Academic Journal of Science and Technology ISSN: 2771-3032 | Vol. 2, No. 3, 2022 94 Workspace Analysis for Humanoid Wrist Mechanism Liang Zhang1, *, Shuzhen Li1, Lei Shi1, Guofang Li1, Jingjing Yu1, Yan Shi2 1College of Mechanical and Electronic Engineering, Hebei Normal University of Science and Technology, Qinhuangdao 066004, China 2Hebei Industrial Manipulator Control and Reliability Technology Innovation Center, Hebei University of Water Resources and Electric Engineering, Cangzhou 061001, China Abstract: Based on the analysis of human wrist, a humanoid wrist joint with 5R parallel mechanism as prototype is proposed. It is an orthogonal spherical two-degree-of-freedom parallel mechanism, which has the advantage of easy to process and manufacture. The inverse position solution of the mechanism is obtained by analytic method, and the velocity Jacobian matrix is established, and the working space of the mechanism is analyzed considering the working constraints of the mechanism. It provides a theoretical basis for its practicality. Keywords: Wrist joint, PARallel mechanism, Workspace, Performance analysis. 1. Introduction Manipulator is a kind of mechanical device created by imitating human arm, which can achieve the purpose of moving the claw to the desired position and bearing the weight of the workpiece and the arm itself [1]. Since the first robotic arm came into being, academia and engineering circles in various countries have invested a lot of manpower and material resources in the research of robotic arm，many breakthroughs have been made，various types of prototypes have been developed successively[2-3]. Because of the advantages of simple structure, simple control and large workspace, the humanoid manipulators on industrial manipulators and humanoid robots mostly adopt the serial mechanism [4-8]. However, the inherent error accumulation effect and poor dynamic characteristics of the serial mechanism limit its application in the case of increasing performance requirements of the manipulator. It is of great practical significance to develop new humanoid joints. Parallel mechanism has many advantages, such as high stiffness, strong bearing capacity, low inertia, no error accumulation, compact structure, low dead load ratio and so on[9]. A humanoid wrist joint based on orthogonal spherical two-degree-of-freedom parallel mechanism is presented in this paper，which is a kind of parallel mechanism with less degrees of freedom. It inherits the advantages of parallel mechanism and has the characteristics of easy manufacture. 2. Mechanism Design Based on Analysis of Wrist Motion 2.1. Motion Analysis of Wrist Fig. 1 shows the motion of the human wrist[10], which is performed by the driving muscles distributed in the upper arm rather than in the wrist itself. From the perspective of mechanism, the wrist is a two-degree-of-freedom mechanism. Figure 1. Movement form of wrist 2.2. Mechanism Design The 5R parallel mechanism has two rotational degrees of freedom, which can achieve motion similar to wrist，and its driving device can be installed on the fixed base, so that the superior motor does not become the lower rotation load，and the inertia of the moving parts can be reduced. This arrangement makes the wrist joint have the advantages of compact structure, strong bearing capacity and good dynamic characteristics. Therefore, the 5R parallel mechanism is suitable as a mechanism prototype of the wrist joint of the anthropomorphic arm. The wrist joint designed with 5R parallel mechanism as prototype is shown in Figure 2. It is an orthogonal two- degree-of-freedom parallel mechanism, which consists of connecting rod, moving platform, ring connecting rod, frame, driving rod and driving connecting rod. Two rods are connected with the revolute pairs, and the axes of all the revolute pairs intersect at one point. The mechanism can simulate the flexion and extension, internal rotation and 95 external rotation of human wrist. 1connecting rod 2moving platform 3 connecting frame rod 4ring connecting rod5 frame 6driving rod 7 driver 8driving connecting rod Figure 2. Prototype of the wrist joint 3. 2 Kinematic Analysis The sketch of wrist mechanism designed with 5R parallel mechanism as prototype is shown in Fig. 3. Two rods are connected with the revolute pair, and the axes of all the revolute pairs intersect the point Ow. The axes of 1wU and 1wV , 3wV and 3wW , 3wW and 3wU are perpendicular to each other, and 1wV and 3wV are perpendicular to each other on the moving platform. In this way, a five-bar mechanism with orthogonal structure is formed. The input of the mechanism is the rotation angle 6 around the Xw axis and the rotation angle 7 around the Zw axis. 1 U driving rodⅠ 2 U connecting rod 3 ring driving rod 4 Moving platform 5 Frame Figure 3. Mechanism of the wrist joint With Ow as the origin of the coordinate system, the coordinate system is established, as