Path Tracking for Electric Mining Articulated Vehicles Based on Nonlinear Compensated Multiple Reference Points Linear MPC

<jats:p>The path tracking control of electric mining articulated vehicles (EMAVs), critical equipment commonly used for mining and transportation in underground mines, is a research topic that has received much attention. The path tracking control of EMAVs is subject to several system constraints, including articulation angle and articulation angular velocity. In light of this, many researchers have initiated studies based on model predictive control (MPC). The principal design schemes for existing MPC methods encompass linear MPC (LMPC) utilizing a single reference point, so named the single reference point LMPC (SRP-LMPC), and nonlinear MPC (NMPC). However, NMPC exhibits suboptimal real-time performance, while SRP-LMPC demonstrates inferior accuracy. To simultaneously improve the accuracy and real-time performance of the path tracking control of EMAV, based on the SRP-LMPC, a path tracking control method for EMAV based on nonlinear compensated multiple reference points LMPC (MRP-LMPC) is proposed. The simulation results demonstrate that MRP-LMPC simultaneously exhibits a commendable degree of accuracy and real-time performance. In all simulation results, the displacement error amplitude and heading error amplitude of MRP-LMPC do not exceed 0.2675 m and 0.1108 rad, respectively. Additionally, the maximum solution time in each control period is 5.9580 ms. The accuracy of MRP-LMPC is comparable to that of NMPC. However, the maximum solution time of MRP-LMPC can be reduced by over 27.81% relative to that of NMPC. Furthermore, the accuracy of MRP-LMPC is significantly superior to that of SRP-LMPC. The maximum displacement and heading error amplitude can be reduced by 0.3075 m and 0.1003 rad, respectively, representing a reduction of 65.51% and 73.59% in the middle speed and above scenario.</jats:p>

section_content: Mining articulated vehicles (MAVs) are specialized vehicles that steer by actively folding the front and rear bodies and differ significantly from other vehicles called articulated vehicles, such as trailer trucks, semi-trailers, and articulated buses. MAVs are commonly used for mining and transporting underground ores and are essential pieces of mining equipment [1]. In recent years, the electrification of mining articulated vehicles has been increasing, and several types of electric mining articulated vehicles (EMAVs) have appeared. For example, Sandvik has introduced the battery power driver loader LH518iB, cable power driver loader LH514iE, and battery power driver truck TH665B. The University of Science and Technology Beijing (USTB) has participated in the development of a 60-ton electric articulated truck. Figure 1 shows the EMAV developed with the participation of have appeared. For example, Sandvik has introduced the battery power driver loader LH518iB, cable power driver loader LH514iE, and battery power driver truck TH665B. The University of Science and Technology Beijing (USTB) has participated in the development of a 60-ton electric articulated truck. Figure 1 shows the EMAV developed with the participation of USTB. EMAVs are more suitable to be used as unmanned platforms compared to traditional MAVs, so more and more research has been conducted on unmanned technologies for EMAVs in recent years. Path tracking is a key technique in unmanned EMAVs or MAVs which aims to control the object to track a reference path. In 2021, Bai et al. provided a review of path tracking control for mining articulated vehicles [2], noting that much research already exists on this topic. Control methods that have been used for path tracking control for mining articulated vehicles include sliding mode control [3] [4] [5], feedback linearization control [6, 7], enforcement learning PID (proportional-integral-differential) control [8], LQR (linear quadratic regulator) control [9], fuzzy control [10] [11] [12], preview control [13], MPC (model predictive control) [14] [15] [16] [17] [18] [19] [20] [21] [22], etc. Each of these control methods has its strengths, but MPC has gained wider attention because of its advantages, such as being able to handle system constraints explicitly [23]. 

