# The Use of Asymmetric Polynomial Profiles for Planning a Smooth Trajectory
## 3 Results  
### Results of the Simulation for a Single Constraint  
The simulation was performed for the rectilinear motion, whereby in all cases the initial and terminal values of displacement values were: \(\mathrm{s_{begin}}=0.3\) m and \(\mathrm{s_{end}}=1.3\) m, but the initial and terminal values of velocity were: \(\mathrm{v_{begin}}=0.2\) m\(\cdot\)s\({}^{-1}\) and \(\mathrm{v_{end}}=0.5\) m\(\cdot\)s\({}^{-1}\). Figure 1 presents the profiles of displacement s(t), velocity s(l)(t), acceleration s(2)(t) and jerk s(3)(t) of the analyzed polynomials at the velocity constraint. The maximal velocity value was assumed to be \(\mathrm{v_{max}}=1\) m\(\cdot\)s\({}^{-1}\).  
In the case of velocity constraint (Figure 1a-d), the time of trajectory realization \(\mathrm{t_{end}}\) was in the range 1.344 to 1.659 s. The shortest time was achieved using the polynomial 2-3-2, but the longest time was for the polynomial 3-1-2. The value of the maximal velocity for all polynomials was equal to the assumed one being \(\mathrm{v_{max}}=1\) m\(\cdot\)s\({}^{-1}\). The largest, absolute value of acceleration (deceleration) was in the range from 1. 537 to 2.887 m\(\cdot\)s\({}^{-2}\). The minimal value was obtained using the polynomial 3-1-2, and the largest one for the polynomial 2-3-3. The maximal absolute value of jerk was in the range from 6.359 to 22.633 m\(\cdot\)s\({}^{-3}\). The smallest value was obtained for the polynomial 2-1-2, and the largest one for the polynomial 2-3-3.  
Figure 2 presents the profiles of displacement, velocity, acceleration and jerk for the analyzed polynomials with the acceleration constraint. The maximal absolute value of acceleration was assumed to be \(\mathrm{\left|\mathrm{a_{max}}\right|=2\) m\(\cdot\)s\({}^{-2}\).  
Figure 1: Profiles of kinematic quantities for the trajectories with the velocity constraint: (**a**) displacement; (**b**) velocity; (**c**) acceleration; (**d**) jerk.  
In the case of the acceleration constraint (Figure 2a-d), the time of trajectory realization was in the range of 1.454 to 1.566 s. The shortest time was obtained using the polynomial 2-1-2, but the longest one in the case of the polynomial 3-1-3. The maximal value of velocity was in the range 0.884 to 1.138 m\(\cdot\)s\({}^{-1}\). The smallest value was obtained for the polynomial 2-3-3, and the largest one for the polynomial 3-1-2. The value of maximal acceleration (deceleration) was equal to the assumed one \(\mid\)a\({}_{\max}\)\(\mid\) = 2 m\(\cdot\)s\({}^{-2}\). The largest absolute value of jerk was in the range 8.412 to 13.683 m\(\cdot\)s\({}^{-3}\). The smallest value was in the case of the polynomial 2-1-2, but the maximal one for the polynomial 2-3-3.  
Figure 3 presents the profiles of displacement, velocity, acceleration and jerk profiles for the analyzed polynomials with the jerk constraint. The maximal absolute value of jerk was assumed to be \(\mid\)j\({}_{\max}\)\(\mid\) = 10 m\(\cdot\)s\({}^{-3}\).  
At the jerk constraint (Figure 3a-d), the time t\({}_{\rm end}\) of trajectory realization was in the range from 1.392 to 1.673 s. The shortest time was obtained using the polynomial 2-1-2, but the longest one for the polynomial 2-3-3. The maximal velocity value was in the range from 0.761 to 1.164 m\(\cdot\)s\({}^{-1}\). The smallest value was obtained for the polynomial 2-3-3, but the largest one for the polynomial 2-1-2. The maximal acceleration was in the range from 1.605 to 2.253 m\(\cdot\)s\({}^{-2}\). The smallest value was when the polynomial 2-3-3 was used but the largest one in the case for the polynomial 2-1-2. According to the assumption the maximal absolute jerk value being \(\mid\)j\({}_{\max}\)\(\mid\) = 10 m\(\cdot\)s\({}^{-3}\) was obtained for all polynomials.  
Figure 2: Profiles of kinematic quantities for the trajectories with the acceleration constraint: (**a**) displacement; (**b**) velocity; (**c**) acceleration; (**d**) jerk.