Path-following control strategy for gantry virtual track train based on distributed virtual driving model

In this paper, a chained architecture of virtual track train (VTT), gantry virtual track train (G-VTT), was proposed. The distributed virtual driving model (DVD model) for G-VTT was proposed to realise the automatic tracking of paths with arbitrary ommon curves. The core of the DVD model is the cascade modular wheel steering angle (WSA) control algorithm based on the optimal lateral acceleration (OLA) of tracking points. A local tracking objective function based on 2-norm was established to find the OLA in the preview window, thereby the optimal local trajectory was realised. The cascade structure of the algorithm gives the DVD model the advantages of modularity and high scalability, facilitating the cascade control of an n-unit G-VTT. The proposed control strategy was verified by co-simulation of G-VTT dynamics model and the DVD model. Results show that the DVD model can effectively improve the path-following ability and small-curve-passing capacity of the G-VTT with remarkable reduction of the lateral deviation (up to about 99.16%) and turning passageway width (about 59.71%); Besides, the yaw and anti-jackknifing stability of the train are also significantly improved, and the DVD model is well adaptable to speed, curve radius, and payload.


Introduction
As a new mode of urban rail transit, the virtual track train (VTT) is designed to integrate the advantages of trams (such as medium capacity, high punctuality, etc.) and buses (such as low cost, high flexibility, etc.) to improve the overall capacity of urban roads and the intelligence level of urban public transport.The basic feature of the VTT is that it replaces the traditional physical track with the virtual digital track.Path-following control technology is one of the key scientific problems of VTT.The aim is to realise an accurate following of each car body module relative to the pre-designed path through the non-contact active multi-wheel/all-wheel steering control technology.Different from the steering control of conventional cars, the motion constraint problem caused by the articulation of multi-car bodies and the cooperative control problem of multi-axle steering needs to be considered in the path-following control of VTT, which is rather complex.According to the published The ART train usually adopts the three-car marshaling of MC1 (the front motor car with a cab) + T (the trailer) + MC2 (the rear motor car with a cab) [Figure 1(d)].Each car is equipped with two steerable single-axle running gears with rubber wheels, and the adjacent car bodies are connected by the hinge mechanism.The DRT train adopts the three-unit marshaling [Figure 1(e)] with steerable double-axle bogies with rubber wheels, and the adjacent car bodies are connected by articulated bogies.The SRT train contains two tailto-tail connected three-axle BRT units, and all the six running gears are steerable singe-axle suspensions [Figure 1(f)].
There are already some studies on all-wheel steering control of VTTs and articulated vehicles.Xiwen et al. [1] introduced the composition of the perception subsystem and the control subsystem of the ART automatic tracking control system, and proposed two path-following control algorithms suitable for the head car: (i) the preview PID (proportion-integration-differentiation) algorithm and (ii) the MPC (model predictive control) algorithm.Jing et al. [2] designed a path-following control algorithm based on the vehicle motion geometry for the autonomous tracking of the second-sixth axle of the three-unit ART train, and carried out real vehicle test verification.The results showed that the maximum lateral deviation was about 0.46 m when the train passed a 20 m-radius circular curve at a speed of 10 km/h.L. Xiao et al. [3,4] proposed an automatic steering control method for an articulated all wheel steered vehicle based on model predictive control with visual navigation technology.The simulation result shows that the designed controller can track the desired trajectory quickly and smoothly.Huang et al. [5] present the design and development of all-wheel steered multiple-articulated rubber-tyre transit and proposed the automatic steering controller to control the first axle, which is used for lane keeping, and the controller can track the reference line smoothly.Yuan et al. [6] introduced an image recognition based automatic steering control principle for all-wheel steered multiple-articulated rubber-tyre transit, and the path-following controller was verified by simulation and real vehicle tests, which exhibited good precise docking and path-following ability.Feng et al. [7] designed a six-axle coordinated automatic all-wheel steering control system with two subsystems of path-tracking controller and path-following controller for all-wheel steering multi-set rubber-tyre transit, and the two controllers were verified by simulation and real vehicle tests.Zhang et al. [8] proposed a novel tracking control method for distributeddrive and active-steering articulated virtual rail train.A new target trajectory generation method was designed to obtain the trajectory information.Zhang et al. [9] designed the feedforward and feedback path-following control method to improve the dynamic performance and riding quality of rubber-tyre trackless train with bogies.Kaneko et al. [10] proposed a steering control strategy based on feedforward and feedback to solve the problem of deviation from the track of the articulated bus.The effectiveness of the control system was verified by vehicle dynamics model simulation and the scale model test, and the influence of the hinge force between adjacent cars was discussed.Nakamura et al. [11] proposed a three-point model of passive steering mechanism for wheeled mobile robots with trailers and found, through theoretical analysis and numerical simulation, that the optimal three-point model with a steering gain of 0.5 is superior to other typical trailer control systems in a passive path-following performance and forward and backward motion stability.Morin et al. [12] proposed a control strategy based on the transverse function method, which was suitable for the unicycle vehicle system with any number of trailers, and studied the stabilisation problem of a class of nonlinear drift-free systems.Jujnovich et al. [13] designed a nonlinear low-speed controller, and a high-speed controller considering trailer sideslip characteristics for articulated heavy vehicles.In the middle-speed range, the speed correlation gain was applied to integrate the low-speed controller with the highspeed controller to realise an accurate path tracking of the trailer in all paths and all normal speeds.Yuan et al. [14] investigated the circular trajectory tracking control problem of a multi-wheel steering tractor-trailer mobile robot, and proposed a control strategy with the nonlinear state feedback as the control input to make the system exponentially stable.Bolzern et al. [15] investigated the path-following problem of an autonomous vehicle with the off-axle hitching consisting of a tractor and n−1 trailers.They established the input-output feedback linearisation model of a vehicle's forward and backward motion, obtained the influence of off-axle distance on the stability of the linearised system by zero dynamic analysis, and derived the multi-wheel steering control law through the closedloop pole assignment of the linearised model.Wagner et al. [16] proposed an all-wheel steering control scheme for the two DOF multi-articulated vehicle based on feedforwardfeedback control based on vehicle inverse model and the tyre inverse model.Bolzern et al. [17] proposed a path-following controller design method based on accurate linearisation, where the original vehicle model was associated with an auxiliary vehicle, which could be accurately linearised.Kim et al. [18,19] conducted research on the automatic steering control of the single articulated all-wheel steering vehicle developed by the Korea Railway Research Institute.A second-order decoupling compensator was designed for the strongly coupled two-input and two-output system and a first-order controller was designed for each decoupling feedback loop using the characteristic ratio distribution method.Lee et al. [20] developed a hardware-in-loop simulation platform for testing and developing the allwheel steering electronic control unit (AWS-ECU) of the two-unit three-axle tram, which verified the effectiveness of the AWS-ECU control algorithm and the vehicle handling.de Bruin et al. [21] deduced a lateral performance model for the all-wheel steering and double-articulated bus with each wheel independently driven and designed a H ∞ robust controller to realise path following of the vehicle under uncertain road conditions.Oskar Ljungqvist et al. [22] proposed a MPC path-following controller is to realise the automatic low-speed steering control of a multi-steered articulated vehicle consisting of a tractor and any number of trailers with passive or active steering.Abhishek Singh Tomar et al. [23] proposed a virtual rigid axle command steering control strategy for dolly steering.The low-and high-speed performance of A-double and LHV-D could be significantly improved compared with the non-steered vehicle.Abbas Ajorkar et al. [24] designed a MPC controller for autonomous vehicle tracking control based on the multi-objective metaheuristic optimisation method, the weighting matrices in the controller is adjusted to minimise the path-following error and RMS value of vehicle lateral acceleration.
This paper proposed a gantry virtual track train (G-VTT) architecture which is similar to that of the on-axle articulated vehicle.However, considering the bi-direction running requirement of urban transit vehicles, the matching design of all-wheel steering and bidirection running is realised by setting a gantry articulation structure between car bodies, and adjusting the locking mechanism between the gantry articulation structure and car body according to the vehicle running direction.Compared with previous steering control methods, the proposed control algorithm has the advantages of modularisation and high scalability, and the chained structure of the algorithm with different number of links could be designed according to different train marshaling.

