectory of the discrete drive manipulator, and the calculated error is less than that of the traditional system, which proves the effectiveness and feasibility of the system.

2 Basic definitions

Discretely-actuated manipulator consists of several movable joints, which allows the system to have motion degrees of freedom in both the horizontal and vertical directions of the three planes, thus completing the various grabbing tasks, as shown in Fig. 1.

Fig. 1

Discretely-actuated manipulator.

The discretely-actuated manipulator mainly consists of the rotating base, arms, elbows, wrists, and pliers that grab the front end. These joints are independently powered by their respective drive motors, and the joints are independent of each other. The driving force of the discretely-actuated manipulator is divided into the main power chain and the branch power chain. The main power train transmits the power of the drive motor to the various joints, and each power branch uses the clutch to draw power from the main power train to drive the corresponding joint. From the perspective of the main power chain, the design can be easily realized by simple gear transmission, which has the advantages of simple structure, high transmission precision and few components. From the point of view of the branch power train, each joint receives power from the main power train through the clutch, which minimizes the dynamic coupling of the joint. In the end, this single-motor driven discretely-actuated manipulator has the same flexible end track as the traditional discretely-actuated manipulator [8].

Through the power branch consisting of a clutch, a worm gear, a worm, a pulley and a gear, each joint acquires power from the main power train of the discretely-actuated manipulator to achieve its own motion. Driving principles of joints are shown in Fig. 2.

Fig. 2

Driving principles of joints.

Discretely-actuated manipulator is a highly coupled multi-input and multi-output strong nonlinearity, and there are uncertainties such as external time-varying disturbances, unmodeled dynamics, parameter perturbation, load time-varying and internal friction in practical engineering applications. It is difficult to establish the accurate kinetic model for it, which leads to poor performance of the trajectory tracking of the discretely-actuated manipulator. The uncertainty of discretely-actuated manipulator mainly includes non-structural uncertainty and structural uncertainty. The latter is mainly caused by load mass error, rod centroid, length error, mass error of discretely-actuated manipulator, high-order unmodeled dynamics, structural vibration and actuator dynamics. The former is mainly caused by the factors of the uncontrolled object itself, such as rounding error, external interference, measurement noise and sampling time lag. These problems will affect the accuracy, stability and real-time performance of discretely-actuated manipulator trajectory tracking [3]. The difficulty of trajectory tracking of discretely-actuated manipulator is that it requires an efficient and reliable and general control method to deal with the trajectory tracking control problem of the discretely-actuated manipulator [17]. Therefore, an appropriate control system must be designed to effectively suppress the nonlinearity and uncertainty of the discretely-actuated manipulator to obtain good trajectory tracking performance.

3 Control system of trajectory tracking of discretely-actuated manipulator

3.1 Composition of system framework

The discretely-actuated manipulator’s control system can control the grabbing of the discretely-actuated manipulator, as well as control its motion positioning, and steadily improve these two functions. A data analysis model of discretely-actuated manipulator is established to constrain the movement and control the small discretely-actuated manipulator to collect the parameter data. Combining the attitude parameter data collected by the small discretely-actuated manipulator arm, the Kalman filter fusion information is processed [14, 15]. The control parameters of the small discretely-actuated manipulator include the motion speed, acceleration and inertia parameters of the manipulator, the object model of the motion control small robot, and the Kalman filter output processing parameters. The DSP is used as the basis for the research and development system, and the hardware modules in the control system of the discretely-actuated manipulator are designed. These include control central unit, executive controller, command output unit, ISA/EISA/microchannel loading and automatic remote control commands. Design host communication module to realize human-computer interaction and upload comprehensive data. In the input control instruction module, the command is loaded to control the mobile system, the execution processor is designed, the transmission module of the bus data is designed, and the control and mobile system of the small discretely-actuated manipulator is realized [19, 22]. The unified bus data transmission is combined with the above overall design to design the structure diagram for moving and controlling the discretely-actuated manipulator system, as shown in Fig. 3.

Fig. 3

The structure diagram for control system of trajectory tracking of discretely-actuated manipulator.

3.2 System hardware

The control system of trajectory tracking of discretely-actuated manipulator communicates with the motor drive controller via a Profinet fieldbus. The corresponding message is sent to control motor motion, such as start, stop, origin setting, speed and position, and to obtain the current motion state of the motor based on the information fed back by the drive. The controller is interconnected with the bus communication module through the dual port memory. The module is interconnected with the drive controller and the driver by Profinet communication, and each driver individually controls the motor action [9]. In addition, some reset circuits, communication circuits, power supply circuits, debug interfaces, etc. are also included, and no specific analysis is performed in this paper.