There are two primary forms of MPC are NMPC (nonlinear MPC) and LMPC (linear MPC). The accuracy of NMPC is better, but its real-time performance is relatively poor [2, 23]. Although researchers have optimized the real-time performance of NMPC by replacing the optimal solution algorithm, adjusting the controller horizon parameters, replacing the online optimal solution process with a neural network, and reducing the solution dimension [24] [25] [26] [27] [28] [29] [30], replacing the optimal solution algorithm, adjusting the controller horizon parameters, and reducing the solution dimension of NMPC still requires a long solving time, and replacing the online optimal solution process with a neural network suffers from the problem of poor generalization performance. LMPC has excellent real-time performance, but the current LMPC scheme is usually used for the path tracking control of unmanned ground vehicles. Its design is generally based on the error model [2, 23, [31] [32] [33] [34] [35] [36] [37], based on which the path tracking controller is designed without considering the information of the reference path in front of the controlled object. It becomes a controller with only one reference point. Thus, this control method can also be called the single reference point LMPC (SRP-LMPC). The accuracy of this controller is good when the curvature of the reference path changes smoothly. Still, the mining articulated vehicle often works in narrower scenarios, where the curvature of the reference path changes in a larger magnitude and faster rate, so it is difficult to achieve a more accurate path tracking control of the mining articulated vehicle based on existing LMPC methods. 

To simultaneously improve the accuracy and real-time performance of the path tracking control of EMAV, the nonlinear compensation quantity obtained through dead reckoning is introduced by changing the linearization expansion point of the SRP-LMPC, and a path tracking control method for EMAV based on nonlinear compensated multiple Path tracking is a key technique in unmanned EMAVs or MAVs which aims to control the object to track a reference path. In 2021, Bai et al. provided a review of path tracking control for mining articulated vehicles [2], noting that much research already exists on this topic. Control methods that have been used for path tracking control for mining articulated vehicles include sliding mode control [3] [4] [5], feedback linearization control [6, 7], enforcement learning PID (proportional-integral-differential) control [8], LQR (linear quadratic regulator) control [9], fuzzy control [10] [11] [12], preview control [13], MPC (model predictive control) [14] [15] [16] [17] [18] [19] [20] [21] [22], etc. Each of these control methods has its strengths, but MPC has gained wider attention because of its advantages, such as being able to handle system constraints explicitly [23]. 

There are two primary forms of MPC are NMPC (nonlinear MPC) and LMPC (linear MPC). The accuracy of NMPC is better, but its real-time performance is relatively poor [2, 23]. Although researchers have optimized the real-time performance of NMPC by replacing the optimal solution algorithm, adjusting the controller horizon parameters, replacing the online optimal solution process with a neural network, and reducing the solution dimension [24] [25] [26] [27] [28] [29] [30], replacing the optimal solution algorithm, adjusting the controller horizon parameters, and reducing the solution dimension of NMPC still requires a long solving time, and replacing the online optimal solution process with a neural network suffers from the problem of poor generalization performance. LMPC has excellent real-time performance, but the current LMPC scheme is usually used for the path tracking control of unmanned ground vehicles. Its design is generally based on the error model [2, 23, [31] [32] [33] [34] [35] [36] [37], based on which the path tracking controller is designed without considering the information of the reference path in front of the controlled object. It becomes a controller with only one reference point. Thus, this control method can also be called the single reference point LMPC (SRP-LMPC). The accuracy of this controller is good when the curvature of the reference path changes smoothly. Still, the mining articulated vehicle often works in narrower scenarios, where the curvature of the reference path changes in a larger magnitude and faster rate, so it is difficult to achieve a more accurate path tracking control of the mining articulated vehicle based on existing LMPC methods. 

To simultaneously improve the accuracy and real-time performance of the path tracking control of EMAV, the nonlinear compensation quantity obtained through dead reckoning is introduced by changing the linearization expansion point of the SRP-LMPC, and a path tracking control method for EMAV based on nonlinear compensated multiple reference points LMPC (MRP-LMPC) is proposed, which is verified by simulation through MATLAB. In addition, the proposed control method is analyzed in comparison with other control methods, such as NMPC and SRP-LMPC.
section_name: Introduction

section_content: MPC must be designed based on a mathematical model. The kinematic model of the mining articulated vehicle is mature [2], and the detailed modeling process has been given in [15, 38], so it can generally be used as a basis for controller design. A schematic of this model is shown in Figure 2. 

