## 3 Model of the Robot
### Dynamics of the Robot
However, to simplify the interpretation of the results, the analysis of the influence of the robot's parameters on the tracking control accuracy and energy efficiency in this paper is limited only to the case of \(L_{f}=L_{b}\) and one selected position of the robot's center of mass. Knowing the normal co-ordinates of the ground reaction forces \({}^{R}F_{Aiz}\) (N) and the wheel slip velocities, one can determine the tangential co-ordinates \({}^{R}F_{Aix}\) (N) and \({}^{R}F_{Aiy}\) (N). In the general case, the slip velocity vector for the \(i\)th wheel is equal to the following:  
\[{}^{R}v_{Si}=\Big{[}{}^{R}v_{Aix}-v_{oi},\ {}^{R}v_{Aiy},\ 0\Big{]}^{T}, \tag{10}\]  
where \(v_{oi}=\omega_{i}\ r_{i}\) (m/s) is the value of the velocity at the wheel circumference, and \({}^{R}v_{Aix}\) (m/s) and \({}^{R}v_{Aiy}\) (m/s) are, respectively, the longitudinal and lateral co-ordinates of the velocity vector of the wheel geometric center. The actual value of longitudinal slip ratio \(\lambda_{i}\) (%) for the \(i\)th wheel is then determined based on the following relationship:  
\[\lambda_{i}=\left\{ \begin{array}{cl}0&\text{if }\text{max}(|{}^{R}v_{Aix}|,\ |v_{oi}|)=0,\\ -\big{(}{}^{R}v_{Aix}-v_{oi}\big{)}/\text{max}(|{}^{R}v_{Aix}|,\ |v_{oi}|)&\text{ otherwise}.\end{array} \right. \tag{11}\]  
The actual value of the lateral slip angle \(\alpha_{i}\) (rad) for the \(i\)th wheel, that is, the angle between the velocity vector \({}^{R}v_{Ai}\) and the wheel plane, is calculated from the following formula:  
\[\alpha_{i}=\left\{ \begin{array}{cl}0&\text{if }\Big{|}{}^{R}v_{Aiy}\Big{|}=0,\\ \text{arctan}2\Big{(}{}^{R}v_{Aiy},\ |{}^{R}v_{Aix}|\Big{)}&\text{ otherwise}.\end{array} \right. \tag{12}\]  
Based on the known longitudinal slip ratio, the actual value of the tire adhesion coefficient on longitudinal direction for this wheel can be determined. For this purpose, one can use the Kiencke tire model described in the book by [PERSON] and [PERSON], 2005 [26] and applied in the automotive industry. However, this model is also sometimes used for modeling of wheeled mobile robots, an example of which can be found in the doctoral thesis by [PERSON], 2009 [27]. As rightly noted in the paper by [PERSON] et al., 2010 [28], the Kiencke tire model provides the best accuracy/simplicity compromise. Due to its effectiveness, its use in the design of control systems for wheeled vehicles is justified. For this reason, a modified version of the [PERSON] tire model is used in this work, that is, in the following form:  
\[\mu_{ix}=\left\{ \begin{array}{cl}\big{(}2\mu_{p}\lambda_{p}\lambda_{i}\big{)} \big{/}\Big{(}\lambda_{p}^{2}+\lambda_{i}^{2}\Big{)}&\text{if }|\lambda_{i}|\leq\lambda_{p},\\ a_{\lambda x}\lambda_{i}+b_{\lambda x}\text{sgn}(\lambda_{i})&\text{ otherwise},\end{array} \right. \tag{13}\]  
where \(\lambda_{p}\) (%) denotes the value of the longitudinal slip ratio corresponding to the value of the maximum tire adhesion coefficient \(\mu_{p}\) (-). Coefficients \(a_{\lambda x}\) (-) and \(b_{\lambda x}\) (-) in the relationship (13) are described by the following formulae:  
\[a_{\lambda x}=\big{(}\mu_{p}-\mu_{k}\big{)}\big{/}\big{(}\lambda_{p}-\lambda_{ max}\big{)},\quad b_{\lambda x}=\mu_{p}-a_{x}\lambda_{p}, \tag{14}\]  
where \(\mu_{k}\) (-) is a coefficient of the kinetic friction between the tire and ground, and \(\lambda_{max}=100\%\). The modified version of the [PERSON] tire model allows for a more precise description of the characteristics of \(\mu_{ix}(\lambda_{i})\) for \(|\lambda_{i}|>\lambda_{p}\), including obtaining \(\mu_{ix}=\mu_{k}\) for \(|\lambda_{i}|=\lambda_{max}\). Knowing the lateral slip angle, it is possible to calculate the actual value of the adhesion coefficient in the lateral direction for the \(i\)th wheel. For this purpose, the following simplified relationship can be used:  
\[\mu_{iy}=-\mu_{ymax}\text{sin}(\alpha_{i}), \tag{15}\]  
where \(\mu_{ymax}\) (-) denotes the maximum value of adhesion coefficient in the lateral direction, which is assumed to be equal to \(\mu_{k}\) (-).