## 3 IMM/EKF-Based Intersection-Target-Intent Estimation  
By receiving the validated measurement for every single target, extended Kalman filters (EKFs) are implemented as the local filter which matches every single mode. This is incorporated using the interacting multiple model (IMM) approach to retain the best estimate of the state vectors and the best matching modes. The algorithm for estimation of intersection target intent is described below. It is noted that the time index is to be omitted unless otherwise needed for clear explanations. _(1). Interaction_ (\(\forall i,j\in\mathbf{M}/\forall n\in\mathbf{T}\)):  
With the \(N_{mode}\) weights of \(\mu_{n}^{i}[k-1]\), the \(N_{mode}\) specifically depicts \(\hat{\mathbf{x}}_{n}^{i}[k-1]\), and the \(N_{mode}\) related covariance \(\hat{\mathbf{P}}_{n}^{i}[k-1]\), which are used to calculate the mixed condition for an initial condition which is used for a filter matched to the state of mode \(j\). It should be noted that if the implemented modes contain different dimensions of state vectors, the lower dimension state of an augmentation with zeros observed may cause a bias toward zero for components of the larger state vector. According to [21], to avoid the issue of 'biasing', a simple procedure is suggested, combined with a proper augmentation of the smaller state's covariance, and it yields unbiased and consistent mixing. _Mode probability with Predictions:_  
\[ \begin{array}{rl}\overline{\mu}_{n}^{j}&\triangleq\Pr\left\{m_{n}[k]=j\big{|} \mathbf{z}_{n}[k-1]\right\}\\ &=\sum\limits_{i}\left\{\mathbf{\Phi}_{ij}[k-1]\mu_{n}^{i}[k-1]\right\}\end{array}  \tag{4}\]  
Figure 1: Eight modes of possible behavior of the target approaching the intersection by using the leftmost lane from the south. _Mixing probability:_  
\[\mu_{n}^{\hat{q}j} \triangleq \Pr\Big{\{}m_{n}[k-1]=\hat{\imath}[m_{n}[k]=j,\mathbf{z}_{n}[k-1] \Big{\}}\] (5) \[=\frac{\mathbf{\Phi}_{ij}[k-1]\mu_{n}^{j}[k-1]}{\overline{\mu}_{n}^{j}}\] _Mixed condition:_  
\[\hat{\mathbf{x}}_{n}^{0j}[k-1]=\sum_{i}\hat{\mathbf{x}}_{n}^{i}[k-1]\mu_{n}^{ \hat{q}j} \tag{6}\]  
\[\hat{\mathbf{P}}_{n}^{0j}[k-1]=\sum_{i}\Bigg{[}\hat{\mathbf{P}}_{n}^{j}[k-1]+ \Bigg{\{} \begin{array}{c}\Big{(}\hat{\mathbf{x}}_{n}^{i}[k-1]-\hat{\mathbf{x }}_{n}^{0j}[k-1]\Big{)}\\ \times\Big{(}\hat{\mathbf{x}}_{n}^{i}[k-1]-\hat{\mathbf{x}}_{n}^{0j}[k-1]\Big{)} ^{T}\end{array} \Bigg{\}}\Bigg{|}\mu_{n}^{\hat{q}j} \tag{7}\]  
_(2). EKF Approach:_  
Every single pair of the \(N_{mode}\) values has weights of \(\hat{\mathbf{x}}_{n}^{0j}[k-1]\), \(\hat{\mathbf{P}}_{n}^{0j}[k-1]\), and it is used to input to an EKF approach which is matched to the state of mode \(j\). _Time update:_  
\[\mathbf{F}_{n}^{j}[k-1]=\left.\frac{\partial\mathbf{f}_{n}^{j}}{\partial \mathbf{x}}\right|_{\hat{\mathbf{x}}_{n}^{0j}[k-1],\hat{\mathbf{u}}[k-1]} \tag{8}\]  
\[\mathbf{x}_{n}^{-j}[k]=\mathbf{f}_{n}^{j}\Big{(}\hat{\mathbf{x}}_{n}^{0j}[k-1 ],\hat{\mathbf{u}}[k-1]\Big{)} \tag{9}\]  
\[\mathbf{\tilde{P}}_{n}^{j}[k]=\mathbf{F}_{n}^{j}[k-1]\cdot\hat{\mathbf{P}}_{n }^{0j}\cdot[k-1]\mathbf{F}_{n}^{j}[k-1]^{T}+\mathbf{W}_{n}^{j}[k-1] \tag{10}\]  
_Filter gain:_  
\[\mathbf{H}_{n}^{j}[k]=\left.\frac{\partial\mathbf{h}_{n}^{j}}{\partial \mathbf{x}}\right|_{\hat{\mathbf{x}}_{n}^{j}[k],\hat{\mathbf{u}}[k]} \tag{11}\]  
\[\mathbf{S}_{n}^{j}[k]=\mathbf{H}_{n}^{j}[k]\cdot\tilde{\mathbf{P}}_{n}^{j}[k] \cdot\mathbf{H}_{n}^{j}[k]^{T}+\mathbf{V}_{n}^{j}[k] \tag{12}\]  
\[\mathbf{K}_{n}^{j}[k]=\tilde{\mathbf{P}}_{n}^{j}[k]\cdot\mathbf{H}_{n}^{j}[k]^ {T}\cdot\mathbf{S}_{n}^{j}[k]^{-1} \tag{13}\]  
_Innovation:_  
\[\mathbf{r}_{n}^{j}[k]=\mathbf{z}_{n}[k]-\mathbf{h}_{n}^{j}\Bigg{(}\mathbf{x}_ {n}^{-j}[k],\hat{\mathbf{u}}[k]\Bigg{)} \tag{14}\]  
_Measurement update:_  
\[\hat{\mathbf{x}}_{n}^{j}[k]=\tilde{\mathbf{x}}_{n}^{-j}[k]+\mathbf{K}_{n}^{j}[ k]\Big{[}\mathbf{r}_{n}^{j}[k]\Big{]} \tag{15}\]  
\[\hat{\mathbf{P}}_{n}^{j}[k]=\Big{(}\mathbf{I}-\mathbf{K}_{n}^{j}[k]\cdot \mathbf{H}_{n}^{j}[k]\Big{)}\tilde{\mathbf{P}}_{n}^{-j}[k] \tag{16}\]  
_(3).