Tracking stability control of industrial vehicles under medium and low adhesion conditions based on LTI/LTV-MPC

Abstract:The working environment of unmanned forklifts is complex, and the structural characteristics of the unmanned forklift without suspension lead to the reduction of stability and accuracy in the process of path following on medium and low attached road surfaces. The adaptive forgetting factor least squares method is used to estimate the transient lateral deviation stiffness. The transient lateral deviation stiffness is equal to zero as the boundary, the tire stable working area and the unstable working area are divided, the current tire sideslip stability state is judged by the estimation results, the tire lateral deviation state stiffness is predicted on the basis of the tire model linearization, the ideal tire force is output and transmitted to the path following controller, the corresponding path following controller module is switched for different working areas of the tire to reduce the complexity of the controller solution, and after the controller receives the ideal tire force, the unmanned forklift drive braking and steering mechanism are adjusted and controlled by solving. to control the tires in a stable state. The results of MTLAB/Simulink and Pre S can co-simulation and hardware-in-the-loop experiments show thatThe proposed path following control strategy can improve the sliding stability of path tracking and maintain the tracking accuracy of industrial vehicles under medium and low adhesion conditions.

Keywords: unmanned forklift; path tracing; model predictive control; Sideslip control; lateral bias stiffness; State stiffness

CLC Number: U270.1

Stability Control of Industrial Vehicle Tracking Sideslip Based on Tire Stiffness

Abstract: The working environment of unmanned forklift truck is complicated, and its stability and accuracy will be reduced in the path tracking process of medium and low adhesion road surface without the structural characteristics of suspension. In this paper, a path tracking controller of unmanned forklift truck based on tire transient lateral stiffness estimation is designed, including transient lateral stiffness estimation module, lateral stiffness prediction module and path tracking control module. The adaptive forgetting factor least-square method is used to estimate the transient lateral stiffness. The stable working area and the unstable working area of the tire are divided with the transient lateral stiffness equal to zero as the boundary, the current tire sidesslip stability state is determined by the estimated results, and the tire lateral stiffness is predicted based on the tire model linearization. The ideal tire force is then output and transmitted to the path tracking controller, and the corresponding path tracking controller module is switched according to different tire working areas. In order to reduce the complexity of controller solution, the controller received the ideal tire force, and then adjusted and controlled the driving braking and steering mechanism of unmanned forklift truck through solving, so as to control the tire in a stable state. The results of MATLAB/Simulink, Carsim and Prescan co-simulation and experiment show that:

Key words: Unmanned forklift truck; Path tracking; Model predictive control; Sideslip control; Lateral stiffness; State stiffness

preface

Industrial vehicles generally refer to various wheeled handling vehicles with cargo loading and unloading, stacking and medium and long distances, because of their strong versatility, flexibility, large range of activities and other characteristics, they play an indispensable and important role in modern industrial production and social services, and at the same time, with the continuous enhancement of the global competitiveness of industry leaders, the product structure and added value of industrial vehicles continue to optimize and improve [^1].。 In addition, for the safety, stability and reliability of industrial vehicles in the working process also put forward higher requirements, according to statistics, nearly 20% of logistics workplace accidents involve industrial vehicles, due to their own structural form and complex operating conditions, resulting in the work process is very prone to horizontal and longitudinal instability, including skidding, rollover, collision, etc^[2]。 The main reasons are: if the road surface adhesion coefficient is low in the path following process, the wheel is easy to slip or side slip during the rolling process, resulting in lateral instability; In addition, compared with passenger vehicles, industrial vehicles have a larger curb weight, a higher height of the center of mass, and are more prone to rollover. Based on the above factors, it is necessary to analyze the tracking accuracy and sideslip stability of industrial vehicles in the path following process.

In terms of vehicle path following control,Nan Kang
In order to solve the problem of path following anti-jamming control of autonomous vehicles with lateral stability, they proposed an improved active disturbance rejection control (IADRC
) control method, which is controlled by an improved extended state observer (IESO
) and based onLQR
error compensator,IESO
The disturbance value is estimated using the output wheel rotation angle and external yaw moment, and the disturbance in the feedback is compensated^[3]，Wu
The results show that the proposed method can intervene in time according to the formulated intervention criteria and effectively improve the stability and accuracy of path tracking^[4]，Malmir与Bacur
et al. proposed a model predictive control method to drive the vehicle to reach the tire attachment limit, aiming to complete the optimality of path following closed-loop control, and the linear time-varying model predictive controller i.eLTV-MPC
Applied to this, it can adapt to the high nonlinearity of the model, and also has low computational complexity, and the proposed control method is verified by experiments^[5]，
Cai Yingfeng et al. from Jiangsu University designed a design based on the problem that the traditional single control algorithm cannot effectively coordinate the lateral control performance under different steering conditionsPID
A hybrid control strategy of control and model predictive control, which is used at low speed and high speedPID
control and model predictive control, and the path following performance, real-time and driving stability of the hybrid control strategy were verified after the experiment^[6]，
In order to improve the lateral stability in the process of autonomous driving, Li Shaosong designed a new linear time-varying (LTV-MPC
Method, pro
The rotation angle optimization sequence of model predictive control is used to predict the steady state of the vehicle, and the simulation results can be verified to improve the lateral stability and tracking accuracy of the test vehicle^[7]，
In order to solve the problem of the reduction of tracking accuracy and stability of active steering when the tire reaches the road adhesion limit, Li Yajun designed a coordinated control strategy of active steering and direct yaw torque based on phase plane theory, and verified the reliability and accuracy of the proposed method in a variety of experiments on road surface^[8]，Xie
In order to solve the problems of rollover on high-attachment road surface and sideslip instability on low-attachment road surface of four-wheel independently driven electric vehicle, a model predictive control algorithm and stability integrated controller were proposed to study the conditions and coordination strategies of vehicle rollover and lateral stability state in the process of vehicle path tracking^[9]，
Zhou Xiaochen designed a trajectory tracking controller that integrates the "side-longitudinal and vertical" coupling characteristics of the vehicle in extreme working conditions, and verified that the control method can achieve the path following effect after testing with different vehicle speeds under the double line shift condition of high and low adhesion coefficients^[10]。

