2. Vehicle Overview
2.3 Function-flow
Fig. 2-4. Function-flow of the upper-level control.
In Fig. 2-4, the function-flow between LK, LC, CC, ACC, and stop-and-go mode in the upper-level control is presented. This system will initially activate LK mode or CC mode, or both simultaneously according to on-line detected traffic condition. In LK mode, the system retrieves real-time sensory information, calculates the deviation from a reference trajectory, and then generates a SW-control command. In CC mode, the subject-vehicle tracks a desired velocity profile set up by a human driver or the limited highway-velocity. As the vehicle
velocity is more than 40 km/h, the control scenario in ACC mode includes both safety-distance and fixed-distance tracking control. The former operation guarantees the safety-spacing keeping from the vehicle ahead while the later operation keeps a constant inter-vehicle spacing for the purpose of increasing the capacity of traffic flow. As a new event is detected, switching from CC mode to ACC mode is automatically activated. The system is defaulted in CC mode for clearance at road-ahead, and switches to ACC mode as a valid-target is detected. A valid-target is defined to satisfy the following conditions:
(a) it is in a designated range which is well defined to the feasible field of the utilized laser range finder;
(b) the velocity of a valid-target is slower than that of the subject-vehicle.
In ACC mode, the safety-distance is derived from constant headway-time policy. The value of fixed-distance can be set according to roadway control [35]. While the vehicle is driven under the velocity 40 km/h, it is reasonably to be assumed that the subject-vehicle is moving in an urban environment or a situation of heavy traffic such that stop-and-go mode will be activated if a valid-target is detected ahead. The preceding-vehicle might come to a complete stop owing to a traffic jam or a stop light. The modes-selection logic scheme is constructed in the upper-level control. The desired reference velocity in each mode will be filtered out, and then passed onto the vehicle-body control.
While the request of LC mode is given by the driver, the system steers the subject-vehicle from the current lane to an adjacent lane. The autonomous changing lane for overtaking a slower vehicle or an obstacle will be further developed in our system. Two schemes of lane-change maneuvers (GPS-guided lane-change and free lane-change) using RTK-DGPS and the vision system, respectively, are developed in our system. In the GPS-guided lane-change scenario, the reference trajectory calculated in the upper-level control is directly added on the lateral position of a specified route on the GPS map. This scheme guarantees the reference path-tracking stability issue, but limits the lane-change maneuver to specific locations where the map must be obtained beforehand. In the free lane-change scenario, the reference trajectory is transformed into the reference steering command that causes the subject-vehicle to track that reference. Without requiring the map information, however, the major difficulty in the free lane-change scenario is the extreme sensitivity of the system performance with respect to sensor noises and parameters variations in vehicle/road model.
Chapter 3
Lateral Control System Design
3.1. Vehicle Lateral Dynamics
In this chapter, the model for vehicle lateral dynamics is introduced to design the lane-keeping and lane-change control. As stated in [16, 32][36], the longitudinal and the lateral dynamics can be separated if the moving velocity does not vary too much. If roll movement is neglected, the vehicle lateral dynamics can be well represented by the so-called
“bicycle model”. The bicycle model which dominates the lateral and yaw dynamics is useful in designing the steering controller to stabilize the vehicle keeping within the lane. As shown in Fig. 3-1, the bicycle model couples two front and two rear wheels together by assuming that the vehicle body is symmetric about the longitudinal plane, and the roll and pitch motion of vehicle are neglected.
Fig. 3-1. Bicycle model for front-steering vehicles.
From Newton’s law, the net lateral forces and the net torque at the center of gravity (CG) of the vehicle can be obtained as
( y x ) yf cos f yr
m v + ⋅v γ =F δ +F (3-1)
z yf yr
I γ=F ⋅ +a F ⋅b (3-2) Based on the assumption of the small steering angle (cosδf ≈ 1) and the linear tire model, the lateral force of tire can be taken as linear proportional to the slip angle with a constant proportionality called cornering stiffness as
yf f f The cornering stiffness of front and rear tire Cf, r considered here is the slope of side force characteristics at the origin on a normal road condition. The slip angles αf and αr can be approximated as the functions of the vehicle’s kinematic parameters
y
The state equation of bicycle model can be rewritten in the following 1 2 1 constant. In reality, it is found that the lateral tire force will initially increase with tire slip angle, and then saturate for a given tire/road friction condition [55-57], as shown in Fig. 3-2.
