Open AccessArticle

Mass Estimation-Based Path Tracking Control for Autonomous Commercial Vehicles

Zhihong Wang

^1,2,3,

Jiefeng Zhong

^1,2,3,

Jie Hu

^1,2,3,*,

Zhiling Zhang

^1,2,3 and

Wenlong Zhao

^1,2,3

Hubei Key Laboratory of Modern Auto Parts Technology, Wuhan University of Technology, Wuhan 430070, China

Auto Parts Technology Hubei Collaborative Innovation Center, Wuhan University of Technology, Wuhan 430070, China

Hubei Technology Research Center of New Energy and Intelligent Connected Vehicle Engineering, Wuhan 430070, China

Author to whom correspondence should be addressed.

Appl. Sci. 2025, 15(2), 953; https://doi.org/10.3390/app15020953

Submission received: 3 January 2025 / Revised: 17 January 2025 / Accepted: 17 January 2025 / Published: 19 January 2025

(This article belongs to the Topic Vehicle Dynamics and Control, 2nd Edition)

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Browse Figures

Figure 1
Vehicle dynamics model. "> Figure 2
Path tracking error schematic diagram. "> Figure 3
Path-tracking controller architecture. "> Figure 4
Vehicle motion force diagram. "> Figure 5
Vehicle roll dynamics model. "> Figure 6
Mass estimation simulation result graph; (a) Unloaded mass estimation; (b) Half-load mass estimation; (c) Full-load mass estimation. "> Figure 7
Comparison of controller simulation results: (a) Path tracking effect comparison; (b) Lateral error; (c) Heading error; and (d) Steering angle. "> Figure 8
EV18 experimental platform. "> Figure 9
Vehicle software architecture diagram. "> Figure 10
Comparative diagram of algorithms: (a) Path tracking effect comparison; (b) Lateral error; (c) Heading error; and (d) Steering angle. "> Figure 10 Cont.
Comparative diagram of algorithms: (a) Path tracking effect comparison; (b) Lateral error; (c) Heading error; and (d) Steering angle. "> Figure 11
Actual vehicle mass estimation diagram. ">

Versions Notes

Abstract

This paper addresses the significant variations in model parameters observed in autonomous commercial vehicles in comparison to passenger cars, with a disparity noted largely due to changes in load. Additionally, it tackles the issue of path tracking inaccuracy caused by external factors such as delays in steering system execution. The proposed solution is a hierarchical control method, grounded in mass estimation and model predictive control(MPC). Initially, to counter the variation in model parameters, a mass estimator is developed. This estimator utilizes the recursive least squares method with a forgetting factor, coupled with M-estimation, thereby enhancing the robustness of the estimation and achieving model correction. Subsequently, an upper-level MPC controller is constructed based on the error model, thereby augmenting the precision of tracking control. To address the delay in the steering system execution common in autonomous commercial vehicles, a lower-level steering angle compensator is designed to expedite the response speed of the execution. The feasibility of the vehicle’s front wheel angle is constrained via the rollover index, thereby enhancing vehicle stability during operation. The efficacy of the proposed control strategy is demonstrated with joint simulations using TruckSim/Simulink and real vehicle tests. The results indicate that this strategy can effectively manage the model mismatch caused by load changes in commercial vehicles and the delay in steering system execution, thereby exhibiting commendable tracking accuracy, adaptability, and driving stability.

Keywords:

autonomous commercial vehicles; mass estimates; steering compensation controller; model predictive control; lateral control

1. Introduction

Over the past few years, the development of autonomous driving technology has garnered significant societal attention. In scenarios like urban logistics, docks, as well as mines, it has begun to play a real role, with autonomous commercial vehicles playing an important role [1]. Autonomous driving technology stands as one of the foremost promising research areas currently [2]. Motion control is a key technology in autonomous driving, mainly divided into decoupled lateral and longitudinal control, as well as coupled lateral and longitudinal control, primarily aimed at achieving high-precision, stable, and comfortable vehicle control.

