Inverse kinematics of a serial-Parallel robot used in hot forging process

Vietnam Journal of Mechanics, VAST, Vol. 38, No. 2 (2016), pp. 81 – 88 DOI:10.15625/0866-7136/38/2/5958 INVERSE KINEMATICS OF A SERIAL-PARALLEL ROBOT USED IN HOT FORGING PROCESS Chu Anh My Le Quy Don University, Hanoi, Vietnam E-mail: mychuanh@yahoo.com Received March 12, 2015 Abstract. In hot extrusion forging process, the use of robot arm for transferring heavy billets can reduce downtime, improve productivity, reduce worker fatigue and optimize the use of energy and manpower. To increase the stiffness of the robot and force the end- effector move in directions parallel with the ground surface, two parallel links are added to a standard serial manipulator. This modification of the structure could make it a bit of challenge in the system modelling and controlling. This paper addresses the inverse kinematics analysis that is the central issue for developing autonomous control modes of the robot application. Keywords: Serial-paralell robot, inverse kinematics, hot extrusion forging. 1. INTRODUCTION Forging is a manufacturing process involving the shaping of metal using localized compressive forces. Forging is often classified according to the temperature at which it is performed:“cold”, “warm”, or “hot” forging.Consider a hot forging process layout in Fig. 1. Fig. 1. Layout of forging station Fig. 2. Forging process c© 2016 Vietnam Academy of Science and Technology 82 Chu Anh My A steel billet weighted about 60 kg is heated up to 1100◦C in the heating furnace, and it is then transferred to the forging machine. The forging machine is a hydraulic press system used for the extrusion process. After the billet located in a die fixed in the machine, it is pressed under the force F by the hydraulic press equipment, to produce the forging part (see Fig. 2). Since billets are hot and heavy, two workers are required to carry and transfer billets one by one for the process. This causes instability, worker fatigue and downtime in the manufacturing. To overcome these disadvantages, an industrial robotics system shown in Fig. 3 is designed to support the workers. Fig. 3. Robot design for hot forging process The robot consists of 6 joints, 5 links and one gripper (the end-effector); and all sys- tem is actuated by hydraulic cylinders. The first joint connecting the manipulator with the chassis is a translation joint; all the other joints are prismatic. The links connecting joint 3 and 4, and connecting joint 4 and 5 are parallel mechanisms. This structural mod- ification accompanied with the hydraulic actuators aim to reinforce the stiffness of the system so that the robot is capable of carrying the heavy billet. In addition, the orien- tation of the end-effector is restricted as desired; the end-link moves in only directions parallel with the ground surface. This advantage could help to reduce the complexity of the control process. Though the hybrid design takes full advantages of flexibility of the serial links and makes full use of the reinforced stiffness of the added parallel mecha- nisms, it possesses complexity in modelling and controlling. For controlling the robot, the kinematical modelling and analysis play a central role that needs to be considered specifically. The kinematic of general manipulator has been the fundamental problem [1], in which the serial chain of links is considered as the stan- dard kinematical model. Either Danavit-Hartenberg method or Craig method is usually used for the modelling. Further studies on the manipulator kinematic could be found in the literature such as the kinematics for the redundancy structures [2], the kinematic of Inverse kinematics of a serial-parallel robot used in hot forging process 83 the parallel robot [3, 4], kinematic performance analysis [5], kinematics of manipulator mounted on a mobile plateform [6], and the kinematics design of manipulator [7]. How- ever, a few researcher interests in modeling and analyzing the hybrid serial - parallel robot structures. This paper addresses the modelling of which the kinematical constraint of the par- allel links is taken into account. A numerical solution to the inverse kinematics problem is proposed. Finally, given the trajectory of the end-effector, the time history of joint vari- ables are determined that can be used for the controlling process. 2. KINEMATICS MODELLING In Fig. 2, we define d1, q2, q3, q4, q5 and q6 are joint variables which determine the configuration of the mechanical system. We denote q = [ d1 q2 q3 q4 q5 q6 ]T. Due to the parallel motion of the two parallel links, the relationship between q3, q4 and q5 can be illustrated in the following Fig. 4. Fig. 4. Geometrical relationship between q3, q4 and q5 As shown in the Fig. 4, the following constrain can be write q5 = −q3 − q4 − pi2 . (1) Following the Denavit-Hartenberg technique, all joint coordinate systems are de- noted as shown in Fig. 2. Also, in Tab. 1, kinematical and geometrical parameters with respect to all links are illustrated. Notice that link number 1’ and 5’ are added to make it easier in writing homogeneous transformation matrixes in the form of the Denavit- Hartenberg’s formulation. O0 ≡ (O0x0y0z0) is denoted as the reference frame. In the same way, the following notations O1,O2, . . . ,O6 are the link frames, correspondingly. The frames O′2,O′3,O′4 are added for the refencing purpose. The homogeneous transformation matrix of the end-effector with respect to the reference frame can be denoted as H0E = [ A0E rE 0 1 ] , (2) 84 Chu Anh My Table 1. Kinematical and geometrical parameters of links Link θi di ai αi 1’ 0 d1 0 0 1” 0 0 0 −pi/2 1 0 a1 0 0 2 q2 d2 a2 pi/2 3 q3 0 a3 0 4 q4 0 a4 0 5’ q5 0 a5 −pi/2 5 0 d5 0 0 6 q6 d6 