In the field of lateral stability control, Basilio Lenzo et al. proposed a parallel control method for yaw angle velocity and side slip angle based on single input single output (SISO) yaw angular velocity controller The controller is able to control the vehicle's sideslip angle within the threshold range ^[11], Wolfgang Sienel of GermanyIt is proposed to estimate important tire parameters such as lateral deviation stiffness by measuring dynamic vehicle parameters such as lateral acceleration and yaw rate, so as to analyze the force saturation of tires under extreme operating conditions ^[12].In order to make full use of the direct yaw control of the four-wheel in-wheel in-wheel motor, the research team proposed an adaptive SMC control scheme, designed a stability evaluation method based on the phase plane of the front and rear tire slip angles, and designed a sliding mode controller to track the vehicle motion, which was verified by CarSim-Simulink co-simulation^[13] Wu Xitao et al. from Beijing Institute of Technology added the stability criterion considering the ultimate performance to the proposed model prediction controller constraints, and used the performance-driven mode to optimize the controller parameters^[14] In order to solve the problem of lateral stability in the process of vehicle path tracking, Lian Yufeng proposed the SISO model, combined with the steering stability constraint and tire lateral stiffness information, and designed the H∞ robust controller. The uncertainty caused by the vehicle parameter perturbation and external lateral interference of the vehicle lateral motion system was suppressed, and the vehicle stability was improved ^[15].

Based on the previous research at home and abroad, it mainly focuses on the research and analysis of the path tracking accuracy and side-slip stability of vehicles, and has made quite important achievements and results, but there are relatively few related studies on industrial vehicles. In addition, there are few studies on vehicle sliding stability under medium and low adhesion conditions, and there are few literatures that take the study of tire characteristics as the starting point. Therefore, when studying the tracking accuracy and sideslip stability in the process of path tracking, it is necessary to integrate some characteristic analysis of the tire itself into it to improve the accuracy and rationality of the research analysis.

In this paper, under the premise of considering the working conditions of industrial vehicles, a three-degree-of-freedom vehicle dynamics model is established, a magic formula tire model is established, the sideslip stability state is judged based on the analysis of tire sideways deviation characteristics and the transient sideways stiffness is used as the basis, the least squares method with forgetting factor is used to estimate it, the state stiffness is introduced to linearize the tire model, and the LTI/LTV-MPC based is proposedThe path-following sideslip control strategy and design the corresponding controller. Co-simulation and hardware-in-the-loop testing will be performed using PreScan and MATLAB/Simulink.

Unmanned forklift vehicle modeling and tire characteristics

Dynamic model of an unmanned forklift

(1) 3-degree-of-freedom sideslip dynamics model

As an engineering vehicle, due to its own structural characteristics and operating environment, the force in the working process of unmanned forklift is different from that of structural road vehicles, so in order to facilitate the study of side-slip characteristics, the following simplifications are made ^[16]: (1) Assuming that the vehicle is driving on a flat road surface, the vertical motion of the vehicle is not considered, that is, the displacement of the vehicle along the Z-axis direction is always zero; (2) Only the translational motion of the vehicle in the horizontal plane is considered, that is, the pitch angle of the vehicle around the Y-axis and the roll angle around the X-axis are zero; (3) ignoring the influence of lateral winds; (4) ignoring the effect of the tire return torque; (5) The product of inertia of the body mass around the X and Z axes is very small and is not considered; (6) Assuming that the vehicle only steers the rear wheels, the steering angle of the front wheels is zero, and the rotation angles of the left and right wheels are constant and equal. A three-degree-of-freedom dynamic model considering the longitudinal, transverse and yaw movements of forklifts was established. Figure 1 shows the side-slip dynamics model of an unmanned forklift.

Figure 1: Side-slip dynamics model of an unmanned forklift

Figure 2: Schematic diagram of the three-degree-of-freedom dynamic model of an unmanned forklift

According to D'Alembert's principle, the center of mass of the unmanned forklift is used as the coordinate origin O, and Figure 2 is established The OXYZ vehicle coordinate system shown in the center of mass of the vehicle, according to Newton's second law, the dynamical equation along the X-axis is as follows:

Where:

where, for the quality of the whole vehicle; for the longitudinal speed of the forklift; for the lateral speed of the forklift; for the longitudinal acceleration of the forklift; is the lateral acceleration of the forklift; is the yaw angle of the vehicle; is the yaw angular velocity of the vehicle; is the yaw angular velocity of the vehicle; is the forklift rear wheel corner; , which is the force in the direction of the X-axis of the forklift; , which is the force in the Y-axis direction of the forklift; is the distance between the front wheelbase centroid; is the rear wheelbase centroid distance; is the moment of inertia of the vehicle around the Z-axis; It is the longitudinal force and lateral force on the front wheel of the forklift; It is the longitudinal and lateral force on the rear wheel of the forklift.

(2) Tire model

Since forklifts are mostly used for picking, loading and unloading operations in factories, logistics or warehouses, the road conditions are relatively flat, and if a certain amount of space is required to install the suspension system, the weight and size of the goods that can be handled by the forklift will be affected, so the forklift only acts as a buffer through the tires. At the same time, as the only part of the forklift in contact with the road surface, the working characteristics of the tire will have a direct impact on the lateral stability and tracking stability of the unmanned forklift, so it is very important to choose an accurate tire model^[17]。 Its general form is as follows:

In the case where the lateral forces are mainly studied in this article, the magic formula is changed to the following form:

where the coefficient is the stiffness factor, where is the camber angle of the tire; is the shape factor,; is the peak factor, which represents the maximum value of the curve. is the curvature factor, which is determined by the vertical load and camber angle of the tire; is an output variable, which can be a longitudinal force, a lateral force, or a correcting moment; The input variables are the deviation angle of the tire or the longitudinal slip rate in different cases.

Wherein, is the side deflection angle of the tire, which is the horizontal drift of the curve; is the vertical drift of the curve, which is the vertical load on the tire.

The above-mentioned model parameters are the fitting parameters, wherein the parameters are obtained by fitting according to the tire force measured by the tire testing machine.

Table 1: Forklift tire parameters A0~A7

The value is derived from Ref. ^[18].

The formula for calculating the deviation angle of the tire is:

In order to understand the accurate side deviation characteristics of tires of unmanned forklifts in the operating environment, we selected a road surface with an adhesion coefficient of 0.6 for testing, and the results are shown in Figure 3Three-dimensional schematic diagram of the lateral deviation characteristics of unmanned forklift tires shown; In addition, in order to enhance the real-time performance of the system as a whole, an unmanned forklift tire lateral force finding table was designed, as shown in Figure 4, to realize the lateral deflection angle of the tire and the adhesion coefficient of the road surfaceCheck the table to get the stiffness of the tire deviation state and the tire deviation force at the current moment ^[19].

