Most of the prior studies on power-split hybrid electric vehicle's (PS-HEV) design focused on the powertrain configuration optimization. Yet, depicting the selected configuration is highly required for further design steps, ultimately manufacturing. This paper proposes an automatic approach to generate all the feasible kinematic diagrams for a given configuration with a single planetary gear (PG) set. While the powertrain configuration, which is the output of prior studies, illustrates the connection of the powertrain components to the PG, the kinematic diagram is a schematic diagram depicting the connections and arrangements of the components. First, positioning diagrams, specifying the position of the components with respect to each other and to the PG, are used to find all the possible arrangements. Then, given that the positioning diagrams have a one-to-one relationship with the kinematic diagrams, the feasible kinematic diagrams are identified using a set of feasibility rules applicable to the positioning diagrams. Finally, few guidelines are introduced to select good kinematic diagrams that best suit the overall vehicle design. Various configurations were investigated, and three of them including Prius and Voltec first-generation single PG configurations are discussed. The study reveals that the kinematic diagrams that have been patented are only a subset of all the feasible kinematic diagrams, and that even some good kinematic diagrams with better manufacturability are identified using this methodology. Thus, this methodology guarantees the search of the entire design space and the selection of kinematic diagrams that best suit the desired vehicle.

## Introduction

The power-split hybrid electric vehicles (PS-HEVs) are one of the most promising solutions for the recent global energy and environmental concerns. The popularity and efficiency of PS-HEVs resulted in numerous studies focusing on employing various control strategies to evaluate and optimize given powertrain configurations in order to find optimal configurations [1–4]. However, a powertrain configuration, which is the common output of prior studies, cannot be directly realized as it only shows the components connections and not their arrangements. Unless the selected powertrain configurations are physically designed and implemented, all the efforts and analyses to find the best configuration can become impractical. The physical design of PS-HEV configurations for manufacturing purposes has not been extensively studied, to the best of our knowledge.

An extensive patent search and review have been conducted to understand the history and development of the kinematic designs for different powertrain configurations. One of the notable findings through the patent review was that many patents have been filed with different kinematic diagrams, but for the same powertrain configuration. For instance, the single PG (1PG) powertrain configuration used for Prius was first introduced in 1997 [5]. The powertrain, which is illustrated in Fig. 1(a), embodies one conventional internal combustion engine (ICE), two electric machines (EMs), and one PG. The small EM (generator) is connected to the sun gear, the ICE is connected to the carrier, and the second EM (motor) is connected to the ring gear. The output (drive) shaft is connected through a differential gear train unit to the ring gear. However, the schematic diagram depicted in Ref. [5] is only one of many possible designs for the same powertrain. Indeed, several variants of this kinematic diagram were developed by Toyota, such as the three kinematic diagrams illustrated in Figs. 1(b)–1(d) [6–8]. Note that the components arrangements are different for each of the kinematic diagrams illustrated in Fig. 1. This suggests that each powertrain configuration has multiple feasible kinematic diagrams. Besides, the packaging and manufacturability of these four patents improved over the time, which could have been avoided if an automated schematic design methodology was used. Thus, such a methodology is necessary to accelerate the design and development process of PS-HEVs and guarantee finding the arrangements that best suit the vehicle.

In academia, significant research has been conducted on the topological analysis of various mechanisms, including automobile transmissions. The graph theory was extensively used for the systematic classification and enumeration of kinematic structures of epicyclic gear type drivetrains. It was first proposed by Buchsbaum et al. and then converted into matrices for computerization purposes [9–11]. Later, several studies applied the graph theory on automotive mechanisms [12–18]. For instance, Hsu and Hsu used it for the systematic generation of kinematic chains of automatic transmissions using two or more epicyclic gear trains [18]. For a better understanding of the use of graph theory for the kinematic diagrams' generation, the readers are encouraged to read Ref. [19]. Nevertheless, our findings revealed that using the graph theory does not guarantee finding all the possible components arrangements as it will be further discussed. Thus, a new approach is required to enumerate all the feasible kinematic diagrams for any given 1PG configuration. The approach introduced in this paper was briefly introduced in Ref. [20], but the proof of the feasibility theorems (FTs), directions to generate the kinematic diagrams, and guidelines to select the good designs are provided in this paper.