World Electr. Veh. J. 2024, 15, x FOR PEER REVIEW 3 of 21 reference points LMPC (MRP-LMPC) is proposed, which is verified by simulation through MATLAB. In addition, the proposed control method is analyzed in comparison with other control methods, such as NMPC and SRP-LMPC.
section_name: Mining Articulated Vehicle Model

section_content: MPC must be designed based on a mathematical model. The kinematic model of the mining articulated vehicle is mature [2], and the detailed modeling process has been given in [15, 38], so it can generally be used as a basis for controller design. A schematic of this model is shown in Figure 2. According to [15, 38], the mathematical form of this model can be written as follows:

where f

x, f y, and f θ are the horizontal coordinate, vertical coordinate, and heading angle of the center of the front axle of the mining articulated vehicle, f v is the speed of the front body, f L are r L the distances from the articulated point to the front and rear axles, γ is the articulation angle between the front body and the rear body, and γ ω is the articulation angle velocity. The distances from the articulation point to the front and rear axles are constant, and the speed for path tracking can usually be considered constant. So, the kinematic model can be further abstracted as follows: According to [15, 38], the mathematical form of this model can be written as follows:

where x f, y f, and θ f are the horizontal coordinate, vertical coordinate, and heading angle of the center of the front axle of the mining articulated vehicle, v f is the speed of the front body, L f are L r the distances from the articulated point to the front and rear axles, γ is the articulation angle between the front body and the rear body, and ω γ is the articulation angle velocity. 

The distances from the articulation point to the front and rear axles are constant, and the speed for path tracking can usually be considered constant. So, the kinematic model can be further abstracted as follows:
section_name: Mining Articulated Vehicle Model

section_content: The proposed MRP-LMPC algorithm is based on the SRP-LMPC algorithm, so the design process of the SRP-LMPC is given first. The design of the SRP-LMPC usually contains three steps, i. e., mathematical model linearization, predictive model design, and optimal objective function design. 

For the SRP-LMPC algorithm, the linearization expansion point is usually assumed to be the key reference point during the mathematical model linearization process, and the key reference point is the point on the reference path closest to the mining articulated vehicle. Therefore, (2) can be linearly expanded as follows:

where ξ ref denotes the information of the key reference point and the subscript 0 denotes the state of the mining articulated vehicle at the start of the control period. 

The prediction model can be further designed based on the linearized model (3). The control period is short, so the differential quantities after linearization can be assumed to vary linearly during the control period:

where t denotes the moment t and T is the control period. 

Then, the differential model can be obtained by associating (3) and (4):

where I is the unit matrix. 

Since the prediction horizon of the controller is usually short, state matrices of the mining articulated vehicle become small during that horizon. Moreover, simplifying the calculation can ensure real-time performance. Therefore, it can be assumed that the state matrices A and B remain unchanged in the prediction horizon during each control period. Then, the state variables in the prediction horizon can be expressed as follows:

. . . 

where c and p are the number of control steps and the number of prediction steps. 

The matrix expression for the prediction model is

Then, to ensure the accuracy and smoothness of the path tracking, the optimization objective function can be designed as follows:

where Q and R are the weight matrices. 

After substituting ( 7 ) into (8), the optimization objective function can be transformed into a standard quadratic form: (9) After adding the system constraints, the SRP-LMPC path tracking control is equivalent to solving the following quadratic form:

where the subscript lim denotes the limit. Solving (10) and taking the first value of the obtained sequence of control inputs as the real control input generated by the controller at the moment t, the path tracking control of the mining articulated vehicle based on the SRP-LMPC can be realized.
section_name: Single Reference Point LMPC

section_content: The design steps of the MRP-LMPC contain the same three steps of mathematical model linearization, predictive model design, and optimal objective function design as those of the SRP-LMPC, but each step is different. 

