Abstract—In this paper, a multivariable nonlinear model predictive control strategy is proposed. The strategy is aimed to control the lateral and longitudinal vehicle dynamics, using as control variables the front wheels steering angle and front and rear wheels rotation speed. The control system is aimed to automatically drive the vehicle along a desired trajectory. For designing the nonlinear predictive controller, a nonlinear vehicle dynamic model is used that captures with higher accuracy the dynamics of a real vehicle. The proposed approach is validated using simulation results and is shown that this approach could provide good performances in practical use.
Today, the complete autonomous vehicle is in the focus of both industry and academic research, being one of the greatest challenges in the automotive domain. The difficulty of bringing this concept to production comes from the fact that the vehicle controllers need to behave as similar as possible to a human driver, in all driving situations. As technology complexity is increasing, all vehicles today are already equipped with advanced systems for driver assistance, comfort and especially safety. One of the tasks in developing the autonomous vehicle is to interconnect and merge all these systems together, and select the best suited control system for fulfilling the driving requirements. In parallel, standards and regulations are starting to be developed with explicit focus on the autonomous vehicles.
Studies for autonomous vehicles have already started for some years and promising concepts were defined, such as car following (or convoying) [1], lane tracking and changing, emergency stopping, and obstacle avoidance [2]. Also in the anticipation of these developments, the automotive industry has announced concepts that make use of the environment such as communication between vehicles and infrastructure [2].
In this work, a multivariable nonlinear model predictive control strategy is developed for the control of vehicle dynamics. At this point, the model predictive control seems to be the most attractive control method for autonomous vehicles, as it incorporates the "predictive behavior", giving the
possibility to take into account the information that is available in front of the vehicle along its path. Studies of applying predictive control methods for autonomous vehicles are already ongoing for some years and interested readers may refer to [3], [4], [5], [6], [7], [8], [9] and [10]. However, the control of both longitudinal and lateral vehicle dynamics is still a challenging task due to the high coupling between them. In [9], the proposed approach is to design a nonlinear predictive controller for vehicle steering, and a separate nonlinear controller for the vehicle longitudinal speed. In [4] and [10] a piecewise linear vehicle model is used, in order to develop a predictive controller for the vehicle dynamics.
In order to automatically control a vehicle along a desired trajectory in a centralized manner, both lateral and longitudinal dynamics need to be controlled. As such, a multivariable control system needs to be developed. The proposed control method is used to control the lateral and the longitudinal vehicle dynamics, using the front wheels steering angle, and the rear wheels speed respectively, showing good performance, despite the high coupling between the longitudinal and lateral dynamics. The proposed solution is considering only constant reference for vehicle speeds.
In Section II the mathematical modeling of vehicle lateral dynamics and longitudinal dynamics is introduced, the nonlinear model based predictive control method of the vehicle dynamics is described in Section III. Simulation results are presented in Section IV, and the conclusions are drown in Section V.
II. VEHICLE MODELLING
Usually, three types of vehicle models are employed in applications for driver assistance, vehicle stability control, path planning and autonomous driving. These model types are defined as: vehicle point-mass models [2], [5], [6], which consider the vehicle reduced to a particle with mass; vehicle kinematic models [5], [10], [11] which are more complex in comparison with the point-mass models, and arise from the geometric relationships of the vehicle's degrees of freedom; dynamic models [3], [4], [5], [7] and [11], which describe the vehicle behavior based on the motion equations and vehicle tires interaction with the road surface. In [5] all these types of vehicle models are presented. Depending on the application,
97
the appropriate type of model needs to be chosen, taking into consideration the performance requirements and computing effort.
In this contribution a vehicle bicycle model (or single track model) is considered, due to its capability to capture the most relevant dynamics of the vehicle. This model considers the front wheels merged into only one front wheel and the same assumption is done for the rear wheels. In Fig. 1 the vehicle bicycle model is illustrated.