Currently, most control strategies are designed based on models with certain accuracy, which can be mainly divided into kinematic models and dynamic models. Kinematic models primarily focus on the geometric relationships of vehicles, describing the variation rules of vehicle position and heading angle over time. Algorithms based on kinematic models, such as pure pursuit algorithms and Stanley algorithms, have good application effects in simple scenarios, but they show obvious adaptability deficiencies when dealing with roads with large curvature, making it difficult to meet the complex and variable control needs in practical applications [3]. In contrast, dynamic models can more comprehensively describe the motion characteristics of vehicles. Algorithms based on dynamic models, such as linear quadratic regulator (LQR) [4] control and model predictive control [5], theoretically provide more precise control effects. However, in actual vehicle systems, due to significant changes in vehicle parameters such as mass, delays in the vehicle actuation system, and sensor noise, challenges arise in vehicle control. Yet, the aforementioned control methods have not proposed effective solutions to these issues. To improve the precision of vehicle control, the impacts brought by the above factors need to be considered. Pan [6] proposed a feedback pure pursuit control method that takes into account the estimated vehicle speed and load to adjust the preview distance. It uses lateral error as a feedback variable for compensation, improving the traditional pure pursuit control method. Amer [7] proposed an adaptive Stanley control method, combined with fuzzy control, to enhance the control method’s responsiveness to model deviations and external disturbances. However, it has poor path adaptability, making it unsuitable for high-speed scenarios that require rapid response. Hu [8] optimized the controller using a path tracking error model. They used fuzzy logic to dynamically adjust the LQR weights, addressing the adaptability challenge of fixed weights at different vehicle speeds. However, if the model is inaccurate, the performance of the controller will significantly decrease, potentially leading to system instability or poor response. It also cannot effectively handle multiple constraint conditions. MPC can deal with optimization problems with multiple constraints, but it relies on model accuracy, is sensitive to external noise, and is prone to model mismatch issues. This is particularly prominent under varying load conditions in commercial autonomous vehicles. Peng [9] proposed robust MPC control in a finite time domain. To address the model mismatch caused by parameter uncertainty, external disturbances, and model time variation, they constructed a linear parameter-varying model and solved it using linear matrix inequalities. This approach overcomes the limitations of infinite time domain MPC control, but increases the difficulty of modeling and solving. Dai [10] proposed an adaptive preview MPC control that adaptively previews the future path curve, improving computational efficiency. Wang [11] proposed a MPC control based on fuzzy adaptive weights that utilizes a fuzzy adaptive control algorithm to dynamically adjust the weights of the cost function within the MPC framework. However, neither paper discussed the impact of parameter changes and vehicle execution delay on control effects.

The vehicle’s motion status will influence the control of autonomous driving. Commercial vehicles have a large load and complex driving conditions, and the vehicle’s mass varies with differing loads. McIntyre [12] introduced a two-phase Lyapunov-based approach to estimate the mass of heavy vehicles, utilizing standard signals accessible within the controller area network for vehicles. An adaptive least squares method is used to estimate the vehicle’s mass. Zhao [13] established suitable operating conditions and devised a confidence factor for mass estimation utilizing fuzzy logic rules and subsequently used recursive least squares for online mass estimation. However, the existing methods fail to consider uncertain factors such as sensor noise interference and error in actual vehicle scenarios. It is essential to improve the robustness of the estimation algorithm against possible disturbances, abnormal errors, and potential faults that may be encountered by the sensor during the driving process of commercial vehicles.

The load variation of vehicles is significant, and the mass parameter has a deeper impact on the control effect. Traditional control algorithms usually set the vehicle mass as a constant, which cannot meet the control requirements of corresponding working conditions well. This paper introduces a model predictive control approach equipped with a mass estimator, referred to as MFFRSL + MPC. First, considering the sensor noise and unstable parameter changes in real vehicle scenarios, the recursive least squares method with a forgetting factor is introduced to dynamically adjust the weight of historical data, reflecting the current state and environmental changes. The joint M-estimation further reduces the influence of sensor noise on the estimation results and enhances the robustness of the algorithm, achieving a robust estimation of vehicle mass. Second, based on the incremental control error model of the steering angle, a model predictive controller is established to improve the control accuracy. A steering angle compensator is established for the execution delay of the steering system. Based on the rollover index, stability constraints are applied to the front wheel steering angle. Ultimately, the efficacy of the introduced control approach is confirmed using collaborative simulations in TruckSim/Simulink under dual lane-changing scenarios and real-world vehicle trials on straight roads integrated with roundabout conditions.

2. Vehicle Model

2.1. Lateral Dynamics Modelling

Several assumptions are put forward as stated in [14]. Specifically, it is presumed that the combination of the car body and suspension constitutes a stiff system, and the vertical motion aspect of the vehicle is disregarded. The present study narrows its focus solely to the lateral movement and yaw motion of the vehicle for the purpose of analysis. Subsequently, a vehicle dynamics model is set up, which is illustrated in Figure 1.

In the context of planar motion, the dynamics model of the vehicle can be described as follows:

{\begin{matrix} m \ddot{y} = F_{y r} + F_{y f} c o s δ - m V_{x} \dot{φ} \\ I_{z} \ddot{φ} = l_{f} F_{y f} c o s δ - l_{r} F_{y r} \end{matrix}

(1)

In the equation,

\dot{φ}

represents the vehicle’s yaw rate,

I_{z}

denotes the vehicle’s vertical rotational inertia, while

l_{f}

and

l_{r}

represent the distances from the vehicle’s center of gravity to the front axle and rear axle, respectively.