This paper introduces a comprehensive automated method to generate all the feasible kinematic diagrams for any given 1PG configuration. The main objective of this study is to allow the fast and accurate depiction of any given 1PG configuration, and the selection of the arrangements that best suit the vehicle's design. The entire process should be automated as the manual enumeration of all the feasible kinematic diagrams for a given 1PG configuration is time-consuming. The remainder of this paper is organized as follows: In Sec. 2, the 12 1PG configurations and the kinematic diagrams are introduced. The proposed automated process to generate the feasible kinematic diagrams is introduced in Sec. 3; first introducing the placement and positioning diagrams, then proposing two theorems to screen out the infeasible kinematic diagrams. The automation of the feasibility theorems and few directions for drawing feasible kinematic diagrams were introduced along with few suggestions to select good diagrams that are potentially more practical for manufacturing. Few case studies are discussed in Sec. 4, and the existing and new methods are compared in Sec. 5. Finally, Sec. 6 introduces the concluding remarks.

## Different Representations of a Single PG Split Hybrid Configuration

The schematic design phase is the intermediate phase between the theoretically conceived configuration and the three-dimensional drawings. This section introduces the powertrain configurations and kinematic diagrams.

### Hybrid Powertrain Configuration.

PS-HEVs have both mechanical and electrical paths connected through a power-split device, usually a PG. The PG, which is the key component of PS-HEVs, consists of a sun gear, a ring gear, planets, and a carrier (see Fig. 2). In this paper, the PS-HEV powertrain configuration is defined as a lever diagram showing the connections of the components to the PG. The PG's structure results in three drive shafts to which components are connected, referred to as three nodes: sun gear, carrier, and ring gear nodes, as illustrated in Fig. 2.

Assuming that the PS-HEV uses one ICE, two EMs, and one PG, Kim et al. assumed that there exist 24 1PG configurations as they found that switching the two EMs in the same configuration affects the performance results [21]. Nevertheless, this study considers only the 12 1PG configurations illustrated in Fig. 3. In fact, the two EMs, denoted EM 1 and EM 2 hereafter, are not distinguished in this work as the schematic design focuses on their physical arrangements rather than their performance. Note that EM 1 is always connected along with either the ICE or the final drive, while EM 2 is connected by itself to the PG.

The 12 1PG configurations illustrated in Fig. 3 are often classified into two types, i.e., input-split and output-split types. Configurations having the engine directly connected to the PG are called input-split, and there are six input-split configurations, e.g., configuration #1. The rest of configurations have the final drive directly connected to the PG, which are called output-split hybrids, e.g., configuration #4. Among the configurations shown in Fig. 3, configuration #9 corresponds to the one used in the commercialized Toyota Prius second-generation and configuration #12 represents the hybrid mode of General Motors' (GM) Chevy Volt (Voltec) first-generation.

### Kinematic Diagrams.

The kinematic diagram, also known as the stick or schematic diagram, is considered the first step toward the production of any given configuration as it is the conceptual description aiming to visualize the new designs. It is a simplified cartoon layout schematically depicting the connections, arrangements, and positions of the main powertrain components, i.e., ICE, two EMs, and the final drive. Each of the kinematic diagrams of the Prius configuration, which are illustrated in Fig. 1, has a unique components arrangement. Note that these kinematic diagrams are few of many existing feasible kinematic diagrams. Nevertheless, the generation of feasible kinematic diagrams, which are arrangements that can be realized without any constraints, is very important and not trivial.

