CONTRIBUTION

Model-Free Control of Longitudinal and Lateral
Dynamics for Automated Vehicles

Professor Brigitte d'ANDRÉA-NOVEL
Centre de Robotique-Ecole des Mines de Paris

The development of “Advance Driver Assistance Systems” (ADAS) will intensify in the future and contribute to
the deployment of Intelligent Transportation Systems (ITS), thus improving the safety and fluidity of traffic, as wellas energy consumption. For that purpose, a new “Model-Free Control” (MFC) approach is presented in the context
of longitudinal and lateral motion of a car, leading to “intelligent controllers”, easy to implement. Simulation
results, obtained on realistic environments, have shown the efficiency of our approach.
Key Words: Automotive control, Longitudinal and lateral control, model-free control, algebraic estimation, ADAS
(Advanced Driving Assistance Systems)

1. Introduction
Our planet is confronted today with 4 major problems:
・Energy: In less than a century, man has almost

exhausted the fossil resources of the planet.
・Pollution: 1 billion cars are running, 60% of which
are in North-America, Europe and Asia (i.e. 15% of
the world population). These cars produce greenhousegases with the consequences we know about global
warming.
・Security: According to WHO data for 2015, we deplore
1.3 million deaths per year (10% in rich countries) due
to road traffic accident. Moreover, 95% of accidents are
due to driver failures.
・Congestion: There is an imbalance between supply anddemand for driving surface, hence the need to improveand regulate the use of roads.
The interest is therefore clear for the development of
new technologies for autonomous cars, or more generally
18)“intelligent transport system” (ITS) .
In Europe, the Eureka PROMETHEUS program(PROgraM for a European Traffic of High Efficiency andUnprecedented Safety) began in 1985 and was completed
in 1994, the date of the first ITS congress in Paris. This
international congress is now organized every year
in various countries of the world, and many such ITS
programs have been conducted at the European level, butalso in the USA (PATH program in California), Japan,
Korea and now in China, where more than 20 million cars
have been produced in 2014.
Moreover, the development of new localization(Global Positioning System) and navigation technologies

2(Geographic Information System and Digital Mapping
System) as well as proprioceptive sensors (accelerometers,
gyrometers, odometers, etc.) and exteroceptive sensors(embedded cameras, radars, lasers, etc.) have made theimplementation of autonomous vehicles and intelligent
transport systems possible.
Therefore, car manufacturers have developed a rangeof ADAS (Advance Driver Assistance System) to assist
the drivers, and subsequently improve safety, comfort,
efficiency and energy consumption. Many studies haveshown the positive impact of ADAS on traffic accidents
13)
and fuel consumption .
Among them, one can cite the airbag and the seat belt
which are “passive” and also the ABS (Antilock Brake
System) and the ESP (Electronic Stability Program)
16)“active”, which are preventing many crashes and which14)
are now mandatory in all new vehicles . Let mention
other optional passive and active ADAS:
・Obstacle and collision warning system: Detecting
obstacles and warning the driver about an imminentcollision (passive).
・Lane-keeping system: Warning the driver when thevehicle leaves the lane unintentionally (passive).
・Emergency braking system: Detecting obstacles
and notifying the driver about an imminent collision(passive). If the collision is deemed unavoidable, thesystem will automatically brake (active, depending on
the driving situation).
・Eco driving support system: Providing the driver with
optimal set point velocity and suitable gear selection tokeep driving in a more environmental friendly manner,
reducing fuel consumption (passive).

