Iterative sliding mode control strategy of robotic arm based on fractional calculus

ZHANG Xin，LU Wenru，MIAO Zhongcui，JIANG Ziyun，ZHANG Jing

(1. School of Automation & Electrical Engineering,Lanzhou Jiaotong University,Lanzhou 730070,China； 2. Gansu Provincial Engineering Research Center for Artificial Intelligence and Graphics & Image Processing，Lanzhou 730070,China)

Abstract：In order to improve the control performance of industrial robotic arms,an efficient fractional-order iterative sliding mode control method is proposed by combining fractional calculus theory with iterative learning control and sliding mode control.In the design process of the controller,fractional approaching law and fractional sliding mode control theories are used to introduce fractional calculus into iterative sliding mode control,and Lyapunov theory is used to analyze the system stability.Then taking a two-joint robotic arm as an example,the proposed control strategy is verified by MATLAB simulation.The simulation experiments show that the fractional-order iterative sliding mode control strategy can effectively improve the tracking speed and tracking accuracy of the joint,reduce the tracking error,have strong robustness and effectively suppress the chattering phenomenon of sliding mode control.

Key words：robotic arm;fractional calculus;iterative learning control;sliding mode control

0 Introduction

The robotic arm is a versatile multi-input and multi-output complex system with non-linear and time-varying uncertainties[1].The various manipulators and the flexible operation are widely used in anti-terrorism,explosion-proof,industrial assembly and other fields.In recent years,China has successively issued “Guiding Opinions on Promoting the Robot Industry”and “Robot Industry Development Plan (2016-2020)”.From the strategic top-level design,the Ministry of Science and Technology has supported the research on robot-related technology through the 863 project.

The iterative learning control mathematical model is accurate,the controller and algorithm are simple and can achieve complete tracking,which is suitable for solving high-precision control problems of objects with high non-linearity,strong coupling,difficult modeling and repetitive motion.However,the system has a robust problem[2].Sliding mode control has the characteristics of fast response and good robustness,which is widely used in such non-linear and uncertain systems,but there is chattering in the control process,which will affect the control accuracy[3].It is found that iterative learning combined with sliding mode control can enhance the robustness of the system and suppress the chattering of the system to improve the control precision of the robotic arm.In Ref.[4],an iterative sliding mode controller was designed,which improved the stability and dynamic response of the control system by making use of the invariance of variable structure control on system parameters and external disturbances and the advantage that iterative learning does not depend on the precise mathematical model of the system.In Ref.[5],a multi-cycle iterative sliding mode control algorithm was proposed to improve the control accuracy of linear motors.In Ref.[6],iterative learning control and sliding mode control were used to study the fault-tolerant control of satellite attitude under external disturbances.

Compared with integer-order calculus,fractional differential and integral processes are more flexible and delicate.And with the development of computer technology,fractional calculus is applied to metallurgy,chemical industry,industrial processes such as machinery[7].The fractional-order control system is widely used in fractional-order sliding integral control and fractional-order iterative learning control which are generated by introducing the fractional calculus operator into traditional sliding mode control and traditional iterative learning control theory for its characteristic of memory and heritability[8].In Ref.[9],the robustness of the PD∂-type fractional iterative learning control algorithm was discussed.When the system is disturbed by external bounded noise,the algorithm can ensure that the tracking error of the system is uniformly bounded with the increase of iterations times.In Ref.[10],an effective fractional sliding mode control method was proposed.The simulation proves the good performance of the designed controller.Compared with the traditional sliding mode approaching law and the traditional sliding mode control law,the fractional sliding mode approaching law has better smoothing characteristics,fractional sliding mode control law has better anti-interference and strong robustness.In Ref.[11],in order to improve the vector control performance of induction motor,a dynamic fractional sliding mode control method was proposed based on the integral sliding mode control and fractional calculus theory.The experimental results show that the control method has better dynamic performance and strong robustness to load disturbances,and effectively suppress chattering of sliding mode control.