## Automated Generation of Kinematic Diagrams

The flow chart of the approach proposed to generate the feasible kinematic diagrams for any given 1PG configuration is illustrated in Fig. 4. First, for a given 1PG configuration, diagrams specifying the position of the components with respect to the PG, referred to as *placement diagrams*, are generated. Second, for each *placement diagram*, *positioning diagrams*, specifying the relative locations of the components, are generated. Third, a set of feasibility theorems are applied on the positioning diagrams, and feasible ones are depicted. Finally, a few guidelines are proposed to sort out the good kinematic diagrams based on few design criteria such as packaging.

### Generation of Placement Diagrams.

*placement diagram*is a powertrain configuration specifying the components' position with respect to the PG, either right or left side of the PG. The total number of placement diagrams is given as follows:

where the product is divided by two to remove the redundancy of symmetric placements. Figure 5 illustrates the eight placement diagrams of the Prius configuration (#9 in Fig. 3). Although the generation of placement diagrams is effortless, the fundamental placement diagram layout shown in Fig. 6 is proposed to facilitate the process and group these diagrams. The six cells on the left and right sides of the PG represent the placement space for components, and the dashed horizontal lines (branches) connect the components to the PG. For further development of the design process, the placement diagrams are categorized into two groups depending on the existence of a branch sharing two components:

- —
*2C-together group*: Two components share a single branch, which results in three branches. - —
*2C-separate group*: None of the four components shares a single branch, which results in four branches.

The number of branches on each side of the PG is different for each placement diagram. The number of branches on each side is used to further classify each group of placement diagrams into two cases as follows:

- —
*2C-together group (three branches)*: - —
*2C-separate group (four branches)*:

The above four cases are summarized in Table 1 illustrating all the possible fundamental placement diagrams. Note that for case 1, there is a unique placement while there are three possible placements for case 2 as only a single branch is placed on one side of the PG, which has three nodes. For cases 3 and 4, there are two possible placements for each, as the components sharing one node do not share the same branch. Hence, each 1PG configuration has in total eight placement diagrams.

### Generation of Positioning Diagrams.

The positioning diagrams are placement diagrams specifying the position of components with respect to each other. The number of positioning diagrams varies with each case due to the number of components placed on each side of the PG. Figure 7 shows an example of positioning diagrams generated for the placement diagram #1 in Fig. 5. The total number of positioning diagrams for each case is summarized in Table 1. Summing up, there exist 32 positioning diagrams generated for any given configuration calculated as follows:

The 32 positioning diagrams of Prius configuration (#9) are illustrated in the Appendix. Our study revealed that not all the 32 diagrams provide feasible kinematic diagrams. Thus, a screening method is required to filter out the positioning diagrams corresponding to infeasible kinematic diagrams.

### Application of Feasibility Theorems.

A set of feasibility theorems are proposed to filter out the infeasible ones. Note that infeasible designs are those that cannot be manufactured due to physical constraints.

#### Groundwork for Feasibility Theorems.

Before stating the two feasibility theorems, the groundwork of the theorems is explained. *First*, as the PG has three nodes to which four components are connected, each 1PG configuration has two components sharing a single node, which is denoted the double-component node. Depending on the double-component node, the naming convention of the 1PG configurations is as follows: (1) *2C-S* if two components share the *sun gear* node, e.g., configuration #1 in Fig. 3, (2) *2C-C* if two components share the *carrier* node, e.g., configuration #5 in Fig. 3, and (3) *2C-R* if two components share the *ring gear* node, e.g., Fig. 3 configuration #9 (Prius).

*Second*, the type of the double-component node is closely related to the feasibility theorems. When two components sharing one node are separated (2C-separate cases), the shafts connected to the double-component node must be extended to both sides of the PG. This extended shaft, which is connected to the double-component node, blocks other shafts from connecting to higher nodes. This results in line intersections, which is equivalent to shafts interference. Figure 8 illustrates an example showing the close relationship between the double-component node and the feasibility of a given design.

