The 14th IFToMM World Congress, Taipei, Taiwan, October 25-30, 2015
DOI Number: 10.6567/IFToMM.14TH.WC.OS20.001
A Simple Strategy for Tyre Force Distribution to Improve the Performance of a Four Wheel
Independent Steer-by-wire Vehicle
Mahek Mody, Anirban Guha and P. Seshu
Department of Mechanical Engineering, IIT Bombay, Mumbai 400076, India
e-mail: [email protected]
Abstract: In a four wheel independent steer-by-wire system, all
four wheels of a vehicle are steered independently by separate
actuators which respond to computer generated signals. This
freedom, if judiciously used, can lead to better vehicle
performance. This work proposes a method of deciding the
steering angles of each wheel at different speeds in a four wheel
independent steer-by-wire vehicle. This is based on obtaining the
distribution of lateral forces of inner and outer tyres through a
sigmoid function which depends on the ratio of cornering
stiffness of outer and inner tyres. Simulations show that such a
system allows considerable improvement over four wheel nonindependent steering and two wheel steering systems. A 10-20%
increase in speed is possible during low speed off-limit cornering
and 40% increase during high speed on-limit cornering.
Keywords: Vehicle dynamics, Steer-by-wire, Four wheel
steering, Independent steering, Tyre dynamics, Slip angle control
1 Introduction
For vehicles moving at high speed, both
responsiveness of the vehicle and stability are
predominant considerations for safety. The handling and
stability of a vehicle are directly affected by tyre forces.
Many different strategies have been proposed over the
years to improve these aspects - four wheel steering
control[1-4], direct yaw moment control[5], optimum tyre
force distribution[6] etc. Steer-by-wire technology, which
enables a vehicle to be steered based on input signals from
the electrical motors at the wheels, is used often to
implement some of these strategies[7-9]. As optimum tyre
force distribution strategy could take into account highly
complex tyre models and attempt to maximize the
utilization of tyre capacity and yet avoid locking, skidding
etc, it has been seen to have immense potential especially
for four wheel independent steering/driving systems.
Noting that little work has been done in the area of
optimization of tyre forces, Naraghi et al[10] proposed an
adaptive optimization strategy. Their method involved a 9
DoF, non-linear vehicle dynamics model, two equality and
four inequality constraints. The objective function had two
terms with linear weighting coefficients – first term to
minimize work load of tyres (i.e. sum of squared
normalized resultant tyre force) and the second term
dealing with squared summation of normalized tyre
longitudinal forces (to adjust the contribution of direct
yaw moment control). Ali Farshad et al[11] proposed a
fuzzy optimization approach to find the appropriate
weighting coefficients for front/rear wheels so that
maximum capacity of the tyres can be utilized.
Kim et al [12] employed a two DoF vehicle lateral
dynamic model and proposed a strategy to distribute tyre
forces for an electric vehicle. Yaw rate control was
implemented using PID and the maximum longitudinal
tyre force was limited by the tyre friction. Simulations
using CarSim software showed improved vehicle stability
and saving of electrical energy under high speed cornering
condition. Wang et al [13] posed the optimum tyre force
distribution as a convex optimization problem and
obtained an analytical solution for a vehicle with four
wheel steering and rear wheel drive. They employed a 7
DoF vehicle model and a “magic formula” based tyre
model.
Mokhiamar and Abe[14] considered a complete steer,
brake and drive by wire system where all the four wheels
can be individually steered, driven and braked. They
proposed an optimization strategy to find the distribution
of longitudinal and lateral forces in the tyres. The
controllers simultaneously addressed both vehicle
responsiveness and stability issues. Simulation results
were presented for lane change motion and the effect of
variation of coefficient of friction and steering wheel
angle amplitude were discussed.
