# Global Time-Varying Path Planning Method Based on Tunable Bezier Curves
## 5 Theoretical Verification  
The process of the GTVP method is described in Sections 3 and 4. In these two sections, the effectiveness and practicability of the GTVP method are proved by theoretical derivation. The path planned by this method can ensure that the moving object will never encounter the obstacle and the distance between the moving object and the obstacle can be adjusted by adjusting the magnification factor. Next, a circular obstacle is taken as an example to describe the proof process.  
If \(O_{0}\) is taken as the origin point, the coordinates of the four points \(E_{1}\), \(E_{2}\), \(E_{3}\) and \(E_{4}\) in Figure 2 are:  
Figure 7: Obstacle avoidance curve in a multi-obstacle environment.  
\[\left\{\begin{array}{l}E_{1}:\left(-\sqrt{r^{2}-y_{0}^{2}},0\right)\\ E_{2}:\left(\frac{-1}{4}\xi,\frac{\sqrt{3}}{4}\xi\right),\xi=\sqrt{4r^{2}-y_{0}^ {2}}-\sqrt{3}y_{0}\\ E_{3}:\left(\frac{1}{4}\xi,\frac{\sqrt{3}}{4}\xi\right),\xi=\sqrt{4r^{2}-y_{0}^ {2}}-\sqrt{3}y_{0}\\ E_{4}:\left(\sqrt{r^{2}-y_{0}^{2}},0\right)\end{array}\right. \tag{34}\]  
where \(r\) is the dimensional information of the circular obstacle: the radius of the circle; \(y_{0}\) is the distance between point \(O_{0}\) (center point) and point \(C\) (fuzzy center of gravity).  
Through adopting the method in the paper, six feature points can be obtained. And then, the expression of the 5-order Bezier curve can be derived:  
\[p(\kappa)=\sum_{i=1}^{N}C_{N-1}^{i-1}(1-\kappa)^{N-i}\kappa^{i-1}P_{i}^{0} \tag{35}\]  
where \(N=6\). Equation (36) can be obtained by expanding the combination number in the above equation.  
\[p(\kappa)=\sum_{i=1}^{N}\bigg{[}\frac{N!}{(i-1)!(N-i-1)!}\bigg{]}(1-\kappa)^{ N-i}\kappa^{i-1}P_{i}^{0} \tag{36}\]  
Put \(N=6\) into Equation (36) and expand to obtain Equation (37).  
\[\begin{array}{l}p(\kappa)=(1-\kappa)^{5}P_{1}^{0}+5(1-\kappa)^{4}\kappa P_{ 2}^{0}+10(1-\kappa)^{3}\kappa^{2}P_{3}^{0}+\\ 10(1-\kappa)^{2}\kappa^{3}P_{4}^{0}+5(1-\kappa)\kappa^{4}P_{5}^{0}+\kappa^{5} P_{6}^{0}\end{array} \tag{37}\]  
Bringing in these 6 points \(P_{1}^{0}\) - \(P_{6}^{0}\), it can be analyzed that, when \(\kappa=0.5\), the distance between the Bezier curve and the obstacle is the closest. Let \(\kappa=0.5\), and yields:  
\[\begin{array}{l}p(\kappa=0.5)=(0.5)^{5}P_{1}^{0}+5(0.5)^{5}P_{2}^{0}+10(0.5) ^{5}P_{3}^{0}+\\ 10(0.5)^{5}P_{4}^{0}+5(0.5)^{5}P_{5}^{0}+(0.5)^{5}P_{6}^{0}\end{array} \tag{38}\]  
Ensure that the curve is outside the obstacle, namely:  
\[p(\kappa=0.5)\geq r-y_{0} \tag{39}\]  
After organization,  
\[10\cdot(0.5)^{3}\cdot\frac{\sqrt{3}}{4}\cdot\xi\geq r-y_{0} \tag{40}\]  
Bringing in \(\xi\), and tidying up, yields:  
\[5\sqrt{3}\bigg{(}\sqrt{4r^{2}-y_{0}^{2}}-\sqrt{3}y_{0}\bigg{)}\geq 16(r-y_{0}) \tag{41}\]  
Let \(y_{0}\)=\(k\cdot r\), where \(k\in(0,1)\). Equation (42) is obtained after simplification.  
\[5\sqrt{3}\Big{(}\sqrt{4-k^{2}}-\sqrt{3}k\Big{)}\geq 16(1-k) \tag{42}\]  
After further organization,  
\[5\sqrt{3}\Big{(}\sqrt{4-k^{2}}\Big{)}+k-16\geq 0 \tag{43}\]Let the left of the inequality sign in the above equation be \(f(k)\), namely:  
\[f(k)=5\sqrt{3}\Big{(}\sqrt{4-k^{2}}\Big{)}+k-16 \tag{44}\]  
When \(f(k)\) is derived and the increase or decrease of the function between the interval (0,1) is analyzed, it can be obtained that the function keeps as the increase function between the interval (0, 1/\(\sqrt{19}\)) and the subtraction function between the interval (1/\(\sqrt{19}\), 1). The two minimum points \(f(k=0)=1.32\) and \(f(k=1)=0\) satisfy the condition of greater than or equal to 0. Through the theoretical derivation in this section, it can be proved that the Bezier curve planned by this method never touches obstacles.