- Email: [email protected]
Lateral Control of Heavy-Duty Vehicles in a Convoy
Copyright @ IF AC Mechatronic Systems, Darmstadt, Germany, 2000
LATERAL CONTROL OF HEAVY-DUTY VEHICLES IN A CONVOY Xavier Claeys *, Carlos Canudas de Wit * , Hubert Bechart ** * Laboratoire d'Autornatique de Grenoble BP 46, 38402 Grenoble, France canudas, claeys @lag.ensieg.inpg.jr ** Plateau Recherche Renault V.!. BP 310, 69802 Saint Priest Cedex hubert. [email protected]
Abstract: This paper presents a kinematic model and a control law design for lateral control of heavy-duty vehicles in a convoy. These results are pertaining to applications where new automatic vehicle feature are to be designed. Control design is based on spatial linearization of the velocity relationships between two vehicle of the platoon. We also present some results obtained from a complete benchmark simulations. Copyright @2000 IFAC
Keywords: Vehicle control, spatiallinearization.
1. INTRODUCTION
(automatic features are legally accepted in longitudinal control problems like ABS, ACC, etc, but not completely accepted in lateral directions).
Automatic guided systems is a domain of intensive research in universities as well as in automotive industries, and state transportation organizations. Some reasons of those developments are due to: a) a large increase in the road transportation load leading to critical problems in terms of saturation of the road networks, pollution, increasing rate of accidents, b) a strong motivation for reducing costs in transportation, in particular by reducing the fuel consumption.
Concerning lateral control two main strategies have been confronted: the road-frame based control and the relative-frame based control. Road-frame based control: The first strategy developed within the PATH program (Berkeley, UCLA, Michigan Univ., ... ) is based on a roadframe based control strategy. The works by Chen and Tomizuka (1995a,b); Tomizuka et al. (1999) proposed control laws to regulate the distance between the center of the rear axis and the road lane. Information are provided by magnetic markers placed on the road. The measurements obtained are very robust but require a special equipment of infrastructure. This technique does not have lookahead capabilities, and thus it work well as far as strong turns are not taken at high speeds.
The concept of ''platooning'' in transportation systems appears as a way to partially cope with some of these problems. In the platoon configuration, the vehicles are electronically linked by radio transmissions and several vision and radar sensors. Each vehicle is equipped with lateral and longitudinal controllers, supervised by the driver and some computer aided system.
Relative frame based control: The strategy uses relative information between two vehicles. This idea is used in the European project "Chauffeur" through several references like Gehrig and Stein (1998); Francke and al (1995) ; Schulze (1996, 1997) (European consortium leaded by Daimler Chrysler and some other participants Iveco, ZF, Bosh, ... ), and several other works such
Lateral servoying of heavy duty vehicle is one of the sub-problems to be solved in the context of platooning. However, it has been revealed to be a crucial one because safety reasons . The crucial point here is that a lateral control can produce severe damage in case of failure. For these reasons lateral (automatic or semi-automatic) assistance systems are still far from the industrial production
127
\ +. \/
............
-"--
':~--,--_
e. more than 20% of the steering angle in worst case. ------ .. - For instance, the steering angle d, of a vehicle moving along a trajectory of a constant radius of curvature (radius of curvature R) is given, in steady state operation, by the following expression:
~
V2 I, + KUBgR R --...,......... '--" geometric angle sliding angle
G,
Ml/;\
(1)
Where I, is the length of the vehicle, V is the longitudinal speed of the vehicle, Kus is a slid(1\ 0.> ing constant (This term depends on the sliding t (1\, coefficients and it is small) and 9 is the gravity L- acceleration. Note that for high speeds the sliding angle can become large. Thus, control designs based on neglecting the sliding angle, should be robust enough to preserve the desired performance Fig. 1. State space variables required for the study of a in presence of sliding lateral velocity. convoy of two vehicles
In the following we derive two kinematics models that can be used as a basis to design controllers
as Haskara et al. (1998) ; Petersen et al. (1996) . In most of these studies the sensors used are camera and telemeters. The camera computes the coordinates of a particular target located at the back of each truck. A lateral offset is thus computed and used to control the vehicle's lateral motion. Advantages of this method are the absence of road equipment, and the prediction of the trajectory due to forward information given by the camera (look-ahead capability) . Some drawbacks are vision sensor failures, lateral string stability problems, etc.
2.1 Full modeling
For seek of simplicity in this presentation, we only consider two vehicles. The leader vehicle states are indexed by i = 0, whereas the follower vehicle states are indexed by i = 1. To find the differential equations relating the time-variation of the lateral and longitudinal distance between two vehicles (see fig. 1), 10 and do, to the ''input'' dl, we write the equations of the target coordinate (the point M describes the target located at the back of the semi-trailer) in the rear vehicle frame, RI. Some coordinates transformations are necessary to express the target coordinates in the suitable frame . For that, we introduce the following notation.
