# Dynamic Modeling and Response Analysis of an Optical Machining Robot with Prismatic Joint Clearance
## 1 Introduction  
Optical processing robots, particularly in the field of optical mirror polishing, have garnered significant attention in recent years. Due to their high mechanical precision and stability, optical machining robots are also employed in high-tech industries such as aerospace, military, and semiconductor manufacturing [1]. In precision operations, these robots must operate within an extremely small error range to ensure the final product's quality. With ongoing technological advancements, optical processing robots have evolvedfrom traditional single-degree-of-freedom manipulators to multi-degree-of-freedom parallel robots, offering greater flexibility and finer control [2]. However, as the DOFs (degrees of freedom) have increased, clearance issues between the robot's joints have emerged, becoming a critical factor affecting its dynamic response, accuracy, and stability [3,4]. Joint clearance has a very pronounced effect on the kinematic and dynamic performance of a robot. Several studies have employed hybrid contact force models to simulate the dynamic behavior of joint clearances. [PERSON] et al. [5] simulated a crank-slider mechanism with multi-clearance joints utilizing a nonlinear contact force model for joint internal force assessment. [PERSON] [6] derived the dynamic equations of a four-bar mechanism with a gap in a microgravity environment and demonstrated experimentally that the gap reduces the accuracy of the mechanism's motion. Friction factors of gap size, relative velocity, and material between moving joints can significantly affect the dynamic response and lead to chaotic behavior [7]. Current research on joint clearance effects primarily focuses on planar or connecting rod mechanisms, where the analyzed clearance typically involves only two-dimensional contact, lacking comprehensive spatial contact collision analysis. Joint clearance in spatial mechanisms significantly impacts their dynamic performance. [PERSON] [8] studied a spatial slider-crank mechanism and found joint clearance caused chaotic kinematic and dynamic responses. [PERSON] et al. [9] compare the effects of different frictional contact models on the dynamical properties of space mechanisms. [PERSON] et al. [10] studied the impacts of clearance, friction, and rod flexibility on a Delta robot through simulation. [PERSON] [11] modeled the dynamics of a compound swing jaw crusher with a gap and investigated the effects of driving speed on the dynamic of the crusher. [PERSON] [12] examined joint clearance in a wine box base assembly robot, formulating the dynamic equation and studying its impact, finding that larger gap sizes affect the end-effector's dynamic performance. Current research has predominantly focused on the contact collision modeling of ball and rotational joint gaps, whereas studies on prismatic joint gaps remain limited. Prismatic joints are prevalent in multibody systems, and contact detection is crucial for understanding and optimizing the system's dynamic behavior. [PERSON] [13] designed a planar clearance-containing mechanism, analyzed basic methods of gap joints, and studied the mechanism's dynamic characteristics via experiments and simulations. [PERSON] [14] simplified the contact pattern of spatial prismatic joints by introducing a gap function. [PERSON] [15] considered joint flexibility, clearance, and friction in a 3-PRS series mechanism, finding that increased joint clearance and friction coefficient raise joint contact forces, affecting the dynamic response and natural frequency. [PERSON] et al. [16] classified the contact patterns in translational joints and proposed a method for contact pattern recognition in planar mechanisms. Most existing studies on the impact of joint gaps on mechanism dynamics have concentrated on spatial prismatic joints within linkage mechanisms. To the best of our knowledge, research on prismatic joint clearance for space-parallel robots with multiple DOFs and complex structures remains highly limited.