Nonlinear Dyn DOI 10.1007/s11071-017-3928-9

ORIGINAL PAPER

Trajectory tracking control for tractor–trailer vehicles: a coordinated control approach Ming Yue Jun Chen

· Xiaoqiang Hou · Renjing Gao ·

Received: 21 January 2017 / Accepted: 4 November 2017 © Springer Science+Business Media B.V., part of Springer Nature 2017

Abstract This article presents a coordinated control approach for a tractor–trailer vehicle such that a satisfactory trajectory tracking performance can be achieved, simultaneously guaranteeing vehicle kinematics restriction and dynamics maneuvers. The coordinated control is consisted of multilevel controllers, each of which is constructed by different algorithms to better clarify their specific advantages and defects, thereby establishing the composition principles of this multilevel architecture. In this regard, on the level of kinematics, linear quadratic regulator and model predictive control (MPC) are used to design the posture controller separately; on the level of dynamics, sliding mode control and global terminal sliding mode control (GTSMC) are introduced to design the dynamic controller for the tracking of the desired velocities generated online. The simulation results suggest that the combination by MPC and GTSMC can offer more favorable control performance for such kind of sophisticated vehicle system.

M. Yue (B) · X. Hou · R. Gao School of Automotive Engineering, Dalian University of Technology, Dalian 116024, Liaoning, China e-mail: [email protected] J. Chen School of Naval Architecture and Ocean Engineering, Harbin Institute of Technology (Weihai), WeiHai 264209, Shandong, China

Keywords Tractor–trailer vehicle · Model predictive control · Linear quadratic regulator · Terminal sliding mode · Coordinated control

1 Introduction Over the past few decades, more and more attention has been paid to the tractor–trailer vehicle for its large cargo capacity, higher transport efficiency, lower cost and fuel consumption, see, e.g., [1–8]. In contrast to the trajectory tracking of traditional vehicles, the control objective of the tractor–trailer vehicle is to drive the midpoint of the two parallel plating wheels for the trailer vehicle to follow an arbitrary path given by the earth-fixed frame. As for this kind of vehicle, the mobile platform is a typical underactuated system since there exist merely two driven wheels as control inputs for tractor, but at least four states reflecting the posture and orientation of tractor and trailer need to be controlled simultaneously. Meanwhile, the tractor–trailer vehicle system must satisfy miscellaneous physical limitations, as well as guarantee the time-vary hitch angle of the platform to be stable during movement. In addition, apart from the nonholonomic constraint for mobile platform, the vehicle system will suffer from unavoidable disturbance, especially in the sophisticated unstructured environment. These particular features are all challenging problems for control community; therefore, finding an effective control strategy for the trajectory tracking of tractor– trailer vehicle is indeed a difficult work.

123

M. Yue et al.

To achieve satisfactory control performance, several studies have focused on various control methods to implement trajectory tracking for this kind of vehicle system. A survey about this kind of connected vehicle system can be found in [5]. As for internal dynamics of the underactuated vehicle system, Maciej revealed that the number and type of interconnections will influence the nonminimum-phase effects for this system in [9], and then, a general trajectory tracking control solution was proposed for truly N -trailer robots comprising an unicycle-like tractor and arbitrary number of passive trailers with sign-homogeneous nonzero hitching offsets [10]. In [2], a Lyapunov-based kinematic control law and a feedback linearization-based dynamic controller were separately reported to make up a robust adaptive dynamic controller, by which a favorable tracking performance can be achieved. Admittedly, these investigations can achieve good control performance for the vehicle system, but finding an appropriate Lyapunov candidate is indeed a difficult matter for the control scholars. After an extensive investigation of control schemes, linear quadratic regulator (LQR) and model predictive control (MPC) catch our attention since they can avoid the selection of Lyapunov candidate. These two mentioned schemes are both constructed on the optimal theory, but there exist some differences for their design and application. The LQR technique achieves the optimal control law with the aid of state linear feedback, by using cheap cost to make the original system obtain a better performance, which is verified to be simple and easy to perform. However, LQR control is excessively dependent on explicit dynamic model, and at the same time, various constraints cannot be applied when it is utilized for practice. In contrast, the MPC approach can overcome these drawbacks readily since the rolling optimization technique is employed [11–13]. In general, the main characteristics of this approach can be summarized as follows: firstly, a forecast model needs to be proposed, which can fit the dynamic characteristics of the controlled system; secondly, MPC optimizes the performance of the finite horizon objective function at each sampling period with the current status and forecast model of the system, to find the optimal control sequence and then the first element of the sequence is applied in the controlled object; last but not least, the feedback correction mechanism is adopted to compensate the model prediction error and other disturbances. To this end, MPC possesses numerous favorable mer-

123

its compared with LQR, thereby leading to its widely application in practical implementation recently [14– 16]. The tractor–trailer vehicle is a sophisticated and underactuated vehicle system, subject to the nonholonomic constraint on the kinematic level, as well as suffer from the uncertainty and the various disturbances on the dynamic level. As far as disturbance rejection is concerned, sliding mode control (SMC) and global terminal sliding mode control (GTSMC) techniques have been paid increasing attention over the past few decades due to their favorable merits, such as parameter insensitivity, simple design procedure and robust capability [17–19]. As a result, it is wildly applied in considerable fields, like rigid spacecraft [20,21], robot finger [22,23], active suspension [24,25] and wheeled mobile robot [26–29]. In this study, for the sake of enhancing the robustness of tractor–trailer vehicle system to overcome the abrupt load or all kinds of interferences, SMC and GTSMC are separately introduced to design the dynamic controller. Besides, now that the control objectives arise from different aspects, namely kinematic and dynamic levels, a coordinated control approach needs to be proposed to achieve multiobjective control effects. After analysis, a typical hierarchical mechanism gets our attention, because it can harmonize the system’s kinematic and dynamic behaviors at the same time. However, how to synthesize the above techniques to establish an effective control approach, in addition to taking the physical limits and uncertain disturbance into consideration, is still an open problem, which motivates this study. To sum up, this paper attempts to present a control strategy consisting of a posture controller and a dynamic one so as to achieve a coordinated control objective. With the presented tracking error dynamics, a MPC-based posture controller is proposed to realize the posture tracking purpose on kinematic level. Besides, a dynamic controller is put forward by employing SMC or GTSMC technique to assure the system robustness to overcome the uncertain disturbance. In summary, the main contributions of this paper lie in: (i) Compared with LQR, the MPC-based controller is adopted for tractor–trailer vehicle to make the midpoint of the two parallel plating wheels for the underactuated trailer follow an arbitrary reference trajectory, as well as satisfy various physical limits, and (ii) a control approach is proposed to achieve a coordinated path tracking control for a tractor–trailer vehicle via a powerful layered

Trajectory tracking control for tractor–trailer vehicles: a coordinated

b

Y

. φP

l

Tractor c

.P

αl

a

αr

ψl

.K

y

Table 1 Nomenclature of tractor–trailer system parameters

.K θ c

d

Trailer ψr

X

x Fig. 1 Schematic diagram of tractor–trailer vehicle system

mechanism, where the MPC and GTSMC techniques are effectively merged together and combined the both merits powerfully. The paper is organized in the following manner. Section 2 briefly describes the dynamic model of a tractor– trailer vehicle. In Sect. 3, a tracking error dynamics for the vehicle system is reported. In Sect. 4, LQR-based and MPC-based posture controllers with respect to the kinematic model are proposed, while SMC-based and GTSMC-based controllers are developed in Sect. 5. In Sect. 6, the simulation results are discussed to validate the effectiveness and feasibility of the proposed control approaches. Conclusions then summarize the paper in Sect. 7.