Figure 3: Three-dimensional schematic diagram of tire lateral deviation characteristics under an adhesion factor

Figure 4: Three-dimensional schematic diagram of the tire lateral force lookup table

(3) Desired path model

Unmanned forklifts are engineering vehicles with slower driving speed and larger load, and the selection of the desired path model is also different from that of ordinary passenger vehicles.

For a common control point, a Bézier point is defined as the following form:

(When,)

where is called the Bernsteinki function.

Based on the 6 Bezier curves, the following desired path can be obtained:

Figure 5 6-degree Bezier curve desired path

The trajectory equation is given by:

where is the desired longitudinal position, is the desired transverse position, and is the desired yaw angle.

Vehicle dynamics model validation

In order to verify the dynamic model of the unmanned forklift established above, this paper compares the established forklift dynamics model with the vehicle model in the Car sim software under two conditions: fixed rear wheel rotation angle input and embankment angle rear wheel rotation angle inputTable 2 [^21].

Table 2 simulates the operating conditions

Figure 6: Comparison of input lateral acceleration at fixed corners

Figure 7: Comparison of yaw velocity under fixed angle inputs

Figure 8: Comparison of lateral acceleration under the embankment corner input

Figure 9: Comparison of yaw angular velocity under embankment corner input

Lateral bias stiffness estimation based on recursive least squares method with forgetting factor

Unmanned forklifts, as construction vehicles, are compared with ordinary forklifts because

There is no human interference of the driver, in the face of some lateral stability of the critical or even instability situation, it is difficult to make accurate judgment and control, in different ground conditions, the change of factors such as the road surface adhesion coefficient will also affect the stability of the unmanned forklift in the driving process, and the tire as the only contact part between the forklift and the road surface, can intuitively judge the side slip stability of the forklift through the side deviation characteristics of the tire, For example, Xiong et al. designed a gain scheduling control system whose gain is updated based on the bias stiffness estimation of the tire ^[22].Fujimoto et al. proposed a side-bias stiffness estimation algorithm in an adaptive direct yaw controller ^[23].However, the above-mentioned methods are all based on the assumption that the lateral deflection force of the front and rear axles of the vehicle is linearly related to the lateral deflection angle, and in the actual operation process of the unmanned forklift, there will be a critical or even unstable situation, and it is in a nonlinear situation at this time, and it is difficult to realize the accurate judgment or control in this state in the above-mentioned mode. Therefore, this paper first analyzes the lateral deviation characteristics of tires, divides reasonable stable working areas into unstable working areas, and judges what working conditions are at the current moment by estimating the transient lateral deviation stiffness of tires [24].。

Analysis of tire deviation characteristics

When the unmanned forklift follows the desired path and drives on the medium and low attached road surface, the center of the wheel will act on the lateral force along the Y axis due to the centrifugal force when driving on the curve, the embankment angle of the road surface or the lateral wind Under the action of lateral deflection force, there will be two situations: 1. When the lateral deflection force does not exceed the lateral adhesion limit between the tire and the ground, there is no slippage (side skid) between the tire and the ground; 2. When the lateral deflection force reaches the lateral adhesion limit between the tire and the ground, the sideslip phenomenon will occur.

As shown in Figure 10, when the tire deviates with the increase of the deviation angle, the slope of the deviation force-deviation curve, i.e., the deviation stiffness, is equal to 0Previously, the lateral deflection force between the tire and the ground did not reach the road surface adhesion limit, and the lateral deflection force was approximately linear with the lateral deflection angle, and when the lateral deflection stiffness was equal to 0, it means that the lateral deflection force reached the road surface adhesion limit, and the lateral deflection stiffness was less than 0。 According to the above analysis, the area where the transient deviation stiffness is greater than zero is set as the stable working area of the tire, and when the transient deviation stiffness is less than or equal to zero, the tire begins to slip or has been sideslipped, and is set as the unstable zone.

Figure 10: Tire side deviation characteristic curve

According to the above delineation of the stable working area and the unstable working area of the tire, the position where the slope of the lateral deviation characteristic curve of the tire is 0 is divided, as shown in Figure 11

Figure 11: Tire work area delineation

Estimation of transient lateral bias stiffness based on recursive least squares method with forgetting factor

In the process of turning and driving, the ground provides the centripetal force required to complete the turning operation to the lateral force of the wheel, with the change of working conditions such as the increase of vehicle speed or the increase of road curvature, the centripetal force required for vehicle turning also increases, but the lateral force does not increase indefinitely, the maximum lateral force depends on the ground adhesion limit, under the condition that the characteristic parameters of the tire itself remain unchanged, The vertical load of the tire and the type of road surface mainly affect the influencing factors of the adhesion limit. Table 3 lists the adhesion coefficient parameters for different pavement situations.

Table 3: Adhesion coefficients for common pavements

The judgment of the stable working area and the unstable working area of the tire in section 2.1 is determined according to the transient lateral stiffness of the tire, so the accurate estimation of the transient lateral deviation stiffness is very important for judging the current stable state of the forklift. Some of the structural parameters of the vehicle are constantly changing during driving, which will also have a certain impact on the lateral stiffness, for example, when driving on a low-adhesion road surface, the steering operation during the process may cause the lateral force of the tire to be nonlinear. In this paper, the recursive least squares method with forgetting factor is used to estimate the lateral stiffness of tires.

The traditional least squares method can not achieve real-time update, although the recursive least squares method can be updated in real time, but will not eliminate the old data in the system, and the reliability of the new and old data is the same, so that with the increase of data volume, the new data will be in a large number of old data, resulting in the identification results can not reach the local optimal solution, and the data is saturated.

The application fields of least squares method are mainly in signal processing and system identification, and its theory is as follows:

In the above equation, it is the input signal; is the output signal. It needs to be obtained through observation, and there will be random interference during the observation process, and the value of the observed signal can be expressed as:

Substituting the formula into the formula gets:

Normally, the statistical properties are unknown, in which case the mean value is 0 for white noise, where:

On this basis, the formula can be rewritten in the following form:

Rewrite part of the equation to:

Set the structure of the system to be identified as follows:

where is the output of the system to be identified, the input of the system to be identified, and the system noise;

where polynomial sum is defined as:

The system is formulated as follows least-squares:

where is the system output observation vector and the system parameter to be identified

In order to estimate the error between the output of the first actual system observation and the estimated value, the residuals are defined:

where is the estimated value of the parameter to be identified based on the previous set of observations

The purpose of the algorithm is to find a vector estimate so that the value minimizes the sum of squares of the model residuals

In the traditional recursive least squares estimation, the new data information will continue to increase and be swallowed up by the old data over time, because the traditional algorithm gives the same confidence to the new and old data, the credibility will decrease with the increasing amount of new data information, and the deviation between the estimated value of the parameter and the true value will gradually increase, and the update and correction function will be lost. In order to solve this kind of data saturation problem, the forgetting factor is used to recursively least squares to weaken the influence of old data, and the core idea is to add a weight in the 0~1 interval before the old data to enhance the role of new data in the predicted time domain.

