Backstepping – sliding mode control for dual arm robot

This paper presents a Backstepping-Sliding Mode Control for dual arm robot in handling, transporting a payload to track a desired trajectory. The control law is based on Backstepping and Sliding mode control technique and Lyapunov theory. A numerical simulation was used to verify the performance and robustness of the controller

BACKSTEPPING – SLIDING MODE CONTROL

FOR DUAL ARM ROBOT

Nguyen Duc Hiep1, Pham Duc Tuan1, Vu Quoc Doanh1,

Bui Van Dan2, Le Xuan Hai1,*

Abstract: This paper presents a Backstepping-Sliding Mode Control for dual arm

robot in handling, transporting a payload to track a desired trajectory The control

law is based on Backstepping and Sliding mode control technique and Lyapunov

theory A numerical simulation was used to verify the performance and robustness of

the controller

Keywords: Dual arm robot, Sliding mode control, Backstepping

1 INTRODUCTION

Dual arm robot (DAR) has an extensively application in a wide area of human life In industrial environment, DAR are able to replace human worker in transferring and assembling devices and component because of ability on handle large object with high precision and reliability with less torque actuator requirement Moreover, many robot systems with dual arm manipulator are increasingly investigated and specially used in hazardous environment that human are not able to approach and work However, applying DAR often have to face with complicated physical analysis and complex controller [1] caused by closed chain system’s kinematic [2]

Many researchers have recently considered and investigated about DAR, various attempts for control of DAR has been proposed and applied [3-8] Uchiyama et al [3] applied the hybrid scheme to the two-arm robot after introducing a unique joint space vector consisting of joint-vectors of the two arms Laroussi et al [4] derived the constraint forces as function of input and state, the inverse plant method and computation of the constraint forces were used to coordinate the control of the system Lin and Huang [5] presented an adaptable fuzzy force control scheme to improve the performance of a dual industrial robotic system then tuned the scaling factor of the fuzzy logic controller X Yun and V Kumar [6] proposed a nonlinear feedback techniques derived from differential geometry is then applied to linearize and decouple the nonlinear model N Xi [7] introduced the event-based planning and a hybrid position/force controllers for multi-manipulator T Yoshikawa et al [8] used a unified controller to grasp and manipulate rigid objects with various contact

Because of the possibility of unexpected disturbances in the working environment, Ensuring desired trajectory tracking and safe transporting become difficult Therefore a controller which is insensitive with disturbance is required Due to the robustness, sliding mode controller (SMC) is widely used in a large range of application that maintaining the tracking accuracy while facing

disturbance is necessary N Yagiz et al [9] present a non-chattering SMC by

deriving dynamic equation of the DAR’s component interaction Y Hacioglu et

al [10] improve [9] by a multiple – input multiple – output fuzzy logic to

eliminate system’s uncertainty

In this study, we proposed a new algorithm based on sliding mode control and

backstepping technique for dual arm robot system in handling and transporting a

payload follow a desired trajectory The controller quality is verified in some

circumstances with unexpected disturbances affecting to the controlled signal The

rest of the paper includes 5 sections In Section 2, the physical model of the dual

arm robotic system is constructed In Section 3, Backstepping – Sliding mode

controller is introduced Section 4 is simulation results and conclusion

2 SYSTEM MODELLING

Fig.1 Physical model of the DAR

Consider a physical model of a planar dual arm robot (DAR) system including

two arms shown in Fig.1 Each arm consists of two links so the robot has four

degrees of freedom (DoF), but when the robot handles the payload, the system’s

DoF is reduced to two due to the constrains Illustrated in Fig 1, mi are the

masses, Ii are the inertial moment, Li are the length of the links of the DAR d1 and

d2 are the width of the rectangular payload and the distance between two joint

which are on the DAR’s platform, respectively ki are the distances from the

inertial center of each link to its preceding joint, are the joint angle of related

joint, bi are the viscous friction efficiencies coefficient on each joints m(t) is used

to represent the mass of the payload, and it can be variable during the operation

Considered actuators are motors mounted on revolute joints

DAR system operates in the horizontal plane and its movement can be

separated to two steps Firstly, it begins from the home position and move to the

payload’s position Then it handles the payload and tracks the designed trajectory