The lateral force exerted by a tire is directly associated with the tire’s slip angle. This relationship is of utmost importance in understanding the vehicle’s handling and stability characteristics, as the lateral force significantly influences the vehicle’s ability to maintain its intended path during maneuvers. Assuming a small-angle approximation, the formula for the tire’s lateral force can be expressed as follows:

{\begin{matrix} F_{y f} = 2 C_{f} (δ - \frac{V_{y} + l_{f} \dot{φ}}{V_{x}}) \\ F_{y r} = 2 C_{r} (\frac{V_{y} - l_{r} \dot{φ}}{V_{x}}) \end{matrix}

(2)

In the equation,

C_{f}

represents the front wheel camber stiffness, and

C_{r}

represents the rear wheel camber stiffness.

2.2. Kinematic Error Model

The vehicle dynamics model exhibits certain limitations when applied to controller design. Consequently, an error model is often incorporated to convert the vehicle dynamics model into a dynamic error model [15], as illustrated in Figure 2.

In the diagram,

e_{d}

indicates the lateral error, and

e_{φ}

denotes the heading error.

R

corresponds to the radius of curvature.

The control objective of the path tracking model is to make the lateral error

e_{d}

and the heading angle error

e_{φ}

converge to zero. Considering the small-angle assumption, combining the path tracking model with the dynamic model, one can further obtain the dynamic error model. This leads to the state–space equation of the dynamic error model.

\dot{x} = A x + B u + C ς

(3)

where

x = {[\begin{matrix} e_{d} & {\dot{e}}_{d} & e_{φ} & {\dot{e}}_{φ} \end{matrix}]}^{T}

;

u = δ

;

ς

is the road curvature.

A = [\begin{matrix} 0 & 1 & 0 & 0 \\ 0 & \frac{C_{f} + C_{r}}{m V_{x}} & - \frac{C_{f} + C_{r}}{m} & \frac{l_{f} C_{f} - l_{r} C_{r}}{m V_{x}} \\ 0 & 0 & 0 & 1 \\ 0 & \frac{l_{f} C_{f} - l_{r} C_{r}}{m V_{x}} & - \frac{l_{f} C_{f} - l_{r} C_{r}}{I_{z}} & \frac{l_{f}^{2} C_{f} + l_{r}^{2} C_{r}}{I_{z} v_{x}} \end{matrix}]; B = [\begin{matrix} 0 \\ - \frac{C_{f}}{m} \\ 0 \\ - \frac{l_{f} C_{f}}{I_{z}} \end{matrix}]; C = [\begin{matrix} 0 \\ \frac{l_{f} C_{f} - l_{r} C_{r}}{m} - V_{x}^{2} \\ 0 \\ \frac{l_{f}^{2} C_{f} + l_{r}^{2} C_{r}}{I_{z}} \end{matrix}]

(4)

2.3. Error Model with Incremental Control

The bilinear discretization method is used, defining the discrete time as Ts. Consequently, the continuous system (3) is converted into a discrete system.

x (k + 1) = A_{d} x (k) + B_{d} u (k) + C_{d} ς

(5)

{\begin{matrix} A_{d} = {((I - A T_{s}) / 2)}^{- 1} ((I + A T_{s}) / 2) \\ B_{d} = B T_{s} \\ C_{d} = C T_{s} \end{matrix}

(6)

where

I

is a 4 × 4 identity matrix.

The selection of the front wheel angle increment as a control variable can effectively mitigate abrupt changes in the angle. This is particularly beneficial when there are significant variations in steering speed, as it can enhance stability during the steering process. By integrating the front wheel angle into the state variables, we can derive the state variables for the control equation.

ξ (k + 1) = [\begin{matrix} x (k + 1) \\ u (k) \end{matrix}]

(7)

ξ (k + 1) = A_{k} ξ (k) + B_{k} Δ u (k) + C_{k} ς

(8)

In the equation

A_{k} = [\begin{matrix} A_{d} & B_{d} \\ O_{1 \times 4} & I_{1 \times 1} \end{matrix}]

;

B_{k} = [\begin{matrix} B_{d} \\ I_{1 \times 1} \end{matrix}]

;

C_{k} = [\begin{matrix} C_{d} \\ O_{1 \times 4} \end{matrix}]

Δ u

denotes the incremental angle of the front wheel at time k,

O

signifies a zero matrix, and

I

represents an identity matrix.

