Vehicle System Dynamics International Journal of Vehicle Mechanics and Mobility
ISSN: 0042-3114 (Print) 1744-5159 (Online) Journal homepage: http://www.tandfonline.com/loi/nvsd20
Robust gain-scheduling automatic steering control of unmanned ground vehicles under velocityvarying motion Jinghua Guo, Yugong Luo & Keqiang Li To cite this article: Jinghua Guo, Yugong Luo & Keqiang Li (2018): Robust gain-scheduling automatic steering control of unmanned ground vehicles under velocity-varying motion, Vehicle System Dynamics, DOI: 10.1080/00423114.2018.1475677 To link to this article: https://doi.org/10.1080/00423114.2018.1475677
Published online: 22 May 2018.
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VEHICLE SYSTEM DYNAMICS https://doi.org/10.1080/00423114.2018.1475677
Robust gain-scheduling automatic steering control of unmanned ground vehicles under velocity-varying motion Jinghua Guoa , Yugong Luob and Keqiang Lib a Department of Mechanical and Electrical Engineering, Xiamen University, Xiamen, People’s Republic of
China; b Department of Automotive Engineering, State Key Laboratory of Automotive Safety & Energy, Tsinghua University, Beijing, People’s Republic of China ABSTRACT
ARTICLE HISTORY
Owing to the fact that unmanned ground vehicles have the features of time-varying, parametric uncertainties and external disturbances, this paper mainly studies robust automatic steering control of unmanned ground vehicles. Firstly, a linear parameter varying lateral model for unmanned ground vehicle is constructed, in which the longitudinal velocity is represented by a polytope with finite vertices. Secondly, a robust gain scheduling automatic steering control scheme based on the linear matrix inequality technique is proposed to deal with the characteristics of time-varying and external disturbances. Finally, Simulation results based on Adams–Matlab joint platform using a nonlinear full vehicle model have demonstrated that the proposed control approach can simultaneously ensure the control accuracy and strong robustness of system.
Received 26 April 2017 Revised 31 October 2017 Accepted 6 May 2018 KEYWORDS
Unmanned ground vehicles; automatic steering control; robust gain-scheduling control; time-varying; lateral motion
1. Introduction As one of the most progressive technology of automotive industry, unmanned ground vehicles have potential applications in many areas, such as executing the deep-space missions and carrying out the routine tasks for industry. Furthermore, it can be used in the intelligent transportation systems to improve vehicle safety and road utilisation. Therefore, unmanned ground vehicles have been attracted more and more attention from industry and academic communities. Automatic steering control is devoted to guarantee that unmanned ground vehicle accurately tracks the expected trajectory based on the onboard sensors. Much research work on the automatic steering control has been carried out in recent years. In order to deal with the parametric uncertainties and nonlinearities of vehicles, an adaptive fuzzy sliding lateral control approach is conducted [1], in which a fuzzy system is replaced with a pure sign function in the conventional sliding mode control law and the control gain is tuned on-line by a fuzzy logic. Xia et al. [2] propose a novel lateral control law based on active disturbance rejection control scheme and differential flatness theory, which can guarantee both control accuracy and strong robustness. In order to track the planned trajectory for collision avoidance manoeuvres, Ji et al. [3] design a path-tracking controller via a multi-constrained model predictive control technique, which can accurately calculate the front steering angle CONTACT Jinghua Guo
[email protected]
© 2018 Informa UK Limited, trading as Taylor & Francis Group
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to prevent the vehicle from colliding with a moving obstacle vehicle. In [4,5], to achieve the accurate and smooth tracking by incorporating the dynamic characteristics of the steering actuation system, an optimal trajectory of the steering command is calculated by applying a quadratic programming optimisation method. Guo presents a genetic-based fuzzy automatic steering control scheme to imitate human analogical reasoning and decision making [6], in which the parameters of rule base and membership functions are given by the intelligent genetic algorithms. A nested proportional-integral-derivative lateral control system of vision-based autonomous vehicles is proposed and experimentally tested to perform path following in the case of roads with an uncertain curvature [7]. Enache et al. [8] construct an assistant steering control system using the synthetically composite Lyapunov theory and the hybrid automata theory, and the practical test verifies the feasibility of this proposed