In this paper we study some control alternatives based on look-ahead visual information. A new control design based on mobile reference frame for lateral control of vehicles is proposed. This simple control structure is tested in a complete simulation benchmark. From these simulations, it is seen that the proposed control law yields suitable results as long as the number of trucks in the platoon is not too big.
The yaw rate of the tractors is defined by 8 i , the front steering wheel angle (direction of the linear speed vector) by CP i, and the articulation angle between the tractors and the semi-trailers by IJ1 i . The vehicles are supposed to be identical. I, stands for the tractor length, lr for the semi-trailer length, and the position of the center of gravity of the tractor and semi-trailer with respect to the front and rear axles are respectively given by: lit, ltz, lrl' lr2 (see fig. 1). Finally, a is the distance between the center of gravity of the tractor and the connecting point between semi-trailer and the tractor. The velocity composition relationship is given by:
2. KINEMATICS MODELS In literature many models have been developed to study the lateral and longitudinal controls of heavy duty vehicles. In many cases, a dynamic model has been derived to study the control of semi-trailer vehicles. In these studies small angle approximation is used and the model is linearized for seek of simplicity. A difficulty with this type of models is that designed control will strongly depend on the dynamics parameters of the vehicle. Nevertheless, we think that a kinematic analysis of the system would be enough to control efficiently the movement of the vehicle as long as the control is sufficiently robust with respect to the external perturbations and the actuator dynamics. Alternatively, kinematic models (modes relating only the vehicle velocities) only depend on geometric parameters but they may not be sufficient to exactly describe the vehicle motion because the sliding velocity component of the model can reach
rl
(v
(RI) M)(R a )
rl
= (v
(RI) OI)(R a )
~
~
rl
(RI)
+ 1l 1\ OIM + (v M)(Rd(2)
where 0 1 is the center of the front wheel axle of the following tractor. Ra is the absolute fixed frame and RI is the mobile frame linked to the tractor of the following vehicle. In the relation (2) the speed of the target point M,
128
Cv M) ~::~
written in frame RI with respect
axles of the vehicle (front and rear axles of the tractor and rear axle of the semi-trailer). Remark 1. As we can see in Fig. 2, the front wheel velocity vector angle ~i' can be approximated by the steering wheel angle 8i when the side slip angle is small (os; ~ 0).
For small angles the expression of the target point M velocity becomes:
v; =
Vo
v; = 0
(9)
Now we write the speed of the target M in the frame of the following vehicle (for small angles): Fig. 2. Front steering axle of the tractor vehicle. We consider a bicycle model where the front two wheels are replaced by a single middle wheel. We represent on this figure the steering wheel angle 6;, the sliding angle of the front wheel cr,; and the velocity angle 4>j.
Finally the model can be written as:
to frame Ra is given by Vo = [v;,v:l and we can write this expression as a function of the speed of the tractor front axle Vo:
to =Vo - VI + l tIo=vot
v: = vocos{"o)cos{"'o) - (vosin{"o) v!
+ (a
(10)
doe
- I, )90) sin{"'o)
and with
= vocos{"o)sin{"'o) + (vosin{ .. o) + (a _ I, )90) cos{"'o) (3) -1.{90
+ ~o)
e
1
(11)
~ 7;81 , it gives
For small angles this expression simplifies to:
(12)
v: = Vo v: =vollto+ (vo4>0+(a-lj)60) -lr(eo+>ito) (4)
This model will be used as a basis for the control design presented next.
Now we calculate the speed of the target point M in the follower vehicle frame. This expression is given by a simple rotation of coordinates:
CV
M
)(Rl) (Ra)
= [C?S(
3. CONTROL DESIGN
-Sin(
cos(
V;
In this section we only address the lateral control problem. We thus assume that suitable control law for the Vi do exist so that the resulting closed-loop longitudinal dynamics is stable.
-I,
The control design should be performed under the following assumptions:
-1.)90 ] (6)
_ I . ~o
Finally, we get: io= Vo - VI + eldo Vo'Po + vollto + vo4>o + (a -Ij -lr)60 (7) -lr>ito - vl4>1 - ello
do =
2.2 Rolling without sliding case
When the sliding (lateral) effect is small we have the following relationships:
(AI) The measured variables are : the two distances li and d i between vehicles and the relative yaw angle between vehicles
e
(8)
The control problem associated to our application is the following. Under assumption (Al)-(A3), and given a desired distance dt , find a control law 8i , (the steering
With 8i the wheel angle. These equations are given by non-holonomic constraints written on the three
129
angle) for (12,) such that small as possible.
Idi
-
dtl
becomes as
A separate problem is to find a suitable dt such that each rear vehicle tracks exactly the same trajectory that the one produced by its corresponding front vehicle. Here we simply assume that dt = 0 is enough for that purpose. Note however that this approximation is valid as long as the the road radius of curvature is large (which is the case in most of the highways). Clearly, this approximation will impose some limitation when this is not the case, and when the number of vehicles in the platoon becomes too large.
