International journal of automotive technology, tập 11, số 2, 2010

Copyright © 2010 KSAE
1229−9138/2010/051−01

International Journal of Automotive Technology, Vol. 11, No. 2, pp. 139−146 (2010)

DOI 10.1007/s12239−010−0019−z

COMPARISON OF TWO INJECTION SYSTEMS IN AN HSDI DIESEL
ENGINE USING SPLIT INJECTION AND DIFFERENT INJECTOR NOZZLES
J. BENAJES , S. MOLINA , R. NOVELLA , R. AMORIM ,
H. BEN HADJ HAMOUDA and J. P. HARDY
1)

1)*

1)

2)

1)

2)

Universidad Politécnica de Valencia, CMT-Motores Térmicos, Valencia 46022, Spain
Renault S.A.S., Rueil Malmaison 92508, France

1)

2)

(Received 28 January 2008; Revised 8 July 2009)

ABSTRACT−The demand for reduced pollutant emissions has motivated various technological advances in passenger car

diesel engines. This paper presents a study comparing two fuel injection systems and analyzing their combustion noise and
pollutant emissions. The abilities of different injection strategies to meet strict regulations were evaluated. The difficult task
of maintaining a constant specific fuel consumption while trying to reduce pollutant emissions was the aim of this study. The
engine being tested was a 0.287-liter single-cylinder engine equipped with a common-rail injection system. A solenoid and a
piezoelectric injector were tested in the engine. The engine was operated under low load conditions using two injection events,
high EGR rates, no swirl, three injection pressures and eight different dwell times. Four injector nozzles with approximately
the same fuel injection rate were tested using the solenoid injection system (10 and 12 orifice configuration) and piezoelectric
system (6 and 12 orifice design). The injection system had a significant influence on pollutant emissions and combustion
noise. The piezoelectric injector presented the best characteristics for future studies since it allows for shorter injection
durations and greater precision, which means smaller fuel mass deliveries with faster responses.

KEY WORDS : Diesel engines, Injection system, Nozzle, Pollutant emissions

NOMENCLATURE

and particulate matter, from petrol and diesel engines. Air
quality, especially in large urban areas, has been impacted
by engine exhaust gases and particulate matter. To reduce
this impact, a stricter emission regulation, Euro 5, began in
2009. Since 1996, when Euro 2 started to tighten emission
limits, new technologies have been developed to meet the
challenge of complying with low emission limits. In the
future, regulations are predicted to become stricter for engine
manufacturers. In the case of Euro 5, there is already a
noticeable reduction in NO and particulate matter compared to Euro 4. It is expected that diesel particulate filters
will be mandatory for all diesel cars by 2011. Euro 6,

which will probably start being enforced in 2014, will
significantly lower NO emission limits from the current
0.180 g/km (Euro 5) to 0.080 g/km. This will force car
manufacturers to invest additional resources into research,
and thus increase the final price of vehicles (EurActive,
2004; The European Commission, 2006a, 2006b).
Part of the Diesel engine’s great advances in performance
and control of pollutant emissions over the last decade can
be attributed to improvements in the injection systems. The
introduction of the high pressure common-rail injection
system has allowed for better control of the combustion
process through flexible, more accurate control of the
injection parameters. This system allows the number of
injection pulses, the time interval between them, the injection duration and the injection pressure (IP) to be precisely

ATDC : after top dead center
BGT : burned gas temperature
BSFC : brake specific fuel consumption
BTDC : before top dead center
CAD : crankshaft angle degrees
CO : carbon monoxide
CO : carbon dioxide
EGR : exhaust gas recirculation
EOI : end of injection
FSN : filter smoke number
HC : hydrocarbons
HRL : heat release law
IMEP : indicated mean effective pressure
MD : mass distribution
NO : nitrogen oxides

RoHR : rate of heat release
SFC : specific fuel consumption
SOI : start of injection
2

X

X

X

1. INTRODUCTION
Many health and environmental problems have been attributed to pollutant emissions, mainly NO (Nitrogen Oxides)
X

*

Corresponding author.

e-mail:
139


140

J. BENAJES

controlled. In newer injection systems, it is also possible to
control the injection rate shape (Robert Bosch GmbH,
2004). Despite the fact that this system increases control of

the fuel injection process, it makes finding the optimum
operating conditions more difficult (Desantes
, 2007).
Due to the need to reduce emissions, extensive research
has focused on in-cylinder control of pollutant formation. It
is well-known that reducing NOX, smoke and HC (Hydrocarbons) emissions at the same time is a very difficult task.
Some strategies focus on split injections as a way to control
emissions. Nehmer and Reitz studied the effects of rateshape and split injection on diesel engine performance and
emissions. They observed that the amount of fuel in the
first injection affected the engine-out emissions and incylinder pressure rise rate, which are directly related to
combustion noise. Higher NOX emissions and lower smoke
production were seen when more fuel was injected in the
pilot injection (Nehmer and Reitz, 1994).
Tow
investigated the effects of multiple injections
on combustion in heavy-duty Diesel engine operation at
medium and low load conditions. Multiple injection strategies reduced NOX emissions, and the dwell time between
injection events was shown to heavily influence combustion process control (Tow
, 1994).
Pierpont, Montgomery and Reitz tested multiple injection strategies involving EGR (Exhaust Gas Recirculation)
in order to reduce NOX emissions without significant penalties on smoke and BSFC (Brake Specific Fuel Consumption). They observed that multiple injections could effectively reduce particulate matter, NOX and combustion noise.
They pointed out that the undesirable EGR collateral effects
of increased particulate emissions might be compensated
for by the use of multiple injections (Pierpont
, 1995).
Montgomery
compared the behaviors of different
nozzles in relation to the flow exit area and number of
orifices and highlighted their influence on combustion.
According to their study, nozzles with shorter spray penetrations produced more particulate matter and lower NOX

(Montgomery
, 1996). Benajes
also investigated
the influence of nozzle orifice number and the use of swirl
in a retarded split injection on gaseous emissions and
combustion noise. They remarked that the low temperature
combustion obtained with a late injection is able to provide
ultra-low NOX emissions and reasonable combustion noise
at medium load conditions. Their results also showed that a
high orifice number is prone to causing very high smoke
emissions due to an undesirable interaction among the fuel
spray jets, which could be intensified by swirl (Benajes
, 2006).
The use of multiple injections in a small diesel engine
was also discussed by Hotta
They investigated how
an early pilot, close pilot and post-injection could affect the
combustion process and pollutant emissions. In their work,
they observed that large early-pilot injections could increase
HC emissions due to a cylinder wall impingement. It was
also found that the use of post-injection helped reduce
et al.

et al.

et al.

et al.

et al.

et al.

et al.

et

al.

et al.

et al.

smoke emissions. However, this phenomenon had previously been noticed (Hotta
, 2005). Desantes et al.
studied the usage of post-injections. They concluded that
post-injections were capable of reducing smoke emissions
considerably with no penalty on NOX emissions. Their study
was focused on post-injection and the phenomenon of soot
oxidation. The results revealed that the post-injection reduced soot. However, it was observed that post-injections did
not interact with the main injection. Consequently, soot
would not be reduced by enhanced soot oxidation caused
by the post-injection. Furthermore, engine-out soot would
be the sum of the soot resulting from the combustion of the
main and pilot injections separately. Thus, the final level of
soot decreased because the main pulse produced less soot
and the post-injection did not produce significant additional
soot (Desantes
, 2007). Finally, Benajes
carried

out an investigation using a small single-cylinder engine
based on a statistical procedure called “Consecutive Screenings”, which showed significant improvements in pollutant
emissions by substantially increasing the EGR rate, retarding the injection event and using variable dwell time (Benajes
, 2007).
The objective of this work was to compare the pollutant
emissions using a piezoelectric injection system and a
solenoid injection system in a light duty engine with a low
compression ratio of 14:1 at constant SFC (Specific Fuel
Consumption), using split injections and running in a low
load engine mode. The influence of dwell times and split
injection mass distribution were also studied, in order to
evaluate the possibilities of each injector in each case.
Usually, in other studies, the influences of pilot- and postinjections on pollutant emissions are investigated, taking
into consideration that the use of split injections can affect
combustion efficiency and IMEP (Indicated Mean Effective Pressure). In this work, the analysis the analysis was
performed with the amount of injected fuel and IMEP held
constant.
et al.

et al.

et al.

et al.

2. EXPERIMENTAL FACILITY AND
EQUIPMENT
The engine used in this work was a single-cylinder research
engine with a displacement volume of 0.287 liters, four valves
and low compression ratio, equipped with a common rail

injection system. This engine corresponds to a 1.2-liter, 4cylinder engine.
The engine was installed in a fully instrumented test cell
with all of the required facilities for the operations and
control of the engine. The required boost pressure (BP) was
provided by a screw compressor, and the intake air was
heated to 40oC. Exhaust gas recirculation (EGR) was kept
constant at 120oC. NOX , CO (Carbon monoxide), HC, CO2
(Carbon Dioxide) and O2 (Oxygen) measurements were
performed with a HORIBA 7100D gas analyzer. Smoke
emissions were measured with an AVL 415 variable sampling smoke meter, which provided results directly in FSN


COMPARISON OF TWO INJECTION SYSTEMS IN AN HSDI DIESEL ENGINE USING SPLIT INJECTION

Figure 1. Engine experimental laboratory set-up.
(filter smoke number). In-cylinder pressure was measured
with a piezoelectric transducer, and additional information,
such as IMEP and combustion noise, could be evaluated
during the tests using this data. The experimental set-up is
presented in Figure 1.
Combustion diagnosis software was used to calculate the
heat release (HRL), rate of heat release (RoHR), burned
gas temperature (BGT) and other valuable information. Data
recorded from 50 consecutive engine cycles with a resolution of 0.2 crank angle degrees (CAD) was used for this
calculation. The model is based on the solution of the energy
conservation equation in the cylinder, with the assumption
of uniform pressure and temperature over the instantaneous
volume. This single-zone model enables the calculation of
the instantaneous average temperature in the burned gas, as
well as the heat released during the combustion. (Lapuerta

, 1999) and (Desantes
, 2004).
et al.

et al.

