# The Use of Asymmetric Polynomial Profiles for Planning a Smooth Trajectory
## 5 Conclusions  
The main advantage of the method of trajectory planning using root multiplicities of polynomials is the continuity of successive derivatives of displacement with respect to time, and obtaining zero values of acceleration and jerk for the time \(t=t_{begin}\) and \(t=t_{end}\). Continuity of displacement, velocity, acceleration, jerk and even snap function is preserved when planning PTP trajectories and obtainable when planning trajectories with intermediate points. Based on the analysis, further conclusions can be drawn:  
1. The acceleration profiles presented in the paper, created using root multiplicities of polynomial, are simple in terms of mathematical description. Determination of other kinematic quantities based on them also does not cause computational difficulties. 2. As a result of the use of asymmetric profiles, for which the assumed initial and final velocity values are arbitrary, it is possible to concatenate the analyzed polynomial profiles, allowing planning of trajectories with intermediate points. 3. Application of all polynomials analyzed in the paper for the formation of motion trajectories passing through via-points, ensures continuity of acceleration and jerk in the whole motion range and using the polynomials 3-1-3 and 3-3-3, it is possible to obtain continuity of snap and at the same time its zero values for the time \(t=t_{begin}\) and \(t=t_{end}\). \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**P Polynomial**} & \multicolumn{3}{c}{**Velocity Constraint**} & \multicolumn{3}{c}{**Acceleration Constraint**} & \multicolumn{3}{c}{**Jerk Constraint**} \\ \cline{2-10}  & & \(t_{end}\) & \(a\) & \(j\) & \(t_{end}\) & \(v\) & \(j\) & \(t_{end}\) & \(v\) & \(a\) \\ \hline \multirow{2}{*}{**2-1-2**} & _min_ & & & [37,38],[*] & [*] & & [37,38],[*] & [37,38],[*] & & \\  & _max_ & & & & & & & [37,38],[*] & [37,38],[*] & \\ \hline \multirow{2}{*}{**3-1-2**} & _min_ & & [*] & & & & & & \\  & _max_ & & & & & & & & \\ \hline \multirow{2}{*}{**2-1-3**} & _min_ & & & & & & & & \\  & _max_ & & & & & & & & \\ \hline \multirow{2}{*}{**3-1-3**} & _min_ & & & & & & & & \\  & _max_ & & & & & & & & \\ \hline \multirow{2}{*}{**2-3-2**} & _min_ & & & & & & & & \\  & _max_ & & & & & & & & \\ \hline \multirow{2}{*}{**3-3-3**} & _min_ & & & & & & & & \\  & _max_ & & & & & & & & \\ \hline \multirow{2}{*}{**3-3-3**} & _min_ & & & & & & & & \\  & _max_ & & & & & & & & \\ \hline \multirow{2}{*}{**2-5-2**} & _min_ & [37,38] & & & [37,38] & & [37,38] & & [37,38] & [37,38] \\  & _max_ & & [37,38] & [37,38] & & & [37,38] & & & \\ \hline \multirow{2}{*}{**4-1-4**} & _min_ & & [37,38] & & & & & & & \\  & _max_ & [37,38] & & [37,38] & & [37,38] & & & & \\ \hline \hline \end{tabular}  
* [*]–in this paper.  
\end{table} 
Table 4: Literature references on the occurrence of extreme kinematic parameters for polynomial profiles with constraints. 4. Exploitation of the analyzed polynomials enables planning the motion paths for which there can be fulfilled (except for the assumed constraints) additional requirements, including, e.g., the shortest time, minimal acceleration or jerk, etc. 5. The possibility of introducing constraints on speed, acceleration and jerk can facilitate, already at the initial planning stage, the adaptation of the trajectory to, for example, the known movement capabilities of the kinematic chain and the properties of the drive system. 6. The acceleration function along with easily derived displacement, velocity and jerk functions are ready-made motion patterns and can be used for trajectory planning: effectors of manipulators, toolheads of the CNC machines as well as mobile robots and autonomous vehicles. It is also advisable to conduct further work on the use of higher-degree polynomials for motion trajectory planning.