- Email: [email protected]
Robust Road Departure Avoidance Based on Driver Decision Estimation
Proceedings of the 18th World Congress The International Federation of Automatic Control Milano (Italy) August 28 - September 2, 2011
Robust Road Departure Avoidance Based on Driver Decision Estimation M. Alirezaei*, M. Corno**, A. Ghaffari*, R. Kazemi* *Mechanical Engineering Department, K. N. Toosi University, Tehran, Iran, (e-mail: [email protected]) ** Delft Center for Systems and Control, Delft University of Technology Delft, Netherlands, (e-mail: [email protected]) Abstract: In this paper a robust road departure avoidance system based on a closed-loop driver decision estimator (DDE) is presented. The goal is that of incorporating the driver intent in the control of the vehicle. The proposed system ensures that the vehicle remains within the road borders with minimal effect on the driver demand. Numerical simulations show the effectiveness and robustness of the proposed approach. A comparison between the presented method and the generalization of methods available in the literature is presented. 1. INTRODUCTION Statistics show that most car accidents are associated with drivers’ errors (Kim et al., 2008). Driver assistance systems (DAS) could therefore improve road safety considerably. DAS can correct driver’s input in case of distractions or errors (Oya et al., 2007). Several systems are being developed: adaptive cruise control (ACC), road departure avoidance (RDA) and driver drowsiness detection are examples of such systems. Highway road departure avoidance (Alirezaie et al., 2010, Rossetter, 2003, Oya et al., 2007) is one of the most interesting applications. Such systems use active steering to prevent the vehicle from departing the road (or lane in some applications). They are particularly promising because on one hand, being developed for highway driving, they can take advance of the highly structured environment (mostly straight roads with well marked lanes, unidirectional traffic and rather uniform surface) and on the other hand, given the high driving velocities, they have a great potential of saving lives. Driving assistance systems can be classified along two axes: sensing scheme and interaction with the user. Two sensing schemes exist (Shladover, 1991): look down where the measurement of the vehicle lateral position relies on buried magnet (or other devices on the road) and look ahead in which the position of the vehicle is measured via visual feedback. The second approach is advantageous for two reasons: it does not require modification of the infrastructure and the look ahead capability yields a better control of the trajectory (Patwardhan et al, 1997). The second axis refers to how the user’s input is accounted for. One of the most daunting tasks for DAS’s is to meet the safety objectives without altering the driving experience when not needed. These difficulties arise from the problem of assessing the driver’s intentions (Boyraz et al. 2009). With this regard two different approaches have been explored. The indirect approach does not consider the driver as part of the control loop (Minoiu et al., 2009). Such systems are switched on when the hazard condition is detected and neglect the driver 978-3-902661-93-7/11/$20.00 © 2011 IFAC
decision. In case of false activation they heavily modify the behaviour of the vehicle and may not be well accepted. The second approach, referred to as the direct approach, accounts for the driver’s intentions (Cerone et al., 2009, Petersson et al., 2005, Rossetter, 2003). Measureable user inputs like steering torque or steering wheel angle are used to interpret his/her intentions. Although the direct approach has the potential of not altering the driving experience, the methods usually rely on an open loop estimation of the driver intention (Cerone et al., 2009). This may introduce the generation of a control action also when not needed. This paper proposes a novel road departure avoidance system. The proposed method is a look ahead method that extends the direct approach by adding a closed loop Driver Decision Estimator (DDE). It is assumed that the vehicle is equipped with a visual system to measure the lateral position of the vehicle and with an active steering system. The DDE yields an unbiased estimation of the driver’s desired path based on the position of the vehicle and control input. The proposed system is constantly active so no on/off strategy is needed. It is shown that when the vehicle is pointing toward a safe region no control action is performed (and thus the vehicle dynamics are not altered), and even in case of aggressive manoeuvres that are erroneously interpreted as dangerous the proposed controller has a minimal effect of the vehicle dynamics. The content of the paper is organized as follows. In Section 2, the vehicle-road model is described. A more precise classification of the available approaches is given in Section 3 and the concept of the DDE is introduced. In Section 4 a robust controller is designed. Simulation results are presented in Section 5. 2. VEHICLE-ROAD MODEL STRUCTURE The vehicle is assumed to be equipped with a vision system and a steering actuator. The vision system measures the lateral offset between the vehicle and the road centreline xla meters away (see Fig. 1). In figure, two reference frames are
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10.3182/20110828-6-IT-1002.02923
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
defined: n-t and nla-tla the n axis being normal to the road centerline and the t axis tangential to it. The origins of the reference coordinates are respectively located at the C.G. of the vehicle and at the intersection between the centerline and the normal to the road through the point xla meters ahead of the vehicle. The distance between this point and the centerline is yla and represents the visual feedback signal. The components of the vehicle velocity in the n-t coordinates are: Vn u sin(ψ V ) + v cos(ψ V ) = (1) Vt = −v sin(ψ V ) + u cos(ψ V )
ψ V is the yaw angle of the vehicle with respect to the centreline. From (1) and defining ψ R as the rotation between
the n-t and nla-tla reference frames, the time derivative of yla is:
= yla Vt sin(ψ R ) + (Vn + xlaψV ) cos(ψ R )
(2)
substituting (1) in (2) one obtains: Fig. 1. Vehicle and road reference frames.
yla = ( −v sin(ψ V ) + u cos(ψ V ) ) sin(ψ R )
(3)
+ ( u sin(ψ V ) + v cos(ψ V ) + xlaψV ) cos(ψ R )
Note that the model, although simple, is suitable for highway traffic. In fact unmodeled dynamics are little excited in highway driving, usually characterized by slow steering action, slow velocity variation and small pitch and roll angles. The nominal parameters used in the model are shown in Table 1.