3.2.1 Monocular stereo sensor

Vision sensors are a direct source of information for the entire machine vision system, consisting primarily of one or two graphic sensors, sometimes with light projectors and other ancillary equipment. The main function of the vision sensor is to obtain enough of the original image to be processed by the machine vision system. Image sensors can use laser scanners, line and area array CCD cameras or TV cameras, or the latest digital cameras. Here, the MTC-73K11 monocular stereo sensor manufactured by Shanghai Mintong Electronic Instrument Co., Ltd. is mainly composed of the monocular camera, the cone, the transparent glass, the deflecting mirror, the collecting lens and the photosensitive chip [4]. The relevant parameters are shown in Table 1.

Table 1
Parameter settings for the monocular camera

Name parameter

Imaging element 1/3 sony ex-view ccd

Effective pixels 1920×1080

Minimum illumination ≥0.01LUX

Horizontal definition 480 line

Signal-to-noise ratio ≥40 dB

Line scan frequency 25 Hz

Field scanning frequency 50 Hz

Frequency band Ultra low bandwidth requirements

dynamic range ≥100 dB wide dynamic

Interface speed 480 MB/S

Connection interface 24 pin1.5mmUSB2.0

support system Windows XP/vista/seven/8.1/Mac

Power Supply 5 V±5%

power Max1 w

Data output type Raw Data10 bits

Video output MPJG/YUY2

Match lens type M12(f = 3.5 MM)

Default shot M12

working temperature –20°C∼+60°C

Working humidity 30% – 90% RH

Name	parameter
Imaging element	1/3 sony ex-view ccd
Effective pixels	1920×1080
Minimum illumination	≥0.01LUX
Horizontal definition	480 line
Signal-to-noise ratio	≥40 dB
Line scan frequency	25 Hz
Field scanning frequency	50 Hz
Frequency band	Ultra low bandwidth requirements
dynamic range	≥100 dB wide dynamic
Interface speed	480 MB/S
Connection interface	2*4 pin1.5mmUSB2.0
support system	Windows XP/vista/seven/8.1/Mac
Power Supply	5 V±5%
power	Max*1 w
Data output type	Raw Data10 bits
Video output	MPJG/YUY2
Match lens type	M12(f = 3.5 MM)
Default shot	M12
working temperature	–20°C∼+60°C
Working humidity	30% – 90% RH

3.2.2 Drive controller

The discretely-actuated manipulator controller is mainly used to realize the functions of servo motor drive, motor position closed-loop control and data processing. The discretely-actuated manipulator controller proposed in this paper is designed based on STM32F429IGT6 embedded microprocessor (DSP), which adopts the design idea of core board and peripheral circuit. The physical map of the discretely-actuated manipulator controller is shown in Fig. 4. The discretely-actuated manipulator controller includes memory, USART interface, timers, JTAG circuits, power circuits, control unit circuits for servo motors, and interfaces to peripherals. In addition, the controller expands the serial FLASH and SDRAM memory to better meet the requirements of the discretely-actuated manipulator control system to store large amounts of data, which greatly enhances its practical performance [6].

Fig. 4

Physical diagram of the controller used to control discretely-actuated manipulator.

ST’s STM32F429IGT6 embedded microprocessor is equipped with a 180 MHz clock, and the Cortex-M4 core features a single-precision floating-point unit that supports all ARM single-precision data processing instructions and data types. The STM32F429 microprocessor integrates up to 2 MB of Flash/256 KB RAM and up to 4 KB of backup SRAM. It is equipped with two APB buses, two AHB buses and one 32-bit multi-AHB bus to provides enhanced I/O interfaces and peripheral resources connected to the bus matrix, such as 12 general-purpose 16-bit timers, 2 general-purpose 32-bit timers, 4 USARTs and 4 DARTs. It can meet some complex control and data computing requirements. Its rich on-chip memory and peripherals make the microcontroller ideal for discretely-actuated manipulator servo motor drive and application control [10].

3.2.3 Servo drive

The motion servo system of the discretely-actuated manipulator is similar to the bones and muscles of the human body. By receiving instructional information from the brain, through the drive of bones and muscles, the corresponding actions are completed, and the changes in the external environment are received through the senses of the human body. Similarly, the motion servo system of the discretely-actuated manipulator receives control commands from the controller. Through the servo system, each joint of the discretely-actuated manipulator is driven to complete the corresponding job task [2]. At the same time, the discretely-actuated manipulator needs to detect various states through sensors. The actual motion of the discretely-actuated manipulator is reflected by the internal sensor signal, and the state and changes of the working environment are detected by the external sensor.

