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Received 3 February 2019

Revised 27 February 2020

Accepted 1 April 2020

Available online 29 May 2020

Battery electric vehicles

Planetary automated manual transmission

Modelling

Gear shifting control

Torque trajectory planning

A B S T R A C T Integrating a multispeed transmission into the electric drive system provides the potential of ulteriorly improving the overall efficiency of battery electric vehicles (BEVs). In this paper, we propose a novel two-speed planetary automated manual transmission (PAMT) which features a more compact mechanical structure than clutchless automated manual transmission (CLAMT) and easier gear shifting control than power-on transmissions. To present the transient behaviours during the gear shifting, a detailed and original dynamical model of the electrified powertrain is developed, including an electric machine (EM), a two-speed PAMT, synchronizer mechanisms, driveline and vehicle, etc. Afterwards, the gear shifting control system is devised, and the gearshift process is orderly divided into five stages based on the proposed control strategy. Next, three alternative planning torque trajectories, i.e., the third-degree polynomial (TDP), the fifth-degree polynomial (FDP) and the seventh-degree polynomial (SDP), are proposed to control the EM torque at the first and the fifth stages of the gear shifting respectively. Subsequently, a series of simulations are performed to validate which candidate torque trajectory is capable of achieving the optimal gear shifting quality in term of reducing the longitudinal vehicle jerk and shift durations. This study provides a beneficial reference for the development of BEVs' transmission system.

(c) 2020 Elsevier Ltd. All rights reserved.

In future, BEVs with their low local-emission status and potential independence from fossil fuels will provide a valuable and sustainable addition to the mobility landscape [1-3]. To encourage the sustained increase in the demand for BEVs, major economies in the world (e.g., United States, European Union, China and Japan) are releasing increasingly financial and policy incentives [4]. Current electrified powertrain solutions normally consist of an electric machine combined with a fixed-ratio transmission and differential gear. Permitting a relatively low level of integration, their hill-climbing ability, accelerating performance and efficiency are limited too. With the increasing demands, BEVs must satisfy without compromise regarding lightweight, recharge mileage and price, while also providing a high level of safety, reliability and ride comfort [5]. To meet these requirements, further exploiting the potential benefits of the electrified powertrain is essential. Many

Fig. 1. Diagram of an electric powertrain adopting 2-speed PAMT.

investigations indicate that integrating a multiple speed transmission into the electric-drive system can make optimum use of high-efficiency motor operating ranges, improve drivability and cut energy consumption [6-10].

Currently, a variety of multispeed transmissions adopted in BEVs are widely studied. However, both advantages and disadvantages of each transmission configuration coexist. These multiple speed transmissions are generally divided into two types, i.e., power-on and power-off [11]. These transmissions with power-on gearshift capacity consist of dual clutch transmission (DCT) [12,13], planetary automatic transmission (AT) [14-16] and torque-assist automated manual transmission (AMT) [17-19]. Power-on capacity transmissions used in BEVs have been widely studied in the current literature. Ref. [20] designed the gearshift schedule using a graphical approach for two-speed DCT pure electric vehicles. Simulation and experiment results validated that this gearshift schedule improves the dynamic and economic performances of the vehicle effectively. Simulation and experimental studies in Ref. [12] demonstrated the transient behaviours of gear shifting for a two-speed DCT used in BEVs. Refs. [14-16] proposed three diverse types of two-speed planetary transmissions applied to BEVs and the gearshift controllers were also devised. Refs. [7,9] studied a two-speed torque-assist AMT. Their studies also proved that this transmission performs better performance than a single speed counterpart. As mentioned above, the most significant feature of these transmissions is no torque interruption during the gear shifting. However, to achieve the uninterrupted torque gearshift, namely clutch-clutch gearshift [21], the gearshift process has to be controlled precisely. This will make the transmission control system considerably complex and expensive, which is the most obvious shortage of this type of transmission. The other style of transmission is without power-on gearshift capacity, i.e., CLAMT regarded as the traditional AMT variant. This transmission characters significantly a simpler mechanical structure, easier gearshift control and lower manufacture cost compared to power-on transmissions. Extensive research has been done on this transmission. In Ref. [22], a robust speed controller was designed to control the motor speed at the third stage of gear shifting. Refs. [23-26] focused on the dynamics and control of gearshift actuators. Torque observers were designed to improve oscillation damping performance and gear shifting quality
. These studies achieve splendid results corresponding to their research topics, but do not consider transient jerk response of the powertrain during the whole gear shifting. This problem was addressed by Walker et al. [29]. Their research demonstrated the motor torque reduction and reinstatement stages have the most significant effect on the longitudinal vehicle jerk response. However, this study does not propose methods to reduce the vehicle jerk.