Inspired by the above literatures,in this paper,an iterative sliding mode control system is designed based on fractional calculus for the application problems of fractional calculus in sliding mode control theory and iterative learning control theory,as well as the traditional sliding mode control chattering problem and the traditional iterative learning control robustness problem.Finally,the two-joint robotic arm control system is taken as control object to verify the effectiveness of the control strategy proposed in this paper by MATLAB.

1 Basic theory

At present,there are three definitions of fractional calculus which are widely used in the control field:Grunwald-Letnikov (GL),Riemann-Liouville (RL)and Caputo[12].The definition of initial conditions of Caputo-type fractional calculus is consistent with that of integer-order calculus and has been extensively studied in engineering applications in recent years.

The Caputo-type fractional calculus is defined as[13]

m-1<α≤m,

(1)

It is well know that[14]

Dαx(t)=f(x,t).

(2)

Ifx=0 is the equilibrium point of the non-autonomous fractional-order system,f(x,t)satisfies the Lipschitz condition.Suppose there is a Lyapunov functionV(t,x(t))that satisfies the following conditions

α1‖x‖≤V(t,x)≤α2‖x‖,

V(t,x)≤-α3‖x‖,

(3)

whereα1,α2andα3are positive numbers.Then system Eq.(2)is progressively stable.

2 Dynamics model of robotic arm

Fig.1 is the physical diagram of a robotic arm,model number is GRB4016-10-23-TC (Kawasaki).As shown in Fig.1,only the two joints marked in the figure are adjusted,and the remaining joints are fixed.The length of link 1 and link 2 are 570 mm and 650 mm.The maximum load is 6 kg and the weight is 140 kg.Working environment:temperature of -20 ℃-60 ℃;humidity of 10%-90%.The corresponding specifications of the joints are shown in Table 1.

Fig.1 Robotic arm

Table 1 Specifications of robot arm

There are usually two methods to analyze the robot dynamics model,which are the basic theory of dynamics and Lagrange mechanics.In this paper,the Lagrange method is used to establish robot dynamics model[3].The structure of the robotic arm is shown in Fig.2.

Fig.2 Robotic arm structure

The Lagrange equationLis the difference between the system kinetic energyKand the potential energyP

L=K-P.

(4)

The Lagrange equation describing the state of the system dynamics is

(5)

The total kinetic energy of the articulated robot system is

(6)

The total potential energy of a multi-joint robot is

(7)

wheremiis the mass of the connecting rod;iriis the center position of gravity of the connecting rod relative to the front joint coordinate system[15].

From Eqs.(4),(6)and (7),we can get

(8)

From Eq.(5)and Eq.(8),we can get

(9)

(10)

The kinetic model is organized and simplified by the above specification as

(11)

3 Design of fractional-order iterative sliding mode controller

3.1 Fractional iterative sliding mode approaching law

There are two components in sliding mode motion:approaching motion and sliding mode motion[16].If a reasonable approaching law can be selected,the approaching speed can be accelerated to a large extent and the system chattering can be reduced,so that the approaching process can obtain better dynamic quality.In the traditional sliding mode control,there are four commonly used approaching laws:constant velocity approaching law,exponential approaching law,power approaching law and general approach law[16].

The fractional-order approaching law chosen in this paper isDαs=-ksign(s),and 0<α<1.The speed of control system state reaching the slip surface can be changed by orderαand coefficientk.

Stability proof of approaching law is as follows:

Step 1:The Lyapunov function is defined as

(12)

Step 2:According to the definition of Caputo-type fractional calculus as Eq.(1),

(13)

Step 3:By deriving both ends of Eq.(12)and combining the selected fractional-order approaching law and Eq.(13),it can be obtained that

(14)

Step 4:From sign(D1-α(-ksign(s)))=-ksign(s)[17],it has

-ksign(sT)sign(s)=-k.

