Trajectory planning and robust tracking control for a class of active articulated tractor-trailer vehicle with on-axle structure

Trajectory planning and robust tracking control for a class of active articulated tractor-trailer vehicle with on-axle structure Recommended by Prof. T Parisini

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Trajectory planning and robust tracking control for a class of active articulated tractor-trailer vehicle with on-axle structure Zhiyuan Liu, Ming Yue, Lie Guo, Yongshun Zhang PII: DOI: Reference:

S0947-3580(19)30317-6 https://doi.org/10.1016/j.ejcon.2019.12.003 EJCON 402

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European Journal of Control

Received date: Revised date: Accepted date:

24 July 2019 3 October 2019 20 December 2019

Please cite this article as: Zhiyuan Liu, Ming Yue, Lie Guo, Yongshun Zhang, Trajectory planning and robust tracking control for a class of active articulated tractor-trailer vehicle with on-axle structure, European Journal of Control (2019), doi: https://doi.org/10.1016/j.ejcon.2019.12.003

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Trajectory planning and robust tracking control for a class of active articulated tractor-trailer vehicle with on-axle structure Zhiyuan Liua , Ming Yuea,b *, Lie Guoa , Yongshun Zhangc a b

School of Automotive Engineering, Dalian University of Technology, Dalian 116024, China

State Key Laboratory of Robotics and System, Harbin Institute of Technology, Harbin 150001, China c

School of Mechanical Engineering, Dalian University of Technology, Dalian 116024, China

Abstract Traditional articulated vehicles exhibit poor steering performance since there exist no direct rotating moment for the underactuated trailers, which cause such vehicles nearly cannot work well in the narrow space. To overcome this shortcoming associated with the traditional articulated vehicles, an active articulated structure is proposed, in which a steering motor is introduced at the articulated joint of tractor and trailer, in addition to utilizing mecanum wheels as trailer wheels. On the basis of this structure, a coordinated control method is designed to ensure the vehicle kinematics constraints and dynamic maneuverability. The coordination control consists of two level controllers. On the level of kinematics, model predictive control (MPC) is adopted as posture controller, which can solve the non-holonomic problem of the whole active articulated tractor-trailer vehicle system; On the dynamic level, a sliding mode control (SMC) is introduced to design a dynamic controller to track the desired velocities generated online, which can increase the robustness of the system. The simulation results show that the proposed active articulated tractor-trailer vehicle system has better maneuverability compared with the traditional one, and the proposed control strategy can ensure the required control effect. Keywords: Active articulated, tractor-trailer vehicle, Trajectory tracking, Model predictive control (MPC), sliding mode control (SMC). 1. Introduction Tractor-trailer vehicles are widely used in real life because of their advantages of cutting down freight transportation costs, large cargo load and decreasing fuel consumption (i.e. reducing the impact on the environment), see, e.g., Refs. [1–7]. Although traditional tractor-trailer vehicles have numerous advantages, they are unsuitable for all road conditions. They are lack of maneuverability and cannot drive on narrow streets, tight corners, and roundabouts [3, 8]. In these areas, the position and direction of the trailer are difficult to control, such that both vehicle flexibility and obstacle avoidance ability are greatly limited. Thus jack-knifing, trailer swinging, and roll-over may also occur when driving at high speed for such kind of tractor-trailer vehicle system [5, 9]. In general, the traditional way of connection between trailer and tractor is usually passive articulation, see, e.g. Refs. [6, 10– 12], but this kind of articulation structure has some disadvantages such as poor maneuverability and difficult to control etc. In view of the shortcomings of traditional articulated vehicle, it is necessary to design a new structure for this kind of vehicle to increase the adaptability under complex unstructured environments. There exist some studies which have already explored the design and improvement of such related structures. ∗ Corresponding

For example, Asghar K., et al. adopted omnidirectional wheel on the trailer [13], while J. Yuan et al. proposed double-steering tractor-trailer structure [14]. In both ways, the same path can be tracked by the tractor and the trailer, which improves the maneuverability of the vehicle, however, at the same time, it is very complicated to design control algorithms for these two structures. To address the above shortcomings, it is necessary to find a more effective design structure. On the other hand, it is worthy to point out that active articulation of snake-like robots in Refs. [15, 16] provided another good idea, but there exist few studies on active articulation applied to tractor-trailer vehicles. Inspired by the design of the snake-like robots, an active articulated tractor-trailer vehicle system is proposed, where a steering motor is introduced at the articulated joint of tractor and trailer. Meanwhile, in order to facilitate the preliminary study, the active articulated vehicle adopts the on-axle structure. Due to the introduction of steering motor, for effective control, trailers need to use wheels that can move laterally. In the industry, mecanum freight vehicles are being applied, which is because the mecanum wheel can move in all directions and cannot need to be rotated to change direction [17]. Mecanum wheel is a wheel with many small rollers obliquely distributed on its rim. These angled peripheral rollers transform part of the wheel steering force into a wheel normal force. Depending on the direction and velocity of each wheel, these forces

author

Preprint submitted to Elsevier

December 31, 2019

are finally combined in any desired direction, thereby ensuring that the platform can move freely in the final resultant direction. Based on the above advantages, mecanum wheels are used for trailer wheels in this study. Furthermore, it is worth mentioning that the proposed active articulation structure is obviously different from the structure designed in references [13] and [14]. The structure proposed in the former reference is that the trailer adopts omnidirectional wheels while the trailer is completely passive. The structure proposed in the latter one is that the connecting rod is connected with the tractor and trailer both in an articulated way, and no steering motor is introduced at the hinge. Moreover, aiming at this active articulated tractor-trailer vehicle system, the kinematics and dynamics models ought to be established at first. There exist much difference between the model of active articulated vehicle and that of passive articulated one. Some scholars, such as Ali K.K., [6] et al. have established the passive articulated vehicle model, in which both the tractor and trailer wheels are subject to non-holonomic constraints. But for this kind of active articulated vehicles, only the tractor wheels are subject to non-holonomic constraints because the trailer wheels, adopting Mecanum wheels, can be moved horizontally and longitudinally. At the same time, the steering motor is introduced to supply the active torgue, thus the system is transformed from under-actuated platform to fullactuated one. It is this feature that has made great changes to the kinematics and dynamics models of the system, which can accurately reflect the above unique movement characteristics. Furthermore, in order to improve the maneuverability of the articulated vehicle, this paper studies the tracking control problem of reducing the sweep-path width during cornering and eliminating the unsafe tail of the trailer. In the active articulated tractor-trailer system, by controlling both the tractor and the trailer to travel along the reference path as much as possible, the sweep-path width can be reduced, meanwhile the steering motor effectively controls the trailer to prevent the tail swing. If the active articulated vehicle provides the possibility to realize the simultaneous control of tractor and trailer, it is necessary to find an appropriate algorithm to achieve this goal. For the control algorithm, many scholars have done some researches based on the passive control of trailer, such as reference [10, 11, 18], but the study of active articulated vehicle, to the best of my knowledge, is rare. On the one hand, the active articulated tractortrailer vehicle system is affected by many control variables or state variables, and there exist non-holonomic constraints, state constraints and mechanism constraints; On the other hand, the active articulated tractor-trailer vehicle has model uncertainties and modeling errors, which affect the control accuracy of the system, and even lead to the system completely out of control. All of the above factors cause the precise tracking control of the mobile platform to be more technically difficult to be implemented. To achieve good tracking control performance of this kind of active articulated vehicle, the control strategy should take into account both the kinematics and dynamics. In addition, because of the kinematics and dynamics involved, the posture and force coordinated tracking control of the articulated tractor-trailer vehicle system will be the optimal