I
••
I
••
I ••
-
Fig. 3. Principle of the simulation program. Vehicle are simulated independently and the communication is handled between processes via software pipe. This program requires a multi-task system to work correctly.
3.1 First order spatial feedback linearization
With the control law:
(h
or equivalent, in spatial coordinates:
I, = -I-I-(k p do 0+ ,
Vo -IPo)
+ Vl
the closed loop equation becomes (as far as 0)
.
d~
(13)
Vl
= Vo -
d~
+ kpdo = 0
Vl
+ -I0+ -I-(kpvldo + volPo) ,
(14)
where d~ = od%so, with dso/dt = Vo. Where we assume only positive velocities. s is the path length of the origin point described by the Vo. The equation (14) is indeed a LSI (linear space invariant) equation. This set-up is much more suitable than the standard time-linearization, that may results in singularities when velocity goes to zero (the first term in the control law (14), the second term can be regularized by making the front vehicle velocity, Vo depends on the rear car velocity Vl, i.e. Vo = Vl v, where v is a new control variable).
3.2 Second order spatial feedback linearization
To cope with potential steady state error that may be caused by bias in the measured signals, it is possible to add an (spatial) integral term to the previously developed control law. Consider now:
In this system the do dynamics equation is stable. Then the zero dynamics can be written (with do = 0) io
= Vo -
(18)
Since the right hand side member of the above equation depends on the state variable do, it is no longer possible to conclude in stability of the closed-loop system. Clearly, to cope with this problem it will be necessary to resort to high-gain controllers, that may be inefficient due to the strong bandwidth limitations that characterizes this loop.
i-
do
10
+ kpdo = VO/Vl IPo(do, t, .. .)
This yields to (20)
Vl
(15) Similar stability consideration follows as before. We next present some simulation results evaluating the proposed laws. The proposed controls have been implemented on the complete truck model including a realistic actuator model and a control law for this actuator. The controller gains have been chosen considering the bandwidth limit of the actuator, which has been shown in works from Canudas-de-Wit et al. (1999); Claeys et al. (1998); Hingwe et al. (1999); Zaremba and Davis (1995) to be limited about 2 - 4Hz.
We may assume here that a Vl (the second input) is designed to make this equation stable. Many choices are possible. In what follows, we thus assume that 10 is bounded. Remark 2. If we do not measure IPo, the relative yaw angle between the two vehicles, then only the first term in (14) can be implemented, i. e.: (16)
4. SIMULATION RESULTS
and the lateral equation are not decoupled anymore. This results in:
do + kpVldo = VolPo (do , t, ... )
A complete simulation benchmark which allows for simulating several vehicles models in a platoon,
(17)
130
Trajedory
Position of the follower with respect to the leader vehicle
250
L...lenIJ 0fT.n
200
150
:[
,.. 100
1
50
n. .-~u_-. . u:-·-·-~-~·~'~--l so
0)0
100
150
1(1)
OL---~--~~--~--~--~--~--~
o
500
1000
1500
2000
2500
3000
Position of the follower with respect to the leader vehicle
3500
x (m)
Fig. 4. Trajectory for evaluation of control strategies. has been implemented. The simulation program is based on three dimensional vehicle models obtained by a complete mechanical analysis, including all possible 3-D forces. Also, different sensors like cameras, telemeters, gyroscopes have been added on each vehicle model and the communication inter-vehicle process has been implemented. All the vehicle programs are executed on a machine as separated processes and the communication is handled between vehicles through sockets or pipe (see, Fig. 3). This architecture is quite convenient and allows to simulate as many vehicles as desired on different computers and to interface this program with other applications (such as driver simulators, etc.). This platform is fully configurable allowing a large amount of different situations can be proposed (noise, and sensor failures, wind disturbances, road vibrations, etc.).
IJF,-_u~-7-~--:::3 -3 0
50
100
ISO
li_(.)
Fig. 5. Time-evolution of the distance do(t), the trajectory offset and the inter vehicle distance, for the control (13) with two different speed of the convoy. On the top v ::::: ISm/sand the controller gains are kp 0.04, kp 0.06, kp 0.08; on the bottom v ::::: 22m/s and the same controller gains are used kp = 0.04, kp 0.06, kp 0.08.
= =
= =
=
road center line (bottom figure) . FUrthermore, this error is amplified as we look back at the end of the platoon (kind of lost of lateral "string" stability); 50cm for the first vehicle and almost 1m for the last one, at the turning points. We remind that this error are obtained at extreme conditions, when the platoon is driving at 90km/hrs with along a curve of 700m radius maximum. This strategy seem thus limited to small platoons (4 - 5 trucks), where this error can be keep within a tolerable bounds.