Observing that very few studies are available, Suzuki
et al[15] studied longitudinal and lateral tyre force
distribution for a full drive by wire electric vehicle. Their
criteria used included minimization of tyre work load (i.e.
square sum of four tyre forces) and reduction of tyre
energy dissipation due to tyre slip (i.e. square sum of
energy dissipations of four tyres). The methodology was
implemented on an experimental vehicle and the
effectiveness demonstrated for a typical lane change
maneuver. A three layer control system was used by
Goodarzi and Mohammad [16] in their study of a four
wheel drive hybrid electric vehicle. The first layer was
used for yaw moment and lateral force, the second layer
for optimum tyre force distribution and the third layer for
active steering, wheel slip, motor torque etc. Simulation
results showed significant improvements in vehicle
handling and specific fuel consumption.
Thus it is observed that significant attention has been
devoted to the study of tyre force distribution strategies
over the past few years and various optimization strategies
have been proposed. However, ease of implementation of
the proposed strategy would improve if it involved a
simple equation. The present work proposes a sigmoid
function, whose shape depends on the ratio of cornering
stiffness of inner and outer tyres, to find the tyre force
distribution. Simulations show promising results and this
could be amenable to real time implementation.
2 Lateral Dynamics
Consider a simple model of the four wheel independent
steer-by-wire vehicle shown in Figure 1. The lateral
cornering force required is to be provided by the four
wheels. During steady state cornering the vehicle speed
(V), the radius of curvature (R), the vehicle side slip angle
(β) and the vehicle yaw rate (r) are constant. Here the F in
the subscript represents the front tyre and R the rear tyre,
similarly O is the outer tyre and I the inner tyre.
developed to estimate this function based on sensory
inputs and vehicle state estimators.[6] Further analysis
makes the assumption that this function is known and goes
on to describe how the four wheel independent steer-bywire system could go on to exploit this.
3
Determination of Force Requirements of
Individual Wheels
To demonstrate the advantages of the four wheel
independent steering system, the requirements during slow
speed off-limit cornering and high speed on-limit
cornering are considered separately. Off-limit cornering
refers to situation in which the lateral force requirement is
such that it is considerably smaller than maximum force
producing capacity of the tyre (less than 20%). On the
other hand, on-limit cornering is when the lateral force
requirement is closer to the force producing limit (greater
than 80%).
Before proceeding it is important to define the
parameter cornering stiffness (C) which is the slope of the
plot between lateral force and slip angle of the tyre
measured at zero slip angle.
Low speed off-limit cornering
Tyre Slip angle
Lateral Tyre Force
Steering Angle
Mass of the Vehicle
Dist. of Front Axle from CoG
Dist. of Rear Axle from CoG
Lateral Acceleration
Figure 1 Schematic for the mathematical model for four
wheel independent steering
The following equations are obtained by force balance
and moment balance:
(1)
(2)
Solving them for front tyre force and rear tyre force:
(3)
(4)
Therefore, the ratio of the sum of front tyre forces and the
sum of rear tyre forces is fixed by the physical dimensions
of the vehicle. The total force to be produced by the two
front wheels is fixed, as is the case for the two rear wheels
but there is room for improved vehicle handling by
choosing the force that each individual wheel produces.
is the lateral force experienced by tyre . It is a
function of α and the function is represents the lateral tyre
characteristics. The tyre characteristics function depends
on many different variables such as camber angle, road
surface, tyre temperature, tyre pressure, etc. These are not
known easily, however many techniques have been
• During this time the outer loaded wheel has more force
producing capability at the same slip angle than the
inner lightly loaded wheel.
• It is best that the bulk of the force production is done
by the outer tyre, as this would decrease the average
slip angle and help with tyre life.
• The bias towards the outer tyre should be a direct
function of the ratio of the outer and inner wheel
cornering stiffness, as it is the factor that determines
the amount of the extra force the outer tyre can
produce at the same slip angle.
• The inner tyre can have a near zero slip angle during
this time which keeps it well within the linear range
of the tyre curve making other control
implementations easier to apply.
High speed on-limit cornering
• High speed on-limit cornering is the state where the
steady state cornering force requirements are
comparable to the maximum cornering force
producing capacity of the tyres.
• It is essential that both the tyres contribute
equivalently when normalized to their force
producing capacity.