To handle the payload, the DAR’s arm effectors apply forces to the surface of the payload Acted force composed of normal forces as F1 and F2, dry friction force

Fs1y and Fs2y along Oy axis, Fs1z and Fs2z along Oz axis

The system dynamic of DAR when operating with the payload can be simplified in vector form as:

[M( )] C( , )  Gu  [ ]J T W 

Fig.2 The forces applying on the pay load

Where [M( )] is a 4x4 mass matrix, C( , )  is a 4x1 vector represented for

coriolis and centrifugal terms, u

is a 4x1 vector including torque input vector, F

is a 4x1 vector including interaction forces, J is the Jacobian matrix and is 4x4, W

is the 4x1 disturbance torque vector,  is the viscous friction forces on all the joint

Define state variables as:

1 ( , 1 2 , 3 , 4 ) ; T 2 ( , 1 2 , 3 , 4 )T

x      x        

And the reference xref  (1ref,2ref,3ref,4ref)T

From the simplified dynamic system equation (1) we obtain state space model

of DAR as:











We set K J F  V Gw, then (2) can be rewritten:

1 2 1 1

2











 (3)

When the DAR handles the payload, the DoF of the robot system reduces to two as:

2 1

m t

m t

(4)

Where (x m t( ),y m t( ))is the position of the payload’s center

According to Newton’s second law of motion, the payload’s motion equations

are given as:

( ) 2 1

( )

m t

s z s z



 (5)

The friction force’s expressions are given as:

( )

2 ( )

2

s y

s y

m t g

m t g





(6)

In order to handle the payload and prevent rotation, the forces acted should be

positive Therefore, friction force should be chosen so F F2, 1 are obtained:

( ) 1

( )

( )

m t

m t

m t

F











if xm t( )  0 (7)

( )

( ) 2

( )

m t

m t

m t

F











if xm t( )  0 (8)

Where (xm t( ),ym t( )) are the accelerations of the payload on the horizontal plane

obtained by differentiating constraint equations:

m

m

3 BACKSTEPPING SLIDING MODE CONTROL

In backstepping – sliding mode controller, the control input is changed according

to predefined rules, which drives, and maintains the system states on a sliding surface Consider the system dynamic mentioned in (3), the controller is designed

by two steps The first is to define the error vector, and the second is to choose the sliding surface

Step 1: The error vector is defined as: z1x1x 1ref (10) Differentiating of z1 by time we obtain: z1x1x1ref x2x1ref (11) Define z2 x2 with  is the virtual control signal, from (11) we obtained:

z1x2x1ref z2x1ref  (12)

Step 2: Chose the sliding surface: sz1Mz2

Similarly, we have its derivative:

1



Theorem: Consider the system dynamic described by the equation (3), if the

control law is designed as

1 2

T

T

sz z







where , c2 are positive and  is a very small positive number, the overall system is asymptotically stable

Proof:

Chose the Lyapunov Candidate Function: 1 1 1 1

2

T

V  z z Differentiating V1along time we obtained:

ref



Chose the virtual control signal:

1 1 1ref

     (16) where c1 is a positive number

Then we obtain:

V  z c z z z (17)

Chose the second Lyapunov Candidate Function: 2 1 1

2

T

V V  s s

Differentiating by time V2 we obtain:

1 2

1 2

T

T

T

T

sz z

s s

sz z

s s









(18)

Substitute the control law (14), the Lyapunov function’s derivative become:

2 1T 1 1 T 2 ( )

V  z c z s c sign s is negatively defined, equivalently the system

errors asymptotically converge to zero as t  

4 SIMULATION RESULTS AND DISCUSSION

For numerical demonstration of the proposed method performance through the

system dynamics of the DAR, the simulation model of the controller and the DAR

were built and verified in MATLAB enironment Firstly, the arm tips of the DAR

move from the initial positions to the object position according to the linear

trajectories in 2 seconds Then the end effectors of the DAR handles the object and

transfers it following a circle trajectory

We set the parameters of the DAR and the controller as follow:

Numerical parameters of the dual arm robot:

2

i i



Numerical parameters of the controller:

