## 3 Results  
To verify the developed algorithms, simulation modeling in the MATLAB Simulink environment was carried out. We used the Euler method with a constant step of \(10^{-2}\) for numerical integration. The results of two experiments using a three-block tracking differentiator (32) are presented below. The first experiment demonstrates the smoothing  
Figure 1: Flowchart of the trajectory planning process. processes of primitive trajectories for the robot's center of mass under different sets of constraints (4). In the second experiment, for one of the smoothed trajectories, a motion trajectory is simulated considering the vehicle dimensions. In the first experiment, the design constraints of two civilian wheeled robots moving in an urban environment were used to set the gains of the tracking differentiator. Based on the inequalities (29), the following gains of the tracking differentiator (32) were adopted for the first robot with a velocity of up to 2.3 m/s:  
\[p_{1}=2.3,\;l_{1}=1.4,\;p_{2}=20,\;l_{2}=0.5,\;p_{3}=250,\;l_{3}=0.3. \tag{43}\]  
For the second robot with a developing velocity of up to 1.9 m/s, the following gains of the tracking differentiator (32) were adopted:  
\[p_{1}=1.9,\;l_{1}=1.4,\;p_{2}=16,\;l_{2}=0.5,\;p_{3}=160,\;l_{3}=0.3. \tag{44}\]  
The robots' motion was planned along a rhombus defined in a fixed Cartesian coordinate system. Thirteen waypoints \((x_{i},y_{i})\) were randomly marked on the sides of the rhombus. Based on them, we generated two sets of 3D waypoint sequences \((x_{i},\;y_{i},\;t_{i})\), \(i=\overline{1,13}\) (38), which differ only in the values of time \(t_{i}\), by which the average velocity of motion on each path segment is defined and limited. In the first sequence, the maximum velocity was limited by 2.5 m/s and in the second by 2 m/s. We purposely used values slightly higher than the above-mentioned robot velocities. Joining neighboring pairs of points by segments, we obtain two primitive non-smooth 3D trajectories (39): for the first sequence at \(v_{\max}=2.5\) m/s in the form of  
\[ \begin{array}{l}\chi_{11}=-t+9,\chi_{12}=t,t\in[0;2);\\ \chi_{11}=-2.5t+12,\chi_{12}=2.5t-3,t\in[2;4);\\ \chi_{11}=-t+6,\chi_{12}=t+3,t\in[4;6);\\ \chi_{11}=t-6,\chi_{12}=t+3,t\in[6;8);\\ \chi_{11}=2.5t-18,\chi_{12}=2.5t-9,t\in[8;10);\\ \chi_{11}=t-3,\chi_{12}=t+6,t\in[10;12);\\ \chi_{11}=t-3,\chi_{12}=-t+30,t\in[12;14);\\ \chi_{11}=2.5t-24,\chi_{12}=-2.5t+51,t\in[14;16);\\ \chi_{11}=t,\chi_{12}=-t+27,t\in[16;18);\\ \chi_{11}=-t+36,\chi_{12}=-t+27,t\in[18;20);\\ \chi_{11}=-2.5t+66,\chi_{12}=-t+24,t\in[20;22);\\ \chi_{11}=-t+33,\chi_{12}=-t+24,t\in[22;24);\\ \chi_{11}=-t+33,\chi_{12}=t-24,t\geq 24,\chi_{ij}[m],t[s];\end{array}  \tag{45}\]  
for the second sequence at \(v_{\max}=2\) m/s in the form of  
\[ \begin{array}{l}\chi_{11}=-t+9,\chi_{12}=t,t\in[0;2);\\ \chi_{11}=-2t+11,\chi_{12}=2t-2,t\in[2;4.5);\\ \chi_{11}=-t+6.5,\chi_{12}=t+2.5,t\in[4.5;6.5);\\ \chi_{11}=t-6.5,\chi_{12}=t+2.5,t\in[6.5;8.5);\\ \chi_{11}=2t-15,\chi_{12}=2t-6,t\in[8.5;11);\\ \chi_{11}=t-4,\chi_{12}=t+5,t\in[11;13);\\ \chi_{11}=t-4,\chi_{12}=-t+31,t\in[13;15);\\ \chi_{11}=2t-19,\chi_{12}=-2t+46,t\in[15;17.5);\\ \chi_{11}=t-1.5,\chi_{12}=-t+28.5,t\in[17.5;19.5);\\ \chi_{11}=-t+37.5,\chi_{12}=-t+28.5,t\in[19.5;21.5);\\ \chi_{11}=-2t+59,\chi_{12}=-2t+50,t\in[21.5;24);\\ \chi_{11}=-t+35,\chi_{12}=-t+26,t\in[24;26);\\ \chi_{11}=-t+35,\chi_{12}=t-26,t\geq 26,\chi_{ij}[m],t[s].\end{array}  \tag{46}\]  
Figure 4: Plots of velocities \(z_{21}(t),z_{22}(t)\) of smoothed trajectory (46). In (**a**), for the tracking differentiator (32), (43). In (**b**), for the tracking differentiator (32), (44). Figure 5: Plots of accelerations \(z_{31}(t),z_{32}(t)\) of smoothed trajectory (45). In (**a**), for the tracking differentiator (32), (43). In (**b**), for the tracking differentiator (32), (44). Figure 6: Plots of accelerations \(z_{31}(t),z_{32}(t)\) of smoothed trajectory (46). In (**a**), for the tracking differentiator (32), (43). In (**b**), for the tracking differentiator (32), (44).