Model Predictive Torque Vectoring Control for a Formula Student Electric Racing Car

Abstract

In this paper we present two torque vectoring control algorithms for an electric racing car with independent all-wheel drive. A nonlinear, two-track vehicle model is used for the design of a linear time-varying model predictive controller and a nonlinear model predictive controller, while the unknown system states are estimated by the unscented Kalman filter. The controller has been tested in various simulation scenarios and the obtained results are compared with the case assuming equal torque distribution, i.e. without torque vectoring. The results show that the proposed optimization-based torque vectoring control strategy may effectively stabilize the vehicle at the limits of handling and in this way increase its track performance.

Full text

Model Predictive Torque Vectoring Control
for a Formula Student Electric Racing Car
Erik Mikul´
aˇ
s, Martin Gulan and Gergely Tak´
acs
Abstract— In this paper we present two torque vectoring
control algorithms for an electric racing car with independent
all-wheel drive. A nonlinear, two-track vehicle model is used for
the design of a linear time-varying model predictive controller
and a nonlinear model predictive controller, while the unknown
system states are estimated by the unscented Kalman filter. The
controller has been tested in various simulation scenarios and
the obtained results are compared with the case assuming equal
torque distribution, i.e. without torque vectoring. The results
show that the proposed optimization-based torque vectoring
control strategy may effectively stabilize the vehicle at the limits
of handling and in this way increase its track performance.
I. INTRODUCTION
The history of electronically controlled torque distribution
goes back to the 1980s, when the first cars with such systems
were released. Direct yaw control was achieved by activating
brakes on individual wheels. Although this technique was ef-
fective for pro-active accident prevention, it was not suitable
for racing cars, where the deterioration of the longitudinal
performance would occur due to braking. The first work
that showed a possibility of directly controlled drive torque
distribution was presented by Sawase et al. [1] who achieved
this by utilizing a special differential with electronically
controlled clutches. However, the true power of such systems
came to spotlight with the development of electric cars with
independently driven wheels. This configuration enables one
to control the driving torque precisely and without additional
energy losses. Such electronic vehicle systems that directly
control driving torque to increase vehicle performance and
safety are generally referred to as torque vectoring (TV)
systems.
The increasing computing power of microcontrollers and
the growing computational efficiency of optimization algo-
rithms has opened new possibilities for the implementation
of more advanced control algorithms into vehicle electronic
systems. A possible field of application for torque vector-
ing systems is, of course, in road cars. A well-designed
control algorithm can enhance vehicle safety with a small
additional cost compared to the improvement of mechanical
parts, which is more expensive and usually can not bring a
significant result. The other field of application for torque
vectoring systems is the racing industry, where most of
*This work was supported by the Slovak Research and Development
Agency (APVV) under the contract APVV-14-0399.
1Erik Mikul´
aˇ
s, Martin Gulan and Gergely Tak´
acs are with the
Institute of Automation, Measurement and Applied Informatics, Fac-
ulty of Mechanical Engineering, Slovak University of Technology in
Bratislava, 81231 Bratislava, Slovakia erik.mikulas@stuba.sk;
martin.gulan@stuba.sk;gergely.takacs@stuba.sk
the cars are already mechanically superb so any electronic
system that can give an edge over rivals is highly appreciated.
The torque vectoring control problem is conventionally ad-
dressed by a proportional-integral-derivative controller (PID)
[2], which provides satisfactory control quality for standard
driving conditions. However, racing cars are often driven
near the limits of handling, where vehicle behavior becomes
highly nonlinear and therefore linear control strategies are
not sufficient. Thus, nonlinear control algorithms are pre-
ferred for racing scenarios. Optimization-based strategies
such as model predictive control (MPC) that employs con-
strained minimization of a quadratic cost function are an
excellent match for vehicles on the edge of performance and
physical possibilities. Thereby, the inherent constraint han-
dling of MPC pushes the performance to the boundaries—
just what a racing car requires. Several works utilize linear
time-varying model predictive controllers (LTV-MPC) for
this reason. An LTV-MPC scheme along with the unscented
Kalman filter (UKF) for state estimation was developed for
vehicles equipped with an independent rear-wheel drive in
[3], [4], [5]. A similar design for all-wheel drive vehi-
cles was presented by Vasiljevic and Bogdan in [6]. Two
nonlinear model predictive control strategies (NMPC) were
presented recently in [7], yet assuming only a rear wheel
driven vehicle. Furthermore, fuzzy logic controllers that are
otherwise seldom used for torque vectoring are presented in
[2] and [8]. In 2013, Mercedes-Benz introduced SLS AMG
Electric Drive [9] as a showcase of what can be practically
achieved by torque vectoring. The control algorithms are not
publicly known yet but the results are remarkable [9]. To
bear on our knowledge MPC-based torque vectoring was not
implemented on formula student race car.
