4. Longitudinal Control System Design
4.3 Collision Warning/Avoidance Maneuver
There are two stages in the CW/CA strategy. The first stage is the computation of warning distance for the normal warning situation. The second stage is the computation of braking distance for the guarantee of collision avoidance. It is possible for a driver to begin a driving collision avoidance maneuver earlier than a braking collision avoidance maneuver. Therefore, the CW/CA strategy should employ a non-conservative warning distance before that the driver begins the steering maneuver, and a conservative braking distance that if less than this distance, then the emergency steering maneuvers is started.
In Fig. 4-12, the dynamics of car-following is illustrated. Initially the headway distance between the preceding vehicle and the following vehicle is dr . The preceding vehicle starts to brake with the deceleration ap and its traveling distance to stop is Vp 2/2ap. After the reaction time Tc of the driver and the brake actuator of the following vehicle, the traveling distance to stop with the deceleration af is Vf Tc + Vf2/2af . Therefore, the anti-collision distance can be
Note that the warning distance (4-59) is also definition of anti-collision distance in ISO 15263 [50] which is specified to CW/CA systems. As to the computation of the braking distance, the time-to-collision (TTC) braking distance is applied. The relative distance dr
dynamics at any time t between two vehicles can be presented as
0 0
0 0 0 0
( ) ( ) ( ( ) ( ))( ) t ( ( ) ( ))
r r p f t t p f
d t =d t + V t −V t t t− +
∫ ∫
τ a s −a s dsdτ (4-60) where t0 denotes the time at which the measurement of TTC is required.When the time t > t0 and dr(t)=0, TTC can be obtained as Tc =t-t0. Since the computation of TTC requires the profiles of deceleration for both preceding and following vehicle, here the assumptions of the maximum deceleration for the preceding vehicle and zero deceleration for the following vehicle are made, i.e., a tp( )=amax− , a tf( ) 0= . In the computation of dbr , TTC Fig. 4-12. Headway distance for car-following.
can be viewed as the total delay which composes of the driver reaction time and the braking
To avoid the discontinuity which exists in the switch between the warning distance and the braking distance, an evaluate signal based on the rational form is given as
r br
Remark 1: The accelerations of the preceding and the following vehicle are both assumed with the same maximum values, and Vp is replaced by Vrel , then the warning distance defined in Eq. (4-59) can be rewritten as be positive for indicating safety while d < dbr . By examining the braking distance in (4-61) and the modified warning distance in (4-63), the low bound of d0 shall be reasonably constrained to the value amax total− τ2 /2 such that dw > dbr is guaranteed.
Remark 2: The values for all parameters used in the CW/CA strategy can be referred to [50]
in which the statistics are specified from many practical tests. The maximum deceleration for the preceding and the following vehicle are both given as 6 m/s2, and the delay time for the human and the braking system is chosen as 0.6 s and 0.2 s, respectively.
The evaluate signal in (4-62) can give a graded display about the event likelihood for the driver. The condition of Iw > 1 corresponds to dr > dw , and it denotes that the headway distance is safe for driving. The condition of 0 < Iw < 1 corresponds to dbr < dr < dw , and this range can be divided into several graded degrees for warnings. If Iw ≤ 0, then this condition corresponds to dr ≤ dbr and the emergency braking should immediately be acted.
The next issue is to define the graded degrees for condition 0 < Iw < 1. In research on traffic conflicts techniques, TTC has proven to be an effective measure for rating the severity of
collision [51]. In general, only encounters with a minimum TTC less than 2 s are considered as a critical situation for drivers. Therefore, here we can compute the Iw corresponding to TTC while the critical situation is encountered. By substituting (4-61) and (4-63) into (4-62), and considering the critical braking for the preceding vehicle (Vp → 0), the warning evaluate signal can be rewritten as
2
By arranging (4-64) into the following equation,
2
The left term is the definition for TTC (as Vp = 0), and we can obtain the relation between TTC and Iw under different velocities, as shown in Fig. 4-13.
Fig. 4-13. The relation between the TTC and the warning evaluate signal with respect to different velocities.
Fig. 4-14. Warning degree in three zones.
With the increased driving velocity, TTC=2 s corresponds to less order of the warning evaluate signal. The order 0.4 which represents TTC=2 s at the high velocity 35 m/s (=126 km/h) is then chosen as the indication from normal situation to critical situation. In order to characterize different warning levels, three zones in Fig. 4-14 are defined as
- Green zone: Iw > 1: the current inter-distance d is larger than the warning distance dw . This is safe range for car-following operation.