Vehicle architecture
The schematic architecture of the G-VTT is shown in Figure 2. The whole vehicle includes the end car body module (ECM), the intermediate car body module (ICM), the end suspension module (ESM), the intermediate suspension module (ISM), and the gantry articulation module (GAM).The ECM is equipped with a cab, and the adjacent car bodies are connected by the standard GAMs.The GAM is mainly composed of the gantry steel frame, hinge, and related components.As the gangway between car bodies, the GAM can effectively reduce the floor height of the car body, improve the curving performance, and facilitate modular marshaling.A common gantry articulation structure with rubber tyres is known as Translohr tram gantry connecting structure [See Figure A1 in Appendix A2].
Within the vehicle, car bodies are connected with the gantry steel frame in GAMs by the upper hinge (free hinge and elastic hinge) and the lower hinge (fixed hinge), forming the train marshaling type of n modules and n + 1 axles.The fixed hinge is used to release the yaw-DOF between the car body and the GAM, and transfer the lateral, longitudinal, and vertical forces.The free hinge acts to release the pitch-DOF to adapt to the vertical curving performance of the train.In addition, two of the four locking mechanisms on a GAM act to constrain the six DOFs between the GAM and the front car body in its moving direction; when the G-VTT runs in the opposite direction, the other two locking mechanisms work.Therefore, the GAM is not only the gangway between the car bodies but also the key component to realise the matching between the switching of the vehicle running direction and the path-following control system.All the suspensions are actively steerable suspensions.The two ESMs are powered suspensions, equipped with the drive motor and mechanical differential, and the other ISMs are non-powered.
The path-following control strategy of the G-VTT is shown in Figure 2. The geometric centre of each axle is taken as the reference point for the path-following control ('tracking point' for short), and the all-wheel steering control scheme is adopted to make the trajectory of each tracking point be roughly the same as the virtual track.The steering control of the front axle of the head car can be designed to be autonomous driving mode or manual driving mode.According to the position and state information of the head car and the reference path information, the optimal WSA of the front axle within the head car could be obtained through the path-following control or driver's handling.The active steering of the other axles is realised through path-following control according to the state of the front car body, the front WSA, and the reference path information.

