Braking force distribution strategy for virtual rail trains based on I-curve

Abstract

The virtual rail trains have attracted much attention in recent years thanks to their high passenger capacity and low construction cost. In this paper, braking force distribution strategy for virtual rail trains with three-car and six-axle was studied. Based on the principle of axle load proportion distribution, braking force distribution method of the four-wheel single car under curve braking was studied by applying the I-curve theory. The distribution laws of centrifugal force between axles during curve braking were analyzed, the lateral force provided by the tires of each axle can be calculated, and the remaining tire adhesion can be set as the upper limit of braking force, which can ensure the maximum utilization of tire adhesion. The method was extended to multi-series trains by analyzing the additional influence of hinge points on the lateral force, and a braking force distribution strategy considering the car body hinge coupling relationship was proposed. Finally, the effectiveness of the strategy was verified by hardware-in-the-loop tests. The research results provide guidelines on the brake control of newly emerged virtual rail trains.

Keywords

Virtual rail train braking force distribution braking stability hardware-in-the-loop test

1. Introduction

As an advanced type of rail transit, virtual rail trains (VRT), which combine the characteristics of buses and rail trains (Ji et al., 2016), are guided by perceiving the virtual track on the ground, with rubber tires as the running part. Besides, the multi-marshaling mode is used to connect multiple carriages. The first VRT, which adopts the form of three-section articulated marshaling, was opened in China (Huang et al., 2019), as shown in Figure 1(a). The VRT is composed of three car body modules in the form of Mc1-M-Mc2 (Mc: motor car with cab, M: motor car), as shown in Figure 1(b). Articulated discs are set between adjacent car bodies, and all wheels adopt full moving wheels.

Figure 1.

(a) VRT; (b) Formation of VRT.

The characteristics of multi-marshaling lead to mechanical coupling between vehicle bodies, and the large number of tires increase the control complexity (Ji et al., 2025; Chen et al., 2025). This leads to a need for differentiated braking force distribution (BFD) strategy between axles and wheels.

For braking control in curve, researchers studied vehicle braking stability by changing the body connection structure (Goodarzi et al., 2008; Yu et al., 2014) and wheel slip ratio (Liu et al., 2015; Wang et al., 2018). Sakai et al., 2002 proposed a four-wheel driving/braking force distribution algorithm: the vehicle’s lateral motion can be controlled by the yaw moment generated by the torque difference between the wheels. Nie et al. (2017) established a simplified model of articulated heavy vehicles with five degrees of freedom and proposed an integrated control strategy based on timely mode switching and the active steering control system of the trailer and differential braking control system. Caizhen et al. (2011) designed the optimal steering controller for semi-trailers. By combining the roll stability parameters with the longitudinal braking-related parameters, the good path-tracking ability of articulated vehicles in steady-state operation is ensured. Esmaeili et al. (2020) coupled the longitudinal and lateral dynamics of an articulated vehicle and used the braking force and the steering angle of the wheels to adjust the vehicle movement. Russo et al. (2007) improved the vehicle stability and course tracking performance during extreme cornering by using the difference between the braking forces of the left and right wheels to generate additional yaw moment. Rieveley and Minaker (2007) proposed a dynamic torque distribution control method based on a 2-DOF linearized model. The vehicle handling stability is improved by adjusting the torque ratio of the front and rear axles and further correcting the torque of the left and right wheels. Goodarzi et al. (2008) believes that the smaller the rear wheel slip ratio during braking, the safer and more stable the vehicle braking. Based on the ideal BFD principle, vehicle stability under different decelerations is improved by reducing the rear wheel slip ratio and increasing the front wheel slip ratio. To analyze the stability of the tractor-semitrailer during braking, Kaneko and Kageyama (2003) constructed the nonlinear motion equation of an articulated vehicle with 8 degrees of freedom. The vehicle transmitted load is applied according to the reaction force of each suspension element, which depends on the roll angle, roll speed, pitch angle, and pitch speed. Ando and Fujimoto (2010) proposed a longitudinal and lateral force distribution method based on the least square solution of longitudinal, lateral, and yaw motion equations to balance the workload of each wheel. Liang et al., (2010) analyzed the drive cycle and motor performance of the hybrid electric vehicle and designed the BFD strategy, which can recover more braking energy, and the vehicle can be more stable when braking. Duong et al. (2018) established the BFD calculation and simulation model based on energy recovery optimization, and a series regeneration control strategy based on hybrid power is proposed.

From the above literature, the current research focuses on the optimization of wheel slip ratio for BFD, and the differential braking force between the left and right wheels for assisted steering. However, the BFD method comprehensively considering the correlation between the vehicle longitudinal braking force and the lateral stability for VRT based on I-Curve was few reported. It does not make full use of the road surface adhesion when adjusting the swing torque through the difference in braking force between the wheels, especially under the low adhesion road surface, where it is easy to lose most of the longitudinal braking force to maintain the lateral stability. Moreover, the research on the BFD of rubber-tired trains is limited to the fixed vehicle type with two-car marshaling, and the problem of all-wheel steering and the expansion of the number of marshaling remains.