In this paper, we propose a novel two-speed PAMT used in BEVs. As illustrated in Fig. 1, the electrified drivetrain is comprised of an electric machine (EM), a two-speed PAMT, final drive, differential, half shafts, and wheels. To be more specific, the two-speed PAMT consists of a single planetary gear train consisting of a sun gear (S), three planet gears (P), a carrier (C), and a ring gear (R). The input of PAMT is
mechanically connected to an EM via a flexible shaft. C is regarded as the output of PAMT. The gearshift actuator is a synchronizer system which is mainly comprised of gear ring 1 (GR1), spline hub (SH), sleeve (SL), gear ring 2 (GR2), etc. Specifically, GR1 is laid out on the input shaft which is also attached to S, GR2 is fixed on the transmission case and SH is connected to R. SL can be controlled to slide along SH. It is well acknowledged that a single-stage planet gear set exists two degrees of freedom at most. So, this transmission can obtain two different gear ratios. When SL is the neutral position, namely never engaging SH with GR1 nor engaging SH with GR2, PAMT will work at the neutral gear. When SL is controlled to engaging GR2 with SH, PAMT will be operated at the first gear. The second gear will be obtained through engaging GR1 and SH with SL. Compared to the existing literature, the first contribution of this paper is that we proposed a novel planet CLAMT, called PAMT, which features more compact mechanical layout

Fig. 2. Schematic diagram of a Lumped powertrain.

than CLAMT and easier gear shifting control than power-on transmissions. Secondly, a detailed and original dynamic model of this powertrain is established, and the corresponding gearshift control system is developed. At last, three alternative planning torque trajectories are proposed for the torque reduction and reinstatement of EM. After comparing a series of simulation results, the optimal torque trajectory is reported, which provides the beneficial reference for the development of transmission controllers.

The rest of this study include the following sections: Section 2 shows the modelling process of the electric-drive system equipped with the two-speed PAMT. Section 3 presents the development process of the gear shifting control system. In Section 4, simulation results and the corresponding analysis are shown in detail. At last, conclusions are presented in Section 5.

The powertrain equipped with the two-speed PAMT is present in Fig. 2. This system achieves the torque transmitting capability through corresponding shaft components and gear friction contacts. In this study, the key drivetrain components are modelled as lumped inertia elements. This model is mainly comprised of an EM, a two-speed PAMT, the gearshift actuator, main reducer, wheel hub and vehicle. Herein, some assumptions should be made in advance to simplify the modelling

a. The rotation elements of the proposed transmission are regarded as a rigid body.

b. There is no backlash during the gear mated process.

c. Stiffness and damping coefficients are linearized.

As well acknowledged, power electronics of EM feature significantly higher transient response frequency than mechanical components. Therefore, the high-frequency vibrations of power electronics can be ignored, which will simplify EM model. In this study, EM is modelled as rotation inertia
and the electromagnetic torque is achieved through a table lookup method. The corresponding EM torque
can be expressed by

where
is the driver's pedal command,
is EM speed and
stands for the maximum EM torque at the corresponding
, captured from experimental data, shown in Fig. 3.

The dynamic model of the electric machine is

where
is the angular acceleration of EM and
is EM inertia.

The input shaft of PAMT is built as a flexible shaft. So,
can be expressed as following

where
is EM torsion angle,
is
torsion angle and
is
rotation speed,
is the stiffness coefficient and
is the damping factor.

Fig. 3. Electric machine torque characteristic.

Fig. 4. Forces and torques added to gear sets.

The main mechanical components of PAMT are represented as free individual bodies, displayed in Fig. 4. Applying the Newtonian mechanics, PAMT model can be developed as followings

where
and
are inertias of GR1, S, SL, R, P and C.
and
represent the meshing force between
and
, and
and
and
are the pitch radius of
and
, and the angular accelerations of
and
are represented by
and
.

Output shaft torque
of PAMT is expressed as following

where
is the torsion spring coefficient of the output shaft,
is the damping coefficient,
and
are torsion angles of
and driving part of the final drive, and
and
are rotation speeds of
and driving part of final drive.