Several experiments have been running with the two different controllers defined in the previous section. The time-evolution of the controlled variable do is used to visualize of the control performance. The experiment were carried out by using the trajectory (path) shown in Fig.4. Figure 5 shows the simulation results for the control law (13). These results show a reasonable regulation of the lateral offset do. The figure at the bottom shows the limitation (due to oscillatory behavior) when the gain kp is increased to its maximum.
A possible way to relax this limitation is to design a new lateral desired offset df that ensures that if d i = df, the road offset is keep small. This idea has been exploited in previous works of Haskara et al. (1998); FUjioka and Omae (1998), but clearly deserve more investigation.
Figure 6 shows the simulation results for the controllaw (19) 2nd order spatiallinearization. Here the gains are kept small for stabilizing the yaw motion. This controller leads to better performances than the previous one, since it allows a error reduction to a maximum of 20cm during the stiff turns (the curvature radius is about 700m and speed of the vehicle about 90km/hrs, so this corresponds to the limit situation on highway for such vehicles) .
5. CONCLUSION We have presented a kinematic model and a control law for the lateral motion of heavy duty vehicles in a platoon configuration. We have first derived a kinematics model for this system, then we have presented two different control alternatives that lead to first and second order spatial linearization, respectively.
A simulation of a platooning of seven vehicles is presented on figure 7. While the lateral offset di remains small (maximum of 20cm) and the generated motion produce a reasonable yaw angle transients (upper figure), there is a magnification of the distance between the point 0 1 , and the
These results have been evaluated on a highly accurate benchmark specially build for studying platooning features. This strategy is sufficient for
131
controlling 4 vehicles in a platoon but further developments need to be performed concerning lateral string stability of a convoy in order to generalize these results to an arbitrarily number of vehicles.
u
S. Gehrig and F. Stein. A trajectory-based approach for the lateral control of car following systems. In Proceedings of the intelligent Vehiucles 98 Symposium, 1998. I. Haskara, C. Hatipoglu, and D. Ozgiiner. Combined decentralized longitudinal and lateral controller design for truck convoys. In IEEE, 1998. P. Hingwe, M. Tai, and M. Tomizuka. Modeling and linear robust control of power steering system of heavy vehicles for ahs. In CCA '99, Hawai, 1999. U. Petersen, A. Riikgauer, and W. Schiehlen. Lateral control of a convoy vehicle system. In Vehicle Dynamics Supplement 25, 1996. M. Schulze. Tow-bar overall function and specific systems definition. Technical report, Daimlerchysler, 1996. M. Schulze. Specification and system concept for electronic steering system. Technical report, Daimler-chysler, 1997. M. Tomizuka, P. Hingwe, M. Tai, and M. Tai. Automated lane guidance of commercial vehicles. In Submitted CCA '99, Hawai, 1999. A. Zaremba and R. Davis. Dynamic analysis and stability of a power assist steering system. In American Control Conference, Seattle, Washintgon, 1995.
Lateral Offsets l..Ma'&I Off.a
:1:
01_ ~:::~~I
50
. .
10
Ibc pnICCIedin, .. dI~k
~: ~l 100
ISO
um£(,)
Launol orraort
to Ihe
lraJOflClOtY
Fig. 6. Time-evolution of the distance do(t), for the control (19) with kp = 0.04 and k/ = 0.0008.
Lateral Offsets
VehideO Vehiele 1 Vehicle2 Vebide 3
-
'i
!
~
_ 2 I
;
0 -I
3
-2
50
100
1>0
limch) UMaaI otfld. 10 Ihe
tnjll!lCUll')'
~:=::~I--r--~---T-"";"'-~----'----'
Vehic:lc6
. . -~. ~;=;"=".,=:::;.;~:~,~.'''"'"'''""'~Z4.~!ii '"
-lO .!---~----;5O!;--~----;-!loo:'--~--~,>o;;-----'
Fig. 7. Time-evolution of the distance do(t) (upper curve) and the distance to the trajectory (below curve) for platoon of 7 vehicles, for the control (19) with kp = 0.04 and k/ = 0.0008 .
References C. Canudas-de-Wit, X. Claeys, and H. Bechart. Stability analysis via passivity of the lateral actuator dynamics of a heavy vehicle. In CCA '99, Hawai, 1999. C. Chen and M. Tomizuka. Dynamic modeling of articulated vehicles for automated highway systems. In Proceedings of the American Control Conference, 1995a. C. Chen and M. Tomizuka. Steering and braking control of tractor-semi-trailer vehicles in automated highway systems. In Proceedings of the American Control Conference, 1995b. X. Claeys, C. Canudas-de-Wit, and H. Bechart. Modeling and control of steering actuator for heavy duty vehicles. In ECC'99, 1998. U. Francke and al. Truck platoon inmixed traffic. In Proceedings of the intelligent Vehicles, page 1, 1995. T. FUjioka and M. Omae. Vehicle following control in lateral direction for platooning. Vehicle System Dynamics Supplement 28,29:422,1998.