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−Model-Free Control of Longitudinal and Lateral Dynamics for Automated Vehicles−・Longitudinal control system: Using exteroceptive andproprioceptive sensors to regulate the car velocity andthe inter-distance with the front vehicle, through controlof the acceleration or the braking pedal (active).
・Lateral control system: Using exteroceptive andproprioceptive sensors to maintain the vehicle on thelane, or to produce a lane change to safely overtake thefront vehicle, through the control of the steering wheel(active).
In this paper, we will focus on the last two ADAS,
since these longitudinal and lateral controls improvesafety, traffic flow, comfort, consumption and reduce
safety distances, thus increasing the flow area and thetraffic flow. The vehicle longitudinal and lateral control
problem has been widely investigated in the literature1), 2), 4), 6), 12), 17), 20), 23), 24). It isvia model-based techniques
obvious that such approaches depend on the calibrationof the corresponding models. Indeed, their performances
are guaranteed whereas the model is valid, which isdifficult to guarantee. This is the reason why we haveprogressively developed a model-free approach, leading tocontrol algorithms easily implementable in an embedded
context.
I n S e c t i o n 2 , w e r e c a l l t h e evo l u t i o n o f o u rdevelopments on longitudinal and lateral vehicle control
problem, first based on a nonlinear 3DoF (Degrees
of Freedom) model. In Section 3, we present a shortsummary on the model-free control approach and then weapply it in Section 4 to the design of a longitudinal/lateralvehicle control. Finally, we give some simulation results
and words of conclusion in Section 5.

2. Previous Developments on Model-based
Longitudinal and Lateral Control
In the case of trajectory tracking implying coupledlongitudinal and lateral dynamics, we have considered thenonlinear 3DoF bicycle model, which is briefly recalledin equation (1):m（V̇ x − ẇV y ）＝ F xf ＋ F xrm（V̇ y ＋ ẇV x ）＝ F yf ＋ F yr⑴

I zẅ ＝ Lf F yf − LrF yr
where:
・m denotes the vehicle mass,
・V x the longitudinal velocity of the car, V y its lateralvelocity,・ẇ the yaw rate,
・I z the yaw moment of inertia,
・F xf and F xr the front and rear longitudinal tire forces,
・F yf and F yr the front and rear lateral tire forces,
・L f and L r the distances from the center of gravity of the

car to the front and rear axles.

JTEKT ENGINEERING JOURNAL English Edition No. 1015E (2018)

The front and rear longitudinal forces can be expressedusing the rotation dynamics of the wheels:

Fxf ＝

1
̇ f ＋Te −Tbf ）（− I rxR

1
Fxr ＝（− I rx
̇ r − Tbr ）R⑵

R denoting the wheel radius, I r the wheel inertia, x f
and x r the rotation velocity of the front and rear wheel,T e the engine torque providing the car acceleration, T bf
and T br being the braking front and rear torques.
Under the assumption of small slip angles, linear
lateral tire forces models can be considered as follows,with front and rear cornering stiffness coefficients C f and
Cr and steering angle d :F yf ＝ C f d −V y ＋ ẇLf
Vx

V y − ẇLr
F yr ＝− C rVx⑶

The vehicle model ⑴-⑶ has been shown to be*1“differentially flat . This property allowed us todesign nonlinear controllers making the tracking error9), 20)exponentially converge to zero .
Unfortunately, this control approach has been shown to
be not sufficiently robust to parameter uncertainties suchas for example tire road forces which are highly nonlinear
and difficult to estimate, as shown in Fig. 1, which
illustrates tire characteristics during an experimental
braking maneuver on a real race track. This maneuver
has pointed out the different nonlinear dynamics ofthe tire forces which cannot be well represented inthat driving situation by linear tire models, and thecornering stiffness coefficients are clearly dependent
on the longitudinal velocity of the car. Moreover, othercritical driving situations exist, for which simplified
vehicle models cannot provide a realistic behavior of theactual car to design efficient model-based controllers,such as the rollover phenomenon due to high values ofload transfer ratio (LTR), under-steering or over-steering
when braking in a turn due to high values of the front
and rear sideslip angles. For these reasons, we havethen developed a vehicle controller based on a “model7)3), 19). Despite that thefree design” but using flat outputs
corresponding numerical results were good, this solution
was suffering from the fact that one of the two flat outputs
was depending on some uncertain physical parameters,*1 Flatness is a system property which characterizes controllabilityof a nonlinear system. In a flat system, the inputs and states
can be computed from the output and a finite number of its
derivatives.

