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A New Approach for Estimating Tire-Road Longitudinal
Forces for a Race Car
Guido Napolitano Dell'Annunziata1,2, Basilio Lenzo1*, Flavio Farroni2, Aleksandr
Sakhnevych2, Francesco Timpone2
1 Sheffield Hallam University, S11WB, Sheffield, United Kingdom
2 Università degli studi di Napoli Federico II, 80125, Naples, Italy
basilio.lenzo@shu.ac.uk
Abstract. In vehicle dynamics, the determination of the tire-road interaction
forces plays a fundamental role in the analysis of vehicle behavior. This paper
proposes a simple yet effective approach to estimate longitudinal forces. The
proposed approach: i) is based on equilibrium equations; ii) analyses the pecu-
liarities of driving and braking phases; iii) takes into account the interactions
between vehicle sprung mass and unsprung mass. The unsprung mass is often
neglected but that might lead to significant approximations, which are deemed
unacceptable in performance or motorsport environments. The effectiveness of
the proposed approach is assessed using experimental data obtained from a high
performance racing car. Results show that the proposed approach estimates tire
longitudinal forces with differences up to 10% when compared against a simp-
ler formulation which uses only the overall mass of the vehicle. Therefore the
distinction among vehicle sprung and unsprung masses, which is likely to be an
easily obtainable piece of information in motorsport environments, is exploited
in this approach to provide significant benefits in terms of longitudinal force es-
timation, ultimately aimed at maximizing vehicle performance.
Keywords: Vehicle Dynamics, Tire-Road Interaction, Longitudinal Forces,
Sprung and Unsprung Masses.
1 Introduction
The accurate estimation of the tire-road interaction forces is of great interest in ve-
hicle dynamics. Only through a precise and accurate calculation of such forces the
real behavior of the vehicle can be thoroughly investigated, understanding whether
the tires are working correctly and at the maximum of their performance.
In recent years, tire technological development has played a fundamental role in
motorsport and in the automotive sector. The availability of reliable procedures and
methods to estimate tire-road interaction forces is a crucial aspect therein.
In the literature several contributions [1-5] deal with this problem, using different
hypotheses and schematizations of the vehicle. In most cases, a single-track model is
introduced, but that is not suitable to calculate the forces on each corner. Other ap-
2
proaches adopt Kalman filters to calculate tire-road interaction forces [6-8]. In [9] the
T.R.I.C.K. tool is presented, which uses a 8 degrees-of-freedom (DOF) rear-wheel-
drive vehicle model to obtain a fairly good estimation of the tire-road interaction
forces. However, only the global mass of the vehicle is considered, without looking in
detail at sprung mass and unsprung mass. In [10], sprung and unsprung masses are
used to develop a tool that calculates the energy released from a vehicle to the road
pavement, but such data are not directly used to estimate the interaction forces.
The main novelty of this paper is to present a formulation for the calculation of
longitudinal forces that takes into account the distribution of the vehicle global mass
into sprung mass and unsprung mass. The availability of such information is not a
difficult task especially in motorsport environments, and it allows to introduce a sig-
nificantly improved accuracy in the estimation of the tire-road longitudinal forces.
Also, a different approach is used when compared to [9], i.e. individual free body
diagrams are studied and exploited in the present formulation, which leads to different
results than [9] even when only the global mass of the vehicle is considered. Addi-
tionally, in most road vehicles a further distinction can be made between the unsprung
mass of the front wheels and the rear wheels, e.g. for a rear-wheel drive architecture,
the rear wheel assembly includes additional components for housing the axle shafts.
Section 2 introduces the equilibrium equations of the individual wheels and of the
whole vehicle, leading to two formulations. Section 3 investigates the general case of
a non-even front-rear distribution of unsprung mass. In Section 4 the proposed formu-
lations are assessed via experimental data acquired on a high-performance race car.
2 Longitudinal Force Formulations
The reference system used to calculate the longitudinal forces is the same as in [1]:
it is centered in the vehicle center of mass, x is the longitudinal axis, positive for-
wards, y is the lateral axis, positive to the left, and z is the vertical axis, positive up-
wards. Table 1 reports the list of symbols used throughout the paper. The vehicle is
assumed to be rear-wheel-drive, also it is assumed >0 and < 0 in the direction
of travel.
Acceleration and braking cases need to be addressed separately because of their in-
dividual peculiarities. Two main formulations are presented:
1. Splitted Mass Formulation (SMF), considering sprung mass and unsprung mass;
2. Overall Mass Formulation (OMF), considering only the total mass of the vehicle.
2.1 SMF - Acceleration
Based on individual free body diagrams, the equilibrium equations can be derived.
