## 3 Target State Prediction with Uncertainty Estimation
### Intention Inference and State Prediction  
Because the behavior of the approaching vehicles can vary widely, it is difficult to express the future behavior with a few models without compromising the continuity of behavior. However, if too many models are used to overcome this problem, the computational load becomes too burdensome to achieve real-time performance. This study utilized the interacting multiple model (IMM) filter to estimate the probability of each behavior model, which is used to fuse each behavior model. A block diagram of the proposed IMM-based target intention inference algorithm is described in Figure 4. The purpose of the IMM filter is to define the fused behavior model by integrating the behavior model of the local extended Kalman filter (EKF) based on probability. The first step of designing the IMM filter is defining the behavior model for local EKF. The 'Cross', 'Yield', and 'Stop' models are defined based on the collected driving data. The data were collected at Midan City, Incheon, South Korea. The velocity profiles of the collected driving data are described in Figure 5. As shown in Figure 5, am asymmetric characteristic of the acceleration usage has been observed, which is biased to braking. Therefore, the IDM has been selected to model the asymmetric acceleration profile [26]. The IDM is composed of two terms, namely the desired velocity tracking term and the car-following behavior term, as follows:  
\[ \begin{array}{l}\dot{x}=v\\ \dot{v}=a_{max}\bigg{(}1-\left(\frac{v}{v_{des}}\right)^{\delta}-\left(\frac{s ^{*}(v,\Delta v)}{s_{des}}\right)^{2}\bigg{)}\\ \text{with }s^{*}(v,\,\Delta v)=s_{0}+v\cdot T+\frac{v\cdot\Delta v}{2 \sqrt{a_{max}}B}\end{array}  \tag{2}\]  
where \(a_{max}\) is the maximum acceleration. \(v\) and \(v_{des}\) are the vehicle velocity and desired velocity. \(s\), \(s^{*}\), and \(\Delta v\) are the clearance, desired clearance, and relative velocity to the preceding vehicle. Parameters \(T\), \(b\), and \(\delta\) are the desired time headway, comfortable braking deceleration, and acceleration exponent. Figure 4: Block diagram of IMM-based target intention inference algorithm. Figure 5: The measured and desired velocity profiles. The collected velocity profiles should be converted to the desired velocity profiles to use the IDM as a process update model of the local EKF. The desired velocity tracking term of the IDM has been rearranged to calculate \(v_{des}\) from the \(v\) profile. The rearranged IDM is as follows:  
\[v_{des,i}(s)=\frac{v_{i}(s)}{\sqrt[]{1-\dot{v}_{i}(s)/a_{max}}} \tag{3}\]  
where \(i\) is the data index of each profile. The generated desired velocity profiles are presented in Figure 5 as a red dotted line. Parameters \(a_{max}\) of 5.0 m/s\({}^{2}\) and \(\delta\) of 4 have been used to calculate the \(v_{des,i}\)(s). The derived \(v_{des,i}\)(s) profiles have been clustered by K-means clustering to define the behavior model as depicted in Figure 6. As mentioned before, these three \(v_{des,i}\)(s) profiles have been named 'Cross', 'Yield', and 'Stop' models and used to determine the process update model of local EKF. The local filters have been designed as an IDM-based EKF to use 'Cross', 'Yield', and 'Stop' models. The state vector \(x\) and measurement vector \(z\) of the IDM-based EKF has been defined as follows:  
\[x=\left[ \begin{array}{cccc}p&v&a\end{array} \right]^{T},\ \ \ z=\left[ \begin{array}{cc}p_{meas}&v_{meas} \end{array} \right]^{T} \tag{4}\]  
where \(p\), \(v\), and \(a\) are the \(d_{DTC}\), velocity, and acceleration, respectively. Parameters \(p_{meas}\) and \(v_{meas}\) represent the measurement of the \(d_{DTC}\) and velocity from the sensor of the autonomous vehicle. The process update of IDM-based EKF is defined as follows:  
\[\begin{array}{ll}\dot{x}_{k|k-1}&=f(\dot{x}_{k-1|k-1},u_{k})\\ &=\left[ \begin{array}{cc}p_{k|k-1}&v_{k|k-1}&a_{k|k-1}\end{array}