control strategy. For the purpose of achieving this goal, a composite control strategy based on MPC-based posture controller and SMC-based dynamic controller is proposed for active articulated tractor-trailer vehicles. The model predictive control (MPC) has the advantages of easy modeling, good dynamic control effect, good robustness of the control system, and easy implementation on the computer, etc. [19–24]. For the characteristics of the active articulated tractor-trailer vehicle, the posture controller adopts the MPC control strategy, which can avoid the search for the complex Lyapulov candidate, and particularly can solve the non-holonomic constraint problem while ensuring various physical limitations. Due to strongly coupled nonlinearities of the active articulated vehicle, there are uncertain dynamics in the system. How to solve the uncertain dynamics in robot system is also an important issue. Chenguang Yang et al. proposed an adaptive fuzzy control scheme in [25], which is a good strategy. However, it is difficult to design dynamic controller with the above scheme in this study. On the other hand, sliding mode control (SMC) has the advantages of parameter insensitivity, simple design process and strong robustness, etc. [26–30]. It is widely used in industrial control, such as robotics [31], ships [32], spacecrafts [33], and electronic circuits [34]. In view of the uncertainty and non-linearity of active articulated vehicle, SMC control strategy is adopted for dynamic controller. It is worth mentioning that for active articulated vehicle, it is very challenging to realize the effective combination of the two algorithms. In summary, the main contributions of this paper are as follows. 1) A new active articulated vehicle system is proposed, which is characterized by introducing a steering motor at the hinge of the tractor and trailer, and adopting mecanum wheels that can move omni-directionally in the trailer. 2) Based on this special structure of the active articulated vehicle, a method of determining the reference trajectory is proposed for the first time. By this method, the tracking path of the tractor and the trailer can be unified. 3) Based on this novel system, a double closed-loop control structure is designed. The standard MPC method is used in the speed control level, and the SMC method is adopted in the dynamics level. Meanwhile, it is worth mentioning that the dynamics controller based on SMC is designed with skillfully decoupling, and finally the motion control can be realized by directly controlling three torques of the active articulated vehicle. The organization of the paper is as follows. In section 2, the proposed active articulated tractor-trailer vehicle system is briefly described, as well as kinematics and dynamics modeling being established. The section 3 describes how to obtain the reference trajectory for this kind of vehicle. In section 4, the tracking error dynamics is derived, upon which an posture controller based on MPC is proposed. The section 5 presents the dynamic controller, which is constructed by sliding mode control technique. In section 6, the simulation is developed to verify the good maneuverability of the proposed active articulated tractor-trailer vehicle as well as the effectiveness and feasibility of the control method. The section 7, the conclusion, summarizes the main points of the thesis. 2

2. System Description and Modeling

The trailer is hinged at point P with tractor by connecting rod, whose hinge mode is active articulation. Particularly, as shown in Fig.1, a steering motor is fixed on the tractor, and the traction pin that is a special internal gear is coaxially mounted with the motor and the connecting rod gear. The inner teeth of the traction pin are simultaneously meshed with the the external teeth of the motor gear and the connecting rod gear. Therefore, the motor torque is transferred to the connecting rod by the traction pin, and then the connecting rod transfers the torque to the trailer. It’s worth mentioning that the connecting rod and the trailer are welded at point Kc . In addition, for convenience, let P and K be the midpoint of the two wheels of the tractor and the trailer respectively. Similarly, let Pc and Kc be the centroid of tractor body and trailer body respectively. Then, the related nomenclatures of the investigated vehicle system are defined in Tab.1.

2.1. System Description

2 3

4

5

8

6 7

1

9

10

a) Side view of active articulated tractor-trailer vehicle

Table 1: Nomenclature of active articulated tractor-trailer vehicle system parameters

b) Top view of active articulated tractor-trailer vehicle

1 Tractor body 2 Steering motor 3 Steering motor gear 4 traction pin 5 FLange 6 Connecting rod 7 Connecting rod gear 8 Trailer body 9 Mecanum wheel 10 Ordinary wheel

Figure 1: The structure of active articulated tractor-trailer vehicle

Parameters

Parameter definition

a b d l r Iω

distance between points K and Kc half of the distance between parallel wheels distance between points P and K distance between points P and Pc radius of wheels mass moment of inertia of tractor wheels about vertical axis mass moment of inertia of trailer wheels about vertical axis mass moment of inertia of tractor about vertical axis mass moment of inertia of trailer about vertical axis mass of the tractor body mass of the trailer body mass of a tractor wheel mass of a trailer wheel kinetic energy coordinates of point P coordinates of point K orientation of the frame attached to the tractor orientation of the frame attached to the trailer torque exerted on the tractor left wheel by actuator torque exerted on the tractor right wheel by actuator torque of steering motor

Itw Y

J1 tr y

J2 m1 m2 m3 m4 T x1 , y1 x2 , y2 ϕ θ τl

Electric motor

tm Mecanum Wheel

Tractor

Pc φ

P

tl Kc

θ

K Trailer X 0

x

Figure 2: Schematic diagram of active articulated tractor-trailer vehicle system

As shown in Figs.1 and 2, Active articulated tractor-trailer vehicle system proposed in this scenario is composed of a twowheeled differential tractor and a trailer that can move in all directions. Standard wheels is adopted by the tractor and the trailer is equipped with mecanum wheels. In this case, assume that the active articulated tractor-trailer vehicle allows movement in an obstacle-free planar environment, and the tractor wheels have pure rolling in the forward direction with no slip along the lateral direction. Because of this assumption, the tractor wheels are subject to nonholonomic constraints. Moreover, mecanum wheels increase the maneuverability and freedom of the system, but cannot impose constraints on the system.