132
LATERAL CONTROL OF HEAVY-DUTY VEHICLES IN A CONVOY Xavier Claeys *, Carlos Canudas de Wit * , Hubert Bechart ** * Laboratoire d'Autornatique de Grenoble BP 46, 38402 Grenoble, France canudas, claeys @lag.ensieg.inpg.jr ** Plateau Recherche Renault V.!. BP 310, 69802 Saint Priest Cedex hubert. [email protected]
Abstract: This paper presents a kinematic model and a control law design for lateral control of heavy-duty vehicles in a convoy. These results are pertaining to applications where new automatic vehicle feature are to be designed. Control design is based on spatial linearization of the velocity relationships between two vehicle of the platoon. We also present some results obtained from a complete benchmark simulations. Copyright @2000 IFAC
Keywords: Vehicle control, spatiallinearization.
1. INTRODUCTION
(automatic features are legally accepted in longitudinal control problems like ABS, ACC, etc, but not completely accepted in lateral directions).
Automatic guided systems is a domain of intensive research in universities as well as in automotive industries, and state transportation organizations. Some reasons of those developments are due to: a) a large increase in the road transportation load leading to critical problems in terms of saturation of the road networks, pollution, increasing rate of accidents, b) a strong motivation for reducing costs in transportation, in particular by reducing the fuel consumption.
Concerning lateral control two main strategies have been confronted: the road-frame based control and the relative-frame based control. Road-frame based control: The first strategy developed within the PATH program (Berkeley, UCLA, Michigan Univ., ... ) is based on a roadframe based control strategy. The works by Chen and Tomizuka (1995a,b); Tomizuka et al. (1999) proposed control laws to regulate the distance between the center of the rear axis and the road lane. Information are provided by magnetic markers placed on the road. The measurements obtained are very robust but require a special equipment of infrastructure. This technique does not have lookahead capabilities, and thus it work well as far as strong turns are not taken at high speeds.
The concept of ''platooning'' in transportation systems appears as a way to partially cope with some of these problems. In the platoon configuration, the vehicles are electronically linked by radio transmissions and several vision and radar sensors. Each vehicle is equipped with lateral and longitudinal controllers, supervised by the driver and some computer aided system.
Relative frame based control: The strategy uses relative information between two vehicles. This idea is used in the European project "Chauffeur" through several references like Gehrig and Stein (1998); Francke and al (1995) ; Schulze (1996, 1997) (European consortium leaded by Daimler Chrysler and some other participants Iveco, ZF, Bosh, ... ), and several other works such
Lateral servoying of heavy duty vehicle is one of the sub-problems to be solved in the context of platooning. However, it has been revealed to be a crucial one because safety reasons . The crucial point here is that a lateral control can produce severe damage in case of failure. For these reasons lateral (automatic or semi-automatic) assistance systems are still far from the industrial production
127
\ +. \/
............
-"--
':~--,--_
e. more than 20% of the steering angle in worst case. ------ .. - For instance, the steering angle d, of a vehicle moving along a trajectory of a constant radius of curvature (radius of curvature R) is given, in steady state operation, by the following expression:
~
V2 I, + KUBgR R --...,......... '--" geometric angle sliding angle
G,
Ml/;\
(1)
Where I, is the length of the vehicle, V is the longitudinal speed of the vehicle, Kus is a slid(1\ 0.> ing constant (This term depends on the sliding t (1\, coefficients and it is small) and 9 is the gravity L- acceleration. Note that for high speeds the sliding angle can become large. Thus, control designs based on neglecting the sliding angle, should be robust enough to preserve the desired performance Fig. 1. State space variables required for the study of a in presence of sliding lateral velocity. convoy of two vehicles
In the following we derive two kinematics models that can be used as a basis to design controllers
as Haskara et al. (1998) ; Petersen et al. (1996) . In most of these studies the sensors used are camera and telemeters. The camera computes the coordinates of a particular target located at the back of each truck. A lateral offset is thus computed and used to control the vehicle's lateral motion. Advantages of this method are the absence of road equipment, and the prediction of the trajectory due to forward information given by the camera (look-ahead capability) . Some drawbacks are vision sensor failures, lateral string stability problems, etc.
2.1 Full modeling
For seek of simplicity in this presentation, we only consider two vehicles. The leader vehicle states are indexed by i = 0, whereas the follower vehicle states are indexed by i = 1. To find the differential equations relating the time-variation of the lateral and longitudinal distance between two vehicles (see fig. 1), 10 and do, to the ''input'' dl, we write the equations of the target coordinate (the point M describes the target located at the back of the semi-trailer) in the rear vehicle frame, RI. Some coordinates transformations are necessary to express the target coordinates in the suitable frame . For that, we introduce the following notation.
