3. Lateral Control system Design
3.3 Lane-change Control Design
Fig. 3-12. Illustration of the lane-change maneuver.
While maintaining lane-keeping at a driving speed, a lane-change maneuver can be viewed as that the vehicle will travel a specified distance along the lateral axis with respect to its body orientation within a finite time period. After the vehicle aligning itself with the adjacent lane at the end of lane-change maneuver, the lane-keeping maneuver will be resumed smoothly.
The lane-change maneuver is illustrated as Fig. 3-12. By considering the vehicle lateral kinematics where ycg denotes the lateral position of the vehicle’s CG with respect to the original lane center.
Initially (t = 0), ycg(0) = vcg(0) = acg(0) = 0 represents the ideal lane-keeping on the original road segment. At the end of the lane-change maneuver (t = tf), ycg(tf) = ydes should be achieved and vcg(tf) = acg(tf) = 0 is required.
Here the proposed autonomous lane-change controller provides the assistance in the generation of the appropriate steering action to achieve the lane-change task without driver interfering. In this controller design, three assumptions are made:
1) only yaw rate and the steering angle are measured;
2) vehicle parameters are known within a bounded vibration with respect to their nominal values;
3) the road curvature does not change significantly throughout the lane-change maneuver.
During the lane-change maneuver, the lateral acceleration should be bounded as well for ride quality consideration. In [41], the most generalized time-optimal control signal that
satisfies the bounds for the lateral acceleration and the lateral jerk is proposed, as shown in Fig. 3-13.
Fig. 3-13. Time optimal lateral jerk reference signal in [41].
According to the lateral jerk reference signal in Fig. 3-13, the final value of ydes can be yielded as Given the desired lateral displacement ydes , the absolute bounds on the lateral acceleration Amax , and the jerk Jmax , the above equation can be solved for ∆1 and ∆2. Firstly, ∆1 can be
In Fig. 3-13, it can be observed that the lateral jerk appears sharp variations from 0 to ±Jmax. There are high-frequency switchings between ±|Jmax| while acg reaches the upper bound. To get smoother response of acg, therefore, we modify the lateral jerk Jcg and propose a modified reference jerk as shown in Fig. 3-14.
Fig. 3-14. Modified lateral jerk reference signal. and then the minimum real positive rood of ∆2 can be solved from (3-25), or zero if there are no positive real roots.
Now the ideal lateral jerk, acceleration, velocity, and the lateral offset trajectory are provided such that the vehicle can perform a lane-change maneuver and keep the ride comfort constraint. Due to the discontinuous availability of valid previewed data from the vision system during lane-to-lane transition, the control input for autonomous lane-change task is the commanded steering angle. Therefore, a reference SW signal is generated by utilizing the required information mentioned above. According to the lateral jerk reference in Fig. 3-14, the ideal lateral acceleration is obtained as
ref ref
cg cg
a =
∫
J dτ (3-27) The changing of lateral offset in the vehicle’s CG can be presented ascg x x
y =v ⋅ + ⋅β v φ (3-28) By assuming that the front-wheel angle δf is small, the vehicle side-slip can be viewed as nearly zero (β ≅ ) and y0 cg can be obtained as
( )
cg x
y =
∫
v ⋅φ dt (3-29) From the bicycle model (3-5), the transfer function from the front-wheel angle δf to the yaw rate γ can be obtainedThe yaw angle signals φ can be determined by using ( ) ( ) ( ) function is studied at various longitudinal velocities. | ( )
Hγ δ− f s | is plotted with vx = [5, 10, …, 30] m/s in Fig. 3-15(a). It can be observed that the dc-gain of | ( )
Hγ δ− f s | is flat over a wide range of velocities in the domain of lower frequencies.
10-1 100 101 102
Fig. 3-15(a). Magnitude plot of front-wheel angel to yaw rate ( ) Hγ δ− f s .