In the instance where the road curvature equals zero, the curvature term becomes negligible. Consequently, we can derive the following linear control system:

ξ (k + 1) = A_{k} ξ (k) + B_{k} Δ u (k)

(9)

3. Path-Tracking Controller

The path-tracking controller architecture is shown in Figure 3. The error calculation module determines error information based on road data and planned trajectories. The mass estimator derives the vehicle’s mass from controlled state parameters and provides this feedback to the controller. The MPC controller calculates the front wheel steering angle, and the steering angle compensation controller computes the compensation amount for the front wheel steering angle.

3.1. Robust Mass Estimation for Commercial Vehicles

3.1.1. Vehicle Longitudinal Dynamics Model

Characterizing the vehicle as a rigid body establishes the foundation for formulating its longitudinal dynamics model. By applying the principles of Newton’s second law, this model (Figure 4) effectively encapsulates the vehicle’s motion characteristics across a range of operating conditions [16].

During the vehicle’s movement, it is subject to various forces, which can be represented by the following formula:

m {\dot{V}}_{x} = F_{T} - F_{j} - F_{w} - F_{i}

(10)

Considering that the slope of the road is typically small, the following assumptions can be made: when the slope angle θ is small, the approximations

s i n α \approx t a n α \approx α

and

c o s α \approx 1

may be applied. Based on these assumptions, Equation (1) can be simplified to the following [17]:

{\dot{V}}_{x} = \frac{T_{t q} i_{0} i_{g} η}{m r} - g f - g α - \frac{C_{d} A ρ V_{x}^{2}}{2 m}

(11)

3.1.2. Robust Mass Estimation Algorithm

By correcting the estimated results with real-time measurement data, the longitudinal dynamics model of the vehicle can be transformed into the form of least squares. Specifically, an approximation method is employed, utilizing the estimate from time k − 1 to predict the output at time k, while assessing the error of the output in the process.

Within the framework of the vehicle’s longitudinal dynamics model, the application of least squares in the mass estimation process can be articulated as follows:

{\begin{matrix} y = (T_{t q} i_{0} i_{g} η) / r - 0.5 C_{d} A ρ V_{x}^{2} \\ φ = a_{x} + g f \\ θ = m \end{matrix}

(12)

where

y

is the output quantity,

φ

is the measurable data,

θ

is the parameter to be observed, and

a_{x}

represents the value measured by the longitudinal acceleration sensor.

a_{x} = {\dot{V}}_{x} + g α

(13)

Considering that the least squares method may lead to data saturation in the data processing stage, to mitigate the adverse effects of data saturation and improve estimation accuracy, a forgetting factor is utilized. To diminish the impact of sensor noise on the estimation results and bolster the robustness of the algorithm, M-estimation is combined with the least squares method that includes a forgetting factor. M-estimation suppresses the influence of outliers and noise by weighting the residuals, thus improving the stability and accuracy of parameter estimation [18].

The objective function can be defined as follows:

f (θ) = ρ (γ)

(14)

γ = (y (κ) - φ^{T} (κ) θ (κ - 1))

(15)

Here, the loss function is defined as follows:

ρ (γ) = {\begin{matrix} γ^{2} / 2 & | γ | \leq h \\ h | γ | - γ^{2} / 2 & | γ | > h \end{matrix}

(16)

where

h

is an adjustable parameter. Differentiating the performance function gives the following:

\frac{\partial f (θ)}{\partial θ} = \frac{d ρ (γ)}{d γ} \cdot \frac{\partial γ}{\partial θ} = 0

(17)

ψ = \frac{d ρ (γ)}{d γ}

(18)

w = \frac{ψ}{γ}

(19)

where

ψ

is the influence factor, and

w

is the weight value.

w = {\begin{array}{l} 1 & | γ | < h \\ \frac{h}{| γ |} & | γ | > h \end{array}

(20)

The application of the least squares method considering the forgetting factor and joint M-estimation in the mass estimation process can be articulated as follows [19]:

(\begin{matrix} \hat{θ} (κ) = \hat{θ} (κ - 1) + K (κ) w [y (κ) - φ^{T} (κ) \hat{θ} (κ - 1)] \\ K (κ) = \frac{P (κ - 1) φ (κ)}{λ + φ^{T} (κ) P (κ - 1) φ (κ)} \\ P (κ) = \frac{1}{λ} [I - K (κ) w φ^{T} (κ)] P (κ - 1) \end{matrix}

(21)

K is the gain matrix, P is the covariance matrix, and

λ

is the forgetting factor.

3.2. MPC Controller

The output of the control system in the predicted time horizon can be expressed in the following form:

{\begin{cases} ξ (k + 1) = Φ_{k} ξ (k) + Θ_{k} Δ U (k) \\ Y (k) = C ξ (k) \end{cases}

(22)

In the formula,

N_{p}

is the prediction domain, and

N_{c}

is the control domain.