Kinematics model
The kinematics diagram of the n-unit G-VTT in the global coordinate system XOY is shown in Figure 3.A nomenclature for this paper is given in Appendix A.1 In Figure 3, (x i , y i ) denotes the coordinates of tracking points, v i velocity of the tracking points, ψ i the heading angle of tracking points (the angle between the moving direction of the tracking point and the X-axis), ϕ i the car body yaw angle (the angle between the longitudinal axis of the car body and X-axis), γ i the folding angle between the car bodies (the difference of the yaw angles of adjacent car bodies, γ i = ϕ i+1 − ϕ i ), δ i the WSA, CG i the centre of gravity of the car body, and L the wheelbase.In Figure 3, the velocity of the tracking point can be expressed as: The lateral velocity constraint of the tracking point is as follows: The following can be obtained by simultaneous Eqs. ( 1) and (2): The longitudinal velocity constraint of the car body is as follows: The yaw constraint of the car body is as follows: According to Eq. ( 5): Assuming that the longitudinal distances between the CG i of the car body and the front & rear tracking point are L f and L r , respectively, there is: The following can be obtained by simultaneous Eqs. ( 4) and ( 7): By simultaneous Eqs. ( 3), ( 4), (6), and ( 8), the yaw angle ϕ i , the heading angle ψ i of the front tracking point, the heading angle ψ i+1 of the rear tracking point, and the velocity v i+1 of the rear tracking point can be calculated according to the steering angle δ i of the front axle wheel, v i as well as the steering angle δ i+1 of the rear axle wheel.Hence, the coordinates of the front and rear tracking points can be obtained.

Forward preview algorithm for arbitrary path
Generally, the reference path consists of a series of discrete coordinate points: Considering the discreteness of the reference path coordinates, the lateral and heading deviations of the tracking point need to be obtained by interpolation, as shown in Figure 4.If the abscissa value of the tracking point satisfies the following conditions: The two adjacent points [x(j), y(j)] and [x(j + 1), y(j + 1)] are selected as the reference coordinate points, and the deviations are calculated with the linear interpolation method.The equation of the lateral deviation is as follows: The heading deviation can be calculated by: It should be noted that the judgment process indicated in Eq. ( 10) requires a series of path points to search for the reference coordinate points.However, as the vehicle moves forward, the stored path points will gradually increase in number and occupy more computer memory, and the search process will consume more time.Therefore, it is necessary to clear the useless historical data and keep only the latest data of l points with the help of shift register and the process is as follows: (1).If j < l: where x store h denotes the x coordinates of the stored h th group path points, and y store h the y coordinates of the stored h th group path points.(2).If j ≥ l: where K = 0 I l−1 0 0 is a l × l-dimension matrix, I l−1 the (l − 1) × (l − 1)-dimension identity matrix, and In the actual vehicle steering control system, due to delay and other factors, it is usually necessary to take the preview deviation as the controller input so that the control system can make judgment in advance.When considering the preview deviation at a certain distance L pre in front of the tracking point, the judgment criteria of Eq. ( 10) should be rewritten as follows: The preview distance L pre could be set as follows: where T p is defined as the preview time.
The preview distance L pre in Eq. ( 16) is a variable related to velocity, which is consistent with the driving habits of real drivers; that is, when the vehicle runs on a straight path or a small curvature path at a faster speed, the driver often needs to observe the road conditions at a farther position in front to make a judgment in advance; on the contrary, when the vehicle runs at a lower speed, the driver needs to observe the position close to the front to make a correct judgment.

Extended Ackermann geometry
When the n-unit all-wheel steering G-VTT passes a curve path, the front-and rear-WSAs satisfy the extended Ackermann geometry shown in Figure 5.The o i denotes the instantaneous velocity centre of the adjacent front and rear tracking points of the i th car body, and R i R i+1 the turning radius of the adjacent front and rear wheels, respectively.
According to the extended Ackermann geometry: where σ fi and σ ri denote the guide angles of the front and rear tracking points of the i th car body, respectively.
The turning curvatures of the two adjacent tracking points can be expressed as follows:

Cascade modular path-following controller
The distributed virtual driving model (DVD model), as the name suggests, refers to the independent virtual driving model at each tracking point that needs the vehicle state information at the local position only as the control input, without the requirement for that of the whole vehicle.Thus, it can greatly simplify the complexity of the control algorithm, especially for the G-VTT with long combination.Herein, the cascade modular WSA control law based on the optimal lateral acceleration (OLA) of tracking points was derived, and the DVD model with cascade structure was established.This can meet the needs of allwheel steering of the G-VTT and realise the cascade adjustment of the control algorithm when the vehicle marshaling length is extended or shortened.

