A redundant steering-by-wire system and multi-axle cooperative steering control algorithm for super autonomous rapid transit

Abstract

In view of the safety problems of the single-channel hydraulic steering system which widely used in the multi-axle cooperative steering system of super autonomous rapid transit (SRT), a multi-axle cooperative steering system based on redundant steering-by-wire is developed. Firstly, the real-time task and system status judgment are placed in the master station of the steering controller for processing, fully ensuring the real-time performance of the task. At the same time, the slave station is set to synchronously process the task and judge the status. The two channels of the steering system work simultaneously, and when one channel fails, the other channel can take over the system in real time. This redundancy strategy ensures the safety of the vehicle. Secondly, a multi-axle cooperative steering control algorithm for SRT is proposed. The steering control system consists of a feedforward and an additional feedback control. The feedforward control is subdivided into three steps. In a first step the angle of axle 2 is calculated with dynamic model of first module and the driven path is estimated. This path is used in a second step to calculate the desired kinematics and finally the desired kinematics are transformed into the desired lateral forces by means of the inverse vehicle model. An additional nonlinear feedback controller stabilizes the tracking behavior on vehicle status level. By establishing the kinematics model of SRT and adopting the PID control of yaw angle deviation, the control error of the vehicle is dynamically compensated, and the control accuracy and robustness of the SRT multi-axle cooperative control algorithm are ensured. The steering system has been applied to the three-module & six-axle SRT, with the functions of steering-by-wire and track following, and can achieve the minimum turning radius of 15 m, and its good control accuracy and stability are verified by experiments.

Keywords

Super autonomous rapid transit steering system steering-by-wire cooperative steering control algorithm

1. Introduction

In recent years, there has been a growing demand for public transportation, transitioning from meeting basic travel needs to pursuing comfort, novelty, and environmental sustainability. The urban public transportation system in China mainly consists of subways and public buses. However, the existing rail transit systems face challenges such as high construction costs, long construction periods, and high maintenance costs. This combination is not the optimal choice for the traffic demand of medium and small cities and imposes a heavy burden on local finances. To address these challenges, there is a need to develop a new type of urban transportation tool that integrates the operation mode of urban rail transit with the technology model of public buses to meet the evolving demands of urban public transportation (Zhang et al., 2022). In response to this need, the Super autonomous Rapid Transit (SRT) has emerged as an innovative solution. SRT can supplement the distribution and expansion of public transportation systems in large and medium-sized cities while also serving as backbone transport in small and medium-sized cities. Compared with traditional urban rail transit systems, SRT offers advantages such as reduced construction costs, minimized damage to urban roads, shortened construction periods, increased passenger capacity, higher operating speeds, and improved environmental performance (Guan, 2020; Chen et al., 2023).

The unique characteristics of SRT vehicles including multiple module formations and long body structures may limit their maneuverability when navigating current city roadways with smaller turning radii (Esmaeili and Kazemi, 2019; Sadeghi K et al., 2020). To address this challenge, a multi-axle cooperative steering control system is essential for ensuring flexible turns while maintaining safety. Research on multi-articulated steering control systems has a long history. Bolzern et al. (1998, 2001) addressed the tracking problem of a train composed of a single tractor and N-1 articulated trailers equipped with off-axis hinges and the control of the train was achieved through closed-loop pole placement in the linearized system. Astolfi et al. (2004) employed Lyapunov-based methods to realize asymptotic stability control for articulated vehicles during both forward and reverse motion. References (Moon et al., 2009; Wagner et al., 2013) utilized an extended Ackermann steering approach to determine the instantaneous center of rotation based on the steering angle of the leading axle, by controlling other wheels’ steering angles to align their instantaneous centers, facilitated tracking for all wheels relative to the midpoint of the first axle successfully. Bruin et al., 2000; Oreh et al., 2016 considered all degrees of freedom associated with train tracking as part of its state variables while treating axle rotations and wheel drive torques as control inputs; they proposed an exhaustive state-tracking control algorithm grounded in vehicle dynamics modeling. Michalek and Marcin (2017) introduced a generalized tracking control method applicable to N-segment wheeled robotic formations, benefiting from modular cascading controllers that allow rapid scalability according to module count while validating effectiveness through principle modeling approaches.

