Fundamental limitations to no-jerk gearshifts of multi-speed

transmission architectures in electric vehicles

Marc-Antoine Beaudoin ∗† Benoit Boulet †

February 17, 2021

Abstract

Multi-speed transmissions can enhance the performance and reduce the overall cost of an elec-tric vehicle, but they also introduce a challenge: avoiding gearshift jerk, which may sometimesprove to be impossible in the presence of motor and clutch saturation. In this article, we introducethree theorems that explicitly define the fundamental limitations to no-jerk gearshifts resultingfrom motor or actuator saturation. We compare gearshifts that consist of transferring transmis-sion torque from one friction clutch to another, to the case in which one of the clutches is a one-wayclutch. We show that systems with a one-way clutch are more prone to motor saturation, causinggearshift jerk to be more often inevitable. We also study the influence of planetary gearsets onthe gearshift dynamical trajectories, and expose the impact on the no-jerk limitations. This workoffers tools to compare transmission architectures during the conceptual design phase of a newelectric vehicle.

Keywords: electric vehicle, multi-speed transmission, transmission architecture, gearshift tra-jectory, gearshift jerk

1 Introduction

Uninterrupted gearshifts are a desirable feature of electric vehicles, but not all multi-speed transmis-sions are capable of it. To provide this capability, vehicle design engineers must select a transmissionarchitecture in which the motor torque can be continuously transferred from one transmission pathto another during gearshifts. But in the presence of motor and clutch saturation, even such a trans-mission can fail to provide an uninterrupted gearshift. In this article, we explore the fundamentallimitations on gearshift performance that originate from actuator saturation.

These fundamental limitations should be considered early in an electric vehicle design process,such as when selecting the transmission type during the conceptual design phase. Established designmethodologies attribute a high importance to the conceptual design phase, as its outcome has alarge influence on the rest of the design project, and ultimately, the product quality [1–3]. We wish toprovide electric vehicle design engineers with clear expectations on the potential gearshift performanceof various transmission architectures, before they delve into resource-intensive detailed modelling.

∗Corresponding author: ma.beaudoin@mail.mcgill.ca†Intelligent Automation Lab, Centre for Intelligent Machines, McGill University, Room 503, McConnell Engineering

Building, 3480 University Street, Montreal, QC, Canada, H3A 0E9

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1.1 Review of powertrain architectures

The literature abounds with powertrain concepts, each with the potential to meet specific clientneeds [4, 5]. Perhaps the simplest architecture is using a single motor and a fixed reduction ratiobetween the motor and the wheels. This concept has an excellent drivability, but it introduces sig-nificant drawbacks on the vehicle design: to meet vehicular performance specifications, the motor isoften oversized, and the resulting powertrain only seldom operates in its optimal efficiency region.This becomes especially problematic for heavier vehicles.

A natural evolution of the single-motor fixed-ratio concept is the introduction of a multi-speedtransmission. A simple concept is the manual transmission [6, 7]. It consists of mounting gears onbearings, and selectively locking different gears to their transmission shafts to achieve different trans-mission ratios. Because electric motors do not need to idle, it is possible to use a manual transmissionwithout a clutch between the motor and the transmission. To shift gears, the motor torque is firstreduced to zero, then the first gear is disengaged, the motor speed is synchronized with the secondgear, the second gear is engaged, and finally the motor driving torque is reapplied. Synchronizersmay also help with the shaft synchronization and gear engagement [8–11]. If properly performed,such a gearshift can have a low jerk level, but a torque gap is inevitable, as the shifting elementshave to be engaged and disengaged when no torque is passed through them. To reduce this torquegap, we can introduce a torque gap filler in the transmission architecture [12, 13]. This is typicallya clutch placed between the motor and the transmission output shaft; it is used only during gearshifts.

Alternatives to manual transmissions used in electric vehicles are dual clutch transmissions [14–16]and automatic transmissions [17–20]. Conceptually, they are almost equivalent. Both consist of of-fering clutching and braking devices that can be modulated such that the transmission’s torque canbe continuously transferred between different transmission paths. The distinction resides in that dualclutch transmissions typically use a parallel shaft architecture and only require two clutches, whileautomatic transmissions typically use a planetary gearset architecture and require more clutching andbraking devices if more than two gear ratios are to be offered. In both parallel shaft and planetaryarchitectures, it can be interesting to replace a friction clutch by a one-way clutch [14, 18, 21]. Aone-way clutch that transfers the transmission torque in a given ratio will automatically disengagewhen a friction clutch of a higher gear ratio is engaged. This automatic disengagement may alsosimplify the gearshift control algorithm, as one fewer clutch needs to be controlled. This concept alsoallows to continuously transfer the torque between the two transmission paths, but with the addedbenefits that a one-way clutch is cheaper and more compact than a friction clutch or a brake.

Instead of using a multi-speed transmission, we could circumvent the drawbacks of having a single-motor fixed-ratio powertrain by using a plurality of motors. The different motors can be mounted ondifferent axles, where the driving torque is shared through the road. Each motor can either power asingle wheel [22] or a front or rear axle [23]. Alternatively, the different motors can also be mounted onthe same axle [24]. These architectures allow the driving torque to be continuously transferred fromone motor to another, thus providing excellent drivability. This is also true for multi-motor architec-tures with multi-speed transmissions. For instance, a planetary gearset architecture can be configuredto receive inputs from two motors [25–28]. These transmission architectures are conceptually indistin-guishable from power-split transmissions used in hybrid electric vehicles [29]. Alternatively, a parallelshaft architecture can be configured to receive inputs from two motors [30–32]. Such a powertrain

2

is capable of perfectly smooth gearshifts: if the driving torque is taken exclusively from one of thetwo motors, the other motor transmits no torque, which allows for an easy gear change on sometransmission shafts. However, a torque gap may still exist, as the torque on one motor has to bereduced to zero and the other motor may not be able to fully compensate this torque decrease.

Finally, electric powertrains can also include mechanical continuously variable transmissions [33].These powertrains provide excellent drivability, as the transmission ratio can be smoothly varied.However, there is a potential concern over a reduction in the powertrain energy efficiency [34].

From this review we conclude that concerns over drivetrain jerk are far greater when using asingle-motor multi-speed transmission concept than any other powertrain concept. However, multi-speed transmissions remain good candidates for electric vehicle designers, as they may offer the besttrade-off between conflicting requirements of cost, volume, weight, and energy efficiency for a givenvehicle design. This motivates the focus of this study on gearshift jerk in single-motor multi-speedtransmission powertrains.

In this article, the terms “no-jerk gearshift” and “uninterrupted gearshift” are used interchange-ably, given that the absence of gearshift jerk implies the absence of a torque gap. In some situationsa torque gap is unavoidable, so engineers are left with balancing a trade-off between minimizing thetorque gap and the gearshift jerk. This is always the case with manual transmissions, as the motortorque must be reduced to zero to allow the gear change. Rapidly reducing the motor torque reducesthe torque gap length, but results in a larger gearshift jerk. The intended vehicle application influenceshow engineers balance the conflicting drivability criteria. However in this article, we do not addressthis tradeoff. Instead, we focus on identifying the situations where gearshift jerk and torque gaps canbe eliminated altogether. Therefore, we further focus this study on dual-clutch and automatic trans-missions, since their clutch torques can be modulated to provide uninterrupted gearshifts. But aswill be demonstrated in this article, clutch nonlinearities and motor saturation result in fundamentallimitations to no-jerk, uninterrupted gearshifts.