Cascade modular control law
According to Eq. ( 19), the lateral accelerations a fi and a ri of the two adjacent tracking points can be expressed as: Then the lateral acceleration of the tracking point in the global coordinate system is: According to Eq. ( 21), the optimal WSA can be derived as follows: Eq. ( 22) shows the cascade modular WSA control law: for the first and the second axle, if the OLA required for each tracking point to approach the reference path is known, the optimal WSA could be obtained by solving the equations.For the 3rd ∼ n th axle, the optimal WSA could be calculated according to the OLA of the tracking point and the front WSA, so the control algorithm has a chained cascade structure.When the train marshaling is extended or shortened, all-wheel steering and path tracking could be achieved by adding or removing the series links in the control algorithm.In addition, the control law of Eq. ( 22) transforms the problem of finding the optimal WSA into the optimisation problem of the OLA of the tracking point.

Local tracking objective function
The principle of local optimisation is adopted to pursuit the OLA of tracking points so that the tracking point could reach the reference path along the optimal trajectory within the preview time T p .Considering the preview result of the preview controller in a neighbourhood U(T p , t) of T p , the preview window width is 2 t.Let t 0 = T p − t and t f = T p + t, and the local tracking objective function could be established: where, where ε i (t) denotes the lateral deviation.According to Eqs. ( 23)-( 25), the mathematical meaning of the local tracking objective function J i is the square of the 2-norm of the deviation signal ε i (t) in the local interval, which represents the 'energy' of the signal ε i (t).ε i (t) can be expressed by Eq. ( 26): where f i (t) denotes the time-domain expression of the reference path at the i th tracking point, y i (t) the lateral position of the i th tracking point, and w(τ ) the deviation weight function; different w(τ ) correspond to different preview strategies.
Assuming that the tracking point approaches the reference path with constant lateral acceleration during the sampling period, the lateral position of the tracking point in the future can be expressed as: Therefore, the tracking objective function J i can be expressed as: Obviously, according to the properties of the 2-norm in the finite dimensional space, J i is a lower bounded continuous differentiable function and satisfies J i ≥ 0. Therefore, the partial derivative of J i can be obtained and the minimum point of J i is the OLA.Let ∂J i /∂ ÿi = 0, then: where ÿ * i (t) denotes the OLA of the i th tracking point.Then: where, where f pi (t) denotes the preview value of the reference path coordinates at the i th tracking point, which is obviously a function related to the deviation weight function w(τ ), the preview time T p , and the preview window width 2 t.In the subsequent controller design, f pi (t) will be used as the input of the tracking controller.

Preview-tracking algorithm
In Section 3.2.1, the design of the preview algorithm is completed, and the influence factors of the preview result f pi (t) are identified [see Eq. ( 32)].In this section, the preview-tracking controller is designed as shown in Figure 6: where P(s) refers to the preview controller, T(s) the tracking controller, f the input of the preview controller (reference path information), f p the output of the preview controller and also the effective input of the tracking controller T(s), and y the output of the system.
To achieve a good path-following performance, for example, y i (t) ⇒ f i (t), the previewtracking controller should meet the following conditions: That is, under certain conditions, the reciprocals of the transfer function P(s) of the preview controller and the transfer function T(s) of the tracking controller are nearly the same.According to Eq. (32): The Taylor series of P(s) is expanded as: where, According to the structure of Eq. ( 35), let: T j s j (37) In order to satisfy Eq. ( 33), T j needs to satisfy the following condition: According to Eq. (37), the value of m determines the order of the tracking controller, while the high-order controller will increase the system errors and even cause system instability since it could increase the oscillation poles in the calculation process, so the value of m is very important.According to Eqs. (34) and (35), the value of m is closely related to the order of the reference path f (s).Considering the time domain meaning of Eq. (35): When the reference path f (t) satisfies the conditions of Eq. ( 40), it is defined as an norder curve: For example, when the reference path f (t) is a second-order curve, the Laplace transform process of Eq. (39) can be expressed as follows: In this case, if T −1 (s) = 1 + P 1 s + P 2 s 2 ,y(s) = f p (s)T(s) = f (s) can be satisfied, then the theoretical optimal tracking of the system output to input can be realised.As indicated, the order of the reference path is the theoretical optimal value of m.Considering that there are few high-order curves (i.e.curves with complex curvatures) in actual virtual track, in order to avoid the system instability caused by the high-order operation, this paper considers the system response within the third-order, and introduces a relaxation factor η to reduce the order of the high-order controller.The tracking controller is designed as shown in Eq. (42): where η ∈ [0, 1].
If η = 0, the tracking controller T(s) is a second-order controller, and the control system has a good tracking behaviour for the second-order reference path.If η = 1, the tracking controller T(s) is a third-order controller, which possesses better tracking behaviour for the reference path of third-order and above.By adjusting the value of η appropriately, the tracking controller T(s) could achieve an ideal tracking performance.In this case, the effective input of the tracking controller T(s) can be expressed as: Therefore, the OLA can be expressed as: According to Eq. ( 44), the OLA of the tracking point is affected by the preview controller P(s), P j , the relaxation factor η, and the reference path input y(s).For a given reference path y(s), when appropriate η and T p values are configured, the deviation weight function w(τ ) determines the characteristics of f p (s) and P j [see Eqs.(36) and ( 43)].The selection of w(τ ), therefore, is very important, which will be discussed next.