control method. Automatic steering control system is designed by the input-output feedback linearisation method [9], but, the accurate model of the plant dynamics needs to be acquired beforehand. Though there are many published works on automatic steering control of unmanned ground vehicles, most of the automatic steering control approaches are focused on the fixed longitudinal velocity case. Because the longitudinal velocity of vehicles usually varies within a wide range, the abovementioned researches focused on a constant velocity have a limited application area. Since the unmanned ground vehicles have the characteristics of parametric uncertainties, time-varying and external disturbance, the difficulty of research of automatic steering control of unmanned vehicles is obviously increased. Robust control is an effective approach to handle with the features of time-varying and parameter uncertainties [10–12]. In this paper, we research the automatic steering control problem of unmanned ground vehicles using the robust gain-scheduling control (RGSC) technique. The main contributions of this works can be summarised as follows: (i) a novel dynamic model which can accurately describe the dynamic features of unmanned ground vehicle is presented, in which a polytope is developed to cover the time-varying longitudinal velocity range. (ii) A robust gain-scheduling automatic steering control approach is proposed, which can effectively deal with the features of uncertainties and time varying of unmanned ground vehicles. (iii) Extensive simulation and experimental studies are used to manifest the automatic steering control performance. The rest of this paper is organised as follows. Section 2 gives a specific description of linear parameter varying (LPV) lateral motion model of unmanned ground vehicles. Section 3 proposes a robust gain-scheduling automatic steering control approach for unmanned ground vehicles under the velocity-varying motion. Simulation and experimental results of the proposed control approach under different operating conditions are discussed in Section 4. Finally, conclusions are drawn in Section 5.
2. Problem formulation 2.1. Notation A matrix polytope can be described as the convex hull of a finite number of matrices Si with the same dimensions, such as l l ai Si : ai ≥ 0, ai = 1 . (1) Co{Si , i = 1, . . . , l} := i=1
i=1
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An LPV continuous-time system can be described in state-space form by the equations x˙ = A (ϑ)x + B (ϑ)u, y = C (ϑ)x + D (ϑ)u,
(2)
where x is the state vector, y is the output vector and u is the input vector, ϑ is a timevarying vector of real parameters, A (ϑ), B (ϑ), C (ϑ) and D (ϑ) denote the state-space matrices, which are the functions of time-varying parameter ϑ. The LPV system (1) has the following properties as: (i) The parameter dependence is affine, the state-space matrices A (ϑ), B (ϑ), C (ϑ) and D (ϑ) depend affinely on ϑ. (ii) The time-varying parameter ϑvaries in a polytope of vertices υ1 , υ2 , . . . , υl , for instance, ϑ ∈ := Co{υ1 , υ2 , . . . , υl } Then, the state-space matrices A (ϑ), B (ϑ), C (ϑ) and D (ϑ) can be written as A i B i A (ϑ) B (ϑ) A (υi ) B (υi ) ∈ Co , i = 1, . . . , l . := C (ϑ) D (ϑ) C (υi ) D (υi ) C i D i
(3)
(4)
2.2. Lateral dynamic model Unmanned ground vehicle is a nonlinear and time-varying system. The main assumptions made in deriving the model are the following: (i) Neglect roll, pitch and vertical motions. (ii) Ignore the difference of tyre cornering properties between the left and right wheel due to the load variation, approximate the tyre model as linearity. (iii) Discount the actuator dynamics (for brake, throttle, and steering). The first assumption is valid without appreciable loss in accuracy under typical to slightly severe manoeuvres for vehicles [13]. As shown in Figure 1, the vehicle lateral dynamics model in the yaw plane can be obtained as 1 (Fyf + Fyr ) − vx r, m 1 r˙ = (lf Fyf − lr Fyr ), Iz
v˙y =
(5)
where vx , vy and r denote the longitudinal velocity, lateral velocity, and yaw rate within a fixed inertial frame, respectively. Iz is the vehicle total inertia. Fyf and Fyr represent the front and rear tyre lateral forces, which are functions of tyre slip angles and can be expressed as Fyf = Cf af ,
Fyr = Cr ar ,
(6)
where Cf and Cr denote the cornering stiffness of front and rear tyres, respectively, af and ar denote the wheel slip angles, which can be obtained as a f = δf −
lf r vy − , vx vx
ar =
lr r vy − , vx vx
(7)
where lf and lr represents the distance of the front and rear axles from the centre of gravity. δ f denotes the front wheel steering angle. Equation (5) can be rewritten as v˙y = a11 vy + a12 r + b1 δf , r˙ = a21 vy + a22 r + b2 δf
(8)
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Figure 1. Vehicle lateral dynamic model.
