Original Article
Robust guaranteed-cost path-following control for autonomous vehicles on unstructured roads
Proc IMechE Part D: J Automobile Engineering 1–13 Ó IMechE 2017 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/0954407017713089 journals.sagepub.com/home/pid
Jinghua Guo1, Jin Wang1, Ping Hu2 and Linhui Li2
Abstract This paper deals with the problem of automatic path-following control for a class of autonomous vehicle systems with parametric uncertainties and external disturbances in cross-country conditions. In the unstructured environments, the unevenness, the discontinuity and the variability of the terrain greatly increase the parametric uncertainties and the external perturbations of autonomous vehicles. To overcome these difficulties, a novel automatic path-following control scheme of vision-based autonomous vehicles is presented by utilizing the guaranteed-cost control theory. First, a new road detection algorithm used for segmenting and extracting the traversable path in unstructured terrains is achieved by using a combination consisting of multiple sensors, and the local relative position information between the vehicles and the desired trajectories can be acquired by the proposed detected algorithm in real time. Then, an optimal guaranteedcost path-following control system is proposed, which can deal with the parametric uncertainties of autonomous vehicles and ensure the stability of the closed-loop control system. Finally, both simulation tests and experimental results demonstrate that the proposed control scheme can guarantee high path-tracking accuracy irrespective of the parametric uncertainties.
Keywords Autonomous vehicles, path-following control, vehicle lateral dynamics, guaranteed-cost control, off-road
Date received: 22 July 2016; accepted: 28 March 2017
Introduction In recent years, autonomous vehicle technology has been investigated worldwide because of its possible broad application in front-line technology. Path following is a key technology for implementing the automatic tracking of the desired trajectories for autonomous vehicle systems, which is provided in real time by road detection algorithms. In unstructured environments, the unevenness, the discontinuity and the variability of the terrain greatly increase the parametric uncertainties and the external perturbations of autonomous vehicles. As a consequence, the study of automatic path-following control of autonomous vehicles is difficult.1,2 At present, growing attention is being paid to the design of a path-following control system. A novel nested proportional–integral–derivative (PID) control architecture was designed for vision-based autonomous vehicles, and the availability of this presented path-following control system was confirmed by several experimental tests.3 A fuzzy path-following strategy was designed by imitating human behaviour.4,5 Furthermore, a feedback
path-following fuzzy gain scheduling controller was proposed for carrying out more human-like steering control.6 A novel lateral cascade architecture was constructed from a fuzzy controller and a PID controller.7 To improve the performances of the lateral fuzzy control system, the membership functions and the rule bases of the lateral fuzzy control system were simultaneously optimized by genetic algorithms.8,9 The dynamics model of a non-holonomic automated vehicle in local coordinates was established,10 and a non-linear control law was designed for the robot AnnieWAY, which took part in the final of the 2007 Defense Advanced Research
1
Department of Mechanical and Electrical Engineering, Xiamen University, Xiamen, People’s Republic of China 2 School of Automotive Engineering, Dalian University of Technology, Dalian, People’s Republic of China Corresponding author: Jinghua Guo, Department of Mechanical and Electrical Engineering, Xiamen University, Xiamen 361005, People’s Republic of China. Email:
[email protected]
2 Projects Agency Urban Challenge competition. A robust H2–HN path-following system for an autonomous snow blower was presented to achieve automatic driving in atrocious work conditions.11 Huand and Tomizuka12 proposed a linear time-varying path-following control system to overcome the influence of a fault in the rear sensors. A model predictive control system for path following was presented by Tsing et al.,13 which can deal with the uncertainties on slippery roads. To achieve lateral driving assistance for lane departure avoidance, the switching steering control system was presented using the theory of hybrid automata and composite Lyapunov functions.14 In addition, an adaptive path-following sliding control system was designed, in which the sliding gains were adjusted by the fuzzy logic.15 From a historical perspective, it is interesting to note that the existing research on the design of a pathfollowing controller has been focused on uniform and smooth road surfaces. In recent years, with the expansion of the applied range of autonomous vehicles, the capacity for cross-country automated driving of vehicles has become increasingly included. Cross-country environment perception plays an important role which ensures that autonomous vehicles achieve the desired dynamic performances of a path-following control system. To deal with both structured roads and unstructured roads, a hierarchical lane detection system was designed,16 and the terrain for unstructured roads was divided into regions by mean-shift segmentation. A fuzzy support self-supervised learning approach was proposed for detection of unstructured roads.17 To improve the overall performance, a novel unstructured road detection which integrates the support vector machine algorithm and the k-nearest-neighbour algorithm was designed.18 A new vanishing-point estimation approach which relies on the joint activities of four Gabor filters was designed for outdoor unstructured road conditions.19 However, because of the strong influence of the complexity of the terrain, the variation in the light and the incompleteness of the information about cross-country terrains, it is difficult to achieve reliable road detection with a visual sensor alone. An automatic path-following control system should have core competencies to follow both rough roads and curvy roads in such terrain. Furthermore, in crosscountry environments, the unevenness, the discontinuity and the variability of the terrain greatly increases the parametric uncertainties and the signal incompleteness of the dynamic systems of a vehicle. Therefore, the primary goal of a path-following control system is to have robustness to the parametric uncertainties. Since robust GCC technology has the capacity to deal with an uncertain plant system, it has attracted much attention. GCC aims to design a feedback controller which can maintain the given quadratic cost function within a certain bound and to guarantee the stability of the system.20,21 In this paper, because the lateral dynamics system of autonomous vehicles in unstructured environments has
Proc IMechE Part D: J Automobile Engineering 00(0) the features of external disturbances and parametric uncertainties, this paper presents an unstructured road detection algorithm based on the combination of a charge-coupled device (CCD) and a laser sensor, and a robust guaranteed-cost path-following feedback control system is constructed to deal with the influences of the parametric uncertainties for autonomous vehicles in unstructured environments. In summary, the main contributions of this paper are as follows. 1.