*Third*, each positioning diagram corresponds to a unique kinematic diagram as it represents a single component arrangement. The relationship between the kinematic diagrams and the positioning diagrams was studied before developing the feasibility theorems. For a given component arrangement, e.g., the positioning diagram in Fig. 8(a), the components are connected to all the possible PG kinematic configurations, which are the kinematic diagrams of the PG illustrating the shafts connections to the nodes. A unique feasible kinematic diagram was obtained for each component arrangement, i.e., positioning diagrams. Therefore, each positioning diagram with a unique arrangement corresponds to a unique kinematic diagram.

#### Feasibility Theorems.

Since the generation of positioning diagrams is much simpler than that of the kinematic diagrams, the proposed feasibility theorems are applied on the positioning diagrams.

*Feasibility theorem #1*: For positioning diagrams having three branches on one side of the PG (case 1 or 3):

- —
*For the 2C-together case (0-PG-3)*: Draw a vertical line starting from the component(s) connected to the ring gear down to the sun gear level: If this line intersects one branch, the positioning diagram is infeasible. Otherwise, it is feasible. - —
*For the 2C-separate case (1-PG-3)*:If the node sharing two components is at the highest (ring) gear, the feasibility theorem of this case is identical to that of case 1 (0-PG-3).

If the node sharing two components is at either sun gear or carrier, draw vertical line(s) down to the sun gear level starting from the node(s) higher than that to which the double-component node is connected. If the lines (line) intersect(s) all the branches connected to the lower node(s), the positioning diagram is feasible. Otherwise, it is infeasible.

*Feasibility theorem #2*: For positioning diagrams having two branches on one side of the PG (case 2 or 4):

- —
*For the 2C-together case (1-PG-2)*: When the single branch is connected to the sun gear or carrier, draw a vertical line starting from the component(s) connected to the ring gear node down to the sun gear node level. If this line intersects one branch, the positioning diagram is infeasible. Otherwise, it is feasible. When the single branch is connected to the ring gear, all the positioning diagrams are feasible. - —
*For the 2C-separate case (2-PG-2)*: Draw a vertical line from the component(s) connected to the ring gear down to the sun gear node level. Note that when the double-component node is at the ring gear, you should draw a line starting from both components in*both sides of PG.*When the double-component node is the ring gear: If the line intersects a branch on each side of the PG, the

*positioning diagram is feasible. Otherwise, it is infeasible.*When the double-component node is either the sun gear or carrier: If the lines intersect one branch on any side of the PG, the positioning diagram is feasible. Otherwise, it is infeasible.

For a better understanding of the above theorems, application examples of each theorem are illustrated in Figs. 9 and 10. The feasibility theorems were applied to the 32 positioning diagrams of configuration #9 (Prius) illustrated in the Appendix. Note that feasibility theorem #1 applies to cases 1 and 3, and feasibility theorem #2 applies to cases 2 and 4 illustrated in Table 1. They also differ depending on the configuration type, i.e., 2C-S, 2C-C, or 2C-R. Thus, one must identify the type of double-component node and the placement diagram case prior to applying the feasibility theorems.

#### Proof of Feasibility Theorems.

For a better understanding of the above theorems, their proof is provided. In order to understand the following, the reader must be aware that if a vertical line is drawn down from one component, there are two possibilities:

If it intersects one branch connected to a lower level, it means that the component connected to that branch is located further from the PG than that from which the line is drawn.

If it does not intersect a branch connected to a lower level, it means that the component connected to that branch is closer to the PG than that from which the line is drawn.

For instance, suppose a vertical line is drawn down from the component connected to the carrier. If it intersects the branch connected to the sun gear, it means that the component connected to the sun is further from the PG than that connected to the carrier. If it does not intersect the branch connected to the sun gear, it means that the component connected to the sun is closer to the PG than that connected to the carrier.