3. METHODOLOGY

The tests were carried out in two stages for each fuel injection system used in this work. The first stage was to define
the engine’s operational conditions. The second stage consisted of the tests comparing dwell time and mass distribution.
3.1. Engine Operational Condition for the Preliminary Tests
In the preliminary tests, a 12-orifice nozzle was used for
the solenoid system and a 6-orifice nozzle was used for the
piezoelectric system. Both nozzles had a conical orifice
shape and similar hydraulic mass flow. The engine was
operated at 1500 rpm and 4.0 bar IMEP, and the dwell time
was set to be 1.0 ms. In this study, the dwell time is
considered to be the time interval between the end of the
first injection event and the beginning of second injection
event.
Ranges for injection pressure and EGR rate were defined
with the aim of performing a parametric study, and final
values are shown in Table 1. The mass distribution (MD) of

141

Table 1. Engine operating conditions of the preliminary
tests and pollutant emission targets.
Engine operating conditions
Solenoid

Piezoelectric
Engine speed
1500 rpm
1500 rpm
IMEP
4.0 bar
4.0 bar
SOI 1
f(SFC, IMEP)
f(SFC, IMEP)
Dwell time
1.0
0.6
Injection pressure 600~1200 bar
900~1300 bar
EGR rate
40%~50%
40%
Nozzle
12 holes
6 holes
Mass fuel
8.0 mg/cc
8.0 mg/cc
MD
50%-50% 20%-80% to 80%-20%
Smoke, noise and pollutant emission targets
Smoke
< 2.00 FSN
Noise

< 80.0 dB
NOX
< 0.25 g/kWh
CO
< 6.50 g/kWh
HC
< 1.50 g/kWh
the split injection is presented in this format: 50/50 MD.
This nomenclature indicates that 50% of the fuel mass is
injected in the first injection event and the remaining 50%
is injected in the second event.
The engine mode and pollutant emission targets are
based on the EURO 4 cycle for a passenger diesel vehicle
with an aftertreatment particulate filter (Table 1). The presence of the particulate filter in the exhaust allows for a
high level of engine-out smoke emissions, as can be seen in
the Table 1.
3.2. Engine Test
The engine test stage was characterized by sweeping the
dwell time for various mass distributions. The engine tests
for both fuel injection systems were performed with two
different nozzles, varying only the number of orifices.
Although the nozzles differ in the number of orifices (10
and 12 orifices), the theoretical hydraulic flow is very
similar. The ISFC was fixed at 250 g/kWh and an IMEP of
4.0 bar was targeted. It is important to point out that the
SOI (Start of Injection) was varied in order to keep the
IMEP at 4.0 bar.
Engine tests using the solenoid and the piezoelectric injection systems were carried out with an injection pressure of
900 bar and a 45% EGR rate, based on the results obtained
in the definition phase. The boost pressure was set at 1.2

bar.
Using the solenoid system, the mass distribution ratio
ranged from 30/70 to 50/50, and the dwell time was swept
from 0.6 to 1.6 ms using both 12-orifice and 10-orifice
nozzles. Dwell times shorter than 0.4 ms would be very
unstable, and thus were not tested. Using the piezoelectric


142

J. BENAJES

injection system, the mass distribution ratio of the split
injection was swept from 20/80 to 50/50, and the dwell
times ranged from 0.2 to 1.4 ms. The piezoelectric injection system responded faster and more accurately, allowing
for shorter injection durations and dwell times. The same
tests were repeated with a constant SOI, instead of a
constant IMEP, in order to separately evaluate the effects of
different engine parameters on engine behavior.

4. RESULTS AND ANALYSIS
4.1. Solenoid Injection System Preliminary Test Results
In the preliminary test phase, some engine tests were carried out to select the most suitable EGR rate and injection
pressure for the next part of this study. The results and the
main observations of this part of the study are presented in
Figure 2(a) and (b). Graph (a) shows the pollutant emissions for each tested case. Graph (b) shows HRL, RoHR
and BGT curves for 900 bar IP EGR swept and 45% EGR
rate injection pressure swept.
As seen in Figure 2(a), for 600 bar IP, the levels of CO
and HC emissions were much higher than the targeted

values. Unexpectedly, the NOX emissions did not seem to
be related to the injection pressure. It is confirmed by the
BGT graph in Figure 2(b) that the burned gas temperature

Figure 2. (a) Noise, soot and pollutant emissions from the
preliminary tests using the solenoid injection system; (b)
HRL, RoHR and BGT vs. Crankshaft angle for solenoid
preliminary tests.

et al.

peak did not change significantly with injection pressure,
while it decreased considerably with increasing EGR rate.
Smoke emissions for the lowest IP (blue circle) were under
the target value but very close to the limit. Smoke was also
higher than observed for the other injection pressures due
to worse mixing conditions. This was not a good result
since it did not leave margin to work on a possible tradeoff. For an injection pressure of 1200 bar, the obtained CO
values were slightly higher. Combustion noise (in the green
circle) was also considered unsuitable, so the small reduction in smoke emissions does not justify its use in the next
phase. Using an injection pressure of 900 bar resulted in
smoke emissions below the target value. Moreover, this
configuration presented lower CO and HC emissions (dotted
red lines) than the other IP values. Thus, 900 bar seemed to
be the most reasonable injection pressure for further development.
Of the tested EGR rates (40%~50%), 40% had the
highest NOX emissions because the higher O2 concentration
led to higher in-cylinder temperatures. Thus, NOX formation
(black arrow) was not inhibited enough to stay under the
target value. On the other hand, increasing the EGR rate to

50% (grey arrows) significantly reduced the O2 concentration, enough to efficiently reduce the in-cylinder temperatures and combustion noise. However, this EGR rate had
unreasonably low combustion, which resulted in high CO
emissions. Furthermore, it was concluded that the next
engine tests using the solenoid injection should be carried
out with the IP and EGR rate set to 900 bar and 45%,
respectively.
In Figure 2(b), the BGT graph shows that the maximum
burned gas temperature did not vary as a function of injection pressure, but decreased when the EGR rate increased.
In order to maintain an IMEP of 4.0 bar, the SOI had to be
advanced when either injection pressure or EGR rate
increased. The effect was that the RoHR of the first combustion was considerably reduced (see red arrows). At an
IP of 1200 bar, it seemed that the combustion of the pilot
and main injections started almost simultaneously, increasing the RoHR slope and justifying the high values of combustion noise observed. At a 50% EGR, the RoHR did not
rise because increasing the EGR rate also caused a significant reduction in the mixing rate and, consequently, in
combustion velocity (see brown arrows).
4.2. Engine Preliminary Test Results Using Piezoelectric
Injection System
The preliminary tests using the piezoelectric injection system
intended to define an appropriate injection pressure and
mass distribution range for the next engine tests. This injection system allows for injection durations as short as 140
µ s, which permits injecting very small amounts of fuel,
such as 20% of the total injected mass (1.6 mg/cc). Thus,
the split injection was swept from 20/80 to 80/20 of the
mass distribution with a fixed dwell time of 0.6 ms (5.4
CAD). There are two different ranges to be considered and


COMPARISON OF TWO INJECTION SYSTEMS IN AN HSDI DIESEL ENGINE USING SPLIT INJECTION

143

Figure 3. Noise, soot and pollutant emissions from the
preliminary tests using the piezoelectric injection system.
observed separately: pilot injection (from 20/80 to 50/50)
and post injection (from 60/40 to 80/20).
Analyzing the results at 1300 bar of IP in Figure 3, the
pilot injection range presented higher levels of HC and CO
emissions than other injection pressures. However, the
emissions tended to decrease when the fuel mass of the first
injection event (green arrow) was increased. However, the
combustion noise increased in the same range because
more fuel mass was burned in the premixed combustion. In
the post-injection range, it is possible to observe very high
combustion noise, up to unacceptable levels. At 1100 bar
of IP, the smoke emissions stayed at very low levels, even
though the HC and CO emissions were high when using a
pilot injection. The smoke emissions were slightly decreased, whereas CO and combustion noise did not increase
much. The use of a 900 bar IP post-injection significantly
reduced smoke formation, as previously known, although
HC and CO emissions increased (blue circles) (Desantes
, 2007; Han
, 1996). Moreover, the use of a pilot
injection at 900 bar of IP kept the CO and HC emissions at
a lower level than the other injection pressures. Although
smoke emissions were higher than the other pressures, they
were still very far below the proposed target. Based on this
analysis, the engine tests using the piezoelectric injector
were carried out with a 900 bar IP and a pilot injection. The
EGR rate was increased to 45% in order to reduce NOX
formation since some of the test points did not meet their

targets.
et

al.

et al.

4.3. Engine Tests Analysis
The engine tests with the solenoid injector were performed
with a 10-orifice and 12-orifice nozzle. The 12-orifice nozzle
tests had a range of mass distribution from 30/70 to 50/50
MD. The same tests were repeated using the 10-orifice
nozzle, except for 30/70 MD because a high level of
combustion instability was found when using the 12-orifice
nozzle under those conditions. The cause was the short

Figure 4. Comparison of (a) 40/60 and (b) 50/50 mass
distributions.
injection duration required to inject only 30% of the injected mass, which caused the injector needle to pulse as fast
as possible. For both nozzles, the dwell time ranged from
0.6 to 1.6 ms.
Engine tests with the piezoelectric injector were carried
out with a 6-orifice and 12-orifice nozzle. The mass distribution range was swept from 20/80 to 50/50, and the dwell
time ranged from 0.2 ms to 1.4 ms.
Figure 4(a) and (b) present a comparison among all the
nozzles used in this study, independently of the injection
system, with mass distributions of (a) 40/60 and (b) 50/50.
A 45% EGR rate was used and the IMEP was isolated at
4.0 bar. The dashed lines are the proposed targets for each
pollutant. It is important to make clear that 0.2 and 0.4 ms

of dwell time were not tested with the solenoid system and
1.6 ms was not tested with the piezoelectric system. The
20/80 and 30/70 mass distribution graphs are not shown
here due to space limitations.
Increasing the dwell time contributed to smoke formation (blue circles) when using the 6-orifice or 10-orifice
nozzles, as seen in Figure 4(a) and (b). The different levels
of smoke emissions depend on the number of orifices of
each nozzle; the 12-orifice nozzles showed lower smoke
emissions than the nozzles with fewer orifices. When using
12-orifice nozzles, more advanced SOI’s were necessary to
reach a 4.0 bar IMEP, and they presented higher combustion noise.


144

J. BENAJES

NOX and smoke levels did not represent a problem since
they stayed below the target in the majority of the tested
points. However, combustion noise, HC and CO emissions
did not fulfill the required limits with both nozzles and all
mass distributions. In general, the 10-orifice nozzles produced more smoke than the 12-orifice nozzles. However, the
12-orifice nozzle presented more combustion noise because
it required a slightly advanced injection timing to maintain
a 4.0 bar IMEP.
As seen in Figure 4(b), the 6-orifice nozzle with reduced
dwell times, such as 0.2 or 0.4 ms, presented unsuitable
levels of combustion noise independently of the mass distribution (black arrow). NO emissions stayed below the
limit. The combustion noise target was achieved for both
mass distributions between 0.6 and 1.0 ms of dwell time.

The 20/80 and 30/70 mass distributions (not shown in the
picture) stayed close to the target. However, HC and CO
emissions are still very high in all the cases. Finally, for the
50/50 mass distribution, the smoke emissions were very
close to the limit of 2.0 FSN.
Figure 5(a) and (b) present two cases in which the
configurations using the piezoelectric injector and solenoid
injector were equal. In Figure 5(a), both injection systems
were tested using 12-orifice nozzles, 45% EGR, 4.0 bar
X

et al.