2.
An unstructured road detection algorithm based on information from a combination of multiple sensors is presented; it can accurately acquire in real time the local relative positions between the vehicles and the desired trajectories. Since the unevenness, the discontinuity and the variability of terrain greatly increase the parametric uncertainties of autonomous vehicles in unstructured environments, optimal guaranteed-cost control (OGC) is established to improve the robust tracking capability.
The rest of this paper is organized as follows. The second section presents an unstructured road detection algorithm based on a combination of multiple sensors. In the third section, an OGC scheme is used for pathfollowing control of autonomous vehicles on unstructured roads.The fourth section shows the simulation tests and the experimental results of the presented path-following control method for a prototype vehicle. Finally, the conclusions are presented in the fifth section.
Unstructured path detection Road detection is an essential prerequisite in the implementation of automatic steering control for autonomous vehicles. To achieve the function of automatic driving for vehicles on unstructured roads, it is necessary to detect the desired trajectory in real time and to acquire accurate information on the local relative positions between the autonomous vehicles and the desired trajectories. Since there are hardly any discernible and invariant features that can characterize the roads or their boundaries,19 it is a challenging task to design a road detection algorithm for unstructured environments. In this section, an unstructured road detection algorithm based on technology using information from a combination of multiple sensors is presented, and the corresponding flow chart is shown in Figure 1.
Calibration The surrounding cross-country road terrains in front of autonomous vehicle systems are gathered simultaneously using the colour camera; to increase the reliability of distinguishing the driving region, the raised obstacles are detected by the laser scanner which can acquire
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3 and the local histogram equalization strategy is proposed for image enhancement to extract the road features; then, the region of the raised obstacles is marked in the image to ensure that the seed points of the subsequent region-growing algorithms can be automatically selected in the non-obstacle regions. For every point (x, y) in the image, the actions performed are as follows. 1.
A histogram in the rectangular area centred around the pixel point (x, y) is obtained as pW ð r k Þ =
nk W2
ð2Þ
where pW() is the probability density function, nk is the number of pixels that have the grey scale rk and W3W are the dimensions of the rectangular region. 2. The cumulative distribution function PW () is calculated as Figure 1. Flow chart of detection algorithms for unstructured roads.
P W ðrk Þ =
k X
pW (i)
ð3Þ
i=0
3.
The pixel point (x, y) is transformed as T( f (x, y)) = 255PW ( f (x, y))
ð4Þ
where T() is a grey-level transformation function.
Textural feature extraction
Figure 2. Calibrated map for the CCD camera and the laser. CCD: charge-coupled device.