##### Proof of feasibility theorem #1.

- (1)
*2C-together (0-PG-3) case*: In this case, all the four components are placed on the same side of the PG. Suppose a vertical line is drawn from the component(s) connected to ring gear down to sun gear level. If this line intersects one branch, then one of the components connected to lower levels (sun and carrier) is closer to the PG than the one connected to the ring gear, while the other is further from the PG than the one connected to the ring gear. This implies that the ring gear component is located between two components connected to the carrier and sun gear. Such a case is infeasible because the component(s) placed between the PG and the component connected to the ring gear must be placed in a hollow shaft as illustrated in Fig. 9(a). This case is infeasible because a huge hollow shaft is needed to embody the component(s) placed between the PG and the ring component. In fact, placing either the engine or EMs inside a hollow shaft is infeasible because both are relatively large and must be grounded to a fixed platform.On the other hand, if this vertical line intersects two branches, it means that the components connected to the carrier and sun gear are further from the PG than the one connected to the ring gear. If this vertical line does not intersect any branch, then the components connected to the carrier and sun gear are closer to the PG (see Fig. 9(b)). In both cases, none of the components connected to either sun gear or carrier is placed in a hollow shaft as none of them is blocked by a higher node's branch; thus, none of the shafts is blocked.

- (2)
*2C-separate (1-PG-3) case*: In this case, three components are placed on the same side of the PG via three different nodes, while one node must be shared by two components placed on each side of the PG. Since the double-component branch can be connected to the sun gear, the ring gear, or carrier, each case is treated separately hereafter.*2C-R (two components sharing the ring gear)*: This case is treated the same as the 0-PG-3 case as the double-component node is the highest node (ring); thus, the above explanation of the previous case is valid (see Figs. 9(c) and 9(d)). The shaft connected to the ring gear extends in both sides of the PG. The transition of the shafts connecting the other components to the carrier and sun gear is restricted similarly as explained in the previous case.*2C-C (two components sharing the carrier) and 2C-S (two components sharing the sun gear)*: Suppose a vertical line is drawn down from the component connected to the ring gear down to the sun gear level. In order for the design to be feasible, this line must intersect both branches connected to the carrier and sun gear, which implies that the component connected the ring gear shall be the closest to the PG. Note that the ring gear level is higher than that of the double-component node (carrier or sun gear). In this case, the branch of the double-component node extends to both sides of the PG. Hence, if the vertical line intersects any branch connected to lower nodes, it blocks the shafts connected to the ring node, which results in either lines intersections or a component blocked by the arms of a gear or carrier.

##### Feasibility theorem #2

- (1)
*2C-together (1-PG-2) case*: In this case, there is always one branch connected to one side of the PG, while the other has two branches. The level of the single branch is very important because it blocks the other branches (see Figs. 10(a) and 10(b)).When the single opposite branch, i.e., the branch connected alone to one side of the PG, is connected to the ring gear, it does not affect the feasibility of the design because the shafts of components connected to lower nodes can be connected either through the left or right side of the PG.

When the single opposite branch is connected to the carrier, the shaft connected to the ring gear is restricted. The vertical line drawn from the ring gear level must intersect the branch connected to sun gear, as the component connected to the ring gear should be the closest to the PG. If not, the component connected to sun gear will be blocked by the ring gear shaft.

When the single opposite branch is connected to the sun gear, the shafts connected to the ring gear and carrier are restricted. The vertical line drawn from the ring gear level must intersect the branch connected to the carrier, which implies that the component connected to the ring gear should be the closest to the PG; otherwise, the component connected to the carrier if blocked between the arms of the ring gear.