However, research on multi-axle cooperative steering systems at home and abroad is still in its early stages, with most efforts focused on achieving multi-axle cooperative steering control functionality using non-redundant hydraulic actuation system (Chen et al., 2024). Such systems are susceptible to single point failures which could compromise lateral vehicle control leading to significant safety risks (Yang and He, 2004).

The concept of steering-by-wire originated in 1950s America, when companies like TRW proposed replacing mechanical connections between steering modules and actuators with electrical signals. This marked the beginning of research into using steering-by-wire technology to improve vehicle steering performance. Subsequently, German engineers Kasselmann and Keranen designed an early steering-by-wire model. However, due to limitations in electronic technology at that time, further research was not pursued (Yuan et al., 2015). Japanese company Koyo Seiko Co., Ltd developed a steering-by-wire system that retained the traditional steering column but incorporated clutch mechanisms allowing for mechanical transmission should steering-by-wire system fail—ensuring system safety (Zong et al., 2013). While some companies have introduced steering-by-wire products for vehicles both domestically and internationally, most are still undergoing research development phases (Cong et al., 2012; Ji et al., 2015). Mi (2021) proposed a dual-system, dual-redundancy control architecture utilizing coupled redundant information. In this framework, both motors output current at a fixed ratio during operation to mitigate issues related to force contention. Yao and Daugherty (2007) implemented coaxial opposition of two motors within the steering mechanism, employing a single controller to drive both motors simultaneously. They utilized two feedback control loops to track a common input and employed synchronization compensators to address angular position discrepancies between the feedback loops, thereby ensuring system synchronization and stability. Zong et al. (2012) presented a cooperative control approach for dual steering motors based on the FlexRay bus. Zou and Zhao (2020) developed a yaw rate controller grounded in hybrid H2/H∞ robust control theory, incorporating sliding mode velocity synchronization strategies based on cross-coupled control structures and sliding mode algorithms.

Despite already in use in the markets, steering-by-wire technologies rely heavily on electronic equipment for information transmission and controls—posing reliability concerns compared against mechanical structures. Therefore, it remains crucially important within ongoing studies surrounding redundant fault-tolerant steering-by-wire systems—to ensure the safety and reliability of the system by designing more dependable solutions.

This paper proposes a redundant steering-by-wire system architecture and multi-axle cooperative steering control algorithm for SRT. Under this redundant steering-by-wire system architecture, the two channels of the steering system work simultaneously, and when one channel fails, the other channel can take over the system in real time. This redundancy strategy ensures the safety of the vehicle. Based on this redundant system architecture, a multi-axle cooperative steering control algorithm for SRT is proposed, which ensured the control accuracy and robustness of the SRT.

2. Steering-by-wire system of SRT

2.1. Super autonomous rapid transit

The next-generation SRT incorporates the redundant multi-axle cooperative steering-by-wire technology. It is equipped with an automatic tracking system that utilizes advanced optical video recognition and 5G satellite positioning navigation technology, eliminating the need for physical track laying by precisely constraining the train to automatically run along a pre-set virtual track line. This innovative approach ensures that the SRT can achieve steering, traction, and braking while intelligently operating on the “pre-set virtual track,” achieving a minimum turning radius of 15 m. Consequently, it can access areas beyond the reach of traditional urban rail transit systems while maintaining high operational safety. Figure 1 shows the SRT series product of CRRC Zhuzhou Locomotive CO., LTD.

2.2. Redundant electric drive steering-by-wire for SRT

As shown in Figure 2, the electric drive steering system uses a double wishbone independent suspension axle. The ends of the axle are connected through steering linkages. The steering motor is installed on the vehicle body and receives network signals from the steering control unit to execute steering commands by rotating and moving its linkages to steer each wheel independently. This system utilizes a steering-by-wire for its steering motor, allowing it to quickly and accurately send and execute steering commands. This guarantees precise execution of steering as well as accurate trajectory tracking while minimizing tire wear on each axle. Additionally, due to its use of an electric drive instead of hydraulic actuators, this system requires no maintenance.

2.3. Redundant network architecture of steering-by-wire system

The redundant steering system network architecture, as illustrated in Figure 3, comprises a dual-channel redundant steering system with four layers of networks. These layers are functionally divided into the vehicle layer, decision layer, acquisition layer, and execution layer.

In the vehicle layer, there are two redundant hot standby vehicle control units (VCU_A and VCU_B). The primary role of these units is to receive network information from the steering control unit (STCU) and transmit relevant vehicle information to the STCU via network loops 2 or 3.

