## 3 Orientation Controller Design
### Determination of the Dynamics of the Steering Train  
An attempt is made to estimate the dynamic behavior of the steering by measuring the robot's orientation. For this purpose, the robot is accelerated to 600 EncImp/s. The steering value PWM, transmitted to the steering servo, is 0 during this time. The robot should therefore drive straight ahead. Then, the steering value is abruptly changed to \(-75\), so the robot should steer strongly to the right. The robot orientation is recorded. This recording is finally evaluated. The expected change in orientation is determined in advance. For this purpose, the steering value of PWM = \(-75\) is used in Equation (16). There is a steering angle \(\delta\) of  
\[\delta=a+b\cdot PWM=9.10034\times 10^{-3}+2.22607\times 10^{-3}\cdot(-75)=-0.15 79=-9.04 \tag{22}\]  
Inserting this into Equation (13), we obtain the expected change in orientation as follows:  
\[\frac{d\phi}{dt}=\frac{v}{\sqrt{\left(\frac{l}{\tan(\delta)}\right)^{2}+l_{h}^ {2}}}=\frac{0.348028}{\sqrt{\left(\frac{0.34}{\tan(-0.1579)}\right)^{2}+(0.135) ^{2}}}=0.1627\frac{\text{rad}}{s}=9.32\frac{\circ}{s} \tag{23}\]  
The block diagram of steering in Figure 2 can be simplified. The quantizer before the steering is omitted because the actuating signal is a constant value corresponding to one quantization level. The subsequent transformation of the PWM value into a steering angle is due to constant amplification for constant control values. The play only has a minor effect because the steering angle is only changed in one direction. The relationship between the steering angle and the change in orientation in the single-track model only represents constant amplification for a constant steering angle. This enables an analytical consideration with which the time constants of the transfer function \(G_{S}(s)\) can be determined. Taking into account the integrator in the single-track model, the step response results for the transfer function are as follows:  
\[G(s)=G_{S}(s)\cdot\frac{K}{s} \tag{24}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Direction** & \begin{tabular}{c} **Circle** \\ **Center (X)** \\ \end{tabular} & \begin{tabular}{c} **Circle** \\ **Center (Y)** \\ \end{tabular} & \begin{tabular}{c} **Circle** \\ **Radius** \\ \end{tabular} & \
\begin{tabular}{c} **Steering** \\ **Angle (rad)** \\ \end{tabular} &
\begin{tabular}{c} **Steering** \\ **Angle (deg)** \\ \end{tabular} \\ \hline Left & \(-0.031\) m & \(-15.547\) m & \(-16.296\) m & \(-0.02087\) & \(-1.20^{\circ}\) \\ \hline Right & \(-1.225\) m & \(17.133\) m & \(16.445\) m & \(0.02067\) & \(1.18^{\circ}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Minimum circle radii when driving straight ahead to determine play. First, the integrator is removed by differentiating the transfer function once in the time domain. The differences between adjacent values must be formed and divided for the measured function by the sampling time. Unfortunately, the quantization distance of the magnetometer is so large that the readings rarely change. Therefore, the differentiated signal resembles strong noise. To control this, the measured values are smoothed by a PT1 element with a time constant of one second. Of course, this must be taken into account in the transfer function \(G_{(s)}\). A modified approach for the transfer function results in the following:  
\[\overline{G_{(s)}}=s\cdot G_{(s)}\cdot\frac{1}{\tau\cdot s-1}=G_{S(s)}\cdot \frac{K}{\tau\cdot s-1}\text{ with }\tau=1\text{ s.} \tag{25}\]  
Figure 4 shows the derived and smoothed measured values with the blue curve. The red curve is a very rough approximation.