When the vehicle is close to the limit working condition, the nonlinear characteristics are very obvious, and when the recursive least squares method of fixed forgetting factor is applied to the identification of lateral deviation stiffness, the fixed forgetting factor is difficult to adapt to the strong nonlinearity of the tire, and it is difficult to converge quickly. Therefore, the forgetting factor with adaptive rules is used to improve the real-time recognition performance of least squares method. To this end, in this algorithm, each term of the model residuals is multiplied by a coefficient, and the algorithm is improved as follows:

 
The recursive formula for recursive least squares with forgetting factor is:

where is the recursive gain, the covariance, and the forgetting factor (value range).The smaller it is, the faster old data is forgotten. The adaptive forgetting factor adjusts its own value in the prediction process according to the variance of the prediction error in a fixed window, which can achieve a relative balance between the estimation bias and the real-time tracking ability of the system. The definition estimate variance update formula is as follows:

When the variance exceeds a certain threshold, the value of the forgetting factor is reduced, and the forgetting factor is calculated as follows:

From Section 1.1 Chinese and Formula:

Substituting the formula into the formula, let, eliminate to get:

Thereinto:

After finding the intermediate variables, the lateral deviation stiffness of the forklift tire can be obtained

Since the forgetting factor, which highlights the updating effect of the old and new data, is an ordinary least squares estimation, the formula can be rewritten as:

The least-squares estimation formula for the adaptive forgetting factor is as follows:

 
In the above two formulas,. The advantage of least squares method is that it can simplify the amount of calculation, reduce the memory occupied by data in the CPU, realize the function of online identification of dynamic features of the system, and add an adaptive forgetting factor to reduce the mutual engulfment of new and old data and improve the estimation accuracy.

In order to verify the actual observation effect of the designed lateral deviation stiffness estimator for unmanned forklifts, the simulation tests of sinusoidal input and double line shift input were carried out, Fig. 13 and Fig 14 are the test results in the case of two simulation experiments. It can be seen from the results that with the change of the operation condition of the forklift, the lateral deviation stiffness also changes, and the estimation method proposed in this paper can complete the stability estimation in time under the two test conditions, and the estimation results converge to a constant value [^25].

Figure 12: Estimating the lateral bias stiffness of the Simulink platform

Fig. 13: Simulation estimation of lateral bias stiffness under sinusoidal condition

Fig. 14: Simulation estimation of lateral bias stiffness under the double-shift line case

Linearization of the tire model based on state stiffness

In previous studies, most of the linearization of tire force was adopted

The Taylor first-order expansion method is used, but the residual lateral force is introduced, which complicates the model in the process of model solving and increases the computational cost of model solving. In this paper, a tire force linearization method based on the lateral deviation state stiffness of the tire is proposed, and the concept of state stiffness is proposed by Academician Guo Konghui, which is defined as the ratio of the lateral force to the slip rate at each lateral slip rate ^[26].。 As shown in Figure 15, the lateral deviation state stiffness is the secant slope at each lateral slip rate, and the expression for the lateral deviation state stiffness is as follows:

Figure 15: Side-biased state stiffness

Figure 15 also gives the lateral stiffness to distinguish between the lateral stiffness and the lateral deviation state stiffness, which is the slope of the tangent at the origin, which is different from the lateral deviation state stiffness, which will change with the change of the slip rate, and when the slip rate is small, the lateral deviation state stiffness is equal to the lateral deviation stiffness. On the basis of the concept of state stiffness, this paper defines the lateral deviation state stiffness as the ratio of the lateral force at the current moment to the lateral deviation angle at each lateral declination angle according to the actual research needs, as shown in Fig. 16, the expression is as follows:

Figure 16: Lateral deviation state stiffness under the new definition

Controller design

Based on the dynamic model of the unmanned forklift, the tire model and the transient lateral stiffness estimator, a path following controller was designed to improve the lateral stability of the unmanned forklift in the path following process and maintain the tracking accuracy in order to solve the lateral stability problem of the unmanned forklift in the path following process. As shown in Figure 17, the overall control strategy is divided into three main parts, starting with the desired pathBased on the unmanned forklift model, the transient lateral deviation stiffness of the tire is estimated by the recursive least squares method with forgetting factor, and the transient lateral deviation stiffness is equal to zero as the boundary, and whether the unmanned forklift is in a stable working area at this time is judged by the transient lateral deviation stiffness equal to zero. Based on the judgment results, the stiffness of the tire side deviation state is continuously predicted, and the chaotic calendar in the predicted time domain is predicted by combining the lateral acceleration and other parameters provided by the forklift model, and is output to the unmanned forklift path tracking controller. The predicted tire force will be output to the LTI-MPC path following controller when it is in a stable working area, and to the LTV-MPC controller when it is in a non-stable working area, so as to reduce the computational complexity, improve the real-time control performance, and ensure the accuracy. At the same time, after receiving the predicted tire force in the time domain, the path following controller will control and adjust the rear wheel rotation angle, pedal force and support cylinder pressure of the forklift, so as to realize the lateral stability control of the unmanned forklift in the path tracking process.

Figure 17: Path following control strategy for unmanned forklifts

LTI-MPC controller design

(1) Linearized vehicle dynamics model

According to the formula, , , The dynamic model can be rewritten into the following form:

According to the longitudinal dynamics, lateral dynamics, yaw dynamics and Newton's second law of motion, the linearized vehicle lateral dynamics model can be obtained as follows:

where the longitudinal and transverse speeds are respectively; Indicates the longitudinal stiffness of the front and rear tires respectively; The lateral stiffness of the front and rear tires is respectively; is the yaw angle of the vehicle.

