## 2 Mathematical Model of the Wheel Loader  
A mathematical model of the loader has been developed to analyze the influence of manipulator drive mechanism parameters on the tribological characteristics of the joints within the kinematic chain. The model comprises the general configuration of the wheel loader's kinematic chain, which consists of the following members: the rear structural-motion member \(L_{1}\) (Figure 1), equipped with the wheels \(L_{11}\) and \(L_{12}\); the front structural-motion member \(L_{2}\), with the wheels \(L_{21}\) and \(L_{22}\); and the manipulator with Z-kinematics that includes the arm \(L_{3}\), bucket \(L_{4}\), double-arm lever \(L_{5}\), coupling rod \(L_{6}\), hydraulic cylinders \(C_{3}\) for the boom drive mechanism, and hydraulic cylinder \(C_{4}\) for the bucket drive mechanism.  
The mathematical model of the wheel loader is based on the following assumptions: (1) the manipulator's kinematic chain is planar in configuration, meaning that the axes of the revolute joints are mutually parallel and the centers of the joints lie within the same plane, (2) the support surface and all members of the loader's kinematic chain are modeled as rigid bodies, (3) the kinematic chain of the loader is characterized by an open configuration, where the final member, the bucket, is subjected to material loading forces \(Wx\), \(Wy\) acting at its center of gravity, (4) the loader's kinematic chain is subjected to gravitational and inertial forces arising from the members of the kinematic chain and members of the drive mechanisms, (5) the coefficient of friction between the revolute joint elements of the manipulator's kinematic pairs is considered constant, while friction within the hydraulic cylinders is neglected.  
The loader model space is defined by the absolute coordinate system \(OXYZ\) (Figure 1), with the unit vectors \(\mathbf{i},\mathbf{j}\), and \(\mathbf{k}\) aligned with the coordinate axes \(OX\), \(OY\), and \(OZ\), respectively. The support surface of the loader lies in the horizontal plane \(OXZ\) of the absolute coordinate system, while the vertical axis \(OY\) coincides with the axis of the \(L_{1}\)-\(L_{2}\) kinematic pair.  
Each member \(L_{i}\) of the loader's kinematic chain is defined in its own local coordinate system \(O_{i}x_{i}y_{i}z_{i}\), with corresponding unit vectors. Each link is characterized by a set of  
Figure 1: Mathematical Model of the Loader Drive Mechanisms.  
vectors and parameters: \(\hat{\mathbf{\epsilon}}_{i}\) is the unit vector along the joint axis at \(O_{i}\), \(\hat{\mathbf{\epsilon}}_{i}\) is the kinematic length of the member \(L_{i}\); \(\hat{\mathbf{\epsilon}}_{i}\) is the position vector from the center of the revolute joint to the center of mass of the member \(L_{i}\), which has the mass \(m_{i}\), \(\hat{\mathbf{J}}_{i}\) is the inertia tensor defined about the center of mass. Vectors expressed in the local coordinate systems of the model are denoted with a hat symbol [31; 32].  
Vectors \(\mathbf{r}_{i}\) represent the positions of the joint centers of the kinematic chain members, while vectors \(\mathbf{r}_{ii}\) denote the positions of the centers of mass of the corresponding members in the loader's kinematic chain.  
\[\mathbf{r}_{1}=\mathbf{r}_{2},\quad\mathbf{r}_{i}=\sum_{j=1}^{i-1}A_{jo}\widehat{s}_{j}\ \forall i=2,\dots,4,\quad\mathbf{r}_{5}=\mathbf{r}_{3}+A_{30}\widehat{s}_{5} \tag{1}\]  
\[\mathbf{r}_{ii}=\mathbf{r}_{i}+A_{io}\widehat{\mathbf{\epsilon}}_{i}\quad\forall i=1, \dots,5 \tag{2}\]  
where \(A_{jo}\) and \(A_{io}\) are the transformation matrices used to convert vectors from the local coordinate systems to the absolute (global) coordinate system of the loader's mathematical model. These matrices are defined as functions of the internal (generalized) coordinates of the kinematic chain model \(\theta=[\theta_{1},\theta_{2},\theta_{3},\theta_{4}]^{\rm T}\).