# Dynamic Modeling and Response Analysis of an Optical Machining Robot with Prismatic Joint Clearance
## 4 Dynamics Simulation  
In the optical mirror processing process, the grinding tool, by a certain trajectory, speed, and acceleration, accurately traverses the mirror surface to be processed on each removal point; when the grinding trajectory is a circle, the grinding tool does not need to change direction frequently in the process of movement, and can make the grinding tool in the process of grinding processing and the mirror surface type match, suitable for the rotary symmetry class of optical component processing. This study examines the impact of joint clearance on the dynamic characteristics of a polishing robot by conducting motion trajectory planning for its moving platform as follows:  
\[\left\{ \begin{array}{l}X=X_{c}+r\cos(\theta_{0}+\frac{vt}{r})\\ Y=Y_{c}+r\sin(\theta_{0}+\frac{vt}{r})\\ Z=Z_{0}\end{array} \right. \tag{21}\]  
where (\(X_{c}\), \(Y_{c}\), \(Z_{0}\)) is the position of the center of the trajectory circle in \(W\)-\(XYZ\), \(r\) is the radius of the trajectory circle with \(r\) = 0.2 m, \(\theta_{0}\) is the initial position of the center of the moving platform on the trajectory circle, and \(v\) is the linear velocity of the moving platform with \(v\) = 0.05 m/s. The structural parameters and the dynamic simulation parameters are shown in Tables 2 and 3, respectively. Figure 7: Dynamic simulation solution flowchart. The simulation time is 0.3 s, with an initial step size of 0.0001 s. The motion curves of the moving platform of the 3-UPRU+UP parallel manipulator are obtained through MATLAB numerical solution and ADAMS simulation. Figure 8 shows that the MATLAB and ADAMS curves share a similar overall kinematic trend. However, due to different solution methods and modeling approaches in MATLAB and ADAMS, the gap simulations are random. This results in varying initial collision positions and peak times for collision forces. If the mechanism is stable, the two curves will be nearly identical, thus validating the selected dynamic model once more. As shown in Figure 8a-c, during the operation of the robot, the joint clearance can lead to displacement errors due to the separation of the elements. Clearance joints have a negligible effect on the displacement of the moving platform: a joint clearance of 0.01 mm results in a displacement error of 0.1 mm. Figure 8d-f shows the velocity of the moving platform. The effect of the joint clearance on velocity is more pronounced than its effect on displacement, being an order of magnitude greater. The velocity swings sharply around the ideal curve due to sudden changes in acceleration. As shown in Figure 8g-i, when joint clearance is present, the acceleration curve exhibits high-frequency fluctuations, indicating that clearance has the most pronounced effect on the acceleration of the parallel manipulator. The acceleration undergoes a sudden change. This is caused by the joint clearance resulting in collision and friction between the guide and the slider. The sudden change in friction and contact force indicates that the contact between the elements has altered, consistent with the results of the point-line-surface contact model used earlier.