This paper focuses on developing BFD strategy of the VRT, and the strategy on straight lines and curves under different road surfaces was studied. First, the realizing distributed braking control function was proposed and its feasibility was analyzed. Then, the train dynamic model was established to analyze the nonlinear characteristics of the tire during curve braking, and the dynamic simulation model was verified offline. Based on the I-curve theory, the difference between single vehicle and multi car body dynamic model was studied, the BFD strategy under the multi car body coupling dynamic model was analyzed, and the strategy was verified by HIL tests.

The rest of this paper is organized as follows. Section 2 gives the details of braking dynamic model. Section 3 describes the BFD strategy based on I-curve. Section 4 presents case analysis results based on HIL test bench. Conclusion remarks are drawn in Section 5.

2. Braking dynamics model for BFD

2.1. Vehicle model

The dynamic model of VRT is a multi-body dynamic system composed of multi-section car bodies, and the relative motion constraints between car bodies are realized by articulated mechanism. Each tire is independent steering with torque control. This paper studies the body stability of VRT under braking force distribution control. The vehicle dynamic model based on longitudinal force, lateral force, and body motion was established. To accurately express the vehicle motion and force in the braking process, it is necessary to analyze and calculate the vehicle motion in detail. This involves many degrees of freedom, which increases the complexity and computational efficiency. Therefore, the model was simplified as follows:

(1) The car bodies are regarded as rigid, which are coupled and connected through articulated mechanisms to limit the relative motion between the car bodies.

(2) In a vehicle dynamics sense, the longitudinal forces of the tire are directly related to the braking force, and the lateral forces are responsible for resisting the centrifugal force at the vehicle’s center of mass during cornering. Therefore, the motion change caused by body roll and pitch is neglected during braking, but the vertical load transfer of the tire in the lateral and longitudinal directions is considered.

(3) The rolling resistance and air resistance during vehicle motion are neglected due to its small value.

In this paper, the vehicle traveling direction was set as the positive direction of the x-axis, the left side of the driver as the positive direction of the y-axis, and according to the right-hand rule, the vertical up direction is the positive direction of the z-axis, and the center of mass of each vehicle body is the origin of the coordinate system. The dynamic model of the VRT has 33 degrees of freedom, including the longitudinal, lateral, and yaw motion of three car bodies, as well as the yaw and normal rotation of 12 wheels.

The basic parameters of VRT are listed in Table 1. The parameters and their physical meanings involved in motion state and force analysis during curve braking are listed in Table 2, in which i represents the ith vehicle body and i

\in

{1,2,3}.

Table 1.

Basic parameters of VRT.

Point	Parameter
Maximum operating speed	70 km/h
Axle load	≤9 t
Average acceleration from 0 to 30 km/h	1.0 m/s²
70 km/h maximum service braking deceleration	4.0 m/s²
Train length	30 m
Braking method	Hydraulic disc brake
Capacity	200 people

Table 2.

Vehicle symbol and meaning.

Symbol	Meaning	Unit
$m_{i}$	Car weight	kg
$a_{i}$	Distance between center of gravity and front axle	m
$b_{i}$	Distance between center of gravity and rear axle	m
$c_{i}$	Distance between center of gravity and articulated point	m
$F_{x i F L / x i F R}$	Left/right front tire braking force	N
$F_{x i R L / x i R R}$	Left/right rear tire braking force	N
$F_{y i F L / y i F R}$	Left/right front tire lateral force	N
$F_{y i R L / y i R R}$	Left/right rear tire lateral force	N
$F_{j x i}$	Articulated mechanism longitudinal force	N
$F_{j y i}$	Articulated mechanism lateral force	N
$ω_{i}$	Yaw rate	rad/s
$u_{i}$	Longitudinal velocity	m/s
$v_{i}$	Lateral velocity	m/s
$z_{x}$	Longitudinal resultant acceleration	m/s²
$α_{i 1 F L / i 1 F R}$	Left/right front tire side slip angle	rad
$α_{i 1 R L / i 1 R R}$	Left/right rear tire side slip angle	rad
$δ_{i j}$	Tire steering angle	rad
$θ i$	Angle between articulated mechanisms	rad
$M_{z i}$	Force moment of the ith car	N·m
R	Instantaneous turning radius	m

As a multi-body dynamic coupling model, the force value at the articulated joint needs to be known. In practice, the motion parameters of the car body are easily measurable, therefore, it is necessary to solve the joint force based on the motion parameters.

The centrifugal force of the ith car is

F_{y i} = m_{i} (\dot{ν_{i}} + μ_{i} ω_{i}) = \frac{m_{i} {u_{i}}^{2}}{R}

(1)

The longitudinal acceleration is set as the target braking deceleration during braking, and the longitudinal force of the car body can be obtained as

F_{x i} = m_{i} (\dot{μ_{i}} - ν_{i} ω_{i}) = m_{i} z_{x}

(2)

According to the force analysis model shown in Figure 2(a), the motion equation can be obtained by analyzing the motion state and force situation.

Figure 2.

(a) Schematic diagram of train coordinate system. (b) Relationship between longitudinal and lateral forces at the hinge point.

