Proceedings of 2018 IEEE International Conference on Mechatronics and Automation August 5 - 8, Changchun, China
Robust Coordination Control of AFS and ARS for Autonomous Vehicle Path Tracking and Stability Xiangkun He, Yulong Liu, Kaiming Yang, Jian Wu, Xuewu Ji* Department of Automotive Engineering Tsinghua University Beijing, China
[email protected];
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[email protected] motion control. Simulink and CarSim-based co-simulation results show that the proposed control method can effectively provide dynamic tracking performance and maintain good maneuverability. In [13], a dual-envelop-oriented path tracking issue is proposed for autonomous vehicle which considers shape of vehicle as inner-envelop and feasible road region as outer-envelop. Then implicit linear model predictive control method is adopted to develop moving horizon path tracking controller in order to track desired path and ensure vehicle stabilization. In [14], a MPC-based control scheme is proposed to mediate among the sometimes conflicting objectives of collision-free path tracking and vehicle stability. In [15], to improve the stability of the autonomous vehicle for high speed tracking, a lateral motion control strategy is designed via multiconstraints model predictive control and unscented Kalman filter. In [16], a learning nonlinear model predictive control for autonomous racing problem that exploits information from the previous laps to improve the performance of the closed loop system over iterations is presented. Although the above research achievements were successful, there are still one main challenge for lateral motion control of autonomous vehicle. Uncertain external disturbance is ubiquitous in practical driving scenarios, which may lead to poor vehicle control performance, and even system instability, especially at the limits of handling. Moreover, these researches generally only use the front steering angle as the input to deal with lateral motion control problem of autonomous vehicle, therefore, there is room for improvement [17]. An autonomous vehicle lateral motion control strategy which coordinated front steering angle control system and direct yaw moment control system is proposed to strengthen the lateral stability and improve the path tracking performance in [7], [18]. However, the direct yaw moment control which differentially brakes or drives the wheels to form a yaw moment, could produce undesired longitudinal deceleration or acceleration. In this paper, a novel robust coordination control scheme based on active front steering (AFS) system and active rear steering (ARS) system is proposed to improve lateral motion control performance of autonomous vehicle under dynamic driving situations at handling limits. In the first stage, reference path model, vehicle dynamics and kinematics model with uncertain external disturbance are established to develop controller. In the second stage, a robust H∞ coordination control strategy based on AFS and ARS is designed using linear matrix inequality (LMI). Finally, an emergency double
Abstract—In this paper, considering uncertain external disturbance due to the frequent variation of running conditions, a novel robust coordination control strategy of active front steering (AFS) system and active rear steering (ARS) system is proposed to simultaneously suppress lateral path tracking deviation while maintaining autonomous vehicle stability under dynamic driving situations at handling limits. Firstly, reference path model, vehicle dynamics and kinematics model with uncertain external disturbance are established. Then, a robust H∞ coordination control scheme is developed based on linear matrix inequality (LMI) by coordinating AFS and ARS. Finally, an emergency double lane change maneuver is carried out via Matlab/Simulink-CarSim co-simulation. The results show that, under uncertain external disturbance, the proposed robust coordination control strategy can provide sufficient path tracking capability as well as stability for autonomous vehicle at or near the physical limits of tyre friction. Keywords—Autonomous vehicle; lateral path tracking; vehicle stability; coordination control; driving limits
I.
INTRODUCTION
With the rapid development of automobile industry and economy, traffic safety issues have become increasingly prominent and severe [1-4]. In 2015, about 1.3 million people around the world were killed in traffic accidents, ranking tenth on the World Health Organization’s list of top causes of death [5]. Among these tragedies, 72% of the traffic accidents can be traced to human error [6]. Because of its great potential on improving traffic safety, congestion and so on, autonomous vehicle has become an emerging research focus in industry and academia [7], [8]. One of the basic and key technologies of autonomous vehicle is lateral motion control which needs to guide vehicles along the desired path and simultaneously maintain vehicle stabilization [9]. In recent years, there are many lateral motion control approach proposed for autonomous vehicle. In [10], a feedback-feedforward steering controller which consists of proportional feedback and feedforward based on vehicle dynamics and kinematics model, is proposed to improve lateral motion control performance of autonomous vehicle at the limits of handling. In [11], a tyre cornering stiffness estimationbased feedforward-feedback control scheme is proposed to simultaneously control autonomous formula racing car to the driving limits and follow the desired path. In [12], the multiconstraints model predictive control strategy is proposed and used to calculate the desired front steering angle for lateral
978-1-5386-6072-0/18/$31.00 ©2018 IEEE
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lane change maneuver is performed via Matlab/SimulinkCarSim co-simulation. The remainder of this paper is organized as follows. Section II gives a detailed description to model for controller design. Section III proposes the robust H∞ coordination control scheme based on AFS and ARS. In Section IV, the co-simulation results are discussed. Section V is the conclusion. II.
