## 2 Intent-Estimation- and Motion-Model-Based Collision Avoidance Method
### Intent Estimation
\[\theta_{lane}=\arg\tan(2k_{2}x+k_{1}) \tag{6}\]  
\[\gamma_{lane}=\frac{2k_{2}}{\left(4k_{2}^{2}x^{2}+4k_{2}k_{1}x+k_{1}^{2}+1 \right)^{\frac{3}{2}}} \tag{7}\]  
Then, the deviation of the vehicle from the lane center can be presented as follows:  
\[D^{2}=\left(X_{lane}-X_{car}\right)^{T}\left(P_{lane}+P_{car}\right)^{-1}(X_{ lane}-X_{car}) \tag{8}\]  
where \(D^{2}\) is the chi-square distance between the two vectors, \(X_{car}\) and \(X_{lane}\). \(P_{lane}\) is the covariance matrix of the lane state vector, and \(P_{car}\) is the covariance matrix of the vehicle state vector. The moving process is continuous, whereas the maneuvering space is discrete. Intent recognition is performed by mapping a set of continuous features into the discrete maneuvering space. In [13], the value of D is used to estimate the maneuvering intent of the vehicle. If the value is small, the vehicle is assumed to maintain its lane, whereas the vehicle is performing a lane change if the value is larger than a fixed threshold. This method faces a challenge, i.e., a fixed threshold would introduce a high false detection rate. To improve the robustness, a feature vector is constructed as follows:  
\[M=\left\{[ \begin{array}{cccc}D&a&yaw&\end{array} ]^{T}\right\}_{n} \tag{9}\]  
where \(n\) is the number of samples that we use in the intent-estimating process and \(a\) represents the acceleration in the direction of the lane. This vector contains the relationship between the vehicle and lane, as well as the moving state of the detected vehicle. As for a lane change, the prediction is successful if the intention of changing lane is estimated before any part of the vehicle enters the target lane. Data in Table 1 are used to illustrate the computation of the feature vector. To eliminate the effect of different dimensions, acceleration is measured with m/s\({}^{2}\) and yaw is measured with degree/s; the feature vector is normalized with Z-score standardization, which is introduced in Appendix A. The third row and the forth row show the means \(\mu\) and standard deviation \(\sigma\) of the features. Two examples, extracted from a right lane change and a normal moving event, are shown in the fourth row and the fifth row. The Z-scores of the samples are shown in the last two rows. To obtain accurate driving intent, a learning method that utilizes Gaussian mixture models (GMM) [14] is applied. GMM consists of a basis Gaussian distribution with a linear method. For a feature vector \(M\) of length \(n\), the probability distribution function is as follows:  
\[P(M|\lambda)=\prod\limits_{t=1}^{n}\sum\limits_{k=1}^{K}\omega_{k}\eta_{k} \tag{10}\]  
where \(K\) is the number of mixture Gaussian distributions and \(\omega_{k}\) denotes the mixture weights of the \(k^{th}\) mixture function, which satisfy the condition \(\sum\limits_{k=1}^{K}\omega_{k}=1\). The GMM can be represented with a set of components \(\lambda_{k}=\{\omega_{k},u_{k},\sum_{k}\}\) and Equation (11):  
\[\eta_{k}(M)=\frac{1}{\sqrt{\left(2\pi\right)^{d}\left|\sum_{K}\right|}}e^{- \frac{1}{2}\left(M-\mu_{k}\right)^{T}\sum_{k}^{-1}\left(M-\mu_{k}\right)} \tag{11}\]  
The expectation maximization algorithm [15] is used to learn the parameters of the GMM, namely, \(\{\omega_{k},\mu_{k},\sum_{k}\},\forall k\in(1,\cdots,K)\). In this paper, we use a GMM of three possible mixtures related to the three mentioned maneuvers of interest, i.e., normal moving, left lane change and right lane change. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Examples** & \(D\) & \(a\) & \(yaw\) \\ \hline \(\mu\) & 0.9 & \(-\)0.1 & 1.91 \\ \(\sigma\) & 0.40 & 1.76 & 0.69 \\ Vector\_1 (left lane change) & 1.1 & \(-\)0.8 & 1.6 \\ Vector\_2 (normal moving) & 0.6 & 0.4 & 0.8 \\ Z-score of Vector\_1 & 0.5 & \(-\)0.82 & 0.45 \\ Z-score of Vector\_2 & \(-\)0.75 & 0.28 & \(-\)1.61 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Means and standard deviations of feature vectors and two feature vector samples. Figure 4: Lane changes.