# Path Planning Based on Obstacle-Dependent Gaussian Model Predictive Control for Autonomous Driving
## 2 Related Work
### Obstacle-Dependent Gaussian Algorithm
#### 3.2.2 Obstacle-Dependent Gaussian Vehicle Model  
Dynamic and static obstacles can be identified through sensors in driving situations, and obstacles must be avoided. The ODG algorithm can be applied to obstacles. Determining whether the obstacle, \(S_{k}\), poses a danger to \(E\), is moving slowly in front of \(E\), or is quickly moving behind \(E\), is essential to apply the ODG algorithm to an obstacle. The ODG is calculated for obstacles that are considered dangerous in the sensor detection area. The obstacle distance and relative speed determine the risk of collision in the ODG against an obstacle. Because the ODG algorithm is inversely proportional to the collision time, \(T_{\text{C}}\), the risk is significantly increased to \(T_{\text{C}}\to 0\), making it dangerous. Therefore, the ODG algorithm calculates the time \(T_{\text{C}}\) to the collision. The way to avoid collisions is by changing lanes or slowing down by maintaining the current lane. For this purpose, the time required for avoidance, \(T_{A}\), is used to define the ratio of \(T_{\text{C}}\) to \(T_{A}\). This definition increases the risk in the event of a collision with the vehicle, which affects the ODG risk. Combining the longitudinal and lateral information determines whether the vehicle should change the lane or reduce speed in the current  
Figure 11: Line obstacle-dependent Gaussian (ODG) with model prediction control (MPC) longitudinal position. Figure 10: Obstacle-dependent Gaussian (ODG) for solid and dotted lines. lane. In the opposite case, it reduces the risk of collision. \(T_{A}\) is a user-defined variable that depends on E performance. If the \(T_{A}\) value is large, a sufficient distance from the obstacle vehicle is ensured, but a robot freezing problem may occur. The ODG algorithm for the obstacle vehicle is expressed through (28)-(30):  
\[O_{v,k}(p_{Y})=\omega\Big{|}\frac{T_{A}}{T_{C}}\Big{|}\exp\left(-\frac{(p_{{}_{ (S,k)_{Y}}}-p_{{}_{Y}})^{2}}{\sigma_{k}^{2}}\right), \tag{28}\]  
\[\sigma_{k}=\left(\frac{W_{E}}{2}+\frac{W_{S_{k}}}{2}+T_{s}\times v_{{}_{(S,k)_ {Y}}}\right)/ERF^{-1}(R), \tag{29}\]  
\[T_{C}=\frac{p_{{}_{(S,k)_{Y}}}-p_{{}_{E_{X}}}}{v_{{}_{(S,k)_{X}}}-v_{{}_{E_{X} }}}, \tag{30}\]  
where \(p_{{}_{(S,k)_{Y}}}\) is the lateral position of the \(k\)th surrounding vehicle, \(W_{S_{k}}\) is the width of \(k\)th surrounding vehicle, and \(v_{{}_{(S,k)_{Y}}}\) is lateral velocity of the \(k\)th surrounding vehicle. \(p_{{}_{E_{X}}}\) and \(v_{{}_{E_{X}}}\) are the longitudinal positions and speeds of \(E\), respectively. In addition, \(p_{{}_{(S,k)_{X}}}\) and \(v_{{}_{(S,k)_{X}}}\) are the longitudinal positions and speeds of the \(k\)th surrounding vehicle, respectively. The value with the minimum risk from the ODG value in the lateral direction along the longitudinal direction is set as a safe position. In this process, the minimum value of the ODG value is calculated based on the lane, and a lateral position having a low risk is searched for in each lane. In order to utilize the reference position information of the MPC, the reference positions have the same size of the horizon in MPC, and are decided by lateral position with minimum ODG value for the lane. In the case of a road departure, it must pass through the dotted line, and in this process there is a further risk of crossing the dotted line. Therefore, for the lanes other than the currently maintained lane, the risk of crossing the dotted line is additionally added. The sum of the minimum ODG values at each horizontal step indicates the risk to the lane as follows:  
\[ODG_{i}=\min_{p_{Y}(1,\cdots,N_{P})}\sum_{h=1}^{N_{P}}[ODG_{Total}(p_{{}_{Y}}(h ))]+O_{i,c}, \tag{31}\]  
where \(ODG_{i}\) is total ODG value of \(i\)-th lane, \(N_{P}\) is the size of prediction horizon in MPC, \(ODG_{Total}(p_{{}_{Y}}(h))\) is minimum ODG value in the i-th lane boundary, and \(O_{i,c}\) is risk of cross the dotted line that is integration of \(O_{l\_d}\). \[O_{i,c}=|i-i_{ref}|\omega_{d}\omega\sqrt{\pi}, \tag{32}\]  
where \(i_{ref}\) is the reference lane to drive.