Front Wheel (Fig. 1a)
─ Moment balance equation (rotation around the wheel center):
+−=
(1)
3
(a) (b)
Fig. 1. Acceleration: (a) front wheel, (b) rear wheel
Table 1. List of symbols
Symbol
Description
Vehicle longitudinal acceleration
Drag force
Rolling resistance
for
wheel ij
Overall tire
-
road l
ongitudinal
interaction force
for
wheel ij
Tire-road longitudinal interaction force for wheel ij, excluding
Vertical load
for
wheel ij
Force transmitted to the hub fo
r
wheel ij
Moment of inertia for wheel i
Braking torque action
for
wheel ij
Overall vehicle
mass
Unsprung mass
Front wheel unsprung mass
Rear wheel unsprung mass
Sprung mass
Effective rolling radius for wheel i
Driving torque
Driving torque acting on a single wheel j
Offset between
contributions due to rolling resistance
Vertical force distribution (
r
ear
l
eft
-
r
ear
r
ight)
Vertical force distribution (
f
ront
-
r
ear)
Angular acceleration for
wheel ij
Subscripts
axle index: 1=front, 2= rear
Side index: 1=left, 2=right
─ Longitudinal equilibrium equation:
− + + =− (2)
4
The reaction force from the road to the vehicle is shifted forwards by with
respect to the wheel center, due to rolling resistance, so that = .
Therefore Eq. (1) can be rearranged as:
=
(3)
The longitudinal force exchanged between tire and wheel 1j is +:
= + =
+ (4)
Combining Eq. (2) and (4), the force transmitted to the hub for wheel 1j is:
= +
+ (5)
Sprung mass (Fig. 2)
Fig. 2. Acceleration: sprung mass
─ Longitudinal equilibrium equation:
+ + − − = − (6)
is assumed applied at the vehicle center of mass, which is an approxima-
tion [1], however that has a negligible effect the estimation of the longitudinal forces.
Equation (6) can be rewritten in the following form, useful to obtain Eq. (11) later:
+ = + + + (7)
The moment balance equation is not studied as it only provides information about
the vehicle pitch motion, with no effect on the estimation of the longitudinal forces.
Rear Wheel (Fig. 1b)
─ Longitudinal equilibrium equation:
+ − = − (8)
─ Moment balance equation (rotation around the wheel center):
+−−= (9)
5
The longitudinal force exchanged between tire and wheel 2j is − +:
= − = −− (10)
Replacing from Eq. (7), using Eq. (5) and introducing =
:
= −−+ +2 + 11
+12
+ + (11)
The overall + contribution is split based on the vertical loads at left and
right side, using , whenever no information is available on the differential, follow-
ing the approach in [9] which accounts for longitudinal and lateral load transfers as
well as downforce.
2.2 OMF and comparison with SMF - Acceleration
The OMF formulation is here introduced, for use when no specific data on sprung and
unsprung mass are available. For the front wheels still Eq. (4) holds, whilst for the
rear wheels the OMF formula is:
= −+ +
+
+ + (12)
The inertia and rolling resistance contributions are the same in Eq. (11) and Eq.
(12). Differences arise in all terms containing a mass: Table 2 compares such terms
for SMF and OMF, for three values of . It should be noted that =+4.
Table 2. Comparison between SMF and OMF for rear wheels - Acceleration
SMF term(s) OMF term(s) Magnitude
0
−
0 |SMF| > |OMF|
0.5
−
2
−
/
2
−
2
−
/
2
|SMF|= |OMF|
1
−
3
−
−
4
−
|SMF| < |OMF|
The OMF introduces an error which depends on the vertical loads on the wheel. In
particular, the error grows when the vertical loads are different from each other,
which is typical of lateral dynamics. If ∈ 0;12
the OMF underestimates the
longitudinal forces. If ∈ 12
;1 the OMF overestimates the longitudinal forces.
2.3 SMF - Braking
Sprung mass (Fig. 3)
─ Longitudinal equilibrium equation:
+ + + + = − (13)
6
which can be rewritten as:
+ + + = − − (14)
Fig. 3. Braking: sprung mass
Front Wheel (Fig. 4)
Fig. 4. Braking: generic wheel
─ Moment balance equation (rotation around the wheel center):
+−− = (15)
─ Longitudinal equilibrium equation:
− + + = − (16)
Introducing =
, from Eq. (14):
=(− −)
(17)
where the factor 2 indicates a 50-50 left-right distribution of the braking effort.
The longitudinal force exchanged between tire and wheel 1j is +:
7
= + = −+(− −)
(18)
Rear Wheel (Fig. 4)
Following from Eq. (18) and from the definition of :
= + = −+(− −)()
(19)
2.4 OMF and comparison with SMF - Braking
OMF does not account for sprung and unsprung masses, therefore:
=(− − )
(20)
=(− − )()
(21)
Table 3. Comparison between SMF and OMF for front wheels - Braking
SMF term(s) OMF term(s) Magnitude
0
−
0 |SMF| > |OMF|
0.5
−
−
/
4
−
−
/
4
|SMF| = |OMF|
1
−
−
/
2
−
2
−
/
2
|SMF| < |OMF|
Table 4. Comparison between SMF and OMF for rear wheels - Braking
SMF term(s) OMF term(s) Magnitude
0
−
−
/
2
−
2
−
/
2
|SMF| < |OMF|
0.5
−
−
/
4
−
−
/
4
|SMF| = |OMF|
1
−
0 |SMF| > |OMF|
Table 3 and Table 4 compare SMF and OMF in braking, for three values of .
