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AIMETA 2019

XXIV Conference

The Italian Association of Theoretical and Applied Mechanics

Rome, Italy, 15-19 September 2019

TORQUE VECTORING CONTROL FOR FULLY ELECTRIC SAE

CARS

Valentina De Pascale1,2, Basilio Lenzo1, Flavio Farroni2, Francesco Timpone2, and

Xudong Zhang3

1 Department of Engineering and Mathematics, Sheffield Hallam University

Howard Street, S1 1WB, Sheffield, UK

e-mail: vdepascale93@gmail.com, basilio.lenzo@shu.ac.uk

2 Industrial Engineering Department, University of Naples “Federico II”

Via Claudio 21, 80125 Naples, Italy

{flavio.farroni, francesco.timpone}@unina.it

3 National Engineering Laboratory for Electric Vehicles, Beijing Institute of Technology

5 South Zhongguancun Street, Beijing, 100081, China,

xudong.zhang@bit.edu.cn

Keywords: Torque Vectoring Control, Driver Model, Fully Electric Vehicle, Formula SAE.

Abstract. Fully electric vehicles with individually controlled powertrains can achieve signifi-

cantly enhanced vehicle response, in particular by means of Torque Vectoring Control (TVC).

This paper presents a TVC strategy for a Formula SAE (FSAE) fully electric vehicle, the “T-

ONE” car designed by “UninaCorse E-team” of the University of Naples Federico II, featur-

ing four in-wheel motors. A Matlab-Simulink double-track vehicle model is implemented, fea-

turing non-linear (Pacejka) tyres. The TVC strategy consists of: i) a reference generator that

calculates the target yaw rate in real time based on the current values of steering wheel angle

and vehicle velocity, in order to follow a desired optimal understeer characteristic; ii) a high-

level controller which generates the overall traction/braking force and yaw moment demands

based on the accelerator/brake pedal and on the error between the target yaw rate and the

actual yaw rate; iii) a control allocator which outputs the reference torques for the individual

wheels. A driver model was implemented to work out the brake/accelerator pedal inputs and

steering wheel angle input needed to follow a generic trajectory. In a first implementation of

the model, a circular trajectory was adopted, consistently with the "skid-pad" test of the FSAE

competition. Results are promising as the vehicle with TVC achieves up to ≈ 9% laptime sav-

ings with respect to the vehicle without TVC, which is deemed significant and potentially cru-

cial in the context of the FSAE competition.

V. De Pascale, B. Lenzo, F. Farroni, F. Timpone, and X. Zhang

1 INTRODUCTION

Formula SAE is an international student design competition which challenges worldwide

students to conceive, design, fabricate and compete with small formula-style racing cars [1].

While the competition was historically based on internal combustion engines (since 1981),

recently there has been an increasing interest towards electric-powered Formula SAE vehi-

cles, with the first Formula SAE Electric competition taking place in 2013 [2]. Most of the

solutions adopted so far include two or four in-wheel electric motors, without differential.

That allows Torque Vectoring Control (TVC), i.e., the individual control of each drivetrain

[3-7]. By imposing an uneven distribution of torque demand between the left and right side of

the vehicle, a direct yaw moment can be generated and appropriately exploited to improve

vehicle performance and, ultimately, reduce laptime.

This paper deals with the development and assessment of a torque vectoring strategy for

the Formula SAE vehicle T-ONE of the UninaCorse E-team from the University of Naples

Federico II (Fig. 1), featuring four in-wheel motors, and with main parameters shown in Table

1. The vehicle model and simulations were implemented in Matlab-Simulink.

Section 2 describes the vehicle model. Details regarding the torque vectoring algorithm are

given in Section 3. Section 4 deals with the reference trajectory and the driver model. Prelim-

inary results are presented in Section 5, and conclusions are in Section 6.

Figure 1: The Formula SAE vehicle "T-ONE".

