## 2 Materials and Methods
### Design of Tracking Differentiators
#### 2.3.3 Reasons for Designing Tracking Differentiators with Different Block Numbers
To fulfill the constraints on the state variables of the differentiator, its corrective influences, and their rate of change (i.e., constraints on the robot's control influences and their rate of change), nominal values \((n+2)\) are required  
\[\left\|y_{1}^{(i)}\right\|\leq Y_{i}=\text{const}>0,t\geq 0,Y_{i}^{2}<Y_{i+1},i= \overline{1,n+2}.\]  
Equations for selection gains of tracking differentiator (35) at which the constraint of its variables is ensured  
\[\left\|z_{1}^{(i)}\right\|\leq Y_{i},t\in[0;T],i=\overline{1,n+2}\]  
are similar to the system of dual inequalities (29) and have the following form:  
\[\left[ \begin{array}{l}2.75/l_{2}+1.25X_{1}<p_{1}\leq Y_{1}-2.2/l_{2},\\ 2.75/l_{i+1}+1.25p_{i-1}^{2}l_{i-1}<p_{i}\leq Y_{i}-2.2/l_{i+1},i=\overline{2, n};\\ 1.25p_{n}^{2}l_{n}<p_{n+1}\leq Y_{n+1};\\ p_{n+1}^{2}l_{n+1}\leq Y_{n+2}.\end{array} \right. \tag{36}\]The iterative procedure for setting the gains of sigmoid functions based on the system of inequalities (36) minimizing the steady-state tracking error (30) is similar to (31):  
\[ \begin{array}{l}p_{n+1}=Y_{n+1},l_{n+1}=Y_{n+2}/p_{n+1}^{2},\\ p_{n}=Y_{n}-2.2/l_{n+1},l_{n}=\left(0.8p_{n+1}-\beta_{n}\right)/p_{n}^{2},\\ p_{i}=Y_{i}-2.2/l_{i+1},l_{i}=\left(0.8p_{i+1}-\beta_{i}-2.2/l_{i+2}\right)/p_{ i}^{2},i=\overline{n-1,1},\end{array}  \tag{37}\]  
where \(\beta_{i},i=\overline{1,n}\) are very small positive constants. Finally, the third factor influencing the choice of the dynamic order of the tracking differentiator is related to the need to filter the external signal if it additively contains parasitic noise \(\overline{\lambda}_{1}(t)=\chi_{1}+\eta(t)\), for example, white noise \(\eta(t)\) with zero mathematical expectation and bounded variance. Note that the noisy signal enters the input of the tracking differentiator (35) and passes through a chain of integrators. Thus, there is natural filtering of the input signal, which improves with increasing number of integrators. Analytical analysis and simulation results have shown that extending the dynamic order of the tracking differentiator is an alternative to installing low-pass prefilters [29]. Two or three integration operations are often sufficient to obtain good performance. For example, if the recovery of the derivatives of the external signal is not a problem, a two or three (32) block tracking differentiator should be used instead of a single-block differentiator (33) for smoothing and filtering the external signal. In turn, instead of a three-block differentiator (32), a four- or five-block tracking differentiator must be used in order to obtain signals of satisfactory quality up to and including the second derivative. Note that, as in the Kalman filter setting, when selecting gains of sigmoidal functions, we have to compromise between tracking accuracy and filtering features of the tracking differentiator [23]. Therefore, another algorithm is needed instead of the procedure for setting gains \(p_{i},l_{i}\) (37). However, a detailed analysis of this problem is beyond the framework of this research.