# Autonomous Vehicles: Vehicle Parameter Estimation Using Variational Bayes and Kinematics
## 1 Introduction  
The behavior of autonomous vehicles is strongly reliant on their sensors' data and the interpretation of the environment [1]. Internal sensors can be used to estimate the vehicle's own movement and assist in their control. However, due to sensor noise, external information needs to be used from perception approaches. These data enable the correction of assumptions based solely on internal sensors, increasing the accuracy and reliability of the active safety systems through knowledge of the vehicle's surroundings (e.g., through object recognition and tracking). One of the main characteristics of an autonomous vehicle is the ability to localize itself. To make localization possible a function \(f:\Delta\Phi\mapsto\mathcal{X}\) is needed (see Equation (1)) which maps for \(m\) wheels the angular rotation \(\Delta\Phi=\left\{\Delta\phi_{0},\Delta\phi_{1},\dots,\Delta\phi_{m}\right\}\) and vehicle parameters \(\Theta\) to a new vehicle pose. In this study, we summarize wheel radii, and distances between wheels used for kinematic models as \"vehicle parameters\". To estimate movement for a two-wheeled vehicle on a two dimensional surface at time \(t\), following general notation can be used:\[ \begin{pmatrix}\Delta x\\ \Delta y\\ \Delta\theta\end{pmatrix} _{t}=f(\Delta\phi_{t},\Delta\phi_{r},\Theta) \tag{1}\]  
Equation (1) calculates pose changes based on the left wheel motion \(\Delta\phi_{t}\), the right wheel motion \(\Delta\phi_{r}\) and vehicle parameter \(\Theta\). Since a vehicle is a machine that transports people or goods from one place to another [2] and a robot is a machine capable of carrying out a complex series of actions automatically [3], we can define autonomous vehicles as vehicular robots. From a mathematical perspective, both terminologies are equivalently described using \(f\), since both models are fully described by analyzing the wheel motion. The function \(f\) is also known as _motion model_. Motion models are crucial for autonomous vehicles since they are utilized in control, perception and planning [4]. The most commonly used motion models are based on either constant turn rate, constant acceleration, constant velocity or on a combination of these [5]. The angular movement of the wheels and the vehicle parameters \(\Theta\) (wheel radii and distance between wheels) are used to calculate the pose of the vehicle. This use of data from the movement of actuators to estimate changes in position over time is defined as odometry. A well documented issue with odometry is imprecision/inaccuracy, since wheels tend to slip on the road's surface [6]. Therefore the previously defined \(f\) function is not adequate to estimate these changes. To alleviate uncertainty (noise) and unknown effects that disturb rather theoretical kinematic models and produce errors [7], probabilistic approaches have been developed to enable localization under uncertainty in real environments. Some of them are summarized in [1]. Applying a probabilistic approach, the function in Equation (1) is able to map the wheel rotation to the pose changes and to merge vehicle parameters, such as wheel radii, as unknown parameters using black box approaches [8; 9]. Probabilistic models are used to determine coherences and marginal distributions [10] and can be described using graphical representations [11]. As a special instantiation, hidden Markov models [12] are the base for Bayes filters [1], which are the main models used for vehicle localization. The goal of a Bayes filter is to fuse information, where a movement model \(p(\vec{x}_{t}|\vec{x}_{t-1},\vec{u}_{t})\), a measurement model \(p(\vec{z}_{t}|\vec{x}_{t})\), the vehicle's last state \(\vec{x}_{t-1}\) at the time slot \(t-1\), as well as the so-called command value \(\vec{u}_{t}\) and measurements \(\vec{z}_{t}\) to the latest vehicle pose \(\vec{x}_{t}\) at the time step \(t\). Compared to simple odometry, Bayes filters have the advantage of being able to represent the uncertainty along a predicted trajectory.