τr τm

2.2. Kinematic Model Based on the previous description, the configuration of the   whole system is given by q(t) = x1 , y1 , ϕ, θ T , where (x1 , y1 ) is the coordinates of tractor point P, ϕ and θ are the orientation of tractor and trailer respectively. The main feature of the system 3

where M(q) ∈ R4×4 is the inertia matrix of the system, C(q, q) ˙ ∈ R4×4 is the coriolis matrix, N(q) ∈ R4×3 is the control input transformation matrix, and τ ∈ R3×1 is the control input vector. The following parts in system (7), i.e.,M(q), C(q, q), ˙ B(q), and τ can be computed as follows:

is that the tractor-wheel has nonholonomic constraints due to no slip and pure rolling, while the tractor-wheel can roll in any direction without nonholonomic constraints. The lateral speed of the tractor is zero, that is, the nonholonomic constraint can be expressed as x˙1 sin ϕ − y˙ 1 cos ϕ = 0

  0 −m1 l sin ϕ A sin θ   m0   0 m0 m1 l cos ϕ −A cos θ  M(q) =  Iϕ 0  −m1 l sin ϕ m1 l cos ϕ A sin θ −A cos θ 0 Iθ   0 0 −m1 lϕ˙ cos ϕ Aθ˙ cos θ 0 0 −m lϕ˙ sin ϕ Aθ˙ sin θ  1  . C(q, q) ˙ =  0 0  0 0 0 0 0 0     cos ϕ cos ϕ 0   τr  1  sin ϕ sin ϕ 0   , τ =  τl  N(q) =    −b −r r  b τm 0 0 r

(1)

The constraint equation can be expressed in matrix form as    x˙1  h i y˙ 1  G(q)q˙ = sin ϕ − cos ϕ 0 0  ˙  = 0  φ  θ˙

(2)

h i where G(q) = sin ϕ − cos ϕ 0 0 is the introduced system constraint matrix. Consider u = [v, w1 , w2 ]T as a system input vector, where v and w1 are linear velocity and angular velocity at point P of the tractor, respectively, and w2 is angular velocity of the trailer. Thus the kinematics model of the system can be expressed as q(t) ˙ = S (q)u   cos ϕ 0 0  sin ϕ 0 0 . Then it’s easy to get where S (q) =  1 0  0 0 0 1 S T (q)GT (q) = 0

PROPERTY 1. M(q) is a symmetric positive definite matrix. ˙ PROPERTY 2. M(q) − 2C(q, q) ˙ is skew symmetric. It should be emphasized that here τm is an active torque, which makes the dynamic model of active articulated vehicle different from the previous one in [6]. The derivation of Eq.(7) is shown in the appendix. Multiplying both sides of (7) by S T (q), and considering the relationship of S T (q)GT (q) = 0, q(t) ˙ = S (q)u(t) and q(t) ¨ = S˙ (q)u(t) + S (q)˙u(t), it yields that

(3)

(4)

¯ u(t) + C(q, ¯ q)u(t) ¯ M(q)˙ ˙ = N(q)τ

3×1

where 0 ∈ R is the zero matrix. Notice that this kinematic model of active articulated vehicle is different from the previous one in [10].

¯ = S T (q)M(q)S (q) ∈ R3×3 , C(q, ¯ q) Where M(q) ˙ = S T (q)(M(q)S˙ (q) 3×3 T ¯ + C(q, q)S ˙ (q)) ∈ R and N(q) = S (q)N(q) ∈ R3×3 . After be¯ ¯ q), ¯ ing calculated, M(q), C(q, ˙ and N(q) can be derived as

2.3. Dynamic Model

   ¯ M(q) =  

 m0 0 −A sin(ϕ − θ)  0 Iϕ 0  −A sin(ϕ − θ) 0 Iθ    0 −m1 lϕ˙ Aθ˙ cos(ϕ − θ)   ¯ q) m1 lϕ˙ 0 0 C(q, ˙ =  .  −Aϕ˙ cos(ϕ − θ) 0 0   0  1 1 1  ¯ N(q) = b −b −r ,  r 0 0 r

According to the first type of Lagrange equation, the system dynamics can be expressed as: d ∂L ∂L ( )− = fk + GT (q)λ, k = 1, 2, 3, 4 dt ∂q˙ k ∂qk

(5)

where fk is the generalized force, λ denotes the Lagrangian multiplier, k represents the number of generalized coordinates, and L can be obtained by L(q, q) ˙ = T (q, q) ˙ − U(q)

(6)

¯ PROPERTY 3. M(q) is a symmetric positive definite matrix. ˙¯ ¯ q) PROPERTY 4. M(q) − 2C(q, ˙ is skew symmetric. If external disturbances are considered, the dynamic system can be expressed as:

where T (q, q) ˙ is the system kinetic energy, and U(q) is the system potential energy. Since the system is almost composed of a number of rigid components, the elastic potential energy can be ignored; in addition, the vehicle is moving on a horizontal plane, and there has no change of gravity potential energy, i.e., U(q) = 0. According to (5) and (6), the following dynamic model based on Euler-lagrange equation can be established as M(q)q¨ + C(q, q) ˙ q˙ = N(q)τ + GT (q)λ

(8)

¯ u(t) + C(q, ¯ q)u(t) ¯ M(q)˙ ˙ + τd = N(q)τ

(9)

where τd = [τd1 , τd2 , τd3 ]T represents the disturbances acting on the vehicle, caused by inevitable factors, such as air resistance, viscous friction between mechanical components, and uncertain external disturbances, etc.

(7) 4

3. Trajectory planning Trajectory planning is the premise of tracking control for wheeled mobile platform. There are many methods of trajectory planning, for example, artificial potential field method is applied in [35] for trajectory planning, which is a good research direction. However, for the consideration of the preliminary study of active articulated vehicle, the reference trajectory is only obtained based on the given path in this paper. In general, both tractor and trailer ought to track the same path, so that the tractor-trailer vehicle has the smallest sweeppath width, which can improve the maneuverability of narrow turning paths. Then the control objective is to make the tractor point P and the trailer point K follow the same path by controlling the driven torques of left and right wheels installed in the tractor and the torque of steering motor. Define xr = xr (t), yr = yr (t) as the reference coordinate of point P of the tractor, vr and w1r as the reference linear velocity of tractor P-point and the reference angular velocity of tractor respectively, and w2r as the reference angular velocity of the p trailer. The reference linear velocity of the tractor is vr = x˙r2 + y˙ 2r and the tractor reference angular can be obtained by ϕr = arctan 2(˙yr , x˙r )

vr

jr p 2

2R0

y¨ r x˙r − x¨r y˙ r x˙r2 + y˙ 2r

Taking the derivative of both sides of (14) with respect to time gives θ˙r = w1r −