The definition of the objective function for the MPC controller can be stated as follows:

J = \min (\sum_{k = 0}^{N_{p}} ξ^{T} (k) Q ξ (k) + \sum_{k = 0}^{N_{c} - 1} Δ u^{T} (k) R Δ u)

(23)

S . T . {\begin{array}{l} ξ_{0} = ξ (0) \\ ξ (k) \in ξ_{b} \\ Δ u \in Δ u_{b} \end{array}

(24)

Q represents the output weight matrix,

Q = diag [Q_{1}, Q_{2}]

Q_{1} = diag [q_{1}, q_{2}, q_{3}, q_{4}]

Q_{2} = diag [q_{5}]

. The greater the weight of

Q_{1}

is, the higher the accuracy of path-tracking control; the greater the weight of

Q_{2}

is, the smoother the change in the front wheel angle of the vehicle. The matrix

R

represents the control weight with

R = diag [r_{1}]

. A larger value of

R

results in a more significant restriction on the increment of the front wheel angle, thereby increasing the suppression of abrupt changes in this angle.

ξ_{0}

is the initial state of the vehicle, and

ξ_{b}

is the state quantity constraint matrix.

Δ u_{b}

is the control quantity constraint matrix, which is determined by the maximum steering wheel rotation speed of the vehicle and the sampling frequency.

The objective function is transformed into the standard quadratic programming form for solving.

J = {[Φ_{k} ξ (k) + Θ_{k} Δ U (k) - Y_{r e f}]}^{T} + \tilde{Q} [Φ_{k} ξ (k) + Θ_{k} Δ U (k) - Y_{r e f}] + Δ U^{T} \tilde{R} Δ U

(25)

Obtain the standard quadratic programming as follows:

J = \frac{1}{2} U^{T} H U + G^{T} U + E^{T} \tilde{Q} E

(26)

\frac{1}{2} H = Θ_{k} \tilde{Q} Θ_{k} + \tilde{R}

(27)

G^{T} = 2 (E^{T} \tilde{Q} Θ_{k} - Y_{r e f} \tilde{Q} Θ_{k})

(28)

The optimal front wheel steering angle control sequence that has been solved takes the first control quantity of the optimal control sequence at time

k

to participate in subsequent calculations.

δ (k) = u (k - 1) + Δ u (k)

(29)

where

δ (k)

is the front wheel corner.

3.3. Steer Compensation Controller

The goal is to enhance steering precision and mitigate the effects of system execution delay; thus, compensation is made for the steering angle. The steering system in commercial vehicles exhibits an inherent execution delay. To enhance the precision of the steering and mitigate the effects of this delay, compensation is made for the steering angle. The difference between the sensor-measured steering angle and the steering angle calculated by the MPC controller is defined as the system’s execution delay

e_{δ}

. The specific formula for steering angle compensation is provided as follows:

u_{f} = K_{a} e_{δ} + K_{b} (e {(k)}_{δ} - e {(k - 1)}_{δ})

(30)

where

K_{a}

is the proportional term, which is used to reduce deviations.

K_{b}

is a differential term used to improve lag.

For the front wheel angle at time

k

, it is composed of three parts:

u (k - 1)

is the front wheel angle from the previous moment,

Δ u (k)

denotes the optimal angle adjustment increment at the present moment, and

u_{f}

is the compensation angle.

δ_{s} = u (k - 1) + Δ u (k) + u_{f}

(31)

The ultimate angle of the vehicle’s steering wheel input is expressed as follows:

δ_{f} = i_{s} * δ_{s}

(32)

In the equation, the variable

i_{s}

denotes the steering transmission ratio.

3.4. Stability Constraints

In order to improve the lateral stability of commercial vehicles, the maximum steering angle is constrained by load and speed. The vehicle roll dynamics model is adopted, as shown in Figure 5. When the vehicle is in a steady-state turn, the risk of vehicle rollover can be represented by the rollover index

R_{L T}

The roll index

R_{L T}

is expressed as follows [20]:

R_{L T} = \frac{2 m_{s}}{m g l_{w}} [H_{s} (a_{y} - h_{s} \ddot{β}) + g h_{s} β]

(33)

In the formula,

H_{s}

is the distance from the sprung mass center to the ground,

h_{s}

is the roll arm length,

L_{w}

is the wheelbase,

β

is the roll angle, and

m_{s}

is the load.

