# Dynamic Modeling and Analysis of a Driving Passenger Vehicle
## 2 Dynamic Vehicle Model  
In this section, a dynamic model of a vehicle that can analyze the behavior of a driving passenger car is presented. This study assumes that the speed of the vehicle is kept constant, and the driving direction of the vehicle is changed by the steering wheel. Under these assumptions, the vehicle does not accelerate in the longitudinal direction (direction of travel), but only in the lateral direction perpendicular to the traveling direction. This lateral acceleration imparts a centripetal force to the vehicle and maintains it on a turning trajectory. The centrifugal force acting on the vehicle during turning is given by the following equation:  
\[F=m_{t}\frac{V^{2}}{R} \tag{1}\]  
where \(m_{t}\) is the total mass of the vehicle, \(V\) is the speed of the vehicle, and \(R\) is the turning radius. When the vehicle is traveling at a constant speed, as long as the mass \(m_{t}\) and speed \(V\) are constant, the centrifugal force \(F\) is determined by the turning radius \(R\). The relationship between the turning radius of the vehicle and the steering angle of the steering wheel was analyzed. First, the relationship between the vehicle's turning radius and the steering angles of the front wheels can be derived from the Ackermann steering geometry [27], as shown in Figure 1. In Figure 1, point \(O\) is the instantaneous turning center, \(G\) is the center of mass of the vehicle, and \(l_{1}\) and \(l_{2}\) are the distances from the center of mass to the front and rear wheels, respectively. \(w\) is the distance between the centers of the left and right wheels, and \(L\) is the distance from the front wheel to the rear wheel. As shown in this figure, the position of the turning center \(O\) coincides with the intersection of the extension line of the rear axle and the vertical lines inside and outside the front wheels. Accordingly, the steering angle of the inner front wheel, \(\delta_{i}\) is formed by the extension line of the rear axle and a line perpendicular to the inner front wheel. Additionally, the steering angle of the outer wheel, \(\delta_{o}\) is formed by the extension line and a line perpendicular to the outer front wheel. The distance \(R\) from the turning center \(O\) to the vehicle center of mass \(G\) is called the turning radius, and the distance from the turning center \(O\) to the center of the rear axle is expressed as \(R_{r}\). Thus, if the instantaneous turning center is determined geometrically, the turning radius \(R\) can be expressed as the inner and outer front wheel steering angles, \(\delta_{i}\) and \(\delta_{o}\). The relationship between the steering angles of the inner and outer front wheels, the distance  
Figure 1: Ackermann steering geometry for the front steer angles of the inner and outer wheels. from the turning center to the center of the rear wheel axle, and the distance between the left and right wheels can be expressed by the following equations:  
\[\tan\delta_{i}=\frac{L}{R_{r}-w/2} \tag{2}\]  
\[\tan\delta_{o}=\frac{L}{R_{r}+w/2} \tag{3}\]  
In general, as the steering angle is not large when the vehicle is traveling, the steering angles \(\delta_{i}\) and \(\delta_{o}\) are assumed to be small, and the turning radius \(R\) is very large compared to the distance w between the centers of the left and right wheels. Therefore, in Equations (2) and (3), \(\tan\delta_{i}\) and \(\tan\delta_{o}\) can be approximated by \(\delta_{i}\) and \(\delta_{o}\), the distance from the turning center to the center of the rear wheel axle, \(R_{r}\) can be replaced by the turning radius \(R\), and the distance between the left and right wheels \(w\) can be neglected. Denoting the average steering angles of the inner and outer front wheels by \(\delta\), that is, \(\delta=(\delta_{i}+\delta_{o})/2\), the turning radius of the vehicle is given by:  
\[R=\frac{L}{\delta} \tag{4}\]  
This equation implies that the turning radius \(R\) is determined by the average steering angle of the front wheels \(\delta\) and wheelbase distance \(L\).