# Improvement of Position Repeatability of a Linear Stage with Yaw Minimization
## 1 Introduction  
Electrical test equipment measures the resistance of a circuit pattern on a printed circuit board (PCB) to inspect any abnormalities. An electrical inspection is performed with thousands or more probe pins concurrently positioned on the upper and lower surfaces of the PCB along the circuit checkpoints. Since the recent miniaturization and thinning of electronic products has resulted in fine-pitch patterned PCBs, there is a need for high-precision alignment of the probe pins. Lowering the manufacturing cost of the equipment is also required. The electrical test equipment has an XY stage that moves the PCB to the position coordinates and precisely aligns the PCB with the probe pins. The stage is a module of significant importance that must meet productivity and precision standards. Figure 1 shows the schematic diagram of the XY stage. The stage has a platform with a central opening to allow the probe pins onto the upper/lower surfaces of the PCB. A coordinate system is chosen such that the x-axis is in the length direction of the platform while the y-axis is in the width direction. The platform is driven by linear ball screw actuator systems offset from the platform's center. Due to the offsets, yaw moments are generated during the operation of the stage. The major contributor to the yaw motion is driving force offset, resulting in deteriorated precision due to the yaw angle [1]. In the stacked XY stage structure, the dynamic behavior of the supported stage (y-stage in Figure 1) is supposed to be affected by that of its supporting stage (x-stage), which is driven by its actuator system. Judging from the studies on single-stage dynamics, the interaction between the stacked stages is expected to be very complicated. In most cases,the supported stage is much lighter than its supporting stage, as depicted in Figure 1. In this work, the subtle dynamics of inter-stage interaction are ignored and left for future study. To reduce the yaw moment, the linear stage may be driven by synchronous control with driving units installed on both sides of the guide. Since the parallelism between the two linear motion guides (LM Guide) on both sides is imperfect, the actuator loads may vary during operation. Repeated load fluctuations may cause structural distortions over time, which ultimately may lead to jamming. Thus, the synchronous control method has many shortcomings, such as increased manufacturing and maintenance costs and complexity of control programs compared to the single motor control method. Typically, a single-axis linear motion stage consists of a pair of LM Guides. Each rail of the configured LM Guides is exposed to shape errors, machining errors on the installation surface, and assembly errors. As a result, unintended 5-degree of freedom (DOF) error motions occur. Among these, errors in the translational motion direction can be corrected to some extent by the averaging effect, whereas errors arising from the angular motions, such as yaw, are difficult to correct [1; 2]. Various studies have been conducted to analyze the error motions caused by the geometrical error of the LM Guide's rail. Major studies have simplified the balls of the LM Guide, which are the most elastic parts deformed by force, to spring models. Such simplification has been used to predict the translational error motions of stages with the LM Guide rail shape error and ball stiffness [3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16]. Linear stages for electrical test machines are designed with high acceleration and deceleration capabilities for productivity. As a result, significant yaw motion and vibration of the platform could occur. Most of the previous studies, however, employed static equilibrium equations for the platform without inertial effects. The position accuracy of the stage may be supplemented by measuring the motion error of the platform and compensating with corrected position coordinates. For this purpose, the position repeatability of the platform is crucial [15; 17; 18].