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Towards Automated Synthesis of Automatic Automated Transmission Designs

The design of automatic transmissions is a challenging task due to complex power flows and high variability of architectures. To overcome those difficulties a methodology is proposed which uses computational design synthesis to generate transmission designs. The methodology is efficiently dealing with the complexity and variability of transmissions while the engineer is focusing on the application of his engineering knowledge. An analysis of 1022 patented transmissions is shown and highlights the trend of increasing complexity in order to achieve higher capability. The generation of innovative transmission designs is presented producing designs with improved properties.


INTRODUCTION
The design process of an automatic transmission is driven by its purpose. The transmissions main purpose is the optimal propagation of the engine power to the driven axis. In the case of an automotive application a transmission is required to have multiple forward and at least one reverse gear. In practice the design of new automotive transmission designs is often based on centuries of experience and existing designs (1) .
Transmissions are designed with multiple requirements in mind. The top requirement is a long service life which in case of automatic transmissions is affected by the design of the planetary gears. Further three of the top five requirements are low production costs, an optimal range of ratios and optimal gear steps (1) . However a low production cost can only be achieved with simple designs, while an optimal range of ratios and optimal gear steps require complex designs. Thus the transmission design process aims to find designs that maximize capability and minimize complexity.  Presented in this work is a methodology able to efficiently generate transmission designs with the maximal capability using a fixed set of components. The presented methodology is applied to the generation of transmission design with complexity 8 (highlighted in Figure 1) with the goal to generate designs with four or more forward speeds and at least one reverse speed. The generation of three novel and one already patented four-speed transmission designs is presented. For each of the novel generated architectures the authors were not able to find patents that include the same designs.

Research Paper 20184115
Henri Karhula et al./International Journal of Automotive Engineering Vol.9, No.4 (2018)  An efficient execution of these steps provides the designer with a set of automatically generated and optimal transmission architectures for an automotive application.
Section 2 provides a brief overview of related work on the subject. Section 3 describes the scientific methodology applied for this research. In Section 4 the model-based design approach is presented. Section 5 explains the topology generation process and presents the evaluation and optimization of the generated topologies in detail. In Section 6 the results of the application of the presented methodology are presented. Section 7 presents the current research towards automatically generated simulation models for drivecycle simulations. Finally Section 8 concludes the paper and describes possible future work.

RELATED WORK
Modeling transmission components or systems such that the resulting power flows can be analyzed is common practice.
Examples of modeling practices are the lever analogy (2) , bond graph modeling (3,4) and canonical graph modeling (5) . Other approaches define kinematic equations to analyze the power flows of a transmission system (6,7) .
Some research also attempts to generate transmission topologies. Köningseder and Shea employ graph grammars to generate transmission topologies (8) . This approach is however restricted by the topologic and parametric rules that drive the generation process. The used algorithm applies these rules iteratively and is not able to guarantee that all relevant solutions are generated in a finite set of executions.
An effort has been made to use canonical graph characteristics to enumerate all possible transmission topologies (9) and automatically analyze the overall transmission ratios of the generated topologies (10) . However canonical graphs of transmission systems do not represent clutches nor brakes.
Therefore a new graph is required for each clutching condition.