5 10 15 20 25 30
A complete signal of front-wheel angle for operating lane-change is in the form of a sine wave. In general, the smooth lane-change will be accomplished within 5 ~ 6 s, and thus the major frequency component will be in the 1.0 ~ 1.2 rad/s range during the transition. Around this frequency, the magnitude of ( )
Hγ δ− f s is essentially the same as the dc-gain for all longitudinal velocities [5, …, 30] m/s. This result can be extended by implementing a velocity-dependent function hγ δ− f ( )vx , as shown in Fig. 3-15(b), without any approximation and using it to generate a yaw rate command profile. Consequently, the equation (3-30) can be approximated as
( ) ( ) ( )
cg t hγ δf vx f d
φ = −
∫
δ τ τ (3-32) Substitute (3-32) into (3-29), the lateral position of vehicle’s CG can be obtained asf( )
cg x x f
y =hγ δ− v ⋅v
∫∫
δ τd dt (3-33) From the idea lateral acceleration in (3-27), the desired lateral offset in the lane-change maneuver for the vehicle CG can be generated asref ref
cg cg
y =
∫∫
a d dtτ (3-34) From (3-33) and (3-34), the referred front-wheel angle can be calculated
and the reference SW angle can be yielded as Note that the generation of reference SW angle in (3-36) is adaptive to the vehicle speed, and by the given reference lateral jerk acgref , the time of lane-change maneuver will be limited to tf which is previously defined.
In our proposed lane-change controlling strategy, the road is assumed to be straight during the entire duration of the lane-change maneuver. Therefore, there is no compensation for variations in the road curvature. The open-loop lane-change’s feasibility will be easily declined by changes in road curvature and level. In order to improve the feasibility of lane-change control, the free lane-change control is divided into two controlling stages:
1) On-set lane-change trajectory: In the first stage, the initial conditions of the lateral data are considered into the generation of reference SW command that the subject-vehicle follows to leave the traveling lane and to perform the free lane-change maneuver. Thus the Eq.
(3-36) is modified to initial conditions x(0). The observer in (3-16) and (3-17) can be used for these initial condition estimations.
2) Lane-catching trajectory: The difference between the actual road reference and the virtual road reference leads to that the lane-change maneuver is likely to end with a lateral offset from the center of the neighboring lane and possibly with an orientation error. This resulting deviation can be corrected by the lane-keeping control which is resumed as soon as the lane-change maneuver is completed. Thus the second stage is to shorten the eventual period of lane-change maneuver, and let the lane-keeping control lead the subject-vehicle to align itself to the target centerline once the vision system re-catches the target lane data.
Note that the early-switch causes that the lane-change maneuver ends with an offset error (with maximum value of half road width); nevertheless, uncomfortable ride can be evitable for the applied FGS which adjusts the lane-keeping steering according to the current velocity and the lateral offset signal.
Chapter 4
Longitudinal Control System Design
4.1. Modeling for Vehicle Longitudinal Dynamics
The true vehicle dynamics is unfortunately complicated and includes high nonlinearity in the interaction between propulsion, roadway interface, and aerodynamics, …, et. However, the vehicle longitudinal dynamics under non-braking conditions can be represented by one nonlinear model in the form of third-order transfer function shown in Fig. 4-1 [42]. This model is employed in this chapter to describe the nonlinearity in velocity dependent dynamics between the voltage applied on throttle valve Vi and the forward velocity of the vehicle V.
( )
Fig. 4-1. Nonlinear model of the vehicle longitudinal dynamics.
The connection between these velocity-dependent parameters and the forces/drags of vehicle/tire is as follows: tp(V) is associated with the throttle actuator and propulsion system and their interaction with the roadway interface, KI(V) is associated with aerodynamic drag and vehicle mass, Kp(V) is associated with the throttle actuator and propulsion system, and
( )V
ξ is associated with interaction between tires and roadway. Reformulate the model and one can obtain with the velocity dependent parameters
( ) ( )
The experimental procedure for model parameters identification is similar to the approach in [42]; nevertheless, instead of estimating each velocity dependent variable (tp , KI , Kp , and ξ), evidently it is more intuitively to identify those parameters ψ, q1, q2, and q3, under varying velocities of the vehicle. The specification of velocity dependent parameters ψ, q1, q2, and q3, are accomplished via a closed-loop model-matching technique such that the responses of the model are matched to those obtained from corresponding real vehicle wherein the same driving controller D(s) is employed. The configuration of closed-loop simulation model is shown in Fig. 4-2, and note that here the characteristics of throttle actuator is considered into the identification procedure, which are presented by saturation scheme for applied voltage Vi
and a transport lag existing in controller command to actuator.
