Forward kinematic analysis of Dobot using closed-loop method

Dobot is a hybrid robot that combines features from parallel and serial robots. Because of this characteristic, the robot excels for is reliability, allowing its implementation in diverse applications. Therefore, researchers have studied its kinematics to improve its capabilities. However, to the extent of our knowledge, no analysis has been reported taking into consideration the closed-loop configuration of Dobot. Thus, this article presents the complete analytical solution for the forward kinematics of Dobot, considering each link. The results are expected to be utilized in the development of a dynamical model that contemplates the dynamics of each element of the robot.

INTRODUCTION Dobot is a 3 Degree of Freedom (DOF) type of commercial serial manipulator that posses numerous attractive features because of its hybrid serial and parallel configuration. One of its principal characteristics is the precision 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-effector remains constant. Therefore, because of these features and its low cost, various researches have implemented Dobot in diverse applications such: 3D printing [1], electrochemical writing [2,3], Pick and place [4][5][6][7][8], education [9], surgery [10], and even electric vehicle charging [11].
Because of the diverse applications of Dobot, researches have conducted different types of analysis on this robot. T. Cheng et al. analyzed the forward kinematics, inverse kinematics, and jacobian matrix of Dobot [11]. Moreover, O. Hock, et al. included an analysis of the pseudoinverse method to compute the inverse kinematics of Dobot numerically [12]. G. Yu et al. modified the end-effector of Dobot to included a gripper and a camera, to implement hand-eye calibration [13]. However, to the extent of our knowledge, no research has conducted the forward kinematics analysis of the linkage that composes the robot.
The forward kinematics analysis permits to obtain the kinematic of each link, which is a requirement to develop a detail dynamical model of Dobot. This type of model allows the implementation of robust control strategies such as inverse dynamics control [14,15], permitting Dobot to be used in the application of high velocity while maintaining its precision [16]. Various researcher has worked in the computation of the direct kinematics for different parallel robots. Yujiong L., et al. presented the computation of the forward kinematics of an H4 parallel robot using a geometric approach [17]. Olaru D. obtained the forward and inverse Ì ISSN: 2089-4856 kinematics of a 5DOF robot, between the relevant results from this research is the increasing of the robot precision due to the implementation of the computation of the forward kinematics in its control [18]. Jin S. K., et al. implemented the kinematic analysis of a 4DOF parallel robots for MRI-Guided percutaneous interventions [19]. Tang et al. study the kinematics of a novel 2R1T parallel mechanism [20]. Therefore, because of the advantages of a complete kinematical model, this research focuses on the computation of the forward kinematics of Dobot, considering the closed-loop configuration of each actuated joint. The results of the algorithm were validated using a CAD model, verifying the accuracy of the results. The organization of the article is the following: Section 2 describes the principal joints of Dobot. Section 3 presents the kinematical diagram of Dobot, and explain the relation between links. Section 4 presents the computation of the forward kinematic of Dobot. Section 5 compares the results of the obtained equations with a CAD model of Dobot, to verify the accuracy of the results. DOBOT 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 effect 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 perpendicular 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 view shows the mechanism inside DOBOT's case, where the variables explained in Section 2 can be related using Figures 2 and 3.

KINEMATICS MODEL
To facilitate the analysis of Dobot the kinematic diagram that presents all the links is obtained, as shown in Figure 3. Note that the actuated 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 q2 , L 7 , and L 6 , that transmit the movement from the base to this joint. The end effector is represent by point P , the orientation of this link is not directly actuated, and is dictated by a Watt's mechanism [21] that is composed for links: L 2 , L 3 , L 4 and L 5 . Then, to verify that the kinematic representation of Dobot is correct, the Gruebler equation [22] is used to compute the DOF of the system, as follows: 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 expected result.

FORWARD KINEMATICS
To obtain the forward kinematics it is used the closed-loop method [23]. This method requires the analysis of closed-loop vector to obtain the position equations. The number of closed-equations is determined by the use of the relation for Li et al. in [17], as follows: 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.

ISSN: 2089-4856
Then, adding the square of the x side and the square of the y side, the term corresponding to θ 6 is simplified. Thus, obtaining an expression that depends on θ 7 , θ q 1, θ q 2, and θ 0 . However, since q 1 and q 2 are entries of the system, and θ 0 is a fix angle, the only variable of interest is θ 7 . Therefore, the resultant expression is expressed in term of θ 7 as follows: Where the terms A i are parameters that depends on the dimensions of Dobot, and known angles. Then, using the half tangent substitution tan θ7 2 = c 1 , with cos θ 7 = , (5) is reduced to an algebraic expression, as follows: Then, using the quadractic equation in (6), the following equation is obtained: Replacing c 1 for the tan θ7 2 , the expression for θ 7 is obtained: Now that the expression for θ 7 is obtained, θ 6 is computed from applying the dot product of 4 with unitary vectorî = [1, 0] T , and solving for θ 6 , generating: Analogously, the second loop equation is presented in the following: The two variables to solve from (10) are θ 2 and θ 3 , whose solutions are obtained applying the same methods for θ 7 and θ 6 , respectively. Therefore, the solution for θ 3 is presented below: −k 12 ± k 2 12 − 4k 11 k 13 2k 11 (11) The values of k 11 , k 12 , and k 13 are presented in the Appendix section. The solution for θ 2 is presented next.

RESULTS
To validate the kinematics model and evaluate its accuracy, a comparison was done using the CAD obtained from DOBOT © . Both, the results from the CAD model and the Kinematic model are presented in Table . When comparing the results it is clear that the computed values and the CAD model present the same results. Therefore, the forward kinematic analysis is accurate. 6. CONCLUSIONS This paper presents the forward kinematic analysis of Dobot, a 3DOF hybrid robot that is posses both, a serial and parallel configuration. The equations were expressed using the closed-loop method and were solved analytically by the application of the tangent half-angle substitution. This result is expected to be utilized in the computation of the dynamics of Dobot, to implement elaborated control strategies such as inverse dynamic control. The accuracy of the equations was also validated 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.