The discretely-actuated manipulator’s AC servo system is based on the discretely-actuated manipulator during the operation and the requirements of the individual joint torque. The discretely-actuated manipulator body uses three different power motors. The servo motor is selected from Panasonic’s MINAS-A4 series servo motor, model number is MSMD012S1, and its basic structure is shown in Fig. 5.

Fig. 5

MSMD012S1 servo motor.

The matching servo drive corresponds to the above three types of servo motors, all of which are Panasonic’s MINAS-A4 series servo drives, model numbers MADDT1205, MADDT1207, MBDDT2210. The parameters of the motor are as follows: rated speed is 3000 (r•min -1), rated power is 200 W, 400 W, 100 W, and the maximum output torque is 0.64 (n•m), which meets the demand of discretely-actuated manipulator. The servo motor used is coaxially equipped with a 17-bit absolute encoder with a resolution of 131072(p•r -1) [7]. When working, the position control method is adopted, and the servo system completes two closed-loop control of speed and position, that is, the AC servo system adopts PI to control the speed. The position adopts PD to control, which has good speed control performance and no oscillation, thus enabling smooth control of discretely-actuated manipulator [18, 13].

3.3 System software

The controller software for discretely-actuated manipulator is mainly STM32 program. It is developed by STM32 MCU and development software KeiluVision5, which is mainly responsible for realizing the functions of data transmission and processing with the discretely-actuated manipulator PC (PC) and servo system motion control [20].

3.3.1 The software of the discretely-actuated manipulator

The main program flow chart of the discretely-actuated manipulator controller is shown in Fig. 6. It mainly includes system initialization and initialization of various functional modules, serial communication, motor drive, position loop adjustment, servo motor position acquisition [21].

Fig. 6

Flowchart of main program of manipulator controller.

The discretely-actuated manipulator motion planning studies the problem that the manipulator obtains a collision-free path from the starting point to the target position. In a given environment, the discretely-actuated manipulator or discretely-actuated manipulator ends can move in a static and fully known environment of one or more obstacles depending on the desired task. The discretely-actuated manipulator system sends the planned location information to the discretely-actuated manipulator controller for execution. During the execution of the task, the external environment often has some interference and self-load disturbance, which is often the source of position error. In order to achieve accurate tracking of the trajectory by discretely-actuated manipulator and improve the stability and reliability of discretely-actuated manipulator motion, the research of control algorithm is especially important [12].

3.3.2 Algorithm of controller

In the control algorithm of the discretely-actuated manipulator, the computed torque method is effective for the control of the discretely-actuated manipulator. The idea of the computed torque method can be understood as the introduction of nonlinear compensation in the inner control loop of the discretely-actuated manipulator control system, by which the mechanical hip is simplified to a linear system to some extent. The kinetic model of the discretely-actuated manipulator is linearized in the case of known precision models. Using the controller of each joint, it is controlled, in this way the discretely-actuated manipulator is converted into a linear control system [11]. The computed torque method guarantees the stability of the trajectory tracking of the discretely-actuated manipulator, and the joints can be controlled separately by the closed-loop control system.

In practical applications, the discretely-actuated manipulator is not only fixed-point control, but also tracks the continuously changing trajectory, so there is no way to guarantee zero for ${\dot{p}}_{a}$ . According to the calculated torque method, a PD control is designed in the control inner loop of the discretely-actuated manipulator. First control is introduced: $t = B_{0} (p) c + D_{0} (p, \dot{p}) + E_{0} (p)$ (1) According to the discretely-actuated manipulator dynamic equation obtained by the general Lagrangian method simplification, the dynamic equation of the discretely-actuated manipulator can be transformed into: $\begin{matrix} B_{0} (p) \ddot{p} +_{0} (p, \dot{p}) \dot{p} + E_{0} (p) = t \\ = B_{0} (p) c + D_{0} (p, \dot{p}) \dot{p} + E_{0} (p) \end{matrix}$ (2)

The above formula can be written as follows. $B_{0} (p) \ddot{p} = B_{0} (p) \ddot{c}$ (3)

Where $\ddot{p} = c$ , B₀ (p) is an irreversible matrix; B₀ (p) 22 is an n × n inertia matrix; $D_{0} (p, \dot{p})$ is a centripetal force matrix; E₀ (p) is a gravity term of n terms; t is the moment of force of the discretely-actuated manipulator; p, $\dot{p}$ , and $\ddot{p}$ represent the desired position, velocity, and acceleration for each joint.

