## 3 Algorithm Description  
The trajectory function obtained by applying the method (IOSTV \((\zeta)_{v}\)-HF) is denoted as \(S_{i(\zeta dt_{s})_{v}}(t)\), where \(i\) represents the number of iterations. While these iterations increase, the trajectory becomes closer and closer to the via point, forming a homotopy with a family of functions with different parameters \(\zeta_{i}\) and \(\zeta_{v_{i}}\) calculated in every iteration until a solution for these two parameters fits to generate the desired via point trajectory. This happens because \(\zeta_{i}\) and \(\zeta_{v_{i}}\) converge when \(i\) tends to infinity (\(i\rightarrow\infty\)) by calculating the sequence \(S_{v_{i}}\) for every iteration, making \(S_{i(\zeta dt_{s})_{v_{i}}}(t)\) pass through the specified via point in the desired via point time (\(t_{v}\)). Then, the function is defined as \(S_{i(\zeta dt_{s})_{v}}(t):[0,t_{s}]\rightarrow\mathbb{R}\) and with the following conditions (\(S_{v_{i}}\) is defined in (9)), each condition is provided with an example:  
\[S_{i(\zeta dt_{s})_{v}}(t)=\left\{\begin{array}{l}q_{0}\;if\;t\leq 0\\ \\ 1 stcond\left\{ \begin{array}{l}\frac{\pi^{t-\zeta_{i}}}{\left|t^{t+\frac{ \theta}{6}-1}-\pi^{t}\right|}+\frac{\frac{\pi^{t-\zeta_{i}}}{6}}{\left|t^{t+ \frac{\theta}{6}-1}-\pi^{t}\right|}+q_{0}\;if\;0<t\leq t_{v}\;and\;q_{0}\leq S _{v_{i}}\leq S_{f}\\ \\ \frac{\pi^{t-\zeta_{i}}}{\left|t^{t+\frac{\theta}{6}-1}-\pi^{t}\right|}+\frac{ \pi^{t_{v}-\zeta_{i}}}{\left|t^{t+\frac{\theta}{6}-1}-\pi^{t}\right|}+q_{0}\; if\;t_{v}<t\leq t_{s}\;and\;q_{0}\leq S_{v_{i}}\leq S_{f}\\ \\ 2 ndcond\left\{-\frac{\pi^{t-\zeta_{i}}}{\left|t^{t+\frac{\theta}{6}-1}-\pi^{t} \right|}+\frac{\pi^{t-\zeta_{i}}}{\left|t^{t+\frac{\theta}{6}-1}-\pi^{t}\right| }+q_{0}\;if\;0<t\leq t_{v}\;and\;q_{0}<S_{v_{i}},S_{f}<S_{v_{i}}\\ \\ -\frac{\pi^{t-\zeta_{i}}}{\left|t^{t+\frac{\theta}{6}-1}-\pi^{t}\right|}+\frac{ \pi^{t_{v}-\zeta_{i}}}{\left|t^{t+\frac{\theta}{6}-1}-\pi^{t}\right|}+q_{0}\; if\;t_{v}<t\leq t_{s}\;and\;q_{0}<S_{v_{i}},S_{f}<S_{v_{i}}\\ \\ \end{array} \right.\;. \tag{7}\]  
For when \(S_{v_{i}}<q_{0}\), the next conditions are followed:  
Figure 1: Trajectory example where \(S_{\zeta dt_{s}}(t)\) is a probability generating function with parameters \(n=2,\;\;t_{s}=3,\;a=0,\;\;S_{f}=4\), and \(q_{0}=0\). \[S_{i(\zeta dt_{s})_{v}}(t)=\left\{ \begin{array}{ll}3 rdcond\left\{\frac{n^{t- \zeta_{i}}}{\left|n^{t+\frac{t}{2}-1}-n^{t}\right|}-\frac{n_{v}^{t-\zeta_{v_{i} }}}{\left|n_{v}^{t+\frac{t}{2}-1}-n^{t_{v}}\right|}+q_{0}\text{ if }0<t\leq t_{v}\text{ and }S_{v_{i}}<q_{0},\ S_{v_{i}}<S_{f}\\ \frac{n^{t-\zeta_{i}}}{\left|n^{t+\frac{t}{2}-1}-n^{t}\right|}-\frac{n_{v}^{t- \zeta_{i}}}{\left|n_{v}^{t+\frac{t}{2}-1}-n^{t_{v}}\right|}+q_{0}\text{ if }t_{v}<t\leq t_{s}\text{ and }S_{v_{i}}<q_{0},\ S_{v_{i}}<S_{f}\\ 4 thcond\left\{-\frac{n^{t-\zeta_{i}}}{\left|n^{t+\frac{t}{2}-1}-n^{t}\right|}- \frac{n_{v}^{t+\frac{t}{2}-1}-n^{t_{v}}}{\left|n_{v}^{t+\frac{t}{2}-1}-n^{t_{v }}\right|}+q_{0}\text{ if }0<t\leq t_{v}\text{ and }S_{v_{i}}<q_{0},\ S_{f}<S_{v_{i}} \\ -\frac{n^{t-\zeta_{i}}}{\left|n^{t+\frac{t}{2}-1}-n^{t}\right|}-\frac{n_{v}^{t -\zeta_{i}}}{\left|n_{v}^{t+\frac{t}{2}-1}-n^{t_{v}}\right|}+q_{0}\text{ if }t_{v}<t\leq t_{s}\text{ and }S_{v_{i}}<q_{0},\ S_{f}<S_{v_{i}} \end{array} \right., \tag{8}\]  
where:  
\(n_{v}\) fixed constant greater to \(1\) (\(n_{v}>1\)). \(a_{v}\) velocity, acceleration, and Jerk modifier parameter (it is fixed). \(t_{v}\) via point time. \(\theta_{v}\) via point. \(S_{v_{i}}\) feedback iterative sequence  
\(\zeta_{i}\) and \(\zeta_{v_{i}}\) converge if the feedback iterative sequence (\(S_{v_{i}}\)) is defined as follows:  
\[S_{v_{i}}=\left\{ \begin{array}{ll}\theta_{v}-\frac{n^{t_{v}- \zeta_{i-1}}}{\left|n^{t_{v}+\frac{t_{v}}{2}-1}-n^{t}\right|}\text{ for the }1 st\text{ and }3 rd\text{ cond}\\ \theta_{v}+\frac{n^{t_{v}-\zeta_{i-1}}}{\left|n^{t_{v}+\frac{t_{v}}{2}-1}-n^{ t}\right|}\text{ for the }2 nd\text{ and }4 th\text{ cond}\end{array}