## 2 Optimization Methodology
### Artificial Potential Field  
The artificial potential field method was firstly defined in [22]. The research was initially applied to the manipulator and mobile robots with static and moving obstacles and target points. In the APF method, a force field is assumed to have a great impact on the planned trajectory of a manipulator. The obstacles were placed on the planned path of the manipulator and experiencing repulsive forces with the manipulator while the target was sending an attractive force to the manipulator. In the APF method, the manipulator's joint moves in a force field. As its name implies, the APF method produces a vector field of forces F(x) derived from the potential (scalar) field U(x) [23]. In the beginning, description of the trajectory path planning by the APF forces will be given, then the APF method will be improved by a differential evaluation algorithm since APF parameters are often adopted empirically, making the process expensive. With the use of algorithms of DE, an improvement in the potential field algorithm is achieved to generate free trajectory collisions. The DE algorithms are conducted for a population of 30 initial individuals (particles) with 50 iterations (generations) with the trajectory of APF using DE showing the smoothest path. It was also possible to check the average error position between the desired end position and the last trajectory position generated by the APF method improved by DE. When comparing computational cost, adopting the DE algorithm gave less execution time regarding parameters calculations of the APF algorithm. Advantages of this hybrid \begin{table}
\begin{tabular}{c c c} \hline
**Contour Type** & **Circular Contour** & **Linear Contour** \\ \hline Starting Point (m) & (0.5,0, pi/3) & (0.49,0.49, pi/3) \\ Duration (s) & 9 & 2 \\ Maximum Joint Speed (rad/s) & 2.291 & 1.280 \\ Ending Point (m) & (0.5,0, pi/3) & (0.69,0.68, pi/3) \\ \hline \end{tabular}
\end{table} 
Table 1: Contour information. algorithm (DE-APF) can be summarized as improvement of the APF algorithm, since its parameters are obtained in optimized form, low computational cost and precision of the points of the trajectories reached. Trajectory Path Planning by APF  
The potential function \(U(\theta)\) resulting from the artificial field is defined as a linear combination of a repulsion potential field \(U_{rep}(\theta)\) which pushes the manipulator away from obstacles, and an attraction potential field \(U_{att}(\theta)\) which pulls the manipulator to the target position [24] as given by the following equation:  
\[U(\theta)=U_{rep}(\theta)+U_{att}(\theta) \tag{9}\]  
Thus, considering the presence of a single obstacle, the coordinates of the obstacle position are defined as \(\theta obs=(\theta obs1,\theta obs2)\) and the target coordinates of target position are defined by \(\theta_{target}=(\theta_{target1},\theta_{target2})\) whereas the position of robot center is located at \(\theta=(\theta_{1},\theta_{2})\). The vector field of forces \(F(\theta)\) is derived from a scalar potential field \(U(\theta)\) according to the following equation [25]. \[F(\theta)=\
abla|U(\theta)| \tag{10}\]  
where \(F(\theta)\) is the resultant force which is also a linear combination of two components; \(F_{att}(\theta)\) is the attractive force that guides the robot's center to the target, and \(F_{rep}(\theta)\) is a force that causes an artificial repulsion from the obstacle surface produced by \(U_{rep}(\theta)\) as given by the following equation  
\[F(\theta)=F_{rep}(\theta)+F_{att}(\theta) \tag{11}\]  
Therefore, the APF algorithm deals with a repulsive force from the obstacles and an attractive force to the target. The trajectory generation using potential fields is a method that can be used in global offline planning, where the robot environment is already defined, or in local online planning where the environment is unknown. The presence of obstacles are being detected by sensors mounted on the robot [26].