## 2 Methodologies
### Collision Avoidance Algorithm  
The widely used CA algorithm is known as the APF due to its simple principle. The APF consists of attractive and repulsive potential fields. For example, a goal position attracts a quadcopter in the attractive potential field while obstacles repel the quadcopter in the repulsive potential field. As mentioned in Section 1, however, the APF has GNRON and local minima issues. The GNRON issue is resolved by introducing an additional term in the repulsive potential function. The total potential function is defined as [20; 21]  
\[U_{t}(^{N}\mathbf{r})=U_{a}(^{N}\mathbf{r})+U_{r}(^{N}\mathbf{r}), \tag{22}\]  
where  
\[U_{a}(^{N}\mathbf{r})=\frac{1}{2}\,k_{a}\,d(^{N}\mathbf{r},\ ^{N}\mathbf{r}_{g})^{2}, \tag{23}\]  
\[U_{r}(^{N}\mathbf{r})= \begin{cases}\frac{1}{2}k_{r}\left(\frac{1}{d(^{N} \mathbf{r},\ ^{N}\mathbf{r}_{o})}-\frac{1}{d_{o}}\right)^{2}\,d(^{N}\mathbf{r},\ ^{N}\mathbf{r}_{g})^{n_{g}}&\text{if}\,d(^{N}\mathbf{r},\ ^{N}\mathbf{r}_{o})\leq d_{o},\\ 0&\text{if}\,d(^{N}\mathbf{r},\ ^{N}\mathbf{r}_{o})>d_{o}.\end{cases}  \tag{24}\]  
Here, \(U_{a}(^{N}\mathbf{r})\) is the attractive potential function, \(U_{r}(^{N}\mathbf{r})\) is the repulsive potential function, \({}^{N}\mathbf{r}\in\mathcal{R}^{3}\) is the quadcopter position in the inertial frame, \({}^{N}\mathbf{r}_{g}\in\mathcal{R}^{3}\) is the goal position, \({}^{N}\mathbf{r}_{o}\in\mathcal{R}^{3}\) is the obstacle's position, \(d(\mathbf{a},\mathbf{b})\) is the relative distance between generic vectors \(\mathbf{a}\in\mathcal{R}^{3}\) and \(\mathbf{b}\in\mathcal{R}^{3}\), \(d_{o}\) is the distance influenced by the repulsive potential function, and \(n_{g}\) is the order of the relative distance term. Furthermore, \(k_{a}\) and \(k_{r}\) are the gain coefficients of attractive and repulsive potential functions, respectively. Due to \(d(^{N}\mathbf{r},\ ^{N}\mathbf{r}_{g})^{n_{g}}\), the total potential function solely becomes zero when the quadcopter arrives at the goal position. This is how the GNRON issue is resolved. Since the potential functions with \(n_{g}=0\) are the same as the original one presented by [PERSON] [7], \(n_{g}\) should be greater than 0. By taking the negative gradient to the potential functions, the potential fields, which are the vector form, are derived as  
\[\mathbf{f}_{a}(^{N}\mathbf{r})=-\
abla U_{a}(^{N}\mathbf{r})=k_{a}\,d(^{N} \mathbf{r},\ ^{N}\mathbf{r}_{g})\frac{\partial\,d(^{N}\mathbf{r},\ ^{N}\mathbf{r}_{g})}{\partial\,N \mathbf{r}}, \tag{25}\]  
\[\mathbf{f}_{r}(^{N}\mathbf{r})=-\
abla U_{r}(^{N}\mathbf{r})= \begin{cases} \mathbf{f}_{r,\rho}(^{N}\mathbf{r})+\mathbf{f}_{r,g}(^{N}\mathbf{r})&\text{if} \,d(^{N}\mathbf{r},\ ^{N}\mathbf{r}_{o})\leq d_{o},\\ 0&\text{if}\,d(^{N}\mathbf{r},\ ^{N}\mathbf{r}_{o})>d_{o},\end{cases}  \tag{26}\]  
where  
\[\mathbf{f}_{r,\rho}(^{N}\mathbf{r})=-k_{r}\frac{d(^{N}\mathbf{r},\ ^{N}\mathbf{r}_{g})^{n_{g}}}{d(^{N}\mathbf{r},\ ^{N}\mathbf{r}_{o})^{2}}\bigg{(}\frac{1}{d(^{N}\mathbf{r},\ ^{N}\mathbf{r}_{o})}-\frac{1}{d_{o}}\bigg{)}\frac{ \partial\,d(^{N}\mathbf{r},\ ^{N}\mathbf{r}_{o})}{\partial\,^{N}\mathbf{r}}, \tag{27}\]  
\[\mathbf{f}_{r,g}(^{N}\mathbf{r})=\frac{1}{2}n_{g}\,k_{r}\,d(^{N}\mathbf{r},\ ^{N} \mathbf{r}_{g})^{n_{g}-1}\bigg{(}\frac{1}{d(^{N}\mathbf{r},\ ^{N}\mathbf{r}_{o})}-\frac{1}{d_{o}}\bigg{)}^{2}\,\frac{ \partial\,d(^{N}\mathbf{r},\ ^{N}\mathbf{r}_{g})}{\partial\,^{N}\mathbf{r}}. \tag{28}\]Note that \(\mathbf{f}_{a}(^{N}\mathbf{r})\in\mathcal{R}^{3}\) is the attractive potential field and \(\mathbf{f}_{r}(^{N}\mathbf{r})\in\mathcal{R}^{3}\) is the repulsive potential field. Also, \(\partial\,d(^{N}\mathbf{r},\ ^{N}\mathbf{r}_{o})/\partial\ ^{N}\mathbf{r}\) is the direction from the quadcopter to the obstacle, and \(\partial\,d(^{N}\mathbf{r},\ ^{N}\mathbf{r}_{g})/\partial\ ^{N}\mathbf{r}\) is the direction from the quadcopter to the goal position.