(10)

where w˙ 1r =

(11)

vr w1r

d(w˙ 1r vr − v˙ r w1r ) s dw1r )2 2v2r 1 − ( 2vr

(16)

... ... y r x˙r − x r y˙ r 2( x¨r x˙r + y¨ r y˙ r )(¨yr x˙r − x¨r y˙ r ) − , and v˙ r = v2r v4r

x¨r x˙r + y¨ r y˙ r . vr From coordinates of the tractor point P(xr , yr ), Eqs.(10) and (14), the reference trajectory is obtained. Notice that the reference trajectory, i.e., qr = [xr (t), yr (t), ϕr (t), θr (t)], and the reference control input and derivatives are continuous and uniformly bounded. Meanwhile, it should be emphasized that the reference trajectory planning method here is different from the previous one, which can more accord with the characteristics of active articulated vehicle. 4. Posture Controller As shown in Fig.4, a double closed-loop control structure is proposed, where MPC algorithm is adopted to construct posture controller, and SMC is employed for dynamic controller. Under this frame, vehicle posture tracking and driving torque control can be realized simultaneously.

(13)

4.1. Tracking Error Dynamics In order to achieve the goal of trajectory tracking, it is necessary to construct a tracking error space on the basis of the current trajectory and reference one, and then the tracking error dynamics used to control the system synthesis is derived. At first, define q¯ = q − qr , and then the tracking error vector q" e = [e x , ey , eϕ#, eθ ]T can be defined as qe = T q, ¯ where T = Γ(−ϕr ) 02×2 is the transformation matrix, with Γ(−ϕr ) = 02×2 I2×2

(14)

Meanwhile, it can be concluded that R0 =

K

Figure 3: Schematic diagram of trajectory planning

From (13), it can be deduced that d 2R0

qr

The reference trajectory

where (x2r , y2r ) is the reference coordinate of the trailer point K. Schematic diagram of trajectory planning is shown in Fig.3, where the turning center of tractor and trailer is the point O, R0 represents the turning radius of the tractor. Assuming that d is small compared with R0 , the turning radius of the tractor is approximately equal to the turning radius of the trailer. Draw a circle with point O as the center and R0 as the radius, then the diagram can be obtained. Since ∆APK is a right triangle, it is easy to get

θr = ϕr − arcsin

d

A

The simple geometry of the connection between tractor and trailer can be governed by " # " # " # x2r x cos θr = r −d (12) y2r yr sin θr

d π cos( − (ϕr − θr )) = 2 2R0

P

o

Also, the reference angular velocity of tractor can be formulated as ω1r = ϕ˙ r =

- (jr - qr )

(15) 5

qr

Tractor-trailer vehicle

Dynamic controller _

vc

ev

n -subsystem

n -subsystem

v

w1 -subsystem

w1

q

+ _

w1c

Posture control

ew1

w1 -subsystem

t r ,t l ,t m

+

w2c _

ew2

+

w2 -subsystem

w2 -subsystem

w2

Kinematic Model qe

Figure 4: Schematic diagram of the control system

"

# cos(ϕr ) sinϕr ∈ R2×2 being the rotation matrix, I2×2 ∈ R2×2 − sin ϕr cos ϕr being the identity matrix, and 02×2 ∈ R2×2 being the zero matrix. Notice that q¯ is the tracking error based on the earth-fixed frame, and qe is the tracking error between the actual vehicle and the reference coordinate system described by the bodyfixed frame. Based on Eq.(3), by taking the derivative of qe with respect to time, a tracking error dynamics can be achieved by    e˙ x1 = v cos(eϕ ) − vr + ϕ˙ r ey       e˙ y1 = v sin(eϕ ) − ϕ˙ r e x (17)    e˙ ϕ = ϕ˙ − ϕ˙ r      e˙ θ = θ˙ − θ˙r 4.2. MPC-Based Posture Controller Define the new control signals     u¯ 1  v cos(eϕ ) − vr      u¯ = u¯ 2  =  w1 − w1r      u¯ 3 w2 − w2r

where Ak,t = I + AT , and Bk,t = BT , with T being the sample interval. The state of the MPC controller is defined as " # qe (k|t) ξ(k|t) = (21) u¯ (k − 1|t) Then, based on Eq.(21), a new state-space expression is obtained as    ξ(k + 1|t) = A¯ k,t ξ(k|t) + B¯ k,t ∆U(k|t) (22)   η(k|t) = C¯ k,t ξ(k|t) " # " # h i Ak,t Bk,t ¯ B ¯ where Ak,t = , Bk,t = k,t and C¯ k,t = In 0m×n 0m×n Im Im with n being the dimension of state and m being the dimension of control variables. To further simplify the calculation, the following assumptions are put forward    A¯ k,t = A¯ t,t , k = 1, · · · , t + N − 1 (23)    B¯ k,t = B¯ t,t , k = 1, · · · , t + N − 1

(18)

The output of the future time of the system can be expressed in the form of matrix as

It can be seen that the original system is a highly nonlinear system, which cannot be directly used for the predictive control of linear time-varying model, and it needs to be linearized at first. After processing, the state space model of the system can be expressed as q˙ e = Aqe + B¯u

Y(t) = Ψt ξ(t|t) + Θt ∆U(t) where

(19)

where   0 −ϕ˙ A =  r  0 0

ϕ˙ r 0 0 0

0 vr 0 0

  0 1  0 0  , and B =   0 0 0 0

0 0 1 0

 0  0 . 0 1

Considering that the MPC algorithm is suitable for discrete systems, the first order difference quotient method is used to discretize the system. Then one can ultimately get that qe (k + 1) = Ak,t qe (k) + Bk,t u¯ (k)

(24)

(20)

 ¯ ¯     Ct,t At,t   η(t + 1|t)   C¯ A¯ 2   η(t + 2|t)   t,t t,t     · · ·     · · ·   ,  , Ψt =  Y(t) =  Nc  C¯ t,t A¯ t,t   η(t + Nc |t)          · · ·  ···    ¯ ¯ Np  η(t + N p |t) Ct,t At,t  C¯ B¯ 0 0 0 t,t t,t   C¯ t,t A¯ t,t B¯ t,t C¯ t,t B¯ t,t 0 0  ..  . ··· ··· ···  N −1 N −2 C¯ t,t B¯ t,t Θt = C¯ t,t A¯ t,tc B¯ t,t C¯ t,t A¯ t,tc B¯ t,t · · ·  C¯ A¯ Nc B¯ ¯ ¯ Nc −1 ¯ ¯ ¯ ¯  t,t t,t t,t Ct,t At,t Bt,t · · · Ct,t At,t Bt,t .. .. .. ..  . . . .  N p −1 N p −2 N p −Nc C¯ t,t A¯ t,t

6

B¯ t,t C¯ t,t A¯ t,t

B¯ t,t

· · · C¯ t,t A¯ t,t

B¯ t,t

        ,    

5. Dynamic Controller

   ∆¯u(t|t)   ∆¯u(t + 1|t)  . ∆U(t) =  ···   ∆¯u(t + Nc |t)

The control input formed by the kinematic model is the amount of velocity, and in the real case, the control input of the system is the motor torque input. At the same time, considering that the system is full of complexity and uncertainty, it is necessary to design a dynamic controller. Moreover, the active articulated tractor-trailer vehicle system is highly nonlinear, strongly coupled and time-varying. For the characteristics of this tractor-trailer vehicle system, a dynamic controller is designed by SMC technology. For convenience, some variables are defined in the Tab.2.

