Parking lock integration for electric axle drives by multi-objective design optimization

Due to safety considerations, electric axle drives (e-drives) are often equipped with a parking lock system, which prevents vehicle movement while parking in redundancy with the parking brake. In order to integrate the parking lock into the e‑drive, various mounting positions inside the e‑drive are eligible, which have a direct influence on the e‑drive packaging. Furthermore, engaging the parking lock may happen at small vehicle velocities and while driving downhill, leading to high loads on the e‑drive components. These loads depend on the mounting position of the parking lock and have to be considered in the design phase to prevent failure of the system. That way, the designs of shafts, gear wheels and bearings of the gearbox are affected by the parking lock integration. A suboptimally integrated parking lock system can thus lead to undesirably high costs and reduced energy efficiency of the entire e‑drive—all alongside the packaging aspect. Consequently, finding the best suitable parking lock integration for a certain e‑drive is a complex task for the design engineers. To reduce the level of problem complexity, an established computer-based system design method for e‑drives by means of a multi-objective optimization is extended to be capable of considering the parking lock integration. The proposed method is applied to a case study and the impact of the parking lock on the optimality of an exemplary e‑drive system is shown.


Introduction
The market for passenger cars continues its shift from internal combustion engines (ICEs) towards electric drive systems. Accordingly, current predictions for electric vehicles (EVs) indicate an exponential growth of global sales by around 1700% in 2030 compared to 2020 [1]. In order to successfully perform the transition towards electrified road mobility, it is a key task for automotive original equipment manufacturers (OEMs) and suppliers to adopt their products and strategies quickly to the upcoming market changes [2]. The optimal designing of the vehicle's powertrain plays a major role in this context.
Although many different electric powertrain architectures exist, the most common one is a purely electric axle drive (e-drive) that consists of the main components "power electronics unit", "electric machine" (EM) and "gearbox", which are schematically depicted in Fig. 1. In the design process, all of these components need to be designed to optimally fulfill the set requirements. Exemplary, minimum costs, maximum efficiency and favorable package integration of the e-drive system are typical concurrent design objectives. In order to find optimal designs regarding all set design objectives, a holistic multi-objective design optimization on system level "e-drive" is required as presented e.g. in [3].
An additional subsystem in e-drives, which is not considered in comparable holistic design methods as described in [3], is a so-called parking lock (exemplary depicted in Fig. 1 on the input shaft of the gearbox). A parking lock might be required due to considerations regarding the vehicle's safety: in redundancy with the vehicle's parking brake, the parking lock prevents vehicle movement while parking by mechanically locking the drivetrain. This function is typically achieved by mounting a parking lock wheel to a shaft inside the gearbox and a mechanically or electrically actuated pawl, which connects the parking lock wheel to the gearbox housing when the parking lock is engaged [4]. That way, an interlocking connection between housing and shaft is made and any rotation of the drivetrain is prevented. A simplified schematic of a parking lock is shown in Fig. 2.
Numerous works have been published that cover the designing of such a parking lock, e.g. [4,9,10]. They give indications about the functional requirements on the parking lock system and mathematical relations needed to design the parking lock wheel, pawl and actuator. However, especially in the context of e-drives, no published work provides a method for finding the most suitable integration of the parking lock in the powertrain. In order to close this research gap, an established design optimization method for e-drives is extended in the present work to be capable of finding the optimal integration of a parking lock. When integrating such a parking lock in the e-drive system, various mounting positions are eligible (e.g. on the input, intermediate or output shaft of a single-speed, twostage gearbox as depicted in Fig. 1). These mounting positions have a direct influence on the installation space demand of the gearbox. Furthermore, special load cases occur inside the drivetrain induced by an engagement of the parking lock at small vehicle speeds and at a certain slope of the road (e.g. speed < 5 km/h and a slope of 30% [4]) or by towing the vehicle when the parking lock is engaged. In general, the former is critical for vehicles with separately driven axles (as common in EVs), the latter for vehicles with permanent all-wheel drives in conventional ICE-based transmissions [4]. The described load cases can result in high torques in the drivetrain (much greater than any nominal operating torque as shown in Sect. 2) and thus impose a b c d Fig. 2 Schematic illustration of a parking lock; a) parking lock wheel, b) pawl, c) actuator, d) pivot connecting the pawl to the gearbox housing K high loads on the mechanical components (e.g. shafts, bearings and gear wheels). These loads need to be considered in the design phase to prevent a failure of the system [5]. Moreover, the loads on the mechanical components depend on the mounting position of the parking lock inside the gearbox, meaning a suboptimally integrated parking lock will result in higher loads and require larger shafts, bearings and gear wheels than the optimum. This in turn can lead to an undesirable increase of the e-drive costs and a decrease of the energy efficiency-all alongside the packaging aspect.