Moving on to the decision layer, it includes two redundant hot standby STCUs (STCU_A and STCU_B). These units receive network information from the VCU and data acquisition information from the acquisition layer. They then send steering commands to the vehicle axle control units (AxCU) in the execution layer through network loop 6.

The acquisition layer consists of steering-by-wire modules SWFE_A and SWFE_B located at both ends of the cabin. Additionally, it contains N-1 articulation control units (ACU_1, ACU_2, …, ACU_N_1) within N vehicle modules. The SWFE’s main function is to receive angle input information from the steering wheel and transmit acquired angle data to the STCU via network loop 6 for controlling either AxCu_1_1 or AxCu_N_2 (depending on whether starting occurs at cabin A or B). The articulation control unit receives signals from VCU and STCU while providing feedback information such as controlled torque and articulation angles through network loop 5.

Finally, in the execution layer there are specific axle control units for each of N vehicles: AxCu_1_1, AxCu_1_2, AxCu_i_1, AxCu_i_2, …, AxCu_N_1, AxCu_N_2. These AxCUs serve as receivers for information coming from both STCUs and SWFEs. Their responsibilities include controlling the steering motors as well as collecting angle data which is then fed back respectively to SWFEs and STCUs through network loops 5 and 6.

3. Multi-axle cooperative steering control algorithm

3.1. Analysis of the multi-axle cooperative steering control algorithm for SRT

SRT is a multi-module train that is controlled by a driver or an automatic driving system along a virtual track, without physical tracks constraining the vehicle laterally, a steering controller is needed to control the lateral movement trajectory of the vehicle modules, so that the vehicle modules can follow the track left by the first vehicle and control the following error small enough. Based on these requirements, the steering controller is designed as follows to enable it to achieve the function of multi-axle cooperative steering control:

(1) Create a desired path from the movement of the first module while driving. Consider this path as the virtual rail and guide the trailing module along it.

(2) Estimate the entire vehicle movement to get all necessary position and orientation information.

(3) Create control errors from the desired path (1) and the estimated vehicle movement (2) and steer all trailing axles in a manner to reduce these control errors.

Figure 4 gives a general survey of the steering control system consisting of a feedforward and an additional feedback control. Firstly, the feedforward control is subdivided into three steps. In a first step the angle of axle 2 is calculated with dynamic model of first module and the driven path s(λ) is estimated. This path is used in a second step to calculate the desired kinematics and finally the desired kinematics are transformed into the desired lateral forces by means of the inverse vehicle model. Secondly, an additional nonlinear feedback controller stabilizes the tracking behavior on vehicle status level. By establishing the kinematics model of SRT and adopting the PID control of yaw angle deviation, the control error of the vehicle is dynamically compensated, and the control accuracy and robustness of the SRT multi-axle cooperative control algorithm are ensured.

3.2. Vehicle model and driven path of first module

In this chapter, the influence of active steering on the lateral motion and yaw motion of the first vehicle module is studied, and the lateral and yaw characteristics of the vehicle are changed by control. Therefore, the first modular ideal vehicle model can be assumed to be an equivalent bicycle model with two degrees of freedom (DOF) in lateral and yaw directions as shown in Figure 5 (Masato, 2012). When establishing a model, it is important to make the model as simple and user-friendly as possible while accurately reflecting the vehicle’s characteristics. The construction of vehicle models is often based on the bicycle model, which involves making the following assumptions:

- Disregarding the vehicle’s motion in the Z-axis direction and only considering motion in the XY horizontal plane;

- Assuming equal steering angles for the left and right wheels, allowing them to be combined into a single tire for easier modeling;

- Ignoring rapid changes in vehicle speed and neglecting load transfer between front and rear axles;

- Assuming that both the body and suspension system of the vehicle are rigid.