Fig. 4-2. Closed-loop configuration employed in the identification of ψ, q1, q2, and q3 .
The identification procedure is repeated several times for each specified command velocity for the vehicle Vc until the consistent response is obtained. Note that the driving controller D(s) set in the configuration of closed-loop model can be chosen different so as to the model parameters are demonstrated to be independent of the specific controller choice [42]. Since the true vehicle dynamics is not known in the beginning, the choice of D(s) must be conservative. In the experiment, it is found that the great amount of gain in the driving controller is inadequate for noise in sensing velocity and bounded input constraint of throttle voltage. Therefore, one simple PI-controller D1(s)=(0.5s+0.12)/s and filter D2(s)
=(0.5s+0.2)/(s+2), respectively, are determined in the closed-loop configuration. Through the recorded throttle voltage Vi and the velocity of vehicle V in these tests, the quantities of velocity dependent parameters ψ, q1, q2, and q3, are appropriately adjusted and assigned values for each condition of driving controller and velocity of the vehicle, so as to match the response of model to the real vehicle. In this way, the velocity dependent parameters ψ, q1, q2, and q3, in the model are all real functions of the longitudinal velocity of the vehicle. The
ψ
driving controller D(s), and the averaged results verses velocity are plotted in Fig. 4-3.
For the interesting range of velocity 0≤V≤30 m/s in Fig. 4-3, the parameters may be presented functionally with velocity V as
20 3.6 Thus, it is clear to see that these parameters are varying functionally with velocity of the vehicle and the upper and lower limits over the range of interest for q1, q2, and q3 are
40.20≤ψ ≤77.03 3.38≤q1 ≤5.86 16.42≤q2 ≤26.00
0.40≤q3 ≤0.74.
Even though the model is identified for small variation in velocity about a given fixed velocity, it is also valid for the case of large signal provided change in velocity is smooth.
This result also agrees with the one in [42]. Here we consider two normal cases of comparison between the response of the real vehicle and the simulation model with the driving controller D(s) = D2(s). The first case is to accelerate the vehicle from initial velocity 5 to 60 km/h; in the second case, the vehicle is accelerated to a specified velocity (about 70 km/h) and kept this velocity. It can be expected that for smooth acceleration command, the applied voltage on throttle actuator need not be large, while the great quantity of input throttle voltage is required
for vehicle in rapid acceleration operation. In Fig. 4-4, two cases show the direct response (open-loop) from the controlled voltage Vi posing to the output velocity of the simulation model and the vehicle platform (Taiwan iTS-1), respectively. For each case, the good correlation exists in these comparisons for the values, variations, and the time of occurrence.
Case 1. Velocity response. Case 1. Controlled throttle voltage.
Case 2. Velocity response. Case 2. Controlled throttle voltage.
Fig. 4-4. The I/O responses of the simulation model and its comparison with the real vehicle.
4.2. Longitudinal Automation System Design
Fig. 4-5. Dual-loop structure of the longitudinal automation system.
As shown in Fig. 4-5, the longitudinal automation system is developed in a dual-loop structure. The adaptive horizontal detection area (HAD) deals with the data from the on-board sensors according to the SW angle and vehicle dynamics. Instead of horizontal detection in a fixed pattern, the adaptive detection area performs the adaptive action in order to guarantee that the preceding vehicle is detected on both straight and curved roads. A main loop supervisory control determines the reference velocity based on the recognized distance from the adaptive HDA and the current velocity of vehicle. A sub-loop regulation control then manipulates the throttle and brake actuators to achieve the desired velocity. By recalling Fig.