Unlike previous works, this paper focuses on the design of
a model predictive torque vectoring controller for a formula
student electric racing car with all-wheel drive. Due to the
utilization of extremely high performance tires and great
driving torque, this car operates on the edge of its perfor-
mance envelope, where the behavior of the tire becomes
highly nonlinear. To this end we propose a LTV-MPC torque
vectoring system, which is subsequently compared to an
alternative NMPC torque vectoring approach.
This paper is organized as follows. Section II introduces a
nonlinear two-track model of the car. Section III presents
torque vectoring fundamentals and objectives. Section IV
presents the proposed LTV-MPC and NMPC strategy with
reference tracking and constraints. Finally, in Section V, the
performance of the proposed control strategies is demon-
strated in two simulation scenarios.
2018 European Control Conference (ECC)
June 12-15, 2018. Limassol, Cyprus
978-3-9524-2699-9 ©2018 EUCA 581

TABLE I
PARAMETERS OF THE SGT-FE18 RACING CAR
Parameter Symbol Value Unit
Weight m235 kg
Front track wf1.22 m
Rear track wr1.19 m
Wheel base l1.57 m
Center of gravity front arm lf0.71 m
Center of gravity rear arm lr0.86 m
Center of gravity height hcm 0.25 m
Wheel radius R0.22 m
Aero drag force at 25 ms−11100 N
Aero down force at 25 ms−1380 N
Max torque per motor 21 Nm
Max power per motor 36 kW
Max combined power (regulated) 144 kW
Transmission ratio i13.9 -
wf
wr
aero
lf
lr
F
FR
x
F
FR
y
F
FL
x
F
FL
y
F
RL
x
F
RL
y
F
RR
x
F
RR
y
x
y
x
z
v
y
x
z
c
c
Fig. 1. Forces acting on the vehicle body.
II. VEH ICLE MODEL
A model was developed to aid simulations and controller
design. The real-world vehicle is a formula student electric
(FSE) racing car named ’SGT-FE18’. The car is currently
being built by FSE team STUBA Green Team at the Slovak
University of Technology in Bratislava, for the 2018 racing
season. The vehicle has a carbon fiber monocoque structure,
lightweight aerodynamic package to create down force and
all four wheels are driven by independent electric motors
through two-stage planetary gearboxes.
In order to simplify the vehicle model, let us assume that
the vehicle is moving only on a flat surface, i.e. the effects
of suspension are omitted; therefore there is no roll and
pitch and the tire rolling resistance is negligible. The model
itself is divided into three parts: equations of motion for
the rigid vehicle body, wheel dynamics and tire model. The
parameters of the vehicle are summarized in Table I.
A. Planar two-track vehicle model
The equations of motion for the vehicle body contain the
forces acting on the vehicle, and these are shown in Fig. 1:
the aerodynamic drag Faero
xand the tire/road contact forces
Fij
k(¨x, ¨y, ¨
ψ, ˙x, ˙y, ˙
ψ), where the superscript istands for the
front or the rear of the vehicle, i.e. i={F,R}, while the
superscript jis used to distinguish the right and the left side,
i.e. j={R,L}, and k={x, y}indicates the direction in
which the variable is defined. The component equations for
longitudinal, lateral and yaw acceleration are as follows [6],
[10]:
˙vx=1
m(FFR
xcos δFR +FFL
xcos δFL −FFR
ysin δFR−
FFL
ysin δFL +FRR
x+FRL
x−Faero
x)−˙
ψvy,(1)
˙vy=1
m(FFR
xsin δFR +FFL
xsin δFL +FFR
ycos δFR+
FFL
ycos δFL +FRR
y+FRL
y) + ˙
ψvx,(2)
¨
ψ=1
Iz
(lF(FFR
xsin δFR +FFR
ycos δFR +FFL
xsin δFL+
FFL
ycos δFL)−lR(FRR
y+FRL
y) + wF
2(−FFR
xcos δFR
+FFR
ysin δFR +FFL
xcos δFL −FFL
ysin δFL)
+wR
2(+FRR
x−FRL
x)),(3)
where all the parameters are according to Tab. I. The steering
angles δF,j are considered only for the front wheels. The toe
angle on the rear wheels is omitted. The rotational dynamics
for each wheel is given by
˙ωij =1
Ii
red
(Mij
p−Mij
b−Fij
xR),(4)
where Mpdenotes the propulsion torque generated by the
motors and Mbis the torque created by the brake.