Parameters

Parameter definition

a

Distance between points K and K c

b

Half of the distance between parallel wheels

d

Distance between points P and K

l

Distance between points P and Pc

x, y

Coordinates of point K

ϕ

Orientation of the frame attached to tractor

θ

Orientation of the frame attached to trailer

αr

Angular displacement of tractor right wheel

αl

Angular displacement of tractor left wheel

ψr

Angular displacement of trailer right wheel

ψl

Angular displacement of trailer left wheel

Ttt

Kinetic energy

m1

Mass of the tractor body

m2

Mass of the trailer body

m3

Mass of a tractor wheel

m4

Mass of a trailer wheel

Iω

Moment of inertia of tractor wheels about vertical axis

Itω

Moment of inertia of trailer wheels about vertical axis

J1

Moment of inertia of tractor about vertical axis

J2

Moment of inertia of trailer about vertical axis

r

Radius of wheels

τr

Torque exerted on tractor right wheel by actuator

τl

Torque exerted on the tractor left wheel by actuator

2 Tractor–trailer vehicle system As shown in Fig. 1, a tractor–trailer vehicle mainly consists of a differentially driven vehicle (i.e., tractor), and a passive following vehicle (i.e., trailer), where both entities are attached by a planar rigid linkage mechanism. In general, the tractor has two driven wheels powered by DC motors, the trailer has two passive wheels, and the one of linkage is flexibly connected to the tractor, while the other end is fixedly hinged to the trailer. In this case, assume that the tractor–trailer vehicle allows movement in an obstacle-free planar environment and the wheels have pure rolling in forward direction with no slip along the lateral direction. In addition, for convenience, let P and K be the midpoint of the two par-

allel plating wheels for tractor and trailer, respectively; similarly, let Pc and K c be the mass centers for tractor body and trailer body, respectively. Then, the related nomenclatures of the investigated vehicle system are defined in Table 1. Consider u = [υ, ω]T as system input vector, where υ is the linear velocity of point P and ω is the angular velocity of tractor. Besides, q = [x, y, ϕ, θ ]T can be introduced to describe the position information for the entire tractor–trailer vehicle system. The underactuated tractor–trailer vehicle system with unknown dynamics and disturbances can be formulated as follows

123

M. Yue et al. Fig. 2 Schematic diagram of the control system

qr υc Posture ω Controller c

eυ

Controller

eω



˙ q(t) = S(q)u (1) ¯ ¯ ¯ ˙ + C(q, ˙ M(q)u(t) q)u(t) + τ d = B(q)τ ¯ ¯ ˙ = where M(q) = S T (q)M(q)S(q) ∈ R2×2 , C(q, q) T ˙ ¯ ˙ + C(q, q)S(q) ∈ R2×2 and B(q) S (q) M(q) S(q) = S T (q)B(q) ∈ R2×2 . Here τ d = [τd1 , τd2 ]T denotes the disturbances acting on the vehicle, caused by the unavoidable factors, like viscous friction between mechanical components, air resistance, external distur¯ ¯ ˙ bance or uncertainty, etc. In this case, M(q), C(q, q), ¯ B(q), S(q) can be derived as ⎤ ⎡ 2 (ϕ − θ ) + Iθ sin2 (ϕ − θ ) 0 cos m ¯ ⎦, M(q) =⎣ 0 d2 0 Iϕ ⎡ ⎤ cos θ cos(ϕ − θ ) 0

⎢ sin θ cos(ϕ − θ ) 0⎥ 1 1 1 ⎢ ⎥ ¯ , , S(q) = ⎢ B(q) = 0 1⎥ ⎣ ⎦ r b −b 1 sin(ϕ − θ ) 0 d

τr 1 −m 1l ϕ˙ ¯ ˙ = τ= , , C(q, q) τl 0 m 1l ϕ˙ where 1 = − m 0 (ϕ˙ − θ˙ ) cos(ϕ − θ ) sin(ϕ − θ ) + Iθ (ϕ˙ − θ˙ ) sin(ϕ − θ ) cos(ϕ − θ ). More details about d2 the formulation of this tractor–trailer vehicle system can be referred to [2], and the derivation process is omitted here for brevity. In terms of system (1) without the nonholonomic constraint, the following properties still hold. Property 1 M¯ is a symmetric positive definite matrix. ˙¯ ¯ ˙ is skew symmetric. Property 2 M(q) − 2C(q, q)

Controller

Tractor–trailer vehicle τυ υsubsystem τω

ωsubsystem

q

υ ω

Kinematic Model

qe

to satisfying various physical constraints, all of which are developed on the system dynamics level. Hence, in order to achieve a coordinated control objective, this scenario adopts a new control structure considering the wheel dynamics, as shown in Fig. 2, to achieve both vehicle posture tracking and driving torque control simultaneously. Considering that various physical limitations including nonholonomic constraint should be guaranteed, MPC or LQR is adopted to construct posture controller because other traditional controller designing methods require to find sophisticated Lyapunov candidates. In this case, by the method of receding horizon optimization, the posture tracking can be readily achieved. Besides, to accomplish high tracking performance for desired velocity, SMC or GTSMC technique is employed to establish dynamic controller, by which the tracking errors can be assured to converge fast and robustly.

3.1 Reference trajectories As mentioned previously, the control objective is to drive the midpoint of the two parallel plating wheels for trailer, i.e., K , to follow the reference trajectory given by the earth-fixed frame, by controlling the driven torques of left and right wheels. Here, if the reference trajectories of point K are defined as xr = xr (t) and yr = yr (t), then by geometrical relationship, the reference linear velocity of point P attached to the tractor can be computed by



2  2 d d (xr (t) + d cos θr ) + (yr (t) + d sin θr ) dt dt

3 Control structure

υr =

Apart from the vehicle needs to realize the posture tracking on kinematics level, the tractor–trailer platform is also required to possess good dynamic performance and disturbance rejection capability, in addition

whereafter, define that θr (t) = arctan xy˙˙rr (t) (t) with θr (t) ∈ (− π, π ]. Obviously, it holds that θ˙r = y¨r x˙2r −x¨r2 y˙r , and

123

x˙r + y˙r

the reference angular of the tractor can be obtained by

Trajectory tracking control for tractor–trailer vehicles: a coordinated ˙

ϕr (t) = θr + arcsin dυθrr afterward. With these preparations, the reference angular velocity of the tractor can be formulated as υr (2) ωr = θ˙r +  υr 2 − d2 θ˙r2 It is worth mentioning that the reference trajectory, namely q r = [xr (t), yr (t), ϕr (t), θr (t)]T , and the reference control inputs and their derivatives are all continuous and uniformly bounded.