Due to this wide-ranging impact on the e-drive system, design engineers are confronted with a complex task when integrating a parking lock into the e-drive system. To reduce this complexity, an established computer-based design method for e-drives, described in [3,6,7], is extended to be capable of integrating the parking lock by means of a multiobjective optimization on e-drive system level. The present work was first published in [15].

Methodology
For holistically designing the e-drive system, the already established design method described in [3,6,7] is briefly outlined in the following. Additionally, extensions to this method for the parking lock integration are suggested. A schematic illustration of the design method based on a multi-objective optimization at e-drive system level is shown in Fig. 3. The method is applied in the early development stages with the aim to quickly find optimal design solutions for a given e-drive design problem. The resulting set of optima then serves as basis for decision-making in the subsequent development steps.
The input to the design method is represented by the defined requirements on the e-drive including the desired optimization objectives (e.g. minimal costs, maximal energy efficiency, favorable package integration of the e-drive in  a given installation space). This information is then passed to a computer-based optimization phase, where certain design parameters of the e-drive are varied. Exemplary, in the case study presented in [7], 13 parameters of the electric machine (including dimensions of stator and rotor as well as magnet dimensions and the magnet arrangement of a permanent magnet synchronous machine), 23 parameters of the gearbox (including shaft dimensions, gear parameters, total center distance and gearbox topology) are directly formulated as optimization parameters and the power electronics unit is considered via optimization constraints. Based on the requirements, the e-drive analysis involves electromagnetic simulations of the electric machine to verify that a sufficient torque-speed characteristic and a sufficiently low heat output for operation combined with the cooling system is present. Furthermore, the load capacity of the e-drive components to achieve the required service life based on a load spectrum is investigated and aspects regarding noise, vibration, harshness (NVH) are considered. Additionally, it is verified that the maximum permissible junction temperatures of the semi-conductor elements in the power electronics unit are not exceeded and the voltage ripple at the DC side is sufficiently low. Besides verifying that the exemplary described requirements are fulfilled, the e-drive analysis involves the calculation of the e-drive properties (e.g. costs and energy efficiency) based on the provided values of the design parameters. After calculating the system properties, the optimization algorithm-following the concept of differential evolution [8]-performs a rating of the design variants and decides about the next parameter sets to be evaluated. The described closed loop of system analysis and design synthesis is repeated until no more improvements in the optimization objectives are observable and converging behavior is present. The output of the method then consists of all found optimal e-drive design variants, which corresponds to the socalled Pareto front in the context of a multi-objective opti- In order to extend the described method with the capability to integrate the parking lock, two major aspects need to be discussed. The first aspect covers the geometrical integration of the parking lock into the e-drive package, the second aspect covers the calculation of the parking-lockinduced load cases and the consequences for sizing the mechanical components in the e-drive.

Geometrical integration of the parking lock
As already mentioned in Sect. 1, various mounting positions of the parking lock in the e-drive system are eligible. In the following, the focus is put on e-drives with a single-speed, two-stage helical gearbox with an offset between EM axis (gearbox input shaft axis) and axle (gearbox output shaft axis), which is also schematically depicted in Fig. 1. Furthermore, the rotor of the electric machine is directly mounted on the gearbox input shaft and this shaft is supported by three bearings. This architecture will later be used in the case study presented in Sect In the following, a given architecture of an electrically actuated parking lock is considered as shown in Fig. 5, which will later be used in the case study presented in Sect. 3. For such a parking lock-independent of the mounting position cPLMount-further degrees of freedom exist when integrating the parking lock. These include the axial placement of the parking lock on the shaft yPL, the mounting angle of the parking lock on the shaft ΘPL, the mounting angle of the actuator Θa and two flipping parameters fPL and fa, which allow to rotate the parking lock and the actuator by 180°around the z-axis respectively. A visualization of all possible combinations of the parameters fPL and fa as well as the definition of the parameters yPL, ΘPL and Θa (Θa is measured from the closest possible position to the shaft) are depicted in Fig. 5. The origin of the shown coordinate system (x, y, z) is defined by the position parameter cPLMount.