The motion equation of the ith car is

{\begin{array}{c} m_{i} (\dot{μ_{i}} - ν_{i} ω_{i}) = \sum F_{x i} \\ m_{i} (\dot{ν_{i}} + μ_{i} ω_{i}) = \sum F_{y i} \\ I_{z i} \dot{ω_{i}} = \sum M_{z i} \end{array}

(3)

The principal vector and principal moment of the external force acting on the ith car are

{\begin{cases} \begin{array}{l} \sum F_{x i} = \sum_{j = F L, F R, R L, R R} [F_{x i j} \cos (δ_{i j} + α_{i j}) + F_{y i j} \sin (δ_{i j} + α_{i j})] \\ + F_{j x i} - F_{j x (i + 1)}^{'} \end{array} \\ \begin{array}{l} \sum F_{y i} = \sum_{j = F L, F R, R L, R R} [F_{y i j} \cos (δ_{i j} + α_{i j}) - F_{x i j} \sin (δ_{i j} + α_{i j})] \\ + F_{j y i} - F_{j y (i + 1)}^{'} \end{array} \\ \begin{array}{l} \sum M_{z i} = \sum_{j = F L, F R, R L, R R} [(F_{x j} \sin (δ_{i j} + α_{i j}) b_{i} + F_{y j} \cos (δ_{i j} + α_{i j})) a_{i}] \\ + (F_{(i + 1) y}^{'} - F_{i y}) c_{i} \end{array} \end{cases}

(4)

Figure 2(b) shows the relationship between the longitudinal and lateral forces at the hinge points.

{\begin{cases} F_{j x i}^{'} = F_{j x i} \cos θ_{i} + F_{j y i} \sin θ_{i} \\ F_{j y i}^{'} = F_{j y i} \cos θ_{i} + F_{j x i} \sin θ \end{cases}

(5)

According to equations (3)∼(5), the resultant force on the car body from the articulating mechanism is

{\begin{cases} \begin{array}{l} {\tilde{F}}_{j x i} = F_{j x i} - F_{j x (i + 1)}^{'} \\ = m_{i} z_{x} - \sum_{j = F L, F R, R L, R R} [F_{x i j} \cos (δ_{i j} + α_{i j}) + F_{y i j} \sin (δ_{i j} + α_{i j})] \end{array} \\ \begin{array}{l} {\tilde{F}}_{j y i} = F_{j y i} - F_{j y (i + 1)}^{'} \\ = \frac{m_{i} {u_{i}}^{2}}{R} - \sum_{j = F L, F R, R L, R R} [F_{y i j} \cos (δ_{i j} + α_{i j}) - F_{x i j} \sin (δ_{i j} + α_{i j})] \end{array} \end{cases}

(6)

During the braking process, the normal load transfer between axles and wheels will occur due to the influence of acceleration, which will affect the slip ratio and sideslip angle of the tires, resulting in changes in the longitudinal and lateral forces of the tires. In terms of braking stability, it is necessary to pay attention to the change in its normal load.

The dynamic change of the normal load of each wheel can be calculated as

{\begin{cases} {\tilde{F}}_{z F L} = (F_{j y i} - F_{j y (i + 1)}^{'}) + m_{i} g \frac{b_{i}}{2 (a_{i} + b_{i})} - m_{i} \dot{u} \frac{h_{i}}{2 (a_{i} + b_{i})} - m_{i} \frac{m_{i} {u_{i}}^{2}}{R} \frac{h_{i}}{t w_{i}} \frac{b_{i}}{a_{i} + b_{i}} \\ {\tilde{F}}_{z F R} = (F_{j y i} - F_{j y (i + 1)}^{'}) + m_{i} g \frac{b_{i}}{2 (a_{i} + b_{i})} - m_{i} \dot{u} \frac{h_{i}}{2 (a_{i} + b_{i})} + m_{i} \frac{m_{i} {u_{i}}^{2}}{R} \frac{h_{i}}{t w_{i}} \frac{b_{i}}{a_{i} + b_{i}} \\ {\tilde{F}}_{z R L} = (F_{j y i} - F_{j y (i + 1)}^{'}) + m_{i} g \frac{a_{i}}{2 (a_{i} + b_{i})} + m_{i} \dot{u} \frac{h_{i}}{2 (a_{i} + b_{i})} - m_{i} \frac{m_{i} {u_{i}}^{2}}{R} \frac{h_{i}}{t w_{i}} \frac{a_{i}}{a_{i} + b_{i}} \\ {\tilde{F}}_{z R R} = (F_{j y i} - F_{j y (i + 1)}^{'}) + m_{i} g \frac{a_{i}}{2 (a_{i} + b_{i})} + m_{i} \dot{u} \frac{h_{i}}{2 (a_{i} + b_{i})} + m_{i} \frac{m_{i} {u_{i}}^{2}}{R} \frac{h_{i}}{t w_{i}} \frac{a_{i}}{a_{i} + b_{i}} \end{cases}

(7)

Observed from equations (4)∼(7), the hinge force is affected by the angle of the vehicle body, the vehicle speed, and the force between each tire and the ground, and substitute the corresponding parameters into the above equation can further lead to the values of each variable through the vehicle motion state. Note that the wheel forces and coupling forces form an underdetermined system (Noori Asiabar et al., 2021).