4 Trajectory planning The effect of outer closed loop is to drive the vehicle to track the reference posture trajectory on the kinematic level. Considering the sophisticated kinematic equation, LQR and MPC are adopted to play this role separately for further analysis.

4.1 LQR-based posture controller Define the control signals

3.2 Tracking error dynamics u¯ =

A tracking error space needs to be constructed based on the current trajectory and the reference one for realizing trajectory tracking purpose, and afterward the tracking error dynamics should be derived for control system synthesis. At the first stage, define q¯ = q−q r , and then, the tracking error vector q e = [ex , e y , eϕ , eθ ]T can be ¯ where T is the transformation obtained by q e = T q, Γ (− θr ) 02×2 matrix as T = , with Γ (− θr ) = 0 I2×2

2×2 sin θr cos θr ∈ R2×2 being the rotation matrix, − sin θr cos θr I2×2 ∈ R2×2 being the identity matrix, and 02×2 ∈ R2×2 being the zero matrix. Notice that q¯ represents the tracking error based on the earth-fixed frame, while q e is the tracking error between the practical vehicle and the reference one described by the body-fixed frame. After that, by differentiating q e with respect to time, a tracking error dynamics can be achieved by ⎧ e˙x = θ˙r e y + υ cos eθ cos(ϕ − θ ) − υr cos(ϕr − θr ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ e˙ y = − θ˙r ex + υ sin eθ cos(ϕ − θ ) (3) e˙ϕ = ω − ωr ⎪ ⎪ ⎪ ⎪ υ υ ⎪ ⎩ e˙θ = sin(ϕ − θ ) − r sin(ϕr − θr ) d d Furthermore, for convenient programming, according to the definition of the system variables and tracking errors, the tracking error dynamics (3) can be further rewritten as follows ⎧ e˙x = θ˙r e y + υ cos eθ cos(eϕ + ϕr − eθ − θr ) ⎪ ⎪ ⎪ ⎪ ⎪ − υr cos(ϕr − θr ) ⎪ ⎪ ⎨ e˙ y = − θ˙r ex + υ sin eθ cos(eϕ + ϕr − eθ − θr ) (4) ⎪ ⎪ e˙ϕ = ω − ωr ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ e˙ = υ sin(e + ϕ − e − θ ) − υr sin(ϕ − θ ) θ ϕ r θ r r r d d



u¯ 1 u¯ 2

υ cos eθ cos(eϕ + ϕr − eθ − θr ) − υr cos(ϕr − θr ) = ω − ωr

(5)

The original system needs to be linearized since it is a highly nonlinear system. Then, it can be obtained that q˙ e = Aq e + B u¯ where ⎡

0 ⎢− θ˙r A =⎢ ⎣ 0 0 ⎡

⎢ ⎢ B =⎢ ⎣

(6)

⎤ 0 0 0 υr cos(ϕr − θr )⎥ ⎥, ⎦ 0 0 3 2 ⎤ 1 0 0 0⎥ ⎥ 0 1⎥ ⎦

θ˙r 0 0 0

1 tan(ϕr − θr ) 0 d

with 2 = d1 υr cos(ϕr − θr )(1 + tan2 (ϕr − θr )) and 3 = − d1 υr cos(ϕr − θr )(1 + tan2 (ϕr − θr )) Subsequently, the linearized state-space model can be described as  q˙ e = Aq e + B u¯ (7) u¯ = − K f q e where K f = [K f 1 , K f 2 , K f 3 , K f 4 ; K f 5 , K f 6 , K f 7 , K f 8 ] is the state-feedback gain vector. Meanwhile, the quadratic performance index function can be chosen as  1 t T ¯ [q Q 1 q e + u¯ T R1 u]dt min J f = 2 0 e (8) s.t. q˙ e = Aq e + B u¯ where Q 1 and R1 are the designed weight matrices.

123

M. Yue et al.

Based on the linear quadratic regulator theory, the feedback control law can be given as

u¯ u¯ = − K f q e = 1 u¯ 2

(9) −(K f 1 ex + K f 2 e y + K f 3 eϕ + K f 4 eθ ) = −(K f 5 ex + K f 6 e y + K f 7 eϕ + K f 8 eθ )

where

Define uc = [υc , ωc ]T as the desired velocity vector, which is generated in real time by LQR-based posture controller. From (5), one can ultimately get that

t = ⎡

uc = ⎡ ⎣



υc = ωc

⎤

−(K f 1 ex + K f 2 e y + K f 3 eϕ + K f 4 eθ ) + υr cos(ϕr − θr ) ⎦ cos eθ cos(eϕ + ϕr − eθ − θr ) −(K f 5 ex + K f 6 e y + K f 7 eϕ + K f 8 eθ ) + ωr

(10)

4.2 MPC-based posture controller Considering that the algorithm is mainly suitable for the discrete system, the system needs to be discretized in the following. Then, one can ultimately get that ¯ q e (k + 1) = Ak,t q e (k) + Bk,t u(k)

(11)

Since the model is discretized by the approximate discretization method, it can be obtained that Ak,t = I + ATs and Bk,t = BTs with Ts being the sample interval. The state of the MPC controller is defined as

q e (k|t) (12) ξ (k|t) = ¯ − 1|t) u(k Hence, the state-space model of the new system is obtained as  ξ (k + 1|t) = A¯ k,t ξ (k|t) + B¯ k,t U (k|t) (13) η(k|t) = C¯ k,t ξ (k|t)



Bk,t Ak,t Bk,t ¯ ¯ where Ak,t = , Bk,t = and 0m×n Im Im C¯ k,t = [In , 0n×m ] with n being the dimension of state and m being the dimension of control variables. For simplicity, the following assumptions need to be given by  A¯ k,t = A¯ t,t , k = 1, . . . , t + N − 1 (14) B¯ k,t = B¯ t,t , k = 1, . . . , t + N − 1 After further deduction, the output expression of the system is obtained as Y (t) = t ξ (t|t) + t U (t)

123

(15)

⎤ ⎡ ⎤ C¯ t,t A¯ t,t η(t + 1|t) ⎢ C¯ t,t A¯ 2 ⎥ ⎢ η(t + 2|t) ⎥ t,t ⎥ ⎢ ⎥ ⎢ ⎢ ··· ⎥ ⎥ ⎢ ··· ⎥ ⎢ ⎥ ⎢ Y (t) = ⎢ N ⎥, ⎥ , t = ⎢ ⎢ C¯ t,t A¯ t,tc ⎥ ⎢ η(t + Nc |t) ⎥ ⎥ ⎢ ⎦ ⎣ ··· ⎣ ··· ⎦ Np η(t + N p |t) C¯ t,t A¯ t,t ⎡

C¯ t,t B¯ t,t 0 ⎢ C¯ t,t A¯ t,t B¯ t,t C¯ t,t B¯ t,t ⎢ ⎢ ⎢ ··· ··· ⎢ ⎢ Nc −1 ¯ Nc −2 ¯ ¯ t,t A¯ t,t ⎢ C¯ t,t A¯ t,t B C Bt,t t,t ⎢ Nc −1 ¯ ⎢ C¯ t,t A¯ Nc B¯ t,t ¯ ¯ Ct,t At,t Bt,t t,t ⎢ ⎢ .. .. ⎢ . . ⎣ N p −1 N p −2 ¯ ¯ ¯ ¯ ¯ Ct,t At,t Bt,t Ct,t At,t B¯ t,t ⎡ ⎤ ¯ u(t|t) ⎢ u(t ¯ + 1|t) ⎥ ⎥. U (t) = ⎢ ⎣ ⎦ ··· ¯ + Nc |t) u(t

0 0 ..