That way, a total of six design parameters (cPLMount, yPL, ΘPL, Θa, fPL, fa) are used to define the geometrical integra- tion of the parking lock in the e-drive system. In order to extend the established design method depicted in Fig. 3 and described in [3,6,7] by the capability to integrate the parking lock, these six parameters are defined as additional optimization parameters for the optimization algorithm. Moreover, certain restrictions apply when geometrically integrating the parking lock. In particular, the body of the parking lock must not clash with any other components in the e-drive system (e.g. with shafts, bearings, gear wheels, rotor and stator of the electric machine). As the e-drive system and the parking lock itself in general show a 3D shape, a clash detection in 3D needs to be performed. This is done by the e-drive analysis, which uses 3D-CAD methods to automatically check for any clashes [13]. That way, it is determined if the integration suggested by the optimization algorithm is valid or not. In case clashes are identified, the design variant is considered invalid and automatically removed from the optimization loop.

Consideration of parking-lock-induced load cases
Aside from geometrically integrating the parking lock and checking for clashes, further constraints need to be satisfied. Specifically, the load cases induced by the parking lock, which are described in Sect. 1, must not lead to a failure of the mechanical components inside the e-drive system. In order to determine if a sufficient load capacity of the mechanical components is present or not, a suitable model of the drivetrain and vehicle to approximately calculate the parking-lock induced loads on the components is required. Based on the loads determined by the model, further criteria need to be applied to determine the load capacity of the mechanical components. For that purpose, the minimal information required is the torque transmitted by both gear pairs and the acting torque on the parking lock.
To calculate these torques, a simple multibody simulation (MBS) model is presented in the following, which requires low computational effort and is thus well suited for the application in the design optimization loop shown in Fig. 3. The mechanical model for mounting positions "INPA", "INTA" and "INTC" is shown in Fig. 6 and represents a rotational system composed of four masses, which are connected by rotational spring-damper combinations. For reasons of simplicity, all kinematic and kinetic quantities are referred to the level of the gearbox input shaft. The actual torques and velocities at the intermediate and output shaft can be calculated by applying the transmission ratios of the two gear pairs. The torques acting in the springs and dampers are given by T c = c '; (2) respectively, where c is the spring rate, d the damping coefficient, Δφ the relative angle between the two end points of the spring and P ' the relative angular velocity between the two end points of the damper. The e-drive with its moment of inertia Jed around the pitch axis of the vehicle is mounted in a suspension with spring rate cS and damping coefficient dS. The parking lock is connected to Jed by its spring rate cPL and damping coefficient dPL. Following the torque paths visualized in Fig. 6, a certain spring rate c1 and damping coefficient d1 accounts for the elasticity and damping of the path between parking lock and the nearest gear wheel. The rotor of the electric machine with moment of inertia JEM is connected to this gear wheel by c2 and d2. The elasticity and damping of the remaining drivetrain connecting the single wheels with moment of inertia Jw and the described gear wheel are considered by crDT and drDT. The gear wheel itself represents an intersection of the three spring-damper combinations and is modelled with moment of inertia Jgw. Analogously, the model for mounting positions "INPB" and "INTB" is defined and shown in Fig. 7. The model is slightly simpler due to the missing spring-damper combi- nation defined by c1 and d1. The intersection of the three spring-damper combinations is now at the parking lock wheel, which is modelled with its moment of inertia JPL. The connection to the longitudinal dynamic of the vehicle is made by the torque rw Ftire acting on the wheels, where rw is the effective rolling radius and Ftire is the longitudinal force at the tire-road contact modelled by the Magic Formula given in [11] (equation (4.49), page 173).
