## 2 Materials and Methods
### Design of Tracking Differentiators
#### 2.3.2 Designing a Three-Block Tracking Differentiator  
Let the reference influences \(\chi_{1}(t)=\left(\chi_{11}(t),\chi_{12}(t)\right)^{\text{T}}\) enter the information and control the on-board system of the robot in real time in the form of a deterministic vector signal. Their analytical description is unknown, and hence, the current values of reference signal derivatives are also unknown. It is assumed that the reference trajectory satisfies the safety requirements, but its realizable conditions (15) and (16) are partially fulfilled. Namely, the time functions \(\chi_{11}(t),\chi_{12}(t)\) are either continuous but non-smooth or piecewise continuous. On the continuity intervals, the derivatives of these functions do not exceed the allowed values (16). The number of special points (junctions or points of the discontinuity of the first kind) is bounded. At these points, there exist bounded left and right first derivatives. Within the framework of the method used, an additional condition of type (14) is introduced:  
\[\left\|\dot{\chi}_{1}(t)\right\|\leq X_{1}<0.8Y_{1},t\in[0;T]. \tag{17}\]  
In real time, the problems of smoothing the vector signal t and restoring its first and second derivatives are posed considering the constraints of the robot under consideration (4). To solve these problems, we propose using a dynamic model (tracking differentiator), which has a canonical form similar to (2):  
\[\dot{z}_{1}=z_{2},\dot{z}_{2}=z_{3},\dot{z}_{3}=w, \tag{18}\]  
where \(z_{1}\in R^{2},z_{2}\in R^{2},z_{3}\in R^{2}\) are vector variables of the differentiator, which are analogs of the robot position \(y_{1}(t)=(y_{11}=x,y_{12}=y)^{T}\), its velocity \(\dot{y}_{1}(t)\), and its acceleration \(\ddot{y}_{1}(t)\), respectively. The analogs of the controls \(u_{1},u_{2}\) of the canonical system (2) are the corrective sigmoidal influences \(w\in R^{2}\), by means of which it is necessary to ensure that the output variables of the differentiator \(z_{1}(t)\) track the external non-smooth signals \(\chi_{1}(t)\) with some accuracy. Tracking accuracy depends on the constraints of a particular robot (4), which need to be ensured in the closed-loop system (18) by choosing the gains of corrective influences:  
\[\left\|z_{1}^{(i)}(t)\right\|\leq Y_{i},t\in[0;T],i=\overline{1,4}. \tag{19}\]  
Under these conditions, the differentiator variables (18) generate the achievable trajectory in signal form and its first and second derivatives. These variables are new reference influences and are used to generate control influences \(u_{1},u_{2}\) in the tracking system of the mobile robot. To design a tracking differentiator (18) with the indicated features by means of non-generated replacement of variables, we proceed to a new coordinate frame, which is formed by the tracking errors and sigmoidal local connections:  
\[e_{1}=z_{1}-\chi_{1},e_{2}=z_{2}+p_{1}\sigma(l_{1}e_{1}),e_{3}=z_{3}+p_{2} \sigma(l_{2}e_{2}), \tag{20}\]  
where \(e_{i}=(e_{1},e_{i2})^{\text{T}},i=1,2,3\); \(p_{i},l_{i}=\text{const}>0\), \(\sigma(l_{i}e_{i})=(\sigma(l_{i}e_{1}),\sigma(l_{i}e_{i2}))^{\text{T}},i=1,2\). Let us also introduce a sigmoidal corrective influence  
\[w=-p_{3}\sigma(l_{3}e_{3}), \tag{21}\]  
where \(p_{3},l_{3}=\text{const}>0\), \(\sigma(l_{3}e_{3})=\left(\sigma(l_{3}e_{31}),\sigma(l_{3}e_{32})\right)^{\text {T}}\), and note the closed-loop system  
\[ \begin{array}{l}\dot{e}_{1}=-p_{1}\sigma(l_{1}e_{1})+e_{2}-\dot{\chi}_{1},\\ \dot{e}_{2}=-p_{2}\sigma(l_{2}e_{2})+e_{3}+\Lambda_{1},\\ \dot{e}_{3}=-p_{3}\sigma(l_{3}e_{3})+\Lambda_{2},\end{array}  \tag{22}\]  
where similarly to (12),  
\[\Lambda_{i}(t)=\frac{d}{dt}(p_{i}\sigma(l_{i}e_{i}))= \begin{pmatrix}0.5p_{i}l_{ i}(1-\sigma^{2}(l_{i}e_{1}))\dot{e}_{i1}&0\\ 0&0.5p_{i}l_{i}(1-\sigma^{2}(l_{i}e_{i2}))\dot{e}_{i2}\end{pmatrix} ,i=1,2. \tag{23}\]The tracking problem is transformed into the problem of stabilization of the virtual system (22), considering the constraints (19).