information on the locations of the scanned subjects relative to the body of the vehicle. In this paper, as shown in Figure 2, one laser point corresponds to five pixel points; therefore, the corresponding relationships between the laser scanning pixel coordinate n and the CCD image horizontal pixel coordinate N are established by the calibration N = 5ðn 100Þ + 160
ð1Þ
Image preprocessing First, the lighting compensation is found for the captured image to eliminate the influences of illumination,
It is well known that the unstructured roads can be distinguished from the other terrains by using the textural features which depict the properties and the spatial relationships of the grey levels of the image; here, the textural feature statistics of the pixels in the cross-country images are generated by the grey-level co-occurrence matrix (GLCM), which can accurately reflect the integrated information on the image grey-scale distribution about the orientation, the local region and the range. The values p(i, j, d, u) in the GLCM denote the probability that two neighbouring resolution cells separated by the distance d occur on the image, one with a grey tone i and the other with a grey tone j, and u is the direction. The GLCM is given as p(i, j, d, 08) (k, l), (m, n) 2 Ly 3Lx 3 Ly 3Lx , =# k m= 0, jl nj= d; f(k, l) = i, f(m, n) = jg ð5Þ p(i, j, d, 458) (k, l), (m, n) 2 Ly 3Lx 3 Ly 3Lx , =# k m = l n = d, f(k, l) = i, f(m, n) = j ð6Þ
4
Proc IMechE Part D: J Automobile Engineering 00(0) p(i, j, d, 908) (k, l), (m, n) 2 Ly 3Lx 3 Ly 3Lx , =# jk mj = d, l n = 0, f(k, l) = i, f(m, n) = j ð7Þ
(f b)(s, t) = max½f(s x, t y) + b(x, y) js x, t y 2 Df , x + y 2 Db
ð13Þ
(f b)(s, t) = min½f(s + x, t + y) b(x, y)
p(i, j, d, 1358) (k, l), (m, n) 2 Ly 3Lx 3 Ly 3Lx , ð14Þ js + x, t + y 2 Df , x + y 2 Db =# k m = d, l n = d, f(k, l) = i, f(m, n) = j The edge of the unstructured road is extracted using ð8Þ the Canny edge detector,22 and the gradient values of where # is the number of elements in the set, Lx is the the input image are calculated by using the 3 3 3 Sobel horizontal spatial domain, Ly is the vertical spatial operator with a very low threshold; thus, the grey-level domain and (k, l) and (m, n) are the pixel coordinates edge magnitude map fm and the Canny edge detector direction map fu can be obtained as in the image. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi In order to analyse and extract the textural features 2 fm = ½ fx (x, y)2 + fy (x, y) from the GLCM, specific statistical measures such as ð15Þ the homogeneity H, the entropy E, the contrast Con ’j fx (x, y)j + fy (x, y) and the correlation Cor are calculated as fy (x, y) fu = ð16Þ k1 X k1 X f p(i, j) x (x, y) H= ð9Þ 2 i = 0 j = 0 1 + (i j) with Con =
k1 X k1 X
p(i, j)(i j)2
ð10Þ
p(i, j) log½p(i, j)
ð11Þ
i=0 j=0
E=
k1 X k1 X
fx (x, y) = ½ f (x 1, y + 1) + 2f(x, y + 1) + f(x + 1, y + 1) ½ f (x 1, y 1) + 2f(x, y 1) + f(x + 1, y 1) ð17Þ
i=0 j=0
Cor =
k1 X k1 X
p(i, j) log½p(i, j)
ð12Þ
i=0 j=0
fy (x, y) = ½ f (x + 1, y 1) + 2f(x + 1, y) + f(x + 1, y + 1) ½ f (x 1, y 1) + 2f(x 1, y) + f(x 1, y + 1) ð18Þ
Region-growing image segmentation First, with advice from the laser sensor, the traversable region in front of autonomous vehicles is determined, and the seed points are randomly selected in this traversable region. Then, the region growth is developed to detect the left road edge and the right road edge, and the segmentation of unstructured terrains is achieved. The algorithm for the implementation of region growing based on the similarity of a local property (intensity level) is given as follows.
Then, the edge chains are searched for from the centreline to the side of image, and information on the length and the dimensions of the edge chains is acquired to recognize the left road boundary and the right road boundary. Finally, the road boundary is described by the quadratic curve-fitting method, and the geometry model of desired trajectory is usually in the form of a polynominal such as y(x) =
n X
cj uj (x)
ð19Þ
j=0
1. 2.
3.
Start with a seed pixel(x0, y0). Append to each pixel (x, y) in the region those eight-connected neighbours which have grey-scale and texture properties that are similar to those of the seed. Stop when the region cannot be made to grow any further.
Trajectory extraction First, the segmented image is processed by the dilation and erosion morphological operations as
where uj(x) are the basis functions and the coefficient values of cj are determined by the recursive weighted least-squares method. Figure 3 shows the detection results of the proposed algorithms in an unstructured environment; it is worth mentioning that the detected unstructured roads are consistent with the actual road edges. The left road edge is selected as the desired trajectory which the autonomous vehicles need to track, and the offset information obtained from the above proposed algorithm can be steadily provided in real time as the feedback signals of the path-following control system.
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5 a powerful approach for overcoming the parametric uncertainties of the dynamic system.21 In this section, to achieve a better tracking performance index in unstructured working conditions, an OGC path-following controller is proposed, which can ensure stability of the closed-loop control system and can reduce the influences of the variations in the parametric uncertainties.
Uncertain vehicle lateral dynamic system
Figure 3. Results detected for an unstructured road.