During the mathematical model linearization, since the linearization can be unfolded at any point, the linearization point can be assumed to be the current position of the mining articulated vehicle:

The differential model can be obtained by referring to the SRP-LMPC:

The difference model only predicts future deviations by current deviations, and therefore, the state quantities need to be picked up:

From ( 13 ), the state of the mining articulated vehicle at the next moment is the sum of three quantities. The first quantity is the linear iteration of the state quantity at the current moment. The second quantity is the linear iteration of the input quantities at the next moment. The third quantity is the difference between the nonlinear iteration and the linear iteration of the mining articulated vehicle according to the reference inputs, and it can also be named the nonlinear compensated quantity. 

Assuming that A and B remain constant over the prediction horizon, as with the SRP-LMPC, (13) can be generalized to the entire prediction horizon:

. . . 

Next, the nonlinear compensated quantities in each iteration step can be presented as follows:

Since the initial linearization point is chosen as the state of the mining articulated vehicle at the current moment, the linearization point for each iteration can theoretically be obtained by nonlinear iteration:

. . . 

Theoretically, the nonlinear compensated quantities obtained by nonlinear iteration are most accurate when the reference input at each moment is the same as the actual input. Still, the actual input is an unknown quantity that needs to be solved by the controller to obtain it. Fortunately, to guarantee real-time performance, the number of control steps is usually chosen to be small [25], and the control period is usually short so that the reference inputs can be assumed to be equal to the actual inputs at the previous moment:

Substituting ( 17 ) into ( 16 ) gives the state of the linearization point for each iteration, from which the amount of nonlinear compensation required for each iteration in the prediction horizon can be calculated. 

Thus, the matrix form of the prediction model can be expressed as follows:

Since the prediction model can generate p and future states of the mining articulated vehicle, the state information of p points on the reference path must also be used as the reference state when designing the optimization objective function:

The distance between every two neighboring points of the p points on the reference path is equal to the sum of the speed and the control period, and the distance between the first of these points and the key reference point is also equal to the sum of the speed and the control period:

Then, to ensure the accuracy and smoothness of the path tracking, the optimization objective function can be designed as follows:

After substituting ( 18 ) and ( 19 ) into ( 21 ), the optimization objective function can be transformed into a standard quadratic form:

After adding the system constraints, the MRP-LMPC path tracking control is equivalent to solving the following quadratic form:

Solving (23) and taking the first value of the obtained sequence of control inputs as the actual control input generated by the controller at the moment t, the path tracking control of the mining articulated vehicle based on the MRP-LMPC can be realized.
section_name: Multiple Reference Points LMPC

section_content: The EMAV is a large vehicle with extremely demanding experimental conditions, so many current studies have used computer simulation for preliminary validation of the proposed control methods [12, 15, 16, 18, 20, 22]. The proposed MRP-LMPC controller is tested by MATLAB R2020a, equipped on a Dell G15 5510 laptop computer. The central processing unit (CPU) of that computer is Intel i5-10200H. The control period of the simulation system is set to 50ms. The parameters of the EMAV are shown in Table 1. The proposed control method is compared with the SRP-LMPC and NMPC. The parameters of these three controllers are shown in Table 2. The parameters of each controller are the best that can be obtained for that controller by manual adjustment. There are four groups of simulation results. The purpose of the top three group simulations is to test the performance of the proposed MRP-LMPC and other control methods for EMAV in different scenarios and at different speeds. The purpose of the fourth group of simulations is to test the performance of the proposed MRP-LMPC in conditions with simulated positioning errors. There are three groups of simulation results. The purpose of the three groups of simulations is to test the performance of the proposed MRP-LMPC and other control methods at low speed, middle speed, and high speed for EMAV in different scenarios.
section_name: Simulation Results

section_content: In this group of simulations, the performance of the proposed MRP-LMPC is tested at a low speed of 3 km/h and compared with SRP-LMPC and NMPC. In the simulation, the reference path is set as a U-bend consisting of straight lines and arcs with a radius of 10 m. The reference path is shown in Figure 3. 