B. Tire model
In order to determine the longitudinal and lateral tire
forces, version 5.2 of the Pacejka’s well-known ‘Magic For-
mula‘ tire model is used [11]. This model can be expressed
as set of semi-empirical functions given in the form of
Fij
x=Fij
x0(κij , F ij
n)Gij
yα(αij κij , F ij
n),(5a)
Fij
y=Fij
y0(αij , F ij
n)Gij
xκ(αij κij , γ ijFij
n)+
SV yκ(αij κij , γ ijFij
n),(5b)
where Fij
x0and Fij
y0denote forces for pure lateral and pure
longitudinal slip, Gij
yα and Gij
xκ are the weighting functions
for combined slip, SV yκ is vertical shift in the lateral force
due to camber and normal force for combined slip and κij
is the longitudinal slip ratio of individual wheels given by
κ=ωR −vx,w
vx,w
.(6)
Furthermore, αij denotes the slip angle of individual wheels
and is given by
α= arctan vy+˙
ψli
|vx−˙
ψwi
2|−δ, (7)
where γis the camber angle of the wheel that is assumed
constant while Fij
nis the normal force acting in the contact
point between the tire and the road for individual wheels,
and is obtained from static load distribution and load transfer
induced by longitudinal and lateral acceleration.
582

C. State-space model formulation
Let us now assume that the dynamics is represented by
the nonlinear state-space model given by Eqs. (1)–(7) as
˙
x(t) = fx(t),u(t),y(t) = hx(t),u(t),(8)
where x(t) = hvx, vy,˙
ψ, ωFR , ωFL, ωRR, ωRLiT
is the state
vector and u(t) = MFR
p, M FL
p, M RR
p, M RL
pTis the input
vector.
For the purpose of controller design the fast wheel dy-
namics was neglected, in order to reduce execution time,
allow for fairly long sampling periods and enhance numerical
stability of controller similarly as in [3]. This simplification
may increase the feasibility of real-time implementation on
computing devices with limited computational performance,
note that even nominal linear MPC required more than 1ms
execution time on microcontrollers. After this simplification
the state vector will be x(t) = hvx, vy,˙
ψiT
and input
vector u(t) = MFR
p, M FL
p, M RR
p, M RL
pT. Subsequently,
the simplified version of the model in Eq. (8) was linearized
and discretized to get the LTV state-space model descibed
by
x(k+ 1) = A(k)x(k) + B(k)u(k),(9)
where the discrete-time state-transition matrix Ak∈R3×3
and input matrix Bk∈R3×4are evaluated for the current
operating point in each sample.
III. CON TROL LER DESIGN
The fundamentals of the proposed LTV-MPC and NMPC
control strategies along with torque vectoring objectives and
a state estimation framework are presented in this section.
A. Torque vectoring principles
A torque vectoring system should manage driving torque
distribution between the driven wheels. In some cases, it is
even possible to apply a negative torque to brake individual
wheels, which is instrumental in small radius cornering. If
the battery management system enables braking energy recu-
peration, it is possible to apply a greater braking power than
driving power. In case energy recuperation is not possible—
like it is assumed here—the total power must be zero or
greater.
The driving torque should be distributed in a way that will
ensure a stable behavior of the vehicle. The torque vectoring
system cannot allow torque resulting in a large tire longitudi-
nal slip ratio. The maximum torque that the tires can transfer
to the road surface is obtained form the mathematical model
of the tire in Eq. (5). Since we neglect wheel dynamics, we
assume that lateral tire forces correspond to the amount of
torque applied to the wheels while the maximum longitudinal
forces that the tires can provide are taken into account on
the level of input constraints for both cases.
