A Simulation Analysis Method for the Trajectory Tracking Model of Overloaded AGV Based on the Koopman Operator

Abstract

The invention discloses a simulation analysis method of a heavy-duty AGV trajectory tracking model based on a Koopman operator, comprising the following steps: step one: data preprocessing; step two: rotating the matrix and converting the coordinate system, converting the body coordinate system of the heavy-duty AGV to the world coordinate system; step three: determining the basis function and the dimension of the increased dimension; step four: simulating and verifying an MPC controller, wherein the method performs data preprocessing according to constraints, solves the rotation matrix, determines the basis function and the dimension of the Koopman model, compares with the Lagrangian model, quantitatively analyzes the accuracy of the Koopman high-dimensional linear model, and indicates that the Koopman model has higher accuracy; finally, through the simulation and experiment of two reference trajectories, a straight line and a circle, it is proved that the MPC controller based on Koopman can enable the heavy-duty AGV to complete the trajectory tracking control well, verifies the effectiveness of the controller, and effectively reduces the complexity of the heavy-duty AGV modeling.

Description

Technical Field

[0001] The present invention relates to the technical field of AGV control, and particularly to a simulation analysis method for a heavy-load AGV trajectory tracking model based on the Koopman operator. Background Art

[0002] Currently, the Koopman operator has been applied to the robot control system, and its application directions include: 1. Using the Koopman operator to obtain a linearizable data-driven model for an unknown dynamic process. The experimental results of the Sphero SPRK robot explore that the increased complexity in the basis function improves the performance of open-loop and closed-loop controllers under various terrains (including sandy land). 2. Using the Koopman operator to design an active learning strategy for a robot system, and then applying the information theory method to the learning process of the Koopman high-dimensional model of the dynamic system, improving the information utilization rate of the data, and deriving an active learning controller for the robot. 3. Using the Koopman operator to propose a data-driven linear embedding method for nonlinear systems, which has an effect on the modeling and control of a tail-driven robotic fish. This is the first experimental verification of Koopman-based LQR control. 4. A method for constructing an explicit linear dynamics model of a soft robot based on the Koopman operator, and implementing a model-based linear control method to control the soft robot. Different from "black box" input-output mappings such as neural networks, this method is data-driven, and a model predictive controller is designed based on this method.

[0003] However, currently, the development of the Koopman operator in the field of heavy-load AGV control is still blank, and there is a lack of a systematic verification method for a heavy-load AGV model constructed based on the Koopman operator, and the effectiveness of this model cannot be verified yet.

[0004] Therefore, how to design a simulation analysis method for a heavy-load AGV trajectory tracking model based on the Koopman operator to verify the effectiveness of this model has become an urgent technical problem to be solved by those skilled in the art. Summary of the Invention

[0005] The purpose of the present invention is to solve the deficiencies of the prior art and propose a simulation analysis method for a heavy-load AGV trajectory tracking model based on the Koopman operator. This method proves through the simulation and experiment of two reference trajectories, namely a straight line and a circle, that the Koopman-based MPC controller can enable the heavy-load AGV to complete trajectory tracking control well, verifies the effectiveness of this controller, and effectively reduces the complexity of heavy-load AGV modeling.

[0006] A simulation analysis method for a heavy-load AGV trajectory tracking model based on the Koopman operator of the present invention includes the following steps:

[0007] Step 1: Data preprocessing;

[0008] Step 2: Rotate the matrix and transform the coordinate system to convert the vehicle body coordinate system where the heavy-duty AGV is stressed into the world coordinate system;

[0009] Step 3: Determine the basis function and the dimension elevation dimension;

[0010] Step 4: Simulate and verify the MPC controller, including:

[0011] (1) Straight-line trajectory

[0012] The straight-line desired trajectory equation is shown in Equation (1):

[0013]

[0014] In the present invention, the initial state of the desired trajectory is selected as: q d0 =(x d (0), y d (0), θ d (0))=(0, 0, π / 6)

[0015] The initial state of the heavy-duty AGV is: q r0 =(x r (0), y r (0), θ r (0))=(0, 0, 0)

[0016] The MPC parameters are set as:

