# Autonomous Vehicles: Vehicle Parameter Estimation Using Variational Bayes and Kinematics
## 4 Results
### Motion Model and Vehicle Parameter Estimation
The estimated parameters, including simulated ground truth values, are summarized in Table 1.  
Figure 7.0 Baseline estimation for both kinematic models using kernel density estimation and maximum (red line). The used bin sizes are 0.002 m for mobile robot baseline (**a**) and 0.001 m for Ackermann steering (**b**). To analyse real-time capability of the presented methodology, time measurements for sGP prediction were done. For 100 inducing points of the mobile robot model, 15.425 ms were measured for a single prediction. For Ackermann steering, 0.284 ms were measured for the 20 inducing point model. Both experiments were performed on an Intel i9-7900X CPU. However, the used sGP implementation [47] is a CPU implementation. Similar to GPU optimized software for artificial neuronal networks such as [53], software for GPU based GP acceleration is under development [54]. We performed additional simulation for the Ackermann kinematic model using different wheel radii for front and rear. Furthermore, we changed the friction coefficient. We used 20 inducing points for the simulations AM2 and AM3 and 200 inducing points for the simulation AM1 (we chose a friction coefficient for rubber and wet concrete based on [[https://hypertextbook.com/facts/2006/[PERSON]](https://hypertextbook.com/facts/2006/[PERSON])]([https://hypertextbook.com/facts/2006/[PERSON]](https://hypertextbook.com/facts/2006/[PERSON]))). The results are summarized in Figure 7.1 \begin{table}
\begin{tabular}{l l l l l l l l l} \hline  & \(\bar{\mathbf{r}}_{(\mathbf{b})}\) & & \(\mathbf{B}\) & & \(\bar{\mathbf{r}}_{\mathbf{v}}\) & & \(\mathbf{T}\) & \\ \hline  & **M** & **GT** & **M** & **GT** & **M** & **GT** & **M** & **GT** \\ \hline Mobile Robot & 3.329 cm & 3.300 cm & 15.96 cm & 16 cm & - & - & - & - \\ \hline Ackermann Steering & 30.79 cm & 31.26 cm & 1.561 m & 1.586 m & 30.797 cm & 31.265 cm & 2.757 m & 2.86 m \\ \hline \end{tabular}
\end{table} 
Table 1: Summarized results of derived vehicle parameters from the sGP motion models (M) and simulated ground truth (GT). \begin{table}
\begin{tabular}{l l l l l l l l l} \hline  & \(\bar{\mathbf{r}}_{(\mathbf{b})}\) & & \(\mathbf{B}\) & & \(\bar{\mathbf{r}}_{\mathbf{v}}\) & & \(\mathbf{T}\) & \\ \hline  & **M** & **GT** & **M** & **GT** & **M** & **GT** & **M** & **GT** \\ \hline AM 1 & 26.890 cm & 26.265 cm & 1.629 m & 1.586 m & 35.428 cm & 36.270 cm & 2.522 m & 2.860 m \\ \hline AM 2 & 29.105 cm & 30.000 cm & 1.534 m & 1.586 m & 23.142 cm & 25.000 cm & 2.437 m & 2.860 m \\ \hline AM 3 & 29.872 cm & 30.000 cm & 1.581 m & 1.586 m & 26.890 cm & 25.000 cm & 2.443 m & 2.860 m \\ \hline \end{tabular}
\end{table} 
Table 2: Summarized results additional simulation results for Ackermann (AM) steering. AM 1 and AM 2 are based on the standard friction of 0.9 for dry concrete. For AM 3, we changed the friction coefficient to 0.45. The histograms for AM 1-AM 3 can be found in the Supplementary Materials. Figure 9: Estimation of \(T\) using the instantaneous center of rotation (ICR). The used bin size is 0.05 m.  
The results based on additional simulation show, that the presented methodology is able to estimate vehicle parameters with different front and read wheel radii. Similar to the initial results (see Table 1), our approach estimates \(T\) not as accurate as the wheel radii and \(B\).