B. State estimation
We have chosen to implement the unscented Kalman filter
(UKF) for state estimation [12]. The UKF enabled to utilize
the full nonlinear vehicle model in Eqs. (1)–(7) while sus-
taining lower computational effort compared to the extended
Kalman filter (EKF). The 6-step Euler method was used for
the discretization of continuous-time dynamics. We assumed
that the vehicle is equipped with wheel speed sensors and an
inertial measurement unit (IMU), the data provided by these
sensors are taken as the output of the system. The simulated
output vector with additive measurement noise is given as
y(k) = h˙vx,˙vy,¨
ψ, ωFR , ωFL, ωRR, ωRLiT
. Consequently,
the state vector estimated by the UKF can be written as
ˆ
x(k) = hˆvx,ˆvy,ˆ
˙
ψ, ˆωFR ,ˆωFL,ˆωRR,ˆωRLiT
.
C. Reference calculation
Both control strategies share the same state reference
formulated similarly as in [6] and calculated according to
the current state, the driver input and the handling limits of
the vehicle as xref =vx,ref vy,ref ˙
ψref T.
The longitudinal speed reference vx,ref is chosen accord-
ing to the current speed, motor torque requested by driver
Mdr,length of prediction horizon npand the maximum
allowable speed vmax that is dependent on the maximal
lateral force that tires can handle Fy,max, determined from
the tire model [11] and the steady-state cornering radius Rss,
which in turn is determined from the car’s geometry and
steering input as
Rss =lF+ lR
δR+δL
2
,(10)
v2
max =RssFy,max
m,(11)
vx,ref = min q|v2
x+v2
y|+iMdr
Rm npTs,q|v2
max −v2
y|.
(12)
Lateral speed reference vy,ref is chosen as the actual lateral
speed being saturated at the value that results in the maximal
allowable vehicle slip angle βpthat is determined from tire
characteristics provided by the tire manufacturer, and is given
by
vy,ref = sign(vy) min (|vy|,tan(βp)vx).(13)
The yaw rate reference ˙
ψref is calculated from the geometry
of the car, actual longitudinal speed, steering angle and the
under-steer coefficient Kus, which can be chosen with respect
to the desired car behavior. Positive values correspond to
under-steer behavior while negative values imply over-steer
behavior. Usually, the desired behavior for racing car is
neutral-steer, then corresponding value of the coefficient will
be Kus = 0. Thus, the reference yaw rate will be
˙
ψref =vxtan(δR+δL
2)
(lR+ lF)+Kusv2
x
.(14)
583

D. Linear MPC
The linear MPC formulatoin assumed here is given by the
following constrained optimization problem
min
U
np−1
X
i=0 kxk+i−xref k2
Q+kuk+ik2
R1,(15a)
s.t.xk=ˆ
x(k),(15b)
xk+1 =Akxk+Bkuk,(15c)
u≤uk+i≤ui= 0,1, . . . , np−1,(15d)
where npis the length of the prediction horizon and the
optimal solution of this problem is the sequence of optimal
future inputs U∗=hu∗
k,u∗
k+1,...,u∗
k+np−1i∈Rnpnu×1.
The current state ˆ
xk—as estimated by the UKF—is chosen
as the operating point for LTV state-space model. The states
are weighted by Q∈Rnx×nxwithin the quadratic objective
Eq. (15a), while inputs are penalized R∈Rnu×nu
1. In our
simulation study, (15a)–(15d) are solved by the qpOASES
solver, in the form of a quadratic programming (QP) problem
[13] in MATLAB/Simulink. Note, that our choice of QP
solver is not arbitrary, as qpOASES have been proven to be
real-time feasible on limited computational hardware before
[14].
Input constraints are recalculated at every sampling period
according to the maximal forces that tires can handle at the
current state and motor limitations. To maintain a predictable
vehicle behavior, there are constraints for the sum of all
torques that cannot exceed the torque requested by the driver.
This ensures that the torque is only distributed between
wheels or reduced to decrease vehicle speed when the
cornering radius is infeasible according to Eq. (12).