The vehicle dynamics can be obtained by applying Newton's second law of motion:
2( ),
2( ),
2( ),
xf xr
yf yr
yf rf
mx my F F
my mx F F
I aF bF
ψψ
ψ
= + +
= − + +
= −
(1)
where xfF , xrF are the front and rear tire longitudinal forces,
and yfF , yrF are the front and rear tire lateral forces. The three
degrees of freedom considered in (1) are the longitudinal position x , the lateral position y and the yaw angle ψ . The parameters m , I , a and b are vehicle parameters and represent the vehicle mass, vehicle inertia, distance from the front axle to the centre of gravity and distance from the rear axle to the centre of gravity, respectively.
The tire forces have the most significant impact in the vehicle dynamics. The longitudinal tire forces xfF , xrF can be
modeled as functions of the longitudinal slip and the lateral tire forces yfF , yrF
can be modeled as functions of the slip angle.
Fig. 1. Vehicle bycicle model
For defining the slip quantities, it is considered that the vehicle is steered only with the front wheels, and driven by the front and rear wheels. It follows that during acceleration, the longitudinal slip of the rear wheel is defined as:
,e r
re r
x R
R
ωκω
−= −
(2)
where rω is the rear wheel rotational speed and eR is the wheel effective radius. Similarly, the longitudinal slip of the front wheel is
,
e ff
e f
x R
R
ωκ
ω−
= −
(3)
where fω is the front wheel rotational speed. It can be seen
form (2) and (3) that the front and rear wheels are considered to have the same radius eR .
The slip angles for the front and rear wheels are given by:
sin ( )costan ,
cos ( )sinfx y a
x y a
δ ψ δαδ ψ δ
− + +=+ +
(4)
tan ,r
y b
x
ψα −=
(5)
where δ in (4) is the front wheel steering angle.
The tire lateral and longitudinal forces can be modeled using the well known Pacejka model or Magic Formula Tire Model [12], [13], which is a semi-empirical model providing high accuracy simulations of the tire-road contact forces. Based on this model, the tire forces are expressed as:
( , , ),xf xf f f zfF f Fα κ=
(6)
( , , ),xr xr r r zrF f Fα κ= (7)
( , , ),yf yf f f zfF f Fα κ=
(8)
( , , ).yr yr r r zrF f Fα κ=
(9)
The forces zfF and zrF in (6) - (9) represent the normal
forces of the front and rear wheels. Assuming that the vehicle is driving on a perfectly flat surface, and neglecting the aerodynamic forces acting on the vehicle body, zfF and zrF
can be written as:
,
2( )zfmgb
Fa b
=+
(10)
98
,
2( )zrmga
Fa b
=+
(11)
The vehicle coordinates in the absolute inertial frame can be obtained using the following equations:
sin cos
cos sin
Y x y
X x y
ψ ψψ ψ
= +
= −
(12)
where X represents the vehicle longitudinal position in the inertial frame, and Y represents the lateral position in the inertial frame.
The vehicle model described by (1) trough (12) can be written in the nonlinear state-space form as:
( , , , ),f rz zφ δ ω ω=
(13)
where [ ], , , , , , ,T
z x x y y X Yψ ψ= is the state vector, and δ ,
fω and rω are the front wheel steering angle, front wheel
rotational speed and rear wheel rotational speed, and φ is a function which describes the nonlinear vehicle dynamics.
The state-space model (13) is used in the vehicle dynamics controller, considering as input signals the steering wheel angle δ , the front and rear wheels rotational speeds fω , rω . For
simplicity the control design was implemented considering
f rω ω= .
III. VEHICLE DYNAMICS CONTROL
As presented above, the proposed control algorithm for the vehicle dynamics is a multivariable nonlinear predictive control strategy.