SMF and OMF coincide only if =0.5. If ∈ 0;12
the OMF underestimates the
longitudinal forces for the front wheels and overestimates them for the rear wheels. If
∈ ]1⁄2;1] the OMF overestimates the longitudinal forces for the front wheels and
underestimates them for rear wheels.
3 SMF with Different Unsprung Masses: SMF*
This paragraph studies presents the case in which front and rear unsprung masses
are different. As discussed before, such a condition is not unlikely, e.g. depending on
the drive architecture, the front and rear wheel assemblies may differ significantly.
This formulation will be indicated as SMF*. It implies =+2 +2.
During acceleration, SMF* and the OMF are identical for the front wheels. For the
rear wheels, replacing with and where appropriate in Eq. (11):
8
= −−+ +2 +
+
+ +
(22)
Table 5 compares SMF* and OMF, for three values of .
Table 5. Comparison between SMF* and OMF for rear wheels - Acceleration
SMF* term(s) OMF term(s) Magnitude
0
−
0 |SMF*| > |OMF|
0.5
−
−
/
2
−
−
−
/
2
−
|SMF*| = |OMF|
1
−
−
−
2
−
2
−
−
2
|SMF*| < |OMF|
During braking, again front and rear wheels need to be analyzed. For the front wheels
Eq. (18) can be modified as:
= −+(− −)
(23)
Table 6 compares SMF* and OMF for the front wheels, for three values of .
Table 6. Comparison between SMF* and OMF for front wheels - Braking
SMF* term(s) OMF term(s) Magnitude
0
−
0 |SMF*| > |OMF|
0.5
−
−
/
4
−
/
2
−
/
4
−
/
2
|SMF*| ≠ |OMF|
1
−
−
/
2
−
−
/
2
−
|SMF*| < |OMF|
For the rear wheels, Eq. (19) changes into:
= −+(− −)()
(24)
Table 7 compares SMF* and OMF, for the rear wheels, for three values of .
Table 7. Comparison between SMF* and OMF - Rear wheels - Braking
SMF* OMF Magnitude
0
−
−
/
2
−
−
/
2
−
|SMF*| < |OMF|
0.5
−
−
/
4
−
/
2
−
/
4
−
/
2
|SMF*| ≠ |OMF|
1
−
0 |SMF*| > |OMF|
When looking at acceleration (Table 5), again SMF* and OMF provide the same
result only if =0.5. Interestingly however, during braking (Tables 6 and 7, last
column) SMF* and OMF provide different results even when =0.5.
9
4 Comparison among T.R.I.C.K. Formulation, SMF and OMF
The SMF and OMF, along with the T.R.I.C.K. formulation [9], were compared by
means of experimental data, obtained on a professional proving ground with a high-
performance vehicle (details on which cannot be disclosed). All the relevant parame-
ters were available for the vehicle, except details on the front and rear unsprung
masses. Only the average value of unsprung mass was available. Therefore it was not
possible to study the performance of the SMF*.
Figure 5 shows a comparison between SMF and OMF considering the longitudinal
forces obtained at each vehicle corner (indicated in the legend). The percentage
error depicted in Fig. 5 is calculated as:
=,,
, 100 (25)
According to Fig. 5, the error between the SMF and OMF reaches up to 10%.
Fig. 5. Percentage error between SMF and OMF for the four wheels
Fig. 6. Percentage error between SMF and T.R.I.C.K. formulation for the four wheels
10
Figure 6 compares, in a similar fashion, the SMF with the T.R.I.C.K. formulation.
The percentage error depicted in Fig. 6 is calculated as:
=,,
, 100 (26)
In this case the error is up to 20%, and in general it is often different from zero.
5 Conclusion
In this paper three different formulations (SMF, SMF* and OMF) were presented
within a new approach for the estimation of the tire-road longitudinal forces of a race
car. The proposed approach is generally based on the equilibrium equations of front
and rear sprung masses, along with the equilibrium equations of the unsprung mass.
Moreover, the knowledge of the vehicle sprung/unsprung mass distribution improves
the estimation of longitudinal forces up to 10% compared to when such specific piece
of information is not available. Also, when comparing the SMF with the T.R.I.C.K.
formulation, differences are up to 20%.
Future studies will involve: i) the availability of specific information on the front
and rear unsprung masses, which allows to experimentally validate the SMF*; ii) the
adoption of advanced sensors on the test vehicle, such as driveshaft load cells, wheel
force transducers, brake pressure sensors. That would allow to further validate the
proposed approach, which ultimately is meant to be an useful tool for motorsport and
race engineers to investigate the vehicle behavior in a simple yet effective fashion.
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