2-3, the supervisory control is the role of reference velocity calculation in the upper-level control, and the regulation control keeps this reference velocity by generating the throttle and brake controlling signal to actuators.
4.2.1. Adaptive HDA
Fig. 4-6. Illustration of the scenario of a vehicle following on curves.
On straight roads, the headway distance can be measured from the forward-looking sensor (FLS) if it can be assumed that there is no failure in the sensor. However, failure to detect a vehicle ahead could be due to curves in the road. In the specification of [43], ACC systems are required to be provided with curve capability; i.e., the system should enable steady-state vehicles to follow with an appropriate headway distance on curves. As illustrated in Fig. 4-6, the preceding vehicle can not be detected by the FLS of the following vehicle if the detection area is too narrow. Such a failure might result in instant acceleration. However, if the detection area is too broad, some unexpected objects near the road might be detected which may cause the throttle to be off. The issue of curve capability, nonetheless, is being considered at present in other longitudinal control systems.
The turning behavior of a bicycle model as depicted in Chapter 3 has been considered using the formula below in which the lateral dynamic from steering angle δf to lateral velocity vy
and yaw rate γ is presented as
In addition to detect the existence of the preceding vehicle, the look-ahead information of the
subject vehicle (also referred to as the following vehicle) must be considered. By assuming the vehicle traveling on a curve with a radius Rf , the equations (similar to (3-8) and (3-9)) capturing the evolution of the point at a look-ahead distance Ld due to the motions of the vehicle and changes in the road geometry are the following
Ld y d f Ld
y =v +L ⋅ +γ V ⋅ε (4-4)
Ld V Rf f
ε = −γ ⋅ (4-5) It should be noted that here the road curvature is assumed to be constant due to the assumption of a steady-state analysis. The reason is that the vehicle will not reach a steady-state condition while traveling on a road with varying curvatures and this makes it difficult to investigate the static relation. By considering a steady-state motion in which the subject vehicle tracks the curved road perfectly at a constant velocity Vf , the variations of the vehicle lateral dynamics (4-3) and the look-ahead motion (4-4) and (4-5) can be set to zero, i.e., vy = =γ yLd =εLd =0. In the following, the subscript of ss denotes the value in the steady-state condition. By calculating the direction, the steady-state steering angle can be obtained The steady-state look-ahead lateral offset can be adopted as [44]
Ldss ss f
y =h −R (4-7) where hss = Rf2 +Ld2 +2R Lf d(−vyss /Vf).
Furthermore, at steady-state the lateral dynamics (4-3) can be represented as
yss
In Equation (4-8) the term on the left-hand side is in the range space of matrix Bs . It should be noted that the fixed values of the yaw rate during the steady-state turning maneuver can be obtained by using
γss= Vf /Rf (4-10) By substituting (4-9) and (4-10) into (4-5), the steady-state look-ahead lateral offset can be
rewritten as
2 2 2
Ldss f d d f
y = R +L + L T −R (4-11) From (4-6) and (4-11), one obtains
2 2 2 It is also reasonable to assume that
2 2
|Ld +2L Td | /Rf á 1 (4-13) and the following approximation via Taylor’s expansion can be obtained as
2
Through (4-14), relation (4-12) becomes
2
It should be noted that P is also the under-steer coefficient of the vehicle [45]. By recalling Fig. 4-6, the following approximation can be applied
Ldss R
y ≅d ⋅ (4-16) θ By substituting (4-15) into (4-16), one can obtain the relation from the steering angle to the adaptive detection angle The steering angle can be substituted with the steering wheel (SW) angle using a constant ratio δSW = isr ⋅δf [32, 46]. Normally the value of isr is between 18 and 22 for passenger vehicles. In (4-17), the resulting feature is an adaptive ratio between the adaptive detection angle and the steering angle which is independent of the road curvature. It can be observed that the adaptive ratio is updated as the vehicle velocity changes. The calculated yLdss refers to the offset between the point at the look-ahead distance and the position through which the vehicle will pass. When the vehicle is turning at a higher velocity, the look-ahead lateral offset increases such that the detection area expands in the same direction as the SW.