2.2 Tire model

The VRT provides the vehicle driving force through the interaction force between the tire and the ground, and the vehicle motion state can be calculated by measuring the tire-ground coupling force. However, the direct measurement of the tire-ground coupling force is more complicated, and the tire force is generally estimated indirectly through the tire’s material properties and motion state. In this paper, the longitudinal force and lateral force of the tire are calculated by using the magic formula. During the actual operation, the vehicle will often encounter curve braking. There is a coupling relationship between the lateral force and the longitudinal force. In the magic tire formula, the relationship between the longitudinal force $F_{x}$ and the lateral force $F_{y}$ and the slip ratio s, the declination angle α and the vertical force $F_{z}$ is

{\begin{cases} {F_{x}}^{'} = \frac{σ_{x}}{σ} F_{x} \\ {F_{y}}^{'} = \frac{σ_{y}}{σ} F_{y} \end{cases}

(8)

where

σ = \sqrt{σ_{x}^{2} + σ_{y}^{2}}

σ_{x} = - \frac{λ}{1 + λ}

σ_{y} = - \frac{\tan α}{1 + λ}

Figure 3 shows the relationship between longitudinal/lateral force and slip ratio when the slip angle is 6°.

Figure 3.

Relationship between longitudinal/lateral force and slip ratio at a slip angle of 6°.

Compared with the longitudinal force, the lateral force is more affected by the slip ratio. With the increase of the slip ratio, the lateral force decreases rapidly, the downward trend decreases, and the range of large lateral force in the whole slip ratio range is small, which explains the susceptibility of the vehicle to lateral instability when tire lock-up occurs. Therefore, the vehicle’s lateral stability can be ensured by reducing the slip ratio or increasing the sideslip angle. The effect of slip ratio on lateral force is greater than that of sideslip angle, and the slip ratio can be controlled by reasonable application of braking force, which can achieve vehicle stability control.

3. BFD strategy based on I-curve

3.1. BFD strategy of four-wheel single car body

In this section, the BFD strategy of the single vehicle was analyzed from the two dimensions of axle control and wheel control. The vehicle is subjected to centrifugal force during the cornering process, lateral acceleration at the center of mass pointing to the outside of the curve generates, and all tires provide lateral force to maintain the vehicle body balance. Therefore, the lateral force of the tire should be considered. The distribution strategy between axles does not distinguish between left and right wheels. To simplify the model, a two degrees of freedom vehicle model commonly used is adopted, as shown in Figure 4(a).

Figure 4.

(a) Two degrees of freedom vehicle model. (b) The first quadrant of the front and rear axle BFD curve under cornering conditions.

Decompose the vehicle along the lateral and longitudinal directions, and set the tire side deflection coefficients of the front and rear axles as k_f, k_r, then the motion differential equations of the two degrees of freedom vehicle are

{\begin{cases} m (\dot{v} + u ω_{r}) = (k_{f} + k_{r}) β + \frac{1}{u} (a k_{f} - b k_{r}) ω_{r} - k_{f} δ \\ I_{z} {\dot{ω}}_{r} = (a k_{f} - b k_{r}) β + \frac{1}{u} (a^{2} k_{f} + b^{2} k_{r}) ω_{r} - a k_{f} δ \end{cases}

(9)

According to the regulations of the coordinate system, the front and rear wheel slip angles are

{\begin{cases} α_{f} = β + \frac{a ω_{r}}{u} - δ \\ α_{r} = β - \frac{b ω_{r}}{u} \end{cases}

(10)

where β is the angle of adjacent cars.

The mechanical equation of the vehicle along the tangent direction of the curve is

z = \frac{F_{x f} \cos (δ - α_{f}) + F_{y f} \sin (δ - α_{f}) + F_{x r} \cos (α_{r} + β)}{G}

(11)

where Z is the braking intensity and G is the vehicle gravity.

The mechanical equation of the vehicle along the direction pointing to the center of the curve is

F_{c} = F_{y f} \cos (δ - α_{f} - β) + F_{y r} \cos (α_{r} + β)

(12)

The moment balance relationship between the centrifugal force F_c on the front and rear axles is

\frac{F_{c f}}{F_{c r}} = \frac{a \cos (δ - α_{f})}{c \cos α_{r}}

(13)

The centrifugal force of the vehicle is

F_{c} = \frac{m u^{2}}{R \cos^{2} β}

(14)

The turning radius R is

R = \frac{L}{δ - (α_{f} - α_{r})}

(15)

where L is the distance between front and rear axles.

The resultant force of the tires in all directions cannot exceed the maximum available ground adhesion $φ F_{z}$

{\begin{cases} F_{x f}^{2} + F_{y f}^{2} \leq {(φ F_{z f})}^{2} \\ F_{x r}^{2} + F_{y r}^{2} \leq {(φ F_{z r})}^{2} \end{cases}

(16)

where φ is the adhesion coefficient of the ground.