There are three groups of simulation results. The purpose of the three groups of simulations is to test the performance of the proposed MRP-LMPC and other control methods at low speed, middle speed, and high speed for EMAV in different scenarios.
section_name: Simulation at a Speed of 3 km/h

section_content: In this group of simulations, the performance of the proposed MRP-LMPC is tested at a low speed of 3 km/h and compared with SRP-LMPC and NMPC. In the simulation, the reference path is set as a U-bend consisting of straight lines and arcs with a radius of 10 m. The reference path is shown in Figure 3. 

The path tracking results are shown in Figure 4, where the MRP-LMPC, SRP-LMPC, and NMPC can track the reference path. Figure 5 shows the displacement error, which is the closest distance from the center of the front axle of the mining articulated vehicle to the reference path. The maximum values of displacement error amplitude for MRP-LMPC, SRP-LMPC, and NMPC are 0. 0496 m, 0. 0255 m, and 0. 0305 m, respectively. This result shows that these control methods can achieve high control accuracy in the lowspeed scenario, and the maximum displacement error amplitude is within 5 cm. Figure 6 shows the heading error, which is the angle between the heading of the front axle of the mining articulated vehicle and the reference path. The maximum value of heading error amplitude for all three control methods is also very small. Figures 7 and 8 show the articulation angle and articulation angular velocity (AAV), which are indicators of smoothness. The articulation angle and AAV generated by each controller are within the system constraints and did not oscillate significantly. The path tracking results are shown in Figure 4, where the MRP-LMPC, SRP-LMPC, and NMPC can track the reference path. Figure 5 shows the displacement error, which is the closest distance from the center of the front axle of the mining articulated vehicle to the reference path. The maximum values of displacement error amplitude for MRP-LMPC, SRP-LMPC, and NMPC are 0. 0496 m, 0. 0255 m, and 0. 0305 m, respectively. This result shows that these control methods can achieve high control accuracy in the low-speed scenario, and the maximum displacement error amplitude is within 5 cm. Figure 6 shows the heading error, which is the angle between the heading of the front axle of the mining articulated vehicle and the reference path. The maximum value of heading error amplitude for all three control methods is also very small. Figures 7 and 8 show the articulation angle and articulation angular velocity (AAV), which are indicators of smoothness. The articulation angle and AAV generated by each controller are within the system constraints and did not oscillate significantly. 

Figure 9 shows the real-time metric, the controller solution time per control period. The maximum value of the solution time is generally considered to be the most crucial real-time indicator because if the maximum value of the controller solution time is too large, it may cause the sum of the solution time and transmission time in a given control period to exceed the control period, resulting in control failure. The maximum and average values of solution time for MRP-LMPC are 5. 8615 ms and 3. 5029 ms. These values are slightly larger than their SRP-LMPC counterparts but smaller than their NMPC counterparts. Compared to NMPC, MRP-LMPC can reduce the maximum value of the solution time by 3. 0002 ms, up to 33. 86%, and can reduce the average value by 0. 4247 ms, up to 10. 81%. 

slightly larger than their SRP-LMPC counterparts but smaller than their NMPC counterparts. Compared to NMPC, MRP-LMPC can reduce the maximum value of the solution time by 3. 0002 ms, up to 33. 86%, and can reduce the average value by 0. 4247 ms, up to 10. 81%. slightly larger than their SRP-LMPC counterparts but smaller than their NMPC counterparts. Compared to NMPC, MRP-LMPC can reduce the maximum value of the solution time by 3. 0002 ms, up to 33. 86%, and can reduce the average value by 0. 4247 ms, up to 10. 81%. slightly larger than their SRP-LMPC counterparts but smaller than their NMPC counterparts. Compared to NMPC, MRP-LMPC can reduce the maximum value of the solution time by 3. 0002 ms, up to 33. 86%, and can reduce the average value by 0. 4247 ms, up to 10. 81%.
section_name: Simulation at a Speed of 3 km/h

section_content: In this group of simulations, the performance of the proposed MRP-LMPC is tested at a middle speed of 6 km/h and compared with SRP-LMPC and NMPC. The reference path is set as a U-bend consisting of straight lines and arcs with a radius of 10 m. 