The mechanical model to describe the longitudinal vehicle dynamic is schematically shown in Fig. 8 for a vehicle with a parking lock mounted on the front axle (for a parking lock mounted on the rear axle, the force Ftire would act on the wheels of the rear axle). The vehicle with body mass mb is modelled on a slope with angle β. At the center of gravity (CoG) the gravitational force mbg is acting. The single wheels with unsprung mass mw are connected to the body by spring-damper combinations representing the longitudinal wheel suspension. The spring rates and damping coefficients for the front axle are given by cwsf and dwsf, the ones for the rear axle by cwsr and dwsr, all of them being where Δx is the spring deflection and P x the relative velocity between the two end points of the damper. That way, the force Ftire couples the longitudinal dynamic of the vehicle and wheels with the rotational dynamic of the drivetrain.
Applying the conservation of momentum to the described models yields equations of motion for all masses, which are numerically solved to obtain the torques acting on the parking lock, gear pair 1 (connecting input and intermediate shaft) and gear pair 2 (connecting intermediate and output shaft). For an exemplary e-drive with parking lock mounting position "INTA", the described model is compared to a reference MBS model, which is implemented in Simcenter Amesim [14]. The reference model is more sophisticated than the suggested model and considers the masses of all gear wheels, shafts (including the axle shafts) and the differential as well as a backlash between parking lock wheel and pawl. Additionally, the vertical movement of the vehicle is modeled and the wheel suspensions are considered by their full compliance in longitudinal and vertical direction. That way, the reference model considers 76 state variables, needs the specification of significantly more parameters and requires a higher computational effort for solving than the suggested model, which only uses 12 state variables. Figure 9 shows the obtained torque curves for an initial vehicle velocity of 3.5 km/h and slope β = 0°from the suggested and the reference MBS model. Mainly due to the modelled backlash, the torques from the reference model tend to oscillate more and phase shifts compared to the curves from the suggested model are noticeable. However, the general torque levels are predicted well by the suggested model, which means it is well-suited for determining the load capacity of the mechanical components with low computational effort and a small number of required model parameters-both properties are of importance for an early-stage design optimization as shown in Fig. 3.
In order to calculate the load capacity of the mechanical components based on the obtained torque curves, the quasistationary forces acting on the gear wheels and the bearing reaction forces are determined from start of the simulation until the vehicle speed reaches zero for the first time. After that, the following criteria are applied to determine whether or not a sufficient load capacity for the parkinglock-induced load case is present: the maximal torque acting on the parking lock must not exceed TPLlim (which is specified for a pre-defined architecture of the parking lock), the maximal nominal torsional stress in the shafts must not exceed a specified permissible stress τlim, the minimal static safety factor of the bearings according to ISO 76 must not be less than a specified value S0min, the minimal static safety factors for pitting and tooth bending of the gear wheels according to ISO 6336 must not be less than specified values SH0min and SF0min respectively.
These conditions are checked by the e-drive analysis and in case an insufficient load capacity is determined, the design variant is considered invalid and removed from the optimization loop.
That way, the established design method described in [3,6,7] for e-drives is extended to be capable of holistically integrating the parking lock by adding the parameters regarding the geometrical integration (cPLMount, yPL, ΘPL, Θa, fPL, fa) as optimization parameters and checking the con-  Figure 10 shows a simplified flow chart of the major steps that are performed by the gearbox analysis: The design parameters of the gearbox, including the parameters regarding the geometrical integration of the parking lock, serve as input. With these parameters, the gear geometry is calculated and the bearings for all shafts are selected first. After that, the parking lock is positioned according to the values of (cPLMount, yPL, ΘPL, Θa, fPL, fa) and the detailed geometry of the shafts is determined. Finally, the gearbox design variant is investigated regarding internal clashes of components and the load capacity of the mechanical components is evaluated (including a fatigue analysis based on a load spectrum and the described investigation of parkinglock-induced load cases).

Results & discussion
The proposed method is applied to a case study that involves the optimization of an e-drive with a permanent magnet synchronous machine and an axially attached power electronics unit. The main requirements are given in Table 1, the available installation space is shown in Fig. 12.
In order to investigate the impact of the parking lock integration on the e-drive, several optimization studies are conducted: First, the e-drive design is optimized without a parking lock; second, separate optimizations are performed for all e-drive mounting positions given by (1). That way, the optimal designs for all discussed parking lock mounting positions can be compared and the consequences of requiring a parking lock in the e-drive system by the vehicles safety concept is shown.