Generally, kinematics of a single stage planet gear set can be expressed as followings [30]

Applying following mathematical operations, the simplified PAMT model can be obtained by removing some internal forces from Eq. (4) to Eq. (7).

According to Eq. (9) and Eq. (10), the following equations can be derived

Substituting Eq. (13) and Eq. (14) into Eq. (11) and Eq. (12) respectively, the following equations can be obtained

where

Solving Eq. (15) and Eq. (16), PAMT model is represented as followings

where

According to the different sleeve position,
can be derived from Eq. (17) and Eq. (18). Specifically,
can be expressed as four types at different cases:

Case 1: Neutral gear

Fig. 5. Typical synchroniser mechanism.

In this situation, SL is on the neutral position, which means that the synchronizer is open. According to Eq. (17) and Eq. (18), this transmission system features two degrees of freedom. So, no torque is transmitted by SL. Hence,

In this case, the synchroniser mechanism makes ring gear engaged with the transmission case,
and
. This transmission system will lose one degree of freedom. Solving Eq. (17) and Eq. (18),
can be expressed as following

where
.

According to Eq.(20), it can be found that
is the reaction torque determined by the transmission input and output torques.

In this situation,
and
are engaged with the synchroniser mechanism. This gearbox is also a single degree of freedom status, i.e.,
.Hence,

where
.

As shown in Fig. 5, typical synchronizer mechanisms are designed to balance loads in the engaging synchronizer process [31]. The engagement of the synchroniser moves through several different stages as the cone clutch is engaged and the sleeve moves forward with the engagement, locking the gear and shaft in preparation for shifting. Many different process descriptions are available with deviations by authors reflecting different research focuses [32,33]. However, this study mainly focuses on that vehicle jerk and shift duration from the first stage and fifth stage is how to be influenced by the alternative change trajectories of motor torque. So, in this paper, the synchroniser mechanism model is developed by using DoubleSided Synchroniser block of Simscape/Driveline which is a high-fidelity driveline simulation tool appropriate for modelling and simulating vehicle transmission systems [34]. In the sliding friction process, the synchronizer model is regarded as friction load torque which eliminates the relative rotation motion between the driving part and the driven of the cone clutches [29]. The friction torque is often obtained as following

where
is the friction torque of the cone clutch,
represents the dynamic friction factor,
is the mean value of the cone clutch radius,
is the force exerted by the gearshift actuator, and
is the cone angle.

As shown in Fig. 3, these sub-models in this section can be expressed as followings

Fig. 6. Vehicle control hierarchy.

where
is the vehicle resistance.

And, it can be expressed as

where
and
stand for inertias of driving part of differential, driven part of differential, wheel hubs and whole vehicle;
and
are utilized to represent the angular replacement, speed and acceleration for the final drive, wheel hubs and tires; torsion stiffness and damping coefficients are depicted as
and
, and
and
by the subscripts of corresponding elements;
is the gear ratio of final drive;
is the whole vehicle mass;
is gravity acceleration;
is incline angle;
is rolling resistance factor;
is air density;
is drag coefficient;
is the tire radius;
is the vehicle frontal area.

In general, the control of vehicle powertrain system is executed through a hierarchical manner, as shown in Fig. 6. The driving and braking operations as well as gear shifting are coordinated by the vehicle control unit (VCU) which also monitors driver inputs and vehicle states, e.g. throttle opening, motor torque, speeds and so on, to determine whether to perform the gear shifting operations. During the driving and braking operations, the gear ratio of transmission is constant. Therefore, there is no effort required in the transmission control unit (TCU). The driver's torque demands are met by the motor control unit (MCU) which controls the power inverter to achieve driver inputs. When gear change events occur, much higher demand is placed on the TCU and VCU will handle coordination control between MCU and TCU.

In this study,
vehicle speed and
throttle opening, and
vehicle speed and
throttle opening are called as the gear shifting points which are decided by dual-parameter gearshift schedule, and [20] shows the gearshift schedule design process in detail. Namely when the vehicle reaches this state, the gear change event will be triggered. And, before the moment of the shift, the VCU will record the throttle value and then send it to TCU. Then, TCU will store it as the initial value for the first stage and the end value for the fifth stage. In the gear shift process, the TCU essentially takes control of the vehicle as it must control both the electric machine and gear shifting actuator to ensure good gear shifting performance. Therefore, the driver's throttle input has been shielded at the moment. In this study, TCU control the motor torque output according to the planning throttle change trajectories in the first stage and the fifth stage of the proposed gear shift strategy.