Bayrak et al. present an approach to design hybrid transmission
architectures which generates all possible lever analogy representations for a set of components and applies bond graph rules to determine viable solutions (11) . This exhaustive enumeration of possible solutions is not constrained by engineering knowledge and may create many solutions, which have to be deemed invalid by the application of bond graph rules. Based on these generated architectures Bayrak et al. present a genetic optimization approach in order to find optimal designs (12) .
Raghavan et al. (13) present an approach which enumerates all possible kinematic combinations of transmission elements and then searches for viable candidates based on the transmission governing equations. This approach has proven to find novel transmission architectures, including the design for the 8L90 transmission produced by General Motors (14) . Their generation requires the designer to decide how many shifting elements have to be engaged at any given time, constraining the design space unnecessarily.
A similar to the approach of is used by Zhang et al. for the generation of hybrid power-split transmissions with one or two planetary gears (15,16) . The process first exhaustively enumerates all possible kinematic combinations of elements and then filters out invalid or inferior designs through analysis and simulation. This methodology is also used by Zhuang et al. for the generation of power-split transmissions with three planetary gears (17) . With three planetary gears there exist 1.67 × 10 11 possible topologies.
However in this application a subset of 19,085,220 topologies were enumerated initially and 14 designs were found through evaluation, that could exceed existing an existing design in terms of performance.
Moreover various research exists which aims to optimize topological and/or configuration parameters of gear trains and/or gear contacts (18,19) . Other research aims to optimize the efficiency of drivetrains with automatic planetary transmissions by optimizing the shifting behavior (20,21) .
The presented work does not alter shifting behavior or detailed gear contact parameters and instead shows that optimal designs for a transmission system can be achieved through the variation of the transmission topology and optimization of planetary ratios. Further this work presents an approach for the generation of transmission topologies based on design knowledge rather than an enumeration of all possibilities.

SCIENTIFIC METHODOLOGY
The aim of the presented methodology is the generation of novel, innovative transmission designs that provide a higher  Table 1 shows all patented designs with complexity 8 of the patent analysis of Figure 1. Again the complexity is defined here as the sum of shifting elements and gear contacts. It is obvious that the most of these simple designs use two single planetary gears, two clutches and two brakes. These designs provide up to four forward speeds and (as it is common in automotives) exactly one reverse gear. Table 1 Patented transmission designs with complexity 8.
Since only one patented four-speed design exists with complexity 8, this data motivated the generation of novel fourspeed transmission designs using two single planetary gears, two clutches and two brakes. The results of this generation process are presented in Section 6 and show that this methodology is able to generate novel designs with maximized capability.
The following Sections 4 and 5 will describe the methodology in detail.

MODEL-BASED TRANSMISSION DESIGN
The presented methodology requires the designer's input solely during the creation of the initial declarative design model to implicitly define the set of automatic transmissions. This model holds the information on the transmission components and the designers intent and expertise.
The transmission components are modeled as blocks with ports that define their connectivity to other components. A detailed representation of the modeling language is given by Nicolai et al. (22) . In this work only necessary explanations of parts of the language will be given.   The declarative design model used in this research has additional constraints that represent engineering knowledge and the designer's intent. Figure 4 shows the visual representation of these constraints. Three different types of constraints are used: Prescriptive connections (P1), limiting connections (L1) and restrictive connections (R1-R5). The concepts behind these constraint types is explained in detail by Nicolai et al. (22)   such that invalid and unfeasible solutions are discarded. This is followed by the optimization of the design parameters (Section 5.3) since the quality of each design should be evaluated based on its optimal configuration.

Topology Generation
For the topology generation a constraint problem is set up, where the solutions are numerical representations of individual topologies. The solving of the constraint problem aims to find a set of connections between components, which results in a connected system. This process is constrained by various constraints inherent to the modeling language.
As the problem is solved, new constraints are added which avoid each topology to be generated again. Thus the problem is solved and further defined iteratively, until the process is stopped or the solution space is empty (22) .
As previously mentioned the solutions define which connections compose a connected system. Based on this solution each system (i.e. topology) is instantiated and can be evaluated for its viability as the architecture of an automatic automotive transmission.

Evaluation
The evaluation of the potential transmission architectures aims to determine the quality of each generated topology, such that the best candidates can be used for further development. The aim of the presented methodology is to determine the quality of each design based on automatically generated simulation results.
However simulations can be expensive compared to quick numerical evaluations. In order to reduce the cost of the evaluation by simulation, two transmission-specific analyses are executed first.
These determine the manufacturability and feasibility for the use in automotives of each generated transmission topology.
These analyses allow a set of topologies to be discarded before simulation, if those solutions are invalid or unfeasible. This can be used to discard topologies that do not support a user-defined number of necessary forward and reverse speeds.
After the execution of this analysis the methodology has produced a set of valid and feasible transmission architectures for an automotive application.

