Inter national J our nal of Robotics and A utomation (IJRA) V ol. 9, No. 3, September 2020, pp. 153 159 ISSN: 2089-4856, DOI: 10.11591/ijra.v9i3.pp153-159 r 153 F orward kinematic analysis of Dobot using closed-loop method J a vier Sanjuan, Mohammad Rahman, Iv an Rulik Mechanical Engineering Department, Uni v ersity of W isconsin-Mil w auk ee, United States Article Inf o Article history: Recei v ed Oct 23, 2019 Re vised Jan 12, 2020 Accepted Mar 4, 2020 K eyw ords: Dobot Dynamical analysis F orw ard kinematics P arallel robots Serial robots ABSTRA CT Dobot is a h ybrid robot that combines features from parallel and serial robots. Because of this characteristic, the robot e xcels for is reliability , allo wing its implementation in di v erse applicati ons. Therefore, researchers ha v e studied its kinematics to impro v e its capabilities. Ho we v er , to the e xtent of our kno wledge, no analysis has been reported taking into consideration the closed-loop configuration of Dobot. Thus, this article presents the complete analytical solution for the forw ard kinematics of Dobot, considering each link. The results are e xpected to be utili zed in the de v elopment of a dynamical model that contemplates the dynamics of each element of the robot. This is an open access article under the CC BY -SA license . Corresponding A uthor: Ja vier Sanjuan, Biorobotics Lab, USR 281, Uni v esirty Services and Research Building, UW -Mil w auk ee, 115 East Reindl W ay , Glendale, WI, 53212, United States. Email: jsanjuan@uwm.edu 1. INTR ODUCTION Dobot is a 3 De gree of Freedom (DOF) type of commercial serial manipulator that posses numerous attracti v e features bec ause of its h ybrid serial and parallel configuration. One of its principal characteristics is the preci sion due to Dobot uses a four -bar linkage to actuate the rotation of each link. Additionally , Dobot utilizes a parallelogram mechanism to ensure that the orientation of the end-ef fector remains constant. Therefore, because of these features and its lo w cost, v arious researches ha v e implemented Dobot in di v erse applications such: 3D printing [1], electrochemical writing [2, 3], Pick and place [4–8], education [9], sur gery [10], and e v en electric v ehicle char ging [11]. Because of the di v erse applications of Dobot, researches ha v e conducted dif ferent types of analys is on this robot. T . Cheng et al. analyzed the forw ard kinematics, in v erse kinematics, and jacobian matrix of Dobot [11]. Moreo v er , O. Hock, et al. included an analysis of the pseudoin v erse method to compute the in v erse kinematics of Dobot numerically [12]. G. Y u et al. modified the end-ef fector of Dobot to included a gripper and a camera, to implement hand-e ye calibration [13]. Ho we v er , to the e xtent of our kno wledge, no research has conducted the forw ard kinematics analysis of the linkage that composes the robot. The forw ard kinematics analysis permits to obtain the kinematic of each link, which is a requirem ent to de v elop a detail dynami cal model of Dobot. This type of model allo ws the implementation of rob ust control strate gies such as in v erse dynamics control [14, 15], permitting Dobot to be used in the application of high v elocity while maintaining its precision [16]. V arious researcher has w ork ed in the computation of the direct kinematics for dif ferent parallel robots. Y ujiong L., et al. presented the computation of the forw ard kinematics of an H4 parallel robot using a geometric approach [17]. Olaru D. obtained the forw ard and in v erse J ournal homepage: http://ijr a.iaescor e .com Evaluation Warning : The document was created with Spire.PDF for Python.
154 r ISSN: 2089-4856 kinematics of a 5DOF robot, between the rele v ant results from this research is the increasing of the robot precision due to the implementation of the computation of the forw ard kinematics in its control [18]. Jin S. K., et al. implemented the kinematic analysis of a 4DOF parallel robots for MRI-Guided percutaneous interv en- tions [19]. T ang et al. study the kinematics of a no v el 2R1T parallel mechanism [20]. Therefore, because of the adv antages of a complete kinematical model, this research focuses on the computation of the forw ard kine- matics of Dobot, considering the closed-loop configuration of each actuated joint. The results of the algorithm were v alidated using a CAD model, v erifying the accurac y of the results. The or g anization of the article is the follo wing: Section 2 describes the principal joints of Dobot. Section 3 presents the ki nematical diagram of Dobot, and e xplain the relation between links. Section 4 presents the computation of the forw ard kinematic of Dobot. Section 5 compares the results of the obtained equations with a CAD model of Dobot, to v erify the accurac y of the results. Figure 1. 3D model representation of DOBO T 2. DOBO T DESCRIPTION The cad model of Dobot is presented in Figure 1. As can be seen, joints 1 and 2 ( q 1 & q 2 ) actuates in the same direction, therefore, the actuation of those joints, and their ef fect on the kinematics of the robot can be analyzed with a plane model. On the other hand, joint 3 rotates the robot in the y direction, which changes the orientation Dobot. Since this rotation is pe rpendicular to the other actuated joints, this actuation is not considered in the analysis. Also, T . Cheng et al. presented a model to study the influence of this joint in the orientation of Dobot [11]. In Figure 2 the sectional vie w sho ws the mechanism inside DOBO T’ s case, where the v ariables e xplained in Section 2 can be related using Figures 2 and 3. Figure 2. Sectional vie w of DOBO T 3. KINEMA TICS MODEL T o f acilitate the analysis of Dobot the kinematic diagram that presents all the links is obtained, as sho wn in Figure 3. Note that the actuate d angles are represented by q 1 and q 2 . Additionally , since dobot posses a parallel configuration, the actuation of the second joint is obtained by a four bar mechanism composed for links: L q 2 ; L 7 ; and L 6 , that transmit the mo v ement from the base to this joint. The end ef fector is represent by Int J Rob & Autom, V ol. 9, No. 3, September 2020 : 153 159 Evaluation Warning : The document was created with Spire.PDF for Python.