1.2 Methodology

The existence of fundamental limitations to no-jerk gearshifts that originate from motor saturationwas hinted in [14], but such limitations were never formulated explicitly. Some studies on gearshiftjerk reduction are framed around gearshift trajectory optimization [35,36]. Typically, authors modela driveline, formulate a cost function that balances vehicle jerk and clutch energy dissipation, thensolve a trajectory optimization problem. The main caveat with this approach is that often the op-timization problem is non-convex, so a global optimum is not guaranteed. Moreover, it is hard totransfer the results to other vehicles or to slight alterations of the transmission. Other studies addressthe design of an optimal gearshift controller [37–40]. But similarly, this approach does not allow togeneralize, as we do not know whether the vehicle jerk is a result of an imperfect controller, or anunavoidable fundamental limitation.

In this article, we take a different approach where we prove that under specific circumstances, mo-tor and actuator saturation will always cause a torque interruption during gearshifts. More precisely,we present theorems that explicitly define the fundamental limitations to no-jerk gearshifts in electricvehicles. We begin this study by defining a general driveline and vehicle model in Section 2. Then,

3

we impose a no-jerk kinematic constraint at the vehicle level, and with the model’s equations, we cansolve for the resulting conditions on the transmission output shaft. This defines conditions for a no-jerk gearshift, which we present in Definition 1. Subsequently, we model various transmission typesand obtain specific system equations. Finally in Section 4, we obtain the theorems by identifying thepossible values of the remaining variables in the specific system equations for which a gearshift wouldmeet the no-jerk conditions on the output shaft, and motor and actuator saturation are avoided. Tofacilitate the visualization of the theorems, we also present example gearshift trajectories in Section 3.We begin Section 3 with a motor selection process for an example vehicle, so that we get realisticmotor limitations. Then, we use the specific system equations to compute example trajectories wherewe impose the no-jerk constraints. This helps illustrate the relation between the remaining variablesin the specific system equations when the no-jerk constraints are met. The objective of Section 3 isfor the reader to build an intuition before being presented with the theorems in Section 4.

In this article, we do not quantify the level of jerk or torque interruption that would result fromsituations where the theorems indicate that a no-jerk gearshift is impossible. This would requirea specific controller design, as well as complete and well calibrated vehicle model, (see, e.g., [41])which also means that the conclusions would only apply to the specific vehicle studied. Also, in theinterest of generality, we work under the assumption of perfect state feedback and a perfect controlof actuator forces. Formally, this translates into Assumption 1. In practice, other system limitationswill influence the dynamics of a gearshift. Nevertheless, the theorems obtained in this article helppredict or identify when motor or actuator saturation is a contributor to gearshift jerk and torqueinterruption.

Assumption 1. Only the following two system limitations can lead to unavoidable gearshift jerk:

1. a limit on the motor torque, which can be characterized both in terms of maximal torque Tmax,or maximal power Pmax;

2. a limit on the torque application rate of a friction clutch, dT/dt.

1.3 Contributions

The contributions in this article include three theorems that explicitly define when an uninterruptedgearshift can be obtained for various architectures of multi-speed transmissions for electric power-trains. More specifically, Theorem 1 provides a necessary and sufficient condition for a no-jerk upshiftwhen the motor operates in the power limited region, and the transmission is a dual-clutch trans-mission with a one-way clutch. Theorem 2 provides a necessary and sufficient condition for a no-jerkgearshift under the same upshift scenario, but for a dual-clutch transmission with two friction clutches.This allows to demonstrate the additional constraints on gearshift trajectories – and therefore, theadditional no-jerk limitations – that are implied by the use of a one-way clutch. Theorem 3 providesa necessary and sufficient condition for a no-jerk downshift when the motor operates in the torquelimited region, also for dual-clutch transmissions. Subsequently, we show that these three theoremscan be adapted to define the limitations to no-jerk gearshifts when using an automatic transmissioninstead.

These theorems can be used by automotive engineers to compare the potential drivability ofvarious transmission types, thereby providing additional information during a transmission selectionprocess for a vehicle conceptual design phase. The theorems can also be used for the design of a

4

𝑇m, 𝜃m

𝐼m

𝑐m

𝐼v

𝑇o, 𝜃v

𝑇v

𝑘

𝑑

𝑇out, 𝜃out

Transmission

𝑐o

𝐼out

𝑇in

Figure 1: General driveline and vehicle model

gearshift controller: prior to initiating a gearshift, we can now predict whether a no-jerk gearshift canbe obtained under the current driving situation, or if an uncontrolled vehicle jerk is likely to occur asa result of sudden motor saturation.

2 Vehicle, driveline, and transmission modelling

In this section, we first present a generic vehicle and driveline model. We also introduce severaltransmission models, which are then embedded in the generic driveline model to compute the gearshifttrajectories of Section 3.

2.1 Generic vehicle and driveline models

We begin with the generalized vehicle model. We assume that the vehicle longitudinal speed v followsthe driving wheel speed θv according to v = θvrw, where rw is the wheel radius. This allows toproject the vehicle mass and the vehicular forces on the wheel coordinate. The vehicle mass mbecomes an equivalent rotational inertia Iv = mr2w. In this model, we consider three vehicular forces:the aerodynamic drag Faero, the tire rolling resistance Ftire, and gravity Fslope. These three forcesbecome an equivalent torque Tv applied on the vehicle wheel as follows:

Tv = rw(Faero + Ftire + Fslope), (1)

Tv = rw(1

2ρAfCdv

2 +mgCr cosα+mg sinα), (2)

where ρ is the air density, Af is the vehicle frontal area, Cd is the aerodynamic drag coefficient, Cr isthe tire rolling resistance coefficient, g is gravity, and α is the road slope.

We use the driveline model shown on Figure 1. We have three rotating bodies: the electric motorIm, the transmission output shaft Iout, and the equivalent vehicle inertia Iv. The three equations ofmotion for the general driveline and vehicle model are

Imθm = −cmθm + Tm − Tin, (3)

Ioutθout = −coθout + Tout − k(θout − θv)− d(θout − θv), (4)

Ivθv = −Tv + k(θout − θv) + d(θout − θv), (5)

where cm is the coefficient of viscous damping on the motor, and co is the coefficient of viscousdamping on the transmission output. The coefficients k and d represent lumped driveline stiffnessand damping, respectively. They cover phenomena such as driveshaft and tire flexibility and damping.

5

In this article, we are interested in finding gearshift trajectories that avoid vehicle jerk. Thus,we impose the kinematic constraint that θv = ar/rw, where ar is a prescribed and constant vehicleacceleration. By extension, we have that θv = (art + vi)/rw, where vi is the initial vehicle speed atthe beginning of the gearshift (t = 0). Moreover, the trajectories we study only take place for a shortamount of time – approximately 0.5 s – so it is fair to assume the vehicular forces Tv are constant.Given the driveline model in Equations 3-5, the constraints introduced on the θv, θv, and Tv variablesalso restrict θout and θout. We can solve for these variables explicitly, as well as reduce the drivelinemodel to a system of only two equations. First, the vehicular forces and acceleration are grouped intoa single constant output torque To = Ivθv + Tv. Substituting To in Equation 5, we get:

To = k(θout − θv) + d(θout − θv). (6)

Taking the time derivative of Equation 6, we obtain:

0 = k(θout − θv) + d(θout − θv). (7)

Substituting θv and θv with their no-jerk constraint in Equation 7, we obtain the following lineardifferential equation:

θout(t) +k

dθout(t) =

k

drw(art+ vi) +

arrw. (8)

We can solve this equation using the initial condition that θout(0) = vi/rw, and we get:

θout(t) =art+ virw

. (9)

We now use the prescribed trajectories on θv, θout, and To to define a no-jerk gearshift as follows.