with a11 = −(Cr + Cf )/mvx a12 = (lr Cr − Cf lf )/mvx − vx a21 = (lr Cr − lf Cf )/Iz vx a22 = −(lf2 Cf + lr2 Cr )/Iz vx
(9)
b1 = Cf /m b2 = lf Cf /Iz .
2.3. Lateral kinematic model Vehicle lateral kinematic model is acquired based on the relative position between the vehicle and the desired trajectory, as shown in Figure 2. Vision system detects the real-time expected trajectory and then calculates the angular error and the lateral error [14,15]. The angular error L denotes the relative angular between the heading of vehicle and the road tangent at a preview look-ahead distance, the lateral error yL denote the lateral distance between the current vehicle position and the expected reference trajectory, which can be expressed as y˙ L = sin(εL )vx − rDL − vy ,
(10)
where DL denote the look-ahead distance. The angular error is assumed as a small value, which is widely used in the lateral motion control of unmanned ground vehicles [16,17], such as sin(εL ) ≈ εL , then, the kinematic model of unmanned ground vehicles can be obtained as y˙ L = vx εL − vy − rDL , ε˙ L = vx KL − r,
(11)
where K L represents the curvature of the reference trajectory at the preview point. Taking the Laplace transform of the combination of Equations (8) and (11), then, the transfer function from δ f to yL can be acquired as M(s) =
M1 (s) yL (s) = , δf (s) M2 (s)
(12)
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Figure 2. Schematic diagram of kinematic model.
where M1 (s) = s2 vx2 Cf I + svx Cr Cf (Lf Lr + L2r ) + Cr Cf vx2 (Lf + Lr ) + DL (s2 mvx2 Cf Lf + svx Cr Cf (Lf + Lr )), M2 (s) = s2 (s2 mvx2 I + svx (I(Cf + Cr ) + m(Cf L2f + Cr L2r )) + mvx2 (Cr Lr − Cf Lf ) + Cf Cr (Lf + Lr )2 ).
(13)
(14)
Remark 2.1: The root locus of system (12) demonstrates that the higher longitudinal velocity vx will damage the stability of vehicle lateral system and the larger look-ahead distance DL will reduce the percentage overshoot of system, however, the image resolution will be degraded by a far look-ahead distance. Therefore, a proper DL value plays an important role in the performance of system. 2.4. LPV model of unmanned ground vehicles The vehicle lateral dynamic equation (8) is combined with the kinematic model (11) to form the following linear state-space equation, which can be represented as: x˙ (t) = Ax(t) + Bu(t) + Ew(t) with
⎡ a11 ⎢a21 A=⎢ ⎣−1 0
a12 a22 −DL −1
0 0 0 0
⎤ 0 0⎥ ⎥ vx ⎦ 0
⎡ ⎤ b1 ⎢b2 ⎥ ⎥ B=⎢ ⎣0⎦ 0
(15) ⎡ ⎤ 0 ⎢0⎥ ⎥ E=⎢ ⎣0⎦, vx
(16)
where the state vector is x = [vy r yL L ]T , the control input vector is u = δ f , and the exogenous disturbance vector is ω = K L .