When the rollover index is

R_{L T} = 0

, the vehicle has no risk of rollover. When

R_{L T} = \pm 1

, the vehicle is in a turn. The outer wheels bear all the vertical load, while the inner wheels interact with the ground. The wheels are at the critical stage of rollover. In order to ensure the safe driving of the vehicle, the rollover index is set to

| R_{L T} | = 0.7

; thus, the stability constraint of the vehicle is expressed as follows:

δ_{L} = \arctan (\frac{R_{L T} m g (l_{r} + l_{f}) + 2 m_{s} h_{s} (l_{r} + l_{f}) (\ddot{β} L_{w} - g β)}{2 m_{s} V_{x} H_{s} L_{w}})

(34)

The roll angle is a small angle during steady-state cornering. Therefore, the maximum front wheel steering angle is mainly constrained by the vehicle’s load and speed. When it exceeds the safety limit, the vehicle stability constraint is more stringent than the actuator physical constraint, causing the feasible region of the front wheel steering angle to shrink.

4. Algorithm Verification

4.1. Simulation Verification

4.1.1. Mass Estimator Simulation Analysis

To verify the accuracy of the quality estimation algorithm, a co-simulation was executed utilizing Simulink and TruckSim. A forgetting factor

λ = 0.98

was incorporated to mitigate the impact of historical data, with an adjustable parameter

h = 150

. The vehicle employed a constant acceleration of 3 m/s² and functioned on a level road. The vehicle used for testing was a light commercial model with a full load mass of 1000 kg. Tests were performed under three distinct conditions: unloaded (0 kg), half loaded (500 kg), and fully loaded (1000 kg). The specifications of the vehicle are presented in Table 1.

It can be observed from Figure 6 that at the beginning of the experiment, due to the large input torque and given that the acceleration has not yet reached the expected level, the experimental value deviates significantly from the real value. This phenomenon reflects the lag of the system response and the instability of initial conditions. However, as the input parameters gradually stabilize, combined with the iterative optimization of the algorithm and the introduction of the forgetting factor, the estimated value begins to slowly and gradually approach the real value of the mass at about 6 s. At 15 s, the estimation process is completed with convergence. Specifically, the vehicle mass, when empty, ultimately converges to 3042 kg, possessing an error margin of 0.39%. In the half-load scenario, the mass converges to 3554 kg, with an error of 0.67%. Under full-load conditions, the mass converges to 4067 kg, exhibiting an error of 0.91%. These results indicate that the designed mass estimator can achieve good estimation accuracy under different conditions. According to the general evaluation criteria, when the estimation error is within 5%, the estimation result can be considered accurate. Therefore, drawing from the preceding analysis, we determine that this mass estimator is capable of efficiently conducting estimations in practical applications and fulfills the genuine requirements of engineering.

4.1.2. Controller Simulation Analysis

Through the joint simulation of MATLAB/Simulink and TruckSim, the effectiveness and superiority of the introduced path tracking controller are confirmed. The simulation vehicle model selected is the large European van, with basic vehicle parameters as shown in Table 2.

The standard double-lane change test condition was selected for testing, where the road was set as a wet and slippery low-adhesion surface. In the actual control program, all controller weight parameters were the optimal parameters from simulation. To evaluate the controller’s performance under rigorous conditions, the vehicle’s speed was set to 50 km per hour within the simulation environment, and the corresponding results are illustrated in Figure 6.

Figure 7a shows the tracking effects of three different controllers on vehicle control under double-lane change conditions. From the magnified view, it can be seen that the model predictive control with mass estimator tracks most accurately and stably follows the reference curve. Both the MPC controller and LQR controller exhibit varying degrees of deviation during steering. Figure 7b represents the lateral errors of the three controllers. The MFFRSL + MPC controller has the smallest lateral error, with a maximum lateral error of only −0.03 m. In contrast, the MPC controller’s lateral error fluctuates significantly during turning, with a maximum lateral error reaching −0.07 m, which is due to inaccurate mass parameters leading to model mismatch. The LQR controller has a maximum lateral error of −0.08 and also exhibits steady-state error at the end. Figure 7c presents the heading errors of the three controllers. The MFFRSL + MPC controller performs best in terms of heading error, with the largest heading error occurring at the first peak, which is 0.04 rad. The LQR controller performs the worst, with a maximum heading error of −0.07 rad. The MPC controller’s maximum heading error is 0.05 rad. Figure 7d illustrates the performance of the steering wheel angle for the three controllers. The MFFRSL + MPC controller maintains a stable steering wheel angle with a quick response. This is because the MFFRSL + MPC controller model takes into account the disturbances caused by load changes and can also achieve feedback correction within the prediction horizon. In contrast, the MPC controller exhibits unstable steering wheel jitter, while the LQR controller responds slowly.

4.2. Real Vehicle Verification

In order to ascertain the practical efficacy of the algorithm, a real-world vehicle test was executed utilizing the EV18 platform. The vehicle under test is outfitted with a steer-by-wire chassis and a domain controller and amalgamates a variety of sensors including LiDAR, millimeter-wave radar, surround-view cameras, and an integrated inertial navigation system, as depicted in Figure 8.

