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
Research 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 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 Journal of Military Science and Technology, Special Issue, No.54A, 05 - 2018 Electronics & Automation 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 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 is simulation results and conclusion 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 N D Hiep, …, L X Hai, “ Backstepping – sliding mode control for dual arm robot.” Research 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 (1) 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: x1 (1 , , , )T ; x2 (1 ,2 ,3 ,4 )T And the reference xref (1ref , ref ,3ref , ref )T From the simplified dynamic system equation (1) we obtain state space model of DAR as: x1 x2 1 1 x2 M (u w) M ( J F V G ) (2) We set K J F V G w , then (2) can be rewritten: x1 x2 1 1 x2 M u M K (3) When the DAR handles the payload, the DoF of the robot system reduces to two as: Journal of Military Science and Technology, Special Issue, No.54A, 05 - 2018 Electronics & Automation d2 d L1 cos 1 L2 cos(1 ) 2 d2 d L3 cos 3 L4 cos(3 ) 2 ym (t ) L1 sin 1 L2 sin(1 ) xm (t ) (4) L sin 3 L4 sin(3 ) Where ( xm (t ) , ym (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: m(t ) xm (t ) F2 F1 m(t ) ym (t ) Fs1 y Fs y (5) m(t ) g Fs z Fs z The friction force’s expressions are given as: m(t ) g ) ( F1 ) 2 m(t ) g Fs y ( ) ( F2 ) 2 Fs1 y ( (6) In order to handle the payload and prevent rotation, the forces acted should be positive Therefore, friction force should be chosen so F2 , F1 are obtained: 2 2 2 mym (t ) m(t ) g F1 mym (t ) m(t ) g F2 m(t ) xm (t ) mym (t ) m(t ) g F1 m(t ) xm (t ) 2 mym (t ) m(t ) g F2 if xm (t ) (7) if xm (t ) (8) Where ( xm (t ) , ym (t ) ) are the accelerations of the payload on the horizontal plane obtained by differentiating constraint equations: xm l1 ( q12 cos q1 q1 sin q1 ) l2 [( q1 q2 ) cos( q1 q2 ) ( q1 q2 )sin( q1 q2 )] ym l3 (q32 cos q3 q32 sin q3 ) l4 [(q3 q4 ) cos(q3 q4 ) (q3 q4 )sin(q3 q4 )] (9) BACKSTEPPING SLIDING MODE CONTROL In backstepping – sliding mode controller, the control input is changed according N D Hiep, …, L X Hai, “ Backstepping – sliding mode control for dual arm robot.” Research 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 x1ref (10) Differentiating of z1 by time we obtain: z1 x1 x1ref x2 x1ref (11) Define z2 x2 with is the virtual control signal, from (11) we obtained: z1 x2 x1ref z2 x1ref (12) Step 2: Chose the sliding surface: s z1 Mz2 Similarly, we have its derivative: s z1 Mz2 z1 M ( x2 ) z1 M ( M 1 K M 1u ) (13) s z1 K u H Theorem: Consider the system dynamic described by the equation (3), if the control law is designed as sz T z (14) u T c2 sign( s ) z1 K H s s where ,c2 are positive and is a very small positive number, the overall system is asymptotically stable Proof: Chose the Lyapunov Candidate Function: V1 z1T z1 Differentiating V1 along time we obtained: V1 z1T z1 z1T ( z2 x1ref ) z1T (c1 z1 c1 z1 z2 x1ref ) (15) V1 z1T c1 z1 z1T z2 z1T (c1 z1 x1ref ) Chose the virtual control signal: c1 z1 x1ref (16) where c1 is a positive number Then we obtain: V1 z1T c1 z1 z1T z2 (17) Chose the second Lyapunov Candidate Function: V2 V1 sT s Differentiating by time V2 we obtain: V2 V1 sT s z1T c1 z1 z1T z2 sT ( z1 K u H ) sz T z V2 z1T c1 z1 sT T c2 sign( s ) c2 sign( s ) z1 K u H s s T sz z V2 z1T c1 z1 sT c2 sign( s ) sT T c2 sign( s ) z1 K u H s s (18) Substitute the control law (14), the Lyapunov function’s derivative become: Journal of Military Science and Technology, Special Issue, No.54A, 05 - 2018 Electronics & Automation V2 z1T c1 z1 sT c2 sign( s ) is negatively defined, equivalently the system errors asymptotically converge to zero as t 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 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: mi 1.5(kg ) ki 0.48(m) bi 110(m) I i 0.18(kgm ) Li 1.2(m) 0.35 d1 0.25(m) d 1.2(m) m 1.5( kg ) Numerical parameters of the controller: c1 372, c2 1082, 23, 1010 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 N D Hiep, …, L X Hai, “ Backstepping – sliding mode control for dual arm robot.” Research Fig 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 Fig.5 External distubance on control torque Similar to the first part of the test, Fig 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 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 Journal of Military Science and Technology, Special Issue, No.54A, 05 - 2018 Electronics & Automation 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) (1988) 414419 [2] A S Al-Yahmadi, J Abdo and T C Hsia, Modeling and control of two manipulators handling a flexible object, Journal of the Franklin Institute, 344 (2007) 349-361 N D Hiep, …, L X Hai, “ Backstepping – sliding mode control for dual arm robot.” Research [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, (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 Robotics and Automation, vol 7, no 5, pp 618–625, October 1991 [7] N Xi, T.-J Tarn, and A Bejczy, “Intelligent planning and control for multirobot 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 multifingered 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 Technology, vol 24, no 5, pp 1177–1184, May 2010 [10] Y Hacioglu, Y Z Arslan, and N Yagiz, “MIMO fuzzy sliding mode controlled dual arm robot in load transportation,” Journal of the Franklin Institute, vol 348 pp 1886–1902, October 2011 TÓM TẮT BỘ ĐIỀU KHIỂN TRƯỢT BACKSTEPPING CHO ROBOT TAY MÁY ĐƠI Trong báo chúng tơi đề xuất điều khiển dựa kỹ thuật Backstepping kết hợp điều khiển trượt lý thuyết ổn định Lyapunov cho hệ thống robot tay máy đôi nhằm di chuyển vật bám theo quỹ đạo đặt trước Bộ điều khiển kiểm chứng môi trường mô điều kiện có nhiễu tác động đến hệ thống Kết mô cho thấy hiệu thuật toán đề xuất việc điều khiển dịch chuyển vật bám quỹ đạo các điều kiệ có khả ứng dụng thực tế Từ khóa: Robot tay máy đôi, Điều khiển trượt, backstepping Received 25th February 2018 Revised 27 th March 2018 Accepted 20 th April 2018 Author affiliations: Hanoi University of Science and Technology; Hung Yen University of Technology and Education *Corresponding author: xhaicuwc.edu.vn@gmail.com Journal of Military Science and Technology, Special Issue, No.54A, 05 - 2018 ... (9) BACKSTEPPING SLIDING MODE CONTROL In backstepping – sliding mode controller, the control input is changed according N D Hiep, …, L X Hai, “ Backstepping – sliding mode control for dual arm robot. ”... the dual arm robotic system is constructed In Section 3, Backstepping – Sliding mode controller is introduced Section is simulation results and conclusion SYSTEM MODELLING Fig.1 Physical model... Hai, “ Backstepping – sliding mode control for dual arm robot. ” Research payload’s position Then it handles the payload and tracks the designed trajectory To handle the payload, the DAR’s arm effectors