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Experimental Validation of a Kinematic Bicycle Model Predictive Control with Lateral Acceleration Consideration

Abstract and Figures

Nowadays, Automated Driving has a growing interest in the scientific and industrial automotive community. The vehicle motion planning is an essential procedure to obtain safe and comfortable trajectories, adapting the longitudinal speed to the road legal limits and mainly to avoid the excessive lateral accelerations along the journey. Typically, the proper speed of the vehicle is intrinsically related to the curvature of the path, requiring a previous approximation of this parameter in the planning stage. In this work, a novel procedure to follow a route trajectory and speed limits considering the lateral acceleration parameter is presented. A lateral jerk equation was developed and introduced into a kinematic bicycle model predictive control formulation. An adaptive speed weight equation that depends on the lateral acceleration is presented to improve the lateral positioning. A vehicle motion control simulation, developed in Dynacar, is validated with some real tests. The results show the capabilities of the proposed approach. An accurate vehicle motion control considers the lateral acceleration to avoid unfeasibility in optimization problem.
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Experimental Validation of a Kinematic
Bicycle Model Predictive Control with
Lateral Acceleration Consideration ?
Jose A. Matute ,∗∗ Mauricio Marcano ,∗∗ Sergio Diaz
Joshue Perez
Tecnalia Research and Innovation, Derio, 48160 Spain (e-mail:
joseangel.matute@tecnalia.com).
∗∗ University of the Basque Country, Bilbao, 48013 Spain (e-mail:
author@ehu.eus)
Abstract: Nowadays, Automated Driving has a growing interest in the scientific and industrial
automotive community. The vehicle motion planning is an essential procedure to obtain safe
and comfortable trajectories, adapting the longitudinal speed to the road legal limits and
mainly to avoid the excessive lateral accelerations along the journey. Typically, the proper
speed of the vehicle is intrinsically related to the curvature of the path, requiring a previous
approximation of this parameter in the planning stage. In this work, a novel procedure to
follow a route trajectory and speed limits considering the lateral acceleration parameter is
presented. A lateral jerk equation was developed and introduced into a kinematic bicycle model
predictive control formulation. An adaptive speed weight equation that depends on the lateral
acceleration is presented to improve the lateral positioning. A vehicle motion control simulation,
developed in Dynacar, is validated with some real tests. The results show the capabilities of the
proposed approach. An accurate vehicle motion control considers the lateral acceleration to
avoid unfeasibility in optimization problem.
Keywords: Intelligent Control, Path Planning, Intelligent Transportation Systems.
1. INTRODUCTION
In Automated Driving, the vehicle motion planning is
an essential procedure to obtain a safe and comfortable
driving experience in automated mode. Researchers have
been putting efforts on this area proposing the use of
smooth curves for obstacle free trajectories, and speed
profiles dependent on curvature to reduce discomfort in
bends and lateral maneuvers (Lattarulo et al., 2018).
However, comfort should not be considered in the planning
stage alone, because it assumes that the controller will
follow the references perfectly. Instead, it must be consid-
ered together with the control aspect of driving. Although
comfort in a dynamic driving task is not formally defined
(Bellem et al., 2018), it is related in practice with variables
such as jerk and acceleration (Bautista, 2017), which are
characteristics of the control stage.
Nonetheless, not all control techniques allow to consider
these vehicle states in the design process, e.g. classical
controllers. In contrast, Model Predictive Control (MPC)
technique allows to combine different objectives functions
?This project has received funding from the Electronic Component
Systems for European Leadership Joint Undertaking under grant
agreement No 737469 (AutoDrive Project). This Joint Undertaking
receives support from the European Union Horizon 2020 research and
innovation programme and Germany, Austria, Spain, Italy, Latvia,
Belgium, Netherlands, Sweden, Finland, Lithuania, Czech Republic,
Romania, Norway. This work was developed at Tecnalia Research &
Innovation facilities supporting this research.
that pursue performance, safety and comfort. In (Matute
et al., 2018), MPC is implemented as a speed controller
with constraints of longitudinal jerk and acceleration,
using a triple integrator model. However, it failed on
considering the lateral actions of the vehicle, which have
a high impact on passenger comfort. Other works (Mata
et al., 2017; Kong et al., 2015), include a kinematic bicycle
model to couple the lateral and longitudinal control of
the vehicle in a single optimization problem. However,
when different objectives functions compete (e.g. increase
tracking performance, reach a speed set point and reduce
steering rate) the optimization problem may be unfeasible,
producing unexpected behaviors on the vehicle motion and
compromising safety.