In this paper we study some control alternatives based on look-ahead visual information. A new control design based on mobile reference frame for lateral control of vehicles is proposed. This simple control structure is tested in a complete simulation benchmark. From these simulations, it is seen that the proposed control law yields suitable results as long as the number of trucks in the platoon is not too big.
The yaw rate of the tractors is defined by 8 i , the front steering wheel angle (direction of the linear speed vector) by CP i, and the articulation angle between the tractors and the semi-trailers by IJ1 i . The vehicles are supposed to be identical. I, stands for the tractor length, lr for the semi-trailer length, and the position of the center of gravity of the tractor and semi-trailer with respect to the front and rear axles are respectively given by: lit, ltz, lrl' lr2 (see fig. 1). Finally, a is the distance between the center of gravity of the tractor and the connecting point between semi-trailer and the tractor. The velocity composition relationship is given by:
2. KINEMATICS MODELS In literature many models have been developed to study the lateral and longitudinal controls of heavy duty vehicles. In many cases, a dynamic model has been derived to study the control of semi-trailer vehicles. In these studies small angle approximation is used and the model is linearized for seek of simplicity. A difficulty with this type of models is that designed control will strongly depend on the dynamics parameters of the vehicle. Nevertheless, we think that a kinematic analysis of the system would be enough to control efficiently the movement of the vehicle as long as the control is sufficiently robust with respect to the external perturbations and the actuator dynamics. Alternatively, kinematic models (modes relating only the vehicle velocities) only depend on geometric parameters but they may not be sufficient to exactly describe the vehicle motion because the sliding velocity component of the model can reach
rl
(v
(RI) M)(R a )
rl
= (v
(RI) OI)(R a )
~
~
rl
(RI)
+ 1l 1\ OIM + (v M)(Rd(2)
where 0 1 is the center of the front wheel axle of the following tractor. Ra is the absolute fixed frame and RI is the mobile frame linked to the tractor of the following vehicle. In the relation (2) the speed of the target point M,
128
Cv M) ~::~
written in frame RI with respect
axles of the vehicle (front and rear axles of the tractor and rear axle of the semi-trailer). Remark 1. As we can see in Fig. 2, the front wheel velocity vector angle ~i' can be approximated by the steering wheel angle 8i when the side slip angle is small (os; ~ 0).
For small angles the expression of the target point M velocity becomes:
v; =
Vo
v; = 0
(9)
Now we write the speed of the target M in the frame of the following vehicle (for small angles): Fig. 2. Front steering axle of the tractor vehicle. We consider a bicycle model where the front two wheels are replaced by a single middle wheel. We represent on this figure the steering wheel angle 6;, the sliding angle of the front wheel cr,; and the velocity angle 4>j.
Finally the model can be written as:
to frame Ra is given by Vo = [v;,v:l and we can write this expression as a function of the speed of the tractor front axle Vo:
to =Vo - VI + l tIo=vo
v: = vocos{"o)cos{"'o) - (vosin{"o) v!
+ (a
(10)
doe
- I, )90) sin{"'o)
and with
= vocos{"o)sin{"'o) + (vosin{ .. o) + (a _ I, )90) cos{"'o) (3) -1.{90
+ ~o)
e
1
(11)
~ 7;81 , it gives
For small angles this expression simplifies to:
(12)
v: = Vo v: =vollto+ (vo4>0+(a-lj)60) -lr(eo+>ito) (4)
This model will be used as a basis for the control design presented next.
Now we calculate the speed of the target point M in the follower vehicle frame. This expression is given by a simple rotation of coordinates:
CV
M
)(Rl) (Ra)
= [C?S(
3. CONTROL DESIGN
-Sin(
cos(
V;
In this section we only address the lateral control problem. We thus assume that suitable control law for the Vi do exist so that the resulting closed-loop longitudinal dynamics is stable.
-I,
The control design should be performed under the following assumptions:
-1.)90 ] (6)
_ I . ~o
Finally, we get: io= Vo - VI + eldo Vo'Po + vollto + vo4>o + (a -Ij -lr)60 (7) -lr>ito - vl4>1 - ello
do =
2.2 Rolling without sliding case
When the sliding (lateral) effect is small we have the following relationships:
(AI) The measured variables are : the two distances li and d i between vehicles and the relative yaw angle between vehicles
e
(8)
The control problem associated to our application is the following. Under assumption (Al)-(A3), and given a desired distance dt , find a control law 8i , (the steering
With 8i the wheel angle. These equations are given by non-holonomic constraints written on the three
129
angle) for (12,) such that small as possible.
Idi
-
dtl
becomes as
A separate problem is to find a suitable dt such that each rear vehicle tracks exactly the same trajectory that the one produced by its corresponding front vehicle. Here we simply assume that dt = 0 is enough for that purpose. Note however that this approximation is valid as long as the the road radius of curvature is large (which is the case in most of the highways). Clearly, this approximation will impose some limitation when this is not the case, and when the number of vehicles in the platoon becomes too large.