Equations (11)∼(16) can be used to obtain the range that the braking force of the front and rear axles can be applied without locking

{\begin{cases} F_{x f}^{2} + F_{y f}^{2} < {(φ G (b + (F_{x f} \cos δ + F_{y f} \sin δ + F_{x r}) h_{g} / G) / L)}^{2} \\ F_{x r}^{2} + F_{y r}^{2} < {(φ G (a - (F_{x f} \cos δ + F_{y f} \sin δ + F_{x r}) h_{g} / G) / L)}^{2} \end{cases}

(17)

When the ground adhesion coefficient is φ, and the tires are locked, equation (17) becomes

{\begin{cases} F_{x f}^{2} + F_{y f}^{2} = {(φ G (b + (F_{x f} \cos δ + F_{y f} \sin δ + F_{x r}) h_{g} / G) / L)}^{2} \\ F_{x r}^{2} + F_{y r}^{2} = {(φ G (a - (F_{x f} \cos δ + F_{y f} \sin δ + F_{x r}) h_{g} / G) / L)}^{2} \end{cases}

(18)

(1) By eliminating the variable φ, the relationship between the braking force of the front and rear axles and the vehicle state parameters becomes

{\begin{array}{c} \frac{F_{x f}^{2} + F_{y f}^{2}}{F_{x r}^{2} + F_{y r}^{2}} = {(\frac{b + (F_{x f} \cos δ + F_{y f} \sin δ + F_{x r}) h_{g} / G}{a - (F_{x f} \cos δ + F_{y f} \sin δ + F_{x r}) h_{g} / G})}^{2} \\ \begin{array}{l} F_{y f} = \frac{b δ m u^{2}}{L^{2}} \\ F_{y r} = \frac{a δ m u^{2}}{L^{2}} \end{array} \end{array}

(19)

The I-curve is obtained by graphing.

(2) When the front wheels are completely locked, there is

{\begin{array}{c} F_{x f}^{2} + F_{y f}^{2} = {(φ G (b + (F_{x f} \cos δ + F_{y f} \sin δ + F_{x r}) h_{g} / G) / L)}^{2} \\ F_{y f} = \frac{b δ m u^{2}}{L^{2}} \end{array}

(20)

Substituting different road adhesion coefficients into equation (20) yields the f-curve.

(3) When the rear wheels are completely locked, there is

{\begin{array}{c} F_{x r}^{2} + F_{y r}^{2} = {(φ G (a - (F_{x f} \cos δ + F_{y f} \sin δ + F_{x r}) h_{g} / G) / L)}^{2} \\ F_{y r} = \frac{a δ m u^{2}}{L^{2}} \end{array}

(21)

Substituting different road adhesion coefficients into equation (21) yields the r-curve.

The parameters in the formula are shown in Table 3, and the three curves obtained are shown in Figure 4(b).

Table 3.

Single vehicle model parameters.

G (N)	h _g (m)	L (m)	a (m)	b (m)	δ
117600	1.3	5.2	2.5	2.7	6°

The BFD of the front and rear axles during curve braking is different from that of the straight-line braking. As shown in Figure 4(b), the I-curve does not exist when the road adhesion coefficient is less than 0.4, and the curve does not pass the origin, which indicates that when the vehicle is driving on a curve and the road adhesion coefficient does not meet the requirements of the I-curve in the first quadrant, the corresponding inter-axle BFD strategy become inapplicable, and the front and rear axles cannot apply braking. The intersection of the I-curve and the x-axis is (12 kN, 0 kN), which indicates that when the vehicle brakes in a curve under the road adhesion coefficient, the maximum allowable braking force of the front axle is 12 kN, and the rear axle cannot apply braking force. Otherwise, the axle will be locked.

The vehicle BFD graphs under the curve braking give the boundary conditions for tire locking during braking based on the range delineation of the road surface adhesion coefficients for the braking force distribution relationship between the front and rear axle. The difference with the straight-line braking is that the tires have a certain turning angle under the curve braking, which changes the running direction. When the turning angle under cornering condition is set to 0, the form of the I-curve is the same as that of the I-curve for straight line braking, indicating that the straight-line braking condition is a special case under cornering braking.

When the vehicle speed is constant, the turning angle is inversely proportional to the turning radius, while the turning radius is negatively correlated with the centrifugal force, and finally the increase of the turning angle leads to the increase of the centrifugal force. The tire needs to provide more friction force in the lateral direction to maintain the lateral stability and complete the turning process. If the road adhesion coefficient cannot meet the requirements of the tire lateral force, the applied braking force will not achieve the lateral stability. On the other hand, if the road adhesion coefficient can cover the lateral force, the remaining adhesion utilization of the road surface can be distributed to the front and rear axles as braking force.

3.2. Secondary allocation between wheels

The braking control mode of the VRT is independently controlled by each tire. Since the vertical load of the left and right wheels is basically the same under the straight road condition, and the force is the same, there is no need to do further work after the BFD between the axles. However, the centrifugal force will redistribute the vertical load of the left and right wheels during curve braking, and the upper limit of the tire-ground coupling force of each tire will be different. By establishing a four-wheel model for dynamic analysis, the control strategy mode of the left and right wheels is analyzed based on the BFD strategy between the axles, as shown in Figure 5(a).

Figure 5.

(a) Four-wheel vehicle model. (b) The BFD diagram between the wheels under different turning angles.

