# Autonomous Vehicles: Vehicle Parameter Estimation Using Variational Bayes and Kinematics
## 1 Introduction
The noise-free prediction of \(y^{*}\) is depicted by \(f^{*}\) and is estimated using the following Equation (2), which has been previously discussed in [25]. \[ \begin{pmatrix}\vec{y}\\ f^{*}\end{pmatrix} \sim\mathcal{N}\left( \begin{bmatrix}\mathbf{K}+\sigma_{n}^{2} \mathbf{I}&\vec{k}_{*}^{T}\\ \vec{k}_{*}&k_{**}\end{bmatrix} \right). \tag{2}\]  
In the equation \(\mathbf{K}\) is an elementwise similarity measurement, \(\vec{k}_{*}\) describes the similarity of a new input (e.g., wheel movement) to the stored values and \(k_{**}\) describes the similarity of the new input to itself. \(\sigma_{n}\) is the variance of noise and \(\mathbf{I}\) is the identity matrix. Thus a prediction is based on the estimation of \(p(f_{*}|\vec{f},\mathbf{X},\vec{x}_{*})\) and is done rearranging Equation (2). For GP-based localization, two GP are used to model \(p(\vec{x}_{t}|\vec{x}_{t-1},\vec{u}_{t})\) and \(p(\vec{z}_{t}|\vec{x}_{t})\). To describe those models a compact notation of Equation (2) is used:\[p(\vec{x}_{t}|\vec{x}_{t-1},\vec{u}_{t}) =\mathcal{N}\left(GP_{\mu}(\vec{x}_{t-1},\vec{u}_{t},\mathcal{D}_{p} ),GP_{\Sigma}(\vec{x}_{t-1},\vec{u}_{t},\mathcal{D}_{p})\right) \tag{3}\] \[p(\vec{z}_{t}|\vec{x}_{t}) =\mathcal{N}\left(GP_{\mu}(\vec{x}_{t},\mathcal{D}_{o}),GP_{ \Sigma}(\vec{x}_{t},\mathcal{D}_{o})\right) \tag{4}\]  
where the functions \(GP_{\mu}(. )=\vec{k}_{*}(\mathbf{K}+\sigma_{n}^{2}\mathbf{I})^{-1}\vec{y}\) and \(GP_{\Sigma}(. )=\mathbf{K}-\vec{k}_{*}(\mathbf{K}+\sigma_{n}^{2}\mathbf{I})^{-1 }\vec{k}_{*}\) are the GP prediction functions [26]. However, even if GP-based localization can outperform classic approaches, physics-based motion models (e.g., whitebox models) remain the most reliable movement models for trajectory prediction and collision risk estimation for road safety. Their reliability as well as the fact that they rely solely on movement properties and do not include any inertial properties has led to their widespread use [27]. Physics-based trajectory prediction algorithms can only be used in low speed situations and are limited to short-term forecasting [27; 28]. Therefore, one of the big challenges of these algorithms is to identify and quantify the potential and limitations of their agents in order to integrate this knowledge into their maneuvers and trajectories [29] for the sake of road safety. To contribute to the field of research, we define the following null hypothesis:  
**H0:**_sGP-based motion models are not able to learn movement based on wheel motion._  
that we will test by converting blackbox models (e.g., GP motion models) to whitebox models, in which the internal processes can be viewed by using simple kinematic models. We estimate the kinematic parameters wheel baseline \(B\), radii \(r\) and distance between rear and front wheels \(T\) which need to be derived from the GP models. The model performance in terms of motion accuracy (X, Y and \(\theta\)) is not focus of this study. We contribute to the state of the art through the extension of a Gaussian process-based [25] localization approach [8] that estimates movement models without vehicle parameters. Based on the movement model, we derive wheel radii and distances and are thereby able to analyze the model behavior using physical plausibility. Thus, we evaluate our approach by comparing the derived vehicle parameters to simulated ground truth data vehicle parameter. To this end, we (i) introduce a Bayesian perspective of a motion model using approximate GP inference and (ii) then deduct the correspondent vehicle parameters as random variables. The non-parametric GP model learns movement models based on data. Explanatory factors such as wheel radii are learned implicitly. Thus, the explanatory factors must be derived from GP models afterwards. To implement this derivation, we extend existing approaches such as [30] by introducing a kinematic model to our derivation. Additionally, as an alternative to approaches based on classic control-theory such as [31], we use recent non-linear methodologies as motion models.