The objective function is defined as J(k) =

Np N c −1 X X ||η(k+i|t)−ηre f (k+i|t)||2Q + ||∆U(k+i|t)||2R +ρ 2 (25) i=1

i=1

where N p is the predictive horizon, Nc represents the conrol horizon. ρ denotes the weighted coefficient, and  is the relaxation factor. Here, Q and R are the designed weight matrices, and the trajectory tracking capability and control constraints of the system are affected respectively by them. The constraints of the control variables are as follows: u¯ min (t + k) ≤ u¯ (t + k) ≤ u¯ max (t + k), k = 0, 1, · · · , Nc − 1; (26) ∆¯umin ≤ ∆¯u(t + k) ≤ ∆¯umax (t + k), k = 0, 1, · · · , Nc − 1

Table 2: Definition of Parameters

Parameters β a1

Notice that u¯ (t+k) = u¯ (t+k−1)+∆¯u(t+k), Ut = 1Nc ⊗u(k−1), and   1 0 · · · · · · 0 1 1 0 · · · 0     ..  . 0 ⊗Im , where 1Nc is the column vector, A = 1 1 1   . . .  .. .. . . . . . 0   1 1 ··· 1 1 | {z }

a2 a3 a4 a5 a6 a7 a8 a9

Nc ×Nc

whose number of rows is Nc . Im is the identity matrix with m being the dimension, and ⊗ represents the Kronecker product. Convert the objective function into a standard quadratic form and combine the constraints to solve the optimization problem as follows

∆Umin

1 ∆U(t)T Ht ∆U(t) + GTt ∆U(t) + ρ 2 2 . (27) ≤ ∆Ut ≤ ∆Umax

Parameters 2

m0 Iθ − A sin (ϕ − θ) A2 ϕ˙ sin(ϕ − θ) cos(ϕ −θ)/β m1 lIθ ϕ/β ˙ −Iθ Aθ˙ cos(ϕ − θ)/β Iθ /βr A sin(ϕ − θ)/β −m1 lϕ/I ˙ ϕ b/Iϕ r −1/Iϕ m0 Aϕ˙ cos(ϕ − θ)/β

a10 a11 a12 a13 b1 b2 b3 b4 b5

values

Am1 lϕ˙ sin(ϕ − θ)/β −A2 θ˙ sin(ϕ − θ) cos(ϕ −θ)/β A sin(ϕ − θ)/βr m0 /β −Iθ /β −A sin(ϕ − θ)/β −1/Iϕ −A sin(ϕ − θ)/β −m0 /β

v˙ =a1 v + a2 w1 + a3 w2 + a4 (τr + τl ) + a5 τm + b1 τd1 + b2 τd3

Umin ≤ A∆Ut + Ut ≤ Umax

(30)

2) w1 -subsystem

where Ht = 2(ΘTt QΘt + R) and Gt = 2ΘTt Qet . Here, et = Ψt ξ(t|t) − Yref (t) represents the tracking error in predictive horizon, where Yref = [ηref (t + 1|t), · · · , ηref (t + N p |t)]T . By solving the above optimization problems, a series of control inputs can be obtained as ∆Ut∗ = [∆¯u∗t ∆¯u∗t+1 · · · ∆¯u∗t+Nc −1 ]T , and then, the first element of the control sequence can be set as the actual control input increment of the active articulated tractor-trailer vehicle system, namely. u¯ (t) = u¯ (t − 1) + ∆¯u∗t

2

At first, (9) can be decomposed into three equations, then the dynamic system (9) can be divided into the following three subsystems: 1) v-subsystem

J(ξ(t), u¯ (t − 1), ∆U(t)) = s.t.

values

w˙ 1 = a6 v + a7 (τr − τl ) + a8 τm + b3 τd2

(31)

3) w2 -subsystem w˙ 2 =a9 v + a10 w1 + a11 w2 + a12 (τr + τl ) + a13 τm + b4 τd1 + b5 τd3

(32)

Through the sliding surface design of three subsystems, a following theorem can be concluded as follows.

(28)

Finally, uc = [vc , w1c , w2c ]T is defined as the desired velocity vector which is generated in realtime by MPC-based posture controller. From (22), it can be obtained that      u¯ 1 + vr    vc       uc = w1c  =  cos eϕ  (29)   u¯ 2 + w1r    w2c u¯ 3 + w2r

Theorem 1. The control law of dynamic system (9) can be expressed as    τr    ¯ uc (t) + M ¯ −1 τd + M ¯ −1Cu(t) ¯ τ =  τl  =N¯ −1 M[˙   (33) τ m

which reflects the movement characteristics of active articulated vehicle at all times.

+ φ(uc (t) − u(t)) + σ tanh(s) + K s]

7

where φ = diag[φ1 , φ2 , φ3 ], σ = diag[σ1 , σ2 , σ3 ], K = diag[k1 , k2 , k3 ], For the w2 -subsystem, one can obtain and s = [s1 , s2 , s3 ]T . φ, σ and K are vectors composed of posiZ t tive design parameters, while s represents the sliding mode surs3 = ε3 (t) + φ3 ε3 (µ) dµ face of the three subsystems. 0

(43)

At the same time, it follows Proof 1. The velocity tracking error signal is defined as     ε1   vc − v      ε(t) = ε2  = w1c − w1      w2c − w2 ε3

s˙3 = −σ3 tanh(s3 ) − k3 s3 The w2 -subsystem can be rewritten as

(34)

w˙ 2c − a9 v − a10 w1 − a11 w2 − a12 (τr + τl ) − a13 τm

For the v-subsystem, the PI-type sliding mode surface is selected as Z t (35) s1 = ε1 (t) + φ1 ε1 (µ) dµ

−b4 τd1 − b5 τd3 + φ3 (w2c − w2 ) + σ3 tanh(s3 ) + k3 s3 = 0

where φ1 is a positive design parameter. Differentiating s1 with respect to time results in