E. Nonlinear MPC
The considered nonlinear MPC scheme is based on solving
the following optimal control problem
min
U,X
1
2
t0+npTs
Z
t0kx−xref k2
Q+ku−uref k2
R2dt, (16a)
s.t.x(t0) = ˆ
x(t0),(16b)
˙
x=f(x,u),(16c)
hineq ≤0,(16d)
where (16d) represent inequality constraints and X∈
Rnpnx×1vector of predicted states. In order to numerically
solve the nonlinear control scheme presented above we
make use of the ACADO Code Generation tool [15], ex-
ploiting direct multiple-shooting, real-time iteration scheme
and sequential quadratic programming using its MATLAB
interface. The mathematical model used in this controller is
the natively nonlinear state-space model in Eq. (8) which is
internally discetized by ACADO. Since the rotational dynam-
ics of the wheels was neglected, there is no way to calculate
κi,j internally, so we used simplified inverted tire model to
calculate κi,j from the current state and the applied torque.
To prevent creating an algebraic loop, normal forces were
assumed constant during the prediction horizon. To ensure a
maximum possible longitudinal performance, we introduced
an additional state ˙s=vx, representing the distance traveled
in the longitudinal direction. The augmented state vector
takes form x(t) = hvx, vy,˙
ψ, siT
To achieve this effect,
we chose the reference distance larger than it is possible to
travel during the prediction horizon. The constraints were
applied to inputs, slip angles αij and to the sum of power.
F. Driver Command
In order to emulate driver inputs, a simple path following
driver model was designed based on PID controllers. This
model controls the accelerator and brake pedals—that di-
rectly represents the amount of requested driving or braking
torque—according to a defined speed reference. Furthermore,
the driver model controls steering input according to the
distance of the car from the track reference and yaw angle.
Subsequently, the steering angle δfor the left and right wheel
is calculated according to the Ackerman geometry:
cot δI= cot δ−l
2wf
,cot δO= cot δ+l
2wf
,(17)
where δIis the inner wheel angle and δOis the outer wheel
angle during cornering.
IV. SIMULATION RESULTS
In this section, we present simulation results and the
comparison of vehicle behavior with torque vectoring and
with equal torque distribution.
The setup of the LTV-MPC controller was the same for all
simulations. The sampling period for the controller and the
estimator was Ts= 0.002 s, the prediction horizon length
was chosen np= 20 steps and the weighting matrices were
set to
Q= diag(100,10,10),
R1= diag(0.002,0.002,0.002,0.002).
In order to ensure a fair comparison of the two torque vec-
toring strategies, most parameters were set in such way that
the effect of them on the closed-loop response is comparable.
This included, horizon length and state penalty, while in the
interest to fine-tune performance the input penalty of the
NMPC controller was chosen as
R2= diag(0.1,0.1,0.1,0.1).
The sampling time of NMPC was chosen as Ts= 0.01 s
to enable the practical tractability of simulations with 20
integrator substeps within one sample. The interested reader
may find more details on the simulation settings, in particular
the UKF, in [16].
A. U-turn scenario
U-turn scenario with 10 m radius at the limit of handling
This scenario demonstrates how torque vectoring pre-
vented vehicle spin, then later during the acceleration after
the apex, helped to maintain the cornering radius. As it can
584

-20 -10 0 10 20 30 40
X [m]
0
5
10
15
20
Y [m]
Reference track
TV on
TV off
Fig. 2. Trajectory for the U-turn with a 10 m corner radius at the limit of
handling.
0 2 4
Time [s]
15
20
25
30
35
vx [ms-1]
(a)
0 2 4
Time [s]
-2
-1
0
vy [ms-1]
(b)
024
Time [s]
-1
0
1
2
v [rad s-1]
(c)
024
Time [s]
-0.1
0
0.1
0.2
0.3
[rad]
(d)
Fig. 3. States for the U-turn with a 10 m corner radius with LTV-MPC at
the limit of handling, (a) longitudinal velocity, (b) lateral velocity, (c) yaw
rate, (d) steering input.
be observed in Fig. 2, the vehicle trajectory with active torque
vectoring (red line) precisely copied the reference track and
the trajectory of the vehicle with equal torque distribution
(dashed blue line) ended up in a spin shortly after entering
the corner. Note that the lateral offset is due to the imperfect
driver model.
Figure 4 shows the torque distribution for the previously
introduced simulation study. The reader may note that be-
tween the time 1.00–1.25 s more torque was distributed to the
left—inner—wheels, while the torque to the right—outer—
wheel was actually negative in order to prevent vehicle spin
while entering the corner. During the first half of the corner,
torque was distributed almost equally and subsequently,
after reaching the apex and when speed started to increase,
more torque was delivered to the outer wheels to reach an
otherwise infeasible cornering radius.