10

The test of the controller includes two parts, in the first part, we verify the

controller performance in ideal condition, without external disturbance affecting

on the robot’s joints

Fig.3 shows the rotating motions of all joints and their designed rotating angles,

it is clear that all joint angles asymptotically approach the references

Fig.3 The reference and actual joint angles

Fig 4 describes the tracjectory tracking of the robot’s end effectors including its desired paths and the real positions with the proposed controller As can be seen clearly, all arm tip’s trajectory approach and track the desired trajectory with high accuracy after a short time

Fig.4 Trajectory of arm tips

In order to test the robustness of the presented control method, in the second part of the test, we add to all joints of the robot a external disturbance torque shown in Fig 5

Fig.5 External distubance on control torque

Similar to the first part of the test, Fig 6 describles the actual angles of all robot’s joints along with their reference It can be seen that all the joints of the robot asymptotically approach the desired angles Additionally, it shows that the

robot’s arms tracked their desire trajectories even with external disturbance

Fig 7 present the real motion of the end effectors of two arms and its desired trajectories when facing with disturbances It is obvious that there is not a significant difference between the actual trajectories in this case and in the first

part of the test that verifies the robustness of the controllers

Fig.6 The reference and actual joint angles

Fig.7 Trajectory of arm tips

In conclusion, the simulation results show that the presented control method is

able to ensure the stability and tracking performance for the Dual arm robot

system Futhermore, the controller provide a high robustness and acuratecy in the

condition that the unknown disturbance is existed

REFERENCES

[1] C R Carignan and D L Akin, Cooperative control of two arms in the

transport of an inertial load in zero gravity, IEEE Transactions on Robotics

and Automation, 4 (4) (1988) 414419

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manipulators handling a flexible object, Journal of the Franklin Institute, 344

(2007) 349-361

[3] [3] M Uchiyama, N Iwasawa and K Hakomori, Hybrid position/force

control for coordination of a two-arm robot, IEEE International Conference

on Robotics and Automation, Raleigh, USA (1987) 1242-1247

[4] K Laroussi, H Hemami and R E Goddard, Coordination of two planar

robots in lifting, IEEE Journal of Robotics and Automation, 4 (1) (1988) 77

[5] S.-T Lin and A.-K Huang, Position-based fuzzy force control for dual industrial

robots, Journal of Intelligent and Robotic Systems, 19 (4) (1997) 393

[6] X Yun and V Kumar, “An approach to simultaneous control of trajectory

and interaction forces in dual-arm configurations,” IEEE Transactions on

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[7] N Xi, T.-J Tarn, and A Bejczy, “Intelligent planning and control for

multi-robot coordination: an event-based approach,” IEEE Transactions on

Robotics and Automation, vol 12, no 3, pp 439– 452, June 1996

[8] T Yoshikawa, “Control algorithm for grasping and manipulation by

multi-fingered robot hands using virtual truss model representation of internal force,” Proceedings of IEEE International Conference on Robotics and

Automation, pp 369–376, 2000

[9] N Yagiz, Y Hacioglu, and Y Z Arslan, “Load transportation by dual arm

robot using sliding mode control,” Journal of Mechanical Science and

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[10] Y Hacioglu, Y Z Arslan, and N Yagiz, “MIMO fuzzy sliding mode

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Institute, vol 348 pp 1886–1902, October 2011

TÓM TẮT

BỘ ĐIỀU KHIỂN TRƯỢT BACKSTEPPING CHO ROBOT TAY MÁY ĐÔI

Trong bài báo này chúng tôi đề xuất một bộ điều khiển dựa trên kỹ thuật

Backstepping kết hợp điều khiển trượt và lý thuyết ổn định Lyapunov cho hệ

thống robot tay máy đôi nhằm di chuyển một vật bám theo quỹ đạo đặt trước

Bộ điều khiển đã được kiểm chứng trong môi trường mô phỏng trong điều

kiện có nhiễu tác động đến hệ thống Kết quả mô phỏng đã cho thấy hiệu quả

của thuật toán được đề xuất trong việc điều khiển dịch chuyển vật bám quỹ

đạo trong các các điều kiệ và có khả năng ứng dụng trong thực tế

Từ khóa: Robot tay máy đôi, Điều khiển trượt, backstepping

2 Hung Yen University of Technology and Education

*Corresponding author: xhaicuwc.edu.vn@gmail.com