## 4 Results: Velocity, Acceleration, and Jerk Function  
The velocity, acceleration, and jerk functions have been presented in [36]. This work requires a retaking of the topic of these functions because it is crucial for a long-life robotic system to exist. As shown in the previous examples, the trajectories presented obtain no continuous velocity, acceleration, and jerk at the via point time (\(t_{v}\)). Nevertheless, every value is bounded, and the gap between the velocity, acceleration, and jerk in \(t_{v}\) can be reduced as much as desired, making a safe motion in the joint-space. Moreover, the IOSTV (\(\zeta\))\({}_{v}\)-HF is not the first method that presents no-continuous velocity, acceleration, or jerk. Some methods mentioned before in the introduction and many others often used currently present no-continuous velocity, acceleration, or jerk. For example, [37] used trapezoidal velocity profiles to generate trajectories and presents a no-continuous jerk profile, which is bounded and ready for implementation. Figure 6: Example of complicated trajectories approaching the via point \(\theta_{v}=25\) at \(t_{v}=2.6\) and initial position \(q_{0}=10\) and \(S_{f}=62\) at \(t_{s}=3\), reducing the spikes of the oscillations changing the parameters \(n_{v},a_{v},\) and \(n,a\). Additionally, Refs. [38; 39] present not-zero velocity at the final point, but this method is a perfect tool for obstacle avoidance, as they wanted to show using IOSTV \((\zeta)_{v}\)-HF. In [36], the via point trajectory taken from [9] has been used to compare it with the IOSTV \((\zeta)_{v}\)-HF method by presenting a trajectory shape with a continuous velocity and another with a continuous acceleration. In [9], the trajectory compared was not optimal, and a significant gap in the velocity was presented. In this work, the via point trajectory taken from [9] is retaken to obtain a better trajectory than the one that was presented in [36] and compared with the result from [35]. The values are \(q_{0}=30\), \(\theta_{v}=180\) at \(t_{v}=1.5\) s, and \(S_{f}=120\) at \(t_{s}=3\) s, and the sixth order polynomial from [9] is:  
\[\theta(t)=-9.22t^{6}+85.19t^{5}-265.56t^{4}+282.22t^{3}+30 \tag{27}\]  
Plotting this trajectory in MATLAB, it is shown that the trajectory obtains 185.4 degrees as a maximum value, and \(t_{v}=1.5\) the trajectory reaches \(\theta_{v}=180\) degrees. As shown in Figure 7, the trajectory presents a maximum value at \(t=1.74\) s. The method proposed in this work tried to reduce this oscillation, so this one would not be greater than 181 degrees. The algorithm IOSTV \((\zeta)_{v}\)-HF was run several times until it obtained some solutions that fulfilled the requirements. First, it tested the trajectory result by finding values on the parameters to obtain a continuous velocity, for example, by using the parameters \(n_{v}=1.91\), \(a_{v}=0.8734\), \(\zeta_{vi}=-6.0657\ldots\), \(n=81.7467\), \(a=2.4\), and \(\zeta_{i}=-0.9204\) in the iteration number 12; the following profiles are obtained and are shown in Figure 8.  
Figure 7: Trajectory obtained from the sixth-order polynomial. Regarding the acceleration and jerk, it can be observed that they were discontinuous in \(t_{v}\). However, these do not present discontinuities at the start and end, like the method presented in [9]. Therefore, the discontinuities have been reduced to one by using IOSTV \((\zeta)_{v}\)-HF, and these are also kept finite. Additionally, the differences between the acceleration at \(t\) approaches to the left-approximation and the right-approximation to \(t_{v}\) are insignificant, about 219.824 degrees/s\({}^{2}\) of difference, and obtaining 180.4 degrees as a leading position in the trajectory. Then, Figure 9 shows a trajectory with a continuous acceleration but a discontinuous velocity and jerk using the following values \(n_{v}=5\), \(a_{v}=0.5865\), and \(\zeta_{v_{l}}=-3.0399\) and \(n=5\), \(a=1.95\), and \(\zeta_{i}=-2.6193\) in the iteration number 28.