Moreover, the imperfection and the limitation of the sim-
ple driver model can be observed in Fig. 3 showing the state
trajectories. The oscillating steering input in Fig. 3(d) caused
corresponding oscillations in the yaw rate shown in Fig. 3(c)
and lateral velocity in Fig. 3(b). The torque distribution was
also affected on a smaller scale and this is demonstrated in
Fig. 4, between the times 1.25–2.00 s.
U-turn scenario with a 10 m radius at 16 ms−1
This simulation compares the LTV-MPC and the NMPC
controlled torque distribution strategies. It is interesting to
observe the effect of sampling speed on the performance of
the controllers: NMPC showed the best results at a sampling
period set to Ts= 0.01 s, while the LTV-MPC controller
was numerically more stable at Ts= 0.002 s. However, to
ensure a fair comparison of the methods, both controllers
were sampled at Ts= 0.01 s for this test.
The reader may note that the trajectories for the two torque
vectoring strategies shown in Fig. 5 are almost identical.
However, one may also see significant differences in the
model states shown in Fig. 6, the torque distribution for the
LTV-MPC in Fig. 7 and the NMPC in Fig. 8.
The NMPC strategy reduced the longitudinal velocity in
Fig. 6(a) more than LTV-MPC, but at the end of the corner it
also increased the speed immensely. Figure 6(b) shows that
the NMPC controller actually caused marginal understeer
behavior rather than the oversteer induced by the LTV-MPC
controller. This is actually beneficial at high-speed cornering,
where understeer is more stable.
The difference between the torque distribution for both
torque vectoring strategies is evident from Fig. 7 (LTV-MPC)
and Fig. 8 (NMPC).
In the case of the LTV-MPC strategy, a greater amount of
torque goes to the inner wheels during the entire duration
of the cornering maneuver. However, in the case of NMPC-
based torque vectoring, more torque is supplied to the inner
wheels in the first part of the cornering maneuver, but the
controller is fluently increasing the torque on the outer
wheels as the vehicle passes through the corner.
B. Steering step response
This simulation scenario demonstrates the torque vectoring
response to a step change of the steering angle. The reference
speed for the driver model is constant, hence any changes of
speed are caused by torque vectoring systems only.
Steering step response at 17.5 ms−1for LTV-MPC
This simulation shows how the LTV-MPC controller re-
acted to a step change of the steering angle, along with a
corresponding change of yaw rate reference. Figure 9 shows
the trajectory of two simulated vehicles, where one may
observe that the vehicle with equal torque distribution would
eventually end up in a spin, while a vehicle with LTV-MPC
torque vectoring will manage to stay on track.
The corresponding torque distribution chart shown in
Fig. 10 demonstrates an immediate response of the controller
that stabilizes the vehicle and subsequently applies negative
torque to decrease the longitudinal speed. The states in Fig.
13 hold another interesting detail. As for the given cornering
radius the tire forces will be less than the centrifugal force
(i.e. grip is lost), the LTV-MPC torque vectoring strategy
must decrease the longitudinal speed reference in order for
585

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Time [s]
-10
-5
0
5
10
15
20
25
Torque [Nm]
Constraints FR
Constraints FL
Constraints RR
Constraints RL
TV on FR
TV on FL
TV on RR
TV on RL
TV off
Fig. 4. Torque distribution for the U-turn with 10 m corner radius at the limit of handling.
0 5 10 15 20 25 30 35 40
X [m]
0
5
10
15
20
Y [m]
Reference track
LTV-MPC
NMPC
Fig. 5. Trajectory for the U-turn with a 10 m corner radius, assuming
LTV-MPC (red) and NMPC (blue dashed) torque vectoring at 16 ms−1.
024
Time [s]
15
15.5
16
16.5
17
vx [ms-1]
(a)
024
Time [s]
-0.5
0
0.5
vy [ms-1]
(b)
024
Time [s]
-1
0
1
2
v [rad s-1]
(c)
0 2 4
Time [s]
0
0.1
0.2
0.3
[rad]
(d)
Fig. 6. States for the U-turn with 10 m corner radius with LTV-MPC
(red) and NMPC (blue dashed) at 16 ms−1(a) longitudinal velocity where
dashed black line is reference for driver, (b) lateral velocity, (c) yaw rate,
(d) steering input.
the tire forces to be less than the centrifugal force. This
behavior can be observed in Fig. 13(a). For the remainder of
the simulation the torque distribution remained steady.