The path tracking results are shown in Figure 10, where the MRP-LMPC, SRP-LMPC, and NMPC can track the reference path. Figure 11 shows the displacement error. The maximum values of displacement error amplitude for MRP-LMPC, SRP-LMPC, and
section_name: Simulation at a Speed of 6 km/h

section_content: In this group of simulations, the performance of the proposed MRP-LMPC is tested at a middle speed of 6 km/h and compared with SRP-LMPC and NMPC. The reference path is set as a U-bend consisting of straight lines and arcs with a radius of 10 m. 

The path tracking results are shown in Figure 10, where the MRP-LMPC, SRP-LMPC, and NMPC can track the reference path. Figure 11 Figure 12 shows the heading error. The maximum values of heading error amplitude for MRP-LMPC, SRP-LMPC, and NMPC are 0. 0360 rad, 0. 1363 rad, and 0. 0541 rad, respectively. This result shows that the maximum values of heading error amplitude of MRP-LMPC and NMPC are close and significantly smaller than that of SRP-LMPC. MRP-LMPC can reduce the maximum value of heading error amplitude compared with SRP-LMPC by 0. 1003 rad, which can be reduced to 73. 59%. Figures 13 and 14 show the articulation angle and AAV. The articulation angle and AAV generated by each controller are within the system constraints and did not oscillate significantly. Figure 15 shows the controller solution time per control period. The maximum and average values of solution time for MRP-LMPC are 5. 9431 ms and 3. 6032 ms. These values are slightly larger than their SRP-LMPC counterparts but smaller than their NMPC counterparts. Compared to NMPC, MRP-LMPC can reduce the maximum value of the solution time by 2. 2894 ms, up to 27. 81%, and can reduce the average value by 0. 4363 ms, up to 10. 80%. Figure 12 shows the heading error. The maximum values of heading error amplitude for MRP-LMPC, SRP-LMPC, and NMPC are 0. 0360 rad, 0. 1363 rad, and 0. 0541 rad, respectively. This result shows that the maximum values of heading error amplitude of MRP-LMPC and NMPC are close and significantly smaller than that of SRP-LMPC. MRP-LMPC can reduce the maximum value of heading error amplitude compared with SRP-LMPC by 0. 1003 rad, which can be reduced to 73. 59%. Figures 13 and 14 show the articulation angle and AAV. The articulation angle and AAV generated by each controller are within the system constraints and did not oscillate significantly. Figure 15 Figure 12 shows the heading error. The maximum values of heading error amplitude for MRP-LMPC, SRP-LMPC, and NMPC are 0. 0360 rad, 0. 1363 rad, and 0. 0541 rad, respectively. This result shows that the maximum values of heading error amplitude of MRP-LMPC and NMPC are close and significantly smaller than that of SRP-LMPC. MRP-LMPC can reduce the maximum value of heading error amplitude compared with SRP-LMPC by 0. 1003 rad, which can be reduced to 73. 59%. Figures 13 and 14 show the articulation angle and AAV. The articulation angle and AAV generated by each controller are within the system constraints and did not oscillate significantly. Figure 15 shows the controller solution time per control period. The maximum and average values of solution time for MRP-LMPC are 5. 9431 ms and 3. 6032 ms. These values are slightly larger than their SRP-LMPC counterparts but smaller than their NMPC counterparts. Compared to NMPC, MRP-LMPC can reduce the maximum value of the solution time by 2. 2894 ms, up to 27. 81%, and can reduce the average value by 0. 4363 ms, up to 10. 80%.
section_name: Simulation at a Speed of 6 km/h

