# Autonomous Vehicles: Vehicle Parameter Estimation Using Variational Bayes and Kinematics
## 5 Conclusion and Future Work  
We rejected the Null hypothesis and accept therefore the alternative hypothesis:  
**H1:**_sGP-based motion models are able to learn movement based on wheel motion_. by converting blackbox motion models to models that describe the model behavior using vehicle kinematics. For the implementation, we chose a sGP-based modelling approach, where we modeled \(p(\vec{x}_{t}|\vec{x}_{t-1},\vec{u}_{t})\) using wheel motion as command input \(\vec{u}_{t}\). Since the input to our parameter estimation is wheel motion and Gaussian distributed motion (see Equations (8) and (10)), any model can be used to produce these data. We selected sGP instead of other models, such as Bayesian neuronal networks [22], due to the non-parametric nature of the defined problem, which decreases user-based bias. Nevertheless, experiments using Bayesian neuronal networks (blackbox model) and Bayesian linear regression (whitebox model) were performed. Both models performed inferior in comparison to sGP models. For sGP motion models, we used the lower bound and Kullback-Leibler divergence to derive the number of auxiliary points for the GP approximation. This approach results in a probabilistic motion model for vehicle behavior. To get insight into the blackbox motion model, we simulated vehicle movement starting from a known position. This movement was used to parameterize the kinematic model, where we used basic properties of probability density functions and kernel density estimation to calculate the vehicle parameters. Using minor additional physical assumptions, we estimated the vehicle's (rear) wheel radii and (read) baseline. It is possible to estimate those parameters between 0.25% and 5.6% prediction accuracy. Our additional simulation results showed that our methodology can estimate vehicle parameters with different front and rear wheel radii. \begin{table}
\begin{tabular}{l l l l l l l l l} \hline \hline  & \(\vec{r}_{(b)}\) & & \(B\) & & \(\vec{r}_{p}\) & & \(T\) & \\ \hline  & **M** & **GT** & **M** & **GT** & **M** & **GT** & **M** & **GT** \\ \hline Mobile Robot & 3.683 cm & 3.300 cm & 17.866 cm & 16 cm & - & - & - & - \\ \hline Ackermann Steering & 28.429 cm & 31.26 cm & 1.436 m & 1.586 m & 28.429 cm & 31.265 cm & 2.582 m & 2.86 m \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Summarized results based on the blackbox model Bayesian neuronal networks of vehicle parameters from the sGP motion models (M) and simulated ground truth (GT). For mobile robot experiments, 15 hidden neurons were used in the hidden layer. For Ackermann steering experiments, 40 hidden neurons were used. Using additional assumptions, namely possible ICR positions and reasonable values, we estimated a \(T\) value close to the simulated ground truth. The results show, that the sGP based motion model is able to learn movement based on wheel motion, being thus able to reject the null hypothesis. The presented methodology proves, that the sGP learns vehicle parameters implicitly. In contrast to that, the major shortcoming of sGP models is the limited real-time applicability. Our results show, that sGP models still has real-time issues and the real-time capability strongly depends on the number of auxiliary points. However, novel software packages such as [54] may solve this shortcoming by GPU acceleration. We assume that real movement data will result in significantly higher variance values. Therefore, future work will address the analysis of real vehicle data as opposed to simulated models and also the evaluation of the derived motion model compared to classic physics-based models as well as dynamical models. The assumptions defined in this study, namely the rigid body assumption and rolling/sliding constraints, were used to derive the vehicle parameters from the sGP models. For real vehicles, those assumptions are typically violated on a small level between the timeslots \(t\) and \(t+1\). This violation results in additional movement noise.