## 2 Materials and Methods
### Design of Tracking Differentiators
#### 2.3.3 Reasons for Designing Tracking Differentiators with Different Block Numbers  
As was shown in the previous subsection, it is sufficient to apply a dynamic model consisting of three blocks (32) to smooth external signals and to recover their derivatives of the first, second, and even third orders. However, the number of blocks of the tracking differentiator can be both reduced and increased depending on the conditions of the problem to be solved. The main factors influencing the dimensionality of the differentiator are as follows:  
1. Methods used to design control in a tracking system of a mobile robot;
2. The considered dynamic order of the control plant;
3. Noisiness/noiselessness of the external signal \(\chi_{1}(t)\). For example, to solve the problem of tracking by the output variables \(x(t),y(t)\) of System (1) of reference signals under the influence of parametric and external disturbances,we can apply a decomposition method for the design of sigmoid feedback, similar to the method we applied in the previous section for the design of a tracking differentiator [24]. In this method, the first derivatives of the reference influences are treated as external disturbances (as in System (22)), and only the reference influences are used to form the static feedback. Another approach is to use mixed-variable observers in tracking systems [27]. In this case, for feedback design, it is not necessary to obtain separate estimates of the derivatives of the reference influences. Thus, in both approaches, we only need to know the reference signals. If these signals do not contain parasitic disturbances, a single-block tracking differentiator with the minimum possible dynamic order can be used to smooth them:  
\[\dot{z}_{1}=-p_{1}\sigma(l_{1}(z_{1}-\chi_{1})),z_{1}(0)=\chi_{1}(0), \tag{33}\]  
where \(z_{1}\in R^{2}\). The inequalities for the choice of gains \(p_{1},l_{1}=\text{const}>0\) considering the constraints on the velocity and acceleration of the mobile robot, similar to (29), (31), are of the form  
\[\left[ \begin{array}{l}1.25X_{1}<p_{1}\leq Y_{1},\\ p_{1}^{2}l_{1}\leq Y_{2};\\ p_{1}=Y_{1},l_{1}=Y_{2}/Y_{1}^{2}.\end{array} \right. \tag{34}\]  
The shapes of the smoothed curves obtained using the tracking differentiators (32) and (34) under the same constraints (4) will be almost identical. The signals of the derivatives of the reference influences are usually needed in program control systems [28] as well as when using the feedback linearization method [3]. The order of the required derivatives depends on the number of dynamic links to be considered and is equal to the relative degree of the model used in the control plant. Let derivatives up to and including \(n\)-th order be required for the design of the tracking system of a wheeled robot. The tracking differentiator, which smooths the external deterministic signal \(\chi_{1}(t)\) and recovers its \(n\) derivatives, consists of \((n+1)\) blocks and has a common order \(2(n+1)\):  
\[ \begin{array}{l}\dot{z}_{i}=z_{i+1},i=\overline{1,n};\dot{z}_{n+1}=-p_{n+1} \sigma(l_{n+1}e_{n+1})=\\ =-p_{n+1}\sigma(l_{n+1}(z_{n+1}+p_{n}\sigma(l_{n}(z_{n}+p_{n-1}\sigma(l_{n-1}( \ldots(z_{2}+p_{1}\sigma(l_{1}(z_{1}-\chi_{1})))\ldots))));\\ z_{1}(0)=\chi_{1}(0),z_{i}(0)=0,i=\overline{2,n+1},\end{array}  \tag{35}\]  
where \(z_{i}\in R^{2},i=\overline{1,n+1}\) are differentiator variables, the vector variable \(z_{1}(t)\) defines the position of the center point of the wheeled platform, and the variable \(z_{i}(t)\) is its \((i-1)\)-th derivative, \(i=\overline{2,n+1}\), \(z_{1}^{(n+1)}=\dot{z}_{n+1}\). To set the tracking differentiator (35), the constraints of a particular mobile robot on the higher derivatives are used.