The model based predictive control (linear or nonlinear) is a well known digital control method, based on calculating an optimal control law, which will drive the controlled system states over a future horizon to a prescribed reference. In order to calculate the optimal control signal, the controller uses an internal model of the process, based on which the future states are predicted. Using these estimates of the future states, the controller iteratively searches for an optimal control signal that will minimize a predefined cost function. These calculations are executed by the predictive controller at each sample time, and only the first value of the obtained control signal sequence is actually applied to the process. A graphical representation of the predictive control strategy for a SISO system is given in Fig. 2. The cost function used for the optimization algorithm usually includes the difference between the reference and the predicted output signals, but it can also include terms containing the control signal and other variables. For details regarding the predictive control algorithm interested readers may refer to [15] and [16].
Fig. 2. The model based predictive control principle
In Fig. 2, t represents the discrete sample time, N represents the prediction horizon over which the future output is estimated, w , y and u are the future reference, the future output and the future optimal control signal. The notation ( | )t k t+ represents the future estimated value of the respective
signal at future sample time t k+ , calculated at present sample time t , and 1,...,k N= . This notation will be used in what follows.
For implementing the model predictive control for the lateral and longitudinal motion of the vehicle, a discrete approximation of the nonlinear state-space model (13) was used, of the form:
( 1) ( ( ), ( ), ( ), ( ))d f rz k z k k k kφ δ ω ω+ = (14)
The control signals that are considered for the multivariable predictive controller design are the front wheel steering angle δ , the front and rear wheel speeds fω and rω (which for
simplicity are considered equal), while the inertial vehicle coordinates X and Y are considered the output signals.
It is assumed that the outputs are measurable by appropriate sensors, also the vehicle future desired trajectory is already known, and methods for resolving these aspects will remain subject to future contributions.
In order to obtain the desired multivariable predictive controller a multi-objective cost function is defined in the form:
2
1
2
1
( , ) ( ( ) ( | ))
( ( ) ( | ))
X
Y
h
refi
h
refj
J X Y X k i X k i k
Y k j Y k j k
=
=
= + − + +
+ + − +
(15)
where refX is the future desired longitudinal trajectory, refY is
the future lateral desired trajectory, Xh and Yh are the prediction horizons of the longitudinal and lateral positions. In what follows, these prediction horizon parameters are considered X Yh h h= = .
99
The cost function (15) is subject to bound constraints defined for the control signals:
[ ] ( 1 | ) [ ]
2 2
0 [ / sec] 40 [ / sec]r
rad k i k rad
rad rad
π πδ
ω
− ≤ + − ≤
≤ ≤ (16)
Due to the high coupling between the vehicle longitudinal and lateral dynamics, which can be seen in (1), the minimization of the cost function (15) subject to (16) can fail causing suboptimal control or even instability. The vehicle dynamics coupling issue is solved by structuring the control signal for the vehicle speed, i.e. rω . This is done by using the
control horizon hω parameter, after which, the control signal will remain constant along the prediction horizon, thus
( | ) ( 1 | ), ,...r rk i k k h k i h hω ωω ω+ = + − = (17)
By considering 1hω = , the optimization algorithm which will minimize the cost function (15), subject to (16) will be forced to search for a solution having ( 1 | )r k i kω + − ,
2,...,i h= constant starting from the next sample in the future, and only ( 1 | )k i kδ + − , 1,...,i h= can vary along the
prediction horizon. Thus, the only change in both rω and δ signals will be done in the sample which is actually applied to the vehicle ( ( | )r k kω and ( | )k kδ ), after which, the optimization algorithm will search for the future values of δ while rω remains unchanged. A graphical representation of the proposed control signal structuring can be seen in Fig. 3. For sufficiently small sample times and prediction horizon, the proposed control signal structuring ensures vehicle lateral and longitudinal dynamics decoupling, while maintaining the optimal control.
The obtained closed loop structure for the proposed control system is presented in Fig. 4.