- (2)
*2C-separate (2-PG-2) case*: In this case, there are two branches located on each side of the PG. Based on the double-component node, each case is treated separately as the double-component node closely affects the feasibility of a given design as explained below.*2C-R (two components sharing the ring gear)*: In this case, two components are connected to ring gear. Thus, two lines are drawn from each component down to the sun gear level. If both lines intersect a lower branch, the design is feasible as the components connected to the ring gear must be the closest to the PG. If the line from any components does not intersect any line, the component connected to the lower node is blocked by shafts of higher nodes (see Fig. 8).*2C-C (two components sharing the carrier) and 2C-S (two components sharing the sun gear)*: The vertical line drawn from the ring gear must intersect the branch connected to lower node. This implies that the component connected to ring node shall be the closest to PG. Note that in this case, only one side of the PG has a component connected to the ring gear. If the component connected to the ring is furthest from the PG, it implies that the component connected to the carrier is blocked by the ring gear shaft.

#### Automation of Feasibility Theorems.

Figure 11 illustrates the flowchart of automation of the proposed schematic design methodology. Note that the placement diagrams are used to help generate and classify the positioning diagrams, while the positioning diagrams guarantee finding all the possible powertrain components' arrangements. As each positioning diagram has a unique kinematic diagram, the feasibility theorems are applied to identify feasible arrangements without actually drawing the kinematic diagrams of each positioning diagram. Another advantage of using the positioning diagrams is that they are represented by matrices when automating the design process.

### Direction for Kinematic Diagrams Generation.

In order to draw the kinematic diagram of a given positioning diagram properly, few mechanical features and constraints need to be considered. Based on these considerations, a few directions are given as follows:

- (1)
The engine is relatively large compared to the other powertrain components and is not symmetrical around the cylindrical axis. The relatively large size and the asymmetric design of the engine have two major implications on the powertrain design:

- (i)
The engine cannot be placed between the other powertrain components. Thus, it should always be located the furthest from all the other components.

- (ii)
The engine can be connected in two different ways to the PG: either directly to the main shaft as illustrated in Fig. 9(d) or to a hollow shaft using a gear train as illustrated in Fig. 9(b). Note that the gear train is not required to connect a component to a hollow shaft, except for the engine.

- (i)
- (2)
EM 1, EM 2, and the PG have a cylindrical structure, which is symmetrical around the central axis. Thus, a compact design is obtained if the three components share the same central axis. Hence, when drawing the kinematic diagram, it is recommended to align the EMs and the PG as this will provide an optimal arrangement.

- (3)
The lines of kinematic diagrams must not intersect with each other because intersection means that two shafts intersect, which is mechanically infeasible (see Fig. 9(a)).

- (4)
The components cannot be placed inside of the hollow shafts because the stator of the EMs has to be fixed to a fixed platform (see Figs. 9(a) and 9(b)).

These directions are applied when generating the kinematic diagrams for each positioning diagram. Note that the size and shape of the powertrain components are the major factors considered.

### Selection of Good Designs.

Using the proposed feasibility theorems, one can generate all the feasible kinematic diagrams for any given 1PG split configuration. In the following, a few guidelines are suggested to help designers identify the good kinematic diagrams and select the best among the feasible ones. Note that good kinematic diagrams are defined as feasible kinematic diagrams that meet certain design criteria, such as manufacturability, packaging, maintenance, and cost.

In order to select the good kinematic diagrams, three design criteria are proposed as follows:

- (1)
Avoid using gear trains to connect the engine to the PG. The less number of gear trains used, the simpler the design and thus, the easier its manufacturability and maintenance. Easy manufacturability implies easy packaging and lower cost.

- (2)
Avoid using two hollow shafts. This implies to avoid connecting three components on one side of the PG. Indeed, such a design complicates the manufacturability as well as packaging, which increases the total fabrication cost.

- (3)
Minimize the number of hollow shafts down to one on each side of the PG, if applicable. This implies that separating two components sharing one node (2C-separate cases) should be avoided. It will further facilitate the manufacturability and packaging which reduce the total cost.