• As far as possible the tyres should be away from their
peak producing powers as these peaks are very
sensitive to small variations in the normal loads and
camber.
• Control over slip angle allows the maximum force
producing capacity of the axle to be the same as the
sum of the maximum forces of both the tyres.
Intermediate Transition
• The intermediate transition should ensure a smooth
transition between the contradicting off-limit
cornering and on-limit cornering conditions.
• Especially at the on-limit cornering end, care should
be taken that the equal contribution is gradually
achieved.
• The outer tyre should not be asked to produce more
cornering force than its maximum force producing
capacity.
Equations 3 and 4 give the front and rear cornering forces
that the front and rear tyres need to produce and let it be
for a given axle. Let the maximum force producing
capacity of the inner and outer tyre be
and
respectively. A parameter µ is defined as:
(5)
µ is a normalized 0 to 1 scale representation of the regime
in which the car is cornering. µ close to zero means the car
is in the off-limit cornering condition and as µ approaches
unity, the on-limit cornering case is achieved.
We define a σ which is a function of μ, a 0 to 1 scale
normalized force contribution of the inner tyre such that
the force to be produced by the inner and the outer tyre is:
Figure 1 The sigmoid function
required as per
considerations (γ=2.3)
The function
estimate.
defined in equation 8 is good empirical
(8)
(6)
(7)
Then, in order to satisfy the above considerations the
function must lie in the domain [0,1] and should
preferably be a sigmoid function, i.e. look like the one
shown in Figure 2. The reason for preferring the sigmoid
shape is as follows. At low lateral force requirements the
outer tyre is made to do the bulk of the work as it can
produce more force with the same slip angle compared to
the inner wheels. At high, close to the limit cornering
conditions this effect is reduced and the contributions are
moved towards being equal such that the tyre remains in
the predictable regime as far as possible. Additionally, it is
beneficial to have low slopes of the µ vs σ curve at both
low and high speed regimes. This is because at a low
speed regime, it is beneficial to let the outer tyre to
contribute to the major portion of the lateral force. This
will allow the inner wheel, which is less loaded, to have a
lower slip angle. The outer tyre is more heavily loaded
and will provide more lateral force for a particular slip
angle. At a high speed regime, as we approach µ=1, we
would prefer the slip angles of inner and outer tyres to be
more evenly distributed over a larger range of µ where the
tyres are close to their limiting lateral forces. This helps in
avoiding overshooting of the lateral forces of the tyres
beyond their limits during the transient phase before
steady state is reached. These obviously require the µ vs σ
curve to be of a sigmoid nature. Moreover, it is preferable
to have the nature of the curve dependent on the ratio
CO
CI
(9)
where
C indicates the cornering stiffness of the tyre
represented by the subscript. The cornering stiffness is the
slope of the lateral tyre characteristic curve as the lateral
force produced tends to zero.
Equation (9) captures the fact that if there is a large
difference in the cornering stiffness of the inner and out
tire at the different loads, then the aggressiveness of the σ
function changes accordingly. This curve ensures that at a
higher cornering stiffness ratio
CO
CI
, the outer tyre is
subjected to higher lateral forces in the low speed regime.
4 Steer Angle Calculations
Since the force requirement is known, the different slip
angles of the four wheels (α) are also known from the
lateral tyre characteristics defined in the Adams tyre
model. To calculate the steady state steering angles to be
set as input the kinematic component of the steering must
be considered. The kinematic component is the steering
angles required for the vehicle to be steered through a
corner of a given radius with velocity tending to zero. This
represents the no slip condition, where the lateral
acceleration is zero and hence slip angles are not required
to induce any cornering forces. These angles are
geometrically determined with the help of figure 3.
.
Figure 2 Kinematic steering angle requirement for no-slip
cornering
To satisfy the no slip condition, the steering angles of the
individual wheels must be such that the perpendiculars
drawn from the direction of the heading of each wheel
must meet at the center of curvature. Therefore using
simple trigonometric relations the following angles
(equations 10 - 13) can be obtained:
(10)
(11)
(12)
(13)
For any other given velocity the tyres will have to produce
lateral cornering forces to counter the centrifugal force for
which slip angles (α) are necessary in addition to the
kinematic steering requirement.