Definition 1. A no-jerk gearshift is obtained if, for the duration of the gearshift,

θout = θv = (art+ vi)/rw, (10)

To = Ivar/rw + Tv. (11)

Finally, the general vehicle and driveline model can be reduced to only two equations of motionas follows:

Imθm = −cmθm + Tm − Tin, (12)

Ioutθout = −coθout + Tout − To. (13)

In the next sections, we transform the general model of Equations 12 and 13 into specific systemsthat depend on the transmission used, thereby replacing Tin and Tout by the relevant clutch, brake,and inertial torques.

2.2 Parallel shaft architecture with two frictional clutches

The first system model we build is that of the architecture illustrated in Figure 2a, when both clutchesare frictional clutches. The equations of motion for this system are

Imθm = −cmθm + Tm − T1 − T2, (14)

Ioutθout = −coθout − To + i1T1 + i2T2, (15)

6

𝐼m

𝐼out

𝑇m

𝑇o

𝑇1

𝑇2

𝑐m

𝑐o

𝑖1

𝑖2

(a) Two-speed transmission with parallel shaft archi-tecture. Short name: dual-clutch transmission.

𝐼m

𝐼r

𝐼s

𝐼c1

𝐼c2

𝑇1

𝑇2

𝑇o

𝐼out

𝑐m

𝑐o

𝑖f

𝑇m

(b) Two-speed transmission with a planetary gearsetarchitecture. Short name: dual-brake transmis-sion [17].

Figure 2: The two powertrain architectures studied in this article.

where the clutch torques T1 and T2 depend on the state of the clutch – i.e., stick or slip. We adoptthe Coulomb friction model and obtain the clutch torques as follows:

T1 =

{Tm − T2 − Imθm − cmθm θm = i1θout

Fn1µdRan sign(θm − i1θout) θm 6= i1θout, (16)

T2 =

{Tm − T1 − Imθm − cmθm θm = i2θout

Fn2µdRan sign(θm − i2θout) θm 6= i2θout, (17)

where Fn is the linear force at the clutch plates, µd is the clutch’s dynamic friction coefficient, Ra isthe mean friction radius, and n is the number of friction surfaces. The clutch starts to slip when thereaction torque at the interface reaches the clutch torque capacity Tcap = FnµsRan, where µs is thestatic friction coefficient.

2.3 Parallel shaft architecture with a one-way clutch

If the first gear clutch is replaced by a one-way clutch in the system of Figure 2a, the same equationsof motion apply, namely Equations 14 and 15. But T1 is a reaction torque in one direction, and it isnull in the other direction:

T1 =

{Tm − T2 − Imθm − cmθm θm = i1θout

0 θm < i1θout. (18)

The one-way clutch also introduces the kinematic constraint θm ≤ i1θout.

2.4 Planetary gearset architecture

In general, planetary gearsets have more degrees of freedom than parallel shaft architectures, thusthey have more equations of motion. A single planetary stage can be seen as a combination of threebodies: a ring gear with inertia Ir, a planet carrier (Ic), and a sun gear (Is). The equations of motionfor each of these bodies in a single stage are

Irθr = Tr − rrF, (19)

Icθc = Tc + rrF + rsF, (20)

Isθs = Ts − rsF, (21)

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𝐼c1

𝐼r1

𝐼s1

𝐼c2

𝐼r2

𝐼s2

𝑇r1

𝑇c1

𝑇s1

𝑇r2

𝑇c2

𝑇s2

Figure 3: General representation of a double planetary gearset, where we show possible connectionsbetween the ring of the first set and elements of the second set. To operate as a transmission, adouble planetary gearset must have three connections in total, which we obtain either by connectingelements together, or by grounding elements to the transmission casing. To change the transmissionratio, we simply change one of these connections.

where T� is the torque applied on either the ring (r), carrier (c), or sun (s), rr is the radius of thering gear, rs is the radius of the sun gear, and F is the tooth force in the gearset. There is also akinematic constraint associated with these equations:

rsθs + rrθr = (rs + rr)θc. (22)

Equations 19-22 are commonly used in analyses of automatic transmissions, see [42] for instance. Itis worth noting that these equations imply approximating as null the rotational inertia of the planetgears. Meanwhile, the mass of the planet gears can be considered in the equations by projecting theminto the rotational inertia for the planet carrier Ic.

It is also common to combine planetary gearsets in series, such as in Figure 3. In this case, anotherset of three equations of motion are added to the system (Equations 19 to 21), as well as another kine-matic constraint (Equation 22). By connecting elements, we reduce the number of degree of freedomin the system. These connections must be done carefully, as the system can become over-constrainedor under-constrained. Transmission designers typically study numerous possible configurations be-fore choosing the most suitable ones – a process named transmission synthesis [43], for which variousmethods exist ranging from using the classic lever analogy [44], to bond graphs [45]. A Ravigneauxplanetary gearset can be seen as a special case of a double planetary gearset with its specific set ofequations.

We are now ready to build the equations for the system in Figure 2b, which consists of a specificinstance of a double planetary gearset architecture [17]. The inertias Im and Ic1 are lumped into asingle mass; we do the same for Iout and Ic2. We obtain the following equations of motion:

Irθr = T1 − rr1F1 − rr2F2, (23)

Imθm = −cmθm + Tm + rr1F1 + rs1F1, (24)

Ioutθout = −coθout − i−1f To + rr2F2 + rs2F2, (25)

Isθs = T2 − rs1F1 − rs2F2, (26)

and the following kinematic constraints:

rs1θs + rr1θr = (rs1 + rr1)θm, (27)

rs2θs + rr2θr = (rs2 + rr2)θout. (28)

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Table 1: Example vehicle parameters

Param. Value Param. Value Param. Value

m 6500 kg Iout 0.05 kg m2 i2 6rw 0.3 m cm 0.02 Nm s/rad Ir 0.03 kg m2

Af 6 m2 co 0.04 Nm s/rad Is 0.03 kg m2

Cd 0.7 k 10 kNm/rad β1 2Cr 0.007 d 75 Nm s/rad β2 4Im 0.3 kg m2 i1 12 if 7.2

Using Equations 27 and 28, we can reduce the four equations of motion (Equations 23 to 26) into aset of two equations of motion. In order to solve this algebraic problem, researchers have assumedthat elements other than the input and output shafts have negligible inertias, which simplifies thereduction process [42]. For the system of Figure 2b, it would mean that Ir = 0 and Is = 0. Whenwe make these assumptions, and introduce parameters β1 = rr1/rs1 and β2 = rr2/rs2, we get a setof equations that is identical in form to that of a parallel shaft architecture, only with differentcoefficients. This can be observed by comparing the equations 29 and 30 to the equations 14 and 15.

Imθm = −cmθm + Tm +1 + β1β1 − β2

T1 −β2(1 + β1)

β1 − β2T2, (29)

Ioutθout = −coθout − i−1f To +

1 + β2β2 − β1

T1 −β1(1 + β2)

β2 − β1T2. (30)

On the other hand, if we do not assume Ir = Is = 0, we get a different system of equations(i.e., Equations 31 and 32), more coupled this time. The constant coefficients C1 to C8 in thesetwo equations are not detailed further since they are rather involved algebraic expressions that onlypertain to the specific architecture of Figure 2b. The important thing to realize is that the motoracceleration is now coupled with the transmission output.