In this sense, (Polack et al., 2017) states that keeping the
lateral acceleration under a threshold of 0.5gguarantees a
feasible solution when a kinematic bicycle model is used.
Considering this, the novelty of the present work relies on
the inclusion of the lateral jerk formula within the MPC
formulation. This consideration allows a new state con-
straint limiting the lateral acceleration to assure a feasible
solution. This also permits the use of simple vehicle model
which require less computational effort. Additionally, an
adaptive speed weight equation that depends on the lat-
eral acceleration is also included to improve the tracking
performance of the vehicle.
This work is organized as follows. In Section II, the control
architecture developed is described in detail. Section III,
Fig. 1. Control architecture
discuss the results obtained in simulation and experimen-
tal tests. Lastly, Section IV presents final conclusions and
future research work.
2. CONTROL ARCHITECTURE
This section describes the algorithm developed for path
tracking applications based on a vehicle predictive model.
Using the state of the vehicle, the trajectory planning
determines the references for the longitudinal and lateral
vehicle motion control, estimating and defining the com-
mand positions for the pedals and steering wheel. This
approach was developed as a general framework for path
following in automated driving applications. The aim is to
follow a predefined route with the maximum level of safety
and desired comfort.
In this work, the same Dynamic Driving Task (DDT) is
defined for both virtual and real situations as depicted in
the Fig. 1. The problem constraints are stated according
with the physical capabilities of the real actuators installed
in the vehicle, considering the maximum displacements
and change rates.
2.1 Trajectory Planning
The references to be followed by the vehicle motion control
are defined performing a strategy of offline and online
stages described by (Matute et al., 2018).
The offline stage contains relevant information about the
predefined map for path tracking. The procedure to gen-
erate the route used in this work is broadly described by
(Lattarulo et al., 2018). The route contains several kinds
of traffic maneuvers very usual in urban environments as
roundabouts, intersections and lane changes. Depending
on the maneuver to be performed, the path is built using
ezier curves of fourth order or higher using hard points
as those depicted in the legend of the Figure 2. From
these hard points, it is possible to obtain smooth routes
with continuous curvatures described by an ordered list
of waypoints which contain relevant information for path
tracking as; locations in Cartesian coordinates (Xand Y),
orientations (Ψ), speed limits (vx), curvatures and traveled
distance from the starting point of the map. Moreover,
physical constraints for vehicle motion control are speci-
fied.
The online stage uses the current and future locations of
the vehicle to project them to the nearest sections of the
route using the map developed in the offline stage. The
location of the vehicle is projected onto the map using dot
Fig. 2. Route for real and simulated tests
Fig. 3. Route references of curvature and velocity
products to obtain the references to perform the lateral
vehicle motion control as shown in the Fig. 3. The speed
reference is generated considering that the lateral accelera-
tion never exceeds 1.0m/s2—, fairly uncomfortable-based
on the ISO 2631-1 Standard (ISO, 1997). The speed limit
of this section is used as reference parameter to perform
the longitudinal vehicle motion control.
It is important to note that the planned route has curva-
tures higher than those generally expected in real traffic
scenarios. A path radius lower than 6.67m (k > 0.15) is
usually considered for slow speed maneuvers. This map is
generated considering the space available for real tests.
2.2 Vehicle Motion Control
This DDT sub-task includes the maintaining of the speeds
below specified limits applying propulsion or braking in-
puts, as well as maintaining an appropriate lateral posi-
tioning through the application of steering inputs.
The sustained regulation of the x-axis and y-axis compo-
nents of vehicle motion depicted in Fig. 4 are performed
using a coupled MPC strategy.
The lateral acceleration is considered as an additional
parameter to improve the lateral positioning, defining also
a desired level of comfort during the path tracking on curve
roads.
