## 1 Introduction  
Chemical mechanical polishing (CMP) is one of the representative semiconductor processes, and is the only process closely related to mechanical motion in semiconductor manufacturing processes [1]. The CMP process involves planarizing the surface of a wafer through the relative motion between the pad corresponding to the tool and the wafer corresponding to the workpiece. The roughness of the pad surface determines the polishing rate of the wafer and the quality of the wafer surface [2; 3]. Therefore, a conditioning process using a diamond conditioner is applied to maintain a constant roughness of the pad surface [4; 5]. The diamond conditioner wears the pad by moving at a constant pressure and by rotating on the pad surface. The conditioning process helps keep the roughness of the pad constant; however, the shape of the pad varies as it wears out. The change in the pad shape during the conditioning process significantly influences the shape of the final polished wafer [6; 7; 8; 9; 10]. Therefore, to ensure the final quality of the wafer surface, reproducibility of the polishing rate, and reproducibility of the wafer shape, it is necessary to maintain the roughness of the pad and the pad shape constant throughout the conditioning process [8]. The roughness of the pad is closely related to its physical properties and the shape of the abrasive of the diamond disc, whereas it is unrelated to kinematic movements such as disc rotation or pad rotation [11]. However, the pad shape is dominantly influenced by disk rotation, pad rotation, and conditioning arm movement. The trajectory of the diamond abrasives that wear the pad is determined by the relative motion comprising the rotational motion of the pad and the conditioner and the sweep motion of the conditioning arm. The trajectory density of the abrasives directly affects the pad shape, and the amount of wear at locations with a high trajectory density is significant. In addition, the conditioning process is divided into several zones in the radial direction of the pad, and the conditioning in each zone is carried out with a difference in the occupancy time of the concierge. The difference in occupancy times causes a difference in the wear amount of the pad. If the conditioning process is performed for a long time under unevenly occupied conditions, the pad shape varies rapidly. In other words, depending on how the occupancy time is set, the pad shape can be maintained as good as that at the beginning or it may end up varying rapidly.  
Although many researchers have investigated the pad cut rate, the pad temperature, conditioning lifetime, and pad deformation during the CMP process due to conditioning and polishing, they did not compare the kinematic predictions with the actual phenomenon [12; 13; 14; 15; 16; 17; 18]. [PERSON] et al. predicted the shape of a pad using simple mathematical modeling [19]. [PERSON] et al. attempted to predict a 1D pad shape based on mathematical calculations. However, these previous studies did not compare the predicted and actual pad wear phenomena and were limited to simple predictions [20]. [PERSON] et al. also predicted the shape of the pad with respect to the diamond disc; however, the condition wherein the pad shape remains constant was not considered [21].  
In this study, we predicted the trajectory of diamond abrasives based on the relative motion of three factors: pad rotation, conditioner rotation, and sweep speed. With this, the relative wear amount of the pad was derived by calculating the trajectory density on the pad, and the pad shape was predicted. In addition, we not only predicted the wear shape of the pad, but also measured it using an in situ pad profiler. The predicted results and actual measured results were found to be in good agreement. Finally, a condition in which the pad could continuously maintain its initial shape was derived, and the maximum height deviation was 10 \(\upmu\)m or less despite performing the conditioning process for more than 2 h.