## 3 Path-Planning Design and Dynamic-Obstacle Avoidance Strategy
### Dynamic-Obstacle-Avoidance Strategy Based on Artificial-Potential-Field Method
#### 3.3.1 Artificial-Potential-Field Method  
The artificial-potential-field method is a virtual-force method proposed by [PERSON], whose method is to define the environment in which the drone is located in terms of the potential field and control the obstacle avoidance driving of the drone by the position information. The basic idea of the artificial-potential-field method is to construct an artificial potential field in which the gravitational field at the target location and the repulsive field around the obstacle act together, and to find collision-free paths by searching for the descent direction of the potential function. Assuming that the current position of the drone is and the position of the target point is \(x_{goal}\), the functional expression of the gravitational potential field is shown in Equation (17).  
\[U_{att}=\frac{1}{2}\eta\rho^{2}(x,x_{goal}) \tag{17}\]  
where \(\eta\) is the proportional gain function of the gravitational field and \(\rho(x,x_{goal})\) denotes the distance vector from the current position of the drone to the position of the target point. The derivative of the gravitational-field function yields the gravitational function \(F_{att}\), whose expression is shown in Equation (18).  
\[F_{att}=-\
abla U_{att}=-\eta\rho(x,x_{goal}) \tag{18}\]  
In the artificial-potential-field method, the factor that determines the repulsive field of the obstacle is the distance between the drone and the obstacle, and the drone is subject to a potential energy of 0 when it is not within the influence of the obstacle. When the drone enters the influence range of the obstacle, the smaller the distance between them, the more potential energy the drone is subjected to, as shown in Equation (19).  
\[U_{req}=\left\{\begin{array}{ll}\frac{1}{2}k(\frac{1}{\rho(x,x_{obs})}-\frac {1}{\rho_{0}})^{2}&0\leq\rho(x,x_{obs})\leq\rho_{0}\\ 0&\rho(x,x_{obs})\geq\ \rho_{0}\end{array}\right. \tag{19}\]  
where \(x_{obs}\) denotes the position coordinates of the obstacle, \(\rho(x,x_{obs})\) denotes the distance vector from the current position of the drone to the position of the obstacle, \(k\) is the positive scale factor of the repulsive field, and \(\rho_{0}\) denotes the maximum distance at which the obstacle exerts a repulsive force on the drone. The corresponding repulsive function Freq is expressed as the negative gradient of the repulsive field, as shown in Equation (20).  
\[F_{req}=\begin{cases}k(\frac{1}{\rho(x,x_{obs})}-\frac{1}{\rho_{0}})\frac{ \
abla\rho(x,x_{obs})}{\rho^{2}(x,x_{obs})}&0\leq\rho(x,x_{obs})\leq\rho_{0}\\ 0&\rho(x,x_{obs})\geq\ \rho_{0}\end{cases} \tag{20}\]  
The drone is in constant motion under the action of a combined potential field consisting of the target point and multiple obstacles. Assuming that the number of obstacles that exert repulsive influence on the drone at the current moment is \(n\), the expression \(U\) for the total potential field can be obtained by superimposing the potential field as shown below.  
\[U=U_{att}+\sum_{i=1}^{n}U_{rep} \tag{21}\]  
The expression of the combined force on the drone is as follows.  
\[F=-\
abla U=F_{att}+\sum_{i=1}^{n}F_{rep} \tag{22}\]