3

−Model-Free Control of Longitudinal and Lateral Dynamics for Automated Vehicles−

rather difficult to identify, such as e.g. the vehicle
mass. This implies that an accurate tracking trajectoryis not guaranteed, especially when the trajectories
are characterized by tight bends which involve highdynamical loads. So, obtaining a “good mathematicalmodel” in general driving situations is a really difficult
task, if not an impossible one. Consequently, we decided
to apply a “model-free approach” using two “natural
outputs” which can be easily computed from onboard
measurements, these outputs being the longitudinalvelocity and the lateral deviation of the car. Let firstrecall the general “model-free control approach” before
presenting the design of the resulting “intelligent
controllers” applied in the context of longitudinal andlateral control of wheeled vehicles.

・v is the time-derivation order,
・au and F have the same order of magnitude.

Until now from our knowledge, in the context ofmodel-free control, in all the existing concrete examples
v = 1 or 2.
Some comments and assumptions on F can be done:
・F is estimated via the measurements of the control
input u and the controlled output z .
・F does not distinguish between the unknown model ofthe system and the perturbations and uncertainties.
Let v = 2 in equation ⑷. It can be rewritten:

z ＝ F＋a u

The corresponding intelligent-Proportional-IntegralDerivative (i-PID) controller, reads:

u ＝−
3 000

C(f, r)1

Lateral forces, N

1 500
1 000
C(f, r)1
C(f, r)2
0

0.5

1

1.5

z ＝ F＋a u

120
100

Vx1
Vx2
0

200

400

⑻

and setting K i =0, we obtain the intelligent-Proportional(i-P) controller, which turns out until now, to be the mostuseful intelligent controller:

80
60

600

800

1 000 1 200 1 400 1 600 1 800

Distance, mu ＝−Fig. 1 Nonlinear behavior of experimental lateral tire force
characteristics

F−z ＋K p ea

3. A Background of the Model-Free
Control Approach
Model-free control was already applied and used quite7), 8), 10), 15). For
successfully in various concrete examples
obvious reasons, let insist here on its applications to5), 20), 24).
intelligent transportation systems
The input-output relation can be expressed in a small
time-interval as follows, where z is the output and u the
input variable:⑷

where a and v are constants, chosen by the practitioner,so that:⑼and combining equations ⑻ and ⑼ leads to:

K pe + e ＝ 0z ( v ）＝ F＋a u⑺

If K i =0, we obtain the intelligent-ProportionalDerivative (i-PD) controller.
Now, if we consider v = 1 in equation ⑷ we have:2

Sideslip angle, deg
Longitudinal speed, km/h

de + K p e + K i∫edt ＋K d e ＝ 0

C(f, r)2

500

4

⑹

a
d

2 000

40

F−z ＋K p e ＋K i∫edt ＋K d ewhere z is the desired signal, e =z –z the tracking
error and K p , K d and K i are the usual PID (Proportional
Integral Derivative) gains. In fact, u given by ⑹ is said an
i-PID controller since the term F is compensated. In fact,
combining equations ⑸ and ⑹ leads to:

2 500

0
−0.5

⑸

⑽

meaning that, as in equation ⑺, the tracking
error exponentially converges to 0. But, it should bepointed out that the term F in the expression of thei-PD controller ⑺ or i-P controller ⑼ should be wellknown to be compensated. But as we have mentioned
before, this term F usually corresponds to un-modeled
dynamics, perturbations or uncertainties which are notknown. Therefore, F has to be estimated. According to
the algebraic parameter identification developed in 10),11), where the probabilistic properties of the corrupting
noises may be ignored, F will be estimated on small timeintervals [t -s, t ] by a piecewise constant function Fest . For
that purpose, let us rewrite equation ⑻ in the operational
domain, where Z , U and U denote respectively thes
Laplace transform of z , u and F which is supposed to be