To capture the saturation property of lateral tire/road friction, several nonlinear tire models were proposed. Bakker and Pacejka proposed a famous “magic formula” which represents that the lateral tire force not only depends on its slip angle but also on vehicle side slip angel αf , r , steering angle δf , and yaw rate γ. Without assuming small angles, the stability condition and bifurcation phenomenon with varying cornering stiffness and different velocities are presented in our previous work [57]. Besides, the front-wheel steering vehicle will become unstable due to the existence of saddle-node bifurcation which is derived in [57] and heavily depends on the rear-tire cornering force characteristics [55].
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
Fig. 3-2. Exampled cornering characteristics of front and rear tires. (Dotted line: high friction road, and dashed line: low friction road)
To validate the bicycle model with the real vehicle dynamics is critical to obtain precise tracking results of steering control. The bicycle model in (3-5) varies with the vehicle speed vx. Notably, the actual input to our vehicle platform is the SW angle δSW rather than the front-wheel angle δf . According to vehicular steering mechanics [37], the SW angle can be expressed as a product of the steering ratio isr and the front-wheel angle δf
SW isr f
δ = ⋅δ (3-6) Because the compliance and steering torque gradients vary with increasing steering angles and load on the front tires, tire pressure, coefficient of friction, etc, in general, isr is not a fixed value for the power steering of the vehicle. However, the constant ratio can be practically used for control design. The steering ratio isr can then be adjusted slightly to yield a response that is more similar to that of the real vehicle platform. The measured SW angle was used as the input to the model. Besides yaw rate, the predicted lateral acceleration from the model can be approximately calculated by
y x y
a ≅ v ⋅ − γ v (3-7) The average errors between the measurements and the model with respect to isr are illustrated in Fig. 3-3, and the minimum point is in the case of isr = 26. Figure 3-4 compares experimental results with the bicycle model predictions for a transient maneuver at around 60 km/h. The lateral acceleration of model agrees with the obtained experimental data in Fig.
3-4(a), and the predicted yaw rate of the vehicle also shows consistent correlation with the
experimental results in Fig. 3-4(b). Several other quantities were also measured and compared with the bicycle model. Results show that the bicycle model (3-5) can faithfully represent the lateral dynamics of the vehicle platform (Taiwan iTS-1).
Fig. 3-3. The relation between the steering ratio and the average error of measurements.
0 5 10 15 20 25 30 35 40 45 50
-0.4 -0.2 0 0.2 0.4
lateral acceleration (g-unit)
0 5 10 15 20 25 30 35 40 45 50
-10 -5 0 5 10
yaw rate (deg/s)
time (sec) (a)
(b)
Fig. 3-4. The states signal for verification between the model and the vehicle in the case of isr
= 26. (solid line: model output; dashed line: measured)
3.2. Lane-keeping Control Design
Fig. 3-5. Vehicle lateral dynamics with respect to road geometry.
The relationship between the lateral dynamics of the vehicle and the desired previewed navigation at a look-ahead distance Ld is plotted in Fig. 3-5. The valid amount of Ld is determined from the vision system [18, 32]. The previewed dynamics can be described as
Ld y d x Ld
y =v +L ⋅ +γ v ⋅ε (3-8)
Ld vx Ld
ε = − ⋅γ ρ (3-9) where the parameters have been defined in Nomenclature.
The bicycle model (3-5) is combined with the previewed dynamics (3-8) and (3-9) to form a linear state-space equation curvature ρLd is viewed as an exogenous disturbance of the system.
The linear system in (3-10) is parameterized with the longitudinal vehicle speed vx . As vx
increases, the poles of the system move toward the imaginary axis, reducing the stability.
Notably, changing the look-ahead distance Ld does not affect the poles location in the transfer function from δf to the previewed lateral offset yLd . If Ld is regarded as being close to the front of the vehicle, then the damping of the zeros in system (3-10) declines drastically and a
high-gain controller drives the closed-loop poles toward the zeros, resulting in a poorly damped closed-loop system. However, Ld can not be chosen too distant from the reliable field of the vision system. The image resolution at far look-ahead distance will be degraded such that collected data includes more errors. As the result of numerous experimental verifications, the reliable value of Ld is chosen as 10 ~ 15 m according to the developed vision system [18].