Imθm = C1(−cmθm + Tm) + C2(−coθout − i−1f To) + C3T1 + C4T2, (31)

Ioutθout = C5(−cmθm + Tm) + C6(−coθout − i−1f To) + C7T1 + C8T2. (32)

3 Gearshift trajectories for an example vehicle

In Section 3.1, we obtain realistic motor limitations by mimicking a motor selection process foran example vehicle. Then in Section 3.2, using the transmission models obtained in Section 2,we illustrate the resulting limitations on gearshift dynamical trajectories. The example vehicle wehypothesize is a commercial vehicle for carrying goods. With a gross vehicle mass of 8500 kg, it wouldbe classified as an N2 commercial vehicle in Europe, and a Class 5 medium-duty truck in NorthAmerica. The vehicle and transmission parameters used are shown in Table 1. Note that the grossvehicle mass of 8500 kg is used for the motor selection process, while the half-payload mass of 6500 kgpresented in Table 1 is used for the gearshift trajectories.

3.1 Motor selection

We set the three design specifications of Table 2 to define the vehicle performance requirements. Thefirst specification consists of an extreme grade; this sets the maximal torque requirement. This is a

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Table 2: Design specifications considered in the motor selection

Design specification v [km/h] α [%] Duration

1: extreme grade 20 20 < 1 min2: highway, cruise speed 110 0 Continuous3: highway, high grade 90 5 < 1 min

0 20 40 60 80 100 120 140

Vehicle speed [km/h]

0

1000

2000

3000

4000

5000

6000

Whe

el to

rque

[Nm

]

80

80

80 85

85

85

90

90

90

90 90

95

95

Peak capacityContinuous capacitySystem efficiencyDesign spec. 1Design spec. 2Design spec. 3

(a) Single speed transmission, ratio of 7.5. Motorrequirements: 200 kW power, 700 Nm peak torque,and 8000 rpm maximum speed.

0 20 40 60 80 100 120 140

Vehicle speed [km/h]

0

1000

2000

3000

4000

5000

6000

Whe

el to

rque

[Nm

]

Peak capacity - Gear 1Peak capacity - Gear 2System efficiency - Gear 1System efficiency - Gear 2

(b) Two-speed transmission, ratios of 12 and 6.Motor requirements: 200 kW power, 450 Nm peaktorque, and 8000 rpm maximum speed.

Figure 4: Vehicle capacity for (a) a single speed transmission and (b) a two-speed transmission.

short duration event, so we allow the powertrain to operate above its continuous capacity limit. Thesecond specification consists of the vehicle cruising on the highway, which sets both a wheel speedrequirement and a continuous power requirement. The third specification happens when the vehicleclimbs a steep but reasonable grade on the highway, which sets the maximal power requirement.

As shown in Figure 4a, if the vehicle is equipped with a single-speed transmission, the vehiclerequirements can be met with a 200 kW motor with 700 Nm peak torque and 8000 rpm maximumspeed, using a fixed total reduction ratio of 7.5. With a two-speed transmission with ratios of 12 and6, we can lower the peak torque requirement to 450 Nm, while keeping the same power and speed limitrequirements. In practice, this would allow vehicle designers to reduce the motor’s active length [46],thereby reducing the motor peak torque capacity while maintaining the same motor power capacity.We display the resulting system capacity in Figure 4b. The vehicle now has a 36 % volumetricallysmaller motor and a larger high-efficiency operating region.

3.2 Gearshift Trajectories

In this section, we present five example gearshift trajectories. In all five examples, the no-jerk condi-tions of Definition 1 are imposed on the transmission output shaft. Our intention with these examplesis to illustrate the implications on the input torques that result from the no-jerk conditions. Morespecifically, we study the three gearshift scenarios presented in Table 3. Scenario 1 is an upshiftduring vehicle acceleration, where the driver torque demand (DTD) at the beginning of the shift is80 % of available torque. This gearshift takes places in the power-limited region of the motor map.

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Table 3: Gearshift scenarios

Scenario Direction Motor quadrant Region vi [km/h] ar [m/s2] DTD

1 Upshift Driving Power-limited 65 1.0 80 %2 Downshift Driving Torque-limited 18 1.0 80 %3 Downshift Braking Torque-limited 45 −1.5 -

0 0.1 0.2 0.3 0.4 0.5

Time [s]

3000

4000

5000

6000

7000M

otor

spe

ed [r

pm]

0 0.1 0.2 0.3 0.4 0.5

Time [s]

-200

0

200

400

Tor

que

[Nm

]

MotorClutch 1Clutch 2

0 0.1 0.2 0.3 0.4 0.5

Time [s]

-200

0

200

400

Pow

er [k

W]

MotorClutch 1Clutch 2

0 2000 4000 6000 8000

Motor speed [rpm]

-400

-200

0

200

400

Mot

or to

rque

[Nm

] End

Start

Figure 5: Example trajectory for the gearshift scenario 1 with a dual-clutch transmission, where thefirst clutch is a one-way clutch.

Scenario 2 is a downshift when the vehicle is accelerating, also with a DTD of 80 %. The downshifttakes place in the torque-limited region of the motor map. This gearshift is justified by the desire tohave a greater wheel torque after the downshift. Scenario 3 is a downshift when the motor is used forregenerative braking, in the torque-limited region.

The example trajectories were computed in matlab™ using the system models obtained in Sec-tion 2: Equations 14-15 when simulating a dual clutch transmission, Equations 29-30 for a dual-braketransmission where we neglect the gear inertias, and Equations 31-32 for a dual-brake transmissionwhere we include the gear inertias. First, the required To is computed considering the vehicular forcesin Equation 2 as well as the vehicle acceleration ar and initial velocity vi, all prescribed by the drivingscenario chosen from Table 3. Then, we compute the motor and clutch torques and speeds at thestart of the gearshift from the model equations. Finally, we solve the system trajectories for eachphase of the gearshift by imposing an arbitrary trajectory for some components, and using the modelequations to solve for the trajectory of the other components. For example in a torque transfer phase,θout follows the conditions for a no-jerk gearshift in Definition 1, To is kept constant, θm follows θoutmultiplied by a given gear ratio, we impose a trajectory for T1, and from the model equations, wecan solve for T2 and Tm.

The trajectory of Figure 5 is an upshift during vehicle acceleration (scenario 1) for the case wherethe transmission is a parallel-shaft architecture, and the first clutch is a one-way clutch. The gearshiftstarts at 0.05 s and ends at 0.50 s. It begins with a torque phase, where the transmission torque is

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0 0.2 0.4 0.6 0.8

Time [s]

3000

4000

5000

6000

7000

8000

Mot

or s

peed

[rpm

]

0 0.2 0.4 0.6 0.8

Time [s]

-200

0

200

400

Tor

que

[Nm

]

MotorClutch 1Clutch 2

0 0.2 0.4 0.6 0.8

Time [s]

-100

0

100

200

Pow

er [k

W]

MotorClutch 1Clutch 2

0 2000 4000 6000 8000

Motor speed [rpm]

-400

-200

0

200

400

Mot

or to

rque

[Nm

] End

Start

Figure 6: Example trajectory for the gearshift scenario 1 with a dual-clutch transmission, where thefirst clutch is a friction clutch.

transferred from clutch 1 to clutch 2, followed by an inertia phase, where the motor is synchronizedwith the gear 2 speed. During the torque phase, we gradually increase T2 starting from zero torque, upto the torque required when the transmission is in gear 2, following an otherwise arbitrary trajectory.This has the effect of gradually reducing the reaction torque T1 on the one-way clutch, down tozero torque at the end of the torque phase. In order to maintain a constant output torque To –a condition for a no-jerk gearshift – we also need to increase the motor torque Tm. Furthermore,because the first clutch is a one-way clutch, this is the only possible trajectory for the system. Sincethe computed motor power exceeds its 200 kW limit in this example, the trajectory is infeasible. Inreality, the vehicle would inevitably experience a torque gap and vehicle jerk. This consists of the firstfundamental limitation we explore in this article, see Theorem 1 in Section 4, whose proof furtherdetails and formalizes the conclusions presented in this paragraph. The rest of the gearshift consistsof the inertia phase, which is not as prone to motor saturation as the torque phase. T1 and T2 are keptconstant, and the motor is synchronized with the gear 2 speed using an appropriate but arbitrarytrajectory. To reduce the variation in motor power during the inertia phase – for instance, to avoidthe motor braking quadrant – the inertia phase can simply be made longer.