Screening out the undesirable designs is a complementary step introduced to reduce the pool of feasible kinematic diagrams. It aims to help a designer select the best kinematic diagram for a given configuration. Thus, more design criteria may be applied to further reduce the number of good designs or vice versa. For instance, the final drive is best connected directly to the PG without using a chain drive, as it is likely to cause the noise vibrations harshness (NVH) issues. Another design criterion that could potentially be applied is that the engine is better placed either alone in one side of the PG or along with one EM. This provides a better packaging.

## Case Study: Configurations #7, #9 (Prius), and #12 (Volt)

The feasibility theorems were applied to several configurations, and the obtained results confirmed that a given 1PG configuration can have multiple feasible kinematic diagrams. Furthermore, the results revealed that the connection of the double-component node, i.e., the node sharing two components, affects the feasibility of the kinematic diagrams. Defining the sun gear as the lowest node, the results showed that the lower the double-component node, the fewer the number of feasible kinematic diagrams. In fact, there exist 12 feasible kinematic diagrams for any configuration type 2C-S, 16 for any configuration type 2C-C, and 18 for any configuration type 2C-R.

Given the huge design space, the feasible and good kinematic diagrams obtained for 3 out of the 12 1PG configurations, i.e., #7, #9, and #12, are illustrated and further discussed in this section. This section focuses on these three configurations as a recent study found that configurations #7, # 9 (Prius), and # 12 (Volt) have the best performance among all the existing 1PG configurations, i.e., the 12 1PG configurations illustrated in Fig. 3 [21]. Note that Volt has four operating modes, two EV and two hybrid modes, employing three clutches. These clutches, however, are not considered in this study, as it focuses more on its hybrid mode. For the design of multimode 1PG split hybrids, readers are referred to the paper by Zhang et al. [22].

Figures 9 and 10 illustrate four of the Prius kinematic diagrams that have already been patented. Note that these four feasible diagrams are just a subset of the 18 feasible kinematic diagrams that are obtained using the proposed method. For instance, Fig. 10(b) is the kinematic diagram used in the current Prius second-generation drivetrain. It has been registered as a patent in the U.S. (Patent No. US6886648) in 2005 (see Fig. 1(d)). Note that the 18 feasible arrangements of Prius configuration are illustrated in the Appendix. Similarly, four examples of the 18 feasible kinematic diagrams of configuration #12 (Volt) are illustrated in Fig. 12. Although several references tried to illustrate the actual Volt kinematic diagram used by GM, so far, to the best of our knowledge, GM has not actually disclosed any information concerning the Volt's components arrangement [23]. Furthermore, configuration #7 could not be found through the patent search. Note that some of these Prius kinematic diagrams have not yet been registered as patents and thus can be used by any other automobile manufacturers. Of course, whether these nonpatented feasible Prius kinematic diagrams are practically good designs need to be answered before further developments. For instance, based on the good design selection criteria proposed in this paper, among the four feasible Prius kinematic diagrams illustrated in Figs. 9 and 10, only the kinematic diagrams illustrated in Fig. 10(b) is considered a good design. Similarly, for Volt configuration #12, only the kinematic diagrams illustrated in Fig. 12(b) are considered a good design. The three kinematic diagrams illustrated in Figs. 10(d), 12(a), and 12(c) are screened out by the second design criteria, while Figs. 10(d) and 12(d) by the third design criteria, and Fig. 9(b) by the first design criteria.

Table 2 summarizes the good kinematic diagrams obtained for configurations #7, #9 (Prius), and #12 (Volt). In the case of Prius configuration, two out of three good kinematic diagrams shown in Table 2 have already been registered as patents. The kinematic diagram #9-1 in Table 2 has not yet been protected by any patent, to the best of our knowledge. For the case of configuration #12, Mi et al. claimed that the second good kinematic diagram, i.e., #12-2 in Table 2, is Volt's [23]. However, from the analysis of the 3D drawings of Volt's powertrain, the authors believe that the first good kinematic diagram, i.e., #12-1 in Table 2, is the actual Volt's diagram. Finally, none of the good kinematic diagrams for configuration #7 could be found through the patent search. Note that all the good kinematic diagrams introduced in Table 2 fall into case #2 of positioning diagram categories. It is mainly due to the second and third good design selection criteria proposed. It was also found that the number of good designs is influenced by which node the engine is connected to. In addition, the results revealed that the number of good designs is influenced by which node the engine is connected to. In fact, for all the 12 configurations, when the engine is connected to the ring gear, there are only two good kinematic diagrams, e.g., configuration #12 in Table 2. On the other hand, when the engine is connected either to carrier or sun gear, there are always three good kinematic diagrams, e.g., configurations #7 and #9 (Prius). This difference in the number of good designs is due to the first good design selection criteria.

