## 2 Preliminary Properties  
### Construction of the Function  
This method presented in [35] has been tested recently to obtain the velocity, acceleration, and jerk with a smaller gap in the via point time (\(t_{v}\)). Although this method presents non-continuous derivatives, these remain finite without showing an inconvenience. Moreover, the resulting trajectory can be changed to avoid or reduce unnatural oscillation, no matter how short or long the distance is. This is possible because of the construction of the function below:  
\[S_{\zeta at_{s}}(t)=\left\{ \begin{array}{c}q_{0}\;if\;t\leq 0\\ \frac{n^{t-\zeta}}{\left|n^{t+\frac{t}{b}-1}-n^{t}\right|}+q_{0}\;if\;0<t\leq t _{s}\;and\;q_{0}\leq S_{f}\\ -\frac{n^{t-\zeta}}{\left|n^{t+\frac{t}{b}-1}-n^{t}\right|}+q_{0}\;if\;0<t\leq t _{S}\;and\;S_{f}<q_{0}\end{array} \right.. \tag{1}\]  
where:  
\(n\) fixed constant greater to \(1\;(n>1)\). \(a\) velocity, acceleration, and Jerk modifier parameter (it is fixed). \(t_{s}\) final trajectory time. \(S_{f}\) final position. \(q_{0}\) initial position. \(\zeta\) objective parameter outcome in the \(i\)th iteration. This function has been shown in [25] and it was used to generate point-to-point trajectories, but in this work, (1) is added to generate via point trajectories as well (this is explained deeply in Section 4). The ideas behind the way this function was constructed are simple. First, we know that any function with the form \(n^{t-\zeta}\), where \(n>1\), can reach any positive or negative point (when this function is negative \(-n^{t-\zeta}\)) monotonically increasing or decreasing, respectively, by just calculating \(\zeta\) at any particular time. Still, it cannot start in any position chosen (when \(t>0\)). Nevertheless, this is solved by multiplying this function with the following \(\left(\left|n^{t+\frac{t}{b}-1}-n^{a}\right|\right)^{-1}\). Then, note that:  
\[\lim_{t\to 0^{+}}\pm\frac{n^{t-\zeta}}{\left|n^{t+\frac{t_{s}}{b}-1}-n^{a} \right|}=0. \tag{2}\]  
Once it is verified that the calculation of the function (1) fulfills the essential requirement to perform a trajectory with any particular start and final position, the parameter \(\zeta\) is required to generate the trajectory starting and ending at a particular point; the formula is presented in [35; 36], but this time the case \(S_{f}=q_{0}\) is added in the definition of \(\zeta\). Then:  
\[\zeta=\begin{cases}t_{s}-\log_{n}\left[\left(S_{f}-q_{0}\right)\left(n^{t_{s} }-n^{a}\right)\right]\;if\;S_{f}>q_{0}\\ t_{s}-\log_{n}\left[\left(q_{0}-S_{f}\right)\left(n^{t_{s}}-n^{a}\right)\right] \;if\;q_{0}>S_{f}\\ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\  
consider the following probability function where \(t\) is any parameter such that \(t\in[0,t_{\rm s}]\) and \(x=1,2,3,\ldots\)  
\[f(x,t)=\left\{ \begin{aligned} 0\ if\ t\leq 0,\\ \left(\frac{n^{(t-l)}}{\left|t\left(n^{t+\frac{k-l}{l}}-n^{t}+n^{t -l}\right)\right|}\right)^{x}\ if\ 0<t\leq t_{\rm s}\end{aligned} \right.. \tag{4}\]  
The following is calculated using the definition of the probability generating function. The parameter \(t\) is considered to be the same parameter \(t\) from \(f(x,t)\) and the definition of the probability generating function:  
\[\sum\limits_{x=0}^{\infty}t^{x}f(x), \tag{5}\]  
Then:  
\[ \begin{aligned} G(t)=\sum\limits_{x=0}^{\infty}t^{x}f(x,t)=\sum \limits_{x=0}^{\infty}t^{x}\left(\frac{n^{(t-l)}}{\left|t\left(n^{t+\frac{k-l} {l}}-n^{t}+n^{t-l}\right)\right|}\right)^{x}\\ =\sum\limits_{k=1}^{\infty}t^{x}\left(\frac{\left(n^{t-l}\right) ^{x}}{\left(\left|t^{x}\right|n^{t+\frac{k-l}{l}}-n^{t}+n^{t-l}\right|}\right) ^{x}\right)\\ =\frac{n^{t-l}}{\left(\left|n^{t+\frac{k-l}{l}}-n^{t}+n^{t-l} \right|}\right)=\frac{n^{t-l}}{\left(\left|n^{t+\frac{k-l}{l}}-n^{t}+n^{t-l} \right|\right)}\.\end{aligned}  \tag{6}\]  
Function (6) is a probability generating function for when \(\zeta=1-\log_{n}\left|n^{t_{\rm s}}-n^{a}\right|\) and because it must fulfill that \(\lim\limits_{t\to 1^{-}}G(t)=1\).