[0017] Q a =0.1*diag(1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1)

[0018]

[0019] R c =1×10 -6 *diag(1, 1, 0.1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1), (2)

[0020] (2) Circular trajectory

[0021] The circular desired trajectory equation is shown in Equation (3):

[0022]

[0023] The initial state of the desired trajectory is selected as:

[0024]

[0025] The initial state of the heavy-duty AGV is as follows:

[0026]

[0027] Moreover, in step one, the preprocessed data includes n pieces of motion trajectory data collected during the simulation process. The sampling time for each trajectory is t, and the sampling period ΔT = 0.01 s, with a total of 500*n data. Among them, 400*n data from 0.8n pieces of motion trajectory data are used to solve the Koopman high-dimensional linear model matrices A, B, and C, and 100*n data from the other 0.2n pieces of motion trajectory data are used to evaluate the approximation degree between the solved Koopman high-dimensional linear model and the true Lagrangian dynamics model; and the original dimension of the heavy-duty AGV is defined as 30 dimensions. The first 15 dimensions are the position information of the x coordinate, y coordinate, inclination angle θ, and the angle of the motor in the heavy-duty AGV and the last 15 dimensions are the velocity information of the x-direction velocity y-direction velocity body angular velocity and the angular velocity of the motor in the heavy-duty AGV

[0028] Moreover, the coordinate system conversion method in step two is as follows:

[0029] In the body coordinate system, the force analysis of the heavy-duty AGV is shown in Equation (6):

[0030]

[0031] where d 1 =-s cosγ 1 +h sinγ 1 -d, d 2 =-s cosγ 1 +h sinγ 1 +d, d 3 =s cosγ 2 +hsinγ 2 -d, d 4 =s cosγ 2 +h sinγ 2 +d, d 5 =-s cosγ 3 -d, d 6 =-s cosγ 3 +d, d 7 =s cosγ 4 -d, d 8 =s cosγ 4 +d, d 9 =-s cosγ 5-h sinγ 5 -d, d 10 =-s cosγ 5 -h sinγ 5 +d, d 11 =s cosγ 6 -hsinγ 6 -d, d 12 =s cosγ 6 -h sinγ 6 +d;

[0032] The rotation matrix from the vehicle body coordinate system to the world coordinate system is:

[0033]

[0034] It is obtained that:

[0035] The rotation matrix C is: Among them,

[0036] Moreover, in step three, a thin plate spline radial basis function with the center located at x 0 is selected as the basis function, that is, ψ(x) = ||x - x 0 || 2 log(||x - x 0 ||); x 0 is a constant vector with elements uniformly distributed in a given interval, and the dimension of the Koopman model is set according to the minimum of the root mean square error and the relative root mean square error.

[0037] The advantages and technical effects of the present invention are as follows:

[0038] A simulation analysis method for a heavy-duty AGV trajectory tracking model based on the Koopman operator according to the present invention preprocesses data according to the constraint situation, solves the rotation matrix, determines the basis function and the dimension elevation dimension of the Koopman model, compares with the Lagrangian model, quantitatively analyzes the accuracy of the Koopman high-dimensional linear model, and shows that the Koopman model has higher accuracy; finally, through the simulation and experiment of two reference trajectories of a straight line and a circle, it is proved that the MPC controller based on Koopman can enable the heavy-duty AGV to complete the trajectory tracking control well, verifies the effectiveness of the controller, and effectively reduces the complexity of the heavy-duty AGV modeling. Description of the Drawings

[0039] Figure 1 It is a graph of RMSE and RRMSE of the heavy-duty AGV in the present invention under different dimension elevation dimensions;

[0040] Figure 2 This is the comparison chart of the straight-line trajectory tracking effect of the heavy-duty AGV of the present invention;

[0041] Figure 3 This is the comparison chart of the straight-line trajectory pose tracking effect of the heavy-duty AGV of the present invention;

[0042] Figure 4 This is the analysis chart of the straight-line trajectory pose channel error results of the heavy-duty AGV of the present invention;

[0043] Figure 5 This is the motor output torque chart of the straight-line trajectory of the heavy-duty AGV in the present invention;