The trajectory of Figure 6 is also an upshift during vehicle acceleration for a parallel shaft ar-chitecture, but this time clutch 1 is a friction clutch. This introduces two new possibilities: we canincrease the motor speed above the gear 1 speed, and we can modulate T1 when the clutch is slipping.We take advantage of these possibilities and craft a new trajectory that results in a no-jerk gearshift.Prior to transferring the clutch torques, the motor speed is increased above the gear 1 speed, up toa prescribed value. Then begins the torque transfer, which has the effect of decreasing the motorspeed. It is important that the torque transfer completes before the motor crosses the gear 1 speed,as this would result in a sign reversal on T1. Recall that when a friction clutch slips, the clutch torqueopposes the clutch slipping velocity, see Equation 16 for instance. A torque reversal on T1 would meanthat T2 has to instantaneously compensate in order to maintain the constant To condition. This is

12

0 0.2 0.4 0.6

Time [s]

1000

1200

1400

1600

1800

2000

2200

Mot

or s

peed

[rpm

]

0 0.2 0.4 0.6

Time [s]

0

100

200

300

400

Tor

que

[Nm

]

MotorClutch 1Clutch 2

0 0.2 0.4 0.6

Time [s]

-50

0

50

100

150

Pow

er [k

W]

MotorClutch 1Clutch 2

0 2000 4000 6000 8000

Motor speed [rpm]

-400

-200

0

200

400

Mot

or to

rque

[Nm

]

End

Start

Figure 7: Example trajectory for the gearshift scenario 2 with a dual-clutch transmission. Thistrajectory is valid for both one-way clutch and dual-friction-clutch architectures.

likely to violate any constraint on the clutch torque application rate. Therefore, this new trajectoryalso contains a fundamental limitation, which is formulated in Theorem 2.

The trajectory of Figure 7 is a downshift during motor acceleration (scenario 2), with a parallelshaft architecture. This trajectory can be obtained for both the cases where the first clutch is aone-way clutch or a friction clutch. This time we proceed with the inertia phase before the torquephase. In the inertia phase, the motor speed is increased using maximal motor power. When themotor reaches gear 1 speed, the torque transfer begins. This time, the fundamental limitation consistsof being able to synchronize the motor speed within an acceptable time while maintaining the outputtorque To constant. This limitation is formalized in Theorem 3.

The trajectory of Figure 8 is a downshift during vehicle deceleration (scenario 3). More specifically,the motor operates in regenerative braking mode. We have that To < 0, which also means that Tm < 0,T1 < 0, and T2 < 0. Because T1 < 0, this trajectory is only possible if clutch 1 is a friction clutch.A one-way clutch can only carry torque in one direction – i.e., the positive direction in our case.Often, transmissions with a one-way clutch are also equipped with a locking mechanism in parallel tothe one-way clutch [14]. This allows for regenerative braking when the transmission operates in firstgear. However, the locking mechanism typically cannot be engaged when there is a significant speeddifference between the mating elements, and it cannot be modulated. Therefore, it is impossible fora dual-clutch transmission with a one-way clutch to provide uninterrupted shifting in regenerativebraking mode. Figure 8 shows that for a dual friction clutch architecture, such a gearshift can beinitiated even when the motor is essentially on the saturation limit. For transmissions with twofriction clutches, the system limitations included in Assumption 1 will not result in gearshift jerk.

In the trajectory of Figure 9, we come back to gearshift scenario 1, but with the dual-braketransmission of Figure 2b this time. In terms of trajectory, we follow the same strategy as that of

13

0 0.1 0.2 0.3 0.4 0.5

Time [s]

2000

3000

4000

5000

Mot

or s

peed

[rpm

]

0 0.1 0.2 0.3 0.4 0.5

Time [s]

-400

-200

0

200

400

Tor

que

[Nm

]

MotorClutch 1Clutch 2

0 0.1 0.2 0.3 0.4 0.5

Time [s]

-100

-50

0

50

100

Pow

er [k

W]

MotorClutch 1Clutch 2

0 2000 4000 6000 8000

Motor speed [rpm]

-400

-200

0

200

400

Mot

or to

rque

[Nm

]

End

Start

Figure 8: Example trajectory for the gearshift scenario 3 with a dual-clutch transmission. Thistrajectory is only possible if clutch 1 is of friction type.

0 0.2 0.4 0.6 0.8

Time [s]

3000

4000

5000

6000

7000

8000

Mot

or s

peed

[rpm

]

0 0.2 0.4 0.6 0.8

Time [s]

-600

-400

-200

0

200

400

Tor

que

[Nm

]

MotorClutch 1Clutch 2

0 0.2 0.4 0.6 0.8

Time [s]

-600

-400

-200

0

200

Pow

er [k

W]

MotorClutch 1Clutch 2

0 2000 4000 6000 8000

Motor speed [rpm]

-400

-200

0

200

400

Mot

or to

rque

[Nm

] End

Start

Figure 9: Example trajectories for the gearshift scenario 1 with a dual-brake transmission, namelythat of Figure 2b. The dotted lines represent the situation where the trajectory is computed withEquations 29-30, thereby assuming Ir = Is = 0. The solid lines represent the situation where weinclude the inertia of Ir and Is and compute the trajectory with Equations 31-32.

14

Figure 6, i.e., increasing the motor speed above the gear 1 speed before proceeding with the torquetransfer. Computing the trajectory with Equations 29-30, which amounts to assuming Ir = Is = 0,we obtain the trajectory in dotted lines. But if we have that Ir = Is = Im/10 and compute thetrajectory with Equations 31-32, we obtain the trajectory in solid lines. The introduction of a smallinertia on Ir and Is has a strong influence on the resulting trajectory. First, we notice that Tmbecomes coupled with T1. In other words, when the motor torque is increased in the first phase ofthe gearshift, T1 must be decreased in order to maintain the same output torque To. This meansTm cannot increase arbitrarily quickly due to the clutch torque application rate limitation. Also, wenotice that the additional inertia increases the time required to attain a given motor speed duringthe first phase. Finally, clutch 2 takes a larger portion of the load during the inertia phase, and themotor takes a smaller portion of the load. From a design perspective, this means the maximal torquerequirement on clutch 2 is higher, which has to be accounted for in the sizing of this component.In conclusion, even small sun or ring gear inertias can have a significant influence on the gearshifttrajectory of a transmission with a planetary gearset architecture. Engineers should take precautionbefore neglecting them in the equations of motion for their systems.

4 Fundamental limitations to no-jerk gearshift

In Section 4.1, we introduce three theorems that define the fundamental limitations to no-jerk gearshiftfor a dual-clutch transmission. Table 4 summarizes the association between the theorems and theirpertaining combinations of gearshift scenarios and transmission types. We do not provide theoremsfor the gearshift scenario 3, as the conclusions follow naturally from Figure 8 and the associateddiscussion in Section 3.2. Then in Section 4.2, we adapt the theorems for transmissions based onplanetary gearsets.