Int J Rob & Autom ISSN: 2089-4856 r 155 point P , the orientation of this link is not directly actuated, and is dictated by a W att’ s mechanism [21] that is composed for links: L 2 ; L 3 ; L 4 and L 5 . Then, to v erify that the kinematic representation of Dobot is correct, the Gruebler equation [22] is used to compute the DOF of the system, as follo ws: Figure 3. Kinematics model N = 3 L 2 J (1) Where N is the number of DOF; L is the number of links, and J is the number of joints. Therefore, the number of DOF is 2, which is the e xpected result. 4. FOR W ARD KINEMA TICS T o obtain the forw ard kinematics it is used the closed-loop method [23]. This method requires the analysis of closed-loop v ector to obtain the position equati ons. The number of closed-equations is determined by the use of the relation for Li et al. in [17], as follo ws: d = J L = 3 (2) Therefore, it is required to use three closed-loop equations to obtain the position of Dobot. The closed-loop used are presented in Figure 4. Figure 4. Closed-loops diagrams F or the first loop, the obtained equation is the follo wing: ~ L q 2 + ~ L 7 + ~ L 63 = ~ L 0 + ~ L q 1 (3) Equation (3) is re grouped in terms of L 6 and e xpressed in v ector form, as follo ws: L 63 cos 6 + 3 sin 6 + 3 = L 0 cos 0 sin 0 + L q 1 cos q 1 sin q 1 L q 2 cos q 2 sin q 2 L 7 cos 7 sin 7 (4) F orwar d kinematic analysis of Dobot using closed-loop method (J avier Sanjuan) Evaluation Warning : The document was created with Spire.PDF for Python.
156 r ISSN: 2089-4856 Then, adding the square of the x side and the square of the y side, the term corresponding to 6 is simplified. Thus, obtaining an e xpression that depends on 7 ; q 1 ; q 2 ; and 0 . Ho we v er , since q 1 and q 2 are entries of the system, and 0 is a fix angle, the only v ariable of interest is 7 . Therefore, the resultant e xpression is e xpressed in term of 7 as follo ws: A 1 cos 7 + A 2 sin 7 + A 3 = 0 (5) Where the terms A i are parameters that depends on the dimensions of Dobot, and kno wn angles. Then, using the half tangent substitution tan 7 2 = c 1 , with cos 7 = 1 c 2 1 1+ c 2 1 and sin 7 = 2 c 1 1+ c 2 1 , (5) is reduced to an algebraic e xpression, as follo ws: k 1 C 2 2 + k 2 C 2 + k 3 = 0 (6) Then, using the quadractic equation in (6), the follo wing equation is obtained: C 1 = k 2 p k 2 2 4 k 1 k 3 2 k 1 (7) Replacing c 1 for the tan 7 2 , the e xpression for 7 is obtained: 7 = atan 0 @ p 2 q L q 1 2 L q 2 2 (cos (2 q 1 2 q 2 ) 1)+2 L q 1 2 sin ( q 1 ) 2 L q 1 L q 2 sin ( q 2 ) 2 L q 1 (L q 1 + L q 1 cos ( q 1) L q 2 cos ( q 2 ) L q 2 cos ( q 1 q 2 )) 1 A (8) No w that the e xpression for 7 is obtained, 6 is computed from applying the dot product of 4 with unitary v ector ^ i = [1 ; 0] T , and solving for 6 , generating: 6 = acos L q 2 cos ( q 2 ) L q 1 cos ( q 1 ) + L q 1 cos ( 7 ) L q 2 (9) Analogously , the second loop equation is presented in the follo wing: ~ L 1 + ~ L 2 + ~ L 31 = ~ L q 2 + ~ L 7 + ~ L 61 (10) The tw o v ariables to solv e from (10) are 2 and 3 , whose solutions are obtained applying the same methods for 7 and 6 , respecti v ely . Therefore, the solution for 3 is presented belo w: 3 = 2atan   k 12 p k 2 12 4 k 11 k 13 2 k 11 ! (11) The v alues of k 11 ; k 12 ; and k 13 are presented in the Appendix section. The solution for 2 is presented ne xt. 2 = acos L q 2 cos ( 1 + 6 ) L q 2 cos ( 1 + 3 )+L q 2 cos ( q 2 ) L q 2 cos ( 1 ) + L q 1 cos ( 7 ) L q 1 (12) Then, using the same procedure as before, the third loop equation is the follo wing: ~ L 3 + ~ L 4 = ~ L 62 + ~ L 51 (13) From solving (13), the solution for 5 is the follo wing: 5 = 2 atan 0 @ p 2 q L 4 2 L q 2 2 (cos (2 3 2 6 ) 1) 2 L q 2 2 sin ( 3 ) + 2 L 4 L q 2 sin ( 6 ) 2 L q 2 (L q 2 L 4 cos ( 6 ) +L q 2 cos ( 3 ) L 4 cos ( 3 6 )) 1 A (14) And for 4 : 4 = acos L q 2 cos ( 1 + 5 ) + L 4 cos ( 6 ) L q 2 cos ( 3 ) L 4  (15) Int J Rob & Autom, V ol. 9, No. 3, September 2020 : 153 159 Evaluation Warning : The document was created with Spire.PDF for Python.
Int J Rob & Autom ISSN: 2089-4856 r 157 Lastly , using the fourth loop as presented in Figure 5, the v alues of x and y are obtained using the follo wing e xpression: L q 1 cos ( q 1 ) + L 4 cos ( 6 ) + L 5 cos ( 5 ) = x L q 1 sin ( q 1 ) + L 4 sin ( 6 ) + L 5 sin ( 5 ) = y (16) Note that (8), (9), (11), (12), (14), (15), and (16) were simplified considering the dimensional parameters presented in T able 1. Figure 5. F ourth loop T able 1. Dimensional parameters on DOBO T P arameter V alues L 63 , L q 2 , L 1 , L 31 , L 61 , L 3 , L 51 43 [mm] L 7 , L q 1 , L 2 135 [mm] L 4 , L 62 147 [mm] L 0 0 [mm] 0 , 1 , 2 , 3 0 [rad] 1 2 : 66 [rad] 1 5 : 1 [rad] 1 0 : 7 [rad] 5. RESUL TS T o v alidate the kinematics model and e v aluate its accurac y , a comparison w as done using the CAD obtained from D O B O T © . Both, the results from the CAD model and the Kinematic model are presented in T able . When comparing the results it is clear that the computed v alues and the CAD model present the same results. Therefore, the forw ard kinematic analysis is accurate. T able 2. T est of the Kinematic Model V ariables CAD measur e [ o ] Kinematic Model [ o ] q 1 44 44 q 2 150 150 2 44 44 3 40.28 40.2823 4 30 30 5 0.28 0.2823 6 30 30 7 44 44 6. CONCLUSIONS This paper presents the forw ard kinematic analys is of Dobot, a 3DOF h ybrid robot that is posses both, a serial and parallel configuration. The equations were e xpressed using the closed-loop method and were solv ed analytically by the application of the tangent half-angle substitution. This result is e xpected to be utilized in the computation of the dynamics of Dobot, to implement elaborated control strate gies such as in v erse dynamic control. The accurac y of the equations w as also v alidated using a CAD model, obtaining an easy to utilize a set of equations to obtain the kinematic of Dobot as a function of the joint angles. F orwar d kinematic analysis of Dobot using closed-loop method (J avier Sanjuan) Evaluation Warning : The document was created with Spire.PDF for Python.
