## 5 Proposed Path Planning Algorithm
### Artificial Potential Field Considering Road Constraints and Multiple Obstacles  
To attract our micro-EV using reference path points, methods leveraging APF are proposed. The traditional attractive force considers only one goal point. On the other hand, our proposed algorithm could consider multiple goal points. In this paper, multiple goals indicate the reference path points. These reference path points are changed to a polynomial to facilitate calculations, as shown in Equation (6). Generally, the Clothoid curve model [22] is applied in vehicle curve fitting. However, this model is generally applied to roads with a road speed of 60 km/h or greater is used. It is not acceptable for our campus environment. Therefore, the Bayesian curve fitting method [23] is applied for curve fitting to this polynomial. To make the cost dimension \(U_{att}\), the polynomial is squared. As a result, this form takes a three-dimensional Cartesian coordinate \(\left(x,y,U_{att_{wp}}\right)\), as shown in Equation (7) (Figure 4a). The \(x_{propa}\) and \(y_{propa}\) denoted in Equation (7) represent the PN \((x,y)\) points. In Equation (7), the PN points converge on the reference polynomial line, wherever it is located, since the cost value increases toward the far direction from the polynomial line region. The reference polynomial line is shown in Figure 4 as a red line. If there is no negative slope toward the far points from the start node, the PNs located on the low-cost valley will not propagate quickly toward the end of the valley, which affects real-time performance. Therefore, the farther away from the starting node, the lower the cost value should be, as shown in Equation (8). In addition, it takes a three-dimensional Cartesian coordinate \(\left(x,y,U_{att_{slope}}\right)\). This equation results in the potential field shown in Figure 4b. The final attractive force results from the sum of Equations (7) and (8), as shown in Figure 4c. The introduced equation is as follows:  
\[y=f(x)=a_{1}+a_{2}x+a_{3}x^{2}+\ldots+a_{n+1}\ x^{n} \tag{6}\]  
\[U_{att_{up}}=k_{1}\big{(}y_{prop}-f\big{(}x_{propa}\big{)}\big{)}^{2},\ \ k_{1}\geq 0 \tag{7}\]  
\[U_{att_{slope}}=-k_{2}\sqrt{x_{propa}{}^{2}+y_{propa}{}^{2}},\ \ k_{2}\geq 0 \tag{8}\]  
Unlike the reference path points, the obstacles have a repulsive force [24]. Additionally, the obstacles are handled in the discrete domain. The obstacles are dots on a grid map at intervals of 10 cm. Each dot point is expressed in the form of an exponential function to reduce the field region's effect that is farther away and improve the closer field region's  
Figure 4: Potential field visualization drawn on three-dimensional Cartesian coordinates. The contour line denotes the height of the cost value. The blue circle denotes the reference path points. The red line, crossing the zero point, denotes the Bayesian polynomial calculated from the reference path points. The zero point (0, 0, 0) mean the center of gravity of the micro-EV. **(a)** Denotes Equation (6). The potential valley is shown through the x-y plane; **(b)** denotes Equation (7). The slope is shown through the x-z plane; **(c)** denotes the sum of Equation (6). effect. This method is beneficial compared to traditional repulsive force functions since the effect region is calculated with only one parameter, \(k_{3}\), without a threshold value (Figure 5). The equation of this repulsive force is Equation (9). \(x_{obsts}\) and \(y_{obsts}\) are the positions of obstacles. The equation is as follows:  
\[U_{rep}=k_{3}exp\left(-\sqrt{\left(x_{obsts}-x_{prop}\right)^{2}+\left(y_{obsts }-y_{prop}\right)^{2}}\right),\ \ k_{3}\geq 0 \tag{9}\]  
The entire proposed cost of the potential field is denoted as the sum of attractive and repulsive forces and is shown in Equation (10). This entire potential field function is applied to the Hybrid A*'s cost model. \[U_{\text{total}}=U_{att_{rep}}+U_{att_{stop}}+U_{\text{rep}} \tag{10}\]  
There are hyper-parameters, \(k_{m}\), in Equations (7)-(9). These are related to the algorithm performance.