0 0 Fig. 5. Kinematical model of serial parallel robot where rE = [ xE yE zE ]T is the position vector of the end-effector; A0E is the rotation matrix of the end-effector with respect to the reference frame. The position and the orientation of the end-effector can be determined via the fol- lowing relationship H0E = H01′ (d1)H1′1H12 (q2)H23 (q3)H34 (q4)H45′ (q5)H5′5H56 (q6) , (3) where Hji is the homogeneous transformation matrix defined for the homogeneous mo- tion of the frame indexed i with respect to the previous frame indexed j. Inverse kinematics of a serial-parallel robot used in hot forging process 85 Hji =  cos θi − sin θi cos αi sin θi sin αi ai cos θi sin θi cos θi cos αi − cos θi cos αi ai sin θi 0 sin αi cos αi di 0 0 0 1  . (4) Substituting all the matrixes Hji yielded in accordance with Tab. 1 into Eq. (3) obtains[ A0E rE 0 1 ] = [ A06 (q) r06 (q) 0 1 ] . (5) Eq. (5) and constraint Eq. (1) describe the forward relationship of the robot. Given a value of q, the position and the orientation of the end-effector can be calculated analyt- ically. If we denote f (q) = r06 (q) as function vector of q, for the position forward kine- matics, the following equation can be yielded rE = f (q) . (6) 3. INVERSE KINEMATICS To control the robot operating as desired, the inverse kinematics computation must be taken into account. Generally, given the path of the end-effector, all the histories of the joint variables must be calculated as q = f−1 (rE) . (7) For the considered robot, the orientation of the end-effector is regardless because of the feature of the parallel structure mentioned previously. In addition, as discussed before, the robot transfers the billet for forging process in discontinuous sequence. There- fore, the continuity of the tool path is not required. The path of the point rE of the end- effector is illustrated in the following Fig. 6. Fig. 6. Required path of the end-effector 86 Chu Anh My As seen in Fig. 6, the path can be separated into 6 arcs (Ac1,. . . , Ac6) representing 6 periods of time that the robot fulfills the required task. To simplify the formulation and implementation of the inverse kinematics, Eq. (7) is solved according to the separated 6 periods of time as follows. Period 1. The robot moves in z0 direction only to locate the end-effector and gets ready for picking up the billet The geometrical parameters are given as: d1 = 0.5 m, a1 = 0.11 m, d2 = 0.25 m, a2 = 0.1 m, a3 = 0.73 m, a4 = 0.63 m, a5 = 0.18 m, d5 = 0.03 m, d6 = 0.43 m. The initial values of joint variables are given as follows: d1 = 0, q2 = 0, q3 = pi 3 , q4 = pi 6 , q6 = 0. The end-effector moves so that xE = 0.4, yE = 0.88 and zE = 0.5t, where t ∈ < is a parameter and t ∈ [0, 1]. In this period, only d1 varies. Finally we obtain d1 = 0.5t. Period 2. The robot transfers the billet to the heating furnace In this period of time, rE can be represented as xE(t) = −0.124t 2 + 0.8t+ 0.4 yE(t) = −0.92t2 + 1.24t+ 0.88 zE(t) = 0.5 Solving Eq. (7) yields d1 = 0.5, q2 = 0, q6 = 0, and q3, q4 and q5 vary as shown in Fig. 7. Fig. 7. Values of q3, q4 and q5 for the period 2 Period 3. The robot picks up the billet heated and moves it outside the furnace In this period of time, rE can be represented as xE = 1.324− 0.95tyE = 1.2zE = 0.5 Solving Eq. (7) yields d1 = 0.5, q2 = 0, q6 = 0, and q3, q4 and q5 as shown in Fig. 8. Period 4. The robot moves the billet up and gets ready to turn transferring the billet to the forging machine Inverse kinematics of a serial-parallel robot used in hot forging process 87 Fig. 8. Values of q3, q4 and q5 for the period 3 In this period of time, rE can be represented as xE = 0.374yE = 0.2t+ 1.2zE = 0.5 Solving Eq. (7) yields d1 = 0.5, q2 = 0, q6 = 0, and q3, q4 and q5 as shown in Fig. 9. Fig. 9. Values of q3, q4 and q5 for the period 4 Period 5. The robot turns 180◦ moving the billet to the forging machine. After that the end-effector turns 90◦ For this period, only q2 varies from 0 to 180◦. After that q6 varies form 0 to 90◦. All other joint variables keep their values as calculated in the last period. Period 6. The robot reaches to the position ready to release the billet on the forging machine In this period of time, rE can be represented as xE(t) = 0.576t 2 − 1.452t− 0.374 yE(t) = 0.12t2 − 0.4t+ 1.4 zE(t) = 0.5 Solving Eq. (7) yields d1 = 0.5, q2 = pi, q6 = pi 2 , and q3, q4 and q5 as shown in Fig. 10. 88 Chu Anh My Fig. 10. Values of q3, q4 and q5 for the period 6 For 6 given segments of the required path, the solution to the inverse position kine- matics is found that is applicable to control the joint actuators. The values of the joint variables change in feasible ranges. 4. CONCLUSION The kinematical model of the serial-parallel robot is formulated. The kinematics equation of the considering robot must satisfy the constraint equation because of the motion constrained of the parallel links. This is different from the typical serial robot. The end-effector moves always parallel with the ground surface that could simplify the control procedure. Moreover, this is to make sure that the griper holds the billet in fixed posture during the transferring time period. The numerical results of the inverse equations are suitable in value and the curves show its proper change in shape and magnitude. REFERENCES [1] N. V. Khang and C. A. My. Industrial robot basics. Vietnam Education Publishing House, (2011). (in Vietnamese). [2] J. Wang, Y. Li, and X. Zhao. Inverse kinematics and control of a 7-DOF redundant manipulator based on the closed-loop algorithm. International Journal of Advanced Robotic Systems, 7, (4), (2010), pp. 1–9. [3] Y. Li and Q. Xu. Kinematic analysis of a 3-PRS parallel manipulator. 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