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Forward and Inverse Kinematics Analysis of Denso Robot Proceedings of the International Symposium of Mechanism and Machine Science, 2017 AzC IFToMM – Azerbaijan Technical University 11-14 September 2017, Baku, Azerbaijan Forward and Inverse Kinematics Analysis of Denso Robot Mehmet Erkan KÜTÜK 1*, Memik Taylan DAŞ2, Lale Canan DÜLGER1 1*Mechanical Engineering Department, University of Gaziantep Gaziantep/ Turkey E-mail: [email protected] 2 Mechanical Engineering Department, University of Kırıkkale Abstract used Robotic Toolbox in forward kinematics analysis of A forward and inverse kinematic analysis of 6 axis an industrial robot [4]. DENSO robot with closed form solution is performed in This study includes kinematics of robot arm which is this paper. Robotics toolbox provides a great simplicity to available Gaziantep University, Mechanical Engineering us dealing with kinematics of robots with the ready Department, Mechatronics Lab. Forward and Inverse functions on it. However, making calculations in kinematics analysis are performed. Robotics Toolbox is traditional way is important to dominate the kinematics also applied to model Denso robot system. A GUI is built which is one of the main topics of robotics. Robotic for practical use of robotic system. toolbox in Matlab® is used to model Denso robot system. GUI studies including Robotic Toolbox are given with 2. Robot Arm Kinematics simulation examples. Keywords: Robot Kinematics, Simulation, Denso The robot kinematics can be categorized into two Robot, Robotic Toolbox, GUI main parts; forward and inverse kinematics. Forward kinematics problem is not difficult to perform and there is no complexity in deriving the equations in contrast to the 1. Introduction inverse kinematics. Especially nonlinear equations make the inverse kinematics problem complex. They may also Robot kinematics specifies the analytical study of the be coupled and have not got unique solutions. Thus motion of a robot manipulator. Formulating the reasonable solutions obtained mathematically may not solve the kinematics models for a robot mechanism is very problem physically [5]. Liu et al. applied geometric important in order to investigate the behaviour of approach for inverse kinematics analysis of a 6 dof robot industrial manipulators. There are two different spaces [6]. Qiao et al. used double quaternions to get solution for used in kinematics modelling of manipulators. They are inverse kinematics problem [7]. Nubiola and Bonev called Cartesian space and Quaternion space. The offered a simple and efficient way to solve inverse transformation between two Cartesian coordinate systems kinematics problem for 6R robots [8]. It is noticed that, can be decayed into a rotation and a translation. There are Artificial Intelligence (AI) methods are frequently used in a lot of approaches to represent rotation like Euler angles, inverse kinematics problem [9, 10, 11] in recent years. Gibbs vector, Cayley-Klein parameters, orthonormal matrices etc. Homogenous transformations based on 4x4 2.1 Forward Kinematics Analysis real matrices have been used dominantly in robotics [1]. Robotics Toolbox contains a lot of functions that are The forward kinematics problem is related between demanded in robotics and addresses fields such as the individual joints of the robot manipulator and the kinematics, dynamics, and trajectory generation. The position and orientation of the tool or end-effector. The Toolbox is convenient for simulation besides analyzing joint variables are the angles between the links for results from experiments with real robots. It is also an revolute or prismatic joints, and the link extension in the effective tool for education. The Toolbox is organized on prismatic or sliding joints [12]. A systematic way of a very common method of representing the kinematics and describing the geometry of a serial chain of links and dynamics of serial-link manipulators. It is described by joints was proposed by Denavit and Hartenberg and is the matrices. These include, in the basic case, the Denavit known today as Denavit-Hartenberg (DH) notation [2]. and Hartenberg parameters of the robot [2]. Any serial- The matrix A representing four movements is found by link manipulator can be configured by the user. A number