IMEP, 1.0 ms dwell time and −11.0 CAD ATDC SOI (). (b)
had the same configuration, except the dwell time and SOI
were changed to 1.2 ms and −11.5 CAD ATDC SOI,
respectively. The injector opening timings of the second
injection event are represented in the graphs by the vertical
lines.
Although the engine test configurations were exactly the
same in each graph, some differences are noticeable when
comparing the HRL, RoHR and BGT curves for both injection systems. The shorter time that the piezoelectric injector
required for opening and closing caused an advance of the
second injection relative to the solenoid injector. Consequently, the combustion process was advanced.
Examining Figure 5, it can be seen that cool flame reactions started before the second injection when using the
solenoid injector. However, those cool flame reactions were
not seen when the piezoelectric injector was used. The
second injections using the piezoelectric injector avoided
the cool flame reactions. In this case, the cool flame reactions were responsible for the temperature increase before
the combustion process. Furthermore, it can be seen that

the maximum BGTs are very similar for both injectors, but
the average temperatures during the combustion were slightly higher using the solenoid injector, which could have
increased NO formation in the beginning and prolonged
smoke oxidation at the end of the combustion process. This
effect is stronger for 1.0 ms of dwell time than for 1.2 ms.
Figure 6 shows the graphs for HRL, RoHR and BGT for
different dwell times corresponding to the tests with 4.0 bar
of IMEP, using the 12-orifice nozzle with a mass distribution of 50/50.
It can be observed that the combustion occurs more
smoothly when the dwell time is increased due to retardation of the center of the combustion. With a dwell time of
0.2 ms, the combustion is very similar to that of a single
injection. When the dwell time is increased to 1.4 ms (12.6
CAD at 1500 rpm) the combustion of both injections seems
to be slower, and the premixed combustion less abrupt.
Dwell times longer than 1.0 ms reduced the slope of the
X

Figure 5. Comparison of HRL, RoHR and BGT vs. crankshaft angle curves for 40/60 MD using solenoid and piezoelectric injectors with 1.0 ms and 1.2 dwell times. The
engine configurations for each dwell time case were
exactly the same, including injection timing.

Figure 6. HRL, RoHR and BGT vs. crankshaft angle for
50/50 mass distributions.


COMPARISON OF TWO INJECTION SYSTEMS IN AN HSDI DIESEL ENGINE USING SPLIT INJECTION

145

Figure 8. HRL, RoHR and BGT vs. crank angle for Iso-SOI

tests with 1.0 ms of dwell time.
Figure 7. Pollutant emissions from the Iso-SOI tests.
RoHR, causing less combustion noise. The maximum BGT
was reduced by increasing the injection dwell time. But the
time period that the BGT remained at NOX formation
temperatures increased with increasing injection dwell time
(Akihama
, 2001). Thus, there was not a significant
change in NOX emissions.
In order to study the isolated effects of mass distribution
and dwell time using the 12-orifice nozzle with the piezoelectric injector, some Iso-SOI tests were carried out. The
chosen SOI was -15.5 CAD ATDC, which was the one at
which the 20/80 mass distribution had an IMEP of 4.0 bar.
Figure 7 represents the complete pollutant emission results
obtained from the Iso-SOI tests. This graph presents all
tested dwell times and mass distributions.
With increasing dwell time, combustion noise was significantly reduced because the combustion started more
smoothly due to retardation of the center of the combustion. NOX formation was greatly reduced by retarding the
SOI. However, there was a significant increase in HC,
smoke and CO emissions, mainly in the mass distributions
with smaller pilot injections.
When the pilot fuel mass was increased (see Figure 8),
the center of the combustion (center of combustion is the
crank angle of 50% of heat release) was advanced towards
the TDC. However, this did not result in a higher IMEP.
The in-cylinder pressure rose earlier for longer pilot injections but also decreased earlier and remained lower during
expansion. The IMEP was kept constant for all mass distributions. In Figure 8, the region where the pressure lines
cross is shown by the red circle. This effect was attributed
to a completely premixed combustion, and changing the
mass distributions did not deteriorate the combustion process. For longer pilot injections, the RoHR is steeper, leading to an increase in combustion noise. Larger pilot injections advanced the entire combustion. However, it is

important to point out that the distance between the peaks
et al.

of the RoHR for the mass distributions of 20/80 and 50/50
was less than 4 CAD. The peak in-cylinder pressure also
increased with larger pilot injections. This led to higher
NOX emissions. However, the 50/50 mass distribution exhibited a reduced slope in the HRL curve.
5. CONCLUSIONS

In this research, two different injection systems (based on a
solenoid and a piezoelectric injector) were investigated.
Preliminary tests were performed to select the best conditions for the main study. In both cases, the chosen injection
pressure was 900 bar and the EGR rate was 45%. These
values were chosen to work with a pilot injection smaller
than 50% of the injected mass per cycle.
Different injection strategies were tested for each injection system. There were two injection events per cycle. The
ISFC was kept constant at 250 g/kW.h during the tests. The
injection strategies were characterized by sweeping the
mass distributions and dwell times. A small sequence of
tests was executed at constant SOI, in order to study the
isolated effects of the mass distribution for different dwell
times or combustion events.
Based on the results from the preliminary test and engine
test, it is possible to conclude:
(1) Independently of the injection system, 900 bar was the
most suitable injection pressure for this engine. The
lowest injection pressure of 600 bar resulted in higher
HC and smoke emissions. This could have been due to
a longer fuel atomization process than the other injection pressures. Higher injection pressures presented
lower smoke but higher combustion noise. This could

be attributed to the fact that more fuel mass is injected
before the combustion process is started, and the atomization of the fuel is better at higher injection pressures.
A 40% EGR rate presented excessive combustion noise
and high NOX emissions due to the fast premix combustion and high in-cylinder temperature. On the other
hand, a 50% EGR rate caused excessive reduction of
O2 concentration, reduced NOX and combustion noise,


146

J. BENAJES

but the HC and CO emissions increased to unacceptable levels.
(2) With the injection pressure at 900 bar, mass distributions with the first injection larger than 50% of the
injected mass present high CO and HC emissions. However, it has been observed that the use of a postinjection smaller than 40% of the total injected mass
significantly reduces smoke formation.
(3) An increase in dwell time with constant SOI produced a
smoother start of combustion and a cooler overall combustion process, reducing noise and NOX emissions.
However, the emissions of HC and CO increased under
these conditions.
(4) Increasing the pilot injection quantity caused an increase
in combustion noise and NOX emissions due to faster,
hotter premixed combustion. The opposite effect was
observed when the dwell time between injection events
was increased.
(5) The solenoid injection system presented unsatisfactory
results due to high HC and CO emissions independent
of the number of nozzle orifices. However, the 10orifice nozzle resulted in levels of combustion noise
close to the target, while NOX remained under the limit.
Smoke emission increased and stayed close to the limit.

(6) The piezoelectric injection system with 6-orifice and
12-orifice nozzles, presented unacceptable results in
terms of HC and CO emissions. The combustion noise
targets were achieved using both nozzles with dwell
times around 1.0 ms. Moreover, the 6-orifice nozzle
showed higher smoke levels than the 12-orifice nozzle.
(7) For the same engine test configuration, the solenoid
injector presented a slightly retarded combustion process compared to the piezoelectric injector due to the
longer time needed for it to open completely. This
behavior slightly changed the in-cylinder temperatures,
favoring NOX formation before the maximum BGT
was reached and soot oxidation at the end of the combustion process.
(8) The piezoelectric injection system presented better
results in terms of pollutant emissions. It also permitted
more accurate control of the injection parameters, including the possibility of injecting very small quantities of
fuel in each injection event.

REFERENCES
Akihama, K., Takatori, Y. and Inagaki, K. (2001). Mechanism
of the smokeless rich diesel combustion by reducing
temperature. SAE Paper No. 2001-01-0655.
Benajes, J., Molina, S., De Rudder, K. M. and Ben Hadj
Hamouda, H. (2006). The use of micro-orifice nozzles
and swirl in a small HSDI engine operating at a late
split-injection LTC engine. J. Automobile Engineering,

et al.

220, 1807−1816.
Benajes, J., Molina, S., De Rudder, K. and Amorim, R.

(2007). Optimization toward low temperature combustion
in a HSDI diesel engine, using consecutive screenings.
SAE Paper No. 2007-01-0911.
Desantes, J., Arrègle, J., López, J. and Garcia, A. (2007). A
comprehensive study of diesel combustion and emission
with post-injection. SAE Paper No. 2007-01-0915.
Desantes, J., Benajes, J., Molina, S. and Gonzales, C. (2004).
The modification of fuel injection rate in heavy-duty
engines Part 2: Effects of combustion. Applied Thermal
Engineering, 24, 2715−2726.
EurActive (2004). Euro 5 Emissions Standard for Cars.
Retrieved 2007, from EurActive.com: activ.
com/en/transport/euro-5-emissions-standards-cars/
article-133325
Han, Z., Uludogan, A. and Hampson, G. R. (1996). Mechanism of soot and nox emission reduction using multipleinjection in a diesel engine. SAE Paper No. 960633.
Hotta, Y., Inayoshi, M. and Nakakita, K. (2005). Achieving
lower exhaust emissions and better performance in an
HSDI diesel engine with multiple injection. SAE Paper
No. 2005-01-0928.
Lapuerta, M., Armas, O. and Hernandez, J. (1999). Diagnosis of DI diesel combustion from in-cylinder pressure
signal by estimation of mean thermodynamic properties
of gas. Applied Thermal Engineering, 19, 513−529.
Montgomery, D., Chan, M., Chang, C., Farrell, P. and
Reitz, R. (1996). Effect of injector nozzle hole size and
number on spray characteristics and the performance of
heavy duty D.I. diesel engine. SAE Paper No. 962002.
Nehmer, D. and Reitz, R. (1994). Measurement of the
effect of injection rate and split injections on diesel
engine soot and NOx emissions. SAE Paper No. 940668.
Pierpont, D., Montgomery, D. and Reitz, R. (1995). Reducing particulate and NOx using multiple injections and

EGR in a D.I. diesel. SAE Paper No. 962002.
Robert Bosch GmbH (2004). Diesel-Engine Management.
3rd edn. SAE. Warrendale. PA.
The European Commission (2006a). Euro 5 and 6 will
Reduce Emissions from Cars. Retrieved 2007, from
EUROPA: />do?reference=MEMO/06/409&format=HTML&aged=
0&language=EN&guiLanguage=en
The European Commission (2006b). Tighter Wmission Limits
for Cars After EP Adoption of Euro 5 and 6. Retrieved
2007, from EUROPA: />Action.do?reference=IP/06/1800&format=HTML&aged
=0&language=EN&guiLanguage=en
Tow, T., Pierpont, D. and Reitz, R. (1994). Reducing particulate and NOx emissions by using multiple injections in
a heavy duty D.I. diesel engine. SAE Paper No. 940897.