Based on the bicycle model and assumptions above, the lateral and yaw dynamic equilibrium equations of a vehicle with front wheel angle $δ_{f}$ and rear wheel angle $δ_{r}$ are

{\begin{cases} m V_{x} (β^{'} + ω) = (C_{f} + C_{r}) β + \frac{l_{f} C_{f} - l_{r} C_{r}}{V_{x}} \cdot ω - C_{f} δ_{f} - C_{r} δ_{r} \\ I_{z} ω^{'} = (l_{f} C_{f} - l_{r} C_{r}) β + \frac{l_{f}^{2} C_{f} + l_{r}^{2} C_{r}}{V_{x}} \cdot ω - l_{f} C_{f} δ_{f} + l_{r} C_{r} δ_{r} \end{cases}

(1)

where

V_{x}

and

V_{y}

are the longitudinal and lateral speeds of the vehicle,

l_{f}

and

l_{r}

are the distances from the front and rear axles of the vehicle to the center of mass,

ω

is the yaw angular velocity,

β

is the side-slip angle of vehicle,

C_{f}

and

C_{r}

are the lateral stiffness of the front and rear wheels,

I_{z}

is the inertia of the vehicle about the Z axis.

In the current research on all-wheel steering control, there are various control laws for determining the rear wheel steering angle, such as rear wheel steering angle proportional to the front wheel steering angle, rear wheel steering angle proportional to the front wheel steering torque and rear wheel steering angle proportional to the yaw rate. All of these belongs feedforward control, and different control laws have different impacts on vehicle performance. The rear wheel steering angle control proportional to the front wheel steering angle can effectively manage the vehicle side-slip angle and is most easily implemented in actual vehicles. Therefore, this paper chooses to investigate the impact of active rear wheel steering on vehicle dynamics based on this particular control law. If the rear wheel steering angle is proportional to the front wheel steering angle, then we can get

δ_{r} = ε δ_{f}

(2)

where

ε

is the ratio of

δ_{f}

δ_{r}

Substituting equation (2) into equation (1), and through Laplace transform, it can be obtained that the response of the side-slip angle of the vehicle to the angle of the front wheel is

\frac{β (s)}{δ_{f} (s)} = \frac{| \begin{array}{l} C_{f} + ε C_{r} & m V_{x} + (l_{f} C_{f} - l_{r} C_{r}) / V_{x} \\ l_{f} C_{f} - ε l_{r} C_{r} & I_{z} s + (l_{f}^{2} C_{f} + l_{r}^{2} C_{r}) / V_{X} \end{array} |}{| \begin{array}{l} m V_{x} s + C_{f} + C_{r} & m V_{x} + (l_{f} C_{f} - l_{r} C_{r}) / V_{x} \\ l_{f} C_{f} - l_{r} C_{r} & I_{z} s + (l_{f}^{2} C_{f} + l_{r}^{2} C_{r}) / V_{X} \end{array} |}

(3)

If the control objective is set to make the side-slip angle of the vehicle zero, then the value of the scale coefficient $ε$ can be obtained, so that the steady state value of the side-slip angle is always zero, that is, the steady state value of the molecule in equation (3) is zero. We can get

ε = \frac{δ_{r}}{δ_{f}} = \frac{m l_{f} V_{x}^{2} / [C_{r} (l_{f} + l_{r})] - l_{r}}{m l_{r} V_{x}^{2} / [C_{f} (l_{f} + l_{r})] + l_{r}}

(4)

In order to make the vehicle side-slip angle not only the steady state value is zero, but the whole response process to the steering input is zero, then the rear wheel angle cannot be regarded as proportional to the front wheel, but has a transfer function relationship.

Therefore, the relation between the rear wheel angle and the front wheel angle can be expressed as

δ_{r} (s) = ε (s) \cdot δ_{f} (s)

(5)

Then replace $ε$ in the numerator of equation (3) with $ε (s)$ , and set the numerator equal to zero, we can get

ε (s) = \frac{δ_{r} (s)}{δ_{f} (s)} = \frac{ε_{0}}{1 + T_{e} s} - \frac{K_{1}}{K_{2}} \cdot \frac{T_{e} s}{1 - T_{e} s}

(6)

where

ε_{0}

is the steady state proportional coefficient, which can be obtained from equation (4), and has

T_{e} = \frac{I_{z} V_{x}}{l_{f} (l_{f} + l_{r}) C_{f} + m l_{r} V_{x}^{2}}

(7)

In summary, if the steering angle control law for the rear wheels can be precisely given in the form of equation (6), the side-slip angle of vehicle’s center of gravity will be zero, thus achieving a constant vehicle heading direction, which also determines the steering angle of the second axle. After determining the steering angles of the front and rear wheels in the first module, a vehicle kinematic model is established based on the bicycle model. Using the sine theorem