. ··· ··· .. . ···

⎤

0 0 ··· C¯ t,t B¯ t,t C¯ t,t A¯ t,t B¯ t,t .. . N p −Nc ¯ ¯ B¯ t,t Ct,t At,t

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

Considering that the real-time performance of the control algorithm is vital for the control of tractor– trailer vehicle system, the incremental control is adopted instead of the control volume in which the relaxation factor is introduced. To a great extent, it not only restricts the control increment directly, but also prevents no feasible solution from appearing in the process of execution. The objective function can be defined as J (k) =

Np 

||η(k + i|t) − ηr e f (k + i|t)||2Q 2

i=1

+

N c −1

|| U (k + i|t)||2R2 + ρ 2

(16)

i=1

where N p is the predictive horizon, Nc is the control horizon, ρ is the weighted coefficient and  is the relaxation factor. Here, Q 2 and R2 are the designed weight matrices. Moreover, if mainly consider the limits of the control variables and control increment in the control process, then the expression of control variables can be ¯ + k) ≤ u¯ max (t + k), k = obtained as u¯ min (t + k) ≤ u(t 0, 1, . . . , Nc −1; while, the expression of control increment constraint can be described as u¯ min (t + k) ≤ ¯ + k) ≤ u¯ max (t + k), k = 0, 1, . . . , Nc − 1. u(t Here, u¯ min and u¯ max are lower and upper bounds of the control input, respectively; u¯ min and u¯ max are lower and upper bounds of the control input increment, respectively.

Trajectory tracking control for tractor–trailer vehicles: a coordinated

To obtain a standard quadratic model with the constraint conditions, some preparations are made to simplify the procedures. Due to the following relationship ¯ + k) = u(t ¯ + k − 1) + u(t ¯ + k), suppose that u(t ¯ − 1), in addition to Ut = 1 Nc ⊗ u(k ⎤ ⎡ 1 0 ··· ··· 0 ⎢1 1 0 · · · 0 ⎥ ⎥ ⎢ ⎥ ⎢ .. ⎢ . 0⎥ A = ⎢1 1 1 ⎥ ⊗Im ⎥ ⎢. . . . .. . . 0⎦ ⎣ .. .. 

1

1

··· 

1

Nc ×Nc

1



where 1 Nc is the column vector, Nc is the number of rows, Im is the identity matrix with m being the dimension, ⊗ denotes the Kronecker product, and u(k ¯ − 1) is ¯ + the control variable of last moment. u¯ min (t +k) ≤ u(t k) ≤ u¯ max (t + k), k = 0, 1, . . . , Nc − 1 can be written as Umin ≤ A Ut + Ut ≤ Umax , where Umin , Umax are the control variables of minimum and maximum values in the control horizon. With these preparations, the objective function can be further transformed into a standard quadratic model with the constraint conditions, which can be described as follows 1 ¯ − 1), U (t)) = U (t)T Ht U (t) J (ξ(t), u(t 2 + G Tt U (t) + ρ 2 s.t. Umin ≤ Ut ≤ Umax Umin ≤ A Ut + Ut ≤ Umax ξ (k + 1|t) = A¯ k,t ξ (k|t) + B¯ k,t U (k|t)

(17)

η(k|t) = C¯ k,t ξ (k|t) k = t, . . . , t + N p Y (t) = t ξ (t|t) + t U (t) where Ht = 2(t T Q 2 t + R2 ) and G t = 2Tt Q 2 et . Also, et = t ξ (t|t) − Yr e f (t) is the tracking error in predictive horizon, where Yr e f = [ηr e f (t + 1|t), . . . , ηr e f (t + N p |t)]T . See Refs. [30,31] for more details about this objective function. By solving (17), a series of control input increment in the control horizon can be obtained as Ut∗ = [ u¯ ∗t , u¯ ∗t+1 , . . . , u¯ ∗t+Nc −1 ]T , and subsequently, the first element of the control sequence can be set as the actual control input increment of the tractor–trailer vehicle system, namely ¯ ¯ − 1) + u¯ ∗t (18) u(t) = u(t Define uc = [υc , ωc ]T as the desired velocity vector, which is generated in real time by MPC-based posture controller. From (5), it can ultimately obtain that

⎡ ⎤

u¯ 1 + υr cos(ϕr − θr ) υc = ⎣ cos eθ cos(eϕ + ϕr − eθ − θr ) ⎦ . uc = ωc u¯ 2 + ωr (19)

Whereafter, the rolling optimization is adopted to achieve the real-time trajectory tracking control of the tractor–trailer vehicle system. Notice that there exist numerous control techniques, such as neural networks [32–34], fuzzy logic [35–37], and their composite application [38–40], to enhance the closed-loop system adaptability. Also, several efforts devote to proposing controller considering various physical limits [41– 44]. However, MPC can deal with adaptive and constrained problem simultaneously by utilizing previous procedures. It is worth pointing out that LQR and MPC are introduced to design posture controller separately in this case; such treatment is just to find a more appropriate combination of multiple control techniques, thereby establishing the composition principle for the presented multilevel architecture.

5 Dynamic controller In this section, a dynamic controller is introduced to enhance the robustness and adaptability of the tractor– trailer vehicle system to overcome the unpredictable uncertainties. Similarly, a SMC-based and a GTSMCbased dynamic controllers are separately established for comparison. At first, it is worth mentioning that the vehicle longitudinal motion is driven by the sum of the left and right wheel torques, while the rotational motion is driven by their difference value. For simplicity, assume that τυ = τr + τl and τω = τr − τl , which represent the corresponding control inputs for the new subsystems. Furthermore, define  = m 0 cos2 (ϕ −θ )+ dIθ2 sin2 (ϕ −

1 l ϕ˙ θ ), f 1 = − 1 1 υ + m ω, g1 =

1 r ,

f2 =

−m 1 l ϕ˙ Iϕ υ,

and g2 = Iϕbr . For control purpose, the reduced order system (1) without considering the inference τ d = [τd1 , τd2 ]T can be partitioned into two subsystems as follows (i) υ-subsystem υ˙ = f 1 + g1 τυ

(20)

(ii) ω-subsystem ω˙ = f 2 + g2 τω

(21)