International Journal of Automotive Technology, Vol. 11, No. 2, pp. 147−153 (2010)

DOI 10.1007/s12239−010−0020−6

Copyright © 2010 KSAE
1229−9138/2010/051−02

DESIGN OF ACTIVE SUSPENSION AND ELECTRONIC STABILITY
PROGRAM FOR ROLLOVER PREVENTION
S. YIM , Y. PARK and K. YI
1)*

1)

2)

3)

BK21 School for Creative Engineering Design of Next Generation Mechanical and Aerospace Systems,
Seoul National University, Seoul 151-742, Korea
Department of Mechanical Engineering, KAIST, Daejeon 305-701, Korea
School of Mechanical and Aerospace Engineering, Seoul National University, Seoul 151-742, Korea
2)

3)

(Received 3 November 2008; Revised 2 September 2009)

ABSTRACT−This paper presents a method for the design of a controller for rollover prevention using active suspension and

an electronic stability program (ESP). Active suspension is designed with linear quadratic static output feedback control
methodology to attenuate the effect of lateral acceleration on the roll angle and suspension stroke via control of the suspension
stroke and tire deflection of the vehicle. However, this approach has a drawback in the loss of maneuverability because the
active suspension for rollover prevention produces in vehicles an extreme over-steer characteristic. To overcome this
drawback of the active suspension based method, ESP is designed. Through simulations, the proposed method is shown to be
effective in preventing rollover.

KEY WORDS : Rollover prevention, Active suspension, ESP, Optimal static output feedback control, Maneuverability

NOMENCLATURE
ay
g
m
ms
mu

Ix
Iz
ks
kt
bs
h
hs
Cf
Cr
tf
lf
lr
r
KB
vx
Kg

ed the number of rollover accidents. For example, in the
USA, there have been 4,045 fatalities and 88,000 injuries
caused by non-collision rollover accidents in 2004 (NHTSA,
2004a). Most rollover accidents are fatal. For instance, in
2003, the portion of rollovers in all crashes was approximately 3%, but 33% of all fatalities were caused by
rollovers (NHTSA, 2003).
As shown in Figure 1, the factors influencing rollovers
are the lateral acceleration ay, the distance from the roll
center to the center of gravity hs, and the lateral tire force
Fy. The untripped rollover occurs due to a large lateral
acceleration generated by excessive steering at high speed.
On a low-friction road or at low speed, the rollover cannot
occur because of insufficient lateral acceleration or lateral

tire force. Based on this observation, to prevent rollovers, it
is necessary to reduce the effect of the lateral acceleration
and the lateral force on vehicles.
Following the aforementioned idea, several control
schemes were proposed to prevent rollovers. The most
common scheme is to reduce the lateral acceleration through
decreasing a reference yaw rate with differential braking or
active front steering to produce a vehicle with under-steer
characteristics (Odenthal et al., 1999; Chen and Peng,
2001; Ungoren and Peng, 2004; Yoon et al., 2006; Schofield and Hagglund, 2008). However, this approach has the
drawback of deteriorated maneuverability, and the yaw rate
tracking performance due to this loss of maneuverability
may cause another accident such as a crash or tripped rollover.
Another approach for rollover prevention is to control
the lateral load transfer with an active suspension, which

: lateral acceleration
: gravitational acceleration constant
: vehicle total mass
: sprung mass
: unsprung mass
: roll moment of inertia about roll axis
: yaw moment of inertia about yaw axis
: stiffness of a suspension spring
: stiffness of a tire
: damping coefficient of a suspension damper
: height of C.G. from ground
: height of C.G. from a roll center
: cornering stiffness of a front tire
: cornering stiffness of a rear tire

: front track width
: distance from C.G. to a front axle
: distance from C.G. to a rear axle
: radius of a wheel
: pressure-force constant
: longitudinal velocity of a vehicle
: gain of sliding mode controller

1. INTRODUCTION
Over the last decade, a widespread supply of SUVs (Sports
Utility Vehicles) with high centers of gravity (C.G.) increas*Corresponding author. e-mail:
147


148

S. YIM, Y. PARK and K. YI

Figure 1. Factors influencing rollovers.
has a direct effect on the rollover (Yang and Liu, 2003;
Duda and Berkner, 2004). In this approach, the precomputed roll moment is dynamically distributed to front
and rear axles through the active suspension. However, the
active suspension is used not for rollover prevention but for
active roll compensation, that is, for roll angle and roll
stiffness at normal driving conditions without rollover danger.
This type of active suspension has some limitations in preventing rollovers because the attenuation of the effect of
lateral acceleration on roll angle or roll rate is weak.
To prevent rollovers, it is necessary to reduce the effect
of the lateral acceleration on the roll angle and rate. Assuming the lateral acceleration as a disturbance, a controller can be designed to attenuate its effect on the roll
angle or roll rate or suspension stroke. For this purpose, an

active suspension controller is designed with the linear
quadratic (LQ) optimal control methodology. With LQ
optimal control, it is easy to design a controller to regulate
a particular state or output variable against a disturbance. In
LQ control methodology, it is assumed that all states are
available. However, in real applications, the full states of a
system are not always available. For practical considerations,
it is desirable to use available sensor signals. Hence, the
optimal static output feedback (SOF) methodology is used
to design an active suspension controller (Levine and Athans,
1970; Toivonen and Makila, 1987).
The active suspension controller designed for rollover
prevention has a tendency to produce an over-steer characteristic in the controlled vehicle, which deteriorates the
maneuverability of the vehicle (Lee
, 1998). To overcome this drawback, it is necessary to design an ESP. ESP
is designed with direct yaw moment control (DYC). In
DYC, the yaw moment control is computed by a sliding
mode control methodology and is distributed to each wheel’s
braking force (Rajamani, 2006).
This paper is organized as follows. Section 2 presents
the design procedure of active suspension for rollover prevention. In this section, LQ static output feedback control
methodology is adopted to design a controller, and simulation is performed on a nonlinear vehicle model based on
commercial multi-body dynamics software, Carsim (Mechanical Simulation Corporation, 2001). ESP is designed with
direct yaw moment control, and simulation is also performed in section 3. Section 4 concludes this paper.

Figure 2. 4-DOF roll-plane model.
The vehicle model for active suspension is a 4-DOF rollplane model, as shown in Figure 2. This model describes
the vertical and roll motion of a sprung mass and the vertical motion of an unsprung mass. The disturbances acting
on a vehicle are the road inputs 1 and 2 and the lateral
acceleration .

Equations of motion for this model can be obtained as
follows:
zr

zr

ay

(1)
In Equation (1), 1 and 2 are suspension forces, defined as
follows:
f

f

(2)
Using the above definitions and the assumption
,
the equations of motion (1) are summarized as follows:
(3)

et al.

2. DESIGN OF ACTIVE SUSPENSION FOR
ROLLOVER PREVENTION
2.1. Vehicle Model

where

Equation (3) can be rewritten in matrix form as Equation

(4).
(4)


DESIGN OF ACTIVE SUSPENSION AND ELECTRONIC STABILITY PROGRAM FOR ROLLOVER PREVENTION 149
Rewriting Equation (4), Equation (5) is obtained as follows:
(5)
where

nearly same values.
LQ SOF control objective is to find a controller with the
form u = −Ky such that the LQ objective function (7) is
minimized (Levine and Athans, 1970). In this paper, the
output y is the roll rate, suspension stroke, and stroke rate,
as shown in Equation (8).
(8)

With the definition of a state, the state-space equation for
the vehicle model is as follows:
(6)
x Ax B1w B2u
where

K (Toivonen and Makila, 1987). However, this problem has

2.2. LQ Static Output Feedback Control
To design a controller for active suspension, LQ SOF control methodology is used. The LQ objective function is given
in Equation (7), containing the terms that emphasize vertical and roll acceleration, roll angle and rate, suspension
stroke, tire deflection, and control input.

Table 2. Suspension parameters of SmallSUV in CarSim.
980/2
I
439.9/2
m
m 1, m 2
40
k 1, k 2
230000
k 1, k 2
20000
b 1, b 2
2181
h
0.66

=

+

+

There have been several methods to compute the optimal

not been proven to have a global optimum. In this situation,
a heuristic search is a good alternative to the classical
gradient-based search (Toivonen and Makila, 1987). For
this reason, the evolutionary strategy, CMA-ES, is used to
find the optimal K (Hansen et al., 2003).

s

x

u

u

s

s

t

t

s

s

s

(7)
For ride comfort, it is necessary to reduce the vertical
and roll acceleration. However, this does not guarantee
rollover prevention. Generally, it is known that it is desirable to reduce roll angle or roll rate to prevent rollover.
However, the reduction of the roll angle or roll rate cannot
guarantee rollover prevention because these parameters
only weakly attenuate the effect of lateral acceleration on
the roll angle or roll rate. To prevent rollover by attenuating

the effect of lateral acceleration on roll angle or roll rate, it
is essential to reduce the suspension stroke and tire deflection. To demonstrate this, the following sets of weights are
proposed, as shown in Table 1. CASE1, CASE2, and CASE3
give large weights to the vertical and roll acceleration, the
roll angle and roll rate, and the suspension stroke and tire
deflection, respectively. The weights of CASE2 and CASE3
are selected such that LQ objective functions have the
Table 1. Three cases for each control purpose.
ρ1

ρ2

ρ3

ρ4

ρ5

ρ6

ρ7

CASE1 1e3 1e3 1e1 1e1 1e1 1e1 1e-4
CASE2 1e1 1e1 1e7 1e7 1e1 1e1 1e-4
CASE3 1e1 1e1 1e1 1e1 5•1e7 5•1e7 1e-4

Figure 3. Bode plots for each case.


150

S. YIM, Y. PARK and K. YI

Table 3. H∞ norms for each case and each input-output
channel.
Passive CASE1 CASE2 CASE3
a →φ
0.0185 0.2680 0.0035 0.0028
a → (z −z ) 0.0124 0.1433 0.0145 0.0011
a →z
0.0014 0.0013 0.0145 0.0018
y

y

s

y

u

u

2.3. Evaluation of Active Suspension System
The parameters of the vehicle model are obtained from
SmallSUV given in CarSim, as shown in Table 2.
For the three cases of weights in Table 1, the LQ SOF
gains are computed by evolutionary strategy, CMA-ES.
With these gains, the Bode plots of the closed-loop system
from the road input and the lateral acceleration to each

output are shown in Figure 3.
As shown in Figure 3(a) and (b), for the road input, the
responses of the vertical acceleration and roll angle are
improved for CASE1. However, the responses of the roll
angle and suspension stroke are deteriorated for CASE2
and CASE3. For the lateral acceleration input, CASE3
shows the best performance for the responses of the roll
angle and suspension stroke at the expense of deteriorating
those of the vertical acceleration and roll angle, as compared with CASE 1.
Table 3 shows the H∞ norms of each controller and each
input-output channel. As shown in Table 3, CASE3 has the
best performance in attenuating the effect of the lateral
acceleration on the roll angle and suspension stroke.
There are several measures to assess the rollover danger,
such as lateral acceleration and lateral transfer ratio.
Among these, the rollover index (RI) based method is very
simple and powerful (Yoon et al., 2006). In this paper, the
RI, calculated as shown in Equation (9), is used to assess
the rollover danger. If the RI is equal to unity, the left or
right wheels lift off.