\frac{\sin (δ_{f} - β)}{l_{f}} = \frac{\sin (\frac{π}{2} - δ_{f})}{R}

(8)

\frac{\sin (β - δ_{r})}{l_{r}} = \frac{\sin (\frac{π}{2} + δ_{r})}{R}

(9)

Combining equations (8) and (9), we can get

(\tan δ_{f} - \tan δ_{r}) \cos β = \frac{l_{f} + l_{r}}{R}

(10)

Therefore, the side-slip angle can be obtained as

β = \tan^{- 1} (\frac{l_{f} \tan δ_{r} + l_{r} \tan δ_{f}}{l_{f} + l_{r}})

(11)

At low speed, the turning radius of the vehicle’s traveling path changes slowly. At this time, we can assume that the direction change rate of the vehicle is equal to the angular speed of the vehicle, then the angular speed of the vehicle is

ψ^{'} = v / R

(12)

Therefore, the angular velocity of the vehicle obtained by coupling equations (10) and (11) is

ψ^{'} = \frac{v \cos β}{l_{f} + l_{r}} (\tan δ_{f} - \tan δ_{r})

(13)

Then, based on the kinematic bicycle model, after the control input is given at a moment, we can calculate the state information (coordinates, yaw angle and speed) of the first vehicle module after the time as follows:

{\begin{cases} v_{i + 1} = v_{i} + a_{i} d t \\ ψ_{i + 1} = ψ_{i} + \frac{v_{i + 1} \cos β_{i + 1}}{l_{f} + l_{r}} (\tan δ_{i + 1, f} - \tan δ_{i + 1, r}) d t \\ x_{i + 1} = x_{i} + v_{i + 1} \cos (β_{i + 1} + ψ_{i + 1}) d t \\ y_{i + 1} = y_{i} + v_{i + 1} \sin (β_{i + 1} + ψ_{i + 1}) d t \end{cases}

(14)

where v is the speed of vehicle, a is the acceleration of vehicle. The subscript i represents the value of each parameter in the current period, and the subscript i+1 represents the value of each parameter in the next period.

3.3. Control algorithm based on kinematics model

Many scholars have studied the multi-axis coordinated steering control algorithm of kinematics model, and the most classic one belongs to the dynamic solution method proposed by Sebastian Wanger based on AutoTram (Wagner et al., 2013). There are some differences between SRT and AutoTram in terms of vehicle structure parameters and axle arrangement, but this method can still be used for dynamic solution. In the Figure 6, the generalized coordinates are shown in red and the tire forces and articulation damping torques are shown in green. The generalized coordinates:

q = {[\begin{array}{l} s_{x 1} & s_{y 1} & \begin{array}{l} ψ_{1} & ψ_{2} & ψ_{3} \end{array} \end{array}]}^{T}

(15)

q

are the coordinates

s_{x 1}

and

s_{y 1}

of the first module as well as the yaw angles

ψ_{1}

ψ_{2}

and

ψ_{3}

of each module. Following the explanations in reference (Wagner et al., 2013) the equations of motion for the SRT can be written as

M (q) \cdot q^{″} + h (q, q^{'}) + f_{P Q} (q, q^{'}) = f_{E} (q) + f_{G} (q)

(16)

In the equation above, $M (q)$ is the symmetrical mass matrix, $h (q, q^{'})$ is the vector of generalized centrifugal and Coriolis forces, $f_{P Q} (q, q^{'})$ is the vector of generalized spring and damper forces, $f_{E} (q)$ is the vector of generalized external forces and $f_{G} (q)$ is the vector of generalized gravitational forces. The numerical solution for

q^{″} = M^{- 1} (q) \times [f_{E} (q) + f_{G} (q) - h (q, q^{'}) - f_{P Q} (q, q^{'})]

(17)

provides the generalized accelerations and by double integration one obtains the generalized velocities

q^{'}

and coordinates

q

The equations of motion for the vehicle considered here are obtained by the so-called Newton–Euler method. For every single body (derived by free cut) of the multi-body system, Newton’s and Euler’s law (momentum and angular momentum law) is applied. The articulation damping forces are calculated by nonlinear functions.