Moreover, with respect to a reduced road curvature (or a longer radius), the look-ahead lateral offset decreases in that the detection area expands less. Table 4-1 depicts this result when Ld = 42 m and dR = 40 m are given (as specified in the experiments).
TABLE 4-1. (δSW, θ) with varying velocities and radiuses of curves.
Rf
Vf
200 m 300 m 400 m 500 m
40 km/h (19.6°, 6.9°) (14.5°, 5.1°) (10.9°, 3.8°) (8.7°, 3.1°) 60 km/h (23.9°, 9.2°) (15.5°, 6.1°) (11.6°, 4.6°) (9.3°, 3.7°) 80 km/h (25.1°, 11.9°) (16.8°, 7.9°) (12.6°, 5.9°) (10.1°, 4.7°) 100 km/h (27.6°, 16.6°) (18.4°, 11.1°) (13.8°, 8.3°) (11.1°, 6.6°)
Two calculations given in ISO 15622 are applied to verify the feasibility of the developed algorithm in (4-17). The HDA formula of ISO 15622 is introduced in Appendix B. The two vehicles cruise on a curve radius of 500 m and 125m with a constant velocity of 31.67 m/s and 16.94 m/s, respectively. The comparative results (the steady-state look-ahead lateral offset and the detection angle) between ISO 15622 and the proposed algorithm are presented in Table 4-2. The evaluation criterion for the curves in ISO 15622 is dependent on the vehicle velocity and the lateral acceleration constraints. Furthermore, this criterion is a minimum requirement for the detection of the forward vehicle on the specified range of curves. The developed maneuver in (4-17) involves the major coefficients of the subject vehicle since they are derived from the lateral dynamics which determine the turning behavior of the vehicle. As a result, the proposed algorithm certainly satisfies the minimum requirement of ISO 15622 and provides a more confident detection range.
TABLE 4-2. Comparative results between ISO 15622 and adaptive HDA.
ISO 15622 Adaptive HDA
yLdss 4.00 m 5.07 m
Example 1
(Rf = 500 m) θ ±3.70° ±4.61°
yLdss 4.60 m 4.83 m
Example 2
(Rf = 125 m) θ ±7.80° ±8.92°
4.2.2. Supervisory control
After detecting road environment and determining the operation modes, we estimate the desired velocity by a two-stages approach. The desired acceleration of each operation mode is determined in the first stage, and then is converted to the desired velocity in the second stage.
This velocity is then issued to vehicle-body control. We here use a desired velocity, instead of acceleration, due to that the subject-vehicle’s velocity can be measured directly. Moreover, high-frequency noise problems from sensed velocity differential can be avoided.
In the first stage, we adopt the sliding mode control (SMC) technique to design the corresponding control laws for various driving modes. A proper surface S is defined as the function of the system states, and the sliding manifold S=0 defined on the closed-loop system is asymptotically stable. To satisfy the reachable and sliding condition, we choose S such that
0
SS< while S ≠ 0 (S =C S( ) is a continuous function of S.). One conventional method of the sliding surface design is as [47]
1 where n is the relative degree of the system, λ is the chosen sliding surface gain, and e presents system error which is desired to drive to zero. In this way the state trajectory is forced to a stable manifold state (S=0) in a finite time interval, and then slides toward equilibrium state. We define the control law as the first derivative of the surface S.
In the CC mode, the desired acceleration is determined such that the subject-vehicle can track any assumed velocity under comfort situation. We define the velocity tracking error as eV = Vf − Vdes . The sliding surface is chosen as SCC = eV and the control law is defined as
CC CC CC
S = −K S (4-19) with constant parameter KCC > 0 for SCC to approach zero. This control law satisfies globally
S = −K S (4-19) with constant parameter KCC > 0 for SCC to approach zero. This control law satisfies globally