Figure 4. Simulation results for each case.

(9)

To demonstrate the effect of the active suspension in
preventing rollovers, simulation is performed with the three
cases of controllers on the vehicle model SmallSUV given
in CarSim. Steering input is the fishhook maneuver with a
maximum angle of 270 degree, as described in NHTSA

(NHTSA, 2004b). Initial vehicle speed is set to 80 km/h,
and there are no controls to maintain a constant speed. The
tire-road friction coefficient is set to 1.1.
Figure 4 shows the simulation results of each controller.
As shown in Figure 4(a) and (b), CASE2 and CASE3 can

Figure 5. Trajectories for each speed.
prevent the vehicle from rolling over. However, CASE2
shows severe chattering in control input (Figure 4(c)).
These results show that the active suspension designed to
reduce vertical/roll acceleration (CASE1) cannot mitigate
rollovers and that the active suspension designed with
CASE2 has severe chattering in control forces. From these
results, it can be concluded that the active suspension
designed with CASE 2 or CASE3 can prevent rollovers.
To check the effect of CASE3, simulations are perform-


DESIGN OF ACTIVE SUSPENSION AND ELECTRONIC STABILITY PROGRAM FOR ROLLOVER PREVENTION 151

Figure 6. 2-DOF bicycle model including the control yaw
moment.
ed at various speeds. Figure 5 shows the trajectories of
vehicles with the active suspension designed by CASE3.
As shown in Figure 5, the controlled vehicle with CASE3
demonstrates severe over-steering because the controlled
vertical force on a tire results in increased lateral force (Lee
et al., 1998). This means that the maneuverability is
deteriorated. The loss of maneuverability can cause other
accidents such as crashes or tripped rollovers. Hence, it is

necessary to design an ESP to maintain maneuverability.

3. DESIGN OF ESP FOR MANEUVERABILITY
3.1. ESP Design
An ESP is a device developed to maintain maneuverability,
that is, yaw rate tracking performance. To design the ESP,
the linear 2DOF bicycle model is used, as shown in Figure
6. Assuming a linear lateral tire force, the equations of
motion for a linear bicycle model are given in Equation
(10).
(10)
In Equation (10), the linear lateral tire force is assumed
to be as given in Equation (11).
(11)
where

For a fixed longitudinal speed v , the reference yaw rate γ
is given in Equation (12) (Rajamani, 2006).
x

d

(12)
To force a vehicle to track the reference yaw rate, the
direct yaw moment control is applied with sliding mode
control theory. To force the error between the reference
yaw rate and actual one to zero, the sliding surface is defined as given in Equation (13). For this sliding surface to
have stable dynamics, condition (14) must be satisfied.
Combining Equations (10), (13), and (14), the control yaw
moment M is obtained as Equation (15) (Uematsu and

Gerdes, 2002).
B

(13)
(14)
(15)
After the control yaw moment is obtained by DYC, it is
necessary to distribute brake pressure to four wheels to
generate the given control yaw moment. The given control
yaw moment is transformed into braking force of the front
wheel as follows:
(16)
The relationship between the braking force F and
brake pressure P
on the front wheel is assumed as
follows:
x,front

B,front

(17)
From Equations (16) and (17), the brake pressures of
each wheel can be obtained for a given control yaw moment
M . The relationship between braking pressures of front
and rear wheels can be obtained as follows:
B

(18)
The brake pressure is applied to only one set of either the
left or the right wheels. For example, if the sign of the control yaw moment is positive, then braking pressure is applied only to the left wheels, and vice versa.

3.2. Evaluation of Active Suspension and ESP
The simulation conditions are identical to those of the previous section except that the active suspension is designed
with CASE3 and that ESP is applied. The parameters of
SmallSUV used in ESP are given in Table 4. Figure 7 shows
the simulation results of the vehicle with an active suspension and ESP. In Figure 7, the legends AS Only, ESP Only,
and AS+ESP indicate a vehicle with an active suspension,
with ESP, and with both, respectively.
As shown in Figure 7(b), the rollover index is over unity
if only the ESP is applied. This means that the vehicle is in
danger of a rollover. In comparison, a vehicle with active
suspension and ESP has a small roll angle and rollover
index, as shown in Figure 7(a) and (b). With active suspension and ESP, a rollover cannot occur at any speed. As the
trajectories of the controlled vehicle show in Figure 7(c),
the vehicle with ESP is not drifted while preventing a
Table 4. Parameters of SmallSUV in CarSim.
1296
m
980
I
C
40000
C
50000
0.88
l
1.32
l
z

f

f

r

r


152

S. YIM, Y. PARK and K. YI

controlled vehicle for various speeds. Contrary to the
results in the previous section, the ESP can maintain the
maneuverability of the controlled vehicle without rollover
at high speeds. From these results, it can be concluded that
the proposed method is effective in preventing rollovers.
4. CONCLUSION

In this paper, a rollover prevention controller was proposed
for vehicles with a high C.G., such as SUVs and vans.
Active suspension with lateral acceleration as a disturbance
was designed. The controller gains were obtained through
the LQ SOF method for several weightings. Through Bode
plot analysis and simulation, the controller, with high emphasis on the suspension stroke and tire deflection, can
effectively prevent the vehicle from rolling over. Despite
the remarkable performance in mitigating the rollover, this
controller resulted in over-steer characteristics for the
vehicle, deteriorating maneuverability. ESP was designed
to maintain the maneuverability of the controlled vehicle.

Through simulations, it is concluded that the proposed
method can effectively prevent rollover at any speed.
As shown in Figure 3, the active suspension designed for
rollover prevention deteriorated the ride comfort. To overcome this drawback, the proposed active suspension should
be activated under rollover situations. To accomplish this, a
switching scheme between normal and rollover situation
will be developed in future research.

ACKNOWLEDGEMENT−This work was supported by the

second stage BK21 Project and the Korea Science and Engineering Foundation (KOSEF) through the National Research
Laboratory Program (R0A-2005-000-10112-0).
REFERENCES

Figure 7. Simulation results for each case.

Figure 8. Trajectories of the controlled vehicle at various
speeds.
rollover. This means that maneuverability is not deteriorated due to the ESP. Figure 8 shows the trajectories of the

Chen, B. and Peng, H. (2001). Differential-braking-based
rollover prevention for sports utility vehicles with humanin-the-loop evaluations. Vehicle System Dynamics 36, 45, 359−389.
Duda, H. and Berkner, S. (2004). Integrated chassis control
using active suspension and braking. Proc. AVEC '04.
Hansen, N., Muller, S. D. and Koumoutsakos, P. (2003).
Reducing the time complexity of the derandomized evolution strategy with covariance matrix adaptation (CMAES). Evolutionary Computation 11, 1, 1−18.
Lee, J. S., Kwon, H. J. and Oh, C. Y. (1998). A study of
effects on the active suspension upon vehicle handling.
Trans. KSME, Part A, 22, 3, 603−610.
Levine, W. S. and Athans, M. (1970). On the determination

of optimal constant output feedback gains for linear
multivariable systems. IEEE Trans. Automatic Control,
15, 44−48.
Mechanical Simulation Corporation (2001). CarSim User
Manual Version 5.
National Highway Traffic Safety Administration (2003).


DESIGN OF ACTIVE SUSPENSION AND ELECTRONIC STABILITY PROGRAM FOR ROLLOVER PREVENTION 153
Motor Vehicle Traffic Crash Injury and Fatality Estimates,
2002 Early Assessment, NCSA (National Center for Statistics and Analysis) Advanced Research and Analysis.

National Highway Traffic Safety Administration (2004a).
Traffic Safety Facts 2004, US Department of Transportation.

National Highway Traffic Safety Administration (2004b).
Testing the Dynamic Rollover Resistance of Two 15passenger Vans with Multiple Load Configurations, US
Department of Transportation.

Odenthal, D., Bunte, T. and Ackermann, J. (1999) Nonlinear steering and braking control for vehicle rollover
avoidance. European Control Conf., Karlsruhe, Germany.
Rajamani, R. (2006). Vehicle Dynamics and Control. New
York. Springer.
Schofield, B. and Hagglund, T. (2008). Optimal control
allocation in vehicle dynamics control for rollover miti-

gation. American Control Conf., Westin Seattle Hotel,
Seattle, Washington, USA, June 11-13, 3231−3236.
Toivonen, H. T. and Makila, P. M. (1987). Newton's method
for solving parametric linear quadratic control problems.

Int. J. Control, 46, 897−911.
Uematsu, K. and Gerdes, J. C. (2002). A comparison of
several sliding surfaces for stability control. Proc. AVEC
2002, Japan.
Ungoren, A. Y. and Peng, H. (2004). Evaluation of vehicle
dynamic control for rollover prevention. Int. J. Automotive
Technology 5, 2, 115−122.
Yang, H. and Liu, Y. U. (2003). A robust active suspension
controller with rollover prevention. SAE Paper No. 200301-0959.
Yoon, J., Yi, K. and Kim, D. (2006). Rollover index-based
rollover mitigation system. Int. J. Automotive Technology
7, 7, 821−826.


Copyright © 2010 KSAE
1229−9138/2010/051−03

International Journal of Automotive Technology, Vol. 11, No. 2, pp. 155−166 (2010)

DOI 10.1007/s12239−010−0021−5

APPROXIMATIONS TO THE MAGIC FORMULA
A. LÓPEZ , P. VÉLEZ and C. MORIANO
*

Industrial Engineering Department, Universidad Antonio de Nebrija, C/Pirineos 55, Madrid 28040, Spain
(Recevied 5 December 2008; Revised 8 June 2009)

ABSTRACT−Pacejka’s tire model is widely used and well-known by the automotive engineering community. The magic
formula describes the brake force, side force and self-aligning torque in terms of the longitudinal slip and slip angle, plus

several corrections. This paper uses approximation theory to obtain different types of approximations to the magic formula:
rational functions (RA) resulting from the Remez algorithm, expansions in a series of Chebyshev polynomials (ACh), a series
of Chebyshev rational polynomials (ARChPs), a series of rational orthogonal functions (ORF) and a series of ARChPs that
result from grade-1 ORFs. The last expansion shows the fastest convergence and most effective computation. Jacobi rational
polynomials can also be obtained to complement this expansion and facilitate fine-tuning in specific areas of the error curve.
This work is complemented by obtaining the original rational approximations to the inverse tangent function, which take
advantage of the curve symmetry to reduce the computation load and provide models that include the influence of the vertical
load. The convergence properties of the development in series and the error values resulting from numeric examples for the
three types of stress are shown. The proposed final ARChP expressions show very low error (1%) compared to the original
magic formula. They can be computed 20 times faster; they can be evaluated, derived and integrated analytically easily; and
their coefficients can be obtained from tests using common least-squares algorithms.
KEY WORDS : Magic formula, Tire model, Approximation theory

1. INTRODUCTION

the origin and to obtain the error in different sections of the
curve. This article covers three types of stress in addition to
simplifications based on curve symmetry.
A new, efficient bivariate expansion is presented. Coefficients are easily obtained from the tests using standard least
squares algorithms.

