## 2 Materials and Methods
### Design of Tracking Differentiators
#### 2.3.4 Some Aspects of Designing Paths and Polygons  
The use of tracking differentiators can greatly simplify the computational aspect of planning achievable trajectories offline on a polygon with stationary obstacles. Let a wheeled robot have a primary operating scenario in the form of a sequence of 3D points defined in a fixed Cartesian coordinate system \(Oxy\), considering time  
\[(x_{i},y_{i},t_{i}),i=\overline{1,m},t_{i}<t_{i+1},t_{1}=0;t_{m}=T. \tag{38}\]  
Connecting all neighboring pairs of point \((x_{i},y_{i},t_{i})\) and \((x_{i+1},y_{i+1},t_{i+1})\) by segments  
\[\frac{\chi_{11}-x_{i}}{x_{i+1}-x_{i}}=\frac{\chi_{12}-y_{i}}{y_{i+1}-y_{i}}= \frac{t-t_{i}}{t_{i+1}-t_{i}},t\in[t_{i};t_{i+1}),i=\overline{1,m-1},\]  
we obtain a primitive non-smooth 3D trajectory and, accordingly, the primary reference influences for both output variables \(x(t),y(t)\) of the system (1):  
\[\left\{ \begin{array}{l}\chi_{11}=\frac{(x_{i+1}-x_{i})t+x_{i}t_{i+1}-x_{i+1} t_{i}}{t_{i+1}-t_{i}},\\ \chi_{12}=\frac{(y_{i+1}-y_{i})t+y_{i}t_{i+1}-y_{i+1}t_{i}}{t_{i+1}-t_{i}},t\in [t_{i};t_{i+1}),i=\overline{1,m-1}.\end{array} \right. \tag{39}\]  
In standard approaches, special analytical approximations are used to smooth the angles of the composite trajectory (39) [11]. In this case, the problem of ensuring the specified constraints on the velocity and acceleration of the robot requires additional algorithmization. If the singular points are sufficiently numerous, analytical smoothing methods provide a large computational load, which makes it difficult to use them in real time. In addition, composite analytical reference influences take up much memory when stored on the on-board computer. The application of dynamic smoothing algorithms for primitive trajectories (39) completely eliminates these problems. Only a sequence of points (38) is stored in the on-board computer memory, and smoothing of the corresponding primitive trajectories (39) is performed in real time using a tracking differentiator (35), (37) of the required dimensionality. Note that this method is not intended for solving terminal problems and does not provide an exact fit to the reference points. However, it can be used for offline simulation on a compressed time scale. We can quickly correct the smoothed trajectory obtained after the first run of the algorithm and achieve the desired result by correcting the coordinates of particular points (38). At the same time, visual results will be obtained quickly and without cumbersome analytical calculations. For such a process, it is sufficient to use a single-block tracking differentiator (33). Let us consider some aspects of planning a safe route for an oversized wheeled vehicle on a polygon with stationary obstacles. The opposite task is to arrange objects on the polygon (factory workshop, warehouse) so that the robot can safely fulfill the operating scenario. Therefore, it is necessary to consider the dimensions of the vehicle when planning the polygon. The developed dynamic smoothing algorithm can be applied to graphically represent the position change not only for the center of mass of the wheeled platform but also for its corner points. In this problem, it is better (but not necessarily) to use a two-block tracking differentiator  
\[ \begin{array}{l}\dot{z}_{1}=z_{2},\dot{z}_{2}=-p_{2}\sigma(l_{2}(z_{2}+p_{1} \sigma(l_{1}(z_{1}-\chi_{1})))),\\ z_{1,2}\in R^{2},z_{1}(0)=\chi_{1}(0),z_{2}(0)=\overrightarrow{0}.\end{array}  \tag{40}\]  
Its parameters are determined from the procedure (37) at \(n=1\) and have the form:  
\[ \begin{array}{l}p_{2}=Y_{2},l_{2}=Y_{3}/p_{2}^{2}=Y_{3}/\gamma_{2}^{2},\\ p_{1}=Y_{1}-2.2/l_{2}=Y_{1}-2.2Y_{2}^{2}/Y_{3}\\ l_{1}=(0.8p_{2}-\beta_{1})/p_{1}^{2}=(0.8Y_{2}-\beta_{1})/\big{[}Y_{1}-2.2Y_{2}^ {2}/Y_{3}\big{]}^{2}.\end{array}  \tag{41}\]  
Let us present the corresponding computations for a symmetrical rectangular wheeled platform whose center of mass is located at the intersection of diagonals in the middle of the center line.