Table 4: Summary of fundamental no-jerk gearshift limitations for the dual-clutch transmission

Scenario One-Way Clutch Dual Friction Clutch

1: Upshift, driving motor Limited by Theorem 1 Limited by Theorem 22: Downshift, driving motor Limited by Theorem 3 Limited by Theorem 33: Downshift, braking motor Impossible Not limited

4.1 Theorems for fundamental limitations with a dual-clutch architecture

Theorem 1 expresses a necessary and sufficient condition for the existence of a no-jerk upshift in thepower-limited region of the motor, when clutch 1 is a one-way clutch. This theorem allows one touse Equation 33 and the assumptions that θout(t) = (art+ vi)/rw and θout(t) = ar/rw to predict if aspecific driving condition allows for a no-jerk gearshift; it is not needed to simulate the gearshift.

Theorem 1. For a dual-clutch architecture where the first gear clutch is a one-way clutch, when themotor operates in a power-limited region, a no-jerk upshift of scenario 1 can be obtained if and onlyif Pm(ttr) ≤ Pmax, where Pm(ttr) is the required motor power at the end of the torque transfer phase,and Pmax is the motor power limit; Pm(ttr) can be evaluated as

Pm(ttr) = i1θout(ttr)((i1Im + i−1

2 Iout)θout(ttr) +

(i1cm + i−1

2 co)θout(ttr) + i−1

2 To

). (33)

15

Proof. (Necessity): The gearshift begins with a torque transfer phase that spans from t = 0 to t = ttr,which is defined as follows: the clutch torques are smoothly varied from T1(0) 6= 0 and T2(0) = 0 toT1(ttr) = 0 and T2(ttr) 6= 0, while θm = i1θout and θm = i1θout for all t ∈ [0, ttr], and To, θout, and θoutfollow the conditions for a no-jerk gearshift as per Definition 1. The condition θm = i1θout is necessarybecause the first gear clutch is a one-way clutch: Equation 18 indicates that T1 6= 0 → θm = i1θout,so this imposes θm = i1θout at least until T1 = 0. The condition θm = i1θout is necessary for a no-jerkgearshift: Equation 18 indicates that if the clutch opens before T1 = 0, which means we would haveθm < i1θout, then the clutch torque immediately drops to 0, and Equation 15 indicates that for To,θout, and θout to follow the conditions for a no-jerk gearshift, then T2 would have to immediatelyjump to a higher value, which violates the limit on clutch torque application rate. Moreover, theone-way clutch imposes the kinematic constraint that θm ≤ i1θout, so if we have that θm = i1θout, wecannot have that θm > i1θout. Thus, we showed that the torque transfer phase as defined above isnecessary for a no-jerk gearshift, as any deviation from it either implies a vehicle jerk, or is physicallyimpossible. We now find the required motor power at the end of the torque phase. With T1(ttr) = 0in Equation 15, we get that

T2(ttr) = i−12

(Ioutθout(ttr) + coθout(ttr) + To

), (34)

which we can substitute in Equation 14 to get the motor torque at ttr:

Tm(ttr) = Imθm(ttr) + cmθm(ttr) + i−12

(Ioutθout(ttr) + coθout(ttr) + To

). (35)

We find the required motor power at ttr using Pm(ttr) = θm(ttr)Tm(ttr), Equation 35, and the con-ditions that θm(ttr) = i1θout(ttr) and θm(ttr) = i1θout(ttr); we obtain Equation 33. Thus a no-jerkgearshift requires a motor power Pm(ttr) as described in (33). Naturally, this requires that the motorbe capable of producing Pm(ttr), therefore a no-jerk gearshift implies that Pm(ttr) ≤ Pmax.

(Sufficiency): Assumption 1 indicates that if a no-jerk gearshift cannot be obtained, it is becauseeither the motor saturates, or the clutch saturates. Further assuming that ttr is large enough such thatthe clutch application rate does not saturate, if we cannot obtain a no-jerk gearshift, it is because themotor saturates. By showing that Pm(ttr) is the maximal required motor power during the gearshift,we can show that if the motor saturates, we must have that Pm(ttr) > Pmax. To do this, let usrearrange Equations 14 and 15 to eliminate T2 and isolate Tm to get the motor power as follows,

Pm = θm

(Imθm + cmθm + T1

(1− i1

i2

)+ i−1

2

(Ioutθout + coθout + To

)). (36)

Assuming that the motor begins to synchronize with the second gear speed the instant the torquetransfer is complete at ttr, we have that the maximal θm(t) is at t = ttr. Furthermore, every term onthe right-hand side of Equation 36 is maximized at ttr. In particular, we notice that since i1 > i2,T1(ttr) = 0 maximizes Pm, as T1 ≥ 0. Therefore, if we cannot obtain a no-jerk gearshift, thenPm(ttr) > Pmax.

In Theorem 2, we express a necessary and sufficient condition such that we can have a no-jerkupshift (gearshift scenario 1) in the power-limited region of the motor, when clutch 1 is a frictionclutch. We follow the strategy detailed in Figure 10. To give the reader a visual appreciation of thelimitations, we also simulated several instances of this strategy, which are displayed in Figure 11.

16

In practice, one can use Theorem 2 to predict whether a specific driving condition allows a no-jerkgearshift. Because Equation 40 makes it such that the equations of motion for the system becomenonlinear, there may not be a convenient closed-form solution to facilitate the process of finding asufficient ∆m. Perhaps the best way to do so is to simulate the first phase of the gearshift up to agiven ∆m, and then validate if this ∆m is sufficient to allow a complete torque transfer before ∆s = 0.If the given ∆m is not sufficient, then the process can be repeated for a higher ∆m, until no higher∆m can be reached, at which point we can conclude a no-jerk trajectory is infeasible.

Theorem 2. Consider a dual clutch architecture with two friction clutches as shown in Figure 2a.Referring to Figure 10, suppose that at t = 0 the motor speed is synchronized with gear 1’s speed andthat it is desired to initiate a gearshift to be completed at time t2. Let ttr > 0 be the set torque transferduration in the gearshift and t1 the time at which the torque transfer is initiated. When the motoroperates in a power-limited region, a no-jerk upshift of scenario 1 can be obtained if and only if

∃ ∆m := θm(t1)− i1θout(t1) ≥ 0, (37)

such that ∆s := θm(t1 + ttr)− i1θout(t1 + ttr) ≥ 0, (38)

θm(t1) ≤ θmax, (39)

where Tm(t) = Pmax/θm(t), 0 ≤ t ≤ t1 + ttr, (40)

T1(t) = 0, t1 + ttr ≤ t ≤ t2, (41)

T2(t) = 0, 0 ≤ t ≤ t1. (42)

Proof. (Necessity): The gearshift begins with a speed phase (0 ≤ t ≤ t1), where the motor speed isincreased above i1θout, with T2 = 0 and Tm = Pmax/θm. Then the gearshift continues with a torquetransfer phase (t1 ≤ t ≤ t1 + ttr), where the clutch torques are smoothly varied from T1(t1) 6= 0and T2(t1) = 0 to T1(t1 + ttr) = 0 and T2(t1 + ttr) 6= 0, while Tm = Pmax/θm. Finally, the gearshiftends with an inertia phase (t1 + ttr ≤ t ≤ t2), where the motor speed is brought down to i2θout,while T1 = 0. During all three phases, the conditions for a no-jerk gearshift in Definition 1 can bemaintained by modulating T1 and T2 as per Equation 15. Motor saturation can be avoided as themotor torque is set to Tm = Pmax/θm for the first two phases, and the motor synchronization ofthe inertia phase requires that Tm < Pmax/θm. The torque application rate on both clutches can bemaintained within limits during the speed phase, as T2 = 0 and the variations on To, θout, and θoutare small enough such that dT1/dt is within limits. The same argument can be made for the inertiaphase, with T1 = 0 this time. During the torque transfer phase, appropriate clutch torque trajectoriesmust be chosen such that limits on dT/dt are respected. Moreover, a torque reversal on clutch 1 mustbe avoided.