The dynamic model of the unmanned forklift path tracking system is obtained by combining the longitudinal and lateral dynamics with the coordinate system conversion model:

Thereinto:

(2) Predictive models

Substituting the formula and the formula, assuming that the yaw angle is small, there is an approximate relationship and the LTI-MPC prediction model can be obtained as:

where the control input and the predicted output are state variables

With the step pair discretization, the incremental discrete model can be obtained

where,

(3) Prediction equations

Based on the formula, according to the control theory of model prediction, the prediction time domain is selected as and the control time domain is , then the model prediction output at the available time is

thereinto

To predict the output sequence, control the input delta sequence

In addition, according to the formula, the reference output sequence can be obtained, where,

LTV-MPC controller design

After LTI-MPC linearizes the tire force at the current moment, the predicted inner deviation state stiffness in the subsequent time domain will remain unchanged, but when the transient deviation stiffness is close to zero, that is, the vehicle is about to slip or has already skidd, the linearization of the tire force adopted by LTI-MPC will produce a large error in the predicted time domain. In the process of path tracing, LTI-MPC can obtain greater lateral force by increasing the front wheel angle by default, but the actual tire force at this time may have reached saturation or even enter the slip zone, and the actual tire force will decrease rapidly when the tire force reaches saturation, so that the vehicle will skid sideways and lose the path following ability.

Therefore, when the tire slip zone and slip zone are about to enter the stage of the tire slip zone and slip zone, this paper designs an LTV-MPC that can be time-varying in the predicted time domain of the tire lateral deviation state stiffness, as shown in Fig. 18As shown, LTV-MPC obtains the approximate linearity of the nonlinear tire force in the predicted time domain by predicting the lateral deviation state stiffness of the future step in the predicted time domain. This tire force linearization method can always obtain the optimal control input near the peak point of the lateral force when it requires a large lateral force to realize the path following type, and avoid the dangerous sideslip phenomenon caused by the lateral force exceeding the road adhesion limit caused by the front wheel rotation with excessive LTI-MPC output.

For the tire force to suddenly enter the saturation region from the linear region due to external uncertainties, the following two situations need to be adjusted when the transient lateral deviation stiffness is equal to the nearby interval:

1) The actual yaw angular velocity does not keep up with the expected value (from steady state to instability), and the phenomenon is that the predicted transient lateral stiffness changes from the actual positive value to a negative value, as shown in Figure 19It can be observed that the deviation angle changes from abrupt to , and the corresponding change of the deviation stiffness changes, and the rate of change is denoted as. At the moment in the prediction time domain, in order to track the expected yaw angular velocity, the controller will look for a feasible solution in the direction of the large lateral force, then the yaw angle is reduced to, and correspondingly, the yaw is reduced byI increase it and become. At the moment in the predicted time domain, the controller continues to look for a feasible solution in the direction of the large lateral force. Suppose that in order to achieve the desired yaw angular velocity, the lateral force should be reached, and the controller will achieve the lateral force at the lateral declination angle along the lateral deflection stiffness, in which case the lateral deflection angle There is a large deviation from the lateral deflection angle corresponding to the lateral force in the Magic Formula tire model, and the tire force is still in the stable zone at this time Adjusted to this, the controller will find a feasible solution at the side deflection angle, force the tire force back to the linear area, and correspondingly, the adjusted value that can be obtained ^[20]。

2) The actual yaw angular velocity is greater than the expected value (from the unstable state to the stable state), and the phenomenon is that the predicted transient lateral deviation stiffness changes from the actual negative value to a positive value, similar to the first case, the controller will look for a feasible solution along the direction of the large lateral force. The specific steps are the same as in the previous case and will not be repeated here.

Figure 18: Predicting tire forces in the time domain

Figure 19: Method for adjusting side-bias stiffness in case of mutation

(1) Prediction of tire side deviation state stiffness

Since the research object of this paper is path following in the case of a reference path, the lateral deviation state stiffness of the tire can be predicted based on the reference path information, and the reference yaw angle and reference lateral displacement specified in the reference path are used as inputs, and the lateral force of the tire is used as the output.

It can be obtained after finishing the formula

Based on the yaw angular velocity and lateral acceleration in the formula, the derivative can be obtained:

Substituting the reference transverse displacement obtained by the equation and its first-order and second-order derivatives to time and the reference yaw angle and its second-order derivatives to time into the equation, we get:

The predicted stiffness of the front and rear tire side deviation states is as follows:

Where, and with the predicted front and rear tire side deviation state stiffness, respectively, andThe moderator denoted as compensating for the effect of the adhesion coefficient is the minimum number to avoid a zero denominator.

The lateral deviation state stiffness in the above is based on the desired path information, which is obtained by inversely solving according to the forklift motion model of the formula and the formula, so it cannot reflect the influence of the pavement conditions such as the pavement adhesion coefficient on the lateral force, so it cannot be used in the formulaTherefore, the pavement adhesion coefficient adjustment factor is introduced in the formula to show the influence of the pavement adhesion coefficient on the stiffness of the tire side-deflection state, where the value of is taken from the adhesion coefficient of the current road The value is obtained by experimental test, and the value range is 0.5~0.8

In addition, in order to avoid the predicted tire force exceeding the road adhesion limit, the following constraints should be met:

Among them, it is the vertical load of the tire and the adhesion coefficient of the lateral road surface, which refers to the front and rear wheels respectively.

Therefore, by taking the expected path data forward, the stiffness of the tire deviation state in the predicted time domain in the future can be obtained

where the function represents the functional relationship between the expression and the formula

The predicted amount of change in the stiffness of the side deviation state can be expressed as

The predicted side bias state stiffness in the time domain is:

where is expressed as the side deviation state stiffness at the current moment

(2) Linear time-varying forklift dynamics model

The formula is expressed as a state-space expression as follows:

Wherein: is an n-dimensional system state variable; is the input quantity of the m-dimensional control of the system; is a collection of system state variables; is a collection of control variables for the system.

When the path tracing system at any time, the state variable and the control variable meet the following relationship:

Perform a Taylor expansion of the system near any point, keeping the first-order terms and ignoring the higher-order terms, and you get:

where is the Jacobian matrix of the system for state variables; A Jacobian matrix for the system to control the input quantities.

Subtracting the above two equations yields:

In order to meet the design requirements of the LTV-MPC controller, the above equation needs to be discretized

where is the identity matrix; is the sampling time for discrete systems.

In the case of considering the amount of control of the system, the combination and transformation can be obtained:

thereinto

where is the control increment

(3) LTV-MPC controller design

Design of the objective function:

The objective function should be able to ensure that the unmanned vehicle can quickly and smoothly track the desired trajectory, so it is necessary to add the optimization of the deviation and control quantity of the system state quantity.

where is the prediction time domain; for the control time domain; is the weight coefficient; is a relaxation factor.