Assuming ψ V and ψ R are small enough i.e. (sin(ψ V ) ≈ ψ V , sin(ψ R ) ≈ ψ R , ψ Rψ V ≈ 0) , yla is determined as: (4) yla = v + uψ VR + xlaψV where, ψ VR = ψ V +ψ R . (5) The time derivative of ψ VR is,
ψVR = ψV +ψ R .
(6) The magnitude of ψ R is defined as the vehicle longitudinal velocity divided by road curvature (u / R ) : (7) ψVR= ψV + (−u / R). Rewriting the classical bicycle model (Gillespie, 1992) augmented with the look ahead distance equations (4) and (7), the vehicle-road model can be described in state space as: x = Ax + Bδ + Ew (8) yla = Cx
The remaining symbols are: v lateral velocity r = ψV yaw rate front wheel angle δ m vehicle mass u longitudinal velocity Iz vehicle moment of inertia around yaw axis
C
a11 a 21 1 0
0 0 1 0] , w [=
a12 a22 xla 1
0 0 0 0 ,B = 0 u 0 0
b11 b 21 = ,E 0 0
Cr
a11
cornering stiffness of rear tires distance between front axle and the C.G. distance between rear axle and the C.G. = −(Cr + C f ) / mu
a12
= (bCr − aC f ) / mu − u
a21
0 0 0 −u
1 R
From (8), the transfer function from the steering angle to yla is: y (s ) m s 2 + m s + m0 G (s ) = la =C (sI − A ) −1 B = 2 2 2 1 δ (s ) s (n 2s + n1s + n 0 ) .
cornering stiffness of front tires
a b
where, v r , A x = = yla ψ VR
Cf
(9)
The two integrators arise from positional states and two other poles determine the vehicle handling.
= (bCr − aC f ) / I z u
a22
= −(b 2 Cr + a 2 C f ) / I z u
b11
= Cf / m
b21
= aC f / I z 3. DRIVER DECISION ESTIMATOR
In this section different approaches to design RDA systems and their characteristics with respect to the driver’s input are explained in details and the novel scheme is proposed. Accounting for driver intents is critical for the user acceptance of the DAS. To better assess the issue consider the uncontrolled case of Fig.2.
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
accurate model of the vehicle is used then the ideal objective is achieved: the control system intervention is limited only to the period when |yla|> yL. The main problem with this method lies in modelling errors. yˆ lad is generated in open loop and any error in the model will modify the driving experience, yielding a control action also when the manoeuvre is safe.
20
yla (m)
15 without control 10 safety limit
5
0 0
1
2
desired
3 Time (sec)
4
5
6
18 16 14
y (m)
12
without control
10 8 road border
6
desired
4 2 0 0
1
2
3 Time (sec)
4
5
Fig. 4. Block diagram of the direct method.
6
Fig. 2. A typical maneuver for a distracted driver.
Here we propose a modification of the direct approach based on the closed loop generation of yˆ lad . Fig. 5 shows the proposed block diagram of the RDA system based on the Driver Decision Estimator (DDE). As before, Gˆ is an estimated transfer function of the vehicle model. According to the block diagram and considering G as a transfer function of the vehicle, yla is computed as δ G (δ C + δ d ) . y= G= (10) la The driver desired lateral offset between vehicle C.G. and centreline at the lookahead distance ( ylad ) is defined as, ylad = Gδ d . (11) According to (10) and (11) ylad is calculated as, ylad = yla − Gδ C (12) . The control signal (δ C ) , is computed as,
In this scenario a driver keeps a constant steering angle of 1 deg for 5 seconds; after this period, he tries to return the vehicle to the road border by steering to the opposite direction. Ideally the objective is to keep the vehicle in the safe region (that yL extend meters to boths side of the center line), one way to do is to force yla to stay in the safe region. So, it is desirable that the system keeps yla in the safety limit; on the other hand it is also important that the systems responds as soon as the driver acts on the steering wheel to correct the trajectory. In the introduction two approaches are outlined: the direct and indirect approaches. The generalized block diagram of the indirect concept is drawn in Fig. 3. δd , δC and GC are the driver input, control input, and transfer function of the controller. The saturation is used to define the safe region for = δ C GC ( yd − yla ) yla as shown in Fig.2. This approach does not take the driver explicitly into account. The main drawback of this approach is it considerably affects the dynamics of the vehicle also when not needed. Consider the above maneuver, during the time period when |yla|> yL an active steering angle is generated; once the driver steers back toward the centerline the control system should be deactivated, but if a deactivation logic is not present, the control action will got to zero with Gc dynamics, thus a “residual” undesired control action will be present.
(13)
Fig. 5. Block diagram of the proposed method. where, yd is the desired lateral offset accounting for the driver’s demand and road limitation, which is defined as, = yd η yˆlad + λ ,
(14)
yˆlad is the estimation of the driver desired lateral offset
between the vehicle C.G. and centreline at the lookahead distance. It is computed in the proposed DDE as,
Fig. 3. Block diagram of the indirect approach. In the direct concept (see Fig. 4), the driver desired path is estimated by using the driver input, for example in Cerone et al., 2009) the steering torque is measured. In general, the driver behaviour is computed based on driver’s input and an estimated vehicle-road model (Gˆ ) . If an
yˆlad = yla − Gˆ δ C .
