# The Use of Asymmetric Polynomial Profiles for Planning a Smooth Trajectory
## 2 Materials and Methods  
### Asymmetric Acceleration Profile  
The calculation of the polynomial coefficients in the form of power expansion comes across some difficulties [9]. Therefore, the polynomial product was applied with one coefficient, for which its profile can be easily formed using the properties of the root multiplicity [33; 37; 38]. The general form of the nth degree polynomial built exploiting the j zero places is as follows:  
\[\mathrm{p}(\mathrm{t})=\mathrm{p}\cdot(\mathrm{t}-\mathrm{t}_{1})^{\mathrm{m} _{1}}\cdot(\mathrm{t}-\mathrm{t}_{2})^{\mathrm{m}_{2}}\cdot\ldots\cdot( \mathrm{t}-\mathrm{t})^{\mathrm{m}_{j}} \tag{1}\] \begin{table}
\begin{tabular}{c c} \hline
**Description of the Trajectory** & **Examples of Publications/Author and Year** \\ \hline Trygonometric splines & [PERSON] 2000 [2], [PERSON] and [PERSON] 2003 [3]. \\ \hline \multirow{4}{*}{Algebraic splines} & [PERSON] et al. 2000 [16], [PERSON] and [PERSON] 2000 [17], [PERSON] 2000 [2], [PERSON] and [PERSON] 2003 [3], [PERSON] and [PERSON] 2003 [5], [PERSON] et al. 2006 [20], [PERSON] i [PERSON] 2008 [22], [PERSON] and [PERSON] 2010 [23], [PERSON] et al. 2020 [6]. \\ \hline \multirow{2}{*}{B-splines} & [PERSON] and [PERSON] 2002 [4], [PERSON] and [PERSON] 2003 [3], [PERSON] and [PERSON] 2007 [21], [PERSON] and [PERSON] 2010 [23], [PERSON] et al. 2013 [28], [PERSON] et al., 2022 [30]. \\ \hline \multirow{4}{*}{S-curve} & [PERSON] 2000 [18], [PERSON] et al. 2005 [19], [PERSON] et al. 2008 [24], [PERSON] et al. 2009 [7], [PERSON] and [PERSON] 2010 [8], [PERSON] et al. 2011 [9], [PERSON] and [PERSON] 2012 [25], [PERSON] et al. 2012 [27], [PERSON] et al. 2014 [10], [PERSON] and [PERSON] 2015 [11], [PERSON] 2016 [12], [PERSON] and [PERSON] 2019 [26], [PERSON] et al. 2019 [13], [PERSON] et al. 2020 [29], Alpers 2021 [14], Alpers 2022 [15], [PERSON] et al. 2022 [31]. \\ \hline \multirow{4}{*}{Polynomials} & [PERSON] and [PERSON] 2000 [32], [PERSON] and [PERSON] 2009 [33], [PERSON] and [PERSON] 2013 [34], [PERSON] 2014 [35], [PERSON] et al. 2015 [36], [PERSON], 2016 [37], [PERSON] et al. 2018 [43], [PERSON] and [PERSON] 2019 [40], [PERSON] et al. 2019 [41], [PERSON] and [PERSON] 2019 [42], [PERSON], 2020 [38]. \\ \hline \end{tabular}
\end{table} 
Table 1: Collation of publications using typical parametrization of the trajectory. whereby the polynomial degree:  
\[n=\sum_{i=1}^{j}m_{i} \tag{2}\]  
where: p--the polynomial coefficient, t--the motion time, t\({}_{1}\), t\({}_{2}\),..., t--the successive zero places, m\({}_{1}\), m\({}_{2}\),..., m\({}_{j}\)--the multiplicity of successive zero places, j--the number of polynomial zero places, n--the polynomial degree. In this paper the product form of the polynomial was used for the formation of the acceleration profile. The displacement function was denoted s(t) and the dependences describing: velocity, acceleration, jerk and snap being successive derivatives in relation to the displacement function time were denoted: s\({}^{(1)}\)(t), s\({}^{(2)}\)(t), s\({}^{(3)}\)(t), s\({}^{(4)}\)(t), respectively. It was assumed that the acceleration profile s\({}^{(2)}\)(t):  
1. Should be in the interval t \(\in\langle 0,t_{\text{end}}\rangle\);
2. Will be asymmetric;
3. Should be composed of two motion phases--the acceleration phase, for which s\({}^{(2)}\)(t) > 0 and the braking phase, for which s\({}^{(2)}\)(t) < 0, whereby the sequence of acceleration and braking phases is optional. The first motion phase is in the interval t \(\in\langle 0,k\cdot t_{\text{end}}\rangle\) but the other one is in the interval t \(\in\langle k\cdot t_{\text{end}},t_{\text{end}}\rangle\). The value of coefficient k, called the asymmetry coefficient, will be placed in the interval k \(\in\langle 0,1\rangle\);
4. Should be tangent to the time axis for the first t = 0 and the last t = t\({}_{\text{end}}\) of the zero place.