The control objectives of this paper mainly include: 1) the vehicle tracks the desired lateral displacement and yaw angle as much as possible; 2) Keep the longitudinal force and rear wheel rotation angle input smooth, i.e. the rate of change of each of them is as small as possible. Therefore, based on the weighted combination of the tracking deviation of lateral displacement and yaw angle and the rate of change of the control input, the objective function is designed as follows:

where, reference the output sequence, where. Controls the output weights, where , controls the input weights, where ,The weight coefficients of lateral displacement, yaw angle, longitudinal acceleration change rate and rear wheel rotation angle change rate are respectively.

Constraint Design:

In the actual control system, some constraints on the state quantity and control quantity of the system should be met to plan the reasonable control of the controller.

For control variables, the constraints are:

For control increments, the constraints are

For control outputs, the constraints are

Combined with the research content of this paper, the constraints are determined as follows:

The formula contains the constraints on longitudinal acceleration and the constraints on the increment of longitudinal acceleration, because the tires will change with the vehicle state during the vehicle movement, so the constraints of the longitudinal force and the longitudinal force increment must be calculated in real time according to the vehicle operating state. The longitudinal force is mainly constrained by the adhesion of the road surface, and the longitudinal force applied to the wheel should be kept within the adhesion range to prevent the wheel from skidding or locking. When the tire reaches the lateral attachment limit, there is still an objective longitudinal force margin in the longitudinal direction of the tire to apply active braking. It can also be seen from the attachment ellipse of the tire force that the longitudinal and lateral force limit trajectories of the tire are not a circle, but an ellipse, as shown in Figure 20The limit of the longitudinal force is greater than the limit of the lateral force. Therefore, the longitudinal braking force limit of the tire can be set as follows:

Figure 20: Attaching an ellipse

The above constrained optimization problem is transformed into the form of quadratic programming QP

Where,; is the Hesse matrix, is the gradient vector, and is the constraint matrix, and the above problem can be usedLMI toolbox in MATLAB

Simulation verification of path tracking of unmanned forklifts

In order to verify the path-following stability of the LTI/LTV-MPC control strategy based on tire stiffness estimation of unmanned forklifts under different working conditions, in

The MATLAB/Simulink platform builds a tire transient lateral stiffness estimator and a LTI/LTV-MPC controller to combine them with the followingIn this paper, several different path following curves are designed to verify the reliability of the control strategy based on the actual application scenarios of forklifts.

Simulation verification of transient lateral deviation stiffness estimation and state stiffness prediction of multi-condition tires

In order to verify the effectiveness of the estimation algorithm proposed above in the multi-condition condition of unmanned forklift, the carsim/Simulink co-simulation was carried out under three typical working conditions. Based on the actual parameters and similarity criteria of the unmanned forklift, the following vehicle model parameters are selected, as shown in Table 4.

Table 4: Basic parameters of unmanned forklifts

(1) Simulation verification of tire transient stiffness estimation under different working conditions

The first working condition is set on the wet asphalt pavement, and the pavement adhesion coefficient is 04, the speed is set to a constant value of 15km / h, the steering wheel step input is 60°, the vehicle state parameters under this condition are shown in the following figure, as shown in Figure 21The rear wheel angular response shown is approximately 3.4°; The longitudinal speed of the vehicle is stable at 15km/h after 2.5s, which is in line with the initial set speed, which is a constant motion of the vehicle; As shown in Figure 23, the lateral acceleration stabilizes at 0.4 g after 1 s, indicating that the pavement is able to provide maximum adhesion without skidding, and the working condition is in a stable working area.

The transient lateral stiffness estimation results are shown in Figure 24, and the lateral bias stiffness begins to converge at about 1.5 s. It is fast and can meet the requirements of real-time vehicle status estimation.

Figure 21: Rear wheel angle (case 1).

Figure 22 Longitudinal velocity (Case 1).

Figure 23: Lateral acceleration (Case 1).

Estimation of the transient lateral stiffness of the attached pavement in Figure 24

The second working condition is set on a slippery ice and snow road with a road adhesion coefficient of 0.25, a speed of 10km/h, and a steering wheel step input of90°, the vehicle state parameters under this working condition are shown in the following figure, such asFigure 25 shows a rear wheel angular response of about 6.5°; The longitudinal speed of the vehicle stabilized at 10km/h after 2s, close to a constant speed. As shown in Figure 27, the lateral acceleration stabilized at 1 s0.48g, which means that the vehicle has slid at this time, and the working condition is in an unstable working area.

The transient lateral stiffness estimation results are shown in Figure 28, and the lateral stiffness begins to converge in about 2s. Its convergence speed is fast and can meet the requirements of real-time vehicle state estimation.

Fig. 25 Rear wheel rotation angle (case 2).

Figure 26 Longitudinal velocity (case 2).

Figure 27: Lateral acceleration (Case 2).

Fig. 28: Verification of transient lateral deviation stiffness of low-adhesion pavement

The third case is to set the variable adhesion coefficient pavement as shown in Figure 29: 0-16s is a pavement with a low adhesion coefficient (), and 16-32s is a pavement with a medium adhesion coefficient (); The steering wheel step input is shown in Figure 30; Set the speed to a constant 12km/h. As shown in Figure 31, the rear wheel angle input changes in sync with the steering wheel step input, and the longitudinal speed is as followsFigure 32 shows that it stabilized at 12 km/h after about 2s, the lateral acceleration includes a stable working area without sideslipping and a non-stable working area with sideslip under different operating conditions.

The transient lateral stiffness estimation results are shown in Figure 34, and the transition process of each two stages converges and tends to stabilize within about 1.5 s. It is shown that the designed transient lateral stiffness estimator can meet the estimation requirements of stable and unstable working areas.

Figure 29: Pavement adhesion coefficient

Figure 30: Steering wheel input

Fig. 31 Rear wheel angle (case 3).

Figure 32 Longitudinal velocity

Figure 33 Lateral acceleration (Case 3).

Fig. 34 Verification of transient lateral deviation stiffness of variable adhesion coefficient pavement

(2) Verification of the stiffness prediction method for the side deviation state

In order to verify the validity of the lateral deviation state stiffness prediction method designed in this paper, the adhesion coefficient of the test vehicle was 0 at a speed of 15 km/h5 road surface, and trajectory tracking at a speed of 10km/h on the road surface with an adhesion coefficient of 0.25, taking the rear wheel of a forklift as an exampleFigure 35 and Figure 36The comparison results between the actual and predicted values of the lateral deviation state stiffness under the two working conditions are given.