η and λ describes the saturation:
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(15)
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
(
)
if yˆ lad < − yL ⇒ η = 0 and if
( yˆ
lad
)
λ = − yL
≤ y= = λ 0 L ⇒ η 1 and
(
)
if yˆ lad > y= and λ yL 0 = L ⇒η
( ( xla + b) / u − mr u / Cr ) 0 u l / u 2 + K us Cˆ ˆ = ˆ = A B = , 0 0 (16) , 1/ u l / u 2 + K us
where yL is the road border. Substituting (13)-(16) into (10), yla can be written as a function of δ d , η and λ , (1 + η GC Gˆ )G yla = δd 1 + GC G + η GC (Gˆ − G ) (17) GC G + λ 1 + GC G + η GC (Gˆ − G ) Depending on the status of the saturation the following cases are given:
(
)
(
)
if yˆ lad < − y= L ⇒ yla
GGC G yL δd − 1 + GGC 1 + GGC
if yˆ lad ≤ yL ⇒ yla = ylad
(
)
if yˆ lad > yL = ⇒ yla
where, l= a + b , mr = ma / l and = K us m(bCr − aC f ) / lC f Cr . The transfer function of the reduced order model is then f s+ f Gˆ ( s ) = 1 2 0 s
( x + b) mr u and f1 la = − f0 . u Cr l / u + K us According to the vehicle nominal parameters (Table 1), mr u / Cr is negligible compared with ( xla + b) / u . So, f1 is determined as,
where, f 0 =
f1 =
Equations (18) show that while the estimation of the driver behaviour is within the lane borders the controller has no effect on the driver demands. Otherwise, if the driver input keeps the vehicle out of the safe region, the controller becomes active. It should be noted the vehicle model (Gˆ ) is only used for generating the condition of equation (18) and has no direct influence on the closed loop behaviour. According to equation (18), when the estimation of driver desired lateral offset is in the safe region, the controller has no effect on the system. In the unsafe area the driver demand is considered as a disturbance for the control system. If a the same analysis is repeated for the direct approach, it is obtained that: GGC G yL δd − 1 + GGC 1 + GGC ˆ 1 + GG ⇒ yla = C ylad 1 + GGC
)
(
)
(
GGC G yL = ⇒ yla δd + 1 + GGC 1 + GGC
if yˆ lad > yL
)
1
2
( xla + b) f0 . u
(21)
Bode Diagram Diagram Bode
Magnitude (dB)
150 4th order model
100
2nd order model
50
0
Phase (deg)
-50 -90
-135
-180 -2 10
-1
10
0 Frequency 0 10(rad/sec)
10
1
10
2
10
Frequency (rad/sec)
if yˆ lad < − y= L ⇒ yla if yˆ lad ≤ yL
(20)
Fig. 6 represents the bode diagram of the original 4th order model and the reduced order. As it can be seen from figure the reduced order model provides a good approximation up to 10 rad/s.
(18)
GGC G yL δd + 1 + GGC 1 + GGC
(
[1 0]
Fig. 6. Bode diagrams of the reduced and original models. (19)
The controller will affect the driver experience also when in the safe region. Equations (19) also point out that if Gˆ = G the both direct and proposed method work similar. 4. ROBUST CONTROLLER DESIGN In this section a robust road departure avoidance closed loop controller is designed. The model derived in Section 2 can be reduced using the balanced truncation method (Skogestad et al., 2007). The state space of the reduced order model is determined as,
The reduced order model is affected by two sources of error: the order reduction and parameters uncertainty. These modelling errors are modelled as multiplicative uncertainty (Skogestad et al., 2007): (22) G (s ) = (1 +W m (s )∆ m (s ))Gˆ (s ), ∆ m ( j ω ) ≤ 1, ∀ω where Gˆ (s ) is transfer function of the nominal reduced order plant, ∆ m ( s ) is a normalized complex function which represents parametric and dynamic modeling errors, and Wm ( s ) is a known function bounding the modeling error. A description of ∆ m ( s ) based on vehicle complex uncertainty is illustrated in Fig. 7. The conservative bound on modelling error is derived as, s+2 Wm ( s ) = s+3
2
(23)
According to (18) the DDW and the estimated model only determines the activation of the road departure avoidance system. When it is active the saturation block generates a reference that the controller tracks. 8430
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
dangerous manoeuvre and then the dynamic behaviours of the proposed strategy are compared to the other approaches.
Table 1. Vehicle parameters with uncertainty. Parameter Uncertainty Value Nominal Value 1100-1500 Kg 1300 Kg m 2550-3450 Kg.m2 3000 Kg.m2 IZ
Cf
50000-100000 N/rad
100000 N/rad
Cr
80000-160000 N/rad
160000 N/rad
a b u xla
15 m/s – 35 m/s -
1m 1.5 m 25 m/s 45 m
vehicle complex uncertainty
5 0
Fig. 8. Interconnection scheme.
Magnitude (dB)
-5
Fig. 9 shows the lateral offset at the look ahead distance, the vehicle trajectory and the control action as a response to a steering angle step input. This simulation is representative of a driver falling asleep, for example. As expected in this case the controller activates as soon as yla exits the safe bounds and the vehicle is successfully kept on the road.