This paper searches for a Chebyshev series expansion of
Pacejka’s tire model in order to obtain a more efficient
mathematical expression with enhanced analytical properties that can be integrated in the series expansion of the
equations that describe vehicular dynamics. The final aim
is to advance toward analytic solutions of those equations
using symbolic computing.
López et al. (2006) has provided an example of expansion in the power series of a simple longitudinal dynamics
of a vehicle. The resulting polynomial expressions facilitate very fast computation of the dynamic equations in real
time. Moreover, pre-computation of answers dependent on

the model entries can be achieved simply with the use of
symbolic computation tools (MAPLE). The need to obtain
simple and accurate formulations for the tire model to be
integrated into the previous dynamic model led to the
publication (López et al., 2007) of a first paper, which in
turn led to the development of RA, ACh and ARChP
approximations for longitudinal stress and bivariate approximations to the magic formula. Now, that report has
been completed and further expanded with the addition of
rational orthogonal functions theory and ORF and ARChP
expansions stemming from ORFs. Expansions in Jacobi
polynomials are also added for exact shift adjustments at

2. REVIEW OF THEORETICAL BASIS
2.1. Approximation of a Function in a Chebyshev Series
(ACh)
Chebyshev polynomials (Fox and Parker, 1968) of the first
kind are defined by
Tn( x ) =cos[n arccos( x ) ]

and are orthogonal to the function w(x)=(1−x )− on the
interval [−1, 1].
To work in different [a, b] intervals, shifted polynomials
must be used:
2

1/2

t=1/2[ ( b – a )x+a+b ]

Their general expression (Abramowitz and Stegun, 1972)

is the following:
n/2

∑

n m
m n m
n=1, 2, 3...; T (x)=1

Tn( x ) = n-2

*Corresponding author. e-mail:

m=0

( –1 )

m(

–
– 1)!
--------------------------- ( 2
!( – 2
)!

0

155

x )n

–2

m

;


156

A. LÓPEZ, P. VÉLEZ and C. MORIANO

where n is the largest whole number less than or equal
to n/2.
They fulfill the following recursive property:
=1, 2,...
(1)
Tn x xTn x Tn x
Chebyshev polynomials can be computed and manipulated
using the MAPLE Orthopoly library.
The expansion of a function in a Chebyshev series
(ACh) has the following form:
/2

+1

(

) =2

f (x )=

(

)−

∞

∑ aT
′

n

–1

(

);

n

x ,

n( )

n=0

The single comma in the summation indicates that the first
term must be divided by 2.
This expansion usually converges faster than the power

series, and the coefficients are described by the following:
an = -2- ∫
π

1

–1

(1 –

x ) f ( x )Tn( x )dx
2

–1/2

If we truncate the series at N degrees, we get an approximation to the function: the accuracy of the approximation
improves as N increases. Because of the properties of
Chebyshev polynomials, truncating the function at N-1
degrees is the best N-1-degree polynomial approximation
to the function with N degrees.
The coefficients n can be assessed with direct integration in some functions, but, in general, this calculation is
not possible, and the previous integral must be approximated by some other quadrature formula. MAPLE uses quadrature algorithms that first analyze the singularities and
then use Clenshaw-Curtis quadrature (Clenshaw and Curtis,
1960; Waldvogel, 2006); if the result is not satisfactory,
Newton-Cotes adaptive formulae are used. All of these
algorithms are carried out in the Chebpade function from
the MAPLE Numapprox library of approximation of functions.
a

2.2. Approximation Using Rational Functions (RA)

RA approximations are more efficient when the function
varies rapidly in some areas but not in others, which occurs
in tire behavior, especially when longitudinal stress is considered.
The Padé approximation provides rational expressions
with their numerators and denominators developed in power
series. They are processed efficiently as a continuous fraction. Chebyshev-Padé developments generate more compact and accurate rational expressions with Chebyshev
polynomials in their numerator and denominator. The
MAPLE Numapprox library also implements the rational
approximations. Its Chebpade function turns the initial
Chebyshev function into a power series, carries out a Padé
approximation and turns the resulting numerator and denominator into Chebyshev series again.
Chebyshev-Padé functions obtain good approximations,
but not those of minimum-maximum error (known as
minimax). To find the latter, the second Remez algorithm

(Remez, 1934) is used, which is a modified ChebyshevPadé approximation; it fine-tunes the result with numeric
iterations and converges to an improved minimax approximation.
The second Remez algorithm produces optimal results
that approximate both rational and polynomial functions.
This function allows the minimum error of any given function ( ) weighted with any weight term ( ) to be calculated. If ( )=1/| ( )| is used, the minimum relative error is
obtained. The minimax approximation with n-degree polynomials in the numerator and m-degree polynomials in the
denominator requires (n+m) additions and (n+m) multiplications for its evaluation, which are indicated as a minimax
approximation [n,m].
These methods are described in many books on approximation theory (Powel, 1981).
In MAPLE, the Remez algorithm is implemented by the
minimax function that is included in the Numapprox library
of function approximations.
f

t

w t

w t

f

t

2.3. Approximation to a Function in a Series of Chebyshev
Rational Polynomials (ARChP)
More recent works (Guo
, 2002; Wynn, 2006) show
the suitability of using rational polynomials, which accelerate convergence when the functions to approximate have
singularities or quick variation areas. These inherit the properties of Chebyshev polynomials and have the form:
et al.

Rn( x ) =Tn⎛⎝ -x----–----1--⎞⎠ =Tn( v ); 0 ≤ x ≤ ∞ ;
x+1
where v = -x---–----1-- ; x = – v----+----1-x+1
v–1

The development of a function in a series of Chebyshev
rational polynomials is:
∞

∞

=0

=0

f ( x ) = ∑ ′βnRn( x )= ∑ ′βnTn⎛⎝ -x----–----1--⎞⎠ =
x+1
n
n
∞
=∑ ′

n=0

for

βn Tn ( v ) ;

–1 ≤

v≤1

Chebyshev polynomials are orthogonal on the interval [−1,
1], but our independent variables (slip K and lateral slip)
vary between 0 and 100 and between −15 and 15 , respectively (because the formula is the same, we will generically
call both of them , and their initial and end points in and
fin, respectively). Thus, the Chebyshev expansion in series
on the original variable cannot be performed because its
domain exceeds the orthogonality of Chebyshev polynomials.
o

x

o

x

x

x

xin ≤ x ≤ xfin ⇒ vin ≤ v ≤ vfin
x –1
x-----–----1-vin= ---in---------- ; vfin= ---fin
xin + 1
xfin + 1

Therefore, shifted polynomials at v (Fox and Parker, 1968,
p. 49), Tnd v Tn u , with the following variable change
must be used:
(

)=

(

)


APPROXIMATIONS TO THE MAGIC FORMULA

[( ) ( )] ⇒
⇒ (( ))

v =1/2 vfin – vin u+ vin + vfin
2v – v + v
u= ----------------in----------fin----vfin – vin

Van Deun gets the coefficients:

;

(

u

f ( x ) ∑ ′β T ( v ) ∑ ′β T ⎛⎝
v
∞
∑ ′β T ( u )
=

∞

s
n

n

=

n=0

2
n ---------------fin – in

n

n=0

=

n

v

⋅ v (vv

)⎞
⎠

vfin – -------------------–
fin v in
in +

y

v

u

x

2.4. Rational Orthogonal Functions (ORF) Theory
2.4.1. Introduction
According to Bultheel
(1999), if A={α , α ..}, is a
sequence of real numbers other than zero, the linear vector
space of n-degree rational functions with poles at {α , …,
α ..} is defined by the space of functions L={b , b …, b },
where the base functions are defined by:
et al.

1

2

1

n

0

) () ()
–1

0

=1;

1

n

1

---αk = α

()

(

) (
)(
) ()

x . α n + Fn 1 – x . α n
En 1 – x . αn
--------- ------------------------- . ϕ n
x
En 1 – x .αn
1–

− (

–1

. 1–

) ()

x . αn .ϕn x
–1

–2

–2

–1

Van Deun
(2004) has obtained the coefficients of the
recurrence relation ( n , n) for the case of Chebyshev ORF
functions with Chebyshev weight functions. A Chebyshev
weight function is a Jacobi weight function of the type:
et al.

E

w(x ) ( x )
( x)
⎧⎪ ( x )
⎪
w(x ) ⎨ ( x )
⎪⎪ ⎛ x⎞
⎩ ⎝ x⎠
δ

1–
= ----------------

γ

1+

1–

=

1–

F

2

–1/2

2

1/2

,
,

case a)
case b)

1–
---------1+

case c)

⎫⎪

⎪⎬
⎪⎪
⎭

1
2

( )

β

2

− αβ
2

;

1–β
If =1, we get:

(

1+β )(1+β

E = 2c ;
Fa = 2 β c ;

n

()

β =α ± α

2

– 1;

n–1

β n – 2)

Eb=2c ;
Ec =2c
Fb= β c ; Fc = 1 – β c

−

1

1

1

where c

1

β
1+β

−

1

1

1

(

1

)

2

1–
1
= ---------------2
1

Examples and application to the magic formula are shown
in section 3.6.
2.4.2. Expansion in ORFs series
The best least-square approximation obtained after truncating the expansion in a series of orthogonal functions ( )
(of any type) of a function ( ) is (Burden and Douglas
1998);
F x

f

x

,

∞

( ) ≈ ∑ a F (x)

f x

k.

(2)

k

n=0

where the coefficients are:
1
j = ---

x2

rj x∫

()() ()

w x .f x .Fj x dx

1

x2

( ) [ ( )]

rj= ∫ w x . Fj x dx
x1

2

The weight function ( ) defines the importance of the
approximation of different sections of the interval [ , ].
For example, the Chebyshev weight function:
w x

x1

()

wx

x2

1
= --------------1–

x

2

has very little influence in the center of the interval and
more influence at its ends.
In this particular case, j ( ) are ORFs. The value of j
represents the norm of the ORF function, and for ORF
functions with Chebyshev weight functions, it takes the
constant value j=π. We recall that, in the case of Chebyshev polynomials j( )= ( , ), this value was j=π /2.
MAPLE does not support any function related to ORFs,
neither the generation nor expansion of functions. Expansion of the magic formula in ORF is presented in section
3.6.
F x

α=J β

+1=0;

2

x

r

r

When the Joukowski transformation is introduced,
α = -- β + z

n

F

where δ y γ = ±1/2

)

( 1 – β n – 1 ) ( β n + β n – 2 ) +2 β n – 1 ( 1 – β n β n – 2 )
1 – β 2n- ---------------------------------------------------------------------------------------------Fn= − ----------------2
2

a

k

If we orthonormalize and assume an interval on the real
line that excludes every pole, then these functions meet the
recurrence relation.
ϕ n x = En. x .

n–1

1

n

x . αk .bk x , b x

β

n

x

1–

2

)(

a

We can see that a double transformation, from the to
domains and from the to u domains, was required. The
function in the domain is approximated by a Chebyshev
development in series, and in the resulting approximate
function, the two previous transformations are undone to
obtain the approximate function in the original domain
(slip or lateral slip).