The software architecture of the vehicle is illustrated in Figure 9. The microcontroller unit (MCU) low-level driver interacts with the vehicle system via controller area network (CAN) signals, while the Simulink algorithm program communicates with the MCU through the user datagram protocol (UDP). The lateral control interface is represented by the steering wheel angle, and the longitudinal control interface comprises drive motor torque and braking deceleration, both operating at a signal transmission frequency of 100 Hz. Testing was conducted on a public road in the economic development zone of Wuhan City, with the desired path defined as a sequence of discrete trajectory points post filtering and smoothing. The actual vehicle parameters are detailed in Table 2.

4.2.1. Controller Effect Comparison Experiment

To test the control effects of LQR, MPC, and MFFRLS + MPC controllers on real vehicles with load changes and execution delay issues, a complex circular path was chosen, with a test speed of 20 km/h. The weight parameters of each controller are the optimal parameters for real vehicles, and the test results are shown in Figure 10.

Figure 10a represents the tracking effect of autonomous commercial vehicles with three different controllers. The tested road is a public road, consisting of straight roads and roundabouts. Under the control of the controllers, the vehicle first passes through the straight road and then enters the roundabout, exiting at the third intersection of the roundabout. From the figure, it can be seen that at the positions where the straight road enters and exits the roundabout, the MFFRSL + MPC controller has the best tracking effect, with the least deviation from the reference path. The LQR controller has the largest deviation. Figure 10b shows the lateral error of the three controllers. The lateral error of the MFFRSL + MPC controller is better compared to the other two controllers, with the maximum lateral error occurring at the entrance to the roundabout being 0.13 m. The MPC controller’s maximum lateral error is 0.15 m, while the LQR controller performs the worst, with a maximum lateral error reaching 0.23 m. Figure 10c represents the heading error of the three controllers. The heading error of the MFFRSL + MPC controller is relatively better, with the maximum heading error being 0.085 rad. The initial lateral and heading errors of the vehicle are not zero because the starting position of the vehicle does not perfectly align with the reference path.

Figure 10d describes the output angles of the three controllers: LQR, MPC, and MFFRSL + MPC. The trends in angle changes during the control process in real vehicle experiments are basically the same, with differences in smoothness. MFFRSL + MPC has better smoothness than LQR and MPC. MFFRSL + MPC can correct model deviations and predictive functions and can converge errors through minor angle adjustments, avoiding overshoot and improving smoothness.

4.2.2. Mass Estimator Experiment

To validate the stability and accuracy of the proposed mass estimator in actual vehicles, a public straight road was chosen. The vehicle traveled at a constant speed of 20 km/h, carrying a full load of 1000 kg. Signals from sensors were collected through the vehicle’s CAN bus, including the motor’s torque, acceleration, and speed. Test results are shown in Figure 11.

The figure displays the testing effects of RSL and MFFRSL. It can be observed that the estimation result of RSL did not converge and exhibited jitters under the influence of sensor noise. It also presented a significant steady-state error. In contrast, the curve of MFFRSL smoothly and stably converged to 4071 kg. It demonstrated strong robustness against noise interference, capable of stable operation in complex environments with an error of 1.01%. Generally, errors below 5% are considered accurate, indicating that the designed mass estimator meets the demands of actual working conditions.

5. Conclusions

This paper addresses the control accuracy issue caused by the load changes in autonomous commercial vehicles and proposes a MPC equipped with a mass estimator. By incorporating a forgetting factor into the recursive least squares method, the influence of historical data on estimation is reduced. In conjunction with M-estimation, the robustness of the estimation is enhanced, suppressing the interference from sensor noise. A model predictive controller is established based on the incremental steering angle dynamic error model, improving control accuracy and increasing the smoothness of steering. To counteract the execution delay of the steering system, a steering angle compensator is introduced to enhance the response speed of steering. In real vehicle experiments, the MFFRSL + MPC tracking performance is the best, closely following the reference path, with smooth and quick steering responses, exhibiting good robustness. The maximum lateral error is 0.13 m, and the heading error is at most 0.085 rad, both of which are less than those of the MPC and LQR controllers. Both simulation and real vehicle test results indicate that this path tracking control method, while ensuring higher control accuracy, possesses better robustness and adaptability compared to LQR and MPC control methods.

This paper considers the design of a path tracking controller for autonomous commercial vehicles, taking into account changes in vehicle load and execution delay. Real-vehicle tests are conducted in low-speed scenarios for light commercial vehicles. Subsequently, tests will be carried out under high-speed conditions and for heavy commercial vehicles to further expand the research, aiming to meet the control needs of more vehicle types and scenarios.

