## 3 Model of the Robot
### Dynamics of the Robot  
The robot is under the action of the following external forces and moments of force:  
* Ground reaction forces \({}^{R}\mathbf{F}_{Ai}= \begin{bmatrix}{}^{R}F_{Aix},\ {}^{R}F_{Aiy},\ {}^{R}F_{Aiz}\end{bmatrix} ^{T}\) acting on each wheel after their reduction to the geometric center of the wheel;
* Moments of force \({}^{R}\mathbf{T}_{Ai}= \begin{bmatrix}{}^{R}T_{Aix},\ {}^{R}T_{Aiy},\ {}^{R}T_{Aiz} \end{bmatrix} ^{T}\) resulting from the forces and moments of force acting at the points of contact of wheels with the ground, also after their reduction to the geometric center of the wheel;
* Gravity force \({}^{R}\mathbf{G}=m_{R}{}^{R}\mathbf{g}\), where \(m_{R}\) (kg) denotes the total mass of the robot. It is assumed that the robot moves on horizontal non-deformable ground, which is associated with a fixed co-ordinate system \(\{O\}\). Reaction forces \({}^{R}\!F_{Aix}\) and \({}^{R}\!F_{Aiy}\), \(i=1,\ \dots,\ n\), acting on the robot are shown in Figure 2.  
Force of gravity vector \({}^{R}\!G\) is a function of the gravitational acceleration vector, that is, \({}^{R}\!g=\left[{}^{R}\!g_{x^{\prime}},\ ^{R}\!g_{y^{\prime}},\ ^{R}\!g_{z}\right]^{T}\), and it is applied at the robot mass center, whose position is described by the vector \({}^{R}\!r_{CM}=\left[{}^{R}\!x_{CM},\ ^{R}\!y_{CM},\ ^{R}\!z_{CM}\right]^{T}\). The Newton-Euler formalism is used to describe the dynamics of the robot. Based on this formalism, the following dynamic equations of motion can be written for the robot in the co-ordinate system associated with the robot \(\{R\}\):  
\[m_{R}{}^{R}\!a_{CM}=\sum\
olimits_{i=1}^{n}{}^{R}\!F_{Ai}+m_{R}{}^{R}\!g, \tag{7}\]  
\[I_{R}{}^{R}\!e_{0}+{}^{R}\!\omega_{0}\times I_{R}{}^{R}\!\omega_{0}=\sum \
olimits_{i=1}^{n}\!\left({}^{R}\!T_{Ai}+\left({}^{R}\!r_{Ai}-{}^{R}\!r_{ CM}\right)\times{}^{R}\!F_{Ai}\right)\!. \tag{8}\]  
where \(I_{R}\)--the inertia tensor of the robot, \({}^{R}\!a_{CM}\)--the acceleration vector of the robot's center of mass, and \({}^{R}\!e_{0}\)--the angular acceleration vector of the robot's mobile platform. In order to determine the robot dynamics model in the simplest possible form, convenient for the synthesis of control systems, the following simplifying assumptions are made:  
* The robot moves on horizontal ground and the tilt of the robot during movement is negligible; hence, \({}^{R}\!g=\left[0,\ 0,-g\right]^{T}\), where \(g\) (m/s\({}^{2}\)) is the value of gravitational acceleration;
* There is no robot's center of mass movement in the vertical direction, i.e., \({}^{R}\!a_{CMz}=0\);
* Points of wheel-ground contact lie in one plane;
* Wheel tires deflect directly proportional to reaction forces acting along a direction normal to the ground and inversely proportional to their stiffness \(k\) (N/m), which is assumed to be constant and the same for all wheels. Figure 2: Reaction forces acting on the robot in the plane of wheel–ground contact. In the case of the analyzed robot, there are a large number of unknown reaction forces related to the contact of the wheels with the ground. However, adopting the simplifying assumptions, as in the article by [PERSON], 2015 [25], it is possible to determine the normal components of the ground reaction forces \({}^{R}F_{Aiz}\) (N) for all wheels. Assuming that \(L_{f}=L_{b}=L\) (m), the solution is obtained in the following form:  
\[ \begin{array}{c}{}^{R}F_{Aiz}=m_{R}\Big{(}g^{R}y_{CM}-{}^{R}a_{CMy}\big{(}r+ {}^{R}z_{CM}\big{)}\Big{)}\text{sgn}\big{(}{}^{R}y_{Ai}\big{)}/(3W)\\ +m_{R}\big{(}g^{R}x_{CM}-{}^{R}a_{CMx}\big{(}r+{}^{R}z_{CM}\big{)}\big{)}\text{ sgn}\big{(}{}^{R}x_{Ai}\big{)}/(4L)+m_{R}g/6.\end{array}  \tag{9}\]  
The general solution, that is, for \(L_{f}\
eq L_{b}\), is much more complex. In practical applications, one can find examples of robots with various \(L_{f}\) and \(L_{b}\) configurations, as well as center of mass positions. In particular, the position of the center of mass may change if the robot is equipped with a manipulator that modifies its configuration.