-10 -15 -20 -25 -30 -3 10
-2
10
-1
10
0
10 Frequency (rad/sec)
1
10
2
10
3
10
Fig. 7. Multiplicative uncertainty of vehicle ∆ m ( s ) and weighting function Wm ( s ) (bold line). Thanks to these considerations the closed loop controller can be designed neglecting the DDE and focusing only on the vehicle dynamics. To guarantee robustness, an H∞ controller is designed based on the interconnection scheme shown in Fig. 8. The reference model design approach is easily recognized from the scheme. The design relies on the definition of several weighting functions: • Gref is the reference model; it defines the desired response of the vehicle. It is a simple first order linear model with a pole at s=-1 and unity gain to guarantee good tracking of yL. • The model matching error is weighted by W s (s ) = 0.01(s + 10) / (s + 0.001) . It penalizes the low- frequency matching error, guaranteeing a small DC-error. • The weighting filter captures the limit on the actuators. The weight W u (s ) =10(s + 1) / (s + 10) limits fast actuator. • Wd models the disturbance acting on the plant. In this scheme the disturbance is the driver’s steering action that can be modelled as a low pass filter (Switkes, 2005) W d (s ) =0.1(s + 10) / (s + 1) . • W n (s ) = 0.01(s + 1) / (s + 100) models sensor noise. It is thus possible to design a robust controller Gc that satisfies the robust stability condition and has a γ < 1. 5. SIMULATION RESULTS In this section the performance of the proposed system is validated in simulation. The system is first validated with a
Fig. 9. Response to a constant steering angle (driver falling asleep). All the approaches previously discussed are able to deal with this situation in a satisfactory way. Analysing a case which shows the effect of each approach on driver’s intent is more interesting. Fig. 10 shows the results of a simulation for the direct, indirect and proposed control schemes. In the simulation, the driver applies constant steering angle till 9 seconds; subsequently he tries to steer the vehicle back to the centre of the road. Notice that the reduced order model is used in the estimation block of both the proposed and direct methods. From figure the following comments can be drawn: (1) all three approaches are safe; none of them causes the vehicle to leave the road. (2) The indirect approach tends to overcorrect; this is due to the intrinsic asymmetry of the
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
architecture. In the presented form the control system can only correct if the vehicle is leaving the road. Once the vehicle has been pointed back to the road the error as seen by the controller is zero and thus the controller action (because of the integrator like behavior) will respond very slowly. A deactivation logic is needed to avoid interfering with the driver (see for example (Minoiu et al., 2009)). (3) As pointed out by (Cerone et al., 2009) the direct approach does not need such deactivation logic. However the figure shows that the performance of the direct method heavily depends on the accuracy of the model. The proposed method is more robust to model uncertainties. Because of modelling errors, the direct approach considerably changes the dynamics of the vehicle by applying a positive steering angle during the first phase of the manoeuvre. Notice further that when the manoeuvre is concluded the proposed approach control action returns to zero while a residual control action is left by the direct method. The proposed method gives a more natural feeling to the driver. 6. CONCLUSIONS
deltad (deg)
The problem of designing a robust road departure avoidance system has been presented. The proposed approach is based on a closed-loop estimation of the driver intentions. It is shown, that the proposed approach successfully prevents the vehicle from leaving the road and has a reduced effect on the driver desired intent which is the main factor in user acceptance of the DAS. 0.2 0
-0.2 0
6
5
2 indirect method
0
direct method
-2 -4 0 6
desired 5
10
Y (m)
2
indirect method
0 -2 -4 0 2
15
proposed controller
4
correction steering angle (deg)
15
proposed controller
4 Y la (m)
10
direct method 5 direct method
1
desired
10
15
indirect method proposed controller
0 -1 0
5
time (sec)
10
Fig. 10. Effects of the driver intervention on the three discussed controllers.
15
REFERENCES Alirezaie, M., A. Ghaffari and R. Kazemi (2010), a SVFC Approach to Design a Robust Lane Keeping Assistance System, International Symposium on Advanced Vehicle Control (AVEC), UK. Boyraz, P., A. Sathyanarayana, J. L. Hansen and E. Jonsson, (2009), Driver Behavior Modeling Using Hybrid Dynamic Systems for ‘Driver-aware’ Active Vehicle Safety, International Technical Conference on the Enhanced Safety of Vehicles (ESV), Germany. Cerone, V., M. Milanese and D. Regruto (2009), Combined Automatic Lane-Keeping and Driver’s Steering Through a 2-DOF Control Strategy, IEEE Transactions on Control Systems Technology, Vol. 17, 135-142. Fletcher, L., A. Zelinsky (2006), Context Sensitive Driver Assistance Based on Gaze – Road Scene Correlation, ISER,06. Gillespie, T. D. (1992), Fundamentals of Vehicle Dynamics, Society of Automotive Engineers, Warrendale, PA. Kim, S. Y. and J.K. Kang (2008), An Intelligent and Integrated Driver Assistance System for Increased Safety and Convenience based on all-around Sensing, Journal of Intelligent Robot System, Vol. 51, 261–287. Katzourakis D., C. Droogendijk, D. Abbink, R. Happee, E. Holweg, (2010) "Force-Feedback Driver Model for Objective Assessment of Automotive Steering Systems", Proceedings of the 10th International Symposium on Advanced Vehicle Control. Minoiu, N., M. Netto, S. Mammar and B. Lusetti (2009), Driver Steering Assistance for Lane Departure Avoidance, Control Engineering Practice, Vol. 17, pp. 642-651. Moore B.C. (1981), Principal Component Analysis in Linear Systems Controllability, Observability, and Model Reduction, IEEE Transactions on Automatic Control, Vol. AC-26, 17-32. Oya, M. and Q. Wang (2007), Adaptive Lane Keeping Controller for Four-Wheel-Steering Vehicles, IEEE International Conference on Control and Automation, China. Patwardhan, S. H. Tan, and J. Guldner, “A general framework for automatic steering control: System analysis,” in Proc. Amer. Control Conf.,1997, pp. 1598– 1602. Petersson, L., L. Fletcher, A. Zelinsky (2005), a Framework for Driver-in-the-Loop Driver Assistance Systems, IEEE Conference on Intelligent Transportation Systems, Austria. Rossetter, E. J. (2003), a Potential Field framework for Active Vehicle Lane Keeping Assistance, a dissertation of doctor of philosophy, Stanford University. Skogestad, S. and I. Postlethwaite (2007), Multivariable Feedback Control, chapter 1, 7, 9, 10, 11, 2nd edition, John Wiley, England. Switkes, J.P. (2005), Handwheel Force Feedback With Lanekeeping Assistance: Combined Dynamics, Stability and Bounding, a dissertation of doctor of philosophy, Stanford University.
