# The Use of Asymmetric Polynomial Profiles for Planning a Smooth Trajectory
## 3 Results
### Simulation Results for a Few Kinematic Constraints  
Simulations were performed for the data presented in the previous subsection, i.e., \(\mathsf{s}_{\text{begin}}=0.3\text{ m}\text{, }\mathsf{s}_{\text{end}}=1.3\text{ m}\text{, }\mathsf{v}_{\text{begin}}=0.2\text{ m}\cdot \mathsf{s}^{-1}\text{, }\mathsf{v}_{\text{end}}=0.5\text{ m}\cdot \mathsf{s}^{-1}\text{, with the simultaneous constraints of: velocity, acceleration and jerk (\(\mathsf{v}_{\text{max}}=1\text{ m}\cdot\mathsf{s}^{-1}\), \(\mathsf{a}_{\text{max}}\mid 1=2\text{ m}\cdot\mathsf{s}^{-2}\), \(\mathsf{i}_{\text{max}}\mid 1=10\text{ m}\cdot\mathsf{s}^{-3}\)). Table 3 presents the time of trajectory realization with the velocity, acceleration and jerk constraints for the analyzed polynomials. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Polynomial**} & \multicolumn{4}{c}{**Time \(\mathsf{t}_{\text{end}}\) [s]**} \\ \cline{2-4}  & **With the Velocity** & **With the Acceleration** & **With the Jerk** \\
**Constraint** & **Constraint** & **Constraint** & **Constraint** \\ \hline
2-1-2 & 1.557 & 1.454 & 1.392 \\ \hline
3-1-2 & 1.659 & 1.510 & 1.504 \\ \hline
2-1-3 & 1.558 & 1.535 & 1.478 \\ \hline
3-1-3 & 1.640 & 1.566 & 1.539 \\ \hline
2-3-2 & 1.344 & 1.467 & 1.543 \\ \hline
3-3-2 & 1.412 & 1.467 & 1.552 \\ \hline
2-3-3 & 1.357 & 1.547 & 1.673 \\ \hline
3-3-3 & 1.412 & 1.533 & 1.542 \\ \hline \end{tabular}
\end{table} 
Table 3: The time of motion with the constraints of velocity, acceleration and jerk. Figure 3: Profiles of kinematic quantities for the trajectories with the jerk constraint: (**a**) displacement; (**b**) velocity; (**c**) acceleration; (**d**) jerk. In the case of the polynomial 2-1-2, the calculated time of trajectory realization with the velocity, acceleration and jerk constraints was: 1.557 s, 1.454 s, 1.392 s, respectively. As the longest motion time \(\mathrm{t_{end}}\) = 1.557 s is obtained at the velocity constraint, the largest velocity is found for the polynomial 2-1-2 but at the acceleration and jerk constraints as well as \(\mathrm{t_{end}}\) = 1.557 s the maximal values are not obtained. The situation is similar in the case of the polynomials: 3-1-2, 2-1-3, 3-1-3. However, for the polynomials 2-3-2, 3-3-2, 2-3-3 and 3-3-3, the longest motion time is obtained at the jerk constraint. Therefore, for these polynomials there will be obtained the maximal jerk but the velocity and the absolute acceleration value will not reach the assumed values. The profiles of position, velocity, acceleration and jerk with the simultaneous constraints of velocity, acceleration and jerk for the analyzed polynomials are presented in Figure 4.  
With the simultaneous velocity, acceleration and jerk constraints (Figure 4a-d), the time of trajectory realization was in the range 1.542 to 1.673 s. The shortest time was obtained when the polynomial 3-3-3, was used and the longest time was in the case of the polynomial 2-3-3. The largest value of velocity being \(\mathrm{v_{max}}\) = 1 m\(\cdot\)s\({}^{-1}\) was obtained for the polynomials 2-1-2, 3-1-2, 2-1-3 and 3-1-3 but in the case of the other analyzed polynomials the maximal value was not obtained. The largest absolute value of acceleration (deceleration) \(\mid\mathrm{a_{max}}\mid\) = 2 m\(\cdot\)s\({}^{-2}\) was not obtained. The maximal value being 1.971 m\(\cdot\)s\({}^{-2}\) was obtained using the polynomial 3-3-3 and the smallest one 1.537 m\(\cdot\)s\({}^{-2}\) in the case of the polynomial 3-1-2. The largest absolute jerk value \(\mid\mathrm{j_{max}}\mid\) = 10 m\(\cdot\)s\({}^{-3}\) was obtained for  
Figure 4: Profiles of kinematic quantities for the trajectories with velocity, acceleration and jerk constraints: (**a**) displacement; (**b**) velocity; (**c**) acceleration; (**d**) jerk. the polynomials 2-3-2, 3-3-2, 2-3-3 and 3-3-3. However, it should be pointed out that this takes place in the acceleration phase for the polynomials 2-3-2, 2-3-3 and 3-3-3 but in the breaking phase for the polynomial 3-3-2.