- Yellow zone: 0.4 < Iw ≤ 1: the current inter-distance dr is smaller than dw . This range indicates the driver that the following distance is not safe and the vehicle should be decelerated.
- Red zone: Iw ≤ 0.4: the current inter-distance dr reaches the braking distance dbr . In this range, the braking should be immediately applied to avoid collision with the preceding vehicle.
Chapter 5
Combined Longitudinal and Lateral Control Design
5.1. Nonlinear Vehicle Longitudinal and Lateral Coupling Dynamics
Fig. 5-1. 3-DOF vehicle model.
By considering a vehicle moving over a flat and level road surfaces, the 3 degrees-of-freedom (DOF) vehicle dynamics is represented by the longitudinal velocity vx , the lateral velocity vy , and the yaw rate γ, as shown in Fig. 5-1. The 3 DOF dynamics, including aerodynamic terms in the vehicle dynamics, is depicted as
cos sin 2
x xr xf f yf f y D x
mv =F +F δ −F δ +m vγ −k v (5-1)
sin cos
y yr xf f yf f x
mv = F +F δ +F δ −m vγ (5-2)
sin cos 2wb ( cos )
z xf f yf f yr xr xf f
I γ=aF δ +aF δ −bF + L ∆F + ∆F δ (5-3)
with ,
, ,
xf xfr xfl xr xrr xrl
xf xfr xfl xr xrr xrl
F F F F F F
F F F F F F
= + = +
∆ = − ∆ = −
Although not shown, the model (5-1)-(5-3) used here also includes quasi-static compensation for the effect of roll and pitch on tire vertical forces [52]. Compensation for roll and pitch effects can be calculated as a function of center of gravity height with lateral acceleration, and longitudinal acceleration, respectively. For simplicity of control design, it is assumed that the vehicle body is symmetric about the longitudinal plane such as the differential forces of tires can be eliminated, i.e., ∆Fxf = ∆Fxr = 0, and equal slip angles on the left and right tires can be By using a linear tire model, the lateral forces are given as
yf f f
F = −C α (5-6)
yr r r
F = −Cα (5-7) In addition, the longitudinal forces of tires can be written as
( 2) Substituting the expressions in (5-4) through (5-9) and making small angle approximation can yield
By assuming a vehicle has throttle, brake, and steer-by-wire capabilities so that steering,
braking and two driving wheels can be controlled, the equations then can be rewritten as ( ) ( , )c
Mq == q +g q u (5-13) where q =[v vx yγ]T and the control vector u =[c FT δf]T; M is the positive definite mass matrix, ( )= q contains the terms without influence of control inputs, and ( , )g q u has the c remaining controlled terms as follows
( , , )z equations can then be used to solve for the control vector, uc.
The dynamics in (5-14) through (5-16) are in terms of absolute states which are defined with respect to a fixed world reference coordinate. However, the interesting aspect of vehicle automation control is the forward velocity, relative lateral offset and heading angle between the vehicle’s CG and the road centerline. The yaw error ψe is the deviation between the heading angle ψ and the desired heading ψd of the local tangent to the roadway with respect to an arbitrary reference direction. The rate of change of ψd can be approximated by the multiplication of longitudinal velocity of vehicle with the reference curvature, that is,
d vx r
ψ = ρ . Note that a straight road is represented as a road with zero curvature (ρr = ). 0 Thus, the rate of change of yaw error can be given by
e vx r
ψ = −γ ρ (5-17) The lateral displacement ye is the distance from the road center to the sensor mounted at a distance ds ahead of the mass center of the vehicle, and can be approximated by
e cg s e
y = y +dψ (5-18) where ycg is the lateral distance between the vehicle’s CG and the centerline of the road.
By differentiating (5-18) and substituting ψ from (5-17), we can obtain e
( )
e cg s x r
y = y +d γ −v ρ
( )
y x e s x r
v vψ d γ v ρ
= + + − (5-19) Here we assume the forward velocity is approximately the longitudinal velocity, and both vy
/vx (this is the sideslip angle of the vehicle CG) and ψe are small. In this chapter, except for the longitudinal velocity vx , the other output measurements of interest are the yaw rate γ and the lateral displacement ye of vehicle.
5.2. T-S Fuzzy Modeling for Nonlinear Vehicle Dynamics
The approach of using sector nonlinearity guarantees an exact T-S fuzzy model construction. Figure 5-2 illustrates the idea of the global sector nonlinearity approach, in which the aim is to find the sector such that x = f(x)∈[a1 a2]x.
