# Kinematic Analysis and Verification of a New 5-DOF Parallel Mechanism
## 4 Workspace Analysis
### Algorithm Simulation of Optimal Workspace  
(1) Based on the influencing factors of workspace, the simulation parameters are changed to obtain the optimal workspace. The parameters are set as follows: the geometric centers of Hook joints on sliders are evenly distributed on the circle with radius \(R=135\) mm. Since the length of link affects the size of workspace, the length of five driving links is set as \(L=165\) mm, 185 mm, 205 mm, and 225 mm respectively, and the projection of workspace on the \(XOY\) plane is shown in Figure 13.  
From the comparison of Figure 13a-d, it can be seen that when the length of the link is changed from 165 mm to 205 mm, the projection of the three-dimensional workspace on the \(XOY\) plane is significantly larger, but there is no obvious change from 205 mm to 225 mm. Figure 13e,f show the projection of the three-dimensional workspace on the \(XOY\) plane when the \(\alpha\), \(\beta\), and \(Y\) coordinates of the moving platform are all zero. After comparison, it is found that when \(L=205\) mm, the workspace has a better \(Z\) direction space. Therefore, we choose the link length of the limb to be 205 mm. (2) After determining the length of the link \(L=205\) mm, the position of spherical hinge is optimized as shown in Figure 14, and the simulation result is shown in Figure 15.  
When the length of the link and distribution radius of the Hook hinge are the same, and the spherical hinge is distributed according to Figure 14a, the projected area of the working space on the \(XOY\) plane is 120 mm \(\times\) 180 mm. When the spherical hinge is arranged according to Figure 14b, the projection area of the corresponding workspace in the \(XOY\) plane is 160 mm \(\times\) 120 mm, and the workspace is heavily biased to the negative direction of the \(Y\) axis, as shown in Figure 15b. Comparing the two schemes of spherical hinge, it is found that Scheme 1 (Figure 14a) has better isotropy in the workspace of \(XOY\) plane, so this scheme will be adopted in the subsequent mechanism construction, and the previous chapters are based on the analysis of this solution. (3) Fixed posture workspace  
As an important form of workspace, fixed posture workspace studies the size of workspace when the moving platform is in a fixed posture. When \(\alpha\) equals \(5^{\circ}\), \(10^{\circ}\), \(15^{\circ}\) and \(20^{\circ}\), the fixed posture workspace is analyzed to study the influence of single posture angle on workspace. The results are shown in Figure 16.  
Figure 13: Limitation of link length interference. Similarly, when \(\beta\) changes from \(5^{\circ}\) to \(20^{\circ}\), the change in the fixed posture workspace is shown in Figure 17.  
Finally, when both two posture angles are \(0^{\circ}\), the size of the three-dimensional workspace is studied on the basis of the optimized length of the link and the position of the spherical hinge. The results are shown in Figure 18.  
From Figures 16 and 17, we can see that the 5-DOF parallel mechanism consists of two degrees of freedom of rotation, and the size of the two posture angles affects the size of the workspace, showing a general rule that the size of fixed posture workspace decreases with the increase of the posture angle and deviates from one direction of the coordinate axis. As to whether any variable attitude curve in space is within the workspace, we can reverse the kinematics of the subdivision points of the target curve to judge if the position of the sliders and the angle of spherical hinges corresponding to each subdivision point are within the range of motion. Figure 14: Distribution of spherical hinges on the moving platform. Figure 15: Workspace diagrams of the distribution of spherical hinges for Schemes 1 and 2.