# Autonomous Vehicles: Vehicle Parameter Estimation Using Variational Bayes and Kinematics
## 4 Results
### Motion Model and Vehicle Parameter Estimation  
For the purpose of this study, our experiments are divided into two parts:  
1. **Motion Model:** We need to fit the sGP motion model to the data. In doing so, we need to optimize parameters and to ensure that the model represents the simulated movement. 2. **Kinematic Parameter Estimation:** The kinematic parameter estimation is based on the result of the motion model and simplified vehicle kinematics. Figure 5: Overview of the setup of the simulation environments, namely the test factory (**a**) and the car environment (**b**). The result of the sGP modelling is shown in Figure 6a for the mobile robot and Figure 6b for Ackermann steering. Based on the methodology proposed in [43], we chose the number of inducing points visually. This is in contrast to VB applications such as [52], where a marginal likelihood maximum occurs. One hundred auxiliary variables were determined, which were used by the model to create 100 pseudo vehicle movements and thereby estimate overall robot behavior. The Ackermann steering model log likelihood for \(GP_{x}\) and \(GP_{y}\) increases significantly after initial convergence. Due to the \(R^{2}\) analyzis of the test data, we assume that 20 auxiliary variables are sufficient to represent the physical system (see Figure 6b). The \(R^{2}\) analyzis is shown in Figure 7. Since there is no significant improvement after 20 auxiliary points, we chose this configuration. Figure 6: Visualization of implemented sparsed Gaussian processe (sGP) models. We select 100 auxiliary points for the mobile robot (**a**). After analyzing the \(R^{2}\) value of the Ackermann motion models, we chose 20 auxiliary points for the car simulation (**b**). Figure 7: Visualization of the \(R^{2}\) values of the sGP model estimation. Based on Equation (8), we defined the probability density function of the mobile robot and Ackermann steering wheel radii (see Equations (11) and (12)). \[\text{Mobile robot wheel radii:}\ \bar{r}\sim\mathcal{N}(\mu_{r},\sigma_{r}^{2})= \mathcal{N}(0.03329,1.327\times 10^{-8}) \tag{11}\] \[\text{Ackermann steering read wheel radii:}\ \bar{r}_{b}\sim\mathcal{N}(\mu_{r},\sigma_{r}^{2})= \mathcal{N}(0.30797,2.680\times 10^{-5}) \tag{12}\]  
In both cases, the uncertainty (variance) is low. We assume that uncertainty will increase in real applications. The mobile robot simulation is based on a real wheel radius of 0.033 m (See [[http://wiki.ros.org/turtlebot3](http://wiki.ros.org/turtlebot3)]([http://wiki.ros.org/turtlebot3](http://wiki.ros.org/turtlebot3))) for robot simulation details, and therefore the error is \(\sim\)2.7 % or 0.029 mm. Similar to that, the simulated ground truth of the Ackermann steering simulation is 0.31265 m (see Section 4.1), leading to an error of \(\sim\)5.6 % or 0.46 mm. The baseline histogram and kernel density estimation of the filtered baseline values \(B\) for both models is shown in Figure 8. For the mobile robot simulation, the peak of the kernel density estimation is at approximately 0.1596 m, the simulation being based on the baseline of 0.16 m (see Section 4.2). This is an error of 0.25 % or 0.04 mm. Similar to that, the baseline value for the Ackermann steering was estimated at 1.5616 m, resulting in an error of \(\sim\)1.5 % or 2.4 cm. The simulated ground truth is 1.586 (see Section 4.1). Additionally, the estimation of the virtual front wheel probability density function \(r_{v}\) resulted in the same probability density function as the mean rear wheel radii \(\bar{r}_{b}\) (see Equation (13))  
\[r_{v}\sim\mathcal{N}(\mu_{r},\sigma_{r}^{2})=\mathcal{N}(0.30797,2.680\times 1 0^{-5}). \tag{13}\]  
We assumed that the radii are equal and thus slipping and rolling constraints apply to motion. Finally, the estimated \(T\) values are depicted in Figure 9. We estimated the peak of the kernel density at \(\sim\)2.7579 m, the simulated ground truth measuring 2.86 m (see Section 4.1). It results in an error of \(\sim\)3.5% or 10.21 cm.