The prototype autonomous vehicle is a front-wheeldriven and a front-wheel-steered system. As illustrated in Figure 4, a simplified lateral dynamic model of the vehicle which describes the essential features can be derived using the following main assumptions.1 1. The pitch, roll and vertical dynamics are neglected. 2. The longitudinal speed is set to a constant value, which has the form 2 3
" Cf #
(Cr + Cf ) lr Cr Cf lf v x vy v_y mvx mvx 4 5 + lfmCf df = l C C l l2f Cf + Cr l2r r r f f r r_ I Iz vx
Iz vx
z
ð21Þ
Figure 4. Simplified vehicle model.
A polynomial saturation function of the longitudinal velocity of the vehicle is used for the desired look-ahead distance L as given by 8 v \ vmin < Lmin , ð20Þ L = t2 v2 + t1 v + t0 , vmin 4v4vmax : Lmax , v . vmax where v is the velocity of the vehicle, t0, t1 and t2 are the polynomial fitting coefficients (given by t0 = 3.5, t1 = 20.41 and t2 = 0.05), Lmin = 4.15 m, Lmax = 10.5 m, vmin = 10 km/h and vmax = 40 km/h.
Path-following controller design The meaningful task of the path-following control system is to guarantee that the autonomous vehicles can track the desired roads in a continuous and smooth manner. In cross-country environments, the unevenness, the discontinuity and the variability of the terrain greatly increase the parametric uncertainties and incompleteness of the dynamic systems of the vehicle. These parametric uncertainties may cause poor performance and instability. The GCC technique is viewed as
where vx is the longitudinal velocity, vy is the the lateral velocity, r is the yaw rate, m is the total mass of the vehicle, Iz is the total inertia, lf is the distance of the front-wheel axle from the centre of gravity, lr is the distance of the rear-wheel axle from the centre of gravity, Cf is the cornering stiffness of the front tyres, Cr is the cornering stiffness of the rear tyres, df is the front-wheel steering angle, af is the side-slip angle of the front tyres, ar is the side-slip angle of the rear tyres, Ff is the longitudinal front-tyre force and Fr is the longitudinal reartyre force. Because the cornering stiffness coefficients of the tyres are very imprecise, which depends on many factors (e.g. the adhesion coefficients), the path-following control system must be robust to the variations in these coefficients. In this section, the uncertainties in the tyre cornering stiffness can be written as23 r 41 Cf = mCf0 = Cf0 1 + Df rf , f ð22Þ Cr = mCr0 = Cr0 ð1 + Dr rr Þ, k rr k41 where m is the adhesion coefficient, Df is the magnitude of the deviation of the front cornering stiffness, Dr is the magnitude of the deviation of the rear cornering stiffness, Cf0 is the front nominal cornering stiffness, Cr0 is the rear nominal cornering stiffness, rf is the front parametric perturbation and rr is the rear parametric perturbation. The visual sensor and the laser sensor capture the images of the cross-country terrains in front of autonomous vehicles and extract informationon the the relative position between the vehicles and the desired trajectories in real time. The lateral error between the
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Proc IMechE Part D: J Automobile Engineering 00(0)
current vehicle position and the desired path is denoted by yL, and the orientation error shaped by the road tangent and the heading of vehicle at a specified lookahead distance L is denoted by eL. yL and eL can be derived as y_ L = vx eL vy rL e_L = vx r r
ð23Þ
where r is the curvature of the desired path. The uncertain lateral dynamics model of autonomous vehicles consisting of the simplified vehicle dynamics model (21) and the kinematic dynamics model (23) is obtained as x_ = (A0 + DA)x + (B0 + DB)u + Ew y = Cx
ð24Þ
Cr0 lr Cf0 lf mvx
Cf0 + Cr0 mvx
6 6 Cf0 lf + Cr0 lr A0 = 6 Iz vx 6 4 1 0 3 2
vx
Cf0 l2f + Cr0 l2r Iz vx
L 1
0 0 0 0
0
3
7 7 07 7 vx 5 0
Cf0
6 m 7 6 Cf0 lf 7 7 B0 = 6 6 Iz 7 4 0 5 0 2 3 0 607 7 E=6 405 vx
0 0 1 0 C= 0 0 0 1
DB = DF½ E1
C D
f0 x f 6 Cmv f0 Df lf 6 D = 6 I z vx 4 0 0 2 rf 0 6 0 rr F=6 40 0 0 0
Cr0 Dr mvx Cr0 Dr lr Ivx
0 0
0 0 1 0
3 0 07 7 05 1
3 0 07 7 05 1
FT (t)F(t)4I
ð26Þ
Robust guaranteed-cost path-following controller design
J=
xT (t)Qx(t) + uT (t)Ru(t) dt
ð27Þ
0
E2
ð25Þ
with 2
0 0 1 0
It can be seen that F(t) satisfies
ð‘