I
••
I
••
I ••
-
Fig. 3. Principle of the simulation program. Vehicle are simulated independently and the communication is handled between processes via software pipe. This program requires a multi-task system to work correctly.
3.1 First order spatial feedback linearization
With the control law:
(h
or equivalent, in spatial coordinates:
I, = -I-I-(k p do 0+ ,
Vo -IPo)
+ Vl
the closed loop equation becomes (as far as 0)
.
d~
(13)
Vl
= Vo -
d~
+ kpdo = 0
Vl
+ -I0+ -I-(kpvldo + volPo) ,
(14)
where d~ = od%so, with dso/dt = Vo. Where we assume only positive velocities. s is the path length of the origin point described by the Vo. The equation (14) is indeed a LSI (linear space invariant) equation. This set-up is much more suitable than the standard time-linearization, that may results in singularities when velocity goes to zero (the first term in the control law (14), the second term can be regularized by making the front vehicle velocity, Vo depends on the rear car velocity Vl, i.e. Vo = Vl v, where v is a new control variable).
3.2 Second order spatial feedback linearization
To cope with potential steady state error that may be caused by bias in the measured signals, it is possible to add an (spatial) integral term to the previously developed control law. Consider now:
In this system the do dynamics equation is stable. Then the zero dynamics can be written (with do = 0) io
= Vo -
(18)
Since the right hand side member of the above equation depends on the state variable do, it is no longer possible to conclude in stability of the closed-loop system. Clearly, to cope with this problem it will be necessary to resort to high-gain controllers, that may be inefficient due to the strong bandwidth limitations that characterizes this loop.
i-
do
10
+ kpdo = VO/Vl IPo(do, t, .. .)
This yields to (20)
Vl
(15) Similar stability consideration follows as before. We next present some simulation results evaluating the proposed laws. The proposed controls have been implemented on the complete truck model including a realistic actuator model and a control law for this actuator. The controller gains have been chosen considering the bandwidth limit of the actuator, which has been shown in works from Canudas-de-Wit et al. (1999); Claeys et al. (1998); Hingwe et al. (1999); Zaremba and Davis (1995) to be limited about 2 - 4Hz.
We may assume here that a Vl (the second input) is designed to make this equation stable. Many choices are possible. In what follows, we thus assume that 10 is bounded. Remark 2. If we do not measure IPo, the relative yaw angle between the two vehicles, then only the first term in (14) can be implemented, i. e.: (16)
4. SIMULATION RESULTS
and the lateral equation are not decoupled anymore. This results in:
do + kpVldo = VolPo (do , t, ... )
A complete simulation benchmark which allows for simulating several vehicles models in a platoon,
(17)
130
Trajedory
Position of the follower with respect to the leader vehicle
250
L...lenIJ 0fT.n
200
150
:[
,.. 100
1
50
n. .-~u_-. . u:-·-·-~-~·~'~--l so
0)0
100
150
1(1)
OL---~--~~--~--~--~--~--~
o
500
1000
1500
2000
2500
3000
Position of the follower with respect to the leader vehicle
3500
x (m)
Fig. 4. Trajectory for evaluation of control strategies. has been implemented. The simulation program is based on three dimensional vehicle models obtained by a complete mechanical analysis, including all possible 3-D forces. Also, different sensors like cameras, telemeters, gyroscopes have been added on each vehicle model and the communication inter-vehicle process has been implemented. All the vehicle programs are executed on a machine as separated processes and the communication is handled between vehicles through sockets or pipe (see, Fig. 3). This architecture is quite convenient and allows to simulate as many vehicles as desired on different computers and to interface this program with other applications (such as driver simulators, etc.). This platform is fully configurable allowing a large amount of different situations can be proposed (noise, and sensor failures, wind disturbances, road vibrations, etc.).
IJF,-_u~-7-~--:::3 -3 0
50
100
ISO
li_(.)
Fig. 5. Time-evolution of the distance do(t), the trajectory offset and the inter vehicle distance, for the control (13) with two different speed of the convoy. On the top v ::::: ISm/sand the controller gains are kp 0.04, kp 0.06, kp 0.08; on the bottom v ::::: 22m/s and the same controller gains are used kp = 0.04, kp 0.06, kp 0.08.
= =
= =
=
road center line (bottom figure) . FUrthermore, this error is amplified as we look back at the end of the platoon (kind of lost of lateral "string" stability); 50cm for the first vehicle and almost 1m for the last one, at the turning points. We remind that this error are obtained at extreme conditions, when the platoon is driving at 90km/hrs with along a curve of 700m radius maximum. This strategy seem thus limited to small platoons (4 - 5 trucks), where this error can be keep within a tolerable bounds.