As long as the additional slip angles are applied in the
manner that the corresponding force produced by the
vehicle satisfies the steady state lateral dynamics
conditions, the vehicle will attain a steady state with the
desired vehicle side slip angle ( ). It is desirable to have
the vehicle side slip angle as close to zero as possible and
hence beyond this point it is assumed that in steady state a
requirement of
is imposed. Mathematically, the
steer angles should be (equations 14 - 17):
(14)
(15)
pinion based model from which the steering angle and in
turn, the vehicle turning radius (R) can be determined.
This, along with the vehicle velocity allows the required
lateral forces at the front and rear pair of wheels to be
computed from equations 3 and 4. These, along with tyre
characteristics lead to µ (as defined in equation 5). The
sigmoid curve (equation 8) now allows σ to be determined
and thus, the force contributions of the inner and outer
tyres can be obtained (equations 6 and 7). This, when fed
to the Adams tyre model, gives the slip angle for each of
the tyres (α). With both R and α being known, equations
14 to 17 are now used to obtain the steer angles of each of
the wheels. These are now fed to the vehicle model of
Adams to obtain the vehicle’s position, velocity and
acceleration. These are again fed to the Adams driver
model which attempts to keep the vehicle on the
prescribed path at the required velocity with a PID
controller. This generates a new rack position for the next
time step and all the previous steps are now repeated.
In the simulation, the vehicle runs for the first 3
seconds in a straight line to get into steady state and up to
speed. The step input is then applied in a linear manner
over 1 second. The vehicle input in ADAMS is given in
the form of a steering wheel turning angle. A linear yaw
map is used. So, the more the steering wheel turns, the
lesser is the target cornering radius. This cornering radius
is independent of the vehicle speed and is achieved by the
control system applying the steering angles as input. The
simulation is run for 25 seconds after the step input is
complete giving substantial time for the vehicle to attain
steady state behavior. The turning radii range from 10
meters to 100 meters. Vehicle speeds selected are starting
from 20 km/hr to the highest attainable speed, in steps of
20 km/hr. Intermediate speeds are only used in simulation
to determine the maximum cornering speed to the nearest
10 km/hr. The steering input given is specific to that speed
such that the vehicle would hold the required turning
radius in the steady state. If the vehicle turning radius isn’t
within 5% of the desired radius after the 25 sec, the
vehicle is said to have exceeded its maximum cornering
speed and is considered to have become unstable.
In order to have a comparison of the four wheel
independent steer-by-wire system, a four wheel nonindependent steering system[17] and a conventional rack
and pinion front wheel steering vehicle is also simulated
for the same turning radius and vehicle speed. The
vehicles are similar in all other aspects.
(16)
(17)
5 Simulations
For the purpose of simulation, ADAMS/CAR is used
with the control system designed and imported from
MATLAB. The standard templates of the suspension and
steering systems are modified to allow for the four-wheel
independent steering system. ADAMS/Controls package
is used to implement the steer-by-wire aspect.
At the start of a simulation, the path information and
vehicle velocity are provided to the driver model in
Adams. This model gives a rack position of a rack and
6 Results
Figure 4 plots the maximum cornering speeds the different
vehicles can manage at different turning radii.
As can be seen in figure 4, the four wheel independent
steer-by-wire system achieves the highest possible
cornering speed at all the different turning radii. Also, the
four wheel non-independent steering consistently
outperforms the conventional front wheel steering system.
The amount of extra cornering speed that the 4 wheel
steering systems provide at high speeds is quite significant
as there are speed gains of about 50-70 km/hr (40%).