Optimization
In order to determine the best candidates from the generated transmission architectures, an optimization process is applied to each design. The optimization aims to minimize an objective function by changing the design parameters. In this application the planetary gear ratios are the only design parameters. A planetary gear ratio is defined as floating point numbers and is not necessarily the ratio of two numbers of gear teeth, but can be approximated by them.
The optimization aims to find optimal values for the planetary gear ratios , such that four-speed configurations are generated.
For this it is assumed that the optimal gear step is = 1.4 , the optimal total range of transmission ratios is = 4 = 3.8416 , the optimal transmission ratio of the highest gear is  (23) . The function ( ) returns a higher value for lower mean service life expectancy.
Further the optimization is restricted by the following inequality constraints: where N fwd is the number of forward speeds, N rev is the number of reverse speeds. The first constraint defines the boundaries of the optimization parameters. The second and third assert that the generated designs support the wanted amount of forward and reverse speeds. The fourth asserts that no transmission ratio is redundant. The last three constraints limit the variation of the reverse gear, lowest gear and total range from their optimal values to 50%.
In order to create a pareto front of optimal design parameters for each generated transmission design the following function ( ) is minimized for a range of values for ∈ [0,1] such that all constraints are considered. Copyright  2018 Society of Automotive Engineers of Japan, Inc. All rights reserved generated simulation models for all generated architectures and is further discussed in Section 7.

GENERATED TRANSMISSION ARCHITECTURES
As described in Section 1 the presented methodology was applied to the automated design of transmission architectures composed of two planetary gears, two clutches and two brakes. The generation of architectures was guided by the declarative design model described in Section 4. For increased efficiency of the generation process all transmission designs with less than four forward speeds and one reverse speed were immediately discarded. In an effort to further compare the generated designs to each other Figure 6 shows the pareto fronts that resulted from the optimization processes for the generated designs four-speed designs. Note that the x-axis is inverted and the maximum gear step sizing error is smaller to the right.
Design b) has a pareto front that consist only of a single point.
This can be explained as this design is only able to conform to all constraints with a single set of planetary gear ratios. In Figure 6 it can be seen that the novel four-speed design a) has a longer mean service life of their planetary gears and gear steps that are closer to the optimal gear step size than the other designs.
This proves that the presented methodology is able to generate designs that are superior to the patented transmission design c) in terms of achievable gear step quality and mean service life.
The exact values of these transmission designs may be improved in future development by a larger optimization loop that includes the evaluation of fuel economy. In order to evaluate the generated designs in terms of fuel economy a simulation is necessary. The following section will present the current efforts towards automatic fuel economy simulations.

SIMULATION MODELS
Such that the generated transmission designs can be evaluated through simulation, each generated transmission topology and its optimized planetary gear ratios have to be represented by a simulation model. To support this process for each generated transmission architecture, the components of the initial declarative design model are augmented with simulation models as described in Section 7.1.
This model augmentation is then used to generate and execute the simulation for each generated transmission architecture as described in Section 7.2.

Model Augmentation
Each can then be used to evaluate each generated architecture (24) .
The augmentation of the components with simulation models is part of the initial creation of the declarative design model. The

Generation and Execution of Simulations
At the current state of development the generation of simulations is not fully automated yet. The process involves the instantiation of the component augmentations in the simulation environment. This is followed by the instantiation of the connections between the simulation models according to each generated architecture. Because the generated simulation models only represent the engine, transmission and driven axle, a World-wide harmonized Light duty Test Cycle (WLTC) simulation cannot be executed yet.
For this a controller system for control of acceleration, deceleration and optimal changing of gears (which is not considered during the generation of transmission architectures) is added to the simulation such that the WLTC is followed.
Finally the simulation results shall be analyzed for the fuel economy of the generated transmission designs, such that the best designs can be chosen for further product development.

CONCLUSION
This