with C f + Cr aC f - bCr ì , a12 = -1 ï a11 = vx m vx2 m ï , í aC f - bCr a 2 C f + b 2 Cr ï , a22 = ï a21 = Jz vx J z î
Cf ì C , b12 = r ïb11 = v m v ï x xm , í ïb = aC f , b = bCr 22 ï 21 Jz Jz î
SYSTEM MODELS FOR CONTROL DESIGN
The diagram of the vehicle-road system model is shown in Fig. 1.
where m is the vehicle mass, Jz is yaw moment of inertia, vx is the longitudinal velocity for the vehicle, β is sideslip angle of vehicle body, γ is yaw rate of vehicle body, d1 and d2 are uncertain external disturbances, δf and δr are front and rear wheel steering angles respectively, a and b are distances from the center of gravity to front and rear axle respectively, Cf and Cr are the cornering stiffnesses of the front-axle tires and the rear-axle tires respectively. C. Vehicle Kinematics Model To focus on path-tracking ability, the state variables of vehicle dynamics are transformed into state variables relevant to the reference path. Generally, it is desirable to both eliminate the lateral error e and heading error Δψ. But only one error could be reduced. In this paper, the projected error ep is adopted to combine the lateral error e and the heading error Δψ. The vehicle kinematics model is formulated by: ì s& = vx cos ( Dy ) - v y sin ( Dy ) ï& ïe = v y cos ( Dy ) + vx sin ( Dy ) , (4) í ïe p = e + x p sin ( Dy ) ïîDy = y -y r
Fig. 1. Schematic of vehicle-road system model.
A. Reference Path Model This paper focuses on how to improve lateral motion control performance of autonomous vehicle at driving limits, so that the reference path will be given directly, without path planning. The reference path model is described in terms of lateral position Yref and yaw angle ψref as a function of the longitudinal position X [19], [20]: dy dy Yref ( X ) = 1 éë1 + tanh ( z1 ) ùû - 2 éë1 + tanh ( z2 ) ùû , (1) 2 2 ìï j ref ( X ) = arctan í d y1 îï -d y2
é ù 1 ê ú ëê cosh ( z1 ) ûú
é ù 1 ê ú ëê cosh ( z2 ) ûú
2
2
æ 1.2 ö ç ÷ ç dx ÷ è 1 ø
æ 1.2 ö üï ç ÷ ç d x ÷ý è 2 ø þï
,
where s is the distance along the reference path, vy is the lateral velocity for the vehicle, ψ is the vehicle heading, ψr is the heading of the reference path, and xp is the constant projected distance. According to small angle approximation for Δψ, and differentiating ep and Δψ in equation (4), the following relations can be derived: ìe& = vx b + vx Dy ï& (5) íe p = e& + x p Dy& . ï Dy& = y& -y& î r
(2)
where z1 = (2.5/25)(X-180)-1.2, z2 = (2.5/25)(X-245)-1.2, dx1=25, dx2=25, dy1=3.76, and dy2=3.76.
With equation (5), and considering uncertain external disturbances, the following relations can be obtained: ìe& p = vx b + x p g + vx Dy - x p Ks& + d3 , (6) í îDy& = g - Ks& + d 4
B. Vehicle Dynamics Model In order to consider vehicle dynamics characteristics in controller design, a two-degree-of-freedom (2DOF) vehicle dynamics model with uncertain external disturbance is adopted to capture the essential vehicle lateral dynamics as follows: ìï b& = a11 b + a12 g + b11d f + b12 d r + d1 , (3) í ïîg& = a21 b + a22 g + b21d f + b22 d r + d 2
where K is the path curvature, d3 and d4 are uncertain external disturbances respectively.
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III.
REALISATION OF CONTROL SCHEME
And then, the robust coordination control law can be designed as: u = Kx , (10)
A block diagram of the control architecture is shown in Fig. 2.
where K is state feedback gain matrix. Select Lyapunov function as: L = x T Px ,
(11)
where P is positive and symmetric Matrix. With equation (9) and equation (10), the differentiation of equation (11) is derived: L& = x& T Px + x T Px& = ( Ax + Bu + Bd e ) Px + x T P ( Ax + Bu + Bd e ) T
++
vx , m
b,g e p , Dy
d ef ( ×) ++
d ef
= ( Ax + BKx + Bd e ) Px + x T P ( Ax + BKx + Bd e ) T
, (12)
= x T Q1T x + x T Q1 x + ( Bd e ) Px + x T P ( Bd e ) T
with Q1 = P ( A + BK ) .
d er
d er ( ×)
Define:
Fig. 2. Architecture of the proposed controller for autonomous vehicle.