() (

1 – β )(1 – β
( 1+ β
) ( 1+ β )
2

n–1

n=0

bk x = x .

1–β

n–1
n
n–1 n
En=2 ------------------------------------------------------------------------2
2

where shifts between −1 and 1. Therefore, the final
development is:
∞

157

β =J

–1

( α)

T

j x

r

2.4.3. Expansion in a series of Jacobi polynomials

Within the families of classic orthogonal polynomials generated from the Sturm-Liouville differential equation, from
which Chebyshev polynomials are also derived, we consider Jacobi polynomials (Totik, 2005). The weight function


158

A. LÓPEZ, P. VÉLEZ and C. MORIANO

in this type of polynomial
δ
w( x )= -(--1----–---x---)---γ
(1 + x)

is controlled by two parameters, δ and γ , which allow the
area of a best approximation in the orthogonality interval to
be chosen. In practice, this is very interesting because it
allows us to improve the error adjustment in any area of the
longitudinal stress, lateral stress, or self-aligning torque
curves, depending on the application in which the approximation is used: for instance, looking either for a more
reduced error in slip values close to zero or for values close
to the maximum stress or the maximum slip point (100%).
The norm j in Jacobi polynomials is not constant, and it
is a function of δ, γ and the degree of the n-polynomial.
r

rj= -----2-------------Γ----(--n----+----δ----+----1---)--.--Γ----(--n----+----γ---+----1---)----n!.( 2n + δ + γ + 1 ).Γ ( n + δ + γ + 1)
δ+γ+1

The recurrence relation seen for the Chebyshev polynomials (1) in section 2.1 is made more general in the case of
Jacobi polynomials:

Pnδ γ ( x )=( an + bb).Pnδ γ ( x )−cn.Pnδ γ ( x ); n=1,2,...
(

,

)

(

+1

,

)

(

,

)

–1

where the recurrence coefficients are now:
2n + 1 + δ + γ )(2n + 2 + δ + γ-)
a =(--------------------------------------------------------------------2(n + 1 )(n + 1 + δ + γ )
(δ – γ )(2n + 1 + δ + γ)
b =----------------------------------------------------------------------------2(n + 1)(2n + δ + γ )(n + 1 + δ + γ -)
(n+δ )(n+γ )(2n + 2 + δ + γ)
a =------------------------------------------------------------------------( n + 1 ) ( n + 1+ δ + γ ) ( 2 n + δ + γ )

Jacobi polynomials can also be computed and manipulated
using the MAPLE Orthopoly library.
The expansion of a function in a series of Jacobi polynomials uses the same expression (2) as in section 2.4.2,
but with a Jacobi weight function. The integral must be
programmed in MAPLE. A library for expansions of functions in Jacobi series is not available.
n

2

2

n

n

2.5. Magic Formula
The well-known model proposed by Bakker
(1987,
1989) and Pacejka (2002), is a semi-empirical tire model
based on the “magic” formula:
Y=D.sin[C.arctan(BX–E.[BX-arctan(BX)])]
The shape of the curve is controlled by four parameters: B,
C, D and E. The equation can calculate the following:
• Lateral forces in a tire, Fy, as a function of the slip angle
of the tire, α, (in degrees)
• Braking force, Fx, as a function of longitudinal slip K (%)
• Self-aligning torque, Mz, as a function of the slip angle α.
B, C, D and E are constants that describe the inclination of
the curve at the origin (BCD), the peak value (D), the
curvature (E) and the basic form (C) for each case (lateral,

braking or self-aligning torque). In addition, the curve can
et al.

have vertical (Sv) or horizontal (Sh) shifts at the origin.
The full expression is:
Y=D.sin[C.arctan(B(X+Sh)–E.[B(X+Sh)
arctan(B(X+Sh))])]+Sv
Coefficients B, D and E are functions of the vertical load in
the tire, Fz:
d=a .Fz +a .Fz; B=BCD/(C.d ); E=a .Fz +a .Fz+a ;
Fz + a F-z ;
= .sin( (arctan( . )));
BCD =a-----------------------−

2

2

1

2

6

7

8

2

3

4

BCD2

a . Fz

1

e5

a3

a4

a5 F

BCD is valid for the longitudinal force and the selfaligning torque with C=1.65 and C=2.4, respectively.
BCD is valid for the lateral force with C=1.3.
The Camber angle γ in the wheel modifies the shifts Sh
and Sv and the stiffness BCD:
∆ S h a γ ∆ Sv ( a F z a F z ) γ ∆ B − a γ B ;
1

2

=

9

.

;

2

=

10

+

.

11

;

=

.

12

E
E = -------------------1–a . γ
0

1

13

is the value modified by the camber angle in the selfaligning torque calculation.
The aforementioned authors published the following
values of the coefficients ... for a given tire:

E1

E

a1

a13

3. APPROXIMATIONS TO THE MAGIC
FORMULA
3.1. Rational Approximations (RA) to the Functions Arctan(x)
and Sin(x)
3.1.1. Arctan(x)
We approximate the function arctan(x) that appears in the
expressions included in the magic formula. We find values
of between 20 and 30 rad approximately, with a function
value which is maximum at its asymptote and equal to π /2.
By using RA minimax [2, 2], deleting the independent
term in the numerator, adjusting and rounding up or down
the coefficients, we obtain the following:
x

−

arctan(x)≈ x (

x)
x + 5.1.x

. 4.66 + 8.
----------------------------------5 + 6.

2

which is valid ∀x , anti-symmetrical, has a very low absolute maximum error ε |ε | < 0.0025, is far more accurate
than other pseudoarctan(x) formulations presented in the
literature , and has “nice” coefficients.
In continuous fraction form, it can be expressed as:
⎛

arctan( )≈sign( ). ⎜⎜
x

x

⎝

0.931719
1.568627 – ------------------------------------------------------------1.762934
– 0.4741 + ------------------------------+ 1.650574

x

x

⎞
⎟
⎟
⎠

The function requires four additions and two divisions
plus the sign. Even lower errors can be obtained by approximations [2, 3] or higher.


APPROXIMATIONS TO THE MAGIC FORMULA

159

3.1.2. Sin(x)
The function sin(x) also appears with −4.1 < x < 4.1 rad. If
we proceed in the same way as with arctan(x), we get:
x.( 0.96 – 0.306. x )
sin( x ) ≈ ----------------------------------------------------------------2 ;
1.025 – 0.357. x +0.1121. x

|ε | < 0.018, 9 Op (5 Mul, 1Div and 3 Add)+2.abs(x)
There are more efficient approximations [2,1] at the
longitudinal and lateral stress ranks:
x.(1.3 – 0.45. x )(−2.3 < x < 2.3) sin(x) ≈ --------------------------------------–1.177+0.154. x

|ε | < 0.014, 6 Op (3 Mul, 1Div 2 Add)
3.2. Direct Approximations ACh to the Magic Formula
In this article, we consider relative error to be the absolute

error divided by the maximum absolute value of the function. This approach is convenient because approximations
with a minimax classic relative error (divided by the
modulus in each x value) give good results in low force
sections of the curve close to 0 (the least interesting
section), but very poor results in the rest of the curve (the
most interesting part). Our definition allows us to compare
errors for different vertical loads easily.
The expansion in the Chebyshev series of the magic
formula does not allow the use of low degree polynomials;
the following table shows the polynomial degree and the
relative error (as defined in the previous paragraph), with a
vertical weight Fz=8 kN.
The high values of the normal weight are those that need
a higher polynomial degree.
Regarding lateral force, the direct ACh of the magic
formula requires n ≥ 5 polynomials at the rank 0 < x < 15o
to cover all the weight values in our sample tire (from 0 to
10 kN). For the rank −15 < x < 15o, we need at least n ≥ 20
degree polynomials: low normal weight values require a
higher polynomial degree.
For the self-aligning torque we need n ≥ 13 for the half
interval and n ≥ 45 for the complete interval 15 < x < 15o.
Therefore, using direct expansions in Chebyshev series
in the magic formula is not a good idea, because convergence is not fast enough.
3.3. Rational Approximations (RA) to the Magic Formula
3.3.1. Longitudinal force
The minimax [2,2] adjustments carried out with the MAPLE
Numapprox library related to our sample tire give relative
error values that increase with the vertical weight Fz from
Rel.Error=0.9% for Fz=1 kN, to 1.36% for Fz=8 kN.

Using minimax approximations [2,3], we get Relative
Error values that fluctuate between 0.36% for Fz=1 kN and
0.79% for Fz=8 kN:
3.1564.106.( x + 13.7157) .( x – 0.01928 )Fx ≈ ------------------------------------------------------------------------------------------( x + 592.4785 ).( x2 + 3.695.x + 22.7886 )

Figure 1. Rational approximation [2,3] of braking force as
a function of slip.

Figure 2. Absolute error (N) at the rational approximation
[2,3] of the braking force, as a function of the slip.
Figures 1 and 2 show both the adjustment of the approximation [2,3] and the error. For the case of [2,2], the curve
has a similar shape but a slightly higher error.
If we accept an acceptable maximum relative error
criterion of less than 1%, we must work with minimax
approximations [2,3].
If we delete the independent term from the numerator in
the previous approximation [2,3], we get a slightly higher
error; however, this error is zero at the origin.
3.3.2. Lateral force and self-aligning torque
If we proceed in the same way, we can see Error < 1% in
minimax adjustments [5,3] for self-aligning torque, where
−15 < x < 15o. The figures show the lateral stress curve and
the error curve with Fz=8 kN.
We will show how to take advantage of the symmetry
using the results of the approximation between 0 and 15o in
section 3.9.
Regarding the self-aligning torque, if we want to ap-


160

A. LÓPEZ, P. VÉLEZ and C. MORIANO

Figure 3. Rational approximation [5,3] of the lateral force
as a function of the lateral slip.

Figure 5. Rational approximation [5,5] of the self-aligning
torque.

Figure 4. Absolute error (N) at RA [5,3] of the lateral force
versus lateral slip.

Figure 6. Absolute error (N.m) at the rational approximation [5,5] of the self-aligning torque.

proximate the whole rank between −15 and 15 with a Rel
Error < 1%, we must use an approximation [5,5].
We check the resulting curves for Fz=3 kN.
If we focus on the rank 0..15, the approximations [2,3]
produce results with a Relative Error < 1%.

n ≥ 8 for Fx with 0 ≤ x ≤ 100, Fy with 0 ≤ α ≤ 15 and
0 ≤ Fz ≤ 8 kN
n ≥ 12 for Mz with 0 ≤ α ≤ 15, and 0 ≤ Fz ≤ 8 kN
It is evident that convergence is faster than in the case of
ACh direct expansions; however, it is still not satisfactory
because the polynomials have at least eight degrees.