Lateral Acceleration Consideration Considering small
time steps for the computation of the vehicle’s motion,
its lateral acceleration ayis approximated as a uniform
circular motion expressed as ay=v2
x/R, where vxis the
vehicle’s longitudinal speed and Ris the path radius.
From the bicycle model assumption considering a front
wheel steered vehicle, it is possible to approximate the
path radius as R=L/δ for small values of slip angle
Fig. 4. Simplified kinematic bicycle model
(β) (Rajamani, 2011). Therefore, the lateral acceleration
equation can be expressed as shown in the Eq. 1.
ay=v2
xδ
L(1)
This formula is later on differentiated to be included as a
differential state variable into the kinematic bicycle MPC
formulation as depicted in the Eq. 2e.
Kinematic Bicycle Model The MPC formulation consid-
ers a kinematic bicycle model of the vehicle as shown in
Fig. 4. The equations are based on geometric relationships
without considering any forces. The two left and right
wheels are represented as two single wheels at points A
and B, representing the front and rear ends of the vehicle
wheelbase L. The center of gravity of the vehicle is at point
Cand defines the location of the vehicle.
The vehicle is assumed to have a planar motion. The
location coordinates and orientation (X, Y , Ψ) describe the
motion of the vehicle. The longitudinal speed (vx) and
lateral acceleration (ay) are parallel and perpendicular
to the longitudinal axis of the vehicle, respectively. The
longitudinal jerk (jx) and the steering change rate (∆δ)
are the two inputs on this model.
˙
X=vxcos(Ψ) (2a)
˙
Y=vxsin(Ψ) (2b)
˙
Ψ = vxtan(δ)
L(2c)
˙vx=ax(2d)
˙ay= (2axδ+vxδ)vx
L(2e)
˙ax=jx(2f)
˙
δ= ∆δ(2g)
MPC Formulation The nonlinear vehicle dynamics for-
mulation described in Eqs. 2a-2e can be described in the
general compact form depicted in the Eq. 3.
dt =f(χ(t), u(t)) (3)
where the differential state and control parameters are
χ= [X, Y, Ψ, vx, ay]Tand u= [ ˙ax,˙
δ]T, respectively.
The control strategy consists to track a desired trajectory
under specific speeds considering the current lateral accel-
erations of the vehicle. The output differential states to be
optimized are defined in the Eq. 4. The term of ayis not
optimized being only used as a constraint.
η(k) = h(χ(k)) =
10000
01000
00100
00010
χ(k) (4)
The cost function for the optimization is in the Eq. 5.
J(χ(t), u(t)) =
H
X
i=1
kηt+i,t ηref
t+i,tk2
Q+kut+i,tk2
R(5)
where η= [X, Y, Ψ, vx]Tand ηref contains the refer-
ences for each of the differential states to be optimized,
all of them related with information provided from the
trajectory planner. At the right side of the Eq. 5, the
first term denoted the penalty on the differential states
and the second one measures the control variables. The
weight matrices are defined intuitively in order to pro-
vide a balance between performance and smoothness as
Q=diag([1,1,1, S ]) and R=diag([1,0.01]), where Qand
Rare the weight matrices for the differential states and
control parameters, respectively. The term Swithin the
matrices is related to the weight of the longitudinal veloc-
ity, which changes according to the value of the current
lateral acceleration as is explained later in the subsection
2.2.4. The finite horizon of the Optimal Control Problem
(OCP) formulation is solved at each step tas:
min
η(·),u(·)J(χt, ut) (6a)
s.t. χk+1,t =f(χk,t, uk ,t) (6b)
ηk,t =h(χk,t ) (6c)
k=t, ..., t +H(6d)
ζf,min ζk,t ζf ,max (6e)
uf,min uk,t uf ,max (6f)
χt,t =χ(t) (6g)
where ηt+i,t is the optimization vector at time t+i
beginning from state χt,t =χ(t). The differential states
prediction horizon is defined as H. The OCP is solved at
each time step and a new His generated in according
to χ(t+ 1). The sequence for the solution is settled as
k=t, ..., t +H. The constraints of the problem are defined
by ζk,t and uk,t.