section_content: In this group of simulations, the performance of the proposed MRP-LMPC is tested at a high speed of 9 km/h and compared with SRP-LMPC and NMPC. The shape of the reference path is modified in this group of simulations because turnarounds are generally not carried out at high speeds. The reference path is set as a lane change path consisting of straight lines and arcs with a radius of 25 m. The reference path is shown in Figure 16. The path tracking results are shown in Figure 17, where the MRP-LMPC and NMPC can track the reference path. SRP-LMPC control fails in this group of simulations. Figure 18 shows the displacement error. The maximum values of displacement error amplitude for MRP-LMPC and NMPC are 0. 1570 m and 0. 1757 m, respectively. The maximum values of displacement error amplitude of MRP-LMPC and NMPC are close. Meanwhile, the displacement error of SRP-LMPC diverges. Figure 19 shows the heading error. The maximum values of heading error amplitude for MRP-LMPC and NMPC are 0. 0517 rad and 0. 0594 rad, respectively. The maximum values of heading error amplitude of MRP-LMPC and NMPC are close. Meanwhile, the heading error of SRP-LMPC diverges. Figures 20 and 21 show the articulation angle and AAV. The articulation angle and AAV generated by each controller are within the system constraints and did not oscillate significantly.
section_name: Simulation at a Speed of 9 km/h

section_content: In this group of simulations, the performance of the proposed MRP-LMPC is tested at a high speed of 9 km/h and compared with SRP-LMPC and NMPC. The shape of the reference path is modified in this group of simulations because turnarounds are generally not carried out at high speeds. The reference path is set as a lane change path consisting of straight lines and arcs with a radius of 25 m. The reference path is shown in Figure 16.
section_name: Simulation at a Speed of 9 km/h

section_content: This group of simulations is designed to test the performance of MRP-LMPC in conditions with simulated positioning errors. For this purpose, we inject simulated positioning errors into the displacement and heading of the controlled EMAV. Considering that existing technologies such as machine vision and differential GPS (global positioning system) [39] can provide positioning accuracies with errors below 1cm, we inject a displacement error of 1 cm and a heading error of 0. 01 rad as a small error condition. Since the
section_name: Simulation with Positioning Errors

section_content: This group of simulations is designed to test the performance of MRP-LMPC in conditions with simulated positioning errors. For this purpose, we inject simulated positioning errors into the displacement and heading of the controlled EMAV. Considering that existing technologies such as machine vision and differential GPS (global positioning system) [39] can provide positioning accuracies with errors below 1cm, we inject a displacement error of 1 cm and a heading error of 0. 01 rad as a small error condition. Since the positioning system may have problems such as drift, we inject a displacement error of 5 cm and a heading error of 0. 04 rad as a large error condition. The path tracking results are shown in Figure 23. MRP-LMPC can control the EMAV to track the reference path in each error condition. Figure 24 shows the displacement error. The maximum magnitude of the displacement error for the small error condition is close to that of the ideal condition. This value is 0. 2675 m for the large error condition, which is still a small value. 

