Rear Wheel Torque Vectoring Model Predictive Control with Velocity
Regulation for Electric Vehicles
Efstathios Siampis
∗
, Efstathios Velenis and Stefano Longo
Centre for Automotive Engineering, School of Aerospace, Transport and Manufacturing,
Cranfield University, Cranfield, UK
(Received 00 Month 20XX; accepted 00 Month 20XX)
In this paper we propose a constrained optimal control architecture for combined velocity,
yaw and sideslip regulation for stabilization of the vehicle near the limit of lateral acceleration
using the rear axle electric torqu e vectoring configuration of an electric vehicle. A nonlinear
vehicle and tyre model are used to find reference steady-state cornering conditions and design
two Model Predictive Control (MPC) strategies of different levels of fidelity: one that uses
a linearized version of the full vehicle model with the rear wheels’ torques as the input, and
another one that neglects the wheel dynamics and uses t he rear wheels’ slips as the input
instead. After analysing the relative trade-offs between performance and computational effort,
we compare the two MPC strategies against each other an d against an unconstrained optimal
control strategy in Simulink an d Carsim environment.
Keywords: torque vectoring; mo del predictive control; combined velocity, yaw and sideslip
control; limit-handling
Notation
α Tyre slip angle
β Vehicle sideslip an gle at its centre of mass
δ Steering angle
ǫ Slack variable
µ
x
, µ
y
Longitudinal and lateral tyre force coefficient
µ
max
Tyre/road friction coefficient
ψ Vehicle yaw angle at its centre of mass
ω Wheel angular rate
a
x
, a
y
Vehicle longitudinal and lateral acceleration at its centre of mass
f
x
, f
y
, f
z
Longitudinal, lateral and normal tyre force
g Constant of gravitational acceleration
i, j Subscripts i = F, R (front, rear), j = L, R (left, right)
ℓ
F
, ℓ
R
Longitudinal distance of centre of mass from the front and the rear track
m Mass of the vehicle
r Wheel radius
s, s
x
, s
y
Total, longitudinal and lateral slip
∗
Corresponding author. Email: e.siampis@cranfield.ac.uk
1
Vehicle System Dynamics: International Journal of Vehicle Mechanics and Mobility,
Volume 53, Issue 11, 2015, pp1555-1579
DOI: 10.1080/00423114.2015.1064972
Published by Taylor and Francis. This is the Author Accepted Manuscript issued with:
Creative Commons Attribution Non-Commercial License (CC:BY:NC 3.0).
w
L
, w
R
Lateral distance of centre of mass from the left and right wheels
B Pacejka’s Magic Formula stiffness factor
C Pacejka’s Magic Formula shape factor
D Pacejka’s Magic Formula peak value
I
w
Wheel moment of inertia of each wheel about its axis of rotation
I
z
Vehicle moment of inertia
N Horizon
N
p
Prediction horizon
N
u
Control horizon
R Vehicle path radius
T Torque
T
s
Sampling time
T
sim
Simulation time
V Vehicle velocity at its centre of mass
1. Introduction
In the p ast few years it has been recognised that active control of the vehicle’s velocity
is not only a very effective strategy in the limits of lateral acceleration but also crucial
in cases of terminal understeer behaviour [22]. The necessity for velocity regulation in
such cases is already mentioned by van Zanten et al. [24] who points out that especially
in the case of J-turns, w here the turning radius is reduced continuously along the trajec-
tory (a scenario typical on highway exits), the Electronic Stability Control (ES C) yaw
moment correction on the lateral dynamics alone is not sufficient. This early remark on
the importance of longitudinal control was later realised as one of the ESC new functions
in [17], wher e correction of terminal understeer behaviour is achieved by superimposing
individual braking of all four wheels on the standard ESC intervention. Similar solutions
applied on a Four Wh eel Drive (4WD) Electic Vehicle (EV) can be found in [15, 16]. In
[15] the torque request from the driver is reduced when the lateral acceleration exceeds
a specific threshold, while in [16] a velocity limit is set as a function of the desired yaw
rate. I n [12] a controller providing decoupled longitudinal force and yaw moment inputs
at the higher level is combined with a static control allocation scheme to calculate forces
and actuator inputs. In [22] a multivariable control architecture to ad dress velocity, yaw
and sideslip regulation in terminal understeer is presented. Simulation results using a
driver model in a U-turn scenario show that the controller not only keeps the vehicle
within the road boundaries, but also allows for a smoother negotiation of the corner with
less steering effort from the dr iver.