Figure 3 Maximum cornering speed obtained from
simulation at different turning radii
It can be clearly spotted in the plot in figure 4 that in
case of the front wheel steering system, Ackermann is
optimized to work well in the 40 m turning radius regime,
similarly the four wheel non-independent steering
Ackermann is optimized for the 50 m turning radius
regime. However, the four wheel independent steer-bywire systems provide more consistency when it comes to
the estimation of maximum cornering speeds at different
radius. This is important because in the real world the
turning radius and speeds are constantly changing in a
normal drive and sudden changes in steering input or
vehicle speeds can cause instability when driving near
limit conditions with conventional steering.
systems. This means that the tyre life is significantly
improved as average slip angles are lower and also rolling
resistance is correspondingly reduced. While the front
wheel steering system and the four wheel dependent
steering system are at 81% and 83% tyre utilization
respectively at their maximum cornering speeds, the four
wheel independent steering can reach much higher values
of greater than 95%. This is the main reason why the four
wheel independent steering system is capable of
significantly higher cornering speeds as compared to the
other two steering systems. The lower percentage
utilization of the four wheel independently steered vehicle
till 170 km/hr indicates that it is likely to be more stable
and safe compared to the other two.
The slip angle distribution is best when the standard
deviation of the individual steady state slip angles of the
four wheels, normalized by their peak load, is minimum.
The figure 6 plots this parameter for different speeds at a
100 m turning radius. It shows that the 4 wheel
independent steer-by-wire system is almost always the
best. More importantly, the four wheel independent steerby-wire system does extremely well at high speeds,
bringing the slip angles close to each other thereby
boosting performance in this regime. It is also interesting
to know that it is also the only steering system that does
not cause an increase in the normalized standard deviation
of slip angle as the speed increases. This means that at the
maximum vehicle speed all the four tyres are at the limit
of traction. The front wheel (i.e. two wheel) steering
system shown here is probably optimized in the 50-100
km/hr region and hence in this range it is able to match up
to the four wheel steering system. But this is at the
expense of performance at other speeds.
Figure 4: Percentage utilization of tyres for different speed
for 100m turning radius
The performance advantage of the four wheel
independent steer-by-wire system is clearly brought out by
the percentage utilization of tyres graph in figure 5. The
utilization of the tyre refers to the ratio of the lateral force
produced by the tyre and its maximum force producing
capacity. The percentage utilization of the tyre is
indicative of the harshness the tyre faces. This is mainly
attributed to the fact that lower utilization results in lower
slip angles which means there is lesser slip within the tyre
which is the main source of graining and permanent
rubber damage. Higher slip angles also contribute to
rolling friction.
At a lower speed the percentage utilization of the tyres
is significantly low as compared to the other steering
Figure 6 Standard deviation of normalized slip angle at
different speeds
The settling time gives an indication of how fast the
steady state conditions are achieved. Figure 7 shows that
the four wheel steering systems have a significant
advantage over the front wheel steering system. The four
wheel independent steering system is only marginally
quicker than the four wheel dependent steering system but
again the difference increases with the operating speed of
the system.
Figure 7 Time taken by vehicle to arrive at steady state
condition for different speeds
7 Conclusions
Four wheel independent steer-by-wire systems have a
distinct advantage over the other steering systems as there
is control available over the individual slip angles of the
vehicle in steady state conditions. This paper presents a
method of handling vehicle performance in both high
speed and low speed regimes. Simulations show a
significant increase in the vehicle’s maximum cornering
speed at all turning radii. The potential cornering speeds
increase is of 10-20% during off-limit cornering and as
high as 40% during on limit cornering. The percentage
utilization of the tyres can attain values of up to 95% in
simulation as compared to a maximum of 83% in case of
conventional steering systems.
The primary contribution of this paper lies in
proposing a very simple strategy for obtaining the lateral
force at each tyre, and thereon the steer angle of each
wheel. The simple equation is expected to allow easier
hardware implementation compared to other elaborate
strategies (involving optimization) reported in literature.
This will be attempted in future and its effect on other
structural members of the vehicle (e.g. the axle) will be
explored.
References
1. Bernard, J. E., Vanderploeg, M. J. and Shannan, J. E.,
Linear Analysis of a Vehicle with Four-Wheel
Steering.
