# Hybrid Model-Based Analysis of Underground Articulated Vehicles Steering Characteristics
## 2 Dynamic Model of UAVs
### Dynamic Model of Vehicle Frames  
A diagram of UAV motion is shown in Figure 1; the motion can be described in the two sets of frames: the ISO (the longitudinal forward direction of vehicle is defined as X axis, vertical downward direction is defined as Z axis, Y axis is in right-hand coordinate system) front frame (X\({}_{1}\)Y\({}_{1}\)Z\({}_{1}\)) and ISO rear frame (X\({}_{2}\)Y\({}_{2}\)Z\({}_{2}\)). In addition, the detailed force information at the articulated point is also shown in Figure 1.  
According to the relationship of the involved forces and moments shown in Figure 1, the motions of the UAV in the XOY plane (the lateral and longitudinal directions) are focused [10]. Based on [PERSON]'s law, the dynamic model of UAV frames in different directions can be expressed as:  
\[\left\{ \begin{array}{l}m_{1}\Big{(}\dot{v}_{x1}-v_{y1}\omega_{z1}\Big{)}=F_{ txf}+F_{x}+F_{sxf}\\ m_{1}\Big{(}\dot{v}_{y1}+v_{x1}\omega_{z1}\Big{)}=F_{tyf}-F_{y}+F_{syf}\\ m_{2}\Big{(}\dot{v}_{x2}-v_{y2}\omega_{z2}\Big{)}=F_{txr}-F_{x}\cos\varphi-F_{y} \sin\varphi+F_{sxr}\\ m_{2}\Big{(}\dot{v}_{y2}+v_{x2}\omega_{z2}\Big{)}=F_{tyr}+F_{y}\cos\varphi-F_{x} \sin\varphi+F_{syr}\\ \end{array} \right. \tag{1}\]  
\[\left\{ \begin{array}{l}I_{21}\dot{\omega}_{z1}=M_{tzf}+M_{zf}+M_{z}\\ I_{22}\dot{\omega}_{z2}=M_{tzr}+M_{zr}-M_{z}\\ \end{array} \right. \tag{2}\]  
where the indexes 1 and 2 denote front and rear frames, respectively; \(\phi\) is the articulation angle; \(m_{1}\) and \(m_{2}\) are the mass of front and rear frames, respectively; \(I_{21}\) and \(I_{22}\) are the moment of inertia in the z direction, respectively; \(F_{x}\), \(F_{y}\), and \(M_{z}\) are internal forces and moment at the central point; \(F_{sxf}\), \(F_{sxf}\), \(F_{syr}\), \(F_{syr}\), \(M_{2f}\) and \(M_{zr}\) are the forces and moments brought by the steering cylinder; and \(F_{tyf}\), \(F_{tyf}\), \(F_{txr}\), \(F_{tyr}\), \(M_{tzf}\), and \(M_{tzr}\) denote the front and rear tire forces and moments in the different x, y, and z directions, which can be expressed as:  
\[\left\{ \begin{array}{l}F_{txf}=F_{txf1}+F_{txfr},F_{txr}=F_{txrl}+F_{txrr}\\ F_{tyf}=F_{tyf1}+F_{tyfr},F_{tyr}=F_{tyrl}+F_{tyrr}\\ M_{tzf}=B\Big{(}F_{txf1}-F_{txfr}\Big{)}-L_{f1}\Big{(}F_{tyf1}+F_{tyfr}\Big{)} \\ M_{tzr}=B(F_{txrl}-F_{txrr})-L_{r1}\Big{(}F_{tyrl}+F_{tyrr}\Big{)}\\ \end{array} \right. \tag{3}\]  
where \(L_{\text{f1}}\) and \(L_{\text{r1}}\) are the distance between the mass centers of the front and rear frames and corresponding axle, respectively; and \(B\) is half of the wheel track. In addition, the longitudinal and lateral tire force can be given by the Magic Formula [21]:  
\[F_{m}=S_{y}+D\sin[\text{Carctan}[B(X_{m}-S_{x})(1-E)+\text{Earctan}B(X_{m}-S _{x})]] \tag{4}\]  
Figure 1: Diagram of underground articulated vehicle (UAV) motion. where \(X_{m}\) denotes the sideslip angle or slip ratio of the tire; \(F_{m}\) denotes the lateral or longitudinal force of tire; the parameters \(Sx\), \(Sy\), \(B\), \(C\), \(D\), and \(E\) are constant, where in this study \(Sx=0\), \(Sy=0\), and other parameters are calculated by equations with vertical load. The slip ratio of different tires can be described as:  
\[\eta_{ij}=\left\{ \begin{array}{l}\left(r_{dij}\omega_{ij}-v_{xij}\right)/r_{ d}\omega_{ij}\times 100\%\quad\quad\text{Drive}\\ \left(r_{dij}\omega_{ij}-v_{xij}\right)/v_{xij}\quad\quad\quad\text{Brake} \end{array} \right. \tag{5}\]  
where \(r_{dij}\) denotes the rolling radius of different tires, \(\omega_{ij}\) denotes the angular velocity of different tires, and \(v_{xij}\) denotes the longitudinal velocity of front and rear frame, which can be written as:  
\[\left\{ \begin{array}{l}v_{xf,lr}=v_{xf}\pm Ba\omega_{zf}\\ v_{xrl,r}=v_{xr}\pm Ba\omega_{xr}\end{array} \right. \tag{6}\]  
The sideslip angle of different tires can be described as:  
\[\left\{ \begin{array}{l}\alpha_{1,2}=\left(v_{y1}-L_{f1}\omega_{z1}\right)/ \left(v_{x1}\pm Ba\omega_{z1}\right)\\ \alpha_{3,4}=\left(v_{y2}-L_{r1}\omega_{z2}\right)/\left(v_{x2}\pm Ba\omega_{z 2}\right)\end{array} \right.