Steering step response at 17.5 ms−1for NMPC
This simulation is performed under the same conditions as
the previous one, but this time using the NMPC controller.
The reader may note that the results are largely similar to
the previous case.
Nevertheless, the most significant difference between the
two proposed strategies is in the torque distribution shown
in Fig. 11, where the initial controller reaction was less ag-
gressive but eventually managed to increase the longitudinal
speed (Fig. 12(a)). This caused oscillations of torque distri-
bution and subsequently the oscillations of lateral velocity
(Fig. 12(b)) and yaw rate (Fig. 12(c)).
V. CONCLUSION
We have presented two model predictive torque vectoring
control systems for a racing car with independent all wheel
drive. One strategy was based on LTV-MPC and the other on
the NMPC framework. A nonlinear two-track mathematical
model of the vehicle with nonlinear tire model was em-
ployed, which was linearized at every sampling period for
the LTV-MPC controller. The reference and constraints were
generated according to handling limits and the capabilities
of the propulsion system for both controllers.
The simulation results showed that the proposed torque
vectoring systems were able to stabilize the vehicle near the
limits of the handling, prevented vehicle spin and enabled
to reach a smaller cornering radius than one possible with
equal torque distribution at given speed. Furthermore, the
steering step response simulation demonstrated a decreasing
speed that was limited by the controller in order to prevent
lateral acceleration hitting the constraints.
While it is clear that implementing a natively nonlinear
TV strategy would be more demanding from the viewpoint
of computational efficiency, the inherent advantage of NMPC
over its linear counterpart is better state trajectory predictions
and ultimately a vehicle behavior that is absolutely on the
586

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Time [s]
-5
0
5
10
Torque [Nm]
Constraints FR
Constraints FL
Constraints RR
Constraints RL
FR
FL
RR
RL
Fig. 7. Torque distribution for the U-turn maneuver with 10 m cornering radius achieved with LTV-MPC at 16 ms−1.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Time [s]
-5
0
5
10
Torque [Nm]
FR
FL
RR
RL
Fig. 8. Torque distribution for the U-turn maneuver with 10 m cornering radius achieved with NMPC at 16 ms−1.
-5 0 5 10 15 20 25 30 35
X [m]
0
5
10
15
20
Y [m]
TV on
TV off
Fig. 9. Trajectory for the steering step response at 17.5 ms−1.
limits of handling. Note, that even though the simulations
presented in this paper show that LTV-MPC apparently
outperforms NMPC but a refinement of the model and the
NMPC framework shall enable us to clearly surpass the
performance achieved by LTV-MPC only.
Future work will optimize the controller algorithm to
increase sampling period and decrease computational effort
in order to enable a real-world deployment on embedded
computing hardware with limited performance. Furthermore,
lap-time simulation using an industry-standard vehicle dy-
0 1 2 3 4 5 6 7
Time [s]
-10
-5
0
5
10
15
Torque [Nm]
Constraints FR
Constraints FL
Constraints RR
Constraints RL
FR
FL
RR
RL
TV off
Fig. 10. Torque distribution for the steering step response at 17.5 ms−1.
namics simulation tool such IPG CarMaker or ADAMS/Car
would help to tune the controllers and show the overall
benefit of the proposed torque vectoring systems.
REFERENCES
[1] K. Sawase and Y. Sano, “Application of active yaw control to vehicle
dynamics by utilizing driving/breaking force,” JSAE Review, pp. 289
– 295, 1999.
[2] B. J¨
ager, P. Neugebauer, R. Kriesten, N. Parspour, and C. Gutenkunst,
“Torque-vectoring stability control of a four wheel drive electric
vehicle,” in 26th Intelligent Vehicles Symposium, 2015, pp. 1018–1023.
587

-10 0 10 20 30
X [m]
0
5
10
15
20
Y [m]
Fig. 14. Trajectory for the steering step response with NMPC at
17.5 ms−1.