T_{a r t, z} = f (ψ_{a r t}^{'}, v_{v e h})

(18)

The articulation damping forces are obtained by the linear tire model depending on articulations velocities $ψ_{a r t}^{'}$ and vehicle speed $v_{v e h}$ . The dependency from vehicle speed is due to the active damping system, which usually increases damping with increasing vehicle speed. The tire forces are obtained with equation (19):

F_{t i r e, y} = c_{t i r e} \cdot α_{t i r e}

(19)

where

c_{t i r e}

is the cornering stiffness and

α_{t i r e}

is the side-slip angle.

The side-slip angle of each tire is calculated by

α_{t i r e} = γ - β

(20)

After inserting the desired kinematics $q$ and $q^{'}$ , the lateral forces $F_{t i r e, y}$ according to equations (19) and (20), the solutions of these equations are the steering angles $γ = (γ_{4}, γ_{6})$ to be set up in the vehicle.

The full trailer axle A3 and A5 can be assumed to be virtual if the forces acting perpendicular to the wheel plane are always zero. This can be achieved using the tire model. Transforming equation (19) in the corresponding wheel fixed CS and applying equations (17) and (20) results in the lateral force at axle A3 and A5:

F_{t i r e, y} = C_{i} (γ - \arctan (v_{y} / v_{x}))

(21)

If the model coincides with reality, this steering angle would cancel the force acting perpendicular to the wheel plane of axle A3 and A5. As a consequence, the overall vehicle dynamics would be equal to a triple articulated bus with semitrailers only. Thus, additional forces in the articulation G1 and G2 induced by axle A3 and A5 are avoided.

3.4. Feedback control based on kinematics

It can be seen from Figure 7 that the lateral stiffness of the tire (absolute value of the slope of the curve) is not a fixed value, but changes with the change of the tire side angle. In the linear zone of the tire, the lateral stiffness is large. When the tire gradually passes to the nonlinear zone, the tire force is saturated and the lateral stiffness is very small and close to zero. Tire side stiffness is not only related to tire side angle, but also to tire vertical load, tire pressure and speed (Ji et al., 2024). However, the feedforward control based on dynamics given in Section 3.3 regards the tire lateral stiffness as a fixed value, and such feedforward control is difficult to make the vehicle track following error reach the expectation. Secondly, in the actual process of driving, the vehicle will inevitably be subjected to some side interference; if there is no feedback control, it is difficult to achieve the desired control effect.

Therefore, the feedforward and feedback control method are selected in this chapter to control vehicle trajectory deviation as small as possible. As shown in Figure 8, The feedback control uses yaw angle errors to guide the vehicle on a virtual rail. The yaw angles $ψ_{2}$ and $ψ_{3}$ are therefore the control variables.

From equation (7), we can know that the yaw angle of the first vehicle module is

ψ_{a c t 1} = \int \frac{V \cos β}{l_{f} + l_{r}} (\tan δ_{f} - \tan δ_{r}) d t

(22)

With the measured articulation angles, one also gets the remaining yaw angles $ψ_{a c t 2}$ and $ψ_{a c t 3}$ of the vehicle modules 2 and 3 according to

{\begin{cases} ψ_{a c t 2} = ψ_{1} - ψ_{a r t 1} \\ ψ_{a c t 3} = ψ_{2} - ψ_{a r t 2} \end{cases}

(23)

The matrix that defines the yaw angle for each vehicle module is

ψ_{a c t} = {(\begin{array}{l} ψ_{a c t 1} & ψ_{a c t 2} & ψ_{a c t 3} \end{array})}^{T}

(24)

It can be seen from equation (17) that the reference yaw angle of each vehicle module obtained by calculation is

ψ_{r e f} = q^{3 : 5} = {(\begin{array}{l} ψ_{r e f 1} & ψ_{r e f 2} & ψ_{r e f 3} \end{array})}^{T}

(25)

Finally, the control errors are simply defined by

ψ_{e r r} = ψ_{a c t} - ψ_{r e f} = {(\begin{array}{l} ψ_{e r r 1} & ψ_{e r r 2} & ψ_{e r r 3} \end{array})}^{T}

(26)

The PID control function between the steering angle of each axle and yaw angle error is established as follows:

φ (γ_{3}, γ_{4}, γ_{5}, γ_{6}) = f (ψ_{e r r}, K_{p}, K_{i}, K_{d})

(27)

The PID parameters need to been determined by closed-loop simulations. The reference path $ψ_{a c t 1}$ of the first module is generated from the yaw angle $ψ_{r e f 1}$ according to the equation (22). The steering angles $γ_{3}$ and $γ_{4}$ are used to reduce the control error of the second module, the steering angles $γ_{5}$ and $γ_{6}$ are used to reduce the control error of the third module.