Extending the two degrees of freedom model to a four-wheel model, and the vertical load of the front and rear axles can be further divided into the vertical load of the left and right wheels

{\begin{cases} F_{z f} = F_{z f l} + F_{z f r} \\ F_{z r} = F_{z r l} + F_{z r r} \end{cases}

(22)

Calculate the moment balance at the center point of the axle

{\begin{cases} c F_{z f l} - c F_{z r l} = h_{g} F_{y f} \\ F_{z f l} + F_{z f r} = F_{z f} \end{cases}

(23)

From equations (22)∼(23), the vertical load ratio j_f and j_r of the left and right wheels of the front and rear axles can be obtained as

{\begin{cases} j_{f} = \frac{F_{z f l}}{F_{z f r}} = \frac{c F_{z f} + h_{g} F_{y f}}{c F_{z f} - h_{g} F_{y f}} \\ j_{r} = \frac{F_{z r l}}{F_{z r r}} = \frac{c F_{z r} + h_{g} F_{y r}}{c F_{z r} - h_{g} F_{y r}} \end{cases}

(24)

According to the ideal BFD theorem, the braking force of each wheel is

{\begin{cases} F_{x f l} = F_{x f} \frac{j_{f}}{1 + j_{f}} \\ F_{x f r} = F_{x f} \frac{1}{1 + j_{f}} \\ F_{x r l} = F_{x r} \frac{j_{r}}{1 + j_{r}} \\ F_{x r r} = F_{x r} \frac{1}{1 + j_{r}} \end{cases}

(25)

Taking the BFD between the left and right wheels of the front axle as an example, the parameters in the formula are shown in Table 3, and the BFD diagram between the wheels under different turning angles is as shown in Figure 5(b).

When the vehicle turns right, the brake force of the left wheel is greater than that of the right wheel. Meantime, as the turning angle increases, the difference between the left and right wheels becomes more and more obvious. Specially, when the steering angle is 0, the braking force of the left and right wheels is equal, which is equivalent to the inter-axle BFD strategy in straight-line conditions.

When the vehicle is driving on a curve with large curvature, the vertical load of the left and right wheels is redistributed due to the generation of centrifugal force, which will affect the lateral stability of the vehicle. If the centrifugal force is directed to the left, part of the vertical load of the right wheel will transfer to the left wheel, the available tire-ground coupling force of the left wheel is larger than that of the straight line, resulting in a differential distribution between wheels. With the increase of centrifugal force, the transfer amount of vertical load between wheels increases, and the proportion of BFD between left and right wheels increases.

3.3. BFD strategy of four-wheel multi-group VRT

The VRT is a highly dynamic model with the coupling of multiple car bodies, and the force change of any part of the car body will be transmitted to the train through the hinge mechanism, causing the motion state to change. Longitudinal acceleration will undergo a sudden change during the process of applying braking force, which will cause the longitudinal coupling relationship between the vehicles to become more complicated. Meantime, the effect of centrifugal force will lead to changes in the lateral coupling relationship between the vehicle body during curve braking, which puts forward higher requirements for the lateral force resistance of the tires.

To reflect the highly dynamic motion characteristics of the VRT under joint working conditions, and to calculate the tire-ground coupling relationship under the target braking intensity, it is necessary to reasonably split the multi-car body model, and to characterize the function properties of the coupling relationship of multi-car body acting on a single car. A single vehicle body dynamics model based on the multi-segment vehicle body coupling model is established, the vertical load transfer and lateral stability requirements are analyzed, and the BFD strategy between axles and wheels are realized.

To split the VRT model into a single-car body, the lateral and longitudinal mechanical coupling relationship is reconstructed based on the single car body model, as shown in Figure 6.

Figure 6.

The split model of the single car body of the VRT.

The calculation method of the longitudinal force $F_{j x i}$ , $F_{j x i + 1}^{'}$ and the lateral force $F_{j y i}$ , $F_{j y i + 1}^{'}$ of the hinge point has been given in equation (6). When the ground adhesion coefficient is ϕ, according to the tire friction circle theory, the ground adhesion utilization limit of the tire is

{\begin{cases} F_{x f i}^{2} + F_{y f i}^{2} \leq {(φ F_{z f i})}^{2} \\ F_{x r i}^{2} + F_{y r i}^{2} \leq {(φ F_{z r i})}^{2} \end{cases}

(26)

{\begin{array}{c} \frac{F_{x f i}^{2} + F_{y f i}^{2}}{F_{x r i}^{2} + F_{y r i}^{2}} = {(\frac{{(φ F_{z f i})}^{2}}{{(φ F_{z r i})}^{2}})}^{2} \\ F_{y f i} + F_{y r i} = \frac{δ m u^{2}}{L} + F_{j y i} + F_{j y i + 1}^{'} \end{array}

(27)

{\begin{cases} c_{i} F_{z i f l} - c F_{z i r l} + c F_{z i r l} - c F_{z i r r} = h_{g i} F_{y i f} + h_{g i} F_{y i r} - F_{j y i} h_{j i} - F_{j y i} h_{j i} \\ F_{z i f l} + F_{z i f r} = F_{z i f} \\ F_{z i r l} + F_{z i r r} = F_{z i r} \end{cases}

(28)

During the braking process, the BFD strategy model needs to monitor physical quantities such as train speed, curve radius, and the relative positional relationship between car bodies, and to detect whether the maximum braking intensity allowed on the road supports the lateral force required for turning and longitudinal force required with target braking intensity. According to the friction circle theory, there is a limit to the lateral and longitudinal force. If the longitudinal braking force takes up too much friction, it will lead to insufficient lateral adhesion, resulting in car body yaw and folding, which will affect the safe operation.