[0044] Figure 6 This is the comparison chart of the circular trajectory tracking effect of the heavy-duty AGV in the present invention;

[0045] Figure 7 This is the comparison chart of the circular trajectory pose tracking effect of the heavy-duty AGV in the present invention;

[0046] Figure 8 This is the analysis chart of the circular trajectory pose channel error results of the heavy-duty AGV in the present invention;

[0047] Figure 9 This is the motor output torque chart of the circular trajectory of the heavy-duty AGV in the present invention;

[0048] Figure 10 This is the experimental flow chart of the heavy-duty AGV in the present invention;

[0049] Figure 11 This is the flow chart of the comparison method between the Koopman model and the Lagrangian dynamics model in the present invention;

[0050] Figure 12 This is the real-time trajectory display chart of the heavy-duty AGV in the ROS rviz module of the present invention;

[0051] Figure 13 This is the motor output torque chart of the heavy-duty AGV of the present invention;

[0052] Figure 14 This is the three-channel trajectory comparison chart of the heavy-duty AGV of the present invention;

[0053] Figure 15 This is the trajectory comparison chart of the heavy-duty AGV of the present invention;

[0054] Figure 16 This is the experimental result comparison chart of the straight-line trajectory of the heavy-duty AGV of the present invention;

[0055] Figure 17 This is the experimental result comparison chart of the straight-line trajectory pose of the heavy-duty AGV of the present invention;

[0056] Figure 18Analysis diagram of the pose channel error of the heavy-duty AGV straight trajectory for the present invention;

[0057] Figure 19 Motor output torque diagram of the heavy-duty AGV straight trajectory for the present invention;

[0058] Figure 20 Comparison diagram of the experimental results of the heavy-duty AGV circular trajectory for the present invention;

[0059] Figure 21 Comparison diagram of the pose experimental results of the heavy-duty AGV circular trajectory for the present invention;

[0060] Figure 22 Analysis diagram of the pose channel error of the heavy-duty AGV circular trajectory for the present invention;

[0061] Figure 23 Motor output torque diagram of the heavy-duty AGV circular trajectory for the present invention. Detailed implementation manners

[0062] The embodiments of the present invention will be described in detail below. The examples of the embodiments are shown in the drawings, in which the same or similar reference numerals denote the same or similar elements or elements having the same or similar functions throughout. The embodiments described below by referring to the drawings are exemplary and are only used to explain the present invention and should not be construed as a limitation to the present invention.

[0063] In the description of the present invention, it should be understood that the orientation or positional relationship indicated by the terms "center", "longitudinal", "transverse", "upper", "lower", "front", "rear", "left", "right", "vertical", "horizontal", "top", "bottom", "inner", "outer", etc. is based on the orientation or positional relationship shown in the drawings, and is only for the convenience of describing the present invention and simplifying the description, rather than indicating or implying that the device or element referred to must have a specific orientation, be constructed and operated in a specific orientation, and thus should not be construed as a limitation to the present invention. In addition, the terms "first" and "second" are only used for descriptive purposes and should not be construed as indicating or implying relative importance.

[0064] A simulation analysis method for a heavy-duty AGV trajectory tracking model based on the Koopman operator of the present invention includes the following steps:

[0065] 1. Data preprocessing and rotation matrix: A total of 50 motion trajectory data are collected during the simulation process. The sampling time for each trajectory is 5s, and the sampling period ΔT = 0.01s, resulting in a total of 25,000 data. Among them, 40 trajectory data (20,000) are used to solve the Koopman high-dimensional linear model (matrix A, matrix B, and matrix C), and the other 10 trajectory data (5,000) are test data used to evaluate the approximation degree between the solved Koopman high-dimensional linear model and the true Lagrangian dynamics model. Since the output torques of 12 motors need to satisfy the above constraints, a straight line and a circle are respectively used to test the approximation accuracy of the model. The original dimension of the heavy-load AGV is 30 dimensions. The first 15 dimensions are position information (the x coordinate, y coordinate, inclination angle θ of the heavy-load AGV, and the angles of 12 motors ), and the last 15 dimensions are velocity information (the velocity of the heavy-load AGV in the x direction the velocity in the y direction the angular velocity of the vehicle body and the angular velocities of 12 motors ).