Several experiments have been running with the two different controllers defined in the previous section. The time-evolution of the controlled variable do is used to visualize of the control performance. The experiment were carried out by using the trajectory (path) shown in Fig.4. Figure 5 shows the simulation results for the control law (13). These results show a reasonable regulation of the lateral offset do. The figure at the bottom shows the limitation (due to oscillatory behavior) when the gain kp is increased to its maximum.
A possible way to relax this limitation is to design a new lateral desired offset df that ensures that if d i = df, the road offset is keep small. This idea has been exploited in previous works of Haskara et al. (1998); FUjioka and Omae (1998), but clearly deserve more investigation.
Figure 6 shows the simulation results for the controllaw (19) 2nd order spatiallinearization. Here the gains are kept small for stabilizing the yaw motion. This controller leads to better performances than the previous one, since it allows a error reduction to a maximum of 20cm during the stiff turns (the curvature radius is about 700m and speed of the vehicle about 90km/hrs, so this corresponds to the limit situation on highway for such vehicles) .
5. CONCLUSION We have presented a kinematic model and a control law for the lateral motion of heavy duty vehicles in a platoon configuration. We have first derived a kinematics model for this system, then we have presented two different control alternatives that lead to first and second order spatial linearization, respectively.
A simulation of a platooning of seven vehicles is presented on figure 7. While the lateral offset di remains small (maximum of 20cm) and the generated motion produce a reasonable yaw angle transients (upper figure), there is a magnification of the distance between the point 0 1 , and the
These results have been evaluated on a highly accurate benchmark specially build for studying platooning features. This strategy is sufficient for
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controlling 4 vehicles in a platoon but further developments need to be performed concerning lateral string stability of a convoy in order to generalize these results to an arbitrarily number of vehicles.
u
S. Gehrig and F. Stein. A trajectory-based approach for the lateral control of car following systems. In Proceedings of the intelligent Vehiucles 98 Symposium, 1998. I. Haskara, C. Hatipoglu, and D. Ozgiiner. Combined decentralized longitudinal and lateral controller design for truck convoys. In IEEE, 1998. P. Hingwe, M. Tai, and M. Tomizuka. Modeling and linear robust control of power steering system of heavy vehicles for ahs. In CCA '99, Hawai, 1999. U. Petersen, A. Riikgauer, and W. Schiehlen. Lateral control of a convoy vehicle system. In Vehicle Dynamics Supplement 25, 1996. M. Schulze. Tow-bar overall function and specific systems definition. Technical report, Daimlerchysler, 1996. M. Schulze. Specification and system concept for electronic steering system. Technical report, Daimler-chysler, 1997. M. Tomizuka, P. Hingwe, M. Tai, and M. Tai. Automated lane guidance of commercial vehicles. In Submitted CCA '99, Hawai, 1999. A. Zaremba and R. Davis. Dynamic analysis and stability of a power assist steering system. In American Control Conference, Seattle, Washintgon, 1995.
Lateral Offsets l..Ma'&I Off.a
:1:
01_ ~:::~~I
50
. .
10
Ibc pnICCIedin, .. dI~k
~: ~l 100
ISO
um£(,)
Launol orraort
to Ihe
lraJOflClOtY
Fig. 6. Time-evolution of the distance do(t), for the control (19) with kp = 0.04 and k/ = 0.0008.
Lateral Offsets
VehideO Vehiele 1 Vehicle2 Vebide 3
-
'i
!
~
_ 2 I
;
0 -I
3
-2
50
100
1>0
limch) UMaaI otfld. 10 Ihe
tnjll!lCUll')'
~:=::~I--r--~---T-"";"'-~----'----'
Vehic:lc6
. . -~. ~;=;"=".,=:::;.;~:~,~.'''"'"'''""'~Z4.~!ii '"
-lO .!---~----;5O!;--~----;-!loo:'--~--~,>o;;-----'
Fig. 7. Time-evolution of the distance do(t) (upper curve) and the distance to the trajectory (below curve) for platoon of 7 vehicles, for the control (19) with kp = 0.04 and k/ = 0.0008 .
References C. Canudas-de-Wit, X. Claeys, and H. Bechart. Stability analysis via passivity of the lateral actuator dynamics of a heavy vehicle. In CCA '99, Hawai, 1999. C. Chen and M. Tomizuka. Dynamic modeling of articulated vehicles for automated highway systems. In Proceedings of the American Control Conference, 1995a. C. Chen and M. Tomizuka. Steering and braking control of tractor-semi-trailer vehicles in automated highway systems. In Proceedings of the American Control Conference, 1995b. X. Claeys, C. Canudas-de-Wit, and H. Bechart. Modeling and control of steering actuator for heavy duty vehicles. In ECC'99, 1998. U. Francke and al. Truck platoon inmixed traffic. In Proceedings of the intelligent Vehicles, page 1, 1995. T. FUjioka and M. Omae. Vehicle following control in lateral direction for platooning. Vehicle System Dynamics Supplement 28,29:422,1998.
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