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Robust Road Departure Avoidance Based on Driver Decision Estimation M. Alirezaei*, M. Corno**, A. Ghaffari*, R. Kazemi* *Mechanical Engineering Department, K. N. Toosi University, Tehran, Iran, (e-mail: [email protected]) ** Delft Center for Systems and Control, Delft University of Technology Delft, Netherlands, (e-mail: [email protected]) Abstract: In this paper a robust road departure avoidance system based on a closed-loop driver decision estimator (DDE) is presented. The goal is that of incorporating the driver intent in the control of the vehicle. The proposed system ensures that the vehicle remains within the road borders with minimal effect on the driver demand. Numerical simulations show the effectiveness and robustness of the proposed approach. A comparison between the presented method and the generalization of methods available in the literature is presented. 1. INTRODUCTION Statistics show that most car accidents are associated with drivers’ errors (Kim et al., 2008). Driver assistance systems (DAS) could therefore improve road safety considerably. DAS can correct driver’s input in case of distractions or errors (Oya et al., 2007). Several systems are being developed: adaptive cruise control (ACC), road departure avoidance (RDA) and driver drowsiness detection are examples of such systems. Highway road departure avoidance (Alirezaie et al., 2010, Rossetter, 2003, Oya et al., 2007) is one of the most interesting applications. Such systems use active steering to prevent the vehicle from departing the road (or lane in some applications). They are particularly promising because on one hand, being developed for highway driving, they can take advance of the highly structured environment (mostly straight roads with well marked lanes, unidirectional traffic and rather uniform surface) and on the other hand, given the high driving velocities, they have a great potential of saving lives. Driving assistance systems can be classified along two axes: sensing scheme and interaction with the user. Two sensing schemes exist (Shladover, 1991): look down where the measurement of the vehicle lateral position relies on buried magnet (or other devices on the road) and look ahead in which the position of the vehicle is measured via visual feedback. The second approach is advantageous for two reasons: it does not require modification of the infrastructure and the look ahead capability yields a better control of the trajectory (Patwardhan et al, 1997). The second axis refers to how the user’s input is accounted for. One of the most daunting tasks for DAS’s is to meet the safety objectives without altering the driving experience when not needed. These difficulties arise from the problem of assessing the driver’s intentions (Boyraz et al. 2009). With this regard two different approaches have been explored. The indirect approach does not consider the driver as part of the control loop (Minoiu et al., 2009). Such systems are switched on when the hazard condition is detected and neglect the driver 978-3-902661-93-7/11/$20.00 © 2011 IFAC
decision. In case of false activation they heavily modify the behaviour of the vehicle and may not be well accepted. The second approach, referred to as the direct approach, accounts for the driver’s intentions (Cerone et al., 2009, Petersson et al., 2005, Rossetter, 2003). Measureable user inputs like steering torque or steering wheel angle are used to interpret his/her intentions. Although the direct approach has the potential of not altering the driving experience, the methods usually rely on an open loop estimation of the driver intention (Cerone et al., 2009). This may introduce the generation of a control action also when not needed. This paper proposes a novel road departure avoidance system. The proposed method is a look ahead method that extends the direct approach by adding a closed loop Driver Decision Estimator (DDE). It is assumed that the vehicle is equipped with a visual system to measure the lateral position of the vehicle and with an active steering system. The DDE yields an unbiased estimation of the driver’s desired path based on the position of the vehicle and control input. The proposed system is constantly active so no on/off strategy is needed. It is shown that when the vehicle is pointing toward a safe region no control action is performed (and thus the vehicle dynamics are not altered), and even in case of aggressive manoeuvres that are erroneously interpreted as dangerous the proposed controller has a minimal effect of the vehicle dynamics. The content of the paper is organized as follows. In Section 2, the vehicle-road model is described. A more precise classification of the available approaches is given in Section 3 and the concept of the DDE is introduced. In Section 4 a robust controller is designed. Simulation results are presented in Section 5. 2. VEHICLE-ROAD MODEL STRUCTURE The vehicle is assumed to be equipped with a vision system and a steering actuator. The vision system measures the lateral offset between the vehicle and the road centreline xla meters away (see Fig. 1). In figure, two reference frames are
8427
10.3182/20110828-6-IT-1002.02923
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
defined: n-t and nla-tla the n axis being normal to the road centerline and the t axis tangential to it. The origins of the reference coordinates are respectively located at the C.G. of the vehicle and at the intersection between the centerline and the normal to the road through the point xla meters ahead of the vehicle. The distance between this point and the centerline is yla and represents the visual feedback signal. The components of the vehicle velocity in the n-t coordinates are: Vn u sin(ψ V ) + v cos(ψ V ) = (1) Vt = −v sin(ψ V ) + u cos(ψ V )
ψ V is the yaw angle of the vehicle with respect to the centreline. From (1) and defining ψ R as the rotation between
the n-t and nla-tla reference frames, the time derivative of yla is:
= yla Vt sin(ψ R ) + (Vn + xlaψV ) cos(ψ R )
(2)
substituting (1) in (2) one obtains: Fig. 1. Vehicle and road reference frames.
yla = ( −v sin(ψ V ) + u cos(ψ V ) ) sin(ψ R )
(3)
+ ( u sin(ψ V ) + v cos(ψ V ) + xlaψV ) cos(ψ R )
Note that the model, although simple, is suitable for highway traffic. In fact unmodeled dynamics are little excited in highway driving, usually characterized by slow steering action, slow velocity variation and small pitch and roll angles. The nominal parameters used in the model are shown in Table 1.
