# Driver Clustering Based on Individual Curve Path Selection Preference
## 2 Methods and Materials  
### Lane Offset Modeling  
The path, and hence the position of the vehicle in the lane, is often described by the position offset to the centerline of the lane (1). \[\delta^{\text{Fr}}(t)\equiv\delta(t) \tag{1}\]  
which is the lane offset value given in the Frenet-frame of the center line. An illustration is shown in Figure 1.  
The lane offset is calculated by the driver model. This is a commonly followed approach by literature solutions [3; 5; 12], where the lane offset is only calculated in nominated node points. Then, a curve is fitted onto these points to formulate the locally planned path (2). \[p^{\text{path}}(t,x)=g(\{p_{i}^{\text{np}}(t)\}_{i=1}^{n},\beta) \tag{2}\]  
where \(\beta\) are the curve parameters, \(p_{i}^{\text{np}}(t)=\left[x_{i}^{\text{np}}(t)\quad y_{i}^{\text{np}}(t) \right]^{\top}\) are the node point coordinates in the local planning frame, with \(0<t\leq n\) and \(n\) is the number of node points. If reference node points are measured on the centerline (e.g., by a video camera), then  
\[p_{i}^{\text{np}}(t)=p_{i}^{\text{np,ref}}+\Delta p_{i}(t) \tag{3}\]  
where  
\[\Delta p_{i}(t)=\left[-\sin(\Theta_{i}^{\text{np,ref}}(t))\delta_{i}(t)\right] \tag{4}\]  
with  
\[d_{i}^{\text{np,ref}}(t)=\|\{p_{i}^{\text{np,ref}}(t)\}_{i=1}^{n}\|_{2} \tag{5}\]  
where \(\Theta_{i}^{\text{np,ref}}\) is the orientation of the lane at node point \(i\), and \(d_{i}^{\text{np,ref}}(t)\) is the distance of node point \(i\) from the origin of the planning frame. Therefore, the path planning problem of (2) is translated to the lane offset estimation problem (6). \[\delta_{i}(t)=f_{i}(u(t),\delta_{i}(t)):\mathbb{R}^{m}\mapsto\mathbb{R} \tag{6}\]  
where \(u(t)\) is the input vector, \(m\) is the number of inputs, and  
Figure 1: Lane offset to the center line. The quantity is given in the Frenet frame of the center line. \[\delta(t)= \begin{bmatrix}\delta_{1}(t)&\delta_{2}(t)&\cdots&\delta_{n}(t)\end{bmatrix}  \in\mathbb{R}^{n} \tag{7}\]  
There are different solutions in the literature to estimate the lane offset. A few approaches propose neural network structures [3; 4; 5]. However, the training of these models is long, and they cannot be used to directly define the reference driver groups. Instead, we chose the Extended Linear Driver Model (ELDM) structure [12]. This is a linear, static parametric model. This model calculates the lane offset in the node points at even time steps. Supposing we have samples of the inputs at even time steps \(T_{s}\), the lane offset is also updated periodically. Equation (6) is rewritten to the discrete time domain at the \(k^{\text{th}}\) step:  
\[\delta_{i,k}=\hat{f}_{i}(u_{k},u_{k-1}\cdots u_{k-n_{s}},\delta_{i,k-1},\delta_ {i,k-2}\cdots\delta_{i,k-n_{s}})+e_{i,k} \tag{8}\]  
where \(\delta_{i,k}=\delta(kT_{s})\) and \(u_{k}=u(kT_{s})\) the samples of the lane offset and the input vector, \(e_{i,k}\) is the fit error and \(\hat{f}_{i}\) is the function approximate of \(f_{i}(\cdot)\), \(n_{u}\) and \(n_{\delta}\) are the time window sizes of the inputs and lane offset, respectively. The LDM model equation is given in (9). \[\mathcal{M}^{\text{LDM}}:\hat{f}_{i}=\delta_{i,k}^{\text{left}}+\delta_{i,k}^ {\text{right}}+\delta_{0}=u_{i,k}^{\text{left}}\text{pleft}+u_{i,k}^{\text{ right}}\text{pright}+\delta_{0} \tag{9}\]  
The straight line distance is denoted by \(\delta_{0}\). \(n_{u}=1\), \(n_{\delta}=0\), and \(n=3\) according to [12]. Thus, \(\text{pleft},\text{pright}\in\mathbb{R}^{3\times 3}\). The node points are nominated at a constant distance from the origin of the planning frame [18]:  
\[d^{\text{mp,ref}}(t)\equiv d^{\text{mp,ref}}= \begin{bmatrix}10.0&39.0&136.0 \end{bmatrix} m \tag{10}\]  
The input of the model is three average curvature values between the origin of the planning frame and the first, second, and third node points. The values are distinguished between left and right curve situations, which are calculated based on the average preview curvature \(\kappa(t)\):  
\[u_{k}^{\text{left}}=