6. Simulation Analysis

s˙1 =ε(t) ˙ + φ1 ε1 (t) =˙vc − a1 v − a2 w1 − a3 w2 − a4 (τr + τl ) − a5 τm

In order to compare this kind of active articulated vehicle with the passive one and verify the proposed control strategy, numerical simulation was carried out in the Matlab/Simulink environment. The simulation parameters of the active articulated tractor-trailer system are shown in Tab.3. Considering the variable curvature and direction of lotus-shaped trajectory, selecting it as reference trajectory can test the maneuverability of active articulated vehicle under complex circumstances very well. Thus, the reference trajectory is selected as follows.     xr = 0.15(10 + cos(16t/50)) cos(4t/50) (46)   yr = 0.15(10 + cos(16t/50)) sin(4t/50)

(36)

− b1 τd1 − b2 τd3 + φ1 (vc − v)

In this case, the sliding mode controller is designed by exponential approach law, then it holds that s˙1 = −σ1 tanh(s1 ) − k1 s1 , where tanh( ) is hyperbolic tangent function. Then the v-subsystem can be expressed as v˙ c − a1 v − a2 w1 − a3 w2 − a4 (τr + τl ) − a5 τm − b1 τd1

−b2 τd3 + φ1 (vc − v) + σ1 tanh(s1 ) + k1 s1 = 0

(37)

The stability of the system is proved by establishing a suitable candidate of positive lyapunov as 1 V = s21 2

Table 3: Parameters of the active articulated tractor-trailer vehicle

(38)

Parameters Nominal values Parameters Nominal values a b d l m1 m2 m3 m4 Iω

Differentiating V with respect to time yields V˙ = −|s1 σ1 tanh(s1 )| − k1 s21

(39)

Obviously, for V˙ ≤ 0 , it can be concluded that all signals on the dynamic level are bounded, and the tracking error will converge to zero as time goes by, according to Lyapunov stability theorem and LaSalle invariance principle. Similarly, for the w1 -subsystem, it follows Z t (40) s2 = ε2 (t) + φ2 ε2 (µ) dµ Meanwhile, it holds that (41)

The w1 -subsystem can be rewritten as w˙ 1c − a6 v − a7 (τr − τl ) − a8 τm − b3 τd2

+φ2 (w1c − w1 ) + σ2 tanh(s2 ) + k2 s2 = 0

0m 0.06 m 0.4 m 0.03 m 1.5 kg 0.75 kg 0.05 kg 0.07 kg 0.00006 kg· m2

Itω J1 J2 r Iϕ Iθ m0 A

0.00006 kg· m2 0.0049 kg· m2 0.0009 kg· m2 0.05 m 0.00673 kg· m2 0.143924 kg· m2 2.49 kg 0.356 kg· m

Furthermore, the initial state of tracking error dynamics is set as qe = [e x (0), ey (0), eϕ (0), eθ (0)]T = [−0.4, −0.2, π/5, π/20]T . The basic parameters of MPC-based controller are set as: the predictive horizon is N p = 15, the control horizon is Nc = 5, the simulation time is 80s and the sample interval is T s = 0.1s. By trial and error of the required performance, other relevant design parameters are selected as follows: Q = 100 × I60 , R = 1 × I15 , ρ = 10, φ1 = φ2 = φ3 = 15, σ1 = σ2 = σ3 = 10, and k1 = k2 = k3 = 1. In addition, the constraints of other

0

s˙2 = −σ2 tanh(s2 ) − k2 s2

(45)

The stability proof of w1 -subsystem and w2 -subsystem is similar to that of v-subsystem, which is omitted and not described here. From (9), (37), (42) and (45), (33) can be obtained, indicating that the theorem is proved

0

=˙vc − v˙ + φ1 (vc − v)

(44)

(42) 8

0.2

2

y1 , y2 , yr

1.5

e1x1 , e2 x1

x1 , y1 x2 , y2 xr , yr

0.01

-0.2

1 0.5 0

-0.01

-0.6

-0.02 30

-0.5 0.3 e1 y1 , e2 y1

-1 -1.5 -1.5

-1

-0.5

0

0.5

1

1.5

2

0.2 0.1 0

x1 , x2 , xr

0

-0.4

0

-2 -2

e1x e2 x11

0

10

35

40

45

50

20 30 40 50 60 70 Abscissa direction tracking error of tractor 0.02

80

e1 y e2 y11

0 -0.02 30

35

40

45

50

-0.1 -0.2 0

Figure 5: Time response of trajectory tracking for the passive articulated tractor-trailer vehicle.

10 20 30 40 50 60 70 80 Longitudinal coordinate direction tracking error of tractor t (s)

2

x1 , y1 x2 , y2 xr , yr

1.5

Figure 7: Time response of posture tracking error (a).

y1 , y2 , yr

1

ulated tractor-trailer vehicle system, the trajectory tracking of passive articulated tractor-trailer vehicle is simulated. The time response of trajectory tracking of the passive articulated tractortrailer vehicle and the active one are shown in Figs.5 and 6. The red solid line in the figure represents the trajectory of the tractor; the black dotted line represents the trajectory of the trailer; the blue dot dash line represents the reference trajectory. By comparing the two figures, it can be found that the actual path of the tractor and the trailer of the passive articulated vehicle always has a deviation, while the actual path of the tractor and the trailer of the active one can be consistent. In the actual road condition, it can be explained that the sweep-path width of the passive articulated vehicle is obviously larger than that of the active one, revealing that its maneuverability is not as good as that of the active articulated vehicle. The simulation results of the whole trajectory tracking process in Fig.6 show that the active articulated tractor-trailer vehicle system can track the reference trajectory well in a short enough time with the initial error. Both point P of the tractor and point K of the trailer can travel along the given path more accurately. In the case of external disturbance, the system can still show good tracking performance. In the process of trajectory tracking, the turning direction of the vehicle has been changed many times, while the trailer of the active articulated vehicle can still track the reference trajectory well, which indicates that the introduction of the steering motor can avoid the occurrence of tail swing. The time response of posture tracking error are shown in Figs.7, 8 and 9, in which e1( ) represents the tracking error of active articulated vehicle and e2( ) means the tracking error of the passive one. It can be seen that the tracking error of both active and passive articulated vehicles can quickly approach zero after a short period of adjustment. when t = 30s and t = 40s,

0.5 0

-0.5 -1 -1.5 -2 -2

-1.5

-1

-0.5

0

0.5

1

1.5

2

x1 , x2 , xr Figure 6: Time response of trajectory tracking for the active articulated tractortrailer vehicle.