123

M. Yue et al.

5.1 SMC controller Define the velocity

error as ε(t) = uc (t) − tracking ε1 υc − υ u(t) = . = ε2 ωc − ω In view of the υ-subsystem, a PI-type sliding mode surface can be given by  T ε 1 (μ) dµ (22) s1 = ε 1 (t) + φ1

As applying SMC, these two subsystems are similar, and thus, the proof procedure for ω-subsystem can be omitted in this case. It is worth noting that that the chattering caused by sign function can be alleviated by introducing hyperbolic tangent function, i.e., tanh(·), in the presented controllers. 5.2 GTSMC controller

0

where φ1 is a positive design parameter. Differentiating (22) with respect to time yields s˙1 = ε˙ 1 (t) + φ1 ε 1 (t) = υ˙ c − υ˙ + φ1 (υc − υ)

(23)

= υ˙ c − ( f 1 + g1 τυ ) + φ1 (υc − υ) Here, the exponential reaching law is adopted to design the SMC dynamic controller, and thus, it holds that s˙1 = − σ1 sgn(s1 ) − k1 s1 . Accordingly, the control law for υ-subsystem can be given by 1 τυ = [υ˙ c − f 1 + φ1 (υc − υ) + σ1 sgn(s1 ) + k1 s1 ] g1 (24) The stability of the closed-loop system can be proved by selecting an appropriate positive Lyapunov candidate as 1 (25) V1 = s12 2 By differentiating V1 with respect to time along the vehicle dynamic system (20), it follows that V˙1 = s1 s˙1 = s1 (−σ1 sgn(s1 ) − k1 s1 ) = − σ1 |s1 | − k1 s12

(26)

= − σ1 |s1 | − 2k1 V1 Obviously, V˙1 ≤ 0. According to Lyapunov stability theorem, it concludes that all the signals in the inner closed-loop system are bounded and the tracking errors will converge to zero as time goes by. Hence, ε 1 = 0 is a uniformly asymptotically stable equilibrium point. Similar to the above procedures, in terms of the ωsubsystem, it can be obtained as  t ε 2 (μ) dµ (27) s2 = ε2 (t) + φ2 0

Besides, let s˙2 = − σ2 sgn(s2 ) − k2 s2 and then the control law of ω-subsystem can be given by 1 τω = [ω˙ c − f 2 + φ2 (ωc − ω) + σ2 sgn(s2 ) g2 + k 2 s2 ] (28)

123

The GTSMC with finite time characteristics has the excellent robustness to the disturbance. Based on the GTSMC theory, system (1) without considering the inference τ d = [τd1 , τd2 ]T can be described as  υ˙ = f 1 + g1 τυ (29) ω˙ = f 2 + g2 τω Define the tracking error as exυ = xυc − xυ and eϕc = ϕc − ϕ, where xυc , ϕc represent the desired longitudinal displacement and the rotational angle for tractor, respectively. Then, it holds that e˙xυ = x˙υc − x˙υ = υc − υ and e˙ϕc = ϕ˙c − ϕ˙ = ωc − ω. In the view of υ-subsystem, the global terminal sliding mode surface can be depicted as q1 p

sxυ = e˙xυ + ϑ1 exυ + β1 exυ1

(30)

In addition, the control law for υ-subsystem can be given by  q q1 p1 −1 1 x¨υc − f 1 + ϑ1 e˙xυ + β1 exυ1 e˙xυ τυ = g1 p1 q2  p (31) + φ3 sxυ + γ1 sxυ2 where q1 , p1 , q2 and p2 are odd positive integers satisfying q1 < p1 , q2 < p2 , φ3 > 0, γ1 > 0, ϑ1 > 0 q1 p

and β1 > 0. Notice that exυ1

−1

in controller (31) can be

q1 p1 −1

replaced by (exυ + 0.001) for avoiding the singularity problem when it is applied in practice. Differentiating (30) with respect to time yields q q1 p11 −1 s˙xυ = e¨xυ + ϑ1 e˙xυ + β1 exυ e˙xυ p1 (32) q q1 p11 −1 = x¨υc − f 1 − g1 τυ + ϑ1 e˙xυ + β1 exυ e˙xυ p1 Furthermore, substituting the control law (31) into (32) results in q q1 p1 −1 s˙xυ = e¨xυ + ϑ1 e˙xυ + β1 exυ1 e˙xυ p0 (33) q2 p

= − φ3 sxυ − γ1 sxυ2

Trajectory tracking control for tractor–trailer vehicles: a coordinated

Therefore, for proving the closed-loop system stability, a positive Lyapunov candidate can be considered as follows 1 (34) V2 = sx2υ 2 By differentiating (34) with respect to time along the υ-subsystem dynamics, it holds that V˙2 = sxυ s˙xυ = − φ3 sx2υ − γ1 s

q2 + p2 p2 xυ

(35)

Obviously, p2 + q2 is an even integer. Therefore, V˙2 ≤ 0. Hence, based on the Lyapunov stability theorem, it concludes that the velocity signal is bounded and the tracking error will converge to zero as time goes by. On the whole, the performance of the control system mainly depends on the designed parameters q1 , p1 , q2 , p2 , ϑ1 , β1 , φ3 and γ1 . As GTSMC is adopted to achieve the tracking of the velocities, these two subsystems are similar, and thus, only a general procedure for the design procedures is performed previously. The global terminal sliding mode surface of the ω-subsystem can be depicted as q3 p

sϕ = e˙ϕc + ϑ2 eϕc + β2 eϕc3

(36)

The control law of the ω-subsystem can be given by  q q3 p3 −1 1 ϕ¨c − f 2 + ϑ2 e˙ϕc + β2 eϕc3 e˙ϕc + φ4 sϕ τω = g2 p3 q4  p (37) + γ2 sϕ 4 where q3 , p3 , q4 and p4 are odd positive integers satisfying q3 < p3 , q4 < p4 , ϑ2 > 0, β2 > 0, φ4 > 0 and γ2 > 0.

6 Numerical simulation study To validate the control strategy, the numerical simulation is developed in the Matlab/Simulink environment, where the simulated parameters of the tractor– trailer vehicle system are listed in Table 2. In this case, the reference trajectory is supposed to be a circle added with a harmonic deviation to construct a more sophisticated path, which is given by  xr = 0.05(100 + 10 cos(36t/50)) cos(6t/50) (38) yr = 0.05(100 + 10 cos(36t/50)) sin(6t/50)