World Electr. Veh. J. 2024, 15, x FOR PEER REVIEW 17 of 21 positioning system may have problems such as drift, we inject a displacement error of 5 cm and a heading error of 0. 04 rad as a large error condition. The path tracking results are shown in Figure 23. MRP-LMPC can control the EMAV to track the reference path in each error condition. Figure 24 shows the displacement error. The maximum magnitude of the displacement error for the small error condition is close to that of the ideal condition. This value is 0. 2675 m for the large error condition, which is still a small value. Figure 25 shows the heading error. The maximum magnitude of the heading error for the small error condition is close to that of the ideal condition. This value is 0. 1108 rad for the large error condition, which is still a small value. Figures 26 and 27 show the articulation angle and AAV. The articulation angle and AAV in each condition are within the system constraints. Figure 28 shows the controller solution time per control period. The maximum solution time is 5. 4580 ms and the average solution time is 3. 5087 ms for the small error condition. Those values for the large error condition are 5. 9580 ms and 3. 5914 ms, respectively. These values are not significantly different relative to the values for ideal operating conditions. positioning system may have problems such as drift, we inject a displacement error of 5 cm and a heading error of 0. 04 rad as a large error condition. The path tracking results are shown in Figure 23. MRP-LMPC can control the EMAV to track the reference path in each error condition. Figure 24 shows the displacement error. The maximum magnitude of the displacement error for the small error condition is close to that of the ideal condition. This value is 0. 2675 m for the large error condition, which is still a small value. Figure 25 shows the heading error. The maximum magnitude of the heading error for the small error condition is close to that of the ideal condition. This value is 0. 1108 rad for the large error condition, which is still a small value. Figures 26 and 27 show the articulation angle and AAV. The articulation angle and AAV in each condition are within the system constraints. Figure 28 shows the controller solution time per control period. The maximum solution time is 5. 4580 ms and the average solution time is 3. 5087 ms for the small error condition. Those values for the large error condition are 5. 9580 ms and 3. 5914 ms, respectively. These values are not significantly different relative to the values for ideal operating conditions. Figure 25 shows the heading error. The maximum magnitude of the heading error for the small error condition is close to that of the ideal condition. This value is 0. 1108 rad for the large error condition, which is still a small value. Figures 26 and 27 show the articulation angle and AAV. The articulation angle and AAV in each condition are within the system constraints. Figure 28 shows the controller solution time per control period. The maximum solution time is 5. 4580 ms and the average solution time is 3. 5087 ms for the small error condition. Those values for the large error condition are 5. 9580 ms and 3. 5914 ms, respectively. These values are not significantly different relative to the values for ideal operating conditions.
section_name: Simulation with Positioning Errors

section_content: Funding: This research is funded by the National Key Research and Development Program of China ( 2023YFC3806603 ) and the China Postdoctoral Science Foundation ( 2022M710354 ).

section_name: Data Availability Statement:

section_content: The original data presented in the study are included in the article, further inquiries can be directed to the corresponding author.
section_name: Data Availability Statement:

section_name: Conclusions

section_content: EMAVs are a kind of vehicle affected by steering angle constraints and steering angle speed constraints. A path tracking control method is proposed based on MRP-LMPC to guarantee the accuracy and real-time path tracking control of EMAVs simultaneously and compared with SRP-LMPC and NMPC. Based on the simulation results, the following conclusions can be drawn:

First, the proposed MRP-LMPC-based path tracking control method for EMAVs has good accuracy and real-time performance. Regarding accuracy, its displacement error amplitude is no more than 0. 2675 m, and its heading error amplitude is no more than 0. 1108 rad. Regarding real-time performance, its maximum solution time in each control period is no more than 5. 9580 ms. 

Second, the proposed MRP-LMPC has an accuracy close to that of NMPC, which is significantly better than that of SRP-LMPC. Under the working scenarios of middle speed and above, MRP-LMPC can reduce the maximum value of displacement error amplitude by at least 0. 3075 m compared with SRP-LMPC, and the reduction is up to 65. 51%. The maximum value of heading error amplitude can be reduced by at least 0. 1003 rad, up to 73. 59%. 

Third, the proposed MRP-LMPC has better real-time performance than NMPC. MRP-LMPC can reduce the maximum value of the solution time in each control period by at least 27. 81%. At the same time, the average value of the solution time under the same scenario can be reduced by 10. 05%. 

EMAVs are large vehicles with extremely demanding experimental conditions, which is why we have not yet completed experiments on real vehicles. We are currently developing an unmanned EMAV prototype, and when conditions allow, we will experimentally validate the proposed method. The problems that may be faced when MRP-LMPC is applied to real EMAV prototypes may include the motor response having nonlinear characteristics, which is what we will focus on in our future work. In addition, some other high-fidelity simulation tools are newly available [40, 41], so we are also considering introducing them to validate our proposed method in future work.
section_name: Conclusions