SAE
Technical
Paper
880643,
doi:10.4271/880643, 1988.
2. Whitehead,
J.
C.
Four
Wheel
Steering:
Maneuverability and High Speed Stabilization, SAE
Technical Paper 880642, doi:10.4271/880642, 1988.
3. Sano, S., Furukawa, Y. and Shiraishi, S., Four Wheel
Steering System with Rear Wheel Steer Angle
Controlled as a Fuction of Steering Wheel Angle. SAE
Technical Paper 860625, doi:10.4271/860625, 1986.
4. Anwar, S. and Zheng, B. Yaw stability control of a
steer-by-wire equipped vehicle via active front wheel
steering, Mechatronics, 19(6), pp. 799-804, 2009.
5. Shibahata, Y., Shimada, K. and Tomari, T.,
Improvement of vehicle maneuverability by direct yaw
moment control, Vehicle System Dynamics, v22, pp.
456-481, 1993.
6. Mokhiamar, O. and Abe, M., How the four wheels
should share force in an optimum cooperative chassis
control, Control Engineering Practice, v14, pp. 295304, 2006.
7. Ackermann, J. Robust Yaw Damping of Cars with
Front and Rear Wheel Steering, IEEE Transactions on
Control Systems Technology, 1(1), pp. 15-20, 1993.
8. Hsu, Y.-H.J. and Gerdes, J.C., Stabilization of a Steerby-wire Vehicle at the Limits of Handling Using
Feedback Linearization, American Society of
Mechanical Engineers, Dynamic Systems and Control
Division (Publication) DSC., 74 DSC (1 PART A), pp.
483-492., 2005.
9. Yamamoto, M., Active Strategy for Improved
handling and Stability. SAE Technical Paper 911902,
doi:10.4271/911902, 1991.
10. Naraghi, M., Roshanbin, A and Tavasoli, A., Vehicle
stability enhancement – an adaptive optimal approach
to the distribution of tyre forces, Proc. of IMechE Part
D: Journal of Automobile Engineering, v224, 2010,
pp. 443-453.
11. Ali Farshad, Naraghi, M. and Samadi, B., Controller
design for vehicle stability improvement using optimal
distribution of tire forces, Proc. of 2nd International
Conference on Control, Instrumentation and
Automation, 2011, pp. 174-179.
12. Kim, S.H., Kim, D.H., Kim, C.J., Ali, M.A., Back,
S.H. and Han, C.S., A study on the improvement of the
tire force distribution method for rear wheel drive
electric vehicle with in-wheel motor, 2012 IEEE/SICE
International Symposium on System Integration,
Fukuoka, Japan, Dec. 16-18, 2012, pp. 390-395.
13. Wang, C.C., Cheng, S.Y., Hsiao, T., and Chou, W.Y.,
Application of optimum tire force distribution to
vehicle
motion
control,
2012
IEEE/ASME
International Conference on Advanced Intelligent
Mechatronics, July 11-14, 2012, Kaohsiung, Taiwan,
pp. 538-543.
14. Mokhiamar, O. and Abe, M., Simultaneous optimal
distribution of lateral and longitudinal tire forces for
the model following control, ASME Journal of
Dynamic Systems, Measurement and Control, 2004,
v126, pp. 753-763.
15. Suzuki, Y., Kano, Y. and Abe, M., A study on tyre
force distribution controls for full drive-by-wire
electric vehicle, Vehicle System Dynamics, 2014, v52,
pp. 235-250.
16. Goodarzi, A. and Mohammadi, M., Stability
enhancement and fuel economy of the 4-wheel-drive
hybrid electric vehicles by optimal tyre force
distribution, Vehicle System Dynamics, 2014, v52, n4,
pp. 539-561.
17. Kreutz M., Horn M. and Zehetner J., Improving
Vehicle Dynamics by Active Rearwheel Steering
Systems, Vehicle System Dynamics, 47(12),
doi:10.1080/00423110802691507, pp. 1551-1564,
2009.