Table 2 Parameters of the tractor–trailer vehicle Parameters

Nominal values

Parameters

Nominal values

a

0m

Itω

0.00005 kg m2

b

0.06 m

J1

0.0039 kg m2

d

0.2 m

J2

0.0007 kg m2

l

0.03 m

r

0.025 m

m1

1 kg

Iϕ

0.005116 kg m2

m2

0.35 kg

Iθ

0.043416 kg m2

m3

0.03 kg

m0

1.47 kg

m4

0.03 kg

A

0.212 kg m

Iω

0.00005 kg m2

The hypothetical reference trajectory can test the longitudinal and steering maneuvers well due to its varied curvature radius and sophisticated shape. Also, the initial states with respect to the tracking error dynamics are set as q e (0) = [1, 1, 0, π/10]T . The basic parameters of MPC controller are set as: the predictive horizon is N p = 15, the control horizon is Nc = 5, the simulation time is 100 s and the sample interval is Ts = 0.1 s. By trail and error for the required performance, the other related design parameters are chosen as follows: Q 1 = I4 , R1 = 1000 × I2 , Q 2 = 10 × I60 , R2 = I10 , ρ = 10, φ1 = φ2 = 15, σ1 = σ2 = 1, k1 = k2 = 1, ϑ1 = 0.2, β1 = 0.1, ϑ2 = 2, β2 = 1, p1 = p3 = 9, q1 = q3 = 5, p2 = p4 = 3, q2 = q4 = 1, φ3 = 20, φ4 = 20, γ1 = 1 and γ2 = 1. Moreover, in view of the physical limits of tractor– trailer vehicle, the constraints of velocity and its control increment can be designed as − 0.05 m/s≤ u¯ 1 ≤ 0.05 m/s and − 0.02 m/s≤ u¯ 1 ≤ 0.02 m/s, respectively. Likewise, the control constraints of angular velocity and its increment can be designed as − 0.1 rad/s≤ u¯ 2 ≤ 0.1 rad/s and − 0.05 rad/s≤ u¯ 2 ≤ 0.05 rad/s, respectively. With the aforementioned formulations, the constraints for the control variables can be expressed as [− 0.05, − 0.1]T ≤ u¯ ≤ [0.05, 0.1]T and [− 0.02, − 0.05]T ≤ u¯ ≤ [0.02, 0.05]T . To further validate the robustness of controller to overcome the external disturbances, the interferences are introduced as: when t < 30 s, τd1 = 0.1 sin(0.05t), τd2 = 0.05 sin(0.05t); when t ≥ 30 s, τd1 = 0.1 tanh(sin(0.1t)), τd2 = 0.05 tanh(sin(0.1t)). First of all, the posture controller is only established to evaluate the performance of trajectory tracking for the reference trajectory, and then, the appreciate con-

123

M. Yue et al.

6 4

1

1

0.5 0 ey

ex 0.5 0

2 y,yr

1.5

-0.5 0

MPC Reference LQR

0

-0.5 10 20 ex(LQR)

0.2

30 40 50 ex(MPC)

-1 60 70 80 90 100 ey(LQR) ey(MPC) 0.5

-2 0 eθ

eφ 0

-4 -6 -6

-4

-2

0 x,xr

2

4

6

-0.2 0

10 20 eφ(LQR)

-0.5 30 40 50 60 70 80 90 100 eθ(LQR) eθ(MPC) eφ(MPC) t(s)

Fig. 3 Trajectory tracking for the tractor–trailer vehicle based on LQR and MPC

Fig. 4 Tracking errors for state variables based on LQR and MPC

trol technique can be found to obtain the better tracking performance. Trajectory tracking for the tractor–trailer vehicle described by the earth-fixed frame is shown in Fig. 3, where the dotted line represents the actual trajectory based on the MPC controller, the solid line stands for the reference one and dash-dotted line plots the actual trajectory based on LQR controller. The simulation results of the entire trajectory tracking suggest that the midpoint of the two parallel plating wheels of trailer will agree well with the reference curve with the aid of the tractor after a transient regulation behavior based on MPC controller; on the contrary, the LQR-based posture controller needs a long period of adjustment to achieve the target tracking. Certainly, the midpoint of the two parallel plating wheels of tractor does not follow the reference one affected by the vehicle mechanical structure, but this is trivial because the control objective is only concerned with the control of trailer. The simulated results clearly display the powerful control capability for MPC approach in dealing with this sophisticated underactuated and nonholonomic mobile system, which cannot be provided by LQR technique. Moreover, the time responses of tracking errors, i.e., ex , e y , eϕ and eθ , are shown in Fig. 4. After a period of adjustment, all tracking errors converge to be within required compact sets including the additional tracking error θ , which is related to the posture information between tractor and trailer. However, the tracking errors controlled by MPC-based controller take fewer time than the errors controlled by LQR-based controller, and the regulating processes based on MPC controller

appear more gentle that is more conducive to perform in practice. As a result, the aforementioned results tell that MPC controller possesses much more favorable control performance compared with the LQR one. On the other hand, in order to validate the dynamic performance provided by the dynamic controller, SMC and GTSMC are both employed to make a comparison. The trajectory tracking for the tractor–trailer vehicle is shown in Fig. 5, where the dash-dotted line represents the actual trajectory based on the SMC controller, the solid line stands for the reference one and the dotted line displays the actual trajectory based on GTSMC controller. The midpoint of the two parallel plating wheels of trailer agrees well with the reference curve with the aid of controlling the tractor after a transient regulation process. Besides, due to the influence of interference τd1 and τd2 , the trajectory has a sudden dramatic change as demonstrated in the selected area, but it is acceptable for the tractor–trailer vehicle system. In order to further exhibit the tracking performance for each variable, the time responses of tracking errors, i.e., ex , e y , eϕ and eθ , are shown in Fig. 6. It shows that during a period about 20 s, all tracking errors including the additional tracking error θ converge to a required compact set. In terms of the interference τd1 and τd2 as t = 30 s, the tracking errors have a sudden dramatic change as demonstrated in Fig. 6, but it will converge to an acceptable bound within 15 s. Obviously, the convergence rate with GTSMC controller is markedly faster than the SMC controller when encountered with mutation interference (t = 30 s), followed that its oscillation

123

Trajectory tracking control for tractor–trailer vehicles: a coordinated

6

1

4

GTSMC Reference SMC

0

Affected by snap load

0.4 0.2

-2

0

-4

-0.2

-6

-0.4

-6

-4

-2

0 x,xr

1 0.8 0.6 0.4 0.2 0 -0.2 -0.4

0.6 υc-υ

y,yr

2

GTSMC SMC

0.8

2

4

6

0

0.1 0.05 0 -0.05 -0.1 29 30 31 32 33 34 35 0 1 2 3 4 5 6 7 8

10

20

30

40

50 60 t(s)

70

80

90 100

Fig. 7 Time response of linear velocity tracking error Fig. 5 Trajectory tracking for the tractor–trailer vehicle based on SMC and GTSMC 1.5

1

0.5 0

-0.5

0.5

-1 60 70 80 90 100 ey(SMC) ey(GTSMC) 1

1.5

0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 28 29 30 31 32 33 34

1 0.5

ωc-ω

-0.5 0 10 20 30 40 50 ex(SMC) ex(GTSMC) 0.5

GTSMC SMC

1 0.5 e 0 y

1 ex

1.5

0 -0.5 0

0

1

2

3

4

5

6

-0.5 eφ 0

0 eθ

-1 -0.5 -1 0 10 20 30 40 50 60 70 80 90 100 e (SMC) e (GTSMC) eφ(SMC) eφ(GTSMC) θ θ t(s)