Three preview strategies
In this section, with the second-order controller as an example (η = 0), the path-following control algorithm with different preview strategies corresponding to different w(τ ) is discussed, and the analytical expressions of optimal WSA are given.

Single-point preview
The deviation weight function w(τ ) is defined as the unit pulse function, i.e.
Eq. (45) satisfies Obviously, as shown in the expression of f pi (t) in Eq. ( 46), when the unit pulse function is adopted as the deviation weight function, the system collects the reference path coordinates f i (t + T p ) at the time T p in front of the tracking point and generates the effective input f pi (t) of the tracking controller T(s), so this preview method is called single-point preview.In this case, the OLA of the tracking point can be derived by the simultaneous Eq. ( 44):

τ 2 window preview
When the deviation weight function w(τ ) = 1 is defined, it means the same importance of each preview deviation in the neighbourhood U(T p , t) in the control algorithm.According to Eqs. ( 32) and (34): Obviously, as shown by the expression of f pi (t) in Eq. ( 48), when w(τ ) = 1 is adopted as the preview weight function, the system processes the reference path f i (t) through the τ 2 window function and generates the effective input f pi (t) for the tracking controller T(s).In this case, the OLA of the tracking point can be derived by combining Eq. ( 44):

Rectangular window preview
When the deviation weight function w(τ ) = 1 τ 2 is defined, it means that the preview deviations in the section [t − T p , t + T p ] close to the tracking point are more important, and those far away from the tracking point are less important.According to Eqs. ( 32) and (34): Obviously, as shown by the expression f pi (t) in Eq. ( 50), when w(τ ) = 1 τ 2 is adopted as the preview weight function, the effective input f pi (t) for the tracking controller T(s) is the mean value of the reference path f i (t) in the section [t − T p , t + T p ].In this case, the optimal OLA of the tracking point can be derived by combining Eq. ( 44):

Cascade control algorithm with heading constraint
The path-following control algorithm discussed in Section 3.2 takes only the vehicle's lateral deviation as the control objective.Although a good tracking performance can be achieved, it is obviously not the optimal strategy, because there are both lateral deviation and heading deviation between the tracking point and the reference path which are coupling variables and cannot be controlled independently [see Eq. ( 3)].Even if the lateral deviation of the tracking point is reduced to close to 0, the tracking point will continue to deviate from the target position due to the existence of heading deviation, resulting in lateral deviation, which requires continuous adjustment of WSA so that both can reach the optimal position at the same time.According to Eq. ( 47), if the single-point preview strategy is adopted, the local optimal trajectory of the tracking point is a parabola [see Figure 7(a)].So, both lateral deviation [f i (t + T p ) − y i (t + T p )] and heading deviation [ ḟi (t + T p ) − tan ψ i (t + T p )] should be considered as the control target to calculate the optimal WSA through the cascade modular control algorithm, then the lateral and heading position of tracking points can reach the optimal position at the same time.So, the control algorithm should satisfy the following two constraints: The derivation of Eq. ( 56) can be obtained: According to Eq. ( 52): Then, the OLA can be derived as follows: Herein, the effective input of the tracking controller T(s) is: After the heading deviation constraint is introduced, the relationship between the local optimal trajectory of the tracking point and the reference path is shown in Figure 7(b).

Compensation of steady-state error
The sideslip effect of the rubber tyre will generate a deviation between the actual heading direction and the WSA direction of the tracking point during the curve manoeuvre of the G-VTT, which will result in steady-state error of the control system.The mathematical relationship between the WSA and the heading sideslip angle is shown in Eq. (58).
where β i denotes the heading sideslip angle of tracking points, i.e. the angle between the actual moving direction of the tracking point and the longitudinal axis of the car body.
When the vehicle enters the circular curve from a straight line, there will be a certain deviation between the tracking point and the reference path, and it will remain at a constant level when the train reaches a steady state on the circular curve because the tyre sideslip angle will be a constant value.

Steady-state error
In order to study the mechanism and influence factors of the steady-state error, the dynamic disturbance signal D(s) is introduced to the original control system structure, as shown in Figure 8.
In Figure 8, D(s) is mainly related to the tyre sideslip angle, so the disturbance action point is set between the nominal optimal WSA δ n * i and the actual WSA δ i .G 1 (s) and G 2 (s) are forward channel transfer functions, representing the output characteristics of the path-following control algorithm and the dynamic response characteristics of the vehicle, respectively.H(s) is the feedback channel transfer function.The expressions of G 1 (s), G 2 (s), and H(s) are shown in Eq. ( 62): where the expressions of G 1 (s) and H(s) are derived from Eq. ( 56), and G 2 (s) is related to the vehicle dynamics parameters.
The steady-state error e ssd is the steady-state component of the error signal; therefore, it can be expressed as: According to the final value theorem of Laplace transform, it can be obtained that: where E D (s) denotes the error transfer function of the system induced by the dynamic disturbance.
To get the steady-state error caused by the dynamic disturbance signal, f p (s) can be set to 0, and the transfer function G D (s) of the output signal to the disturbance input signal is: where y D (s) denotes the output signal of the system under the action of a dynamic disturbance input signal.Therefore, the error E D (s) of the system under disturbance is as follows: Assume that the disturbance signal is a step function D(s) = A s , and A denotes the magnitude of the disturbance.By substituting it into Eq.(64), it can be obtained that: Normally G 1 (0)G 2 (0)H(0) 1, so: As shown in Eq. ( 66), the steady-state error caused by the dynamic disturbance is positively correlated with the magnitude of disturbance A; that is, under the condition of large curvature and high speed, the steady-state error will increase when the tyre has a large sideslip angle.Meanwhile, the steady-state error is negatively correlated with the forward channel transfer function G 1 (s) before the disturbance action point.Therefore, the steadystate error can be reduced by increasing the gain of G 1 (s), but it is obviously impossible to eliminate the steady-state error fundamentally by this means.

