## 3 Orientation Controller Design
### The Single-Track Model  
First, the single-track model by [PERSON] for small velocities and accelerations was considered. The slippage of the vehicle due to centrifugal forces was neglected. Figure 3 shows the geometric relationships of the single-track model used here. Figure 3: Simplified single-track model from [PERSON] and [PERSON]. Figure 2: Robot orientation controller. The single-track model ([PERSON] and Schunck) permits the approximation of the vehicle lateral dynamics, as shown in Figure 3. Utilizing it makes some simplifications possible without influencing the fundamental analysis of mobile robot behavior for lateral dynamics. Readers are encouraged to study for a more detailed implementation of the [PERSON] and Schunck model [56]. In the proposed case, we consider the static obstacles. The single-track model analysis explains the analysis of handling the mobile robot during the ride from the doctor's office to the patient ward, and the behaviors of the wheels are observed during the track. By employing this model, it is easy to translate it into the simulation tool. Secondly, it is mostly employed because of simplicity and linear dependency between the wheel forces and the slip angle. The utilization of this model in the proposed research problem is based on some simplification about the slip angle, wheel forces, wheel-load distribution and longitudinal forces. Figure 3 shows the geometric relationships of the single-track model. Figure 3 describes the following parameters: \(\delta\), \(r\) and \(b\). The sideslip angle denotes the vehicle's deviation direction from the center of gravity. The current velocity \(v\) of the front wheels is denoted over a fixed coordinate system \(b\) and the steering angle \(\delta\). The vehicle's movement relates to the vehicle's center of gravity SP. When cornering with a constant steering angle \(\delta\), it describes a circular path with the curve radius \(r\). The relationship between the steering angle \(\delta\) and the curve radius \(r\) is sought. For this purpose, the right-angled triangle with sides \(r\), \(a\) and \(l_{h}\) is considered first. Side \(a\) can be determined using the Pythagorean theorem. \[a=\sqrt{r^{2}-l_{h}^{2}} \tag{1}\]  
Now, the right-angled triangle consisting of sides \(a\) and \(l_{h}+l_{v}\) and the angle \(\zeta\) is considered. The tangent of angle \(\zeta\) can be expressed using the two well-known legs of the triangle. \[\tan\zeta=\frac{a}{l_{h}+l_{v}} \tag{2}\]  
By inserting Equation (1) for \(a\), we obtain  
\[\tan\zeta=\frac{\sqrt{r^{2}-l_{h}^{2}}}{l_{h}+l_{v}}. \tag{3}\]  
The auxiliary angle \(\zeta\) and the steering angle \(\delta\) add up to a right angle \(\pi/2\). Thus, the steering angle \(\delta\) as  
\[\zeta=\frac{\pi}{2}-\delta \tag{4}\]  
Substituting the value of \(\zeta\) in Equation (3). \[\tan\Bigl{(}\frac{\pi}{2}-\delta\Bigr{)}=\frac{\sqrt{r^{2}-l_{h}^{2}}}{l_{h}+ l_{v}} \tag{5}\]  
Finally, Equation (5) can still use two identities  
\[\tan\Bigl{(}\frac{\pi}{2}-\delta\Bigr{)}=\cot(\delta) \tag{6}\]  
and  
\[\frac{1}{\cot(\delta)}=\tan(\delta) \tag{7}\]  
to  
Figure 2:0Now, there are two levels of approximation. First, \(\tan^{-1}\) can be neglected for the small steering angles \(\delta\). Figure 2:1  
If the curve radius is very large compared to the distance \(l_{h}\) between the center of gravity and the rear axle, the term is simplified as follows:  
Figure 2:2  
According to Equation (10), the robot implements simple approximation because it delivers significantly better results than strong approximation. It should also be noted that only positive curve radii and steering angles are considered here. The information for steering to the left and right can be stored in the sign of the curve radius \(r\) and the steering angle \(\delta\). For the robot, a positive sign means steering to the right, and a negative sign means steering to the left. Now, the change in orientation \(\frac{d\phi}{dt}\) is of interest as its integration gives the orientation angle \(\phi\).