Fig. 6 Tracking errors for state variables based on SMC or GTSMC

amplitude is relatively small. Therefore, the resulting closed-loop trajectory tracking control system is verified to be stable with good performance. As previously mentioned, the MPC-based posture controller produces desired velocities, while the SMC-based and GTSMC-based dynamic controller are employed to realize the tracking for the generated velocity values, thereby leading to a coordinated control effect. For further illustrating this property, the time responses for velocity tracking errors are plotted in Figs. 7 and 8, in which the dash-dotted line and dotted line represent the SMC-based and GTSMC-based tracking errors for velocity, respectively. The fact that there exists drastic regulation process on the start point;

0

10

20

30

40

50 t(s)

60

70

80

90 100

Fig. 8 Time response of angular velocity tracking error

after the initial 0–2 s, the tracking errors for the velocity signals will perfectly converge to zero. Due to the influence of τd1 and τd2 , the tracking errors for the velocity signals have a sudden dramatic change as demonstrated in the drawing of partial enlargement about 30 s, which is converged to an acceptable bound within 2 s. Also, the tracking performance of the SMC-based controller for the linear velocity has a violent shake as illustrated in Fig. 7, while the tracking process of the SMC-based controller for the angular velocity will have some slight fluctuation. On the whole, the tracking performance for GTSMC-based controller is far better than the performance for SMC-based controller. In addition, the differential of the orientation of tractor and trailer, named hitch angle (i.e., ϕ − θ ), can directly reflect the lateral stability and vehicle han-

123

M. Yue et al.

0.2

GTSMC

Reference

0.15

1

SMC

GTSMC SMC

τυ

0.5

φ-θ,φr-θr

0.1

0.15 0.1 0.05 0 -0.05 -0.1 -0.15 -0.2 26

0.05 0

-0.05

-0.5 0

27 28 29 30 31 32 33 34 35

20

30

40

50

60

70

80

10 20 30

90 100

0 -0.5

-0.15 -0.2 0

10

0.5 τω

-0.1

0

40 50 60 t(s)

70 80 90 100

-1 0

GTSMC SMC 10

20

30

40

50 60 t(s)

70

80

90 100

Fig. 9 Actual and desired hitch angle for tractor–trailer vehicle

Fig. 10 Control inputs for tractor–trailer vehicle

dling, which is also plotted in Fig. 9. The desired hitch angle is computed by the reference trajectory in prior as described by solid line, while the time response of actual hitch angle based on the SMC and GTSMC dynamic controller is separately depicted by the dashdotted line and the dotted line, respectively, to highlight the control effects. With the adjustment after the initial time, the actual hitch angle can follow the desired one rapidly, which can exhibit favorable tracking capability and better dynamic behaviors of the subsequent trailer. However, affected by the snap load (τd1 , τd2 ), the actual hitch angle has a dramatic mutation about t = 30 s as illustrated in the drawing of partial enlargement, which is also converged to an acceptable bound within 2 s. Moreover, the control inputs are shown in Fig. 10, which reveals that all the related control inputs satisfy the constraints setting according to the requirements. The continuity of the control signals guarantees the feasibility of the proposed controllers.

ward, with the help of MPC-based posture controller, SMC and GTSMC are adopted to design the dynamic controller, respectively, and the simulation results confirm that with the GTSMC method, the tracking errors can be assured to converge to compact sets fast and robustly. The investigated results reveal that the presented control approach can deal with various problems from both kinematics and dynamics levels of vehicle system, and thus, a coordinated trajectory tracking performance is achieved.

7 Conclusions In this paper, a coordinated control approach is proposed for tractor–trailer vehicle with nonholonomic constraint and underactuated trailer, and varied control techniques are employed to construct the system controllers so as to exhibit their advantages and disadvantages. Firstly, LQR and MPC are used to design the posture controller, respectively, and the simulation results demonstrate that the tracking performance of MPC controller is better than the LQR controller. After-

123

Acknowledgements This research was supported by Grants from the National Natural Science Foundation of China (Nos. 61573078, 51475115, 11372063 and 11572073) and the Natural Science Foundation of Liaoning Province of China (No. 20170540171).

References 1. Divelbiss, A.W., Wen, J.T.: Trajectory tracking control of a car-trailer system. IEEE Trans. Control Syst. Technol. 5(3), 269–278 (1997) 2. Khalaji, A.K., Moosavian, S.A.A.: Robust adaptive controller for a tractor-trailer mobile robot. IEEE/ASME Trans. Mechatron. 19(3), 943–953 (2014) 3. Kayacan, E., Kayacan, E., Ramon, H., Saeys, W.: Learning in centralized nonlinear model predictive control: application to an autonomous tractor-trailer system. IEEE Trans. Control Syst. Technol. 23(1), 197–205 (2015) 4. Yuan, J., Sun, F.C., Huang, Y.L.: Trajectory generation and tracking control for double-steering tractor-trailer mobile robots with on-axle hitching. IEEE Trans. Industr. Electron. 62(12), 7665–7677 (2015) 5. Dorrell, D., Vinel, A., Cao, D.P.: Connected vehiclesadvancements in vehicular technologies and informatics. IEEE Trans. Industr. Electron. 62(12), 7824–7826 (2015)