Fig. 3. Control signal structuring with control horizon 1hω =
Fig. 4. Block diagram of closed loop system
IV. SIMULATION RESULTS
As previously presented, the vehicle model used internally in the predictive controller is (14), and the considered cost function is (15), subject to (16) and (17). The prediction horizon was considered 10h = , and the sample time was set to
0.01 [sec]sT = . No modeling errors were considered, i.e. the vehicle model is considered identical with the controlled vehicle. Also no external disturbances such as road bumps or side wind were considered.
The objective is to automatically control the vehicle longitudinal and lateral motion, such that a desired trajectory is followed as close as possible. In Fig. 5, an autonomous obstacle avoidance maneuver is presented. The reference trajectory is generated for constant, vehicle speed
11 [ / ]V km h= . It can be seen that the vehicle is following closely the desired trajectory. The initial wheel speed is considered to be 10 [ / sec]r radω = , which is equivalent with
a vehicle speed of 11 [ / ]V km h= .
0 5 10 15 20 25Time [sec]
0
0.5
1vehicle trajectory reference trajectory
0 5 10 15 20 25Time [sec]
-0.05
0
0.05
0 5 10 15 20 25Time [sec]
0
5
10
Fig. 5. Obstacle avoidance maneuver at 11 [ / ]V km h=
In Fig. 6, Fig 7, and Fig 8 similar maneuvers are illustrated, where the reference trajectory is considered for higher vehicle speed values, i.e. 17.5 [ / ]V km h= , 22.3 [ / ]V km h= and
37 [ / ]V km h= , respectively. Also for higher vehicle speed it
100
can be seen that the proposed controller is showing very good trajectory tracking performance.
0 5 10 15 20 25Time [sec]
0
1
2
3vehicle trajectory reference trajectory
0 5 10 15 20 25Time [sec]
-0.4-0.2
00.2
0 5 10 15 20 25Time [sec]
0
10
20
Fig. 6. Obstacle avoidance maneuver at 17.3 [ / ]V km h=
0 5 10 15 20 25Time [sec]
0
2
4vehicle trajectory reference trajectory
0 5 10 15 20 25Time [sec]
-0.1
0
0.1
0 5 10 15 20 25Time [sec]
0
10
20
Fig. 7. Obstacle avoidance maneuver at 22.3 [ / ]V km h=
In Fig. 6, the initial vehicle wheel speed was considered 10 [ / sec]r radω = , and is increased to 17 [ / sec]r radω = ,
thus the predictive controller is stable, even if the initial value of the vehicle speed is not optimal, but close enough to the optimal value. Small oscillations can be observed in the steering wheel angle control signal, but no constraints violation occurs.
In all presented simulation experiments, the reference trajectory was generated, such that the steering wheel angle δ would have the same optimal evolution, in order to check if the proposed algorithm can find this value always, at different vehicle speed. As it can be seen, all results presented in Fig. 5 to Fig. 8, show that the steering wheel angle δ has a similar evolution and differs only in the oscillations that result at the beginning of the maneuver. No evaluation of the multivariable
predictive controller performance was done considering variable vehicle speed.
0 5 10 15 20 250
5
10
Time [sec]
Y [
m]
0 5 10 15 20 25-0.1
0
0.1
Time [sec]
Ste
erin
gan
gle
[rad
]
0 5 10 15 20 250
20
40
Time [sec]
Wh
eel
spee
d[r
ad/s
ec]
vehicle trajectory reference trajectory
Fig. 8. Obstacle avoidance maneuver at 37 [ / ]V km h=
V. CONCLUSIONS
In this paper, a multivariable nonlinear model predictive control strategy is proposed for controlling the vehicle longitudinal and lateral dynamics. For predicting the future states of the vehicle, a nonlinear vehicle model was used with three degrees of freedom (the longitudinal position x , the lateral position y and the yaw angle ψ ) and for generating the lateral and longitudinal tire forces, which determine the vehicle motion, the well known Magic Formula Tire Model was integrated in the selected vehicle model.