Figure 35 shows that the adhesion coefficient on the pavement is 0At a speed of 4,1 5km/h, the actual value of the tire side deviation state stiffness is 1 7.79sIt reaches a maximum near -52983N/rad, and the predicted value reaches a maximum around 17.99s, which is about-56008N/rad。 There is a phase difference of about 0.2 s between the actual value and the predicted value and a peak deviation of 3025 N/rad. From 18.2s, the phase difference and the numerical difference between the actual value and the predicted value gradually decreased. Figure 36 shows that the adhesion factor on the pavement is 0.25,1At 0km/h speed, the actual value of the stiffness of the tire side deviation state reaches the maximum around 19.80s, which is about -49173N/ rad, the predicted value reaches a maximum around 20.40s and is about -52544N/rad, there is a phase difference of about 0.6 s and a peak deviation of 3371 N/rad, which is 2At the beginning of 1.6s, the phase difference and the numerical difference between the two gradually decreased, and the actual value of the forklift fluctuated in 2 4.4s during operation due to the occurrence of side slip under this condition. The above results show that the error between the predicted value and the actual value of the state stiffness is small, which meets the requirements of tire force linearization in the predicted time domain.

Fig. 35: Estimation of state stiffness with an adhesion coefficient of 0.4 and a vehicle speed of 15 km/h

Fig. 36: Estimation of state stiffness with an adhesion coefficient of 0.25 and a vehicle speed of 10 km/h

Path tracing simulation verification of unmanned forklifts under different working conditions

(1) Path following simulation analysis of different vehicle speeds under low adhesion road surface

Due to the influence of factors such as road adhesion and forklift speed, unmanned forklifts will be in different working conditions during actual operation, so three working conditions are designed in this paper: 1. Adhesion coefficient0.3, vehicle speed 5km / h; 2, adhesion coefficient 0.3, speed 10km / h; 3. The adhesion coefficient is 02. The vehicle speed is 5km/h to verify the sideslip stability of the unmanned forklift in the process of path tracking on the low-attachment road surface.

Figure 37 Case-1 path following curve

Figure 38: Case 2 path following curve

Figure 39: Three-path trace curve for working conditions

Figure 40 Transverse displacement deviation

Figure 41: Lateral velocity deviation

Figure 42: Yaw angular velocity deviation

From Figure 37-Figure 39It can be seen that the control method proposed in this paper has a good tracking effect on the desired path under different working conditions, and the unmanned forklift can follow the desired path and quickly stabilize, and the tracking effect does not fail due to the gradual complexity of the working conditions. In addition, according to the lateral displacement deviation shown in Figure 40, it can be seen that the unmanned forklift does not have a large displacement deviation and maintains good tracking stability. Fig. 41 shows the lateral speed deviation, which is in the range of (-0.3m/s, 0.25m/s), indicating that the proposed control method ensures the good driving stability of the unmanned forklift. Figure 42 shows the yaw angular velocity deviation, which is at (-0.1 rad/s, 0.05 rad/s). ), it shows that the vehicle maintains good handling stability under several working conditions, and the tire does not have obvious side-slip instability, and the proposed control method can effectively control the unmanned forklift to be in a stable state and avoid the possible occurrence in the process of path following on the low-adhesion road surfaceSideslip instability. Therefore, the control method proposed in this paper has good stability in the process of road path tracking of unmanned forklifts, and can make the target vehicle follow the desired path well on the low-attachment road surface.

(2) Path following simulation analysis under different control methods

In order to further verify the reliability of the control method proposed in this paper, the following three working conditions are designed: the first working condition is 10km/h, and the adhesion coefficient is 0.4； Working condition two: the speed of 10km / h, adhesion coefficient 03； Working condition three: the speed of the vehicle is 10 km/h, and the adhesion coefficient is 02； LTI-MPC control, NMPC control and the LTI/LTV-MPV control in this paper were used to trace the path of unmanned forklifts on low-attachment roadsComparative simulation tests were conducted.

Figure 43: Comparison curve of path tracing under working conditions

Figure 44: Comparison curve of path tracing under case 2

Figure 45: Three-path tracing comparison curve for working conditions

Under the three working conditions, the tracking effect under the action of different path tracking controllers is shown in the above three figures, and the three control methods under the condition of one condition complete the tracking of the desired path, but the tracking effect of LTI-MPC is significantly worse than that of the other two control methodsUnder the conditions of working conditions 2 and 3, the LTI-MPC control has an obvious deviation at the longitudinal displacement of the desired path of 50 m, and the tracking has not been realized. It is shown that at a longitudinal displacement of 50 m, backslip instability occurred due to the inability of the road surface to provide sufficient lateral adhesion for the unmanned forklift, and NMPC withIn the LTI/LTV-MPC control method, the tracking of the desired path is realized under the simulation conditions of case 2 and case 3, but there is a certain phase difference between the two. This is due to the complexity of the calculation of N PMC, and the time delay generated by the controller in the solution process, resulting in a delay in path tracing. In order to further compare the control effects of LTI/LTV-MPC and NMPC controllers, the path tracing of the two under the above three working conditions was comparedStability parameters and accuracy parameter deviations are compared, Figure 46-Figure 49 is the deviation curve of each parameter under the working condition5. Analysis of path following parameters under working conditions; Figure 50 - Figure 53is the deviation curve of each parameter under case 2, and Table 6 is the analysis of path following parameters under condition 2. Figure 54 - Figure 57It is a diagram and table of the deviation curves of each parameter under the third working condition 7 is the analysis of path following parameters under three working conditions.

Fig. 46 Condition 1 Lateral Displacement Deviation Curve

Fig. 47: Condition 1 heading angle deviation curve

Figure 48: Lateral velocity deviation curve for one case

Fig. 49 Condition 1 yaw angular velocity deviation curve

In order to accurately analyze the tracking effect of NMPC and LTI/LTV-MPC control methods in the path tracing process, the above is takenThe coordinates of the peaks and valleys of the respective parameters of the control method in the path tracing process are shown in the following table.

