## 2 Basic Path Diagram (BPD)
### Bpd  
To provide a systematic method to simplify the FPG, another graphical method called the BPD is proposed in this paper. There are only seven kinds of powertrain elements, named \"basic units\", as listed in Table 1. Other elements that are represented in the FPG will be ignored or combined with basic elements during the process of conversion from FPG to BPD.  
The main purpose of the BPD is to represent the available power flow paths of a powertrain system and to exclude unrelated units present in the FPG. The conversion from FPG to BPD can be achieved by performing the following steps:  
1. Rename the power input unit in the FPG as IN\({}_{1}\), IN\({}_{2}\), IN\({}_{3}\)..., etc., in sequence of their maximum power output if there are multiple power sources.
2. Ignore the independent subsystems that have no connection with other transmission paths and are driven by a single power source. For example, the rear wheels of a vehicle are independently driven by an e-Axle module. This e-Axle module is purely powered by electric motor(s) and has no connection with other powertrain units such  
\begin{table}
\begin{tabular}{c c c c} \hline
**Unit** & \multicolumn{2}{c}{**Symbol**} & **Explanation** \\ \hline Power I/O units & IN & Power input, such as an ICE, M/G, and flywheel (FW). \\ \cline{2-4}  & OUT & OUT & Power output, such as an individual wheel or final drive unit including the differential. \\ \hline Transmission unit & CVU & CVU & Transmission unit with continuously variable ratio. \\ \hline Transmission & 2 DOF-3E & Two-degree-of-freedom (2 DOF) & Two-degree-of-freedom (2 DOF) \\  & 2 DOF-4E & 2 DOF & gearest with three elements, such as \\  & (DOFs) & & \\ \cline{2-4}  & & & \\ \cline{2-4}  & & & \\ \cline{2-4}  & & & \\ \cline{2-4}  & & & \\ \cline{2-4}  & & & \\ \hline Connection unit & Connection line & Power transmission & \\ \end{tabular}
\end{table}
Table 1: List of basic units in the basic path diagram (BPD).  
as the engine and the transmission. Hence, this e-Axle module needs to be completely ignored in this step.
3. Ignore the auxiliary power input units that have no mechanical connection to the wheel or are unable to drive the wheel directly. Examples include electric motors that accelerate the flywheel to recharge its kinetic energy and superchargers that are driven by electric motors.
4. Ignore all of the clutches, brakes, couplings, and torque converters.
5. Ignore transmission units that have constant ratios or have step-variable ratios. These include gear trains, chain drives, belt drives, manual transmissions (MTs), automatic transmissions (ATs), and dual-clutch transmissions (DCTs).
6. If there are gearsets with multiple DOFs, such as planetary gearsets, differentials, and Ravigneaux gearsets, replace them with 2 DOF-3E, 2 DOF-4E, and MDOF.
7. Locate the transmission paths that are only used for reversing and combine them with the transmission paths for low-gear forward drive.
8. Combine the coupling points into adjacent basic units, such as IN, OUT, and CVU.
9. Combine the remaining coupling points with adjacent ones.
10. Adjust the orientation of the basic units, and avoid overlap and intersection of the connection lines that are parallel in the original FPG.  
Here, the hybrid system proposed by Eindhoven University of Technology (Technische Universiteit Eindhoven, TU/e) [11,12] is given as an example of the BPD. A schematic diagram of the mechanism is shown in Figure 1. According to [35], the FPG for this system is shown in Figure 2a. There are seven gear drive units and six clutches, including a sliding gear mechanism for reverse that is considered as one of the clutches. The BPD for this system is shown in Figure 2b, with the transmission path for the reverse drive (\(C_{r}\) and \(G_{r}\)) merging with that for the low-speed forward drive (\(C_{ep}\), \(G_{e}\), and \(G_{fp2}\)).  
Figure 1: Schematic diagram of the hybrid system proposed by TU/e [11,12].