## Comparison of Existing and Proposed Kinematic Diagrams Generation Methods

Using the graph theory, which was so far the commonly used method to generate the kinematic diagrams for any given geared system, two feasible kinematic diagrams can be obtained for configuration #9 (Prius), and they are illustrated in Figs. 9(a) and 10(b) [15,24]. On the other hand, using the methodology introduced in this paper, 18 feasible kinematic diagrams can be found, among which the few feasible examples are illustrated in Figs. 9 and 10. Besides, among the two feasible kinematic diagrams obtained using the graph theory, one is considered good based on the proposed selection criteria. However, as illustrated in Table 2, using this new method results in three good designs that can be easily manufactured and packaged. Thus, the newly proposed methodology to enumerate feasible kinematic diagrams for any given lever configuration guarantees considering the entire design space, finding all the feasible arrangements, and selecting the best designs.

This methodology and the approach introduced in this paper to generate the kinematic diagrams become more attractive when considering systems using more than one PG. In fact, this work is the groundwork for the next level of this study, which is the kinematic diagrams generation of split hybrid configurations using more than one PG. The authors expect that the kinematic diagrams would then play a key role in the selection of optimal configurations, as some optimal configurations might be not good for packaging and manufacturability.

## Conclusion

This paper proposes an automatic process to generate all the feasible kinematic diagrams for a given 1PG powertrain configuration. First, the placement diagrams, specifying the position of components with respect to the PG, are introduced to facilitate the generation of positioning diagrams, which specify the relative position of each component. Second, thanks to the two feasibility theorems, the infeasible positioning diagrams are filtered out. Based on the one-to-one relationship between positioning and kinematic diagrams, feasible kinematic diagrams can be easily generated from the feasible positioning diagrams. Since many feasible kinematic diagrams are available for a designer to choose from, a few design criteria are proposed to select the good kinematic diagrams. The proposed process is applied to three different configurations including two famous ones: Prius and Voltec first-generation. These case studies revealed that a few Prius and Volt kinematic diagrams have not yet been registered as patents. Toyota filed multiple Prius patents over the last 15 years, which suggests that the process was based on intuition and engineering insights. Using the proposed process, the patented designs along with new designs can be generated in few hours, if done manually, and in seconds, if programed. Two configurations (#7 and #12 (Volt)) that potentially have high performance have been also investigated, and the authors found that all the good kinematic diagrams of configuration #7 are still not protected by any registered patent. This research focused on the PS-HEV using a single PG as it is the simplest case. However, depicting the two PG split configurations, which is more useful but challenging, will be studied. The authors expect that this method will shine more when dealing with more complex systems.

## Acknowledgment

This work was supported partially by Basic Science Research Program through the National Research Foundation of Korea (NRF) by the Ministry of Science, ICT and Future Planning under Grant No. 2013R1A1A1060397 and by Technology Innovation Program by the Ministry of Trade, Industry, and Energy under Grant No. 10051876.

### Appendix: Application of Feasibility Theorems on the 32 Positioning Diagrams of Configuration #9 (Prius)

The feasibility theorems were applied to the 32 positioning diagrams of configuration #9 (Prius) as shown in Fig. 13. The feasibility of each positioning diagrams is determined using the feasibility theorems #1 and #2.