2 VEHICLE MODEL

A double-track vehicle model was implemented. The Adapted ISO sign convention [8] and

the vehicle reference frame and schematic in [9] were adopted in this study. Hence, the -axis

represents the forward direction, the -axis indicates the lateral direction (positive to the left),

the -axis is vertical (positive upwards). The longitudinal and lateral components of the veloc-

ity of the centre of mass of the vehicle are respectively u and v, while r is the vehicle yaw

rate.

and

are, respectively, the longitudinal and lateral forces at the corner , where

1,2 for front and rear axles, and 1,2 for left and right sides. The wheel steering angle,

, is assumed to be the same for both front wheels.

V. De Pascale, B. Lenzo, F. Farroni, F. Timpone, and X. Zhang

Quantity Symbol Value and unit

Wheel radius 0.26 m

Front semi-wheelbase 0.990 m

Rear semi-wheelbase 0.660 m

Vehicle mass 350 kg

Moment of inertia along the vertical axis 400 kg

Wheelbase l 1.650 m

Track t 1.200 m

Height of the centre of mass h 0.32 m

Table 1: Main parameters of the Formula SAE vehicle T-ONE.

The longitudinal equilibrium equation is

1

2

(1)

which includes the aerodynamic drag, where is the air density, the drag coefficient, and S

the frontal area of the vehicle.

The lateral equilibrium equation is

1

2

(2)

where is the lateral drag coefficient.

The moment balance equation in the direction leads to:

+ (3)

where is the yaw moment generated via the TVC (see Section 3).

The congruence equations, under the assumption of small sideslip angles, read

2 (4)

2(5)

2 (6)

2(7)

V. De Pascale, B. Lenzo, F. Farroni, F. Timpone, and X. Zhang

where is the tyre slip angle at the corner .

The constitutive equations were implemented using a PAC2002 Pacejka formulation, start-

ing from the .tir file of the used tyre, i.e. Hoosier 13''. The adopted formulation provides the

lateral forces as functions of camber angle, , vertical load, , slip angle, , and

wheel radius, , in pure lateral conditions. , instead, were obtained with an even distribu-

tion among the four wheels of the overall desired longitudinal force, , provided by the driver

model (see Section 3). The vertical loads are

(8)

(9)

(10)

(11)

where the downforce and longitudinal load transfer contributions are

2

(12)

2

(13)

and the lateral load transfers are

1

(14)

1

(15)

where h is the height of the centre of mass, d the height of the roll centre below the centre of

mass, and the front and rear aerodynamic lift coefficients, e the front

and rear relative roll stiffness values,

and

the static front and rear

vertical loads are, the gravity acceleration, and the vehicle longitudinal and lateral

accelerations.

3 TORQUE VECTORING CONTROL

The developed TVC strategy is based on the scheme proposed in [10]. A reference yaw

rate value, , is generated through a look-up table which takes as input the wheel steering

angle, , and the vehicle velocity, . The look-up table was built considering steady-state

conditions and a desired vehicle cornering response (a.k.a. understeer characteristic), shaped

as in Equation 17. With respect to the baseline vehicle, i.e. the vehicle without TVC, the cor-

nering response is designed so as to: i) decrease the understeering gradient; ii) extend the re-

gion of linear relationship between dynamic steering angle, , and lateral acceleration, up

to

∗; iii) increase the maximum lateral acceleration achievable, ,, which is very im-

portant in the interest of laptime minimisation. Specifically, the look-up table was built by

defining vectors of and , then using the following relationships:

(16)

V. De Pascale, B. Lenzo, F. Farroni, F. Timpone, and X. Zhang

Then, to relate the dynamic steering angle to the overall steering angle, the kinematic steering

angle (Ackermann angle), , was obtained as

(18)

and added to the dynamic contribution to obtain the total wheel steering angle:

(19)

Finally, the table was inverted in order to have wheel steering angle and vehicle velocity as

input, and reference yaw rate as output.

A PID controller was implemented to track the yaw rate, specifically taking as input the er-

ror between the reference yaw rate and the current yaw rate, , and giving as output the value

of yaw moment, , to be generated.

