# Individual Drive-Wheel Energy Management for Rear-Traction Electric Vehicles with In-Wheel Motors
## 5 Vehicle Dynamics Modeling
### Kinematics  
The basis for the EDS is the geometry implicit in the Ackermann equations. These equations describe the geometric and dynamic relationships that need to be achieved to guarantee that a no-slipping condition occurs in the drive wheels during a turning maneuver. The base equations were taken from Jazar [30], and their development to obtain the drive wheels' angular speed is shown below. The symbols used in the equations are in Appendix A. Figure 7 shows the existing relationship between the parameters of a car making a turn with a radius \(\mathrm{R_{1}}\). The angles \(\delta_{\mathrm{i}}\) and \(\delta_{\mathrm{o}}\) must be such that the line coming from the turning center-point \(O\) is perpendicular to the left and right wheel's lateral plane, which guarantees that both turning wheels are in full rolling behavior, not slipping. The rear axle is of interest to establish the correspondent geometric and dynamics equatio  
Figure 5: Acceleration and a top speed of the test car in chassis dynamometer. Figure 8 shows the same rear-drive car with more parameters, like the distance \(a_{2}\) between the center of mass \(C\) and the rear axle, the distance \(R\) from the turning center-point \(O\) to the center of mass, the angular speed of each rear wheel \(\omega_{i}\) and \(\omega_{o}\), and the longitudinal speed vector of the inner and outer rear wheel \(V_{i}\) and \(V_{o}\), respectively. The relationship between these parameters allows the construction of the following equations:  
\[R=\sqrt{a_{2}^{2}+l^{2}\ cot^{2}\ \delta} \tag{1}\]  
Equation (1) shows the vehicle's turning radius as a function of \(\delta\), where \(\delta\) is the cot-average of the inner and outer steer angles and 1 distance between the front and rear axles. Equation (2) shows the equivalent angle \(\delta\) when the front wheels have different values. \[cot(\delta)=\frac{cot(\delta_{o})+cot(\delta_{i})}{2} \tag{2}\]  
The test car's steering mechanism characteristics classify it as a parallel steer (Figure 9). This means that the steering angle of both front wheels is equal (Equation (3)). The effects of having a steering mechanism that does not adhere to the Ackerman geometry are increased tire scrub during cornering, tire wear, and increased steering effort. The magnitude of not having Ackerman steering is variable and depends on the steering angle or the radii of the turning maneuver being performed. For all the tests considered in this work, the steering angles were at or below 16 (deg), so the deviation from the ideal curve was minimal, and the effects can be neglected. \[\delta_{i}=\delta_{o}=\delta \tag{3}\]  
Figure 6: Torque and power characteristic curves of the test car power train. Figure 7: A front-wheel-steering vehicle and the Ackermann condition ([PERSON], 2008). From the geometric relations in Figure 8, we obtain Equation (4). \[Cot\ \delta=\frac{R_{1}}{l} \tag{4}\]  
The dynamics equations for each rear wheel's longitudinal speed (\(V_{i}\) and \(V_{o}\)) and the geometric proportions of a vehicle turning left can be written as shown in Equations (5) and (6) for the inner wheel and the outer wheel, respectively. The wheel's effective radius is denoted by \(R_{w}\) and the Yaw-rate of the vehicle by \(Y_{R}\). \[V_{i}=Y_{R}\Big{(}R_{1}-\frac{w}{2}\Big{)}=\omega_{i}R_{w} \tag{5}\]  
\[V_{o}=Y_{R}\Big{(}R_{1}+\frac{w}{2}\Big{)}=\omega_{o}R_{w} \tag{6}\]  
Replacing Equation (4) with Equations (5) and (6) and solving the angular speed of the inner (\(\omega_{i}\)) and outer (\(\omega_{o}\)) wheels of the rear axle of a rear-drive car, we obtain Equations (7) and (8). \[\omega_{i}=\frac{Y_{R}\big{(}l*Cot(\delta)-\frac{w}{2}\big{)}}{R_{w}} \tag{7}\]  
\[R=\sqrt{a_{2}^{2}+l^{2}\ cot^{2}\ \delta} \tag{1}\]0  
The former equations are the base for the EDS and are programmed in a series of successive code lines that calculate the required speed for each of the rear wheels as a function of the car's instantaneous turning angle and velocity.