Adaptive Speed Weight The highest errors in path fol-
lowing occur before, during and after the vehicle makes a
turn. An adaptive speed weight in the OCP formulation
highly contributes to a more accurate trajectory tracking
on this case. The aim is to reduce the weight of the speed as
the lateral acceleration is increasing. This strategy permits
to give more importance to the location of the vehicle,
reducing the lateral and angular errors when the vehicle is
taking a curve. A proposal to obtain an adaptive weight
(n= 2) is presented in the Eq. 7.
S= 1 |ay|n
max(ay)(7)
where ayis the current laterals acceleration of the vehicle
and max(ay) is the maximum lateral acceleration permit-
ted and settled for driving constraints. The value remains
always among 0 and max(ay).
Driving Constraints The bounds for the differential
states (ζk,t) and control (uk,t ) are selected considering the
physical capabilities of the vehicle and actuation devices,
as well as comfort level to be perceived by the passenger.
The constraint values are depicted in the Table 1.
Table 1. Driving constraints
Parameter Min. Max. Unit
States ay
k,t -2 2 m/s2
vx
k,t 0vref
k,t m/s
ax
k,t -10 1 m/s2
δk,t -0.69 0.69 rad
Controls jx
k,t -0.50 0.50 m/s3
δk,t -0.50 0.50 rad/s
According to (Polack et al., 2018), values of lateral ac-
celeration below 0.5gavoid the appearance of unfeasible
results in the solution of the OCP when the kinematic
bicycle model is employed in vehicle motion control.
MPC Solver The open-source ACADO Toolkit is used
to solve the OCP (Quirynen et al., 2015). The OCP is
reformulated to an approximate nonlinear program (NLP)
using a direct multiple shooting discretization method.
The generalized Gauss-Newton approximation iterates by
solving the Sequential Quadratic Program (SQP) algo-
rithm to solve the NLP. The SQP is then solved by the
dense linear algebra solver qpOASES3. A continuous out-
put Implicit Runge-Kutta of Gauss-Legendre integrator of
order 2 is exported by the code to simulate the system with
20 integration steps. The Hestimation is parameterized to
obtain 10 elements discretized at 300ms.
2.3 Test Vehicle
The test vehicle selected for real and simulated exper-
iments is a Renault Twizy E80, an electric quadricycle
which technical specifications are depicted in the Table 2.
A multi-body model is developed in the software Dynacar
and is employed to simulate the behavior of the vehicle
as explained in (Lattarulo et al., 2017). The power-train,
brakes and steering are characterized to resemble the real
devices as detailed in (Marcano et al., 2018).
Instrumentation and Data-acquisition The vehicle has
been instrumented as shown in the Fig. 5 to carry out
the experiments. A Global Navigation Satellite System
plus an Inertial Navigation System device (GNSS+INS)
is used to obtain Real-Time Kinematic positioning with
a position accuracy of 2cm. A high-performance personal
computer (PC) runs the control architecture software un-
der MATLAB/Simulink environment sending the control
commands to an industrial Programmable Logic Con-
troller (PLC).
The PLC is connected through a CAN bus network to
the low-level control of the vehicle. The throttle pedal
Fig. 5. Test vehicle instrumentation
has a parallel connection that simulates the pedal position
for propulsion signals when the vehicle is in automated
mode. The brake pedal is physically connected with a
rotary servo motor through a steel cable. The steering
wheel is also physically connected with a rotary servo
motor through a synchronous belt though. Both brake and
steer have position controllers that receive and execute the
control from position commands delivered by the PLC.
An encoder is connected to the rotary servo motor of the
steering wheel to deliver the position feed-back to the PLC
being useful as an MPC differential state.
The sampling process for all the measurements in the real
platform are synchronized with the GNSS+INS with a
frequency 100Hz, the same used in the simulated platform.
3. RESULTS AND DISCUSSION
The same tests were performed in both the simulator
and the physical vehicle, following the route depicted on
Fig. 2. This test circuit represents several traffic urban
conditions, with different curvature radius and reference
speeds (see Fig. 3). In this section, the behavior of the
kinematic bicycle MPC under such conditions is evaluated
and validated both in simulation and in actual vehicle
tests. Note that the same controller is used in both cases.