0123456
Time [s]
-10
-5
0
5
10
Torque [Nm]
FR
FL
RR
RL
Fig. 11. Torque distribution for the steering step response with NMPC at
17.5 ms−1.
0246
Time [s]
15
16
17
18
vx [ms-1]
(a)
0 2 4 6
Time [s]
-2
-1
0
vy [ms-1]
(b)
0 2 4 6
Time [s]
-1
0
1
2
v [rad s-1]
(c)
0 2 4 6
Time [s]
0
0.1
0.2
[rad]
(d)
Fig. 12. States for the steering step response with NMPC at 17.5 ms−1
(a) longitudinal velocity, (b) lateral velocity, (c) yaw rate, (d) steering input.
0 2 4 6
Time [s]
14
16
18
20
vx [ms-1]
(a)
0 2 4 6
Time [s]
-2
-1
0
vy [ms-1]
(b)
0246
Time [s]
-1
0
1
2
v [rad s-1]
(c)
0 2 4 6
Time [s]
0
0.1
0.2
[rad]
(d)
Fig. 13. States for the steering step response at 17.5 ms−1(a) longitudinal
velocity, (b) lateral velocity, (c) yaw rate, (d) steering input.
[3] E. Siampis, M. Massaro, and E. Velenis, “Electric rear axle torque
vectoring for combined yaw stability and velocity control near the
limit of handling,” in 52th Conference on Decision and Control, 2013,
pp. 1552–1557.
[4] E. Siampis, E. Velenis and S. Longo, “Predictive rear wheel torque
vectoring control with terminal understeer mitigation using nonlinear
estimation,” in 54th Conference on Decision and Control, 2015, pp.
4302–4307.
[5] E. Siampis, E. Velenis, and S. Longo, “Model predictive torque
vectoring control for electric vehicles near the limits of handling,”
in 14th European Control Conference, 2015, pp. 2553–2558.
[6] G. Vasiljevic and S. Bogdan, “Model predictive control based torque
vectoring algorithm for electric car with independent drives,” in 24th
Mediterrian Conference Control and Automation, 2016, pp. 316–321.
[7] E. Siampis, E. Velenis, S. Gariuolo, and S. Longo, “A real-time
nonlinear model predictive control strategy for stabilization of an
electric vehicle at the limits of handling,” IEEE Transactions on
Control Systems Technology, pp. 1–13, 2017.
[8] J. Park, H. Jeong, I. G. Jang, and S.-H. Hwang, “Torque distribution
algorithm for an independently driven electric vehicle using a fuzzy
control method,” Energies, pp. 8537–8561, 2015.
[9] S. L. J. Feustel and M. Hand, “The electric super sports car SLS AMG
electric drive,” ATZ worldwide, pp. 4–10, 2013.
[10] S. Antonov, A. Fehn, and A. Kugi, “Unscented Kalman filter for
vehicle state estimation,” Vehicle System Dynamics, pp. 1497–1520,
2011.
[11] H. B. Pacejka, Tyre and Vehicle Dynamics. Butterworth Heinemann,
2006.
[12] E. A. Wan and R. V. D. Merwe, “The unscented Kalman filter for
nonlinear estimation,” in Proceedings of the IEEE 2000 Adaptive Sys-
tems for Signal Processing, Communications, and Control Symposium
(Cat. No.00EX373), 2000, pp. 153–158.
[13] H. Ferreau, C. Kirches, A. Potschka, H. Bock, and M. Diehl,
“qpOASES: A parametric active-set algorithm for quadratic program-
ming,” Mathematical Programming Computation, vol. 6, no. 4, pp.
327–363, 2014.
[14] G. Batista, G. Tak´
acs, and B. Rohal-Ilkiv, “Application aspects of
active-set quadratic programming in real-time embedded model pre-
dictive vibration control,” in 20th World Congress of IFAC, Jun 2017,
pp. —.
[15] B. Houska, H. Ferreau, and M. Diehl, “ACADO Toolkit – An Open
Source Framework for Automatic Control and Dynamic Optimization,”
Optimal Control Applications and Methods, vol. 32, no. 3, pp. 298–
312, 2011.
[16] E. Mikul´
aˇ
s, “Model predictive torque vectoring control for the SGT-
FE17 racing car,” Master’s thesis, Slovak University of Technology in
Bratislava, Bratislava, Slovakia, 2017.
588