Fig. 5-2. Global sector nonlinearity.
Sometimes it is difficult to find global sectors for general nonlinear systems, therefore, the local sector nonlinearity is considered reasonably as the variables of physical systems are always bounded. Besides, the coupling effect between the inputs of traction force and steering angle is also an arduous problem to the modeling procedure. To overcome this difficult, here we adopt a decomposition transformation so called virtual-inputs instead of real inputs of vehicle dynamics as presented as
( 2) cos y sin
By considering the nonlinear vehicle dynamics (5-13) with the virtual inputs and the augmented sensory kinematics, equations (5-13) with (5-17) through (5-20) can be rewritten as
( ) ( , )
s s s s T r
q == q +g δ Tδ +Dρ (5-22) where qs = [q T, ye , ψe]T ∈R5×1, is the augmented fifth order state vector,
( )
and the remaining terms are defined the same as in the previous section. Here the road curvature in (5-22) can be viewed as an external disturbance to the vehicle system.
To obtain the corresponding T-S fuzzy model to the vehicle model (5-22) without consideration of disturbance, the three premise fuzzy variables are defined as
1
We name the membership functions “Max”, “Min”, “Positive”, and “Negative”, respectively, and Fig. 5-3 shows these membership functions. Thus, the T-S fuzzy model for the vehicle system in (5-22) is given by the following eight-rule fuzzy model:
Plant Rule 1:
If z1 is “Min” and z2 is “Negative” and z3 is “Min”,
By the defuzzification process, the overall fuzzy model is inferred as
( )
where hi is the weight of the i-th rule and is calculated by the membership values as1( ) 11( ( ))1 21( ( ))2 31( ( ))3
7( ) 11( ( ))1 22( ( ))2 32( ( ))3
h t =h z t ×h z t ×h z t
8( ) 21( ( ))1 22( ( ))2 32( ( ))3
h t =h z t ×h z t ×h z t . Note that the values of hi is in the range of [0, 1].
5.3. Fuzzy Automated Driving Control Design
Before designing the automated driving controller, it should be noted that the order of the inputs (δT, Tδ) in the vehicle fuzzy model is too larger than states X. Thus it is necessary to scale the order of state vector and input vector appropriately so that which elements have similar numerical variations. Referring to the T-S fuzzy system (5-25), we define
' x
X = S X and 'U =S UU
and choose SX =diag{0.1, 1, 1, 1, 1} and SU =1e-5. Substituting the amplitude-scaling relations into the fuzzy system (5-25), it yields each subsystem
Plant Rule i:
The problem of optimal fuzzy vehicle tracking control is to design a rule-based nonlinear fuzzy inference form of
Ri: If z1 is F1i and z2 is F2i and z3 is F3i ,
then U(t) = ri (t), i=1,…, 8. (5-29) such that the quadratic cost function,
0 1 nonnegative symmetric matrix with appropriate dimensions; XT(t)L1X(t) is the state trajectory penalty to produce smooth response; UT(t)SU(t) is fuel consumption; and the last term in (5-30) is related to tracking error cost.
Moreover, the performance index in (5-30) can be rewritten as
0
where L=L1 + CTL2C and the desired state trajectory Xd (t)=CT(CCT)-1Yd (t).
By defining the error state eX = X–Xd , its variation with regard to the fuzzy subsystem can be obtained as
Here the desired states Xd is assumed to be the solution of
8 Therefore, the tracking problem to the desired states can be converted into the regulation problem of the error state. Note that this assumption is based on that the vehicle dynamics should be taken into account of the specification of the desired states. If the given desired states cannot satisfy the condition of (5-33), it would be improper for the tracking targets.
The optimal global decisions U*(t) from t = t0 to t = ∞ can be regarded as a series of optimal global decision based on the following successively on-going local quadratic optimal issue with the initial state resulting from the previous decision. Given the fuzzy system as
Ri: If z1 is F1i and z2 is F2i and z3 is F3i ,
then e tX( )= A e ti X( )+B r ti i( ) , i=1,...,8 (5-34) with the initial error state resulting from the previous decision, i.e., eX (0)=e*X (0),
(1) find the optimal local decision r*i(t), for minimizing the cost function,
0
( ( )) ( ( ) ( )iT i X( )T X( ))
J u⋅ =
∫
t∞ r t Sr t +e t Le t dt (5-35) (2) obtain the optimal global decision u*(t), for minimizing the cost function J(u(.)) in (5-31)by fuzzily blending each local decision, i.e.,
* 8 * Note that the next decision initial state is obtained by
* 8 * * since there exists the one-to-one relationship between each fuzzy subsystem and the corresponding fuzzy controller. Furthermore, solving the optimal fuzzy control problem for each fuzzy subsystem can be achieved by simply generalizing the classical linear quadratic
theorem from the deterministic case to fuzzy case. Therefore, the following optimal fuzzy control design for the T-S fuzzy system (5-34) can be employed.