Gear shifting control of PAMT requires a coordination strategy which can combine displacement control for the synchroniser mechanism and speed and torque controls for EM. Regarding a traditional AMT, the primary clutch should be completely disengaged before gear shifting. This decouples the ICE from the remainder of the powertrain in the gear change process. Therefore, the load force exerted on the synchroniser will be reduced to a large degree during its disengagement and engagement process. Compared to ICE, EM features the better speed and torque controllability, which can achieve precise speed matching before engaging the target gear. Therefore, the cone friction clutches applied to minimise any relative slip speed between the driving cone clutch and the driven one may ideally be not required for gear shifting [35].

Schematically, the shifting coordination control tactic of the gear shifting process is shown in Fig. 7. This tactic is mainly comprised of five stages, including the torque and speed control of the electric machine and the displacement control of synchronizer mechanisms. Stage 1: Reduce electric machine torque to the threshold value; Stage 2: Release synchronizer to the neutral position; Stage 3: Regulate electric machine speed to the target value; Stage 4: Engage synchronizer mechanism to the target gear; Stage 5: Reinstate electric machine torque to the target value. To be specific, at the first stage, we require that
should be reduced to 0 when this stage finished and the throttle opening will be controlled to recover the level

Fig. 7. Flow chart of gear shifting strategy.

Fig. 8. EM Control schedules.

before the gear shifting. It should be also noted that the control modes of driving electric machine switches according to the different gear shifting stages, as shown in Fig. 8.

In order to evaluate the gear shifting quality, some metrics should be given. In this study, longitudinal vehicle jerk and shifting time are chosen as the evaluation metrics. Since the relative slip speed difference between the cone clutch driving side and the driven has been adjusted to a small range at the third stage, the friction dissipation energy from the cone clutches is not considered as a metric.

(1) Longitudinal vehicle jerk is rate change of acceleration observed by the driver. Minimising jerk will offer a reference of improving the gear shifting quality, and what factors will impose impacts on gear shifting quality. Ref. [29] demonstrated that the largest level of jerk induced at the first stage and the fifth stage. It should be worth noting that the standard level of jerk in German is
, while in China it is
[12, 36]. In this study, the longitudinal vehicle jerk can be calculated as following

where
is the longitudinal vehicle jerk (
), and
is the longitudinal vehicle acceleration (
).

(2) Gearshift duration measures the entire period when gearshift happens, and it can be observed by the diver. In this study, it is calculated by the sum of the time consumption in each stage of the shifting process. The entire gear shift time is able to be reduced through minimising each stage duration. However, this will be at the expense of increased jerk [29].

According to the description of Section 3.1, the gearshift execution components include the driving electric machine and synchroniser mechanisms. In this study, three alternative torque trajectories are proposed to control
change rate at the

Fig. 9. Schematic diagram of
controller.

Fig. 10. Schematic diagram of
controller.

Fig. 11. Schematic diagram of the displacement controller.

first stage and the fifth stage. And, adopting an on-off manner to control synchroniser mechanisms achieves the gear change by a PID controller.

According to PAMT dynamic equations, this transmission is a single degree of freedom system in the first and fifth stages of the gear shifting process. If using the vehicle jerk and shift durations as optimal objectives in these two stages, the torque control problem of EM can be formulated as a multiple-objective optimisation problem. To solve this kind of optimization problem, Pontryagin's minimum principle and Dynamic programming [37] are often applied to generate the optimized control variable trajectories. However, these two methods are so computationally intensive that they are unable to be utilised online in real time, especially in high transient gear shifting operations. To address this issue, this study proposes three alternative empirical polynomials to plan the motor torque change rate at the first stage and the fifth stage. These alternative polynomials are respectively the third-degree polynomial (TDP), the fifth-degree polynomial (FDP) and the seventh-degree polynomial (SDP). They are ideal for real-time applications due to the low amount of computation. The torque control system diagram is shown in Fig. 9.

According to Eq. (1),
can be determined by
when knowing
. Base on control strategies of the first and the fifth stages, these desired candidate
polynomials should meet the following requirements

where
represents the desired
at the first and the fifth stages,
and
is the start timing and end timing of actuation operations,
and
are respectively the start and end values of pedal opening. At the first stage,