Combining equation (3) and equation (6), a multiple-input multiple-output (MIMO) linear system with uncertain external disturbances can be described as: ì x& = Ax + Bu + w , (7) í îy = x with é a11 a12 0 0 ù éb11 b12 ù êa ú êb a22 0 0 ú b ú , B = ê 21 22 ú , A = ê 21 ê vx x p 0 vx ú ê0 0 ú ê ú ê ú 1 0 0û 0 û ë0 ë0
T
λ = éë x T
d eT ùû ,
(13)
λT = éë x T
d eT ùû .
(14)
then
With equation (13) and equation (14), then the following equation can be obtained: x T Q1T x + x T Q1 x = x T Qx é Q 0ù é x ù d eT ùû ê (15) úê ú, ë 0 0û ë d e û é Q 0ù = λT ê úλ ë 0 0û PB ù é x ù é 0 Px + x T P ( Bd e ) = éë x T d eT ùû ê T B P 0 úû êë d e úû ë .(16) PB ù T é 0 =λ ê T úλ ëB P 0 û = éë x T
where x = [ β γ ep Δψ ]T is the state vector of system, y is the measurement output of system, u = [ δf δr ]T is the control input of system, w = [ d1 d 2 d 3 - x p Ks& d 4 - Ks& ]T is uncertain
( Bd e )
external disturbance. As shown in equation (7), sideslip angle β or yaw rate γ can directly affect projected error ep and heading error Δψ, and to take into account random road input’s influence on lateral motion control performance of autonomous vehicle, the equivalent uncertain external disturbance is defined as: we = Bd e , (8) with de = [ def der ]T,
T
Combining equation (15) and equation (16), equation (12) can be described as: PB ù é Q 0ù é 0 λ + λT ê T L& = λT ê ú úλ ë 0 0û ëB P 0 û . (17) PB ù T é Q =λ ê T úλ ëB P 0 û
where def and der are equivalent uncertain external disturbances acting on front-axle and rear-axle respectively. Therefore, w is replaced with we, equation (7) can be transformed into the following form: ìï x& = Ax + B ( u + d e ) . (9) í ïî y = x
Select H∞ performance index as:
ò
t
0
t
y T ydt £ r 2 ò d e T d e dt , 0
(18)
where ρ is prescribed attenuation level, and ρ > 0. With equation (9), the following equation can be derived:
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y T y - r 2 de T de = x T x - r 2 d e T d e = éë x T
0 ùé xù éI d eT ùû ê ú. 2 úê ë0 - r I û ë d e û
é AP -1 + BKP -1 + P -1 AT + P -1 K T B T ê BT ê ê P -1 ë where é P -1 0 0 ù ê ú S = ê 0 I 0ú . ê 0 0 Iú ë û
(19)
0 ù éI = λT ê λ 2 ú ë0 - r I û Combining equation (17) and equation (19), the following equation can be obtained: éQ + I PB ù L& + y T y - r 2 d e T d e = λT ê T λ. (20) 2 ú ë B P -r I û
then
PB ù < 0, - r 2 I úû
F = KP -1 , R = P -1 .
(22)
Integrating equation (22), the following relation can be derived: t
t
0
0
L £ r 2 ò d e T d e dt - ò y T ydt + L ( 0 ) t
£ r 2 ò d e T d e dt + L ( 0 )
.
(23) Combining LMI (31) and LMI (32), the state feedback gain matrix of the robust coordination controller based on AFS and ARS can be obtained. The numerical values of parameters related to the controller are given in Table 1.
With equation (11) and equation (23), the following relation can be obtained: 2
t
£ x T Px £ r 2 ò d e T d e dt + L ( 0 ) . 0
(24)
TABLE I CONTROLLER PARAMETERS
With equation (24), the following relation can be derived: t
x
2
£
r 2 ò d e T d e dt + L ( 0 ) 0
Pmin
.
(31)
Considering that P is positive and symmetric matrix, the second LMI can be designed as: R > 0. (32)
0
Pmin x
(29) (30)
And then, the first LMI can be designed as: é AR + BF + RAT + F T B T B Rù ê ú T -2 B -ρ I 0 ú < 0 . ê ê R 0 - I ûú ë
(21)
L& + y T y - r 2 d e T d e £ 0 .
0
Define:
Let:
éQ + I ê BT P ë
P -1 ù ú 0 ú < 0 . (28) - I úû
B - ρ-2 I
(25)
Therefore, it can be inferred that, under finite time horizon and bounded disturbance, the system state is bounded. According to the Schur complement theorem [21], equation (21) can be transformed into the following form: PB I ù é Q ê B T P - ρ-2 I 0 ú < 0 . (26) ê ú êë I 0 - I úû
Parameter
Value
Units
Parameter
Value
Units
xp
5
m
Jz
0.2
kg m2
a
1.192
m
Cf
42000
N/rad
b
1.598
m
Cr
95000
N/rad
m
1528.13
kg
ρ
1
null
IV.