Trajectory tracking control for tractor–trailer vehicles: a coordinated 6. Kayacan, E., Ramon, H., Saeys, W.: Robust trajectory tracking error model-based predictive control for unmanned ground vehicles. IEEE/ASME Trans. Mechatron. 21(2), 806–814 (2016) 7. Ritzen, P., Roebroek, E., De Wouw, N.V., Jiang, Z.P., Nijmeijer, H.: Trailer steering control of a tractorctrailer robot. IEEE Trans. Control Syst. Technol. 24(4), 1240–1252 (2016) 8. Yuan, J.: Hierarchical motion planning for multisteering tractor-trailer mobile robots with on-axle hitching. IEEE/ASME Trans. Mechatron. 22(4), 1652–1662 (2017) 9. Michałek, M.M.: Non-minimum-phase property of N -trailer kinematics resulting from off-axle interconnections. Int. J. Control 86(4), 740–758 (2013) 10. Michałek, M.M.: Cascade-like modular tracking controller for non-standard N -trailers. IEEE Trans. Control Syst. Technol. 25(2), 619–627 (2017) 11. Xiao, H.Z., Li, Z.J., Yang, C.G., Zhang, L.X., Yuan, P.J., Ding, L., Wang, T.M.: Robust stabilization of a wheeled mobile robot using model predictive control based on neurodynamics optimization. IEEE Trans. Industr. Electron. 64(1), 505–516 (2017) 12. Zhang, L.X., Zhuang, S.L., Braatz, R.D.: Switched model predictive control of switched linear systems: feasibility, stability and robustness. Automatica 67, 8–21 (2016) 13. Sun, X.M., Wu, D., Liu, G.P., Wang, W.: Input-to-state stability for networked predictive control with random delays in both feedback and forward channels. IEEE Trans. Industr. Electron. 61(7), 3519–3526 (2014) 14. Li, Z.J., Xiao, H.Z., Yang, C.G., Zhao, Y.W.: Model predictive control of nonholonomic chained systems using general projection neural networks optimization. IEEE Trans. Syst. Man Cybern. Syst. 45(10), 1313–1321 (2015) 15. Chen, H., Fang, Y.C., Sun, N.: A swing constraint guaranteed mpc algorithm for underactuated overhead cranes. IEEE/ASME Trans. Mechatron. 21(5), 2543–2555 (2016) 16. Li, Z.J., Yang, C.G., Su, C.Y., Deng, J., Zhang, W.D.: Visionbased model predictive control for steering of a nonholonomic mobile robot. IEEE Trans. Control Syst. Technol. 24(2), 553–564 (2016) 17. Niu, B., Zhu, J., Su, Y.K., Li, H.Y., Li, L.: Tracking control of uncertain switched nonlinear cascade systems: a nonlinear H∞ sliding mode control method. Nonlinear Dyn. 73(3), 1803–1812 (2013) 18. Bayat, F., Mobayen, S., Javadi, S.: Finite-time tracking control of nth-order chained-form non-holonomic systems in the presence of disturbances. ISA Trans. 63, 78–83 (2016) 19. Wang, W., Huang, J.S., Wen, C.Y., Fan, H.J.: Distributed adaptive control for consensus tracking with application to formation control of nonholonomic mobile robots. Automatica 50(4), 1254–1263 (2014) 20. Zhou, N., Xia, Y.Q., Wang, M.L., Fu, M.Y.: Finite-time attitude control of multiple rigid spacecraft using terminal sliding mode. Int. J. Robust Nonlinear Control 25(12), 1862– 1876 (2015) 21. Hu, Q.L., Xiao, B.: Adaptive fault tolerant control using integral sliding mode strategy with application to flexible spacecraft. Int. J. Syst. Sci. 44(12), 2273–2286 (2013) 22. Sun, W.C., Pan, H.H., Zhang, Y.F., Gao, H.J.: Multiobjective control for uncertain nonlinear active suspension systems. Mechatronics 24(4), 318–327 (2014)

23. Zhou, Q., Li, H.Y., Shi, P.: Decentralized adaptive fuzzy tracking control for robot finger dynamics. IEEE Trans. Fuzzy Syst. 23(3), 501–510 (2015) 24. Ning, D.H., Sun, S.S., Li, H.Y., Du, H.P., Li, W.H.: Active control of an innovative seat suspension system with acceleration measurement based friction estimation. J. Sound Vib. 384, 28–44 (2016) 25. Pan, H.H., Sun, W.C., Gao, H.J., Jing, X.J.: Disturbance observer-based adaptive tracking control with actuator saturation and its application. IEEE Trans. Autom. Sci. Eng. 13(2), 868–875 (2016) 26. Guo, Z.Q., Xu, J.X., Lee, T.H.: Design and implementation of a new sliding mode controller on an underactuated wheeled inverted pendulum. J. Frank. Inst. 351(4), 2261– 2282 (2014) 27. Cui, R.X., Guo, J., Mao, Z.Y.: Adaptive backstepping control of wheeled inverted pendulums models. Nonlinear Dyn. 79(1), 501–511 (2015) 28. Ou, M.Y., Du, H.B., Li, S.H.: Finite-time formation control of multiple nonholonomic mobile robots. Int. J. Robust Nonlinear Control 24(1), 140–165 (2014) 29. Yang, C.G., Li, Z.J., Li, J.: Trajectory planning and optimized adaptive control for a class of wheeled inverted pendulum vehicle models. IEEE Trans. Cybern. 43(1), 24–36 (2013) 30. Falcone, P., Borrelli, F., Asgari, J., Tseng, H.E., Hrovat, D.: Predictive active steering control for autonomous vehicle systems. IEEE Trans. Control Syst. Technol. 15(3), 566– 580 (2007) 31. Camacho, E.F., Alba, C.B.: Model Predictive Control. Springer, Berlin (2013) 32. Chen, C.P., Liu, Y.J., Wen, G.X.: Fuzzy neural networkbased adaptive control for a class of uncertain nonlinear stochastic systems. IEEE Trans. Cybern. 44(5), 583–593 (2014) 33. Wang, N., Er, M.J., Han, M.: Dynamic tanker steering control using generalized ellipsoidal-basis-function-based fuzzy neural networks. IEEE Trans. Fuzzy Syst. 23(5), 1414–1427 (2015) 34. Sun, C.Y., He, W., Ge, W.L., Chang, C.: Adaptive neural network control of biped robots. IEEE Trans. Syst. Man Cybern. Syst. 47(2), 315–325 (2017) 35. Li, Y.M., Tong, S.C., Liu, Y.J., Li, T.S.: Adaptive fuzzy robust output feedback control of nonlinear systems with unknown dead zones based on a small-gain approach. IEEE Trans. Fuzzy Syst. 22(1), 164–176 (2014) 36. Tong, S.C., Zhang, L.L., Li, Y.M.: Observed-based adaptive fuzzy decentralized tracking control for switched uncertain nonlinear large-scale systems with dead zones. IEEE Trans. Syst. Man Cybern. Syst. 46(1), 37–47 (2016) 37. Liu, Y.J., Gao, Y., Tong, S.C., Li, Y.M.: Fuzzy approximation-based adaptive backstepping optimal control for a class of nonlinear discrete-time systems with deadzone. IEEE Trans. Fuzzy Syst. 24(1), 16–28 (2016) 38. Yang, C.G., Li, Z.J., Cui, R.X., Xu, B.G.: Neural networkbased motion control of an underactuated wheeled inverted pendulum model. IEEE Trans. Neural Netw. Learn. Syst. 25(11), 2004–2016 (2014) 39. Wang, T., Gao, H.J., Qiu, J.B.: A combined adaptive neural network and nonlinear model predictive control for multirate

123

M. Yue et al. networked industrial process control. IEEE Trans. Neural Netw. Learn. Syst. 27(2), 416–425 (2016) 40. Yue, M., Wang, S., Sun, J.Z.: Simultaneous balancing and trajectory tracking control for two-wheeled inverted pendulum vehicles: a composite control approach. Neurocomputing 191, 44–54 (2016) 41. He, W., Ge, S.S.: Vibration control of a flexible beam with output constraint. IEEE Trans. Industr. Electron. 62(8), 5023–5030 (2015)

123

42. Liu, Y.J., Tong, S.C.: Barrier lyapunov functions-based adaptive control for a class of nonlinear pure-feedback systems with full state constraints. Automatica 64, 70–75 (2016) 43. Sun, N., Fang, Y.C., Chen, H.: Adaptive antiswing control for cranes in the presence of rail length constraints and uncertainties. Nonlinear Dyn. 81(81), 41–51 (2015) 44. Li, Y.M., Tong, S.C., Li, T.S.: Hybrid fuzzy adaptive output feedback control design for uncertain mimo nonlinear systems with time-varying delays and input saturation. IEEE Trans. Fuzzy Syst. 24(4), 841–853 (2016)