# The Use of Asymmetric Polynomial Profiles for Planning a Smooth Trajectory
## 4 Discussion  
Table 4 presents a comparison of the results included in papers [37,38] and in this paper [*] concerning the extreme values of the time realization of the trajectories modelled by means of higher degree polynomials. In papers [37,38], there were obtained the results of trajectories planning for which the initial velocity \(\
u_{\rm begin}\) and the final velocity \(\
u_{\rm end}\) were equal to zero. There were analyzed the polynomials: 2-1-2, 2-3-2, 3-1-3, 2-5-2, 3-3-3 and 4-1-4. In this paper, the velocities: initial \(\
u_{\rm begin}\) and final \(\
u_{\rm end}\) were different from zero and the analyzed polynomials were: 2-1-2, 3-1-2, 2-1-3, 3-1-3, 2-3-2, 3-3-2, 2-3-3, and 3-3-3. Based on the modelling results available in [37,38] as well as in this paper [*], it can be stated that at the constraint of both velocity and acceleration for the polynomial 2-1-2 there were obtained the minimal values of jerk. However, in the case of jerk as constraint there is obtained the shortest time \(t_{\rm end}\) for the analyzed polynomial; however, then the extreme values of velocity and acceleration are greater (than those of the polynomials). Similar statements can be made for the other polynomials, whereby they can refer to the results presented in [37,38] or only in this paper [*].  
In the case of polynomials concatenation during the trajectory planning with the intermediate points (Figure 5a-e), the maximal velocity \(\
u_{\rm max}=1\) m\(\cdot\)s\({}^{-1}\) was obtained in the interval t\(\in\)\(\langle\)t\({}_{1}\), t\({}_{2}\)\(\rangle\) in which the polynomial 3-1-3 was used. The maximal value of acceleration (deceleration) \(\mid\)a\({}_{\rm max}\)\(\mid\) = 2 m\(\cdot\)s\({}^{-2}\) was obtained in the interval t \(\in\)\(\langle\)0, t\({}_{1}\)\(\rangle\) when the polynomial 2-1-3 was used and in the interval t \(\in\)\(\langle\)t\({}_{2}\), \(t_{\rm end}\)\(\rangle\) when there was applied the polynomial 2-1-2. The largest, absolute jerk value \(\mid\)j\({}_{\rm max}\)\(\mid\) = 10 m\(\cdot\)s\({}^{-3}\) was not obtained. Its maximum was 9.858 m\(\cdot\)s\({}^{-3}\) being in the first-time interval t \(\in\)\(\langle\)0, t\({}_{1}\)\(\rangle\). The displacement, velocity, acceleration profile is continuous in the whole motion range. However, the snap course is discontinuous for the time t = t\({}_{2}\). This is due to the use of the polynomial 2-1-2, whose multiplicity of the first root m\({}_{1}\) = 2, for t \(\in\)\(\langle\)t\({}_{2}\), \(t_{\rm end}\)\(\rangle\). For preservation of snap profile continuity, the polynomial was chosen from: 3-1-2, 3-1-3, 3-3-2 or 3-3-3 for which the multiplicity of the first root m\({}_{1}\) = 3. Moreover, in order to obtain a zero values snap at the motion beginning and end, the first root multiplicity of the polynomial used in the interval t \(\in\)\(\langle\)0, t\({}_{1}\)\(\rangle\) should be equal to m\({}_{1}\) = 3 and the third root multiplicity of the polynomial used in the interval t \(\in\)\(\langle\)t\({}_{2}\), \(t_{\rm end}\)\(\rangle\) should be equal to m\({}_{3}\) = 3.  
Figure 7: Profiles of kinematic quantities for the polynomial 2-1-2 and AS-curve: (**a**) displacement, (**b**) velocity, (**c**) acceleration, (**d**) jerk, (**e**) snap.  
Comparing the corresponding courses of the polynomial 2-1-2 and AS-curve, there were observed slight differences in the course of displacement, velocity and acceleration (Figure 7a-c). The greatest differences were found during the change of motion phase or in the terminal motion stage. Application of the AS-curve linear function for the jerk profile resulted in the discontinuities in the snap course as its profile is the staircase function (Figure 7e).