SIMULATION RESULTS
In this section, the satisfying performance of the robust coordination control scheme is verified via Matlab/SimulinkCarSim co-simulation. In CarSim, a B-segment class sedan model is selected, and the parameters of Brilliance® vehicle are given in [22]. Considering that the vehicle can easily become unstable as it makes a sharp turn at high speed, a double lane change maneuver is carried out on a dry asphalt pavement with road adhesion coefficient µ = 1.0 and longitudinal velocity vx = 108 km/h. Moreover, in this work, the equivalent uncertain external disturbances acting on front-axle and rear-axle are shown in Fig. 3.
Combining equation (12), equation (15) and equation (26), the following relation can be obtained: é PA + PBK + AT P + K T B T P PB I ù ê ú T -2 B P - ρ I 0 ú < 0 . (27) ê ê I 0 - I úû ë Multiplied S by both sides of above inequality, the following relation can be derived:
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(b)
Fig. 3. Equivalent uncertain external disturbance.
The results in Fig. 4 show that the peak lateral accelerations of vehicle controlled by the AFS scheme, by the proposed scheme without de(·), and by the proposed scheme with de(·) are about -8.28 m/s2, 8.24 m/s2 and 8.26 m/s2, respectively. Hence, it can be inferred that, during the path tracking, the tyres of vehicle in the different schemes worked in their nonlinear region, and the vehicle operates at its driving limits. (c) Fig. 5. Path tracking control performance. (a) Global trajectory of the path following, (b) path tracking error, (c) heading error.
It can be seen from Fig. 6(a) that, the peak yaw rates of vehicle controlled by the AFS scheme, by the proposed schemes without de(·) and with de(·) are about -39.37 deg/s, 33.39 deg/s and 33.53 deg/s. Fig. 6(b) shows that the peak sideslip angles of the vehicle controlled by the AFS scheme, by the proposed schemes without de(·) and with de(·) are about -7.88°, -5.79° and -5.93°. Therefore, compared with the AFS scheme, the proposed schemes without de(·) and with de(·) exhibit superior stability for the controlled vehicle.
Fig. 4. Vehicle lateral acceleration.
As shown in Fig. 5(a), compared with the AFS scheme, the proposed scheme without de(·) and the proposed scheme with de(·) show superior performance in tracking the reference path. Fig. 5(b) shows that the peak values of lateral path tracking error for vehicle controlled by the AFS scheme, by the proposed scheme without de(·) and by the proposed scheme with de(·) are about -0.71 m, -0.59 m and -0.61 m, respectively. The results in Fig. 5(c) show that the peak values of heading error in the AFS scheme, in the proposed scheme without de(·) and in the proposed scheme with de(·) are about -8.83°, -7.87° and -7.96°, respectively.
(a)
(a) (b) Fig. 6. Dynamics control performance. (a) Vehicle yaw rate, (b) vehicle sideslip angle.
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[4]
It can easily be found from Fig. 4 - Fig. 6 that, the proposed scheme can effectively resist against uncertain external disturbances. In addition, the front wheel steering angle and the rear wheel steering angle of the vehicle controlled by the proposed scheme without de(·) and with de(·) are shown in Fig. 7.
[5] [6]
[7]
[8]
[9]
[10] Fig. 7. Front wheel steering angle and rear wheel steering angle for autonomous vehicle.
V.
[11]
CONCLUSIONS [12]
In this paper, considering uncertain external disturbance, a robust H∞ coordination control scheme based on AFS system and ARS system is proposed to improve lateral motion control performance of autonomous vehicle under dynamic driving situations at handling limits. The satisfying performance of the proposed strategy is verified via Matlab/Simulink-CarSim co-simulation on the dry asphalt pavement (μ = 1.0). The results show that, compared with the AFS scheme, both the proposed schemes without de(·) and with de(·) can provide sufficient path tracking capability as well as stability for autonomous vehicle at or close to the driving limits. Moreover, the proposed algorithm exhibits superior robustness to uncertain external disturbances during emergency maneuver. In future work, longitudinal and lateral motion coordination control method will be researched for autonomous vehicle.
[13]
[14]
[15]
[16]
[17]
ACKNOWLEDGMENT [18]
The authors greatly appreciate the support from the National Natural Science Foundation of China (grant numbers U1664263, 51375009), the Independent Research Program of Tsinghua University (grant number 20161080033).
[19]
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