Figure 7 shows the BFD strategy for the VRT under the combined condition of curve braking.

Figure 7.

Flowchart of BFD strategy.

If the road adhesion coefficient cannot provide enough friction force to simultaneously meet the requirements of lateral force and longitudinal braking force, the lateral force distribution is given priority, and distribute the remaining friction to the longitudinal braking force. The corresponding calculation process is shown in Figure 8.

Figure 8.

Four-wheeled vehicle body multi-group BFD calculation process.

4. Case analysis under multi-scenario working conditions based on HIL test bench

In order to verify the effectiveness of the proposed BFD strategy, multi-scenario working conditions tests were conducted on a HIL simulation platform, as shown in Figure 9. The platform included controller, data transmission, and virtual vehicle model. The controller is an implementation of the BFD strategy in an industrial PC with its hardware and human machine interface (HMI). National instrument A/D modules are used for data transmission. The virtual vehicle model, that is, the dynamic model of the VRT (Figure 10), is also implemented in an industrial PC with its hardware and HMI.

Figure 9.

HIL simulation platform.

Figure 10.

Dynamic model of the VRT.

4.1. Braking stability analysis with different BFD strategies

The bogie-controlled strategy of the rail train and the wheel vertical load transfer strategy of the automobile are used to be analyzed against the BFD strategy based on I-curve proposed in this paper, and the effect of different strategies on different dynamic parameters, including train speed, longitudinal and lateral acceleration, and wheel speed are analyzed.

The road adhesion coefficient is set as 0.35, the curve radius is 50 m, and the radian is π/2. The initial vehicle speed is set as 40 km/h. It enters the curve from the 7th s and remains in the turning state during the subsequent operation, and at the 10th s the controller starts to apply a braking torque with a target braking intensity of 0.2 g until the vehicle stops. This working condition requires high braking control ability, the braking is performed on the road with low adhesion coefficient, and the target braking intensity is close to the adhesion limit, which is prone to lateral instability.

The simulation results are shown in Figure 11. The results show that no wheel lock phenomenon with BFD strategy based on I-curve occurs. The longitudinal and lateral acceleration changes are low, the motion states of the three cars are consistent, and the following performance between the front and rear car bodies is good.

Figure 11.

Simulation results of different BFD strategies on low-adhesion curved roads. (a) Braking torque. (b) Velocity and longitudinal/lateral acceleration. (c) Tire speed.

It can be seen from Figure 11(b) that the train does not control the braking deceleration according to the target braking intensity at the initial stage of braking, but increases nonlinearly, and maintain the target braking intensity at the end of braking. In the prospect of physics, due to the low road adhesion coefficient, most of the tire-ground coupling force is used to balance the lateral force required to corner, and the remaining tire-ground coupling force is provided to distribute the braking force, so the proposed strategy is preferred. The priority is to ensure that the train successfully completes the cornering function, and part of the tire-ground coupling force required for longitudinal braking is sacrificed to ensure lateral stability, which is reflected in the longer braking time.

All wheels are locked during braking with bogie-controlled strategy, and the longitudinal and lateral accelerations of the three car bodies show divergent oscillations. The lateral acceleration of Mc1 changes suddenly at 15 s and tends to move in the opposite direction with other car bodies, indicating that Mc1 acts as a forward-oriented car body and has a phenomenon of decreased directional control stability.

All wheels are locked during braking with wheel vertical load transfer strategy. From Figure 11(b), the train does not decelerate according to the target braking intensity during the entire braking process. Meantime, it can be seen that the Mc2 car body has the largest lateral variation trend, and the middle car body is prone to fold.

4.2. Multiple operating parameters influence analysis

In order to further validate the feasibility and robustness of the proposed strategy, two key operating parameters of braking intensity and curve radius are investigated and discussed.

4.2.1. Braking intensity

The road adhesion coefficient is set as 0.6, curve radius is 50 m, and curve radian is π/2. The braking intensity is set as 0.1 g, 0.2 g, 0.3 g, and 0.4 g, respectively, so as to get the lateral acceleration under different braking intensities.

It can be seen from Figure 12 that the maximum value of lateral acceleration of each car body changes to different degrees with the increase of braking intensity. The change of lateral acceleration of M is the most obvious among the three, the lateral acceleration increases with the braking intensity. Mc2 is the least affected, but its value is the largest.

Figure 12.

Lateral acceleration under different braking intensities. (a) Mc1; (b) M; (c) Mc2.

4.2.2. Curve radius

The road adhesion coefficient is set as 0.6, braking intensity is 0.2 g, and curve radian is π/2. The curve radius is set as 25 m, 30 m, 40 m, and 50 m, respectively, so as to get the lateral acceleration under different curve radius.

It can be seen from Figure 13 that as the curve radius increases, the lateral acceleration changes towards a decreasing trend, and Mc2 is the most affected.

Figure 13.

Lateral acceleration under different curve radius. (a) Mc1; (b) M; (c) Mc2.