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From Equation (15), it can be seen that the time-varying parameters vx and 1/vx exist in the system model matrix [18], because the longitudinal velocity vx is usually bounded and varying in the range [v, v], ¯ and 1/vx varies in the range of [1/v, ¯ 1/v], v and v¯ denote the minimum and maximum value of velocity. In this section, a polytope which contain 22 = 4 vertices is applied to descript all possible choices for the parameter pair (vx , 1/vx ). The coordinates of vertices can be obtained as ¯ ηˆ 1 = v, ηˆ 2 = v,
(17)
¯ η˜ 2 = 1/v. η˜ 1 = 1/v,
(18)
Therefore, the time-varying parameters vx and 1/vx can be represented by a linear combination of the vertices as follows vx =
2
ρˆi (t)ηˆ i ,
1/vx =
i=1
2
ρ˜i (t)η˜ i ,
(19)
i=1
where ρˆi and ρ˜i denote the weighting factors, which can be written as ρˆ1 (t) =
v¯ − vx vx − v , ρˆ2 (t) = , v¯ − v v¯ − v
(20)
1/v¯ − 1/vx 1/vx − 1/v , ρ˜2 (t) = . (21) 1/v¯ − 1/v 1/v¯ − 1/v Based on Equations (15) and (19), an LPV lateral dynamic model of unmanned ground vehicles can be rewritten in the polytypic form as ρ˜1 (t) =
x˙ (t) =
4
ρi (t)[Ai x(t) + Bi u(t) + Ei ω(t)]
i=1
(22)
= A(ρ)x(t) + B(ρ)u(t) + E(ρ)ω(t) with ρ1 = ρˆ1 ρ˜1 ; ρ2 = ρˆ1 ρ˜2 ; ρ3 = ρˆ2 ρ˜1 ; ρ4 = ρˆ2 ρ˜2 T ρ = ρ1 ρ2 ρ3 ρ4 . T The measured output vector y = r yL εL is defined as y(t) = Cx(t) with
⎡ 0 C = ⎣0 0
1 0 0
0 1 0
(23)
(24) ⎤ 0 0⎦ . 1
Combining Equations (22) and (24), the LPV model of unmanned ground vehicles can be expressed as x˙ (t) = A(ρ)x(t) + B(ρ)u(t) + E(ρ)w(t), y(t) = Cx(t).
(25)
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3. Robust gain-scheduling automatic steering controller design The task of automatic steering control problem is to construct a robust control law u(t) such that the close-loop system is asymptotically stable and has the following H ∞ performance index γ to attenuate the effect of the external disturbances, such as ||y||2 ≤ γ ||w||2 .
(26)
In order to facilitate the analysis of H ∞ performance and stability of the closed-loop system, the following useful lemmas are introduced. Lemma 3.1 [19]: Consider a polytopic LPV plant featured by the state-space equation ¯ ¯ x˙ = A(p)x + B(p)u,
(27)
¯ y = C(p)x with
¯ A(p) ¯ C(p)
¯ B(p) A¯ i ∈ = Co 0 C¯ i
where Co
A¯ i C¯ i
B¯ i , 0
i = 1, . . . , n =
n i=1
B¯ i ,i 0
A¯ pi ¯ i Ci
= 1, . . . , n ,
B¯ i : pi ≥ 0, 0
n
(28) pi = 1 ,
i=1
¯ ¯ ¯ B(p) and C(p) at the vertices pi of the parameter A¯ i , B¯ i and C¯ i denote the values of A(p), polytope. Then, the following statements are equivalent: (i) This LPV system is stable with quadraticH ∞ performance index γ1 . ¯ ¯ A(p) B(p) (ii) There exists a single matrix P > 0 for all ¯ ∈ , such that the following C(p) 0 condition holds: ⎡ T ⎤ ¯ ¯ PB(p) C¯ T (p) A¯ (p)P + PA(p) ⎣ (29) ∗ −γ1 I 0 ⎦ 0 satisfying the following linear matrix inequalities (LMIs), such as ⎤ ⎡ T C¯ iT A¯ i P + PA¯ i PB¯ i ⎣ (30) ∗ −γ1 I 0 ⎦ < 0, i = 1, 2, . . . , n, ∗ ∗ −γ1 I where the symbol * denotes the symmetric element of the matrix, is a convex set.