The control objective for vehicle lane-keeping is to regulate the offset at the look-ahead yLd
to zero. Moreover, the controller is well anticipated to ensure that the vehicle lateral acceleration does not exceed 0.4g (g is 9.8 m/s2) during the control process, such that smooth responses and the comfort of the passengers can be achieved. Given the vehicle model as (3-10), the state feedback control seems to be naturally applied with u =− Kfb x where
[ y Ld Ld]
fb v y
K = k kγ k kε . Here the pole-placement design approach is adopted to consider that the control effort required is related to how far the open-loop poles are moved by the feedback. The objective of pole-placement aims to fix specifically the undesirable aspects of the open-loop response, and avoids either large increases in bandwidth or efforts while poles are moved [38]. Moreover, it typically allows smaller gains and thus smaller control efforts by moving poles that are near zeros rather than arbitrarily assigning all the poles. The closed-loop poles for the system with high order (>2) can be chosen as a desired pair of dominant second-order poles with the rest poles which correspond to sufficiently damped modes, so that the system will mimic a second-order response with the reasonable balance between system errors and control effort. The closed-loop bandwidth for the look-ahead lateral offset is chosen at 5.35 rad/s to mimic human responses [16]. As for comfort requirement, the corresponding closed-loop poles are chosen to ensure that the lateral acceleration above 0.5 Hz will not be amplified during the steering path. Besides, the pole-selection can also be specified by the bandwidth requirement with regard to the transfer function yLd (s)/ρLd (s) with the maximal allowable yLd to reasonable step changes of ρLd [3-6].
From the computer simulations for the closed-loop system response with the step change in curvature, it is found that the complex poles with a damping ratioζ =0.707 will meet the constraint of lateral acceleration. Therefore, the natural frequency ωn of the prototype second-order system can be determined by [39]
1 / 2
2 4 2
BW =ωn⎡⎣(1 2− ζ )+ 4ζ −4ζ +2⎤⎦ , (3-11) and then we have the conjugate dominant poles as s1,2 =−3.58±3.58j. The other two poles are chosen as those in the original system. Notably, increasing the speed will reduce the stability
of the closed-loop system since the poles move close to the imaginary axis.
Recall the system matrix A in (3-10) is time-varying with the longitudinal vehicle velocity vx. The feedback control is supposed to be designed under the highest speed of interest such that the stability for lower velocities can be guaranteed by applying the convex nature of the lateral vehicle dynamics. The following proposition summarizes this approach of using full state feedback control design.
Proposition 1: If a constant feedback control Kfb is chosen such that
min min 0
for a matrix P > 0, than the closed-loop system is stable for varying velocities of the range vxmin ≤ vx ≤ vxmax .
Proof:
The closed-loop matrix at a velocity vx can be represented as a convex combination of ACL(vxmin) and ACL(vxmax), namely
whereµis a parameter whose values depend on the vehicle velocity vx and at the range 0≤µ( ) 1vx ≤ .
By choosing Lyapunov candidate as V = x PxT , we can obtain its derivative as
min min max max
Furthermore, the transport lag is also emphasized in the lateral controller design. The transport lag is caused by the SW motor which arises while the desired command is sent to force the actuator, and also includes image processing delay. The further phase lags are added over the range of frequencies, and severely destabilizing the overall system. Thus a
pure transport lag element e-sτ is included into the designed lateral controller. The input can
By combining the system (3-10) with (3-14), we obtain the augmented system:
2 2
The stability of the transport-lagged system is determined by the characteristic roots of the system matrix in (3-15); consequently, the feedback controller Kfb is designed to guarantee the stability of the closed-loop system (3-15) with a transport lag at the highest velocity (145 km/h in our system).
Remark 1: Due to the presence of nonzero term ρLd , the states in (3-15) will not all converge to zero when the vehicle is traveling on a curve even through the closed-loop matrix ACL is asymptotically stable.
Remark 2: The transport lag comes from the transmission delay in two latencies: 0.04 s for the complete processing of the vision system and measurements available to the controller, and 0.52 s for the average duration gathered by the step response and sine wave tracking between the command and the reaction of servo motor. We choose the transport lag 0.6 s in stable controller design.
3.2.1. Observer Design
Based on the state space model of vehicle lateral dynamics, the full-state feedback control strategy, i.e., all the states must be measurable, is applied in the previous section. However, it is difficult to measure the lateral velocity vy of vehicles directly from general available sensors.
In addition, it is not expected for rapid response in control due to the sensor noise in high
frequency. Thus, the observer design is naturally adopted for the lateral control system.