158 r ISSN: 2089-4856 APPENDIX The follo wing equations present the v alues of k 11 , k 12 , and k 13 . k 11 =2L q 2 2 cos ( 1 q 2 ) 2L q 2 2 cos ( 1 1 ) + 2L q 2 2 cos ( 1 6 ) 2L q 2 2 cos ( q 2 1 ) + 2L q 2 2 cos ( q 2 6 ) 2L q 2 2 cos ( 1 6 ) + 4L q 2 2 2L q 1 L q 2 cos ( 1 7 ) + 2L q 1 L q 2 cos ( 6 7 ) + 2L q 1 L q 2 cos ( 1 7 ) + 2L q 1 L q 2 cos ( q 2 7 ) (17) k 12 =4 L q 2 2 sin ( 1 q 2 ) 4 L q 2 2 sin ( 1 1 ) + 4 L q 2 2 sin ( 1 6 ) + 4 L q 1 L q 2 sin ( 1 7 ) (18) k 13 =2L q 2 2 cos ( 1 1 ) 2L q 2 2 cos ( 1 q 2 ) 2Lq2 2 cos ( 1 6 ) 2L q 2 2 cos ( q 2 t 1 ) + 2L q 2 2 cos ( q 2 t 6 ) 2L q 2 2 cos ( t 1 t 6 ) + 4L q 2 2 2L q 1 L q 2 cos ( 1 7 ) + 2L q 1L q 2 cos ( 6 7 ) 2L q 1 L q 2 cos ( 1 7 ) + 2L q 1 L q 2 cos ( q 2 7 ) (19) REFERENCES [1] L. W u, J. Y ang, X. Zhang, and Y . Chen, “Mult i manipulator cooperati v e 3d printing based on dobot manipulator , in IOP Confer ence Series: Materials Science and Engineering , v ol. 382, no. 4. IOP Publishing, pp. 042040, 2018. [2] X. Zhai, X. Zou, J. Shi, X. Huang, Z. Sun, Z. Li, Y . Sun, Y . Li, X. W ang, M. Holmes et al. , Amine- responsi v e bilayer films with impro v ed illumination stability and electrochemical writing property for visual monitoring of meat spoilage, Sensor s and Actuator s B: Chemical , p. 127130, 2019. [3] R. Guo, X. Sun, B. Y uan, H. W ang, and J. Liu, “Magnetic liquid metal (fe-e g ain) based multifunc- tional electroni cs for remote self-healing materials, de gradable electronics, and thermal t ransfer printing, Advanced Science , 2019. [4] I. Md Rasedul, R. Md Arifur , M. A. uz Zaman, and H. R. Mohammad, “Cartesian trajectory based control of dobot robot, in Pr oceedings of the International Confer ence on Industrial Engineering and Oper ations Mana g ement . IEOM Society International, pp. 1507–1517, 2019. [5] F . Nag ata, Y . Seda, K. Hamada, S. Suzuki, A. Otsuka, T . Ik eda, H. Ochi, K. W atanabe, M. K. Habib, and T . K usano, “Outline font handler for industrial robots, in 2018 IEEE International Confer ence on Mec hatr onics and A utomation (ICMA) . IEEE, pp. 1823–1828, 2018. [6] P . Jask ´ olski and K. Nadoln y , “Characteristic of process flo w in modular didactic production system for gear trains, J ournal of Mec hanical and Ener gy Engineering , v ol. 3, no. 2, pp. 115–120, 2019. [7] P . Urhal, A. W eightman, C. Di v er , and P . Bartolo, “Robot assisted additi v e manuf acturing: A re vie w , Robotics and Computer -Inte gr ated Manufacturing , v ol. 59, pp. 335–345, 2019. [8] R. Chen, R. Song, Z. Zhang, L. Bai, F . Liu, P . Jiang, D. Sindersber ger , G. J. Monkman, and J. Guo, “Bio-inspired shape-adapti v e soft robotic grippers augmented with electroadhesi on functionality , Soft r obotics , 2019. [9] D. Scaradozzi, L. Screpanti, and L. Cesaretti, “T o w ards a definition of educational robotics: a classifi- cation of tools, e xperiences and assessments, in Smart Learning with Educational Robotics . Springer , pp. 63–92, 2019. [10] T . Cheng, W . Li, C. S. H. Ng, P . W . Y . Chiu, and Z. Li, “V isual serv o control of a no v el magnetic actuated endoscope for uniportal video-assisted thoracic sur gery , IEEE Robotics and A utomation Letter s , v ol. 4, no. 3, pp. 3098–3105, 2019. [11] M. Behl, J. DuBro, T . Flynt, I. Hameed, G. Lang, and F . P ark, Autonomous electric v ehicle char ging system, in 2019 Systems and Information Engineering Design Symposium (SIEDS) . IEEE, pp. 1–6, 2019. [12] O. Hock and J. ˇ Sedo, “F orw ard and in v erse kinematics using pseudoin v erse and transposition method for robotic arm dobot, in Kinematics . IntechOpen, 2017. [13] G. Y u, Y . Liu, X. Han, and C. Zhang, “Objects grasping of robotic arm with compliant grasper based on vision, in Pr oceedings of the 2019 4th International Confer ence on A utomation, Contr ol and Robotics Engineering . A CM, pp. 62, 2019. [14] P . K. Jamw al, S. Hussain, M. H. Ghayesh, and S. V . Rogozina, Adapti v e impedance control of parallel ankle rehabilitation robot, J ournal of Dynamic Systems, Measur ement, and Contr ol , v ol. 139, no. 11, pp. 111006, 2017. Int J Rob & Autom, V ol. 9, No. 3, September 2020 : 153 159 Evaluation Warning : The document was created with Spire.PDF for Python.
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