In effect, the clutch torque T1 is necessarily positive when the motor is driving the vehicle throughthe first gear and clutch 1 sticks. When clutch 1 slips, the direction of T1 is dependent on the slipdirection, as described in Equation 16. If θm < i1θout, then the torque T1 suddenly reverses directionand becomes negative. In order to maintain the conditions for a no-jerk gearshift, Equation 15 indi-cates that the torque reversal on T1 must be instantaneously compensated by an increase in T2, whichnecessarily violates any application rate limitation. Consequently, we must have that θm ≥ i1θout aslong as T1 6= 0. Once the torque transfer phase begins, we have θm(t) < θm(t1), as i1 > i2, whichcan be seen from Equations 14 and 15. Therefore, the torque transfer needs to start at a sufficientlyhigh ∆m such that ∆s ≥ 0. Moreover, ∆m must correspond to a motor speed that is within its limitθmax. However, nothing guarantees the system can reach such a ∆m. If it fails to reach a satisfactory∆m, then one of the conditions for a no-jerk gearshift will not be satisfied due to system limitations

17

𝑡tr

𝑇2 = 0

𝑇1 = 0

𝜃m

𝑖1𝜃out

∆m

∆s

𝑡1

𝑡 = 0

Figure 10: Strategy for a power-on upshift with a dual-friction clutch transmission. We first increasethe motor speed to an increment ∆m above the gear 1 speed. Once ∆m is reached, we begin thetorque transfer, which takes a set time ttr. At the end of the torque transfer, the motor will havedecelerated to a different speed whose increment over gear 1 speed we define as ∆s. In order to avoidtorque reversal at clutch 1, which would imply an unavoidable jerk, we must have that ∆s ≥ 0.

0 0.5 1 1.5 2 2.5 3

time [s]

0

20

40

60

80

100

120

m [r

ad/s

]

(a) Even when using the maximum available motorpower, not all ∆m can be reached.

0 20 40 60 80

m [rad/s]

0

20

40

60

80

s [rad

/s]

ttr = 0.05 s

ttr = 0.10 s

ttr = 0.15 s

ttr = 0.20 s

(b) The resulting ∆s after the torque transfer is afunction of the starting ∆m and the transfer time ttr.

Figure 11: Example of limitations for a power-on upshift with the example vehicle and a dual-friction-clutch transmission, when the gearshift is initiated at vi = 65 km/h, and ar = 1.0 m/s2

– there will be jerk. This proves the necessity part by contraposition.

(Sufficiency): By construction, the actuation strategy described in Equations 40 to 42 is sufficientfor a no-jerk gearshift, given that Assumption 1 holds.

This actuation strategy is not unique, but any deviation would only make it harder to achieve asufficiently high ∆m. Consequently, if a no-jerk gearshift can be obtained using such a deviation fromEquations 40 to 42, it can also be obtained using the actuation strategy described in these equations.Therefore, if a no-jerk gearshift cannot be obtained, there is no ∆m such that ∆s ≥ 0 when followingEquations 40 to 42.

Comparing Theorems 1 and 2, we conclude that transmissions with a one-way clutch are moreprone to motor saturation. In effect, the constraint on the maximal motor power imposed in Theo-rem 1 does not exits in Theorem 2, and as a result, transmissions with two friction clutches have awider set of possible no-jerk gearshift trajectories. For example, both gearshift trajectories of Figure 5(one-way clutch) and Figure 6 (two friction clutches) are initiated under the same driving scenario,namely an upshift at 80% DTD in the motor’s power-limited region. The trajectory of Figure 5 results

18

in motor saturation and Theorem 1 would indicate that it is inevitable, meanwhile the trajectory ofFigure 6 completes without motor saturation and a no-jerk gearshift is obtained.

In Theorem 3, we express a necessary and sufficient condition such that we can have a no-jerkdownshift (gearshift scenario 2) in the torque limited region of the motor. This limitation is the samefor both when the first clutch is a one-way clutch or a friction clutch. The limitation is illustrated inFigure 12. We simulated several instances of gearshift scenarios 2, where we varied the initial vehicleacceleration ar. We recorded the required time for the motor to synchronize with the first gear speed,which we label ts. When ar is too high, either ts is impractically long, or the motor never synchronizeswith the first gear speed.

Theorem 3. For a dual-clutch architecture with two friction clutches, when the motor operates in atorque-limited region, assuming a constant T2 during the inertia phase, a no-jerk power-on downshiftof scenario 2 can be obtained if and only if

∃ ts > 0 s.t. 0 = −i1vi + artsrw

+ exp

(−cmImts

)i2virw

+Tmax − T2

cm

[1− exp

(−cmImts

)]. (43)

Proof. (Necessity): The gearshift begins with an inertia phase (0 ≤ t ≤ ts), where θm is acceleratedfrom gear 2 synchronization speed to gear 1 synchronization speed, while Tm = Tmax and T1 = 0. Thegearshift ends with a torque phase (ts ≤ t ≤ ts + ttr), where the transmission torque is transferredfrom clutch 2 to clutch 1. During both phases, the conditions for a no-jerk gearshift in Definition 1can be maintained by modulating T1 and T2 as per Equation 15. Motor saturation can be avoided byrestricting the motor power to Tm ≤ Tmax. Clutch torque application rate saturation can be avoidedby using an appropriate torque transfer trajectory during the torque phase. However, when usingthese restrictions together, nothing guarantees that the motor will eventually synchronize with gear1 speed at some time ts. We now find an equation that describes the synchronization time ts whenthe no-jerk gearshift conditions are maintained. With Tm = Tmax and T1 = 0 in Equation 14, we getthe motor acceleration

θm = I−1m

(−cmθm + Tmax − T2

). (44)

Due to the reasonable assumption that T2 is constant during the inertia phase, Equation 44 is a linearordinary differential equation. We solve Equation 44 to obtain the evolution of θm(t). We impose theinitial condition that θm(0) = i2θout(0), and we get

θm(t) = exp

(−cmImt

)i2θout(0) +

Tmax − T2cm

[1− exp

(−cmImt

)]. (45)

When the motor synchronizes with the gear 1 speed, we have that θm = i1θout. Substituting thiscondition into Equation 45, and using the fact that θout(t) = (vi + art)/rw, we obtain Equation 43.For the motor to synchronize with gear 1 speed, Equation 43 must have a solution. Therefore, ifEquation 43 has no solution for ts, then a no-jerk gearshift cannot be obtained, which proves thenecessity part by contraposition.

(Sufficiency): By construction, the actuation strategy described in the first paragraph of this proofis sufficient for a no-jerk gearshift, given that Assumption 1 holds.

This actuation strategy is not unique. We could have Tm < Tmax, but it would only make itharder to synchronize the motor with gear 1 speed, which can be seen from Equation 44. If clutch

19

0 0.2 0.4 0.6 0.8 1 1.2

ar [m/s2]

0

0.5

1

1.5

2

t s [s]

Figure 12: Example of limitations for a power-on downshift with the example vehicle, when thegearshift is initiated at vi = 18 km/h, and ar is varied. When the desired vehicle acceleration ar isincreased, the time ts required for the inertia phase to complete also increases.