The above analysis shows that the BFD strategy can be adapted to different braking intensity and curve radius. Meanwhile, the increase of braking intensity and the decrease of curve radius will lead to the increase of the lateral acceleration during curve braking. Therefore, decreasing the target intensity and increasing the curve radius of the planned route are conducive to providing the lateral stability.

5. Conclusion

This paper studied the BFD strategy of VRTs based on I-curve. The dynamic distribution of braking force, wheel anti-lock, lateral stability optimization, braking instability boundary perception, and braking intensity active takeover under different road environments were realized and verified by digital simulation and HIL test bench.

Based on the ideal BFD law of the four-wheel single car body dynamics model, the characteristic law of the I-curve which can reveal the BFD relationship between the front and rear axles under curve braking was analyzed. Based on the relationship, the secondary distribution of the braking force between the left and right wheels was carried out. The method was extended to multi-series trains by analyzing the additional influence of hinge points on the lateral force, and a BFD strategy of the VRT was proposed.

The VRT dynamic simulation model was modeled and a HIL test bench was built to verify the effectiveness of the proposed strategy under different working conditions, which proves that the strategy is able to improve the braking stability.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: Sponsored by the National Key R&D Program of China (2023YFB4301605), the science and technology innovation Program of Hunan Province (2024RC4021), and Shanghai Collaborative Innovation Research Center for Multi-network & Multi-model Rail Transit.

ORCID iDs

Pengfei Diao

Jingxian Ding

References

Ando

Fujimoto

. (2010) Yaw-rate control for electric vehicle with active front/rear steering and driving/braking force distribution of rear wheels. In: Proceedings of the 11th IEEE International Workshop on Advanced Motion Control, Nagaoka, Niigata, Japan, 21-24 March: 726–731.

Caizhen

Roebuck

Odhams

, et al. (2011) High-speed optimal steering of a tractor–semitrailer. Vehicle System Dynamics 49(4): 561–593.

Chen

Liu

Rao

, et al. (2025) Explicit speed-integrated LSTM network for non-stationary gearbox vibration representation and fault detection under varying speed conditions. Reliability Engineering & System Safety 254: 110596.

Duong

TTD

Nguyen

. (2018) Research on braking force distribution in regenerative braking system apply to conventional vehicle. In: Proceedings of 2018 4th International Conference on Green Technology and Sustainable Development, Ho Chi Minh City, Vietnam, 23-24 November: 48–52.

Esmaeili

Kazemi

Tabatabaei Oreh

(2020) Design of a new integrated controller (braking and steering) to maintain the stability of a long articulated vehicle. Proceedings of the Institution of Mechanical Engineers - Part C: Journal of Mechanical Engineering Science 234(5): 981–1013.

Goodarzi

Behmadi

Esmailzadeh

(2008) An optimised braking force distribution strategy for articulated vehicles. Vehicle System Dynamics 46(sup1): 849–856.

Huang

Yuan

, et al. (2019) Modeling and simulation of all-wheel steered multiple-articulated rubber-tire transit for autonomous driving control. In: 2019 IEEE Vehicle Power and Propulsion Conference, Hanoi, Vietnam, 14-17 October: 1–6.

Ren

Wang

(2016) Contact model of guiding wheel/rail of translohr tram. Journal of Mechanical Engineering 52(20): 111–119.

Huang

Zeng

, et al. (2025) A physical‒data-driven combined strategy for load identification of tire type rail transit vehicle. Reliability Engineering & System Safety 253: 110493.

10.

Kaneko

Kageyama

(2003) A study on the braking stability of articulated heavy vehicles. JSAE Review 24(2): 157–164.

11.

Liang

Shang

Fang

, et al. (2010) Braking Force Distribution Strategy for HEV Based on Braking Strength. Proceedings of 2010 International Conference on Measuring Technology and Mechatronics Automation, Changsha City, China, 13-14 March. IEEE, 759–764, In press.

12.

Liu

Wei

Zhu

, et al. (2015) Optimal allocation of tire force based on vehicle stability. Journal of Mechanical & Electrical Engineering 32(3): 433–438.

13.

Nie

Wang

, et al. (2017) Integrated control strategy of articulated heavy vehicle based on timely mode switching under medium/high speed conditions. Journal of Traffic and Transportation Engineering 17(6): 135–149.

14.

Noori Asiabar

Gordon

Gao

, et al. (2021) On the control allocation performance of longer combination vehicles. In: Proceedings of the 27th Symposium of the International Association of Vehicle System Dynamics, St. Petersburg, Russia, 17-19 August: 757–766.

15.

Rieveley

Minaker

(2007) Variable torque distribution yaw moment control for hybrid powertrains. SAE Technical Paper.

16.

Russo

Terzo

Timpone

(2007) Software-in-the-loop development and validation of a cornering brake control logic. Vehicle System Dynamics 45(2): 149–163.

17.

Sakai

Sado

Hori

(2002) Dynamic driving/braking force distribution in electric vehicles with independently driven four wheels. Electrical Engineering in Japan 138(1): 79–89.

18.

Wang

Yang

(2018) Research on vehicle stability control based on optimal tire force distribution. Agricultural Equipment & Vehicle Engineering 56(2): 1–6.

19.

Liu

Xiong

, et al. (2014) Control strategies of handling improvement of distributed drive electric vehicle. Journal of Tongji University 42(7): 1088–1095.