Assuming ψ V and ψ R are small enough i.e. (sin(ψ V ) ≈ ψ V , sin(ψ R ) ≈ ψ R , ψ Rψ V ≈ 0) , yla is determined as: (4) yla = v + uψ VR + xlaψV where, ψ VR = ψ V +ψ R . (5) The time derivative of ψ VR is,
ψVR = ψV +ψ R .
(6) The magnitude of ψ R is defined as the vehicle longitudinal velocity divided by road curvature (u / R ) : (7) ψVR= ψV + (−u / R). Rewriting the classical bicycle model (Gillespie, 1992) augmented with the look ahead distance equations (4) and (7), the vehicle-road model can be described in state space as: x = Ax + Bδ + Ew (8) yla = Cx
The remaining symbols are: v lateral velocity r = ψV yaw rate front wheel angle δ m vehicle mass u longitudinal velocity Iz vehicle moment of inertia around yaw axis
C
a11 a 21 1 0
0 0 1 0] , w [=
a12 a22 xla 1
0 0 0 0 ,B = 0 u 0 0
b11 b 21 = ,E 0 0
Cr
a11
cornering stiffness of rear tires distance between front axle and the C.G. distance between rear axle and the C.G. = −(Cr + C f ) / mu
a12
= (bCr − aC f ) / mu − u
a21
0 0 0 −u
1 R
From (8), the transfer function from the steering angle to yla is: y (s ) m s 2 + m s + m0 G (s ) = la =C (sI − A ) −1 B = 2 2 2 1 δ (s ) s (n 2s + n1s + n 0 ) .
cornering stiffness of front tires
a b
where, v r , A x = = yla ψ VR
Cf
(9)
The two integrators arise from positional states and two other poles determine the vehicle handling.
= (bCr − aC f ) / I z u
a22
= −(b 2 Cr + a 2 C f ) / I z u
b11
= Cf / m
b21
= aC f / I z 3. DRIVER DECISION ESTIMATOR
In this section different approaches to design RDA systems and their characteristics with respect to the driver’s input are explained in details and the novel scheme is proposed. Accounting for driver intents is critical for the user acceptance of the DAS. To better assess the issue consider the uncontrolled case of Fig.2.
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
accurate model of the vehicle is used then the ideal objective is achieved: the control system intervention is limited only to the period when |yla|> yL. The main problem with this method lies in modelling errors. yˆ lad is generated in open loop and any error in the model will modify the driving experience, yielding a control action also when the manoeuvre is safe.
20
yla (m)
15 without control 10 safety limit
5
0 0
1
2
desired
3 Time (sec)
4
5
6
18 16 14
y (m)
12
without control
10 8 road border
6
desired
4 2 0 0
1
2
3 Time (sec)
4
5
Fig. 4. Block diagram of the direct method.
6
Fig. 2. A typical maneuver for a distracted driver.
Here we propose a modification of the direct approach based on the closed loop generation of yˆ lad . Fig. 5 shows the proposed block diagram of the RDA system based on the Driver Decision Estimator (DDE). As before, Gˆ is an estimated transfer function of the vehicle model. According to the block diagram and considering G as a transfer function of the vehicle, yla is computed as δ G (δ C + δ d ) . y= G= (10) la The driver desired lateral offset between vehicle C.G. and centreline at the lookahead distance ( ylad ) is defined as, ylad = Gδ d . (11) According to (10) and (11) ylad is calculated as, ylad = yla − Gδ C (12) . The control signal (δ C ) , is computed as,
In this scenario a driver keeps a constant steering angle of 1 deg for 5 seconds; after this period, he tries to return the vehicle to the road border by steering to the opposite direction. Ideally the objective is to keep the vehicle in the safe region (that yL extend meters to boths side of the center line), one way to do is to force yla to stay in the safe region. So, it is desirable that the system keeps yla in the safety limit; on the other hand it is also important that the systems responds as soon as the driver acts on the steering wheel to correct the trajectory. In the introduction two approaches are outlined: the direct and indirect approaches. The generalized block diagram of the indirect concept is drawn in Fig. 3. δd , δC and GC are the driver input, control input, and transfer function of the controller. The saturation is used to define the safe region for = δ C GC ( yd − yla ) yla as shown in Fig.2. This approach does not take the driver explicitly into account. The main drawback of this approach is it considerably affects the dynamics of the vehicle also when not needed. Consider the above maneuver, during the time period when |yla|> yL an active steering angle is generated; once the driver steers back toward the centerline the control system should be deactivated, but if a deactivation logic is not present, the control action will got to zero with Gc dynamics, thus a “residual” undesired control action will be present.
(13)
Fig. 5. Block diagram of the proposed method. where, yd is the desired lateral offset accounting for the driver’s demand and road limitation, which is defined as, = yd η yˆlad + λ ,
(14)
yˆlad is the estimation of the driver desired lateral offset
between the vehicle C.G. and centreline at the lookahead distance. It is computed in the proposed DDE as,
Fig. 3. Block diagram of the indirect approach. In the direct concept (see Fig. 4), the driver desired path is estimated by using the driver input, for example in Cerone et al., 2009) the steering torque is measured. In general, the driver behaviour is computed based on driver’s input and an estimated vehicle-road model (Gˆ ) . If an
yˆlad = yla − Gˆ δ C .