The computational time of the control cycle was 1ms
in average both in simulation and real-life experiments,
proving the applicability of the algorithm aa well as to be
efficiently enough for real-time applications.
Table 2. Vehicle technical specifications
Parameter Value Unit
Mass 611.50 kg
Dimensions 2.34 x 1.23 x 1.45 m
CG location -0.93 x 0.00 x 0.49 m
Wheelbase 1.69 m
Track-width 1.09 m
Inertia 243.18, 430.17, 430.17 kg-m2
Front wheel radius 0.27 m
Rear wheel radius 0.28 m
Steering ratio 14.27:1 -
Traction torque 57 N-m
Transmission ratio 1:9.23 -
Braking torque 500 N-m
Fig. 6. Command signals to actuation devices
Pedal and steering wheel actuation is portrayed in Fig.
6. First, note the similarity between simulations and ex-
periments. However, it is also noted that a significantly
stronger brakes application is required prior to curve en-
trances for the experimental case. This might be related to
the fact that brakes are simulated with “as new” parame-
ters, while in the experimental bench brakes are actually
nearing the end of their life-cycle. It is also apparent that,
in the straights, the experimental setup requires more
acceleration (i.e. torque) than the simulation. Steering
wheel actuation in the actual vehicle closely resembles
that obtained from the simulations, both in amplitude and
phasing.
Some oscillations not seen on the simulation do appear
on the experiments, both for the steering and pedal
positions. It is speculated that such oscillation might
be related to the behavior of the low-level control and
actuators, particularly on the pedals, which might present
delays, free play, and hysteresis not fully modeled in
the simulations. Recall that the lateral and longitudinal
motions are coupled both by the vehicle dynamics and by
the implemented MPC.
Fig. 7 shows longitudinal and lateral accelerations. Again,
the experiments follow closely the simulations, fully com-
plying with the dynamic and comfort bounds. The con-
troller is able to adjust pedal inputs on the actual vehicle
to compensate for the extra braking and motor torque re-
quirements (as discussed above) in order to follow the cir-
cuit defined path with the same dynamic behavior than the
simulated vehicle (showing similar longitudinal and lateral
accelerations). As expected, the oscillations observed on
the actuators for the experimental bench are also evident
on the accelerations. As shown by the present tests, it is a
bound effect and the system can acceptably work with it.
Further research will be performed to reliably determine
the source of this oscillation and to establish the most
appropriate means to deal with it. It will be investigated
whether an improved actuator design and/or a purposely
tuned control can reduce the reported oscillation.
Lateral and heading errors along the circuit are shown
in Fig. 8, and Fig. 9 depicts statistical parameters of
the magnitude (i.e. absolute value) of such errors. Lateral
error is generally smaller for the simulated case, as the
Fig. 7. Longitudinal and lateral accelerations
Fig. 8. Errors and speed references
experimental case is affected by the aforementioned os-
cillations. Bias is inverted, as the simulated vehicle tends
to turn on the inside of the reference trajectory (lateral
error of the same sign of the lateral acceleration) while
the actual vehicle tends to turn on the outside of the
reference (lateral error of opposite sign to the lateral ac-
celeration). On the other hand, heading error (estimated
as the vehicle attitude angle with respect to the reference
trajectory) is generally smaller for the experimental test
than for the simulation. The oscillation phenomenon is
again clearly observed in the experimental case, with the
angular error fluctuating around zero. The behavior on
the higher curvature turns exposes the most notorious dif-
ference. On the simulated tests, the vehicle heading error
grows significantly with the curvature, with the vehicle
pointing outwards of the trajectory, as expected for an
under-steering vehicle. Whereas, for the experimental tests
the heading error oscillates around zero, a behavior closer
to a neutral steering (or less under-steering) vehicle. It is
also worth noting that, when the tires are more loaded
laterally, a high frequency component does appear on the
heading error both for the simulation and the experimental
tests.
Fig. 9. Lateral and angular errors
Fig. 8 also shows speed limits for comfort and the actual
speed profiles for simulation and experiments. When seen
alongside the acceleration profiles of Fig. 7, it can be
concluded that the proposed control strategy can satis-
factorily fulfill speed, safety and comfort requirements in
both plants (the simulated and the experimental one).