(15)

According to the dynamic model of the manipulator,qd(t)is taken as the ideal position of the joint,q(t)is the actual position of the joint,and the tracking error of each joint position is defined as

e(t)=qd(t)-q(t).

(16)

By deriving Eq.(16),it has

(17)

Take the sliding surface as

(18)

By deriving Eq.(18),it has

(19)

From Eqs.(11),(17)and (19),we can get

(20)

The approaching law adopts the fractional approaching law designed above,

Dαs=-ksign(s).

(21)

By deriving Eq.(21),it has

(22)

By combining Eq.(20)with Eq.(22)and simplifying,the control law can be obtained as

(23)

Stability proof of control law is as follows

Step 1:Select the Lyapunov function

(24)

Step 2:Take the derivation at both ends of Eq.(24)and substitute Eqs.(11),(19)and (23)into Eq.(24)

s[-kD1-αsign(s)]≤0.

(25)

3.2 Fractional iterative sliding surface

According to the position tracking error defined above,design a fractional sliding mode surface as

s=Dαe+ce.

(26)

By deriving Eq.(26),it has

(27)

Select the exponential approaching law as

(28)

The combination of Eq.(28)and Eq.(27)is simplified as

D1-αks)+H-f.

(29)

Proof of stability is as follows

Step 1:The Lyapunov function is defined as

(30)

Step 2:Take the derivative of Eq.(30)and substitute it into Eq.(27)and Eq.(29),

(31)

Step 3:According to the Lyapunov stability theory,the system is progressively stable.

The system control block diagram is shown in Fig.3,whereuk(t)anduk+1(t)are the signals of the previous control and the current control;yr(t)andyk+1(t)are the input signals and feedback signals;ek(t)is the error .

Fig.3 System control block diagram

4 Verification of controller validity

4.1 Simulation parameters

The simulation examples in Ref.[2] are used to verify the effectiveness of the proposed control method,and the simulation experiments are carried out using MATLAB software.Numerical simulations are performed using Fractional Modeling and Control Toolbox of FOMCON[18].

Using the model of the robotic arm of Eq.(12),the machine parameters are selected as

f(t)=3sin(2πt),

wherev=13.33,q01=8.98,q02=8.75,g=9.8.

The initial state of the two-joint manipulator system is given as:[q1q2q3q4]=[0 3 1 0],c1=c2=40,ε=0.5,k=50,0<α<1.Set the iterations times as 10,and the two joint position commands areq1d=sin(3t),q2d=cos(t).

The following control methods are used to compare and analyze the simulation:

Method 1:Fractional-order iterative sliding mode approaching law;

Method 2:fractional-order iterative sliding surface;

Method 3:Fractional sliding mode approaching law;

Method 4:fractional sliding mode surface;

Method 5:Integer-order iterative sliding mode control.

4.2 Experiment results and analysis

Using the control algorithm proposed in 4.1 and the above simulation parameters,the following simulation results are obtained.Fig.4 shows the position trajectory tracking curves of joints 1 and 2 and the partial amplification curves of methods 1-4.

The comparison shows that the control algorithm proposed in this paper can soften the motion trajectory of the system,and the controller has good smoothing characteristics,which makes the tracking performance better.That is to say,the fractional-order iterative sliding mode control strategy is better than the fractional sliding mode control strategy for location tracking after 10 iterations.

(a)Joint 1

Fig.5 is the control input of the system and its partial amplification curve under the control strategy of method 1 and method 3.It can be seen that the fractional-order iterative sliding mode approaching law control strategy can suppress chattering better than the fractional sliding mode approaching law control algorithm,and the generated control input signal is smoother.Fig.6 shows the control input of the system under control strategy of method 2 and method 4.Compared with the fractional sliding surface control strategy,the chattering suppression effect of the fractional-order iterative sliding surface control strategy is better.

