## 2 Dual-State Logic Drive
### Analysis of Rotational Speed and Speed Ratio of Single-Row Coplanar Gear Planetary Array  
In the study of the graph theory model of the transmission system, the basic circuit is usually used to calculate and analyze the speed, torque or power flow of the planetary gear. The basic circuit is a closed circuit composed of three components, which is composed of two meshing gears and the planet carrier between its rotating shaft. From this, we can see the characteristics of the basic circuit: a closed circuit composed of three points (components), one gear pair edge (dotted line) and two rotary pair edges (solid line). The number of basic circuits are equal to the number of gear pairs. In a planetary basic gear train, there are two basic circuits which can form two basic circuit equations. In Figure 3d, the coplanar gear planetary array contains \((p_{1},s)\) c and \((p_{2},r)\) c, two basic circuits, and the basic circuit equation is as follows [23]:  
\[\omega_{s}^{\rm I}-\omega_{c}^{\rm I}=i_{sp1}(\omega_{p1}^{\rm I}-\omega_{c}^ {\rm I}) \tag{1}\]  
\[\omega_{p2}^{\rm II}-\omega_{c}^{\rm II}=i_{p2}(\omega_{p}^{\rm II}-\omega_{c}^ {\rm II}) \tag{2}\]  
\[\omega_{p1}^{\rm I}=\omega_{p2}^{\rm II} \tag{3}\]  
\(\omega_{s}\), \(\omega_{p1}\), \(\omega_{p2}\), \(\omega_{c}\) and \(\omega_{r}\), respectively, represent the rotational speed of the corresponding components in the basic circuit, namely, the sun gear, the planet gear 1, the planet gear 2, the planet carrier and the gear ring, where \(\omega_{p1}=\omega_{p2}\), \(i_{sp1}=z_{p1}/z_{s}=k_{1}\) and \(i_{p21}=z_{r}/z_{p2}=k_{2}\), respectively, represent the tooth number ratio of the two internal meshing gears of the  
Figure 3: Schematic diagram of positive planet array. (**a**) Coplanar gear planetary row 1. (**b**) Coplanar gear planetary row 2. (**c**) Coplanar gear planetary row 3. (**d**) Graph theory expression model.  
Figure 2: Schematic diagram of planetary array. (**a**) Ordinary single planet row. (**b**) Ordinary double planet row. (**c**) Graph theory expression model.  
planet gear 1 and the sun gear, and the planet gear 2 and the gear ring. Then, the whole coplanar gear characteristic parameter \(k=k_{1}\times k_{2}\) is rewritten with \(k_{1}\) and \(k_{2}\) for the above basic loop equation, the circuit mark is removed, \(\omega_{p1}\) and \(\omega_{p1}\) are replaced with \(\omega_{p}\) to obtain the following equation:  
\[\omega_{{}_{s}}-k_{1}\omega_{p}-(1-k_{1})\omega_{c}=0 \tag{4}\]  
\[\omega_{{}_{p}}-k_{2}\omega_{r}-(1-k_{2})\omega_{c}=0 \tag{5}\]  
Written in the form of a transmission ratio matrix, the following equation is obtained:  
\[A\cdot\omega=\begin{vmatrix}1&-k_{1}&0&-(1-k_{1})\\ 0&1&-k_{2}&-(1-k_{2})\end{vmatrix}\begin{bmatrix}\omega_{s}\\ \omega_{p}\\ \omega_{r}\\ \omega_{c}\end{bmatrix}=0 \tag{6}\]  
That is, \(A\times\omega=0\), where \(A\) is the transmission ratio matrix, \(\omega\) is the coefficient of the angular velocity of Equations (4) and (5), and Equation (6) represents the case of a single planet of a coplanar gear lining up two loops, which is the rotational speed characteristic matrix of a single planetary row of a coplanar gear. The following establishes the corresponding speed characteristic equation (matrix) for a gearbox composed of five coplanar gear planetary rows and one ordinary planetary row connected in series.