shown in Figure 3. In the fixed coordinate system Ow-XwYwZw, the axes of Xw and Zw coincide with the axes of revolute pairs 1wU and 3wU respectively, the direction of Yw axis is determined by the right-handed screw rule. In the moving coordinate system Ow- xwywzw, the axes of xw and zw coincide with the axes of the revolute pairs 3wV and 1wV respectively, the direction of yw axis is determined by the right-handed screw rule. The directions of coordinate axes are shown in Figure 3. In the initial attitude, the fixed coordinate system coincides with the moving coordinate system. The size parameters of the mechanism are defined as 1w1w wlO V , 53w wlwO V , 33w wlwO W , 21w wlwO U , 43w wlwO U , 2 4w wl l The method of Euler angle is used to describe the transformation matrix between the moving coordinate system and the fixed coordinate system[11]. The transformation matrix is given as     cos sin 0 , , cos sin cos cos sin sin sin sin cos cos w w w w w w w w w w w w w w w w w x z                         R R R (1) where w and w are the attitude angles of the moving platform. 96 3.1. Inverse Position Solution Using eq. (1), the direction cosine of each axis wiV relative to the fixed coordinate system on the moving platform can be given as w wiwiV R v (2) where wiv is the coordinate of the revolute pair connected to the moving platform in the moving coordinate system. According to the orthogonal characteristic and rotation relationship of the mechanism, the vectors of the axes of the rotating pairs connected to the input axes relative to the fixed coordinate system w w w wO X Y Z are given as T 1 6 6(0 sin cos )w   V (3) T 3 7 7( sin cos 0 )w    W (4) According to the geometric relationship of the mechanism, the kinematics constraint equation by using staggered angle theorem[12] for the mechanism can be written as 3 3 0w w W V (5) From eqs. (2)-(5), the inverse solution of position is given as eq. (6), the forward solution of the position is given as the eq. (7). 6 7 arctan(tan cos ) w w w          (6) 7 6 6 arctan(tan cos )w w          (7) 3.2. Structural Constraint (1) 6max and 6min are defined as the maximum and minimum rotation angles of the actuator 1wU respectively, 7max and 7min are the maximum and minimum rotation angles of the actuator 3wU respectively, and the constraint conditions of no interference between the fixed platform and the connecting rod can be expressed as 6min 6 6max 7min 7 7max             (8) (2) The constraint conditions for the mechanism to avoid singular posture can be expressed as 6 7 90 90 90 90                (9) (3) It is assumed that section of the rods is round, and its diameter is d, the shortest distance between the same side of the connecting rod and the rod connected to the frame is set iD , the constraint conditions that there are not interference between two rods can be expressed as iD d (10) 4. Workspace Analysis The structural parameters of the mechanism are shown in Table 1. The point  0 -20 0 T w p on the moving platform of the mechanism is selected as reference point for the workspace. Given the constraints of eqs. (10)-(12), 60 60w    , 45 45w    , and the step length is 5w w      , then, the three-dimensional search method is used to simulate and analyze the workspace of the mechanism with MATLAB. The results are shown in Fig. 4- 5. Compared with the range of human wrist motion, the mechanism can basically meet the motion requirements of the human wrist joint. Table 1. Structural parameters and inertial parameters Structural Parameters Value w wa b length 1 / mmwl 20 w wb c length 2 / mmwl 35 w wd e length 3 / mmwl 30 w wg h length 3 / mmwl 30 w we f length 4 / mmwl 35 w wf g length 5 / mmwl 20 ( mm)iD 10 7max ( ) � / 7min ( ) � 60/-60 6max ( ) � / 6min ( ) � 60/-60 Figure 4. Three-dimension graph of the workspace -20 0 20 -20 -10 0 10 20 -20 -18 -16 -14 -12 z/mmx/mm y/ m m 97 Figure 5. Boundary map of the workspace 5. Conclusion Based on the analysis of the motion form of human wrist joint, a humanoid wrist joint based on orthogonal spherical two-degree-of-freedom parallel mechanism is proposed. Based on the inverse kinematics solution of the mechanism, the result of workspace analysis shows that the mechanism can basically meet the motion requirements of human wrist joint. Acknowledgment This work is supported by Teaching case project in Hebei Normal University of science and technology, Project number: 2021AL02，and Teaching Reform Project of Hebei Normal University of Science and Technology, Project number: JYYB202153, and Hebei Professional Degree teaching case base project, Project number: KCJSZ2021074，Project of Hebei Industrial Manipulator Control and Reliability Technology Innovation Center (Hebei University of Water Resources and Electric Engineering), Project number: JXKF2108 and JXKF2107. References [1] Tianmiao Wang, Yong Tao. 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