η and λ describes the saturation:
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
(
)
if yˆ lad < − yL ⇒ η = 0 and if
( yˆ
lad
)
λ = − yL
≤ y= = λ 0 L ⇒ η 1 and
(
)
if yˆ lad > y= and λ yL 0 = L ⇒η
( ( xla + b) / u − mr u / Cr ) 0 u l / u 2 + K us Cˆ ˆ = ˆ = A B = , 0 0 (16) , 1/ u l / u 2 + K us
where yL is the road border. Substituting (13)-(16) into (10), yla can be written as a function of δ d , η and λ , (1 + η GC Gˆ )G yla = δd 1 + GC G + η GC (Gˆ − G ) (17) GC G + λ 1 + GC G + η GC (Gˆ − G ) Depending on the status of the saturation the following cases are given:
(
)
(
)
if yˆ lad < − y= L ⇒ yla
GGC G yL δd − 1 + GGC 1 + GGC
if yˆ lad ≤ yL ⇒ yla = ylad
(
)
if yˆ lad > yL = ⇒ yla
where, l= a + b , mr = ma / l and = K us m(bCr − aC f ) / lC f Cr . The transfer function of the reduced order model is then f s+ f Gˆ ( s ) = 1 2 0 s
( x + b) mr u and f1 la = − f0 . u Cr l / u + K us According to the vehicle nominal parameters (Table 1), mr u / Cr is negligible compared with ( xla + b) / u . So, f1 is determined as,
where, f 0 =
f1 =
Equations (18) show that while the estimation of the driver behaviour is within the lane borders the controller has no effect on the driver demands. Otherwise, if the driver input keeps the vehicle out of the safe region, the controller becomes active. It should be noted the vehicle model (Gˆ ) is only used for generating the condition of equation (18) and has no direct influence on the closed loop behaviour. According to equation (18), when the estimation of driver desired lateral offset is in the safe region, the controller has no effect on the system. In the unsafe area the driver demand is considered as a disturbance for the control system. If a the same analysis is repeated for the direct approach, it is obtained that: GGC G yL δd − 1 + GGC 1 + GGC ˆ 1 + GG ⇒ yla = C ylad 1 + GGC
)
(
)
(
GGC G yL = ⇒ yla δd + 1 + GGC 1 + GGC
if yˆ lad > yL
)
1
2
( xla + b) f0 . u
(21)
Bode Diagram Diagram Bode
Magnitude (dB)
150 4th order model
100
2nd order model
50
0
Phase (deg)
-50 -90
-135
-180 -2 10
-1
10
0 Frequency 0 10(rad/sec)
10
1
10
2
10
Frequency (rad/sec)
if yˆ lad < − y= L ⇒ yla if yˆ lad ≤ yL
(20)
Fig. 6 represents the bode diagram of the original 4th order model and the reduced order. As it can be seen from figure the reduced order model provides a good approximation up to 10 rad/s.
(18)
GGC G yL δd + 1 + GGC 1 + GGC
(
[1 0]
Fig. 6. Bode diagrams of the reduced and original models. (19)
The controller will affect the driver experience also when in the safe region. Equations (19) also point out that if Gˆ = G the both direct and proposed method work similar. 4. ROBUST CONTROLLER DESIGN In this section a robust road departure avoidance closed loop controller is designed. The model derived in Section 2 can be reduced using the balanced truncation method (Skogestad et al., 2007). The state space of the reduced order model is determined as,
The reduced order model is affected by two sources of error: the order reduction and parameters uncertainty. These modelling errors are modelled as multiplicative uncertainty (Skogestad et al., 2007): (22) G (s ) = (1 +W m (s )∆ m (s ))Gˆ (s ), ∆ m ( j ω ) ≤ 1, ∀ω where Gˆ (s ) is transfer function of the nominal reduced order plant, ∆ m ( s ) is a normalized complex function which represents parametric and dynamic modeling errors, and Wm ( s ) is a known function bounding the modeling error. A description of ∆ m ( s ) based on vehicle complex uncertainty is illustrated in Fig. 7. The conservative bound on modelling error is derived as, s+2 Wm ( s ) = s+3
2
(23)
According to (18) the DDW and the estimated model only determines the activation of the road departure avoidance system. When it is active the saturation block generates a reference that the controller tracks. 8430
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
dangerous manoeuvre and then the dynamic behaviours of the proposed strategy are compared to the other approaches.
Table 1. Vehicle parameters with uncertainty. Parameter Uncertainty Value Nominal Value 1100-1500 Kg 1300 Kg m 2550-3450 Kg.m2 3000 Kg.m2 IZ
Cf
50000-100000 N/rad
100000 N/rad
Cr
80000-160000 N/rad
160000 N/rad
a b u xla
15 m/s – 35 m/s -
1m 1.5 m 25 m/s 45 m
vehicle complex uncertainty
5 0
Fig. 8. Interconnection scheme.
Magnitude (dB)
-5
Fig. 9 shows the lateral offset at the look ahead distance, the vehicle trajectory and the control action as a response to a steering angle step input. This simulation is representative of a driver falling asleep, for example. As expected in this case the controller activates as soon as yla exits the safe bounds and the vehicle is successfully kept on the road.