control variables can be expressed as [−1 − 1.2 − 1.2]T ≤ u¯ ≤ [1 1.2 1.2]T , [−0.5 − 1 − 1]T ≤ ∆¯u ≤ [0.5 1 1]T . In order to further verify the robustness of the controller against disturbance, disturbance is introduced into the articulated vehicle system. For making the applied disturbance closer to the actual situation, the form of the disturbance takes into account the three cases of mutation, gradual change and randomness, which can be represented by the following formula.     0.1+0.2 sin(0.05t)+0.02rand(1, 1)           0.05+0.1 sin(0.05t)+0.01rand(1, 1) , 30s ≤ t ≤ 40s   τd1          τ = 0.05+0.1 sin(0.05t)+0.01rand(1, 1)  d2       τd3       0 ∈ R3×1 , otherwise

(47)

To demonstrate the advantages of the proposed active artic9

0.4

e1x2 e2 x2

0 0.01

-0.2

0

-0.4

-0.01

-0.6

-0.02 30

0

10

20

30

0

-0.2 40

40

-0.4

50

50

60

70

-0.6 0

80

Abscissa direction tracking error of trailer

-0.2 -0.4 0

10

20 30 40 50 60 70 Orientation angular tracking error of tractor

0.4

e1 y2 e2 y2 0.05 0

0.2

0

0.1

-0.05 30

80

e1q e2q

0.05

0.3 e1q , e2q

e1 y2 , e2 y2

0.2

0

e1j e2j

0.2

e1j , e2j

e1x2 , e2 x2

0.2

40

50

0

-0.05 30

40

-0.1

50

-0.2 0

10 20 30 40 50 60 70 80 Longitudinal coordinate direction tracking error of trailer t (s)

10 20 30 40 50 60 Orientation angular tracking error of trailer t (s)

70

Figure 8: Time response of posture tracking error (b).

Figure 9: Time response of posture tracking error (c).

a sudden change of disturbance occurs to the articulated vehicle system. The figures shows that the active articulated vehicle has stronger disturbance rejection ability than the passive one. In order to more clearly compare the accuracy of the tracking error between the active and passive articulated vehicle, a tracking error evaluation function is introduced, i.e.

Table 4: Posture tracking error evaluation index

   e f 1 = g1 |e1x1 | + g2 |e1y1 | + g3 |e1ϕ |       er1 = g4 |e1x2 | + g5 |e1y2 | + g6 |e1θ |    e f 2 = g1 |e2x1 | + g2 |e2y1 | + g3 |e2ϕ |      er2 = g4 |e2x | + g5 |e2y | + g6 |e2θ | 2 2

(48)

where e f 1 and er1 are the evaluation indexes of tractor and trailer posture tracking error of active articulated vehicles respectively, e f 2 and er2 represent the evaluation indexes of tractor and trailer posture tracking error of passive articulated vehicles respectively, gi (i = 1, 2, 3, 4, 5, 6) is weight coefficient, and |( )| represents the absolute value of the area between the tracking error curve i.e., ( ) and the coordinate axis. Let g1 = g2 = g4 = g5 = 100, and g3 = g6 = 10. By trail and error, the parameters of (48) can be obtained as shown in the Tab.4. From the Tab.4, compared with passive articulated vehicle, the accuracy of tractor posture tracking error of active one is improved by 50.4%, and that of trailer posture tracking error is improved by 38.1%. The time response of velocity tracking error of active and passive articulated tractor-trailer vehicle is shown in Fig.10, in which e1( ) represents the velocity tracking error of active articulated vehicle and e2( ) means the velocity tracking error of the passive one. The figure shows that the velocity tracking performance of both vehicles is good, but in the face of disturbance, the chattering of active articulated vehicle is smaller, indicating

|e1x1 | 0.1838

ef1 |e1y1 | 0.4659 67.585

|e1ϕ | 0.2615

|e1x2 | 0.1020

er1 |e1y2 | 0.5126 61.561

|e2x1 | 0.5356

ef2 |e2y1 | |e2ϕ | 0.7868 0.4062 136.302

|e2x2 | 0.6234

er2 |e2y2 | 0.3310 99.468

80

|e1θ | 0.0101

|e2θ | 0.4028

that its anti-disturbance ability is better. Similarly, to compare the accuracy of the tracking error between the active and passive articulated vehicle more clearly, the velocity tracking error evaluation function is introduced, as shown in (49).    em = g7 |e1v | + g8 |e1w1 | + g9 |e1w2 | (49)   en = g7 |e2v | + g8 |e2w1 | + g9 |e2w2 |

where em and en are the velocity tracking error evaluation indexes of active articulated vehicle and passive one respectively, gi (i = 7, 8, 9) is weight coefficient, and |( )| represents the absolute value of the area between the velocity tracking error curve and the coordinate axis. Let g7 = g8 = g9 = 100. By trail and error, the parameters of (49) can be obtained as shown in the Tab.5. From the results, compared with passive articulated vehicle, the accuracy of tractor velocity tracking error of active one is improved by 7.6%. It shows that active articulated vehicle has better trajectory tracking performance, which explains why its posture tracking error accuracy is higher than that of passive

10

0.8

e1v e2v

0 0.02

-0.5

0.6

0

-1

-0.02

-1.5 0

30

10

20

30

40

0.4

50

40

50

60

80

70

Linear velocity tracking error of tractor

0.2

e1w e2w11

e1w1 , e2 w1

1.5 1 0.5 0 -0.5 -1 -1.5 0

0

-0.2

10

20 30 40 50 60 Angular velocity tracking error of tractor

70

0.5 e1w2 , e2 w2

j -q jr - qr

j - q ,jr - qr

e1v , e2 v

0.5

-0.4 0

80

e1w2 e2w2

0

10

20

30

40 t (s)

50

60

70

80

Figure 11: Time response of velocity tracking error.

0.04 0

-0.5

ior, but it tends to be stable soon. When t = 20s, the input torque oscillates violently, which is because of the characteristics of formula (10). Despite the oscillation, it tends to be stable in a very short time. When t = 30s and t = 40s, the system disturbances suddenly appear and disappear respectively, while the input torque changes slightly and remains stable from 20s to 30s. In general, the continuity of control signals ensures the feasibility of the proposed controller.

-0.04 -0.08 30

-1 0

10

40

50

20 30 40 50 60 Angular velocity tracking error of trailer

70

80

t (s)

Figure 10: Time response of velocity tracking error.

one.

