## 4 Initial Trajectory Generation
### Comparison of Several Trajectory Techniques  
In the preceding subsections, several types of trajectory parameterization techniques were considered. We have selected three different types of trajectory parameterization techniques for this comparison: piece-wise-polynomials technique, fitting based on a sequence of points, and the third is uniform B-spline-based technique. The objective of piece-wise-polynomials is to find optimal polynomial coefficients [53] or end-derivatives [35] of consecutive segments, whereas the objective of the third technique is to find a set of points satisfying the provided constraints [57]. A comparison of how velocity, acceleration, jerk, and snap are varied for selected techniques in terms of mean, standard deviation (std), min, and max for the same a set of control points and knot vector is present below. The considered knot vector and control points are:  
\[\begin{split} p_{ctrl}&=[[0.011,-0.0329,2.017],[1.867,3.408,1.6],[7.514,5.715,3.735],\\ &[8.410,0.911,1.600],[6.902,-5.531,4.306],[1.899,-6.680,3.082],\\ &[-2.302,-0.611,5.375]]\\ \end{split} \tag{36}\]  
\[\begin{split} p_{knot}&=[0.0,5.0,12.0,18.0,26.0,31. 0,40].\end{split}\]  
Each approach has its own set of parameters to fine-tune for obtaining an optimal trajectory. The generated trajectories are shown in Figure 5 with different configuration setups (with different parameter sets). Figure 6 shows how the derivatives up to the fourth change over time in each direction, i.e., x, y, z, separately for each technique. When looking at the derivatives of each method, it is clear that smoothness, which is the main point to be considered for motion planning, is higher in both B-spline and minimum-snap compared to CHOMP. Since uniform B-spline is used in this comparison, smoothness changes of each derivative between B-spline and minimum-snap cannot be compared directly due to time allocation when generating the trajectories. Hence, minimum-snap trajectory smoothness  
Figure 4: Trajectory generation using uniform B-spline. The smoothness of the curve is dependent on the degree of the B-spline. The trajectory passes precisely through the given control points at the degree equal to 1, as depicted in light blue color. The smoothness of the trajectory increases with the order of the B-spline.  
can be changed, optimizing the time allocation process [35]. On the contrary, such a time allocation process is not necessary for a uniform B-spline. However, control points are interpolated appropriately to generate a continuous and smooth trajectory.  
We varied the parameters for each approach appropriately and estimated the mean, standard deviation, maximum, and minimum of the velocity, acceleration, jerk, and snap profiles. The results are shown in Table 1. The results clearly show that the consistency of the trajectory depends on the parameters used to parameterize the trajectory. Therefore, selecting the appropriate parameter set for a given task is of utmost importance, as can be seen by looking at the statistical properties (mean, standard deviation, minimum, and maximum) of the higher-order derivatives, such as velocity, acceleration, jerk, and snap. As described in the previous paragraph, the time allocation process directly affects the parameter selection for minimum-snap. Further, the optimal polynomial coefficients process depends on time allocation, as given in (13). On the other hand, the Poly-traj [35] generation process has fewer parameters to be optimized, since it uses free-end derivatives of each segment. Hence, the latter technique is faster than minimum-snap.  
Figure 5: Generated trajectories using three different approaches for a given sequence of control points and knot vector (36).  
Figure 6: Changes of position, velocity, acceleration, jerk, and snap profiles over time for the provided control points sequence and knot vector (36).