## 2 Identifying Steering Feel Factors
### Analysis of the Steering Feel Mechanism  
In this section, the free-body diagrams from the steering wheel to the tire are presented as three frames. We utilized them to analyze the factors affecting steering feel and in what direction our vehicle should be designed to achieve the steering feel of the target vehicle [24,25]. Figure 4 shows the components from the steering wheel to the pinion. The equation of the upper steering system is presented in Equation (1), which comprises the driver torque input \(\tau_{sw}\), steering wheel inertia \(J_{sw}\), torsion bar stiffness \(K_{tb}\), and steering wheel and column angles \(\delta_{sw}\), \(\delta_{c}\). Assume that the column damping is negligible. Figure 4: Free-body diagram: from the steering wheel to the pinion. Figure 3: Steering feeling test methods (ISO 13674-1, 2). The reaction force from the rack to the pinion affects the pinion, as shown in Equation (2). The motor torque \(\tau_{m}\) applied at the lower column assists the driver's steering intention in conjunction with the EPS control system. \(\tau_{\mathit{fric,~{}c}}\), \(r_{p}\) are the friction torque at the column and the radius of the pinion gear, respectively. \[\tau_{\mathit{saw}}=J_{\mathit{sw}}\ddot{\delta_{\mathit{sw}}}+K_{\mathit{ tb}}(\delta_{\mathit{sw}}-\delta_{c}) \tag{1}\]  
\[K_{\mathit{tb}}(\delta_{\mathit{sw}}-\delta_{c})+\tau_{m}-\tau_{\mathit{fric,c }}-J_{c}\ddot{\delta_{c}}=F_{\mathit{reaction}}\cdot r_{p} \tag{2}\]  
Second, the relationship between the pinion and tie rod is shown in Figure 5. As shown in Equation (3), the reaction force applied to the rack \(F_{\mathit{reaction}}\) results in the movement of the rack \(x_{r}\) against friction and inertia \(F_{\mathit{fric,~{}r}}\) from the rack and wheel. \[F_{\mathit{reaction}}=m_{r}\ddot{x_{r}}+F_{\mathit{fric}}\cdot cos\ \beta+F_{\mathit{fric,r}} \tag{3}\]  
Lastly, the tie rod to the tire is shown in Figure 6. The axis of steering in the tire is mainly determined by the caster angle, which is presented as a red dot in the figure. The point of application of the tie rod and the angle between the wheel rotation axis and tie rod are presented as a blue dot and \(\beta\), respectively. Equation (4) represents the force from the tie rod \(F_{\mathit{fie}}\) to the tire lateral force \(F_{\mathit{t}}\). \(J_{\alpha}\), \(\tau_{\mathit{fric,~{}s}}\) are the inertia of the wheel and the friction at the suspension, respectively. \(l_{e}\) is the effective arm length, which denotes the distance between the steering axis and the tie rod application point. \(l_{t}\) is the trail arm that indicates the distance between the steering axis and the point of action of tire lateral force. \[F_{\mathit{tie}}\cdot l_{e}=J_{\alpha}\ddot{\theta_{t}}+F_{\mathit{t}}l_{t}+ \tau_{\mathit{fric,~{}s}} \tag{4}\]  
Figure 5: Free-body diagram: from the pinion to the tie rod. Figure 6: Free-body diagram: from the tie rod to the tire. Equation (5) can be obtained from Equations (1)-(4). Combining the three aforementioned frames, we can derive an equation for the mechanism of constructing the driver's steering torque. Figure 3:0  
Assuming that the angular acceleration of the steering wheel and column are the same, Equation (6) can be established. Figure 3:1  
As shown in Equation (6), the linearity of the tire lateral force significantly affects the steering feel. Additionally, designing minimal friction at the column, rack, and suspension contributes toward a linear steering feel. This finding aligns with the results of previous studies, which emphasize the influence of friction occurring at various locations in the steering system on the handling characteristics [7]. Particularly, in on-center handling situations with small steering angles, the role of friction cannot be ignored [2]. If friction cannot be minimized, the EPS control system can compensate for the friction elements to improve on-center handling. Furthermore, the angular acceleration term of the steering wheel can be used to derive the characteristics of steering feel during transient steering.