# A Geometric Model for a Shield TBM Steering Simulator
## 1 Introduction  
There are two main types of tunnel excavation: conventional tunneling by drilling and tunnel boring machines (TBMs), which are widely used in urban tunnels where rapid excavation is required with high safety standards. As there are many underground structures within a city, such as utility tunnels and building foundations, new tunnels are often a combination of straight and curved alignments. Therefore, TBM steering systems are critical in successfully producing tunnel curves. [PERSON] and [PERSON] [1] presented a theoretical model for TBM excavation based on the dynamic loading concept with which they simulated TBM operation, evaluated shield behavior, and undertook sensitivity analysis of the model parameters used for shield behavior in both straight and curved line excavations. Based on the simulation, they confirmed that shield movement could be affected by ground properties, with the influence on shield behavior being more marked in curved excavation than in straight line excavation. [PERSON] et al. [2, 3] introduced geometrical and mathematical models to determine the three-dimensional orientations of a TBM during its advance, and they derived a transformation matrix to calculate the attitude of TBM position in terms of yaw, pitch, and roll as the coordinates of one tracking point of TBM position during advance. [PERSON] and [PERSON] [4] introduced three operational factors (articulation angle, shield jack length difference, and copy cutter stroke) that are necessary for curved excavation and derived calculation methods for those factors and methods for forming curved alignments, including circular and spiral curves in two-dimensional space. They also compared three operational factors obtained from theoretical calculations with those measured in a real guidance system. [PERSON] et al. [5] and [PERSON] et al. [6] described the kinematic behavior of a TBM in soft ground, defining TBM shield position and orientation with two reference pointsand introducing the horizontal and vertical tendencies of the TBM calculated from the deviations of two reference points caused by the yaw and pitch of the shield. Based on these tendencies, they compared the relative positioning of the TBM with its planned tunnel alignments. [PERSON] et al. [7] undertook a numerical simulation to evaluate the structural stability of the TBM shield during a tightly curved excavation, identifying the load transfer mechanism from articulation and shield jacks to the shield skin and the effect of load transfer on the structural stability of the shield skin. [PERSON] et al. [8] developed a 1:17.7 scale model experiment to describe the behavior of thrust pressure based on different shield jacking force systems and articulation angles for curved excavation. Considering the uneven pressure, they proposed a guideline for the distribution of thrust force depending on articulation angle. Recently, [PERSON] and [PERSON] [9] described the overall TBM steering system and the related procedure required for tightly curved excavation, and they presented field case histories of TBM design and operational technology recognized as being successful in extremely tight curved excavation. These previous studies highlight limitations in designing the steering system for the TBM simulator. [PERSON] and [PERSON] [1] presented various factors related to shield behavior; [PERSON] and [PERSON] [4] introduced operational factors for TBM curved excavation, and the TBM navigation system (the critical part of the steering system) was partially covered in both studies. [PERSON] et al. [2,3] suggested determining the TBM position and its deviation caused by pitch, yaw, and roll, but considered only one reference point when calculating TBM position, whereas at least three reference points must be considered for a navigation system in three-dimensional space. [PERSON] et al. [5,6] considered two reference points (front and rear) to calculate horizontal and vertical tendencies. However, the navigation system was considered in two-dimensional space.