For the T-S fuzzy system in (5-34) and the fuzzy controller in (5-29), if (Ai , Bi ) is C.C. and (Ai , Ci ) is C.O., for i=1,…, 8, then fuzzy controller is designed as
Controller Rule i:
If z1 is F1i and z2 is F2i and z3 is F3i ,
then r ti*( )= −BiTπice*X( ) ,t i=1,...,8 (5-38) then, fuzzily blending global optimal fuzzy controller is formed as
* 8 * which minimizes J(u(.)) in (5-30), where πic is the unique symmetric positive semi-definite solution of the steady-state Riccati equation (S.S.R.E.)
1 0
T T
c i i c c i i c
P A + A P −P B S B P− + = (5-40) L The detailed description for the utilized control theorem can be referred to [26, 27].
5.3.2. Fuzzy Observer Design
The fuzzy control design introduced above requires all states feedback in each local linear controller. In practice, all states of the system can not be directly measured, lateral velocity and yaw error especially, such that a fuzzy observer is developed to observe states of T-S vehicle fuzzy model.
The objective of fuzzy observer requires
( ) ˆ( ) 0 ,
X t − X t → t→ ∞
where ˆ ( )X t denotes the state vector which is estimated by a fuzzy observer.
For the fuzzy observer design, it is assumed that the fuzzy system (5-28) is locally observable, i.e., (Ai , Ci ) is C.O., for i=1,…, 8. The fuzzy observer is designed as
Observer Rule i:
If z1 is F1i and z2 is F2i and z3 is F3i ,
then X tˆ( )= A X ti ˆ( )+BU ti ( )+L Y toi( ( )−Y tˆ( )),
ˆ( ) ˆ 1,...,8
Y t =CX i= (5-41)
where L (i =1, 2, …, 8) are the local gains to be determined by the design of the fuzzy oi observer, Y and ˆ( )Y t are the final output of the fuzzy system and the fuzzy observer, respectively.
By applying the separation property, the fuzzy observer can be designed in a way similar to the fuzzy control for each rule of the T-S fuzzy system to provide an optimal estimate of the state ˆ ( )X t . The T-S fuzzy system is supposed to be a plant with a known control input U(t), a measured output, Y(t), and white process noise and measurement noise, w(t) an v(t), with know power spectral densities, W (positive semi-definite matrix) and V (positive definite matrix). The fuzzy observer gain can be can be obtained as
1
o o T
i i
L =π C V− (5-42) where πio is the covariance matrix which satisfies the following steady-state Riccati equation (S.S.R.E.)
1 0
T T T
o i i o o o
P A + A P −P C V CP− + FWF = (5-43) where F is the noise coefficient matrix.
As shown in (5-41), the fuzzy observer has the linear state observer’s laws in each consequent rule. The overall fuzzy observer is presented as
8
and the final output of the fuzzy observer is
8 Note that the same weight hi as the weight of i-th rule of the fuzzy system (5-26) and controller (5-38) is used. observer-based fuzzy controller can be represented as
8
The actual control inputs to the vehicle system are the net force FT of traction/braking extended on four tires and the steering angle δf . Given U =[ ,δT Tδ]T, the actual controlling
By substituting Eq. (5-6) into above equation, and applying small angle approximation (cosδf
≈1 and sinδf ≈δf), Eq. (5-50) can be yielded as
After substitution of Fxf , we can obtain
( 2)
T T f b L x
F =δ +Tδδ + f a b+ mg k v− (5-53) Therefore, u =[c FT δf]T can explicitly be calculated from using (5-51) and (5-53).
5.4. Numerical Simulations
In this section, simulation results of our proposed TS-fuzzy driving controller under a standard double-lane-change (DLC) maneuver are presented. The software used here is CarSim, a general software to simulate the dynamic behavior of a road vehicle and provide the interface programmed by Matlab/Simulink and animation [28]. The nominal values of vehicle parameters which are required in simulation are given as Table A.3 in Appendix A.