where x = [vy r yL eL]T is the system state vector, u = [df ] is the control input vector and w = [r] is the exogenous disturbance vector. It is assumed that the time-varying uncertain matrices DA and DB are norm bounded and can be described as ½ DA
rf 0 6 0 rr E1 = 6 40 0 0 0 2 3 vx 6 0 7 7 E2 = 6 4 0 5 0
The design requirement of a vehicle path-following control scheme is to track accurately the desired path while ensuring the stability of an autonomous vehicle.1 GCC is an effective approach for dealing with a problem with external distractions, parametric uncertainties and multiple objectives.21,24 In this subsection, an optimal guaranteed-cost path-following control approach that can ensure the stability of the path-following control system for autonomous vehicles with parameter uncertainties is proposed, and the quadratic performance index with system (24) is given as
with 2
2
0 0 1 0
0
3
7 07 7 05 1
where Q and R are positive definite matrices. The performance index function is constructed with the state variables vy, r, yL and eL and the control input df which can ensure that the path-following control system not only satisfies a certain level of energy consumption but also is asymptotically stable. The structure of the GCC law can be designed as u = Kx(t)
ð28Þ
Substituting equation (28) into equation (24), the closed-loop path-following control system is obtained as _ = ½A + BK + DFðE1 + E2 KÞx(t) + Ew x(t)
ð29Þ
Next, an OGC is designed not only to make the uncertain closed-loop automated path-following control system asymptotically stable but also to define a minimum upper bound of the performance index given by J4J*, where J* is a given constant. Remark 1. The external disturbance w is the smaller value, and the influence of w cannot destroy the stability of the closed-loop control system;6,8 therefore, the parameter w is neglected in the design of the pathfollowing controller.
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The stability condition for the uncertain closed-loop path-following control system (29) is given in the following theorems. Theorem 1. The closed-loop path-following control system (29) of autonomous vehicles with the feedback control law (28) is asymptotically stable, if there exists a positive definite matrix P = PT . 0 and a feedback gain K satisfying the matrix inequality Q + KT RK + P½A + BK + DFðE1 + E2 KÞ + ½A + BK + DFðE1 + E2 KÞT P \ 0
ð30Þ
Theorem 2. For the closed-loop path-following control system (29), there exists matrices P = PT . 0 and K, such that the matrix inequality (30) holds if and only if there exist scalars e . 0, and matrices W and X = XT . 0, such that the linear matrix inequality is satisfied as 2 3 N0s ðE1 X + E2 WÞT X WT 6 E1 X + E2 W eI 0 0 7 6 7\0 1 4 X 0 Q 0 5 W 0 0 R1 ð37Þ with
for all admissible uncertainties.
N0s = AX + BW + (AX + BW)T + eDDT
Proof. Defining the Lyapunov function as V(x) = xT Px
ð31Þ
the time derivative of V(x) along the trajectory of the closed path-following control system (23) is given by _ = xT fP½A + BK + DFðE1 + E2 KÞ V(x) o + ½A + BK + DFðE1 + E2 KÞT P x
ð32Þ
From equation (30), the inequality established for all admissible uncertainties is _ = xT Q + KT RK x V(x) ð33Þ \0 Based on the Lyapunov stability theory, the closedloop path-following control system of autonomous vehicles is asymptotically stable. Furthermore, by integrating both sides of the inequality (33) from 0 to N, and by means of the Lyapunov theorem, it can be obtained as ð‘ J=
T x (t)Qx(t) + uT (t)Ru(t) dt ð34Þ
0
ð38Þ
If the matrix inequality has a feasible solution (e, W, X), thus, a candidate controller gain can be given by K = WX1
ð39Þ
Equation (39) is the state feedback guaranteed-cost path-following control law of autonomous vehicles, and the corresponding upper bound of the system performance is J4Tr(X 1) = J
ð40Þ
Proof. If we define the equation Y = Q + KT RK + P(A + BK) + (A + BK)T P
ð41Þ
then the matrix inequality can be given as Y + PDFðE1 + E2 KÞ + ðE1 + E2 KÞT FT (PD)T \ 0 ð42Þ
By means of Lemma 1, the above matrix inequality holds for all uncertain matrices F satisfying FTF4I, if and only if there exist constants e . 0, such that Y + ePDDT P + e1 ðE1 + E2 KÞT ðE1 + E2 KÞ \ 0
4V(x(0))
ð43Þ
= xT0 Px0
On applying the Schur complement, inequality (43) is equivalent to 2 3 N1s ð E 1 X + E 2 WÞ T I KT 6 E1 + E2 K eI 0 0 7 6 7\0 1 4 I 0 Q 0 5 K 0 0 R1 ð44Þ