The power electronics unit, electric machine and gearbox are subject to optimization and the minimization objectives for all optimizations are defined as e-drive costs, degree of losses of the e-drive based on the WLTC [12] and package metric.
The e-drive costs consider the main cost drivers of power electronics unit, electric machine and gearbox. They are calculated with a cost model according to Table 2: The listed cost criteria are those properties of the e-drive design that have the strongest influence on the total system costs. The degree of losses is the ratio of the e-drive system's loss energy to the vehicle propulsion energy required for driving the WLTC driving cycle [12]. The loss calculation considers conduction losses and switching losses of the power electronics unit, copper losses and iron losses of the electric machine as well as frictional losses in the gearbox originating from the bearings, shaft seals, meshing of the gear wheels and oil splash losses. The package metric is the protruding volume of the installation space by the e-drive in its most favorable position, meaning it takes a value of zero in case the e-drive completely fits inside the given installation space. Furthermore, the parking-lock-induced load case considered in the optimization is defined by a vehicle  Fig. 11. The relative costs are the e-drive costs as a ratio of those of a selected e-drive solution without a parking lock (denoted as "Sel. w/o PL" in Fig. 11). This solution thus shows relative costs of 100%. Furthermore, the costs of the parka b c d Fig. 11 Projections of the obtained Pareto front approximations ing lock itself are excluded to allow a direct visualization of the cost increase caused by the necessity to adapt the e-drive system for a parking lock integration. The upper two charts in Fig. 11 show the projections of the obtained Pareto front approximations for a parking lock mounted on the input shaft and for an e-drive without parking lock (denoted as "w/o PL"). The increase in costs compared to a system without parking lock is significant, especially in the region of low costs and higher losses. However, mounting position "INPA" performs slightly better than mounting position "INPB" in dimensions costs and losses and performs significantly better when comparing the package metric. The mounting positions on the intermediate shaft are shown in the lower two charts of Fig. 11. These positions in general are slightly more favorable than the ones on the input shaft, especially in the region of low costs and higher losses. Mounting positions "INTA" and "INTC" show particular strength when comparing them to the solutions without a parking lock. Similar good objective values are achievable for some regions of the Pareto fronts in the shown 2D-projections (but not simultaneously in all three dimensions).
In the following, one design solution without a parking lock is selected from a region representing a balanced tradeoff between costs and losses (denoted as "Sel. w/o PL" in Fig. 11). This solution shows relative costs of 100%, a package metric of zero (meaning it completely fits inside the installation space) and losses of around 10.08%. Furthermore, a design solution with parking lock that approximately shows the same losses and also has a package metric of zero is selected (denoted as "Sel. w PL" in Fig. 11). This e-drive uses mounting position "INTA" and shows relative costs of around 101%. In other words, an e-drive design with parking lock mounting position "INTA" was found that shows a marginal cost increase of 1% (excluding the costs of the parking lock itself) compared to a similar optimal e-drive without parking lock and otherwise equal objective values. That way, this design solution might be of high interest for the detailed subsequent development phases. A CAD visualization of both selections that is automatically created by the design optimization method is shown in Fig. 12.

Summary & conclusion
An extension to an established multi-objective design optimization method for e-drives is proposed, which enables to find the optimal integration of a parking lock in the e-drive system context. Various degrees of freedom in the geometrical integration of a given parking lock architecture are considered and additional optimization parameters are derived. Moreover, constraints regarding clashes between components and the load capacity of the mechanical system for parking-lock-induced load cases are discussed.
The method is applied to a case study and the various mounting positions of the parking lock inside the e-drive are investigated. The obtained Pareto front approximations show the impact on the system costs, energy efficiency and package integration compared to optimal e-drives without parking lock. Furthermore, in the region of a balanced tradeoff between costs and losses, the results indicate that the integration of the parking lock can be achieved with only a marginal cost increase and otherwise same objective values compared to a system without parking lock. This information is of great value for the subsequent development steps and also enables to assess the consequences of requiring a parking lock by the vehicle's safety concept.
Funding Open access funding provided by Graz University of Technology.
Conflict of interest D. Lechleitner, M. Hofstetter, M. Hirz, C. Gsenger and K. Huber declare that they have no competing interests.
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