4. Simulation

To verify the proposed multi-axle cooperation algorithm, a SRT model is built. This model is created with Matlab/Simulink 2018b and the shown in Figure 9. The SRT has 3 modules and 6 axles. Each vehicle module is the same, the wheelbase is 7 m, the articulation length is 1.8 m, the distance between 2 and 3 axes is 2.8 m, and the total length of the vehicle is 29.4 m. In the simulation, we tested the algorithm based on a circular curve with a radius of 15 m, and verified the effectiveness and robustness of the algorithm through different control algorithms.

The simulated operating condition follows a 15-m radius circle curve, with the vehicle speed linearly accelerating from 0 to 10 km/h on the straight segment and then maintaining a speed of 10 km/h within the curve. The input for the first axle angle is fixed at 14.5°. Based on this input condition, simulations were conducted for the algorithm under both open-loop control (without feedback control) and closed-loop control (with feedback control) conditions. Under closed-loop control conditions, the motion trajectories of the vehicle’s axles and articulations obtained are shown in Figure 10. It can be observed from Figure 10 that after traveling through a curved trajectory with a radius of 15 m, the motion trajectories of the vehicle’s axles and articulations coincide, indicating that this control algorithm enables SRT to achieve trajectory tracking functionality.

The simulation results for steering control under conditions of open-loop control and closed-loop control are presented in Figures 11 and 12, respectively. In Figure 11, the horizontal axis represents time, while the vertical axis displays relevant control data for a 15-m radius circular curve simulation. These include the steering angles of axle 1 to 6 and lateral errors of each axle relative to the first axle. As shown in Figure 11(a), the input angle for axle 1 is fixed at 14.5°, with axle 2 to 6 following corresponding changes based on a multi-axis cooperative steering control algorithm. These results align with expectations. From Figure 11(b), it can be observed that under open-loop control condition, lateral deviations for axle 2 to 6 are controlled within ±0.3496 m. Meanwhile, from Figure 12(b), it can be seen that under closed-loop control condition these deviations are controlled within ±0.1251 m. This demonstrates that after implementing feedback control into the multi-axle cooperative steering control algorithm significantly enhances SRT’s trajectory-following capabilities and effectively minish lateral deviations of follower modules, which providing substantial evidence supporting the effectiveness of feedback control.

5. Experiments

In this paper, the redundant electric drive steering system based on steering-by-wire is put forward for vehicle test on a three-module&six-axle SRT. The specific arrangement on the train is shown in Figure 13, and the static test of fault safety and the dynamic test of 90° right angle turn are carried out.

5.1. Redundant steering system layout

Figure 13 illustrates the symmetrical arrangement of redundant hot standby STCU and VCU in module one and module three. Each vehicle module consists of two axles, with AxCu_1 and AxCu_2 positioned separately, totaling 6 AxCus. The SWFE is symmetrically placed at both cabins. The three-module SRT includes two articulations used to connect the car body modules, each containing an ACU, amounting to a total of 2.

5.2. Fail-safe operation testing

During the test, the two channels were in normal mode after the vehicle was powered on and ignited, and the low-voltage power supply of each controller was disconnected one by one to simulate a single failure. A total of 8 different working conditions were tested, and the results were shown in Table 1.

Table 1.

The results of fail-safe operation testing.

No	Simulated condition	Channel 1 status	Channel 2 status	Steering system status
1	All controllers are powered on properly	Normal	Normal	FULL_OPERATION
2	The low-voltage power supply of STCU_A is disconnected	Error	Normal	FAIL_SAFE_OPERATION
3	The low-voltage power supply of STCU_A is disconnected	Normal	Error	FAIL_SAFE_OPERATION
4	The low-voltage power supply of channel 1 of AxCU1 – AxCU6 is disconnected	Error	Normal	FAIL_SAFE_OPERATION
5	The low-voltage power supply of channel 2 of AxCU1 – AxCU6 is disconnected	Normal	Error	FAIL_SAFE_OPERATION
6	The low-voltage power supply of SWFE_A is disconnected	Error	Normal	FAIL_SAFE_OPERATION
7	The low-voltage power supply of SWFE_B is disconnected	Normal	Error	FAIL_SAFE_OPERATION
8	The power supply of the dual-channel controller is disconnected simultaneously	Error	Error	ERROR

The steering system status are defined as follow:

FULL_OPERATION: Steering control system Full Operation without error.