Therefore, Equation (3) can be equivalent to a linear steady-state system. In the control system, the position signal can be obtained through feedback from the system, and the speed signal can be obtained by one differential, but the deviation of the two differentials is amplified. When the desired trajectory is known, 1 and 2 are available, so the PD control can be introduced on the problem of seeking ${\ddot{p}}_{a} (t)$ , which is expressed as follows: $\ddot{p} = {\ddot{p}}_{a} - k_{a} (\dot{p} - {\ddot{p}}_{a}) - k_{b} (p - p_{a})$ (4)

Where k_a and k_b are positive definite matrices; p (t), $\dot{p} (t)$ , $\ddot{p} (t)$ respectively represent the position, velocity, and acceleration of discretely-actuated manipulator.

According to Equation (3) and Equation (4), after simplification, the error formula can be obtained, and a stable closed-loop system is obtained. $\ddot{f} = k_{a} \dot{f} + k_{b} f = 0$ (5)

Where $\dot{f} = \dot{p} - {\dot{p}}_{a}$ , f = p-p_a. According to the stability of the PD controller, the error of the closed-loop system can converge to 0, that is, $(p, \dot{p}) \to (p_{a}, {\dot{p}}_{a})$ .

By substituting Equation (5) into the manipulator kinetic equation obtained by the general Lagrangian method simplification, the input of the discretely-actuated manipulator system can be obtained: $\begin{matrix} t = B_{0} (p) ({\ddot{p}}_{a} - k_{a} \dot{f} - k_{b} f) \\ + D_{0} (p, \dot{p}) \dot{p} + E_{0} (p) \end{matrix}$ (6)

It can be seen from Equation (6) that t can be obtained by ${\ddot{p}}_{a} - k_{a} \dot{f} - k_{b} f$ . The dynamics of the discretely-actuated manipulator are divided into positive kinematics and inverse kinematics. Forward kinematics solves the motion of the discretely-actuated manipulator through the moment acting on the discretely-actuated manipulator; inverse kinematics solves the moment acting on the discretely-actuated manipulator through the motion of the discretely-actuated manipulator. Therefore, the computed torque method is also called “inverse kinematics”, which belongs to the feed forward control method.

4 Results

In the Matlab/Simulink environment, the kinematics, dynamics and control system simulation model of the loader working device were established. The model output from the ADAMS software was imported into the Simulink module of Matlab.

4.1 System test platform

In the above, the hardware and software of the control system of trajectory tracking are designed. Combined with the discretely-actuated manipulator designed by the members of the laboratory team, the whole experimental platform of the control system was built. The discretely-actuated manipulator was mounted on the tape machine. The discretely-actuated manipulator control system hardware was wired and the power source was installed in the control cabinet, as shown in Fig. 7.

Fig. 7

System test platform.

The task of the discretely-actuated manipulator control module is to accept the information processed by the vision module, such as the position of the target object, and to change the received information into corresponding instructions to control the discretely-actuated manipulator in order to complete the crawling task.

4.2 The test of the discretely-actuated manipulator

The research object in this paper is IRB120 discretely-actuated manipulator with 6 degree of freedom. The discretely-actuated manipulator is a small multi-functional discretely-actuated manipulator launched by ABB, which has the advantages of compact structure, light weight and high agility. It has 6 degrees of freedom and a maximum load of 3 kg. The entire discretely-actuated manipulator system consists of a discretely-actuated manipulator and an IRC5 controller. Joints are shown in Fig. 8.

Fig. 8

IRB120 discretely-actuated manipulator with 6 degree of freedom (DOF).

During the simulation, the carrier posture of the discretely-actuated manipulator and the end claws simultaneously completed the desired motion, and the carrier attitude angle was adjusted from 0.108 rad to (π/4) rad after the motion is completed. The initial values of the carrier posture of the discretely-actuated manipulator and the three wrists gestures were taken as 0.200 rad, 2.500 rad, and 3.500 rad. The entire tracking process took 20.3 seconds.

4.3 Tracking trajectory

Fig. 9 shows the expected trajectory curves of three wrists changing with time. In order to test the trajectory tracking effect of different methods from multiple angles, three wrists are selected as tracking targets, which are respectively represented as wrist 1, wrist 2 and wrist 3. The closer the trajectory tracking curve is to the curve in Fig. 9, the better the tracking effect is.