The proposed multivariable predictive controller was designed to use the front steering wheel angle δ and the wheels rotational speed rω as control signals, while the controlled outputs are the longitudinal and lateral positions in the absolute inertial plane, i.e. X and Y . By structuring the control signals, the implemented controller coped very well with the high coupling between lateral and longitudinal vehicle dynamics, and provides very good performance at constant vehicle speed. Further work will include studies of the disturbances influences, such as road bumps and front and side winds acting on the vehicle body.
VI. ACKNOLEGEMENTS
The work of R. C. Rafaila was supported by a grant of the Romanian National Authority for Scientific Research and Innovation, CNCS – UEFISCDI, project number PN-II-RU-TE-2014-4-0970.
REFERENCES
[1] M. Amoozadeh, H. Deng, C. N. Chuah, H. M. Zhang, D. Ghosal,
Platoon management with cooperative adaptive cruise control enabled by VANET, Vehicular Communications, Vol. 2, Issue 2, 2015
101
[2] Ü. Özgüner, T. Acarman, and K. Redmill, Autonomous ground vehicles, Artech House, 2012
[3] T. Keviczky, P. Falcone, F. Borrelli, J. Asgari, and D. Hrovat, Predictive Control Approach to Autonomous Vehicle Steering, American Control Conference, Minneapolis, pp. 4760-4675, 2006
[4] P. Falcone, M. Tufo, F. Borrelli, J. Asgari, and H. E. Tseng, A Linear Time Varying Model Predictive Control Approach to the Integrated Vehicle Dynamics Control Problem in Autonomous Systems, 46th IEEE Conference on Decision and Control, New Orleans, Los Angeles, USA, pp. 2980 - 2985, 2007
[5] A. Carvalho, S. Lefévre, G. Schildbach, J. Kong, and F. Borrelli, Automated driving: The role of forecasts and uncertainty - A control perspective, European Journal of Control, Vol. 24, pp. 14–32, 2015
[6] R.C. Rafaila, and G. Livint, Predictive control of autonomous steering for ground vehicles, 9th International Symposium on Advanced Topics in Electrical Engineering, Bucharest, Romania, 2015
[7] R.C. Rafaila, and G. Livint, Nonlinear Model Predictive Control of Autonomous Vehicle Steering, 19th International Conference on System Theory, Control and Computing, Cheile Gradistei, Romania, 2015
[8] C. F. Caruntu, A. E. Balau, M. Lazar, P. P. J. van den Bosch, and . Di Cairano, A predictive control solution for driveline oscillations damping, Hybrid Systems: Computation and Control, Chicago, USA, 2011.
[9] R. Attia, R. Orjuela, M. Basset, Combined longitudinal and lateral control for automated vehicle guidance, Vehicle System Dynamics, Taylor & Francis, 52 (2), pp.261-279, 2014
[10] A. Katriniok, J. P. Maschuw, F. Christen, L. Eckstein, D. Abel, Optimal Vehicle Dynamics Control for Combined Longitudinal and Lateral Autonomous Vehicle Guidance, European Control Conference, Zürich, 2013
[11] R. Rajamani, Vehicle Dynamics and Control, chapters 2 and 4, Springer Verlag, 2nd ed. 2012.
[12] H. Pacejka, Tire and Vehicle Dynamics, chapters 1,2,4, Elsevier Ltd, 3rd ed, 2012
[13] H. B. Pacejka, E. Bakker, and L. Nyborg, Tyre Modelling for Use in Vehicle Dynamics Studies, Society of Automotive Engineers, 1987
[14] C. M. Kang, S. H. Lee, and C. C. Chung, Comparative Evaluation of Dynamic and Kinematic Vehicle Models, 53rd IEEE Conference on Decision and Control, Los Angeles, California, USA, 2014
[15] E. F. Camacho, and C. Bordons, Model Predictive Control. Springer Verlag, 2004
[16] L. Grune, and J. Pannek, Nonlinear Model Predictive Control: Theory and Algorithms, Springer Verlag, 2011
