# Autonomous Vehicles: Vehicle Parameter Estimation Using Variational Bayes and Kinematics
## 3 SGP Motion Model Implementation
### Vehicle Parameter Estimation  
Throughout this paper a few assumptions have been made to derive the explanatory factors from the GP models, namely the car consists of a rigid body construction, the constant speed of the Ackerman vehicle and the sliding and rolling constraints. Where the rigid body assumption is one of the fundamental theories for kinematic models, which were used for wheel radii and distance estimation. Since physical characteristics of the vehicle must be described, this assumption has to be made. The speed of the car was set to a constant speed for the purpose of simplicity to generate data. Since this assumption was not done in mobile robot experiments, we do not assume that variance in speed effects the parameter estimation. Finally, the sliding and rolling constraints are used to derive the kinematic equations which are the fundamental building stone of the GP algorithm. Initially, we estimate \(\tilde{r}\) by rearranging Equation (8). \[\rho=\sqrt{\Delta X^{2}+\Delta Y^{2}}=\frac{(\Delta\phi_{t,r}+\Delta\phi_{t,l} )\cdot\tilde{r}}{2}. \tag{8}\]  
The baseline is estimated using \(\Delta\theta\) but it is affected by singularities that are related to a straight movement. As with the mobile robot kinematic, the car movement refers to the center of the rear baseline \(P\). Thus, we can use the same procedure for back wheel radius and baseline estimation. The radius of the virtual front wheel \(\tilde{r}_{v}\) can be calculated easily using the wheel movement. Finally, \(T\) (distance between front and rear axles) is determined using the ICR and basic kinematics, where we assume a stiff chassis and wheel motion. This estimation is done using \(R\) (distance/radius to ICR) and the baseline \(B\). In the approach presented in this paper we used a simulation set up that included noise for both vehicle models. Therefore we had to face multiple challenges, including difficulty estimating the baseline values for both models in a straightforward maneuver. This is due to the fact that the \(\Delta\theta\) becomes zero, and unrealistically large baselines arise due to computational problems. Based on this and due to the fact that the distance to the ICR is significantly affected by noise, we assume a noisy and hard to estimate \(T\). Furthermore, we currently use a point estimate of \(B\) and the mean wheel radii. Since we do not assume a Gaussian distribution for \(B\), this will disturb the \(T\) calculation. For mean wheel radius \(\tilde{r}\) estimation of the mobile robot, we performed a single prediction for \(GP_{x}\), \(GP_{y}\) and \(GP_{\theta}\) using the mean value of the input (wheel movement \(\Delta\phi_{t,l}\)). The sGP models estimate the inherent \(\Delta X\), \(\Delta Y\) and \(\Delta\theta\). Since both \(\Delta X\) and \(\Delta Y\) are normally distributed according to sGP prediction and \(\Delta\phi_{t,l}\) are constants, we assume that \(\tilde{r}\) is also Gaussian. Due to singularities for the baseline estimation (see \(\Delta\theta\) calculation in Equation (7)), we used the whole test dataset of the sGP prediction. We excluded the motion data that fulfilled \((\Delta\phi_{r}-\Delta\phi_{l})<0.1[rad]\) to exclude straightforward movement. We also deleted unrealistic baseline values by excluding estimated baselines that were smaller than 5 cm and larger than 0.5 m, where the ground truth is 16 cm. For each resulting motion, we estimated a possible baseline length. We applied a histogram-based approach including kernel density estimation [48] to estimate the baseline value. For Ackermann steering, we followed the same procedure for the estimation of both the rear wheel radius \(\tilde{r}_{b}\) and the baseline \(B\). We applied additional data filtering for faulty time stamps, where samples with a time differences above 0.1 seconds were deleted.