4

Table 5: Velocity tracking error evaluation index

|e1w2 | 0.1939

|e2v | 0.8176

en |e2w1 | 0.5087 152.5

3

0.4 0.2

|e2w2 | 0.1987

2

t r ,t l ,t m

|e1v | 0.4935

em |e1w1 | 0.7219 140.93

tr tl tm

0 -0.2

1 -0.4

In addition, the difference of the orientation between the tractor and the trailer of the active articulated vehicle, known as the hitch angle (i.e., ϕ − θ), is an important variable to directly reflect the lateral stability and vehicle maneuverability, which is also drawn in the Fig.11. The blue dotted line in the figure is the desired reference hitch angle during the system motion, while the red solid line represents the actual hitch angle. As can be seen from the figure, starting from 0s, after a short adjustment, the actual hitch angle will approach the desired reference hitch angle, indicating the good following ability and dynamic performance of the trailer. At t = 30s and t = 40s, due to the change of external disturbance, there exists a mutation in the hanging angle, but the change value ranges from 0 to 0.2rad, which is within an acceptable range, and the actual hanging angle will track the expected value within 2s. Finally, the control inputs of the active articulated vehicle are shown in the Fig.12. As shown in the figure, at the beginning, the control input has a short and drastic regulation behav-

30

35

40

45

30

40 t (s)

50

60

0

-1

-2

0

10

20

70

80

Figure 12: Time response of velocity tracking error.

7. Conclusion Aiming at the disadvantage of traditional tractor-trailer vehicle facing narrow turning and other complex paths, this paper presents an active articulated tractor-trailer vehicle system. First, the reference trajectory is planned based on the given path 11

curve, whose design method is different from that of the passive articulated vehicle. Then, based on the derived kinematics and dynamics model, the double closed-loop control structure is designed, and the MPC control algorithm is used to construct the posture controller, and the SMC strategy is used for the dynamic controller. Finally, motion simulation is carried out in matlab/simulink environment. The results show that under the designed control strategy, even with external disturbance, the system can track the trajectory well, that is, both the tractor and the trailer can travel along the same path. Compared with the traditional passive articulated vehicle, the active one has a smaller sweep-path width when turning, and the tail swing of the trailer can be avoided due to the introduction of steering motor. Thus it can be considered that the proposed active articulated tractor-trailer vehicle system has good maneuverability, which is more suitable for transportation in unstructured road environment. It can be predicted that active articulated vehicles have more potential applications. The future work will consider the implement of experiment and obstacle avoidance research.

(52) 4) The kinetic energy of the trailer wheels is 1 d T 4 = m4 [( (x1 − d cos θ + b sin θ))2 2 dt d 1 + ( (y1 − d sin θ − b cos θ))2 ] + Itω θ˙ 2 dt 2 d 1 + m4 [( (x1 − d cos θ − b sin θ))2 2 dt d 1 + ( (y1 − d sin θ + b cos θ))2 ] + Itω θ˙ 2 dt 2 2 2 2 ˙2 2 ˙2 =m4 ( x˙1 + y˙ 1 + d θ + b θ ) + 2m4 x˙1 dθ˙ sin θ − 2m4 y˙ 1 dθ˙ cos θ + Itw θ˙ 2

Therefore, the entire kinetic energy of the system can be described as T =T 1 + T 2 + T 3 + T 4 1 1 1 (54) = m0 ( x˙12 + y˙ 21 ) + Iϕ ϕ˙ 2 + Iθ θ˙ 2 2 2 2 ˙ x˙1 sin θ − y˙ 1 cos θ) + m1 lϕ(˙ + Aθ( ˙ y1 cos ϕ − x˙1 sin ϕ)

Appendix The Euler-Lagrangian dynamic model of articulated tractortrailer vehicle is derived by using Lagrangian equation (6). The kinetic energy of the whole vehicle system can be divided into four parts, namely, the kinetic energy of the tractor body and the trailer body, and the kinetic energy of the tractor wheel and the trailer wheel. The derivation of Eq.(7) is as follows 1) The kinetic energy of the tractor body is

where m0 = m1 + m2 + 2m3 + 2m4 , Iϕ = m1 l2 + J1 + 2m3 b2 + 2Iw , Iθ = m2 (d − a)2 + J2 + 2m4 d2 + 2m4 b2 + 2Itw , and A = m2 (d − a) + 2m4 d. The potential energy changes to zero as the vehicle moves on a horizontal plane, so it holds that T = L. From Lagrange’s equation, we get d ∂L ∂L ( )− dt ∂ x˙1 ∂x1 = m0 x¨1 − m1 lϕ¨ sin ϕ + Aθ¨ sin θ + Aθ˙ cos θθ˙ − m1 lϕ˙ cos ϕϕ˙

d d 1 1 T 1 = m1 [( (x1 + l cos ϕ))2 + ( (y1 + l sin ϕ))2 ] + J1 ϕ˙ 2 2 dt dt 2 1 2 2 2 2 = m1 ( x˙1 + y˙ 1 + l ϕ˙ ) + m1 lϕ(˙ ˙ y1 cos ϕ − x˙1 sin ϕ) 2 1 + J1 ϕ˙ 2 2 (50)

=

τl τr cos ϕ + cos ϕ + λ1 sin ϕ r r

(55)

∂L d ∂L ( )− dt ∂˙y1 ∂y1 = m0 y¨ 1 + m1 lϕ¨ cos ϕ − Aθ¨ cos θ + Aθ˙ sin θθ˙ − m1 lϕ˙ sin ϕϕ˙

2) The kinetic energy of the trailer body is 1 d T 2 = m2 [( (x1 − (d − a) cos θ))2 2 dt d 1 + ( (y1 − (d − a) sin θ))2 ] + J2 θ˙ 2 dt 2 1 2 2 2 ˙2 = m2 ( x˙1 + y˙ 1 + (d − a) θ ) 2

(53)

=

τr τl sin ϕ + sin ϕ − λ1 cos ϕ r r

(51) d ∂L ∂L ( )− dt ∂ϕ˙ ∂ϕ

1 + m2 x˙1 (d − a)θ˙ sin θ − m2 y˙ 1 (d − a)θ˙ cos θ + J2 θ˙ 2 2

τr τl = Iϕ ϕ¨ + m1 l¨y1 cos ϕ − m1 l x¨1 sin ϕ = b − b − τm r r

3) The kinetic energy of the tractor wheels is 1 d d 1 T 3 = m3 [( (x1 + b sin ϕ))2 + ( (y1 − b cos ϕ))2 ] + Iω ϕ˙ 2 2 dt dt 2 d d 1 1 + m3 [( (x1 − b sin ϕ))2 +( (y1 + b cos ϕ))2 ]+ Iω ϕ˙ 2 2 dt dt 2 =m3 ( x˙12 + y˙ 21 ) + m3 b2 ϕ˙ 2 + Iω ϕ˙ 2

d ∂L ∂L ( )− = Iθ θ¨ + A x¨1 sin θ − A¨y1 cos θ = τm dt ∂θ˙ ∂θ

(56)

(57)

(58)

In summary, formula (7) can be deduced from formula (53), (54), (55) and (56). 12

8. Acknowledgment

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Conflict of Interest The authors declared that they have no conflicts of interest to this work. We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted.

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