1 is a one-way clutch, then we must have T1 = 0 until the motor synchronizes with gear 1 speed,as per Equation 18. If clutch 1 is a friction clutch, then it is possible to activate T1 before ts. Butfrom Equation 16, we notice that we would have T1 < 0, so Equation 15 dictates that T2 increasesby |T1|(i1/i2) to respect the no-jerk conditions, and since i1 > i2, θm would again be smaller than ifT1 = 0. Therefore, if a no-jerk gearshift can be completed with a different actuation strategy thanthe one presented in the necessity part of the proof, it can also be completed with this strategy. Asa result, the existence of a solution to Equation 43 is a sufficient condition for the possibility of ano-jerk gearshift.

4.2 Theorem adaptations for planetary gearset architectures

In this section, we adapt Theorems 1 to 3 to planetary gearset architectures described by the generalEquations 31 and 32.

4.2.1 Theorem 1 adaptation

With the new system equations, the motor power at the end of the torque transfer phase – originallydescribed by Equation 33 – now becomes

Pm(ttr) = i1θout(ttr)

(cmi1θout(ttr) +

(C1 −

C4C5

C8

)−1[(i1Im −

C4

C8Iout

)θout(ttr)

+

(C2 −

C4C6

C8

)(coθout(ttr) + i−1

f To)])

. (46)

The proof for the necessity condition remains valid, as it it based on the limitations imposed by theone-way clutch, and these limitations still apply. The proof for the sufficiency condition requiresto demonstrate that Pm(ttr) is the maximal motor power required during the gearshift, which nowultimately depends on the coefficients C1 to C8, as can be seen by adapting Equation 36 into a new

20

expression for the motor power:

Pm = θm

(cmθm +

(C1 −

C4C5

C8

)−1[Imθm −

C4

C8Ioutθout

+

(C2 −

C4C6

C8

)(coθout + i−1

f To)−(C3 −

C4C7

C8

)T1

]). (47)

In particular, for T1 = 0 to maximize Pm, we must have −(C3 − C4C7/C8)(C1 − C4C5/C8)−1 < 0.

For the architecture in Figure 2b, this expression reduces to −(β1 + 1)/β1, so that indeed T1 = 0maximizes Pm since −(β1+1)/β1 < 0. Assuming that the other variables – i.e., θm, θm, θout, θout, andTo – remain approximately constant during the torque phase, we have that Pm(ttr) is the maximalmotor power required during the gearshift for the case of the architecture in Figure 2b.

4.2.2 Theorem 2 adaptation

The existence of a sufficient ∆m such that ∆s ≥ 0 remains a necessary and sufficient condition for thepossibility of a no-jerk gearshift. The only difference is that we cannot have Tm jumping to Pmax/θmat t = 0, as prescribed in Equation 40. In effect, Tm now appears in the second equation of motionof the system, i.e., Equation 32, so to maintain To, θm, and θm such that the no-jerk conditions inDefinition 1 are met, a sudden increase in Tm implies a sudden increase in T1 (as we impose T2(0) = 0),which necessarily violates any torque application rate limitation. The effect of an increase in Tm onT1 can be seen in Figure 9. So Theorem 2 remains valid, but we must ramp up Tm such that dT/dtis within limits.

4.2.3 Theorem 3 adaptation

The proof for Theorem 3 remains valid, but the necessary and sufficient condition described byEquation 43 must be adapted to the new system equations. First, we get θm during the inertia phaseby imposing T1 = 0 in Equations 31 and 32; we get

θm = I−1m

(−γcmθm + γTm − τ

), (48)

where γ =

(C1 −

C4C5

C8

), (49)

τ =

(C2 −

C4C6

C8

)(coθout + i−1

f To

)+C4

C8Ioutθout. (50)

Following the argument in Section 4.2.2, we cannot have Tm jump to Tmax at t = 0, as this wouldimply a jump in T2, which would violate any dT2/dt limitation. Therefore, Tm should be increasedgradually from Tm(0) to Tmax at the beginning of the gearshift. For this proof adaptation however,we neglect the effect of gradually ramping Tm on the synchronization time ts – we assume the rampis completed very quickly, so Tm ≈ Tmax. Further, we assume that θout(t) = θout(0) and To(t) = To(0)for the duration of the inertia phase, which we also assumed in Theorem 3 by imposing T2 constantfor the duration of the inertia phase. We now have a linear ordinary differential equation of thesame form as in Theorem 3, which we also solve imposing θm(0) = i2θout(0). We obtain an adaptedcondition for the possibility of a no-jerk gearshift as follows:

∃ ts > 0 s.t. 0 = −i1vi + artsrw

+ exp

(−γcmIm

ts

)i2virw

+γTmax − τ

γcm

[1− exp

(−γcmIm

ts

)]. (51)

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4.3 Theorem adaptation for other vehicle models and kinematic conditions

The theorems presented in this article are based on the general driveline model described in Section 2,which does not include tire slip and other nonlinear behaviors likely to occur on real-world vehicles.Should one desire to adapt the theorems to other vehicle models, they would need to solve for therequired transmission output torque To(t) and velocity θout(t) using the new equations for the vehiclemodel and the no-jerk constraints they wish to impose. Effectively, this would result in differentconditions for a no-jerk gearshift than what we present in Definition 1. With these new conditions forTo(t) and θout(t), the theorems can be adapted accordingly, perhaps with additional mathematicalcomplexity as more terms may become time-dependent.

Similarly, one could adapt the theorems to other kinematic conditions than the no-jerk conditionwe imposed (

...θ v = 0), such as small and constant vehicle jerk level for instance. This also requires

solving for new conditions on To(t) and θout(t), and adapting the theorems accordingly. The sameremark holds: this may increase the mathematical complexity as more terms may become time-dependent.

5 Conclusion

In this article, we presented theorems that describe the fundamental limitations to no-jerk gearshiftin the presence of motor and clutch saturation. We showed that transmissions with a one-way clutchhave stronger limitations than their friction clutch counterparts. This means that a one-way clutchtransmission will fail to provide an uninterrupted gearshift under a wider set of driving conditions.We also showed that transmissions with a planetary gearset architecture have different dynamics thantransmissions with parallel shaft architectures, which requires slight adaptations of the theorems forfundamental limitations of no-jerk gearshifts.

This work has important implications for automotive engineers. The theorems are tools to quicklyevaluate if a no-jerk gearshift is possible given a vehicle description, driving scenario, and transmissionarchitecture. This can be used to motivate the choice of a transmission type over another during theconceptual design phase of a new vehicle. For example, if a vehicle is expected to regularly performupshifts close to saturation in the power-limited region of its motor map, and Theorem 1 indicatesthat a gearshift jerk would often be inevitable, then the vehicle designers should consider using a dual-friction-clutch architecture instead of a one-way clutch. Also, the theorems can be integrated in atransmission control unit: when a gearshift is desired, the unit quickly evaluates if a no-jerk gearshiftis possible, and then decides if the driving torque should be smoothly reduced prior to initiatingthe gearshift, or if the gearshift can be initiated with the current DTD without substantial risks ofsaturating the motor and obtaining a large driveline jerk. This work also has important implicationsfor academic researchers. The theorems present conditions where gearshift jerk is unavoidable, andany attempt at eliminating jerk with a new controller design would be futile. Moreover, this workhelps to identify when two transmission architectures are mathematically equivalent, and thereforewill result in the same fundamental limitations.

Funding sources

This work was supported by Mitacs and Quebec’s Fonds de recherche Nature et technologies.

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