In this paper we propose a constrained optimal control architecture that s tabilizes
the vehicle near the limits of lateral acceleration while accounting for the important in
such cases system constraints. MPC, a control strategy tracing its origins in the chemi-
cal p rocesses industry [19], has been increasingly popular in the industry and academia
due to its ability to naturally handle multivariable system constraints. Looking in the
automotive active system applications, a variety of MPC solutions can be found. For
example, in [4] we find a Linear Time Vary ing MPC (LTV-MPC) strategy for control-
ling the lateral dynamics of the vehicle u sing independent b raking of the four wheels.
Simulation results show that the LTV-MPC controller successfully completes the sine
and dwell test but with a consid er ab le decrease in speed due to the braking strategy
2
used. An example of an MPC application for active lateral dynamics control utilizing
the steer-by-wire system of a Rear Wheel Drive p rototype vehicle can be found in [5].
For the MPC formulation velocity dependent bounds are imposed on the yaw rate an d
sideslip angle in a way similar to the envelope control concept from the aerospace in-
dustry. Simulation and experimental results using a slalom manoeuvre show that the
controller restricts the steering command from the driver when the envelope bounds are
violated. In [7] two explicit MPC formulations for a yaw s tability controller using Active
Front Steering (AFS) an d differential wheel braking are presented. Experimental results
using the less computationally expensive switched MPC strategy show that the controller
can successfully stabilize the vehicle in a fast double lane change on a slippery road by
constraining the tyre slip angles within their limits. Another example of an explicit MPC
law can be found in [6], where a yaw control strategy using a rear active d ifferential is
presented. Here the Nonlinear MPC (NMPC) problem is solved offline using the near-
est point ap proach. Simulation results show a good agreement between the p roposed
approach and a nominal NMPC controller but with some chattering, which could be
potentially corr ected with higher number of offline computed points but at the expense
of higher memory and computational requirements. A different app roach can be found in
[10], where an MPC strategy for roadway departure prevention using AFS and braking
utilizes futu re road information and a driver model to ensure that the current vehicle
state belongs to the set of states th at will evolve to a desired final set [8]. For the MPC
formulation the cost function penalizes only the control effort, while the road boundaries
are set as constraints. Simulation tests show that the controller can successfully keep the
vehicle within the lane bou ndaries in the case of overspeeding through a curve.
In this work we propose an MPC strategy for combined velocity, yaw and sideslip
regulation for s tabilization of the vehicle near the limits of lateral acceleration using the
rear axle electric torque vectoring configuration of an electric veh icle. While using an
MPC strategy has obvious advantages for the specific application at hand, it also has
distinct disadvantages, which are mainly connected to the computational time needed to
construct and solve the resulting optimization problem. Based on this observation, the
goal of this paper is not only to develop an appropriate MPC strategy for the demanding
task of stabilizing the vehicle near the limits of handling in the best possible way, but also
one that can be implemented in real time. The structure of the paper is as follows: after
introducing the nonlinear vehicle and tyre models along with the steady-state cornering
analysis used to generate the reference for th e controller to follow, the basic MPC prob lem
is explained. Since the main disadvantage of using an MPC strategy is the computational
effort attached to it, two MPC strategies of different complexity are then constructed:
1) one using the full vehicle mod el and 2) a simpler one that neglects the wheel speed
dynamics. The effect of varying the sampling time and the horizon on the performan ce
and the computational load of each strategy are then analysed. Finally, the two strategies
are compared against each other in Simulink environment and against a Linear Quadratic
Regulator (LQR) strategy [22] in CarSim environment.
2. Vehicle Model and Reference Generation
In this section we introduce th e vehicle and tyre models u sed in this paper. The formu-
lation is similar to th e one found in [22, 26], where the interested reader can refer to for
more details.
3
2.1. Vehicle Model
f
RLy
f
RLx
f
F Ly
f
F Lx
δ
δ
f
F Rx
f
F Ry
V
β
w
L
w
R
f
RRy
f
RRx
CM
ℓ
R
ℓ
F
˙
ψ
Figure 1. The four-wheel vehicle model.
