## 4 Obstacle Avoidance Planning
### Improved Method
#### 4.2.1 Improvement of Target Point Unreachable Defect
So, the improved resultant repulsion force \(F_{r\
eqq}\) becomes smaller and the direction tends toward the target point. The improved resultant repulsion force \(F^{*}_{r\
eqq}\) is superimposed on the original attraction force, which makes the improved resultant force \(F^{*}\) closer to the target point and solves the detection of the unreachable target. The different values of the adjustment factor \(n\) will have different effects on the improved repulsion function \(F^{*}_{r\
eqq}\). The different values of \(n\) are analyzed below. Figure 7: Schematic diagram of obstacle distance. (1) When \(0<n<1\), the mathematical characteristics of \(d^{n}(X,X_{g})\) in \(F_{np1}^{*}\) and \(d^{n-1}(X,X_{g})\) in \(F_{rep2}^{*}\) are analyzed, and the results are shown in Figure 8.  
Analyzing the data in Figure 8, it can be founded that when \(X\to X_{g}\), \(d^{n}(X,X_{g})\to 0\), \(d^{n-1}(X,X_{g})\rightarrow\infty\), then:  
\[\left\{ \begin{array}{l}\lim_{X\to X_{g}}\|F_{rep1}^{*}\|=\lim_{X\to X_{g}} \biggl{[}\varepsilon(\frac{1}{d(X,X_{obs})-r}-\frac{1}{d_{0}})\frac{1}{(X-X_{ obs})^{2}}d^{n}(X,X_{g})\biggr{]}=0\\ \lim_{X\to X_{g}}\|F_{rep2}^{*}\|=\lim_{X\to X_{g}}\biggl{[}\exists \varepsilon(\frac{1}{d(X,X_{obs})-r}-\frac{1}{d_{0}})^{2}d^{n-1}\left(X,X_{g} \right)\biggr{]}=\infty\end{array} \right., \tag{33}\]  
It can be found that when the pickup manipulator gradually approaches the target point, the improved repulsion component \(F_{np1}^{*}\) approaches zero. The compensated attraction component \(F_{np2}^{*}\) approaches infinity. Under the action of the resultant potential, the pickup manipulator can smoothly move to the target point. (2) When \(n=1\), the mathematical characteristics of \(d^{n}(X,X_{g})\) in \(F_{np1}^{*}\) and \(d^{n-1}(X,X_{g})\) in \(F_{rep2}^{*}\) are analyzed, and the results are shown in Figure 9.  
Analyzing the data in Figure 9, it can be found that when \(X\to X_{g}\), \(d^{n}(X,X_{g})\to 0\), \(d^{n-1}(X,X_{g})\to c\), then:  
\[\left\{ \begin{array}{l}\lim_{X\to X_{g}}\|F_{rep1}^{*}\|=\lim_{X\to X_{g}} \biggl{[}\varepsilon(\frac{1}{d(X,X_{obs})-r}-\frac{1}{d_{0}})\frac{1}{(X-X_{ obs})^{2}}d^{n}(X,X_{g})\biggr{]}=0\\ \lim_{X\to X_{g}}\|F_{rep2}^{*}\|=\lim_{X\to X_{g}}\biggl{[}\exists \varepsilon(\frac{1}{d(X,X_{obs})-r}-\frac{1}{d_{0}})^{2}d^{n-1}\left(X,X_{g} \right)\biggr{]}=c\end{array} \right., \tag{34}\]  
It can be found that when the manipulator gradually approaches the target point, the improved repulsion component \(F_{rep1}^{*}\) approaches zero. And the compensated attraction component \(F_{np2}^{*}\) approaches  
Figure 8: Mathematical characteristics when \(0<n<1\). Figure 9: Mathematical characteristics when \(n=1\). the constant c. The manipulator can smoothly move toward the target point under the action of the resultant potential. (3) When \(n>1\), the mathematical characteristics of \(d^{n}(X,X_{S})\) in \(F^{*}_{rep1}\) and \(d^{n-1}(X,X_{S})\) in \(F^{*}_{rep2}\) are analyzed, and the results are shown in Figure 10.  
Analyzing the data in Figure 10, it can be founded that when \(X\to X_{S}\), \(d^{n}(X,X_{S})\to 0\), \(d^{n-1}(X,X_{S})\to 0\), then:  
\[\left\{ \begin{array}{l}\lim_{X\to X_{S}}\|F^{*}_{rep1}\|=\lim_{X\to X_{S}} \Bigl{[}\varepsilon(\frac{1}{d(X,X_{\text{obs}})-\tau}-\frac{1}{d_{0}})\frac{1 }{(X-X_{\text{obs}})^{2}}d^{n}(X,X_{S})\Bigr{]}=0\\ \lim_{X\to X_{S}}\|F^{*}_{rep2}\|=\lim_{X\to X_{S}}\Bigl{[}\frac{n}{2} \varepsilon(\frac{1}{d(X,X_{\text{obs}})-\tau}-\frac{1}{d_{0}})^{2}d^{n-1}(X,X _{S})\Bigr{]}=0\end{array} \right., \tag{35}\]  
It can be found that when the manipulator gradually approaches the target point, the improved repulsion component \(F^{*}_{rep1}\) and the compensated attraction component \(F^{*}_{rep2}\) approaches to zero at the same time. The manipulator can still move smoothly to the target point under the influence of the original potential attraction force \(F_{att}\). According to the above three cases, different values of the adjustment factor \(n\) can make the pickup manipulator move smoothly to the target point.