## 2 Automatic Transmission Modeling
### Kinematics and Power Flows of a Transmission Structure
#### 2.1.1 Planetary Gear Train Kinematics  
Kinematics of a planetary gear train can be analyzed by solving a linear matrix formed by the generalized formulation approach. The generalized formulations consist of three groups of equations, which are (1) kinematic equations of planetary gear sets, (2) constraint equations and (3) an input equation. As shown in Figure 3, two types of planetary gear sets are generally used to build a planetary gear train. In Figure 3a, one is a single-planet (simple) planetary gear set which includes the sun gear \(s\), ring gear \(r\) and carrier \(c\) supporting \(N_{P}\) number of planet gears \(p\). Central members of the planetary gear set form external and internal gear meshes. The external and internal gear meshes mean conjugations of the sun-planet gears and ring-planet gears, respectively, with a carrier connection. These external (\(s\)-\(p\)-\(c\)) and internal (\(r\)-\(p\)-\(c\)) gear meshes hold corresponding gear mesh kinematic equations which are given as  
\[ \begin{split}\omega_{s}Z_{s}+\omega_{p}Z_{p}-\omega_{c}\big{(}Z_{s }+Z_{p}\big{)}&=0,\\ \omega_{r}Z_{r}-\omega_{p}Z_{p}-\omega_{c}\big{(}Z_{r}-Z_{p}\big{)} &=0,\end{split}  \tag{1}\]  
where \(\omega\) and \(Z\) denote the absolute angular velocity and the number of teeth of a gear. Subscript \(s\), \(r\), \(p\) and \(c\) mean the sun, ring and planet gears and the carrier, respectively. The other type shown in Figure 3b is called a double-planet planetary gear set which consists of the sun and ring gears and the carrier holding inner planet gears \(pi\) and outer planet gears \(po\). Two external and one internal gear meshes are formed by central members with inner and outer planet gears. Kinematic equations representing the double planet planetary gear set are given as  
Figure 2: A schematic diagram of the transmission structure [28]. \[ \begin{split}\omega_{s}Z_{s}+\omega_{pi}Z_{pi}-\omega_{c}\big{(}Z_{s}+Z _{pi}\big{)}=0,\\ \omega_{r}Z_{r}-\omega_{po}Z_{po}-\omega_{c}\big{(}Z_{r}-Z_{po} \big{)}=0,\\ \omega_{pi}Z_{pi}+\omega_{po}Z_{po}-\omega_{c}\big{(}Z_{pi}+Z_{po} \big{)}=0.\end{split}  \tag{2}\]  
where subscript _pi_ and _po_ denote an inner and outer planet gear, respectively. Constraint equations involve two cases of central member connections. The first case means that amongst central members of several decks of planetary gear sets, two member \(i\) and \(j\) of them, which are connected by a rigid connection, have the same angular velocity. The other is a fixed connection where central member \(k\) is connected to a transmission housing by brakes and does not rotate. Lastly, an input equation for transmission input member \(m\) needs to be specified with a designated input value \(\omega_{in}\). These constraint and input equations are represented mathematically as  
\[\omega_{i}-\omega_{j}=0\text{ for }i\
eq j, \tag{3}\]  
\[\omega_{k}=0, \tag{4}\]  
\[\omega_{m}=\omega_{in}. \tag{5}\]  
All linear algebraic equations including kinematics of planetary gear sets, constraints, and an input can be developed into a matrix form \(AX=B\) with a matrix \(A\) involving the whole equations, an unknown vector \(X\) comprised of angular velocities, and a vector \(B\) having one non-zeros component and the others being zero. For a given schematic diagram in Figure 2 along with the actuation scheme in Table 1, rotational speed ratios of central members of planetary gear sets are analyzed for 8 forward and 1 reverse drives with respect to an input of \(\omega_{c2}=1\), and the results of the kinematic analysis are presented in Table 2a.