postmultiplying the four matrices giving four movements of examples are provided for well-known robots such as to reach frame {j-1} to frame {j} in Figure 1. the Puma 560 and the Stanford arm [3]. Constantin et al. 71 Proceedings of the International Symposium of Mechanism and Machine Science, 2017 AzC IFToMM – Azerbaijan Technical University 11-14 September 2017, Baku, Azerbaijan The following table shows DH parameters of the Denso robotic arm necessary to derive the kinematics of the robot. Gripper is not included in the analysis. Table 1. DH Parameters of the Denso Robotic Arm Joint Limits Joint i i di ai i (degrees) 1 q1 d1 0 /2 160, 160 q 120, 120 Fig. 1. DH representation of a general purpose joint-link 2 2 0 a2 0 combination q Transformation between two joints in a generic form [3] is 3 3 0 a3 /2 20, 160 given in Eq. (1). 4 q4 d4 0 cosj sin j cos j sin j sin ja j cos j sin cos cos cos sin a sin 5 q5 0 0 j1 j j j j j j j (1) Aj 0 sinj cos jd j 6 q6 d6 0 0 360, 360 0 0 0 1 where = 0.125m, a2 =0.21m, a3 =-0.075m, = 0.21m Denso Robot is a 6 Degrees-of-Freedom (DOF) robotic and = 0.07m. manipulator. The link lengths are given in Figure 2(a). World frame and joint frames used in calculations and Transformation matrix for each joint can be obtained home position of Denso robot are shown in Figure 2b and by using Eq. (1). The parameters given in Table 1 are Figure 2c, respectively. substituted into Eq. (1) to find each of them. Six transformation matrices are presented in Eq. (2). CS1100 C2 S 20 a 2 C 2 SC00 S C0 a S 11 2 2 2 2 A1 A 2 0 1 0 d 0 0 1 0 1 0 0 0 1 0 0 0 1 CS00 C0 S a C 44 3 3 3 3 SC4400 S30 C 3 a 3 S 3 A4 A3 0 1 0 0 0 1 0 d4 0 0 0 1 0 0 0 1 CS00 (a) Link lengths (b) World frame and joint frames 55 CS 00 66 SC5500 SC6600 A5 A 0 1 0 0 6 0 0 1 d6 0 0 0 1 0 0 0 1 (2) where Cos and Sin are abbreviated to C and S, respectively. The total transformation between the base of the robot and the hand is; R (c) Home position TAAAAAAH 1 2 3 4 5 6 (3) Fig.2. General overview of DENSO 72 Proceedings of the International Symposium of Mechanism and Machine Science, 2017 AzC IFToMM – Azerbaijan Technical University 11-14 September 2017, Baku, Azerbaijan Transformation matrices for six axes given in Eq. (2) are To find the inverse kinematics solution for the 1st joint th postmultiplied in an order which is given in Eq. (3). This 1 as a function of the known elements, the 6 link equality is shown in Eq. (4). transformation inverse is postmultiplied as follows in Eq. nx o x a x P x (9). n o a P (4) y y y y nx o x a x P x AAAAAA1 2 3 4 5 6 nz o z a z P z ny o y a y P x A A A A A A A1 xA1 (9) 0 0 0 1 1 2 3 45 66 n o a P 6 z z z x 0 0 0 1 where n (normal), o (orientation), a (approach) elements are for orientation and P (position) elements are position 1 where AA I . I is identity matrix. In this case, the elements relative to the reference frame [12]. The 6 6 above equation is resulted in Eq. (10). elements of the matrix shown in left hand side of Eq. (4) are given in Eqs. (5, 6, 7 and 8). nx o x a x P x n SCS( SCCC ( CSS )) CCSS ( ( CCCC ( ny o y a y P x 1 x 641 4123 123 6514 4123 A A A A A xA 1 2 3 45 n o a P 6 CSSSCCSCCS1 2 3)) 5 ( 1 2 3 1 3 2 )) z z z x n SCC( SSSS ( CCS )) CCCS ( ( CSSS ( 0 0 0 1 y 614 4123 231 6514 4123 CCSSCSSCSS)) ( )) (10) 2 3 1 5 2 1 3 3 1 2 n CSCC( ( SS ) CCCS ( CS )) SSCS ( CS ) z 6523 23 4523 32 4623 32 The required multiplication in Eq. (10) is carried out and (5) it yields as Eq. (11). '' '' '' C1 nCoSx 6 x 6 oC x 6 nS x 6 a x Pad x x 6 ox SCSS6514( ( CCCC 4123 ( CSS 123 )) SCCS 5123 ( CCS 132 )) '' '' '' S1 nCoSy 6 y 6 oC y 6 nS y 6 a y Pad y y 6 (11) CCSSCCCCSS6( 4 1 4 ( 1 2 3 1 2 3 )) '' '' '' '' nC oS oC nS a Pad z6 z 6 z 6 z 6 z z z 6 0 0 0 1 0 0 0 1 oy CCC614( SSSS 4123 ( CCS 231 )) SCCS 6514 ( ( CSSS 4123 ( CCSSCSSCSS)) ( )) 2 3 1 5 2 1 3 3 1 2 st th oz SSCC6523( ( SS 23 ) CCCS 4523 ( CS 32 )) CSCS 6423 ( CS 32 ) It is noticed that the elements located in 1 row & 4 column (abbreviated to (1, 4)) and 2nd row & 4th column (6) (abbreviated to (2, 4)) can be used in defining (these abbreviations are also used in the remaining part of the inverse kinematics analysis). All elements in the left-hand ax SSS514( CCCC 4123 ( CSS 123 )) CCCS 5123 ( CCS 132 ) a SCS( CSSS ( CCS )) CCSS ( CSS ) side of the Eq.
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