Therefore, u = Kx(t) is a guaranteed control law for the path-following control system, and J = xT0 Px0 is an upper bound of the performance index for the closed-loop system. This completes the proof. Lemma 1. Given appropriately dimensioned matrices N, L and M, with NT = N and FTFT4I, then T T
T
N + LFM + M F L \ 0
ð35Þ
holds, if and only if there exists e . 0, which satisfies N + eLLT + e1 MT M \ 0
ð36Þ
with N1s = P(A + BK) + (A + BK)T P + ePDDT P
Multiplying both sides of equation (44) by diag P1 , I, I, I
ð45Þ
ð46Þ
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Proc IMechE Part D: J Automobile Engineering 00(0)
and using X = P21 and W = KP21 yield the matrix inequality in equation (37), and a candidate controller gain can be given as K = WX21. Theorem 3. For the lateral dynamic system (24) of autonomous vehicles with the cost function (27), ~ X~1 x(t) is an optimal state feedback control u(t) = W ~ X, ~ M ~ for the optilaw, if there exists a solution ~e, W, mization problem min ½Tr(M) 2 N0s 6E X + E W 6 1 2 s:t: (i)6 4 X e, W, X, M
W
(ii)
M
I
I
X
ðE1 X + E2 WÞT eI
X 0
0 0
Q1 0
3 WT 0 7 7 7\0 0 5 R1
Figure 5. Profile of the desired trajectory.
dfb = Kb x
.0
ð47Þ
~ X, ~ ~ Proof: If ~e, W, M is asolution of the optimization ~ X, ~ M ~ is a solution of constraints problem (47), ~e, W, (i) in problem (47); based on Theorem 2, ~ X~1 x(t) is a GCC law of the path-following u(t) = W control system (24). By using the Schur complement, constraints (ii) of problem (47) can be equivalent to M . X1 . 0; therefore, the minimum of Tr(M) can ensure the minimum of the upper bounds of the performances of the system, because both the objective function and the constraints of problem (47) are the convex function of the variables; thus, a convex optimization problem is formulated to design the optimal guaranteed-cost pathfollowing controller, and the goal minimum can be guaranteed.
Performance verification The proposed control scheme in the third section is evaluated using simulation tests and experimental tests in different working conditions.
Simulation results First, the MATLAB–ADAMS co-simulation experiments are conducted to evaluate the dynamic characteristics of the presented path-following controller. In ADAMS, a virtual 14-degrees-of-freedom vehicle with a Dugoff tyre model is constructed and the tyre–ground adhesion features are configured. In addition, a linear quadratic regulator (LQR) feedback path-following controller is compared with the above-presented control scheme, and it is designed on the basis of the linear dynamics model consisting of equations (21) and (23). The LQR control strategy, which minimizes the performance index,25 is given as
ð48Þ
where Kb = [20.055 20.256 0.224 1.322], and the performance index is given as ð‘ JLQR =
xT QLQR x + uT RLQR u dt
ð49Þ
0
The weighting matrices QLQR and RLQR are selected as 2 3 10 0 0 0 6 0 10 0 0 7 6 7 QLQR = 6 7 4 0 0 10 0 5 0
0
0
10
RLQR = 2
A simulation test path is constructed which consists of several straight sections and several curved sections, as shown in Figure 5. First, in the initial stages, the curvature of the reference path is zero, which is utilized to analyse the behaviours of the proposed control method on a straight road. Second, the reference path is time varying, which is applied to analyse the behaviours of the proposed method on a road with various curvatures. Then, the vehicle negotiates two roads with opposite constant curvatures of 0.0015 m21 and 20.01 m21. The initial lateral error and the initial orientation error are 0.3 m and 23° respectively, and the speed is set as 60 km/h. As demonstrated earlier in the third section, the cornering stiffness of the tyres is sensitive to the surface adhesion coefficient. As a consequence, the proposed lateral controller should guarantee sufficiently high performance and robustness for the variable cornering stiffness. To illustrate this condition, simulation tests are conducted for different adhesion coefficients. Figure 6 shows the comparisons of the responses of the proposed control method and the LQR controller on a dry road with an adhesion coefficient of 0.8. As
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9
Figure 6. Response results for m = 0.8: (a) response of the lateral error; (b) response of the orientation error; (c) response of the yaw rate;(d) response of the steering angle. OGC: optimal guaranteed cost; LQR: linear quadratic regulator.