Compensation method
In Section 4.1, the mechanism of the steady-state error of the system is described, which, in the final analysis, is the principle error caused by the mismatch between the control algorithm based on the moving geometry model and the vehicle dynamics system model.The main factor leading to the model mismatch is the tyre sideslip angle.Therefore, this section considers eliminating the steady-state error from the perspective of error compensation and offsetting the disturbance term by introducing the steady-state error compensator (SEC) (see Figure 9) to achieve the optimal tracking performance and realise the zero-error control of the system.
In Figure 9, C(s) denotes the SEC.To offset the dynamic disturbance term, the condition shown in Eq. ( 67) should be satisfied: Here, the SEC is designed in the form of integral compensation, i.e.: The expression of the compensation gain K c can be derived as follows by substituting Eq. (68) into Eq.(67): The design of DVD model based on cascade modular path-following control algorithm is now completed.

Co-simulation model
Dynamics topology of the G-VTT is shown in Figure 10.Considering the symmetry of the G-VTT, the ECM1, ESM1, GAM1, and ICM1 are taken as examples to introduce the topological structure and force state, and the other modules are the same as them.What is included in the diagram are the degree-of-freedoms of the main rigid bodies, the mechanical relationship between the rigid bodies, the main structural parameters of the vehicle, etc.The meanings of other specific parameters, design values, and symbols of the rigid body degree-of-freedoms are shown in Tables A1 and A2 in Appendix A3.
A four-unit G-VTT dynamics model was established in the software SIMPACK, including five suspensions, four car bodies, and three GAMs.The suspension was composed of  the axle, wheels, and trapezoidal steering mechanism.The trapezoidal steering mechanism was simplified as a four-bar linkage mechanism, and the magic formula model was adopted for the tyre model.The outputs were the vehicle state parameters, which were used as the inputs for the DVD model.The DVD model was established in the MATLAB/Simulink software, including the preview controller, the tracking controller, and the steady-state error compensator (Figure 11).

Results
Figure 12 compares the lateral deviation of the G-VTT in four simulated cases.The simulation was conducted in the condition of a 50m-radius circular curve, and the speed was 20 km/h.The single-point preview strategy was adopted.The results shown in Figure 12 is summarised in Table 1.The comparison of results in Figure 12(a) and Figure 12(b) reproduce the steady-state error of the path-following control system and verify the mechanism of steady-state error.The result in Figure 12(c   Figure 13 compares the simulation results of the heading deviation whether the heading constraint was introduced to the path-following control algorithm or not, and the simulation condition was the same as that in Figure 12.The results show that when the vehicle enters and leaves the circular curve, the heading deviation of each axle has a peak value.The maximum value of the whole vehicle heading deviation is about 0.024 rad without the heading constraint [see Figure 13(a)], and about 0.011 rad with the heading constraint, and the convergence speed of the deviation is significantly faster than that without the heading constraint [see Figure 13  Figure 14 compares the WSA and the lateral deviation of the vehicle passing a small radius curve under three different preview strategies.The simulation condition was a 25mradius circular curve, and the speed was 15 km/h.The path-following control algorithm with the heading constraint was adopted.T p = 0.4s, 2 t = 0.2s, and K c = 15.
The simulation results show that when the single-point preview strategy is adopted, the lateral deviation is smaller [Figure 14(b) (left plot)]; however, the variation of each WSA is sharper [Figure 14(a) (left plot)], and there is obvious overshoot in the WSA curves and fluctuations in lateral deviation curves.When the τ 2 or the rectangular window preview strategy is adopted, the simulation results are similar, the variation of each WSA is gentler [Figure 14 Figure 15 compares the simulation results of the vehicle turning passageway width.Figure 15(a) and (b) shows the sweeping area of the vehicle and the trajectories of the tracking points.Figure 15(c) shows the width of turning passageway.When the vehicle is not controlled, the turning passageway width increases gradually when the vehicle enters the circular curve, and decreases gradually when the vehicle leaves the circular curve, with a maximum value of about 6.696 m.When the vehicle is controlled, the turning passageway width reaches the peak value when the vehicle enters and leaves the circular curve, with a maximum value of about 2.698 m.
Figure 16 compares the simulation results of the lateral deviation, heading deviation, folding angle between the adjacent car bodies, yaw rate of car bodies and sideslip angle of car bodies.In particular, when the vehicle is not controlled, the lateral deviation of the first-fifth axles and the sideslip angle of each car body increase successively when the vehicle enters the circular curve, and there appears steady non-zero values of the two indices when the vehicle is on the circular curve.When the vehicle is controlled, however, the peak values of lateral deviation of each axle and each car body sideslip angle appear when the vehicle enters and leaves the circular curve, and they are close to 0 when the vehicle is on the circular curve [see Figure 16  The results in Figure 16 are summarised in Table 2.It shows that the DVD model can effectively improve the path-following ability of the vehicle with the lateral deviation reduced by about 99.16%; the maximum values of the folding angle, yaw rate of car body, and the sideslip angle of car body are significantly reduced, indicating that the yaw and  lateral stability of the vehicle are significantly improved.In addition, the width of turning passageway is reduced by about 59.63%, indicating that the small-curve-passing capacity of the vehicle has been significantly improved.With the increase of the speed or the decrease of the curve radius, the maximum lateral deviation, and the maximum the turning passageway width will increase at different degrees.Especially, under the simulation condition of 20m-radius circular curve and speed With the increase of the speed, the variation of payload has no significant effect on the maximum value of the lateral deviation, but the maximum value of the turning passageway width will gradually increase with the increase of payload, and the influence is more significant when the speed is greater than 5 m/s.This is because with an increase of the payload, the centrifugal force on car bodies increases when the vehicle passes the curve, leading to the increase of the roll angle of car bodies and the width of turning passageway.When the payload is 16t and the speed is 8 m/s, the maximum turning passageway width increases by about 2.67% compared with the low speed (3 m/s), and increases by 0.88% compared with the low payload (6t) [see Figure 17