FAIL_SAFE_OPERATION: Degraded operation due to lost availability redundancy OR control loop out of limits. Driver/VCU must bring the vehicle in to a fail-safe state by initiating a smooth braking to a safe vehicle speed and stop at a safe place.

ERROR: System failure OR control loop failure. Driver/VCU must bring the vehicle in to a fail-safe state by initiating an emergency brake to stop immediately.

The results presented in Table 1 demonstrate that each channel operates redundantly in hot standby mode and all channels are actively utilized. When both channels are in normal status, the steering system status is FULL_OERATION. When a single failure occurs in one of the two channels, the steering system status is FAIL_SAFE_OPERATION. When both channels are in error status, the steering system status changes to ERROR. Consequently, this design effectively resolves safety concerns associated with single failure within the steering network and mitigates potential risks arising from single-channel data acquisition, thereby significantly enhancing the reliability of the entire steering system.

5.3. 15-m radius quarter-circle curve testing

As shown in Figure 14, a quarter-circle with a radius of 15 m is delineated on the test field to serve as the designated track for experimental SRT runs. Reflective strips are affixed along the quarter-circle and its extended tangent lines to mark the trajectory. Ample distance (exceeding 100 m) is ensured at both ends of the circular arc, allowing for outer wheels of all axle to follow along the tangent line. Additionally, near points C, D, and E around the center of the turning radius, reflective tape is applied within a range of −2 m to 2 m from each center point (OC = OD = OE = 15 m). Scale lines are drawn along straight lines OC, OD, and OE to record movement trajectories when outer tires pass near points C, D, and E.

As shown in Table 2, the tests were performed at two speed conditions (5 km/h and 10 km/h). Based on the test results, it can be observed that the Axle4 and Axle6 display minimal errors, whereas the Axle3 and Axle5 display significant errors. The maximum error for each axle of SRT occurs during curve entry (measured at point D), while the minimum error is observed when driving in a steady state through the curve (measured at point C). This is because the path of the test condition is from the straight-line section to the curve section without excessive curves, the curvature radius is not continuous change, there is a mutation phenomenon, resulting in a certain difference between the curve fitted by the curve passed by the first axis and the actual path. Experimental results indicate that when SRT passes a 15-m radius quarter-circle curve, maximum error amounts to 0.35 m.

Table 2.

Experiment results.

The difference values between the passing distance for outside wheel of each axle and the Axle1 (m)
Velocity	Position	Axle1-2	Axle1-3	Axle1-4	Axle1-5	Axle1-6
5 km/h	D	−0.21	0.32	−0.09	0.35	−0.01
	C	0.01	0.02	−0.01	−0.04	0.03
	E	0.15	−0.09	0.11	−0.17	0.06
10 km/h	D	−0.20	0.26	−0.10	0.32	−0.01
	C	0.01	0.03	0.01	−0.03	−0.08
	E	0.27	−0.18	0.16	−0.26	0.08

6. Conclusion

This paper provides a detailed introduction to the multi-axle steering system based on redundant steering-by-wire, including the redundant steering-by-wire architecture and control algorithms. The hardware and software of the system have been successfully developed independently, and the on-vehicle test has been completed based on this. A 15-m radius circular curve simulation result shows that under closed-loop control condition these deviations are controlled within ± 0.1251 m. Experimental results indicate that when SRT passes a 15-m radius quarter-circle curve, maximum error amounts to 0.35 m. The steering-by-wire system uses the electrical steering motor driven, which has obvious technical advantages compared with the hydraulic actuator:

(1) The mechanical connection between the steering wheel and the steering actuation module is eliminated, and the steering is completely driven by the motor, freeing the traditional steering system from various limitations.

(2) The system can quickly and accurately send and execute the steering control command, thus minimizing the wear of the tires on each axle.

(3) The use of motor drive has the advantage of maintenance-free compared with the hydraulic actuator.

(4) The application of redundant steering system network architecture ensures the reliability and safety of the system.

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: this research is financially supported by Hunan Innovation Platform and Talent Plan - Huxiang Youth Talent Project 2023 (Grant No. 2023RC3244).

ORCID iD

Hongda Liu

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