Table 5 Path Tracing Parameters for Condition 1

Fig. 50 Condition 2 lateral displacement deviation curve

Figure 51: Condition 2 heading angle deviation curve

Figure 52: Lateral velocity deviation curve for case 2

Fig. 53 Case 2 yaw angular velocity deviation curve

Table 6: Path Tracing Parameters for Case 2

Fig. 54 Condition 3 lateral displacement deviation curve

Fig. 55 Condition 3 Heading Angle Deviation Curve

Figure 56: Lateral velocity deviation curve for three operating conditions

Fig. 57: Deviation curve of yaw angular velocity in case 3

Table 7: Path Tracing Parameters for Condition 3

From Figure 46 - Figure 57and Tables 5-7 show that The observed parameters are all important parameters to characterize the stability, reliability and accuracy of the target forklift in the path following process. It can be obtained from the diagram of case 1, under the condition 1 simulation condition, under the control of NMPC Compared with the LTI/LTV-MPC control, the peak-to-valley value of the parameter is delayed by 1-1.8 sThe numerical deviation is in the range of 2%-14%, and the real-time effect and stability of path tracing are reduced compared with the control method proposed in this paper. According to the correlation chart of case 2, it can be seen that the peak and trough values of the parameters under the control of NMPC are compared with those of LTI/LTV-MPC under the simulation conditions of case 2The control delay is 1-3s, and the deviation of the parameter peak-to-valley value is in the range of 1.4%-30%, which is further increased compared with the range of working conditions, and the path tracking stability and accuracy are reduced. However, in the simulation of case 3, the peak and trough values of various parameters under the control of NMPC are delayed by 1 compared with those controlled by LTI/LTV-MPC.5-2.5s, the numerical deviation is 3.4%-21.5%, and it is worth noting that compared with case 2, the peak-to-valley deviation of more parameters is 20%, which indicates that the accuracy and stability of the control at this time are further reduced.

In summary, the simulation results and parameter analysis obtained from the three working conditions show that the LTI/LTV-MPC control method proposed in this paper can be used for low-attachment road path following in unmanned forkliftsThe stability parameters and accuracy parameters of the target vehicle can be controlled within a reasonable range, and good path tracking stability and high tracking accuracy can be maintained.

Hardware-in-the-loop test verification and analysis

The software part of the bench is mainly composed of PreScan and MATLAB/Simulink, and the hardware part is mainly composed of corner sensors, host computers, and CANTruck and steering column composition; Among them, PreScan provides vehicle information and scene building environment for path tracing, and the mixing program written in C++ language is compiled in MATLAB to implement PreScanThe corner signal interacts with the CAN card, and the CAN card converts the corner signal into a CAN signal and transmits it to the controller. The controller runs a control strategy to control the rotation of the steering string to achieve control of accuracy and stability during path tracing.

Figure 58: Hardware-in-the-loop system and signal interaction

(1) Test condition setting

In order to reflect the effect of hardware-in-the-loop test in the path tracing process, the road attachment conditions are determined by CarSim, the vehicle modeling and environmental scene construction are provided by PreScan, the control object is provided by the hardware-in-the-loop test bench, and the vehicle speed is maintainedDriving at a constant speed of 10km/h, the road adhesion coefficient of 0.2 and 0.4 is set respectively, and the LTI-MPC is applied under this working condition, NMPC and the LTI/LTV-MPC control strategies and algorithms in this paper were tested, and the test results were compared and analyzed.

Lateral displacement deviation

Yaw angular velocity deviation

Figure 59: Hardware-in-the-loop comparison test with an adhesion factor of 0.4

Because the control effect of LTI-MPC has been obviously unstable under the condition of medium adhesion coefficient of 0.4, it is even better for NMPC under the condition of low adhesion coefficient of 0.2It was compared with the hardware-in-the-loop test of LTI/LTV-MPC.

Lateral displacement deviation

(b) Yaw angular velocity deviation

Figure 60: Hardware-in-the-loop comparison test with an adhesion factor of 0.2

(2) Analysis of experimental results

As can be seen from (a) in Figure 59, compared to LTI-MPCcontroller, the control strategy proposed in this paper is significantly lower than the former in terms of deviation peak and trough, and it can be found that the longitudinal displacement of the former is about 85m, and the lateral displacement is significantly higher than the peak value of the LTI/LTV-MPC controller and continues to maintain a high position, and the tracking accuracy has been lost. Compared with the NMPC controller, although the control strategy in this paper is not much different in terms of peak and valley values, it is significantly better than the former in terms of calculating the response delay, and the phase of the two peaks and one trough is about 1/5 earlier than the test cycle time of the former. It shows that the control strategy proposed in this paper has certain advantages in terms of computational speed and response. In addition, according to (b) in Figure 59, LTI-MPC andThe peaks of the NMPC controller were 0.072 and 0.058, respectively, while the peak of the LTI/LTV-MPC controller was0.049, which is improved by 32% and 15.5% respectively in terms of stability control, and it can be seen that the control strategy proposed in this paper is better than the tracking accuracy and stability control in the path following hardware-in-the-loop test under the medium attachment condition Both the LTI-MPC and NMPC controllers improve the control effect to varying degrees, which verifies the rationality and feasibility of the LTI/LTV-MPC controller proposed in this paper.

According to (a) in Figure 60, compared with the NMPC controller, In this paper, the peak lateral displacement deviation of the LTI/LTV-MPC controller is about 0.38m, and that of the NMPC controller is 0.51m, the path tracing accuracy is improved by 25.4%, and the control strategy in this paper also has an advantage over NMPC in terms of calculation speed, and according to the curve trend analysis in (b), it can be seen that In terms of vehicle stability control for path tracking, the LTI/LTV-MPC controller can maintain a relatively stable yaw angular velocity deviation, which can be clearly seen in the peak and valley value of the deviation, combined with the above analysis, it can be shown that the control strategy proposed in this paper can achieve good control effect in both accuracy and stability.

conclusion

(1) A three-degree-of-freedom dynamic model of an intelligent industrial vehicle was established, and a magic formula tire model was established to study the influence of tire force on the vehicle path following process. In addition, based on the characteristics of tire deviation, the transient deviation stiffness is used as the basis for judgment to divide the stable and unstable working areas to achieve accurate judgment of the stable state of the vehicle, and the adaptive forgetting factor least squares method is used to estimate the transient deviation stiffness, and the concept of state stiffness is introduced to linearize the tire model to achieve accurate prediction of tire force.

(2) Based on the divided stable and unstable working areas, the LTI-MPC and LTV-MPC controllers were designed respectively, and the comprehensive design of the two was proposedThe LTI/LTV-MPC controller realizes targeted calculation and solving, and proposes the processing method in the case of transient lateral deviation stiffness abrupt change to ensure the accuracy of the controller.

(3) Build a hardware-in-the-loop test platform, based on PreScan, CarSim and MATLAB/ The signal interaction between Simulink software and PCIe-CAN card, host computer, steering bench and other hardware level realizes the implementation of the control strategy in the hardware-in-the-loop test, which verifies the reliability and accuracy of the path following stability control strategy for intelligent industrial vehicles proposed in this paper.

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