## 4 Obstacle Avoidance Planning
### Improved Method
#### 4.2.1 Improvement of Target Point Unreachable Defect  
The target point is unreachable; that is, the obstacle is distributed near the target point. So the pickup manipulator will never move to the target point due to the repulsive force of the obstacle. In order to solve the problem of an unreachable target point in the special case, the repulsion potential field function of the traditional artificial potential field method is improved. In order to enable the pickup manipulator to reach the target point in the special case, the relative distance term \(d^{n}(X,X_{g})\) between the pickup manipulator and the target point is increased based on the original repulsion potential field function. \(n\) is adjustment factor of the distance and the repulsion potential field can be adjusted according to the relative distance between the pickup manipulator and the target point. In addition, the influence of the obstacle boundary is considered, and the obstacle radius r is introduced into the repulsion potential field function on the obstacle avoidance planning. Figure 7 shows that under the influence distance \(d_{0}\) the obstacle, the obstacle boundary is used as the collision condition. The distance \(d(X,X_{obs})\) of the current point of the pickup manipulator to the obstacle boundary is calculated, and the repulsion potential field is constructed under this distance. Thus, the mathematical expression of the improved repulsive potential field function is:  
\[\left\{ \begin{array}{l}U^{*}_{r\
eqq}(X)=\frac{1}{2}\varepsilon(\frac{1}{d(X,X_ {obs})-r}-\frac{1}{d_{0}})^{2}d^{n}(X,X_{g}),X-X_{obs}\leq d_{0}\\ U^{*}_{r\
eqq}(X)=0,X-X_{obs}>d_{0}\end{array} \right., \tag{29}\]  
The improved repulsion force is generated by a negative gradient of the improved repulsion potential field function. The function expression relationship is:  
\[F^{*}_{r\
eqq}(X)=-\
abla U^{*}_{r\
eqq}(X)=\left\{ \begin{array}{l}F^{*}_{r \
eqq1}+F^{*}_{r\
eqq2^{\prime}},X-X_{obs}\leq d_{0}\\ 0\hskip 28.452756 pt,X-X_{obs}>d_{0}\end{array} \right., \tag{30}\]  
When \(X-X_{obs}>d_{0}\), it means that the distance between the end-effector and the obstacle exceeds the boundary affected by the repulsive potential field. The repulsion force is set to zero. When \(X-X_{obs}<d_{0}\), the pickup manipulator is within the influence of the repulsion potential field. At this time, the repulsion force generated by the obstacle is composed of the repulsion component \(F^{*}_{r\
eqq1}\) and the compensating attraction component \(F^{*}_{r\
eqq2}\). The functional expressions are:  
\[\left\{ \begin{array}{l}F^{*}_{r\
eqq1}=\varepsilon(\frac{1}{d(X,X_{obs})-r}- \frac{1}{d_{0}})\frac{1}{(X-X_{obs})^{2}}\frac{\partial d(X,X_{obs})}{\partial X }d^{n}(X,X_{g}),X-X_{obs}\leq d_{0}\\ F^{*}_{r\
eqq2}=-\frac{\eta}{2}\varepsilon(\frac{1}{d(X,X_{obs})-r}-\frac{1}{ d_{0}})^{2}d^{n-1}(X,X_{g})\frac{\partial d(X,X_{g})}{\partial X},X-X_{obs}\leq d_{0} \end{array} \right., \tag{31}\]  
In the formula, \(F^{*}_{r\
eqq1}\) represents the repulsion force generated by the obstacle to the manipulator, pointing from the obstacle direction to the manipulator. \(F^{*}_{r\
eqq2}\) is the compensation of attraction force for the target point unreachable, pointing from the direction of the pickup manipulator to the target point. The modes of the respective vectors of \(F^{*}_{r\
eqq1}\) and \(F^{*}_{r\
eqq2}\) are:  
\[\left\{ \begin{array}{l}\|F^{*}_{r\
eqq1}\|=\varepsilon(\frac{1}{d(X,X_{obs})- r}-\frac{1}{d_{0}})\frac{1}{(X-X_{obs})^{2}}d^{n}(X,X_{g}),X-X_{obs}\leq d_{0}\\ \|F^{*}_{r\
eqq2}\|=\frac{\eta}{2}\varepsilon(\frac{1}{d(X,X_{obs})-r}-\frac{ 1}{d_{0}})^{2}d^{n-1}(X,X_{g}),X-X_{obs}\leq d_{0}\end{array} \right., \tag{32}\]  
Compared with the original repulsion force \(F_{r\
eqq}\), the improved repulsion component \(F^{*}_{r\
eqq1}\) becomes smaller, but the direction is unchanged. The increased compensating attraction component \(F^{*}_{r\
eqq2}\) points to the target point.