Conclusions
In this paper, a new virtual track train architecture, the gantry virtual track train, was proposed, and a cascade modular path-following control strategy called distributed virtual driving model was proposed, which can achieve the automatic tracking of arbitrary path.The specific research content and the conclusions are as follows: 1.The G-VTT adopts a modular marshaling type.The standard GAM between car bodies is adopted to realise the modular marshaling through hinges, forming a symmetrical and chained structure, making the vehicle more flexible and easier to pass small-radius curve commonly seen in urban block.2. The cascade modular control law was derived, and the DVD model with a cascade structure was constructed.Considering the tyre sideslip effect, the mechanism of steady-state error of the control system was analysed, and a steady-state error compensation method was provided.

Disclosure statement
No potential conflict of interest was reported by the author(s).

Figure 4 .
Figure 4. Schematic diagram for the calculation of the lateral and heading deviations.

Figure 7 .
Figure 7. Local optimal trajectory of the tracking point under single-point preview strategy: (a) with lateral deviation constraint; (b) with both lateral and heading deviation constraint.

Figure 8 .
Figure 8. Schematic diagram of the control system structure with dynamic disturbance.

Figure 9 .
Figure 9. Schematic diagram of the control system structure with a SEC.
) indicates that the SEC can effectively reduce the steady-state error of the control system.The comparison of the simulation results in Figure12(b) and (c) verify the effectiveness of the SEC.The comparison results in Figure12(c) and (d) verify the effectiveness of the heading constraint to reduce the lateral deviation and improve the convergence speed of the deviation.
Figure13compares the simulation results of the heading deviation whether the heading constraint was introduced to the path-following control algorithm or not, and the simulation condition was the same as that in Figure12.The results show that when the vehicle enters and leaves the circular curve, the heading deviation of each axle has a peak value.The maximum value of the whole vehicle heading deviation is about 0.024 rad without the heading constraint [see Figure13(a)], and about 0.011 rad with the heading constraint, and the convergence speed of the deviation is significantly faster than that without the heading constraint [see Figure13(b)].