-10 -15 -20 -25 -30 -3 10
-2
10
-1
10
0
10 Frequency (rad/sec)
1
10
2
10
3
10
Fig. 7. Multiplicative uncertainty of vehicle ∆ m ( s ) and weighting function Wm ( s ) (bold line). Thanks to these considerations the closed loop controller can be designed neglecting the DDE and focusing only on the vehicle dynamics. To guarantee robustness, an H∞ controller is designed based on the interconnection scheme shown in Fig. 8. The reference model design approach is easily recognized from the scheme. The design relies on the definition of several weighting functions: • Gref is the reference model; it defines the desired response of the vehicle. It is a simple first order linear model with a pole at s=-1 and unity gain to guarantee good tracking of yL. • The model matching error is weighted by W s (s ) = 0.01(s + 10) / (s + 0.001) . It penalizes the low- frequency matching error, guaranteeing a small DC-error. • The weighting filter captures the limit on the actuators. The weight W u (s ) =10(s + 1) / (s + 10) limits fast actuator. • Wd models the disturbance acting on the plant. In this scheme the disturbance is the driver’s steering action that can be modelled as a low pass filter (Switkes, 2005) W d (s ) =0.1(s + 10) / (s + 1) . • W n (s ) = 0.01(s + 1) / (s + 100) models sensor noise. It is thus possible to design a robust controller Gc that satisfies the robust stability condition and has a γ < 1. 5. SIMULATION RESULTS In this section the performance of the proposed system is validated in simulation. The system is first validated with a
Fig. 9. Response to a constant steering angle (driver falling asleep). All the approaches previously discussed are able to deal with this situation in a satisfactory way. Analysing a case which shows the effect of each approach on driver’s intent is more interesting. Fig. 10 shows the results of a simulation for the direct, indirect and proposed control schemes. In the simulation, the driver applies constant steering angle till 9 seconds; subsequently he tries to steer the vehicle back to the centre of the road. Notice that the reduced order model is used in the estimation block of both the proposed and direct methods. From figure the following comments can be drawn: (1) all three approaches are safe; none of them causes the vehicle to leave the road. (2) The indirect approach tends to overcorrect; this is due to the intrinsic asymmetry of the
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
architecture. In the presented form the control system can only correct if the vehicle is leaving the road. Once the vehicle has been pointed back to the road the error as seen by the controller is zero and thus the controller action (because of the integrator like behavior) will respond very slowly. A deactivation logic is needed to avoid interfering with the driver (see for example (Minoiu et al., 2009)). (3) As pointed out by (Cerone et al., 2009) the direct approach does not need such deactivation logic. However the figure shows that the performance of the direct method heavily depends on the accuracy of the model. The proposed method is more robust to model uncertainties. Because of modelling errors, the direct approach considerably changes the dynamics of the vehicle by applying a positive steering angle during the first phase of the manoeuvre. Notice further that when the manoeuvre is concluded the proposed approach control action returns to zero while a residual control action is left by the direct method. The proposed method gives a more natural feeling to the driver. 6. CONCLUSIONS
deltad (deg)
The problem of designing a robust road departure avoidance system has been presented. The proposed approach is based on a closed-loop estimation of the driver intentions. It is shown, that the proposed approach successfully prevents the vehicle from leaving the road and has a reduced effect on the driver desired intent which is the main factor in user acceptance of the DAS. 0.2 0
-0.2 0
6
5
2 indirect method
0
direct method
-2 -4 0 6
desired 5
10
Y (m)
2
indirect method
0 -2 -4 0 2
15
proposed controller
4
correction steering angle (deg)
15
proposed controller
4 Y la (m)
10
direct method 5 direct method
1
desired
10
15
indirect method proposed controller
0 -1 0
5
time (sec)
10
Fig. 10. Effects of the driver intervention on the three discussed controllers.
15
REFERENCES Alirezaie, M., A. Ghaffari and R. Kazemi (2010), a SVFC Approach to Design a Robust Lane Keeping Assistance System, International Symposium on Advanced Vehicle Control (AVEC), UK. Boyraz, P., A. Sathyanarayana, J. L. Hansen and E. Jonsson, (2009), Driver Behavior Modeling Using Hybrid Dynamic Systems for ‘Driver-aware’ Active Vehicle Safety, International Technical Conference on the Enhanced Safety of Vehicles (ESV), Germany. Cerone, V., M. Milanese and D. Regruto (2009), Combined Automatic Lane-Keeping and Driver’s Steering Through a 2-DOF Control Strategy, IEEE Transactions on Control Systems Technology, Vol. 17, 135-142. Fletcher, L., A. Zelinsky (2006), Context Sensitive Driver Assistance Based on Gaze – Road Scene Correlation, ISER,06. Gillespie, T. D. (1992), Fundamentals of Vehicle Dynamics, Society of Automotive Engineers, Warrendale, PA. Kim, S. Y. and J.K. Kang (2008), An Intelligent and Integrated Driver Assistance System for Increased Safety and Convenience based on all-around Sensing, Journal of Intelligent Robot System, Vol. 51, 261–287. Katzourakis D., C. Droogendijk, D. Abbink, R. Happee, E. Holweg, (2010) "Force-Feedback Driver Model for Objective Assessment of Automotive Steering Systems", Proceedings of the 10th International Symposium on Advanced Vehicle Control. Minoiu, N., M. Netto, S. Mammar and B. Lusetti (2009), Driver Steering Assistance for Lane Departure Avoidance, Control Engineering Practice, Vol. 17, pp. 642-651. Moore B.C. (1981), Principal Component Analysis in Linear Systems Controllability, Observability, and Model Reduction, IEEE Transactions on Automatic Control, Vol. AC-26, 17-32. Oya, M. and Q. Wang (2007), Adaptive Lane Keeping Controller for Four-Wheel-Steering Vehicles, IEEE International Conference on Control and Automation, China. Patwardhan, S. H. Tan, and J. Guldner, “A general framework for automatic steering control: System analysis,” in Proc. Amer. Control Conf.,1997, pp. 1598– 1602. Petersson, L., L. Fletcher, A. Zelinsky (2005), a Framework for Driver-in-the-Loop Driver Assistance Systems, IEEE Conference on Intelligent Transportation Systems, Austria. Rossetter, E. J. (2003), a Potential Field framework for Active Vehicle Lane Keeping Assistance, a dissertation of doctor of philosophy, Stanford University. Skogestad, S. and I. Postlethwaite (2007), Multivariable Feedback Control, chapter 1, 7, 9, 10, 11, 2nd edition, John Wiley, England. Switkes, J.P. (2005), Handwheel Force Feedback With Lanekeeping Assistance: Combined Dynamics, Stability and Bounding, a dissertation of doctor of philosophy, Stanford University.
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