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−Model-Free Control of Longitudinal and Lateral Dynamics for Automated Vehicles−

constant on [t -s, t ]25) :

sZ ＝

U

s

＋aU ＋z（0）⑾We get rid of the initial condition z (0) by leftdifferentiating both sides of equation ⑾ with respect to s ,which gives:

Z ＋s

U
dZ
dU
＝− 2 ＋adsdss

u 2 ＝−

Then, noise attenuation is achieved by left-multiplying−2
both sides by s , leading to integral expressions in thetime domain (using inverse Laplace transform):（ r））d r＋ar（ s − r）u⑿

The extension to the case v = 2 is straightforward, andthe resulting estimation of F is given by the following
25)
expression :60（ s2＋ 6r 2 − 6sr ）z（ r）
Fest（t ）＝− s 5 ∫dr22（ s − r）r u（ r）d r− s5 ∫⒀

Let us now develop this model-free control approach inthe context of vehicle control.

4. Longitudinal and Lateral Model-Free
Control of the Car
As mentioned in the previous section, an appropriate
choice of inputs and outputs is required to take advantageof the model-free control approach. To avoid any
modeling problem and ensure a desired tracking forlongitudinal and lateral motions, the following input andoutput variables are selected.
・The first input u 1 is the acceleration or braking torque,and the first output y 1 is the longitudinal velocity V x ofthe car.
・The second input u 2 is the steering wheel angle, and thesecond output y 2 is the lateral deviation of the car (withrespect for example to the center of the road).
It is obvious that the second output, which is the lateraldeviation of the car, can be expressed from V x , V y and theyaw rate ẇ . This allows to include some coupling effects
between the longitudinal and lateral motions. According
to the background on the model-free setting, the above
inputs/outputs, and Newton second law, the following two
local models are considered:Longitudinal local model：ẏ 1 ＝ F1 ＋ a 1 u 1⒁

ÿ 2 ＝ F2 ＋ a 2 u 2⒂

Lateral local model：

JTEKT ENGINEERING JOURNAL English Edition No. 1015E (2018)

F 2 −y d2 ＋K p2 e 2 ＋K d2 e 2a2⒃

and equation ⒁ will be closed by an i-P controller ofdthe form ⑼ with e 1 = y 1 – y 1 :u 1 ＝−

6（（ s − 2r）z（ r）
Fest（t ）＝− s 3 ∫30aEquations ⒁ and ⒂ seem to be decoupled, but the
coupling longitudinal and lateral effects are included in
the terms F 1 and F 2. Let us also point out that equation ⒂is an order 2 formula with respect to the time derivative ofy 2, this is due to Newton second law. So equation ⒂ will
be closed by an i-PD of the form ⑹ with K i = 0 and e 2 =dy 2 – y 2:

F 1 −y d1 ＋K p1 e 1
a1⒄

Let us notice that F 1 in equation ⒄, respectively F 2 inequation ⒃, will be estimated using equation ⑿ with z = y 1
and u = u 1, respectively using equation ⒀ with z = y 2 andu = u 2.
Remark: In the above development, the inputs
have been assumed as ideal actuators (instantaneous
response to any command). For example, in a vehicle,
the steering wheel angle is controlled and consequently
has a finite time response. However, this response isassumed much faster than that of the yaw dynamics ofthe vehicle. Typically, the angle control of the steeringsystem is developed by the steering supplier. JTEKT has
developed an angle control strategy, based on a similar
control approach than that presented in this paper, where
unknown dynamics is estimated and compensated for
22)
ensuring high level of robustness .