¯ A(p) In Lemma 3.1, it can be seen that the condition (29) will hold for all ¯ C(p) A¯ B¯ i for i = 1, . . . , n. if and hold if it holds at the vertices ¯ i Ci 0
¯ B(p) ∈ 0
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Lemma 3.2 [20]: The following statements are equivalent. (1) T I I < 0, W V V
(31)
˜ such that the following condition holds: (2) There exists a matrix G T N ˜T ˜ W+ G +G V −I
−I < 0,
(32)
where I is the identity matrix, V and W are arbitrary matrices. ˜ proLemma 3.2 can produce an expanding condition (32) and the slack variable G vides additional flexibility for verifying the LMI conditions, therefore, the system matrix and Lyapunov function can be decoupled by the Lemma 3.2. Theorem 3.1: The LPV system (27) is stable with the quadratic H∞ performance index γ2 , if and only if there exists a symmetric positive-definite matrix F such that GT VG < 0, where
⎡
I ⎢ 0 G=⎢ ⎣ 0 A¯ T (p)
0 I 0 C¯ T (p)
⎤ 0 0⎥ ⎥; I⎦ 0
⎡
0 ⎢ 0 V=⎢ ⎣B¯ T (p) F
(33)
0 −γ2 I 0 0
¯ B(p) 0 −γ2 I 0
⎤ F 0⎥ ⎥. 0⎦ 0
(34)
Proof: ⎡
⎤T ⎡ ¯ I 0 0 0 0 B(p) ⎢ ⎢ ⎥ 0 I 0 0 −γ I 0 2 ⎥ ⎢ GT VG = ⎢ T ⎣ ⎣ 0 ⎦ ¯ 0 I 0 −γ2 I B (p) F 0 0 A¯ T (p) C¯ T (p) 0 ⎡ ⎤ I 0 0 ⎢ 0 ⎥ I 0 ⎢ ⎥ ⎣ 0 0 I⎦ A¯ T (p) C¯ T (p) 0 ⎤ ⎡ T ¯ ¯ F C¯ T (p) B(p) PA¯ (p) + A(p)F ⎥ ⎢ ¯ =⎣ 0 ⎦ < 0. C(p)F −γ2 I B¯ T (p) 0 −γ2 I
⎤ F 0⎥ ⎥ 0⎦ 0 (35)
Therefore, based on the Lemma 3.1, this LPV system (25) is stable with the quadratic H∞ performanceγ2 .
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Theorem 3.2: The LPV system (27) is stable with the quadratic H∞ performance γ2 , if and only if there exists n symmetric positive-definite matrices Fi (i = 1, . . . , n) and a matrix G such that the following condition holds: ⎡ ⎤ A¯ i G + GT A¯ Ti GT C¯ T B¯ i Fi − GT ⎢ ∗ −γ2 I 0 0 ⎥ ⎢ ⎥ < 0. (36) ⎣ ∗ ∗ −γ2 I 0 ⎦ ∗ ∗ ∗ 0 Proof: From the Lemma 3.2 and Theorem 3.1, defining ˜ T1 = G 0 0 0 G
(37)
⎤ ¯ A(p) ¯ ⎦ N T = ⎣C(p) 0 ⎡
⎡
0 ⎢∗ M=⎢ ⎣∗ ∗
0 −γ2 I ∗ ∗
¯ B(p) 0 −γ2 I ∗
(38) ⎤ F 0⎥ ⎥, 0⎦ 0
˜ 1 is a slack variable, N and M are arbitrary matrices. where G Substituting Equations (37)–(39) into Equation (32), then ⎡ ⎤ ¯ 0 0 B(p) F ⎢∗ −γ2 I 0 0⎥ ⎢ ⎥ ⎣∗ ∗ −γ2 I 0 ⎦ ∗ ∗ ∗ 0 ⎤ ⎡ T⎤ ⎡ ¯ A(p) G ⎢C(p) ¯ ⎥ ⎢ 0 ⎥ T ⎥ ⎢ ⎥ ¯ ¯T +⎢ ⎣ 0 ⎦ G 0 0 0 + ⎣ 0 ⎦ A (p) C (p) 0 −I < 0 0 −I ⎡ ⎤ ¯ 0 0 B(p) F ⎢∗ −γ2 I 0 0⎥ ⎥ =⎢ ⎣∗ ∗ −γ2 I 0 ⎦ ∗ ∗ ∗ 0 ⎤ ⎡ T T ⎡ ⎤ ¯ A(p)G 0 0 0 G A¯ (p) GT C¯ T (p) 0 −GT ⎢ ⎢C(p)G ¯ 0 0 0 0 ⎥ 0 0 0⎥ ⎥+⎢ ⎥