The Equations Of Motion (EOM) for the four-wheel vehicle model with front wheel
steering (Fig. 1) are
m
˙
V = (f
F Lx
+ f
F Rx
) cos(δ − β) − (f
F Ly
+ f
F Ry
) sin(δ − β) (1a)
+ (f
RLx
+ f
RRx
) cos β + (f
RLy
+ f
RRy
) sin β,
˙
β =
1
mV
[(f
F Lx
+ f
F Rx
) sin(δ − β) + (f
F Ly
+ f
F Ry
) cos(δ − β) (1b)
− (f
RLx
+ f
RRx
) sin β + (f
RLy
+ f
RRy
) cos β] −
˙
ψ,
I
z
¨
ψ = ℓ
F
[(f
F Ly
+ f
F Ry
) cos δ + (f
F Lx
+ f
F Rx
) sin δ] − ℓ
R
(f
RLy
+ f
RRy
) (1c)
+ w
L
(f
F Ly
sin δ − f
F Lx
cos δ − f
RLx
) + w
R
(f
F Rx
cos δ − f
F Ry
sin δ + f
RRx
)
I
w
˙ω
ij
= T
ij
− f
ijx
r, i = F, R, j = L, R. (1d)
where the r elevant variables and parameters are as defined under the Notation section
at the beginning of the paper.
The tyre forces f
ijx
and f
ijy
in the above EOM are found as functions of the tyre slip
using Pacejka’s Magic Formula (MF) [3]. In particular, we obtain the resultant tyre force
coefficient as a function of the resultant slip at each tyre fr om the MF:
µ
ij
(s
ij
) = MF(s
ij
) = D sin(Catan(Bs
ij
)),
where s
ij
=
q
s
2
ijx
+ s
2
ijy
is the resultant tyre slip with s
ijx
and s
ijy
the theoretical
longitudinal and lateral slip quantities respectively [3], and D = µ
max
is the tyre/road
friction coefficient. Then using the friction circle equations
µ
ijx
= −
s
ijx
s
ij
µ
ij
(s
ij
), µ
ijy
= −
s
ijy
s
ij
µ
ij
(s
ij
),
4
we obtain the tyre force coefficients µ
ijx
and µ
ijy
in the longitudinal and lateral direction.
Neglecting th e pitch and roll rotation along with the vertical motion of the sprung
mass of the vehicle, the vertical force f
ijz
on each of the four wheels can be calculated
using the static load distribution and the longitud inal/lateral weight transfers under
longitudinal/lateral acceleration [26]. The longitudinal and lateral tyre forces are then
given by
f
ijx
= µ
ijx
f
ijz
, f
ijy
= µ
ijy
f
ijz
.
The values for the above vehicle an d tyre parameters used in this paper correspond to
a compact family car and can be found in Table 1.
Table 1. Vehicle and tyre parameters.
Parameter Value Parameter Value
m (kg) 1420 ℓ
F
(m) 1.01
I
z
(kgm
2
) 1027.8 ℓ
R
(m) 1.452
I
w
(kgm
2
) 0.6 r (m) 0.3
w
L
(m) 0.81 B 24
w
R
(m) 0.81 C 1.5
h (m) 0.55 D 0.9
2.2. Reference Generation
In order to derive feasible targets for the controller to follow, steady-state cornering
analysis of the four-wheel vehicle model (1) is used. Neglecting the wheel s peed dynamics
and enforcing the steady-state cornering conditions
˙
V = 0,
˙
β = 0,
¨
ψ = 0,
we obtain three algebraic equations with six unknowns, namely the equilibriu m state
(V
ss
, β
ss
, R
ss
= V
ss
/
˙
ψ
ss
) and input (δ
ss
, s
ss
RLx
, s
ss
RRx
). Providing three of the six unknowns
(in this work the triplet (V
ss
, R
ss
, δ
ss
)), we can then solve the steady-state equations using
the fsolve function in MATLAB (a nonlinear equation solver) to obtain the rest of the
unknowns.
Based on the above steady-state analysis, we next examine the feasibility of the vehicle
path r ad ius as requested by the driver. Similar to common practice in vehicle stability
control [21] we obtain an estimate of the driver’s intended path using a neutral steer
linear bicycle model under steady-state cornering. We ther efore set
R
kin
= (ℓ
F
+ ℓ
R
)/tan(δ
ss
).
The desired path radiu s may or may not be feasible depending on the vehicle’s velocity.
Consider for example the steady-state conditions for a fixed δ
ss
and a range of V
ss
in Fig. 2. In all three cases the desired R
ss
= R
kin
is around 14m, according to th e
steering command of δ
ss
= 10deg. Then, for a vehicle velocity of V
ss
= 10.75m/s (green
curve) the vertical red dashed line corresponding to R
kin
intersects the curve of the
calculated steady-state conditions, hence the requested R
kin
is feasible. On the other
hand, if the vehicle velocity is V
ss
= 11.25m/s (purple curve) the R
kin
is smaller than
the minimum achievable R
ss
and not longer feasible. In this case the controller will select
a steady-state condition such that the desired R
kin
becomes feasible by reducing the
5