3.4. Approximations ARChP to the Magic Formula with
Constant Fz
Applying the expansion described in section 2.3, if we

include the suggested double transformation, we can see
that the ARChPs of the form:
f (x)=

∞

∑

n=0

′

x–1
βnTns⎛⎝ -----------⎞⎠
x+1

give results with a Relative Error < 1% with polynomials:

3.5. ARChP Approximations from ORFs
The convergence speed can be improved if we expand in a
Chebyshev series of rational functions of the type:
f (x)=

∞

∑

n=0

′

x
φ nTns ⎛⎝ -----------⎞⎠
x+b

The optimal factor b in each case varies with Fz.


APPROXIMATIONS TO THE MAGIC FORMULA
We obtain Relative Error < 1% in the following expansions:
n ≥ 4 with b=4 for Fx with 0 ≤ x ≤ 100 and 0 ≤ Fz ≤ 8 kN
n ≥ 4 with b=4 for Fy with 0 ≤ α ≤ 15 and 0 ≤ Fz ≤ 8 kN
n ≥ 9 with b=3.5 for Mz with 0 ≤ α ≤ 15 and 0 ≤ Fz ≤ 8 kN
The specified values of b guarantee a RelError < 1% for
0 ≤ Fz ≤ 8 kN. However, we can improve the error for
every value of Fz by modifying b slightly.
In Fx, for 4 ≤ Fz ≤ 7 kN, the result is n ≥ 3 with Error < 1%.
This expansion calculation in MAPLE is performed with
the Minimax-Remez function, which produces more accurate results than the Chebpade function, as has already been
stated.
These results are excellent: for example,
Fz=6 kN (constant)
C=1.65; D=6097.2; B=0.2064; E=0.606; BCD=2076.600
a1=-21.3; a2=1144; a3=49.6; a4:=226; a5=0.69e-1;
a6=-0.6e-2; a7=0.56e-1; a8=.486
xin=0; xfin=100; vin=0; vfin=100/104; b=4
Original equation (Magic formula)
Fx=6097.2.sin(1.65.arctan(0.0813 x+0.606.arctan(0.2064 x)))
Approximation
Fxap=

∞

∑

n

=0

′

βT
n

⎛
n

⎝

2
----------------- .
fin –
in

v

v

v +v
v – ------------------⎞⎠

v –v
in

fin

fin

in

Fxap=3557.1888.T (0,v)+2587.3377.T (1,v)−
1536.3996.T (2,v)−515.9311.T (3,v)
------ .v−1)
being: T (n,v)=T(n, 52
25
In addition, according to the Horner normal form:
Fxap=−50.61+(8506.27+(13491.33−18571.27.v).v).v
x
where v=--------x+4
If we execute expansions in x for different values of Fz,
s

s

s

s

s

161

Figure 8. Absolute Error (N) versus Longitudinal slip in a
Chebyshev series of the rational function x/(x+b).
look for the optimal b in each case and calculate the regression of b=f(Fz), we increase the convergence speed; in
the case of our sample tire, we can get:
n ≥ 3 with b=5.4629-0.2829.Fz
for Fx with 0 ≤ x ≤ 100 and 0 ≤ Fz ≤ 8 kN.
3.6. Monopole ORF Approximations to the Magic Formula
When applying the results given in section 2.4, we can get
the base of the monopole ORF functions from the values of
b.
As an example, the three ORFs and the approximation
for the braking force are shown for the same tire data and
maximum force (6 kN).

.
In this case, the error curve is very similar to that shown in
Figure 8, although it fluctuates between −60 N and +60 N.
This approximation is less accurate and requires more
computation than the previous one because the previous
approximation was a minimax (this can be seen when
checking the maximum local error leveled in Figure 8), and
this approximation is a minimum squared expansion,
which is less precise.
Figure 7. Braking Force (N) versus Longitudinal slip in
Chebyshev series of the rational function x/(x+b).

3.7. Bipole ORF Approximations to the Magic Formula
Using the same tire data as in the previous example, starting from pole 4, the third pole increases and the initial
bipole decreases until the minimum error is observed:

162

A. LÓPEZ, P. VÉLEZ and C. MORIANO

In this case, the error is better than the previous ones, and
because it is similar to Figure 8, it fluctuates between −40
N and +40 N.
This bipolar ORF approximation has the same error as
RA minimax [2,3], although the minimax has complex poles.
Similar ORF expansions can be performed for lateral force
and self-aligning torque.
3.8. Approximations in a Series of Jacobi Rational Polynomials
If we use the expressions in section 2.4.4., we can start
from the values δ =−1/2 and γ =1/2, which correspond to
the Chebyshev weight function (which, in turn, is a particular case of Jacobi polynomial). Increasing both values
reduces the error in the central area of the curve and
increases it at the ends. The error can also be adjusted to
zero at the ends by keeping one of the two parameters fixed
and changing the other, while keeping the maximum error
values constant for the whole curve. Thus, for instance, the
value of shifts from the origin to the values Sv and Sh
indicated in section 2.5 can be adjusted.
For example, for the same tire with Fz=6 kN, an adjustment with a null error is shown at the origin (Sv=Sh=0)
with the values δ =−1/2 and γ =−0.4685/2. The resulting
approximation (with b=3.85) is the following:
Fxap=(7369.26+(15916.3-19867.52.v).v).v
x where v=--------------x+3.85
The error curve looks very similar to that shown in Figure

8, which is also between ±50 N, but the current error curve
has a null error at the origin.
We can adjust the null error at the end of the curve or at
its maximum using this method.
We can also adjust the slope at the origin (the value BCD
in the original Pacejka formula) to obtain an exact value or,
with a moderate error, to achieve global maximum error
values around 1%. For example, the following approximation (δ =−0.16 and γ = −0.68, b=3.85):
Fxap=−62.86+(7904.7+(14744.57−19130.30.v).v).v
x being v=--------------x+3.85

has a slope at the origin of BCDap=2053.
(BCD original=2076). (Error=1.1%)
The approximate peak value is Dap=6135
(D original=6097) (Error=0.6%), with a maximum global
error of 67 N (Error=1.09%)
In this type of approximation:
Fap=A0+(A1+(A2+A3.v).v).v
x
v= ---------x+b
The value of A1/b is the derivative at the origin (BCD).
3.9. Use of Symmetry
As we have expanded the lateral force Fy and the selfaligning torque in the current semi-axis, we must look for
valid formulations for the entire real line to take advantage
of the performed computation with a lower degree polynomial.
To calculate the approximate expression we do the
following:
(1) Calculate the approximation to the original function
without shifts with a null error at the origin in the
interval 0..vfin using Jacobi polynomials.

(2) Calculate the valid expression for the interval -xfin..xfin,
which passes through the origin.
(3) Apply Sv and Sh shifts to the approximate expression.
The following is an example for the lateral force Fy:
a1=−22.1; a2=1011; a3=1078; a4=1.82; a5=0.208; a6=0;
a7=−0.354; a8=0.707;
D=5270.4; BCD=1076.149; B=0.1571: E=−1.417
Sh:=−0.126:Sv:=−181:Fz=6 kN.
(1) Approximation using a series of Jacobi rational polynomials of the function with Sv=Sh=0 and a null error
at the origin (calculated with α =−1/2 and β =0.71).

Figure 9. Lateral force (N) versus slip angle in a 3-degree
symmetric Jacobi approximation, with shifts Sv and Sh.


APPROXIMATIONS TO THE MAGIC FORMULA

163

3191 -=2.842; and A3′=--------------–3828-=−3.409.
A2′= ---------------

1122.7
1122.7
In addition, using the original peak value factor,
D=a1. Fz +a2Fz; a1= −21.3; a2=1144; and
D =D =1122.7 N
The approximate shape factor giving minimum error is:
Fs=5.956−0.5181.Fz+0.0255.Fz
The stiffness factor with minimum error is:

A’ =−0.00102906.Fz +0.0092337.Fz+1.104
The slope at the origin is D.A’ /F , where D the approximate peak value.
Figures 11 and 12 show the error in this approximation.
If we use the original BCD value to calculate A’ , where
A’ =Fs.BCD/D, we can use existing data; however, the
error is three times greater.
Clearly, we can integrate the factor D into the A’i, coefficients by finding the products D.A’ and D.A’ , and then
reducing the product D.A’ to two degrees using MinimaxRemez [2,0]. We can also consider A’ and A’ to vary with
Fz for a longer period of time to obtain more accurate
expressions: shorter expressions of D.A’ (Fz) have a larger
error There are many possibilities. One of the simplest is
the following:
Fxap=Fz.(B +(B +B .v).v).v
x - ; B =const.
v= -----------x + Fs
2

1

(Fz =1)

2

2

1

1

S

1

1

Figure 10. Absolute error (N) versus slip angle in a degree
3 symmetrical Jacobi approximation, with shifts Sv, Sh.

3

1

.

(2) A 3-degree expression valid for -xfin ≤ x ≤ xfin
F xap=sign(x).(5166.26+(24265.833-30220.43.v ).v ).v
1

1

1

x -; sign(x)=----xv1=----------------x
x + 6.5

(3) Shifted final approximation
F xap=F xap(x+Sx)+Sv
The resulting curves are shown in Figures 9 and 10.
2

3

2

The maximum absolute error is 68 N:
Fxap=(5166.26+(24265.833-30220.43.v).v).v
x v= -----------x+6.5
1

2

1

1

4. INFLUENCE OF THE VERTICAL LOAD

1

2

3

i

with a maximum RelError=3.2% and the following values
of Bi:
B =930; B =3910; B = 4250
and working with the same shape factor:
Fs=5.956−0.5181.Fz+0.0255.Fz

For the lateral force Fy, the shape of the curve changes
1

2

3

−

2

For the longitudinal force, we approximate the influence of
the vertical load from the curve Fap obtained for 1 kN by
adding the peak value factor D(Fz) that coincides with that
of the original formula (now it gives the approximate peak
value) and a second shape factor F . Both factors are functions of the vertical load Fz. We associate the linear coefficient A’ with stiffness at the origin. We calculate the regression with optimal values of Fs and A’ for each value of
Fz.
Fxap=D.( A + A + A .v .v).v
1

f

1

1

1

′

(

2

′

3

′

)

A2- ; A ′= ----A3x - ; A ′=----v= -----------x + Fs 2 D1 3 D1

We show an example of the braking force with three degrees
and a maximum relative error of 1.1% for 1 kN ≤ Fz ≤ 8 kN:
From the Jacobi approximation with Fz=1;
δ =−1/2; γ = −1.98/2;
b=5.5; A =1249; A =3191; A =−3828;
1

2

3

Figure 11. Braking force versus longitudinal slip for Variable vertical load (1 kN ≤ Fz ≤ 8 kN).