Figure 15 .
Figure 15.Simulation results of the turning passageway width: (a) the sweeping area and trajectories of the tracking points without control; (b) the sweeping area and trajectories of the tracking points with control; (c) turning passageway width.
Figure14compares the WSA and the lateral deviation of the vehicle passing a small radius curve under three different preview strategies.The simulation condition was a 25mradius circular curve, and the speed was 15 km/h.The path-following control algorithm with the heading constraint was adopted.T p = 0.4s, 2 t = 0.2s, and K c = 15.The simulation results show that when the single-point preview strategy is adopted, the lateral deviation is smaller [Figure14(b) (left plot)]; however, the variation of each WSA is sharper [Figure14(a) (left plot)], and there is obvious overshoot in the WSA curves and fluctuations in lateral deviation curves.When the τ 2 or the rectangular window preview strategy is adopted, the simulation results are similar, the variation of each WSA is gentler [Figure14(a) (middle & right plot)], and the fluctuations in the lateral deviation curves are smaller compared with those of the single-point preview strategy.However, the maximum value of the lateral deviation is larger [Figure 14(b) (middle & right plot)].Figures 15 and 16 compare the curving performance indices of the vehicle with and without path-following control.T p = 0.4s, K c = 15.The single-point preview strategy was adopted.The simulation case was a U-shaped bend with a 25 m-radius circular curve and the speed was 15 km/h.Figure15compares the simulation results of the vehicle turning passageway width.Figure15(a) and (b) shows the sweeping area of the vehicle and the trajectories of the tracking points.Figure15(c)shows the width of turning passageway.When the vehicle is not controlled, the turning passageway width increases gradually when the vehicle enters the circular curve, and decreases gradually when the vehicle leaves the circular curve, with a maximum value of about 6.696 m.When the vehicle is controlled, the turning passageway width reaches the peak value when the vehicle enters and leaves the circular curve, with a maximum value of about 2.698 m.Figure16compares the simulation results of the lateral deviation, heading deviation, folding angle between the adjacent car bodies, yaw rate of car bodies and sideslip angle of car bodies.In particular, when the vehicle is not controlled, the lateral deviation of the first-fifth axles and the sideslip angle of each car body increase successively when the vehicle enters the circular curve, and there appears steady non-zero values of the two indices when the vehicle is on the circular curve.When the vehicle is controlled, however, the peak values of lateral deviation of each axle and each car body sideslip angle appear when the vehicle enters and leaves the circular curve, and they are close to 0 when the vehicle is on the circular curve [see Figure16(a) and (e)].The above result shows different steady- Figure14compares the WSA and the lateral deviation of the vehicle passing a small radius curve under three different preview strategies.The simulation condition was a 25mradius circular curve, and the speed was 15 km/h.The path-following control algorithm with the heading constraint was adopted.T p = 0.4s, 2 t = 0.2s, and K c = 15.The simulation results show that when the single-point preview strategy is adopted, the lateral deviation is smaller [Figure14(b) (left plot)]; however, the variation of each WSA is sharper [Figure14(a) (left plot)], and there is obvious overshoot in the WSA curves and fluctuations in lateral deviation curves.When the τ 2 or the rectangular window preview strategy is adopted, the simulation results are similar, the variation of each WSA is gentler [Figure14(a) (middle & right plot)], and the fluctuations in the lateral deviation curves are smaller compared with those of the single-point preview strategy.However, the maximum value of the lateral deviation is larger [Figure 14(b) (middle & right plot)].Figures 15 and 16 compare the curving performance indices of the vehicle with and without path-following control.T p = 0.4s, K c = 15.The single-point preview strategy was adopted.The simulation case was a U-shaped bend with a 25 m-radius circular curve and the speed was 15 km/h.Figure15compares the simulation results of the vehicle turning passageway width.Figure15(a) and (b) shows the sweeping area of the vehicle and the trajectories of the tracking points.Figure15(c)shows the width of turning passageway.When the vehicle is not controlled, the turning passageway width increases gradually when the vehicle enters the circular curve, and decreases gradually when the vehicle leaves the circular curve, with a maximum value of about 6.696 m.When the vehicle is controlled, the turning passageway width reaches the peak value when the vehicle enters and leaves the circular curve, with a maximum value of about 2.698 m.Figure16compares the simulation results of the lateral deviation, heading deviation, folding angle between the adjacent car bodies, yaw rate of car bodies and sideslip angle of car bodies.In particular, when the vehicle is not controlled, the lateral deviation of the first-fifth axles and the sideslip angle of each car body increase successively when the vehicle enters the circular curve, and there appears steady non-zero values of the two indices when the vehicle is on the circular curve.When the vehicle is controlled, however, the peak values of lateral deviation of each axle and each car body sideslip angle appear when the vehicle enters and leaves the circular curve, and they are close to 0 when the vehicle is on the circular curve [see Figure16(a) and (e)].The above result shows different steady-

Figure 17 compares
the simulation results of the maximum lateral deviation and the maximum turning passageway width under conditions of different speeds (v = 3 − 8 m/s), different curve radii (R = 20 − 50 m), and different payloads (m c = 6 − 16 t).

Figure 17 .
Figure 17.Adaptability of the DVD model: (a) the maximum lateral deviation under different speed and curve radii, m c = 12t; (b) the maximum turning passageway width under different speed and curve radii, m c = 12t; (c) the maximum lateral deviation under different speed and payload, R = 50 m; (d) the maximum turning passageway width under different speed and pay load, R = 50 m.

3 .
The DVD model was verified by simulation.(i) Compared with the train dynamics model without path-following control, the DVD model could effectively improve path-following ability and curve-passing capacity, as well as yaw and anti-jackknifing stability; (ii) The introduction of SEC to the control algorithm can effectively eliminate the steady-state error.The introduction of a heading constraint can effectively reduce the maximum lateral and heading deviation, and improve the convergence speed of the deviations; (iii) The DVD model has good adaptability to speed, curve radius, and payload.

Table 1 .
Summary of results in Figure12.Max.lat.deviation refers to the maximum value of lateral deviation when the vehicle enters and leaves the circular curve; (ii) Steady lat.deviation refers to the steady-state value of lateral deviation when the vehicle is on the circular curve.

Table 2 .
Summary of results in Figure16.