Two remarkable examples are compared with our control algorithm to highlight the feature of our approach.
Comparison 1: In Autopia program, fuzzy logic control is used to design the steering control and the speed control separately [10-12]. As shown in Fig. 5-4, the left vehicle is controlled by using Autopia approach, while the right vehicle is controlled by our controller. The control objective is to follow the desired lateral offset trajectory (to vehicle’s CG, ds = 0 m) of DLC and maintain a reference velocity of 90 km/h. Figures 5-5(a) and 5-5(b) depict the two vehicles’ lateral offset and reference velocity response, respectively. The Autopia approach fails to track the desired lateral offset trajectory, and its velocity response varies more promptly than our proposed controller. It can be concluded that even though fuzzy control is applied in vehicle speed and steering controller, the coupling effect still can not be handled due to the separation design of vehicle longitudinal and lateral control.
Autopia approach T-S fuzzy driving control Fig. 5-4. Animation window of Comparison 1.
Fig. 5-5(a). Lateral offset comparison of the proposed controller against the Autopia approach.
Fig. 5-5(b). Velocity comparison of the proposed controller against the Autopia approach.
Comparison 2: The next comparison is with the MacAdam preview model which is most widely referred human steering model [53], and verified experimentally to predict average human steering motion accurately. The speed control with MacAdam preview model is a PI controller designed by CarSim. The animation plot is shown in Fig. 5-6. The control objective is the same as a DLC maneuver. Here the sensor location for the lateral offset is at a fixed preview distance ds = 10 m. This distance corresponds to the preview time 0.4 s while the vehicle velocity is 90 km/h (25 m/s). In Figs. 5-7(a) and 5-7(b), it can be seen that our controller is approximate to the MacAdam model with preview time 1 s in the DLC trajectory tracking performance. This preview time implies that the required previewed distance is 25 m which is difficult to be implemented by vision systems as mentioned previously. As shown in Fig. 5-7(b), the velocity tracking performance of the proposed T-S fuzzy driving system is better than the PI speed controller.
MacAdam model with the preview time = 1 s and PI speed controller.
T-S fuzzy driving with the preview distance
= 10 m Fig. 5-6. Animation window of Comparison 2.
Fig. 5-7(a). Trajectory comparison of the proposed controller against the MacAdam model with preview time 1 s.
Fig. 5-7(b). Velocity comparison of the proposed controller against the PI speed controller.
Table 5-1 summarizes the DLC performance for the proposed TS-fuzzy automated driving controller with preview distances [0, 5, 10] m. The maximum and average errors of distance between C.G. of the vehicle from the road centerline are denoted as |ey_max| and |ey_ave|, respectively. The maximum and average errors of velocity between the referenced and the measured vehicle velocity are denoted as |evx_max| and |evx_ave|, respectively. The vehicle velocity is at 90 km/h. It can be seen that the longer preview distance improves the tracking accuracy of lateral offset trajectory but does nothing to velocity tracking performance.
Consequently, the value 10 m for the preview distance is adequate to be implemented by the vision system.
TABLE 5-1. DLC performance with different preview distances.
ds |ey_max| (m) |ey_ave| (m) |evx_max| (m/s) |evx_ave| (m/s)
0 m 0.2723 0.0652 0.0761 0.0466
5 m 0.1327 0.0328 0.0747 0.0462
10 m 0.0927 0.0233 0.0754 0.0467
Chapter 6
Experimental Results
6.1. Test-track Testing
The system implemented on Taiwan iTS-1 was firstly tested at the Automotive Research and Testing Center (ARTC) with respect to different weather and lighting conditions. The missions of automated lane-keeping with varying velocities on 2 testing-tracks are summarized in Table 6-1. Some events like rain which can not occur usually are denoted as N/A.
TABLE 6-1. Testing Conditions of Different Environmental Sets.
Sunny Cloudy Night Rainy
Weather Testing Track
CDTT 0 ~ 145 km/h 0 ~ 145 km/h 0 ~ 90 km/h N/A
NVHSTT 0 ~ 130 km/h 0 ~ 120 km/h 0 ~ 90 km/h N/A CDTT: Coast down test track.
NVHSTT: Noise vibration & harshness surface test track.
Lane-keeping mode was initially evaluated on the test track at ARTC, and the experiments of regulation were undertaken to examine the stability of the overall system at various speeds.
Initially the system was tested without FGS compensation. In this case, the automated
Initially the system was tested without FGS compensation. In this case, the automated