By considering the system (3-10) again and feeding back the error between the measured and estimated outputs, the equation for the observer scheme can be described as
( )
The measurements in previewed lateral offset yLd and angle εLd can be obtained from the real-time vision system [18], while the yaw rate γ can be provided from optic gyro.
The observer gain matrix Lo∈ℜ4x3 is chosen to achieve satisfactory error characteristics in a number of ways. Here, the pole-placement technique is adopted for the pole-selection convenience of feedback control. As with the control design, the best estimator design keeps the balance between good transient response and low-enough bandwidth such that sensor noise does not significantly impair actuator [38]. Hence, the observer pole locations are selected to be slower than two times the controller poles, and this yields the overall system with lower bandwidth and more noise reduction.
3.2.2. Fuzzy Gain Scheduling
Although the static feedback control strategy suffices to meet the requirements of vehicle lateral control, it is sensitive to the parameters of the system, such as vehicle mass, cornering stiffness, and road curvature, and thus solid reflected by feedback signals. The desire to steer the vehicle in a more human fashion and to provide a smooth automated steering control process, motivates the adoption of a fuzzy inference scheme as part of the lateral controller designing strategy. Based on fuzzy set theory, an adaptation scheme using fuzzy gain scheduling (FGS) is proposed to improve the lateral controller of the vehicle.
Fig. 3-6. Block diagram of the proposed auto-tuning lateral control system.
As presented in Fig. 3-6, FGS is designed to auto-tune the lateral controller. The kernel of the proposed FGS is the inference rule base, which constitutes a natural environment in which engineering judgment and human knowledge can be applied to the vehicle steering controller.
FGS supports more human-like driving behavior during the process of keeping to the lane.
The system mimics humans’ driving more aggressively at low speeds, and more gently at high speed, even when the deviation between the vehicle and the centerline of the road is large. Accordingly, the linguistic input variables are the immediate velocity of the vehicle and the lateral offset from the centerline at the look-ahead distance. FGS yields the proper tuning gain, based on the following rules
i-th rule: If vx is A and yLd is B , then ∆i is C (3-18) where A, B, and C are corresponding linguistic terms,
A ={LOW, MED, HIGH}
B ={NB, NS, ZO, PS, PB}
C={S, M, L}
where the notation NB: negative big, NS: negative small, ZO: zero, PS: positive small, PB:
positive big, S: small, M: medium, L: large.
Table 3-1 shows the rule base of FGS. These parameters of the membership functions for prior and consequent expressions are tuned manually to ensure satisfactory steering performance. The shapes chosen in FGS are trapezoidal for vx, and triangular for yLd and ∆fg ,
respectively, as shown in Fig. 3-7. In the defuzzification strategy, the center of area (COA) method is adopted to determine the gain
15 Finally, the terminal front-wheel steering quantity is thus obtained by
δc = −∆fg⋅ Kfb x (3-21)
As mentioned in Remark 1, during a vehicle steers on curves, the road curvature serves as an exogenous input to the previewed dynamics (3-8)-(3-9). In straight roads (zero curvature) case, both the look-ahead lateral offset and heading angle will be regulated to zero. For the
case of non-zero curvature, these two states will be stabilized to non-zero steady-state values as in Fig. 3-5. Although the curvature information at a look-ahead distance is useful to improve the steering performance while entering/existing curves, it is difficult to obtain the correct information of road curvature in practice. An additional curvature estimator is proposed for feedforward control [16] to improve the transient behavior as the vehicle enters and exits curves. By examining (3-8) and (3-9), changes in curvature ahead can be anticipated by the varying lateral offset ahead. Therefore, FGS is able to compensate the effect from this unknown curvature information: if the lateral offset ahead of the vehicle increases, then the steering control will be increased. Figure 3-8 shows that yLd response with respect to step change of ρLd (300-1 m-1 within 3 ~ 14 s) at a look-ahead distance 15 m is improved by FGS as compared with the pure feedback design. It can be seen that FGS possesses comparable performance with the curvature feedforward approach in [16].
Fig. 3-8. The performance of the proposed pure feedback and FGS-feedback as compared to curvature-feedforward approach in [16].
Another important factor that influences closed-loop performance of the lateral controller is the variation of the cornering stiffness. When the vehicle turns, the mass would transfer onto the external wheels such that the tires pressure increase which leads to variations in the cornering stiffness. Stephant et al. have presented that the variations caused by this factor are