Fig.7 and Fig.8 respectively are the convergence process of the maximum absolute value of the position error and the maximum absolute value of the velocity error.The following conclusions can be obtained by the comparison between Fig.7 and Fig.8.

(a)Control input of system under method 1

(a)Control input of system under method 2

(a)Joint 1

(a)Joint 1

1)Under the two control methods proposed in this paper,the maximum absolute value of initial position (speed)error and the final position (speed)error of two joints are smaller than that under the control strategy of method 5.

2)The control strategy of method 5 makes the absolute value of the final speed error of the two joints slightly fluctuate compared with the results obtained by method 1 and method 2.The two control strategies proposed in this paper make the dispersion degree of the joint speed error generally smaller.

Table 2 and Table 3 show the comparison of the maximum and minimum values of the maximum absolute value of the position error and the maximum absolute value of the velocity error of the two joints under three control methods.It can be seen that the control strategies of method 1 and method 2 can obtain smaller position error,smaller speed error and higher control precision than control strategy of method 5.

Table 2 Comparison of position error between two joints

Table 3 Comparison of speed error between two joints

To further verify the effectiveness of the controller,take the step response as the desived trajectory,the following results are obtained.Fig.9 is the unit step response curve under the five methods,and Table 4 is the unit step response performance index of the two joints under the five methods.As seen from Fig.9 and Table 4,the method proposed in this paper can satisfy the dynamic performance metrics of unit step response well.

Fig.9 Unit step response of five control methods

Table 4 Unit step response performance indicators

In order to judge tracking performance of each control method more clearly,the root mean square error of the position error and the angular displacement adjustment time are selected as judgment reference value.

The comparison results of the obtained data are shown in Table 5.It can be seen from Fig.10 and Table 5 that the control strategy proposed in this paper can speed up the joint angular displacement adjustment time.After the joint is adjusted,the position tracking error fluctuation is smaller and the tracking effect is relatively better.

Table 5 Data comparison of five control methods

Fig.10 Comparison of position tracking errors of five methods

In order to verify the control performance of the control algorithm proposed in this paper,the trajectory tracking experiment was performed on the first two joints of the GRB4016 robotic arm,and the other joints were locked.The experimental platform is a 6-DOF robot arm developed by Gugao Company.The control system is shown in Fig.11.

Fig.11 Experimental platform

Fig.12 Tracking error curves of joints

Its hardware system mainly includes GRB4016 robotic arm,robot electric control cabinet,host computer and serial port RS232.The software system is developed by the high-tech company and can be fully compatible with programming development platform OtoStudio in MATLAB.Users can adjust the internal controller in the simulink environment or independently design the controller,and automatically generate executable files through the software.System control process is that:measuring the angular position of the joint motor,transmitting it to the data acquisition module of the robot electric control cabinet through a dedicated cable,and then transmitting it to the computer through RS232,calculating the output control torque through the host computer controller,and sending it to the driver through the data acquisition module.Drive the motion of the arm joint to complete the control task of robotic arm.The experimental results are shown in Fig.12,which verifies the effectiveness of the control strategy designed in this paper.

5 Conclusions

By comparing the proposed control strategy with fractional sliding mode control and integer-order iterative sliding mode control strategy,the following conclusions are obtained:

The two methods proposed in this paper can suppress chattering to some extent compared with fractional sliding mode control,and the tracking performance is better.Compared with the integer-order iterative sliding mode control,the fractional-order iterative sliding mode control strategy proposed in this paper has a certain reduction in the maximum absolute value of the position error and speed error of the two joints.In the input step response,method 1 and method 2 have a certain reduction in overshoot,rise time and adjustment time compared with method 3,method 4 and method 5,which improves the control accuracy.In addition,compared with other methods,the angular displacement adjustment time and the root mean square error of the two joints of the manipulator are reduced under the method proposed in this paper.Therefore,the method proposed in this paper greatly enhances the following speed and follow-up effect of each joint.And the effectiveness of the control strategy designed in this paper is verified by experiments.