illustrated in Figure 6(a), it is interesting to note that both the proposed optimal guaranteed-cost scheme and the LQR scheme can direct the steady-state lateral errors to be bounded by 60.1 m and 60.18 m, and both the maximum lateral errors occur in the section of the reference path with the largest curvature of 0.015 m21. The dynamic responses of the orientation errors are exhibited in Figure 6(b). It can be seen that both the proposed optimal guaranteed-cost scheme and the LQR scheme have a steady-state orientation error within 60.9°; furthermore, the overshoot and the oscillations produced by the proposed scheme are smaller than those of the LQR scheme. Figure 6(c) demonstrates the dynamic response results for the yaw rate; it is worth noting that the yaw rate can be converted to the expected value by the proposed optimal guaranteed-cost approach and the LQR approach, that the response speed is increased and that the oscillations are decreased by the presented optimal guaranteed-cost scheme. Figure 6(d) demonstrates the contrasting results of the front-wheel steering angles provided by the proposed optimal guaranteed-cost scheme and the LQR scheme; it can easily be seen that the control inputs
of the proposed optimal guaranteed-cost scheme are smoother than those of the LQR. Figure 6 indicates that the presented optimal guaranteed-cost scheme has a better dynamic tracking precision than does the LQR scheme. In the same way, Figure 7 shows the comparisons of the responses of the proposed optimal guaranteed-cost scheme and the LQR controller on an icy road with an adhesion coefficient of 0.2. During automatic steering, the values of the lateral error and the orientation error do not increase very much, even though the vehicle is turning swiftly. From Figure 7, it is worth noting that the dynamic performances of both the schemes are degraded to different extents but, in comparison with the LQR scheme, the stability and the tracking performance can be controlled by the optimal guaranteedcost scheme in an acceptable range. One of the reasons why the system performances become degraded is that deviations of the working conditions occur on the icy road. Furthermore, the proposed optimal guaranteedcost approach takes into account the parametric uncertainties of autonomous vehicles beforehand; however, the LQR scheme does not.
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Proc IMechE Part D: J Automobile Engineering 00(0)
Figure 7. Response results for m = 0.2: (a) response of the lateral error; (b) response of the orientation error; (c) response of the yaw rate; (d) response of the steering angle. OGC: optimal guaranteed cost; LQR: linear quadratic regulator.
Figure 8. Tested autonomous vehicle. PC: personal computer; ECU: electronic control unit.
Guo et al.
Figure 9. Detected path in the experimental tests.
Experimental results The proposed automatic path-following controller was verified on a prototype autonomous vehicle, and the configuration of the prototype autonomous vehicle is shown in Figure 8. A central processing system based on a personal computer (PC) and three CCD cameras together form the real-time visual system, which has a processing time of less than 20 ms/frame. A LSM211 laser installed on the vehicle sends a 9600 Bd signal via the serial interface to the connected PC and, in the 180°/0.5° resolution mode, there are 361 measurement values per scan. The combination of a visual system and a laser
11 system acquires information on the external environment, such as the positive and negative static obstacles and the relative position between the vehicle and the desired trajectory. The actuator electonic control unit regulates the steering system to track the desired commands. In order to test the validity of the control law for given initial conditions in real cross-country circumstances, a practical trial is conducted, and the results are compared with those of the LQR controller and the PID controller3 at the same speed; the desired road edge that was used to guide the vehicle autonomously is reviewed as quasi-straight, as shown in Figure 9. During the tests, the vehicle speed is assumed to remain constant by a longitudinal controller.26 The practical comparison results of the proposed optimal guaranteed-cost scheme, the LQR controller and the PID controller at the actual velocity of 25 km/h are graphically depicted in Figure 10.The initial lateral error and the orientation error are 20.4 m and 4° respectively. The response of the proposed optimal guaranteed-cost scheme is denoted as the solid curve, and the responses of the LQR controller and the PID controller are denoted as the dashed curve and the dotted curve respectively. Figure 10(a) demonstrates that the maximum overshoots of the lateral error controlled by the LQR and PID controller are almost 8% and 21% respectively, but the overshoot of the lateral error controlled by the optimal guaranteed-cost scheme is close to zero. Figure 10(b) shows the response of the orientation errors; it can be seen that the overshoot of
Figure 10. Experimental results: (a) response of the lateral error; (b) response of the orientation error;(c) response of the yaw rate;(d) response of the steering angle. OGC: optimal guaranteed cost; LQR: linear quadratic regulator; PID: proportional–integral–derivative.