Author Contributions

Writing—original draft preparation, Z.W. and J.Z.; writing—review and editing, J.Z., Z.Z. and W.Z.; supervision, J.H.; project administration, J.H.; visualization, J.Z. and Z.Z.; funding acquisition, J.H. and W.Z. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by Major Research Projects of Hubei Province (grant number (JD)2023BAA017).

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The data will be made available by the authors on request.

Conflicts of Interest

The authors declare no conflicts of interest.

References

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Figure 1. Vehicle dynamics model.

Figure 2. Path tracking error schematic diagram.

Figure 3. Path-tracking controller architecture.

Figure 4. Vehicle motion force diagram.

Figure 5. Vehicle roll dynamics model.

Figure 6. Mass estimation simulation result graph; (a) Unloaded mass estimation; (b) Half-load mass estimation; (c) Full-load mass estimation.

Figure 7. Comparison of controller simulation results: (a) Path tracking effect comparison; (b) Lateral error; (c) Heading error; and (d) Steering angle.

Figure 8. EV18 experimental platform.

Figure 9. Vehicle software architecture diagram.

Figure 10. Comparative diagram of algorithms: (a) Path tracking effect comparison; (b) Lateral error; (c) Heading error; and (d) Steering angle.

Figure 11. Actual vehicle mass estimation diagram.

Table 1. Vehicle parameters.

Parameter	Value	Unit
Vehicle mass (unladen) $m$	3030	kg
Vehicle load	0/500/1000	kg
Gravitational acceleration $g$	9.8	m/s²
Air density $ρ$	1.206	Kg/m³
Windward area $A$	6	m²
Air resistance coefficient $C_{d}$	7
Transmission efficiency $η$	0.9
Transmission gear ratio $i_{g}$	9.5

Table 2. Main kinematic parameters of the vehicle.

Parameter	Value	Unit
Vehicle length	5.995	m
Vehicle width	2.18	m
$Moment of inertia I_{z}$	4245	kg·m²
$Distance from center of mass to front axle l_{f}$	1.20	m
$Distance from center of mass to rear axle l_{r}$	2.108	m
$Front wheel camber stiffness C_{f}$	170,000	N/rad
$Rear wheel camber stiffness C_{r}$	170,000	N/rad
$Discrete time T_{s}$	0.01	s
$Steering ratio i$	22
$Prediction domain N_{p}$	40
$Control domain N_{c}$	40
State vector weight matrix	Diag[3,0,40,0,10]
Control weight	8

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Wang, Z.; Zhong, J.; Hu, J.; Zhang, Z.; Zhao, W. Mass Estimation-Based Path Tracking Control for Autonomous Commercial Vehicles. Appl. Sci. 2025, 15, 953. https://doi.org/10.3390/app15020953

AMA Style

Wang Z, Zhong J, Hu J, Zhang Z, Zhao W. Mass Estimation-Based Path Tracking Control for Autonomous Commercial Vehicles. Applied Sciences. 2025; 15(2):953. https://doi.org/10.3390/app15020953

Chicago/Turabian Style

Wang, Zhihong, Jiefeng Zhong, Jie Hu, Zhiling Zhang, and Wenlong Zhao. 2025. "Mass Estimation-Based Path Tracking Control for Autonomous Commercial Vehicles" Applied Sciences 15, no. 2: 953. https://doi.org/10.3390/app15020953

APA Style

Wang, Z., Zhong, J., Hu, J., Zhang, Z., & Zhao, W. (2025). Mass Estimation-Based Path Tracking Control for Autonomous Commercial Vehicles. Applied Sciences, 15(2), 953. https://doi.org/10.3390/app15020953

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Mass Estimation-Based Path Tracking Control for Autonomous Commercial Vehicles

Abstract

1. Introduction

2. Vehicle Model

2.1. Lateral Dynamics Modelling

2.2. Kinematic Error Model

2.3. Error Model with Incremental Control

3. Path-Tracking Controller

3.1. Robust Mass Estimation for Commercial Vehicles

3.1.1. Vehicle Longitudinal Dynamics Model

3.1.2. Robust Mass Estimation Algorithm

3.2. MPC Controller

3.3. Steer Compensation Controller

3.4. Stability Constraints

4. Algorithm Verification

4.1. Simulation Verification

4.1.1. Mass Estimator Simulation Analysis

4.1.2. Controller Simulation Analysis

4.2. Real Vehicle Verification

4.2.1. Controller Effect Comparison Experiment

4.2.2. Mass Estimator Experiment

5. Conclusions

Author Contributions

Funding

Institutional Review Board Statement

Informed Consent Statement

Data Availability Statement

Conflicts of Interest

References

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