5. Results and Conclusion
Embedding ADAS on cars is known to be a difficult
task, from a technical point of view, but also dueto the public perception and acceptance of possible
interventions as well as in reliability and safety of thesystem. In this context, simulation is a necessary stepfor validation before actual vehicle implementation. For
that purpose, many platforms have been developed, suchas the interconnected pro-SiVIC/RTMaps prototyping
platform described for example in 21). Simulation allows
to approach the reality of a situation by playing forexample with weather factors or studying the influenceand the impact of physical parameters on the robustnessof the functions used in embedded applications. Difficult
or dangerous tests are also easily accessible such as
emergency braking due to obstacle detection or roadfriction problems or road departure warning.
In this study, the simulation stage is carried out
according to the block diagram of Fig. 2, using a precise
10DoF nonlinear vehicle for simulation model, andactual data, previously recorded on a race track with a5−Model-Free Control of Longitudinal and Lateral Dynamics for Automated Vehicles−

1. Flat nonlinear control
2. Classical PID controllers
3. Model-free control

VehicleOutputsu1

Longitudinaland lateral u
2
vehicle control

ActualPID controllers
Flat control
MFC

110

10DoF
Nonlinearvehicle modelOr
Simulation
platforms
SiVIC/RTMaps

Comparison
performanceof controllers

Longitudinal speed, Km/hReference
outputs

120

100
90
80
70
60

Fig. 2 Block diagram of vehicle controllers, reference
trajectory reconstruction and vehicle models

50
40

0

200

400

600

800 1 000 1 200 1 400 1 600 1 800
Distance, m

350

Errors, m

0
−0.5
−1

0

200

400

600

800

1 000 1 200 1 400 1 600 1 800
PID controllers
Flat control
MFC

Errors, deg

4
2
0
−2
−4

0

200

400

600

800 1 000 1 200 1 400 1 600 1 800
Distance, mFig. 5 Tracking trajectory errors on lateral deviation andyaw angle

1 000
0−1 000
0

200

400

600

800 1 000 1 200 1 400 1 600 1 800
ActualPID controllers
Flat control
MFC

150

200
150
100

100
50
0
−50
0

50

200

400

600

800 1 000 1 200 1 400 1 600 1 800
Distance, m

0
−50
−200

0.5

Angle, deg

Y, m

250

1

−2 000

ActualPID controllers
Flat control
MFC

300

Fig. 4 The actual longitudinal speed versus the closed-loop
simulated longitudinal velocities

Torque, Nm

car prototype, as reference trajectories which should be
followed by the car.
The efficiency and relevance of the model-free control
as well as its performances are compared to those of
nonlinear flat control and classical PID controllers.
Two simulation scenarios are conducted for two valuesof the road friction coefficient: the first one for dryasphalt l = 1 and the second one for wet asphalt l = 0.7.
These scenarios mean that the adhesion capability of theground is reduced, thus, the vehicle maneuverability andcontrollability become more difficult.
For dry road, results are shown in Figs. 3, 4, 5 and6. These simulations demonstrate that the model-freecontrol (MFC) gives quite satisfying results, especiallyon Fig. 5 showing that the tracking errors on the lateraldeviation and on the yaw angle outputs produced by the
MFC are much better compared to those produced by theother controllers. These errors are less than 10 cm and 0.5
deg in the case of MFC. It should be noticed that the test
track which has been considered implies strong lateraland longitudinal dynamical requests. This track involves
different types of curvatures linked to straight parts, andall these configurations represent a large set of driving
situations. Finally, Fig. 6 shows that the control signals
are quite close to the actual ones provided by the driver

−100

0

100

200

300

400

500

X, m

Fig. 6 Wheel torques and steering angles control signals:
actual and simulated

Fig. 3 The reference trajectory versus the closed-loop
simulated trajectories

6

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−Model-Free Control of Longitudinal and Lateral Dynamics for Automated Vehicles−

along the track race.
Moreover, in the case of wet road, all normalized
errors are deteriorated as expected, but using the modelfree control, the maximum normalized error values areless than 3.5%, which is much better than with the othercontrollers.
Consequently, the relevance of this new model-freecontrol approach, even in extreme driving conditions, hasbeen validated. Let us also point out that the resulting
control algorithms are easy to implement and that theyuse measurements which are available on most vehicles.

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