Once the value of desired overall force and yaw moment are known, a "control allocator"

block [4,11] calculates the four wheel torque demands, , as:

2

4 (20)

2

4 (21)

4 REFERENCE TRAJECTORY AND DRIVER MODEL

Among the Formula SAE dynamic tests, this study selected the Skid-pad test [12], in

which the car goes through a figure-of-eight shaped track including two circles with diameter

15.25 m. The car performs two laps in one of the circles, then it moves to the other circle and

it performs other two laps. The best laptime is selected between the second attempt at each

circle. Hence, in a first implementation of this work, it is sufficient to design a circular trajec-

tory with radius R, to be negotiated twice. Specifically, the vehicle starts in (0,0) so the circle

has centre in (0, R). The equations for the reference position are thus:

cos/(22)

sin/(23)

where s is the curvilinear abscissa, which can be calculated as:

(24)

The driver model used in this study is inspired to [13]. It calculates: i) the wheel steering

angle, , through a Proportional controller based on errors on position and orientation of the

vehicle; ii) the acceleration/brake inputs, i.e. the overall longitudinal force demand, , to

achieve the maximum possible vehicle speed.

The reference trajectory is obtained via Equations 22, 23 and 24. The reference orientation

of the vehicle, _, is taken after a speed-dependent "visual" distance, , defined as

∗

∗,

∗ln,

∗,

∗

(17)

V. De Pascale, B. Lenzo, F. Farroni, F. Timpone, and X. Zhang

2 (25)

where depends on the driver's behaviour (herein assumed as 0.3 s) and V is the vehicle

speed, i.e. √. Denoting the current position with (x, y), the position error is

cossin (26)

and the orientation error is

_

d

2(27)

where the constant

is needed to guarantee the use of consistent reference frames. Finally,

(28)

where and are calculated according to [13].

The maximum, i.e. target, vehicle speed, , depends on the maximum allowable lateral

acceleration,,:

, (29)

The target longitudinal acceleration, ,, can be worked out as a function of the maxi-

mum allowable longitudinal acceleration, ,:

,,1 ||

, (30)

The overall longitudinal force demand, , is composed of a feedforward contribution,

,

, to improve the driver promptness (the sign in front of the aerodynamic

drag is positive in acceleration and negative during braking), and a feedback contribution

which is a Proportional Integral controller based on the error . Due to the specific

electric motors used in this project, the individual motor torques are saturated to 21 Nm.

5 PRELIMINARY RESULTS

Based on the vehicle model described in Section 2 integrated with the TVC algorithm de-

scribed in Section 3, and on the driver model presented in Section 4, simulations were per-

formed in Matlab-Simulink to assess the performance of the proposed control strategy. The

circumference radius to be followed by the centre of mass of the car was set to 8.3 m, as it

takes into account the size of the vehicle.

Figure 2 shows the reference trajectory and the actual trajectory during the second lap. The

reference trajectory is perfectly followed, demonstrating the effectiveness of the driver model.

Figure 3 shows the curvilinear abscissa and the yaw rate (negative as the vehicle is negotiat-

ing a right turn, according to the adopted sign conventions) as a function of time for the base-

line vehicle and the TVC vehicle. With the baseline vehicle, the time taken to complete the

trajectory is 4.26 s. By activating the TVC, the laptime decreases to 3.84 s. So, there is a lap-

time improvement of 9% by using TVC with respect to the baseline vehicle.

V. De Pascale, B. Lenzo, F. Farroni, F. Timpone, and X. Zhang

Figure 2: Actual and reference trajectory, which coincide thanks to the driver model.

Figure 3: Comparison between baseline vehicle and TVC vehicle: (top) curvilinear abscissa as a function of

time; (bottom) yaw rate as a function of time.

V. De Pascale, B. Lenzo, F. Farroni, F. Timpone, and X. Zhang

6 CONCLUSIONS

In this paper, a Torque Vectoring Control strategy was presented for a Formula SAE elec-

tric vehicle. Matlab-Simulink was used to implement a double track vehicle model featuring

Pacejka tyres, and a driver model providing the steering angle and the acceleration/braking

input. The implemented Torque Vectoring Control strategy allowed a time saving of around

9% during a skidpad test. Further developments will include the improvement of the simula-

tion model adopted (e.g. by including tyre combined interaction), the assessment of the bene-

fits of the proposed technique along a simulated lap, and the experimental validation on the T-

ONE vehicle.

REFERENCES

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May 2019.

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