Fig. 9

Expected trajectory curves of three wrists changing with time.

Analysis of Fig. 10 shows that whether the wrist 1, wrist 2 or wrist 3, the wrists trajectory tracking curve of this method is closer to the expected trajectory curve of the wrists, which indicates that the tracking result of the method is more reliable. This is because the system in this paper introduces the calculation moment method in the control of the discrete drive manipulator. The calculation moment method can ensure the stability of the trajectory tracking control of the discrete drive manipulator, and can control each joint separately through the closed-loop control system.

Fig. 10

Wrists trajectory tracking curves.

4.4 Error analysis

Table 2 shows the comparison results of the trajectory tracking errors of discretely-actuated manipulator with different methods.

Table 2
Errors in trajectory tracking

Track tracking control system Wrist 1 Wrist 2 Wrist 3

Calculation moment method 0.0568 0.0347 0.0044

PD control 0.0742 0.0684 0.0148

Variable structure control, 0.0845 0.0715 0.0156

Adaptive control 0.0785 0.0847 0.0071

Track tracking control system	Wrist 1	Wrist 2	Wrist 3
Calculation moment method	0.0568	0.0347	0.0044
PD control	0.0742	0.0684	0.0148
Variable structure control,	0.0845	0.0715	0.0156
Adaptive control	0.0785	0.0847	0.0071

It can be seen from the above table that when the system is stable, the maximum error of the wrist 1 of the discretely-actuated manipulator simulated by the calculated torque method is 0.0568 rad; the maximum error of the wrist 2 is 0.0347 rad after the system is stable; the maximum error of the wrist 3 is 0.0044 rad. All of the above data are superior to the application results of three trajectory tracking systems of discretely-actuated manipulator based on PD control, variable structure control and adaptive control. This shows that the computed torque method can make the trajectory tracking effect of each joint of the discretely-actuated manipulator better. This is because the system in this paper uses the position control method to complete the two closed-loop control of speed and position, which has good speed control performance. At the same time, the method of calculating moment is used to control the discrete drive manipulator, thus reducing the control error of the system.

5 Conclusions

With the rapid development of modern science and technology, control objects, controllers, and control tasks have become increasingly complex, and the control of discretely-actuated manipulator has become more demanding. The control effect directly affects the overall performance of the discretely-actuated manipulator, so the more advanced and effective control methods have always been the core of the control system of trajectory tracking of discretely-actuated manipulator. According to the analysis in the paper, the environment will affect the fixed position of the discretely-actuated manipulator and the corresponding grab operation. Adjusting the steady-state difference will also affect itself, which leads to the operation accuracy of the discretely-actuated manipulator. The steady-state error will be relatively large. To solve this problem, a DSP-based intelligent discretely-actuated manipulator was designed. In this paper, the hardware of the discretely-actuated manipulator control system was designed to analyze the movement of the discretely-actuated manipulator and control the overall design and technical specifications of the system. The hardware of the control system was designed and developed in the DSP environment. Finally, the test results show that the maximum errors of each joint of the discrete drive mechanical arm and wrist are 0.0568rad, 0.0347rad and 0.0044rad respectively, which are lower than the traditional system, indicating that the control and movement system of the small discrete drive mechanical arm proposed in this paper has higher accuracy, smaller error and better convergence. In order to improve the application performance of this system, the control efficiency will be the main research object in the future.

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Control system of trajectory tracking of discretely-actuated manipulator based on computed torque method

Abstract

Keywords

1 Introduction

2 Basic definitions

3.1 Composition of system framework

3.2.1 Monocular stereo sensor

3.3.1 The software of the discretely-actuated manipulator

4.1 System test platform

Table 2 Errors in trajectory tracking Track tracking control system Wrist 1 Wrist 2 Wrist 3 Calculation moment method 0.0568 0.0347 0.0044 PD control 0.0742 0.0684 0.0148 Variable structure control, 0.0845 0.0715 0.0156 Adaptive control 0.0785 0.0847 0.0071

References

Table 2
Errors in trajectory tracking

Track tracking control system Wrist 1 Wrist 2 Wrist 3

Calculation moment method 0.0568 0.0347 0.0044

PD control 0.0742 0.0684 0.0148

Variable structure control, 0.0845 0.0715 0.0156

Adaptive control 0.0785 0.0847 0.0071