4. CONCLUSIONS AND FUTURE WORK
A novel procedure for the design of a safe and comfortable
vehicle motion controller, considering parameters such
as jerk and accelerations has been presented. The main
contribution of this work is the MPC controller based
on the extended kinematic bicycle model which achieves
real time performance in urban conditions. Specially, the
lateral jerk formula is a novelty in this work allowing
to include the lateral acceleration as a new constraint.
This addition has been shown to be beneficial to avoid
unfeasible solutions in the optimization problem.
Our approach has been validated in a high-fidelity dynamic
simulator (Dynacar) and an instrumented platform (Re-
nault Twizy) in real urban scenarios. This controller gave
good results for different maneuvers, including: intersec-
tions, roundabouts and lane change.
The lateral and heading errors show a good performance
in both simulation and real tests. The method proposed
in the paper, can also be employed to resolve optimal
controllers in an efficient manner using available MPC
solvers for Matlab/Simulink. The easy integration of the
controllers first tested in simulation and then in the vehicle
platform shows that this controller is suitable for real
vehicles with available commercial sensors.
Although small differences are present between simulations
and experiments, e.g. oscillations and command signals
amplitude (see Fig. 6). It is worth noticing that the
proposed control architecture based on MPC is available
to handle platforms even with small behavior differences,
while achieving the objectives of performance, safety and
comfort (see Figs. 7-8).
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... It provides appropriate performance at low speed (less than 5 m/s) when tire deformations are small and slips angles on the wheels can be neglected [11]. Experimental validations have shown a good performance of model-based controllers at low speeds [12,13]. However, when the lateral forces on tires increase (e.g., while turning at high speeds), its accuracy is compromised [14,15]. ...
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Model-based trajectory tracking has become a widely used technique for automated driving system applications. A critical design decision is the proper selection of a vehicle model that achieves the best trade-off between real-time capability and robustness. Blending different types of vehicle models is a recent practice to increase the operating range of model-based trajectory tracking control applications. However, current approaches focus on the use of longitudinal speed as the blending parameter, with a formal procedure to tune and select its parameters still lacking. This work presents a novel approach based on lateral accelerations, along with a formal procedure and criteria to tune and select blending parameters, for its use on model-based predictive controllers for autonomous driving. An electric passenger bus traveling at different speeds over urban routes is proposed as a case study. Results demonstrate that the lateral acceleration, which is proportional to the lateral forces that differentiate kinematic and dynamic models, is a more appropriate model-switching enabler than the currently used longitudinal velocity. Moreover, the advanced procedure to define blending parameters is shown to be effective. Finally, a smooth blending method offers better tracking results versus sudden model switching ones and non-blending techniques.
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We study the use of kinematic and dynamic vehicle models for model-based control design used in autonomous driving. In particular, we analyze the statistics of the forecast error of these two models by using experimental data. In addition, we study the effect of discretization on forecast error. We use the results of the first part to motivate the design of a controller for an autonomous vehicle using model predictive control (MPC) and a simple kinematic bicycle model. The proposed approach is less computationally expensive than existing methods which use vehicle tire models. Moreover it can be implemented at low vehicle speeds where tire models become singular. Experimental results show the effectiveness of the proposed approach at various speeds on windy roads.
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Comfort in automated driving: An analysis of preferences for different automated driving styles and their dependence on personality traits
  • H Bellem
  • B Thiel
  • M Schrauf
  • J F Krems
Bellem, H., Thiel, B., Schrauf, M., and Krems, J.F. (2018). Comfort in automated driving: An analysis of preferences for different automated driving styles and their dependence on personality traits. Transportation research part F: traffic psychology and behaviour, 55, 90-100.
Urban motion planning framework based on n-bézier curves considering comfort and safety
  • R Lattarulo
  • L Gonzalez
  • E Marti
  • J Matute
  • M Marcano
  • J Perez
Lattarulo, R., Gonzalez, L., Marti, E., Matute, J., Marcano, M., and Perez, J. (2018). Urban motion planning framework based on n-bézier curves considering comfort and safety. Journal of Advanced Transportation, 13 pages. doi:https://doi.org/10.1155/2018/6060924. Article ID 6060924.