12 orientation error controlled by the optimal guaranteedcost scheme is extremely small. In addition, it is interesting to note that the increasing times of the lateral errors and the orientation errors controlled by the optimal guaranteed-cost scheme are smaller than those of the LQR controller and the PID controller. Furthermore, the proposed optimal guaranteed-cost scheme has less overshoot and smaller oscillations than do the automatic LQR controller and the PID controller, and the steady-state lateral error and the steady-state orientation error of the proposed optimal guaranteed-cost scheme are close to zero. The comparison results of the yaw rate are illustrated in Figure 10(c); it is well known that the optimal guaranteed-cost scheme can guarantee the stability and the ride comfort of the vehicle. Figure 10(d) shows the contrasting results of the control input in the experimental tests; it is clear to see that the front-wheel steering angle provided by the optimal guaranteed-cost scheme is acceptable within certain limits. It is clear that the path-following control law developed in this work can reasonably be considered to be valid to drive an autonomous vehicle as precisely as a human can. As a conclusion, the automatic path-following control mission not only is accurately performed but also yields a ride of high quality.
Conclusions This paper has proposed an optimal guaranteed-cost lateral control scheme for vision-based autonomous vehicles in unstructured environments. An unstructured road detection algorithm based on the combination of a visual sensor and a laser sensor is constructed, which can accurately detect in real time and extract the desired trajectory in complex cross-country working conditions. Additionally, an optimal guaranteed-cost pathfollowing control scheme of autonomous vehicles is proposed to guarantee high performances and considers the parametric uncertainties in the cornering stiffness of the tyres. Furthermore, the simulation results and the experimental results demonstrate that the presented path-following scheme has good tracking performances even in adverse driving conditions. Acknowledgements The authors gratefully thank the reviewers for their valuable suggestions. Declaration of conflict interest The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article. Funding The author(s) disclosed receipt of the following financial support for the research, authorship, and/or
Proc IMechE Part D: J Automobile Engineering 00(0) publication of this article: This work was supported by the Natural Science Foundation of Fujian Province (grant number 2017J01100), the National Basic Research Project of China (grant number 2016YFB0100900), the National Natural Science Foundation of China (grant number U1564208) and the Fundamental Research Funds for the Central Universities (grant number 20720160102). References 1. Guo J, Li K, Luo Y et al. Coordinated control of autonomous four drive electric wheels for platooning and trajectory tracking using a hierarchical architecture. Trans ASME, J Dynamic Systems, Measmt, Control 2015; 137(10): 1–18. 2. Li K, Tao C, Luo Y et al. Intelligent environment friendly vehicles concept and case studies. IEEE Trans Intell Transpn Systems 2012; 13(1): 318–328. 3. Marino R, Scalzi S and Netto M. Nested PID steering control for lane keeping in autonomous vehicles. Control Engng Practice 2011; 19(12): 1459–1467. 4. Raimondi F and Melluso M. Fuzzy motion control strategy for cooperation of multiple automated vehicles with passengers comfort. Automatica 2008; 44(11): 2804–2816. 5. Kayacan E, Kayacan E, Ramon H et al. Towards Agrobots: trajectory control of an autonomous tractor using type-2 fuzzy logic controllers. IEEE/ASME Trans Mechatronics 2015; 20(1): 2912–2924. 6. Wu S, Chiang H, Perng J et al. The heterogeneous systems integration design and implementation for lane keeping on a vehicle. IEEE Trans Intell Transpn Systems 2008; 9(2): 246–263. 7. Perez J, Milanes V and Onieva E. Cascade architecture for lateral control in autonomous vehicles. IEEE Trans Intell Transpn Systems 2011; 12(1): 73–82. 8. Guo J, Hu P, Li H et al. Design of automatic steering controller for trajectory tracking of unmanned vehicles using genetic algorithms. IEEE Trans Veh Technol 2012; 61(7): 2913–2924. 9. Onieva E, Naranjo J and Milanes V. Automatic lateral control for unmanned vehicles via genetic algorithms. Appl Soft Comput 2011; 11(1): 1303–1309. 10. Kammel S, Ziegier J, Pitzer B et al. Team AnnieWAY’s autonomous system for the 2007 DARPA Urban Challenge. J Field Robotics 2007; 25(9): 615–639. 11. Tan H, Bu F and Bougler B. A real-world application of lane-guidance technologies – automated snowblower. IEEE Trans Intell Transpn Systems 2007; 8(3): 538–548. 12. Huang J and Tomizuka M. LTV controller design for vehicle lateral control under fault in rear sensors. IEEE/ ASME Trans Mechatronics 2005; 10(1): 1–7. 13. Tsing Q, Hong C and Dong C. Switching-based stochastic model predictive control approach for modelling driver steering skill. IEEE Trans Intell Transpn Systems 2015; 16(1): 365–375. 14. Enache N, Mammar S and Netto M. Driver steering assistance for lane-departure avoidance based on hybrid automata and composite Lyapunov function. IEEE Trans Intell Transpn Systems 2010; 11(1): 28–39. 15. Guo J, Li L, Li K et al. An adaptive fuzzy-sliding lateral control strategy of automated vehicles based on vision navigation. Veh System Dynamics 2013; 51(10): 1502–1517.
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