[0066] 2. The data used during the experiment are the input torques of 12 motors, the current state and the next state of the heavy-load AGV in the same coordinate system. The selection of training data needs to satisfy being in the same coordinate system, which can effectively avoid uncertain factors caused by changes in data directions. The current state and the next state of the heavy-load AGV are both represented in the world coordinate system, and the input torques of 12 motors are in the vehicle body coordinate system. Therefore, in the present invention, the world coordinate system is selected as the unified coordinate system, and the input torques of 12 motors in the vehicle body coordinate system need to be converted into control inputs in the world coordinate system.

[0067] In the vehicle body coordinate system, the force analysis of the heavy-load AGV is shown in Equation (6):

[0068]

[0069] where, d 1 =-s cosγ 1 +h sinγ 1 -d, d 2 =-s cosγ 1 +h sinγ 1 +d, d 3 =s cosγ 2 +hsinγ 2 -d, d 4 =s cosγ 2 +h sinγ 2 +d, d 5 =-s cosγ 3 -d, d 6 =-s cosγ3 +d, d 7 = s cosγ 4 -d, d 8 = s cosγ 4 +d, d 9 = -s cosγ 5 -h sinγ 5 -d, d 10 = -s cosγ 5 -h sinγ 5 +d, d 11 = s cosγ 6 -hsinγ 6 -d, d 12 = s cosγ 6 -h sinγ 6 +d;

[0070] The rotation matrix from the vehicle body coordinate system to the world coordinate system is:

[0071]

[0072] It is obtained that:

[0073] The rotation matrix C is: Among them,

[0074] 3. Determination of basis functions and determination of the dimension of dimensionality increase

[0075] For the selection of basis functions, the commonly used basis functions at present are Hermite polynomials and radial basis functions. Hermite polynomials are generally used for data subject to normal distribution, and radial basis functions are generally used for problems defined on irregular domains. Therefore, the thin plate spline radial basis function centered at x 0 is selected as the basis function, that is, ψ(x) = ||x - x 0 || 2 log(||x - x 0 ||). x 0 is a constant vector with elements uniformly distributed within a given interval.

[0076] To determine the dimension of dimension elevation during the experiment, relevant experiments were designed, and simulation data was used for system testing. The test dimension range was 0 - 20 dimensions (corresponding to the actual Koopman model dimension of 30 - 50 dimensions). When exceeding 20 dimensions, both the Root Mean Squared Error (RMSE) and the Relative Root Mean Squared Error (RRMSE) exceeded 100%, and the accuracy of the system model decreased significantly, so it was not adopted. During the experiment, each dimension was tested 10 times respectively, and the RMSE and RRMSE of the Koopman model and the real model with different dimensions were tested. Finally, the average value of 10 times was calculated. The results showed that the RMSE and RRMSE increased with the increase of the test dimension. When the test dimension was 1 dimension, that is, when the actual Koopman model dimension was 31 dimensions, the RMSE and RRMSE were the smallest. Therefore, the dimension of dimension elevation was set to 1 dimension, and the actual Koopman model dimension was 31 dimensions. The RMSE and RRMSE of the heavy-load AGV under different dimensions of dimension elevation are as Figure 1 shown.

[0077] 4. Simulation verification of the MPC controller:

[0078] (1) Straight-line trajectory

[0079] The straight-line desired trajectory equation is shown in Equation (1):

[0080]

[0081] In the present invention, the initial state of the desired trajectory is selected as: q d0 =(x d (0), y d (0), θ d (0))=(0, 0, π / 6)

[0082] The initial state of the heavy-load AGV is: q r0 =(x r (0), y r (0), θ r (0))=(0, 0, 0)

[0083] The MPC parameters are set as:

[0084] Q a =0.1*diag(1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1)

[0085]

[0086] R c =1×10 -6*diag(1, 1, 0.1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1), (2)

[0087] The simulation results of the heavy-duty AGV's straight-line trajectory are as follows Figures 2 - 4 shown. From Figures 2 - 4 it can be seen that the MPC controller based on the Koopman model can well complete the tracking control of the heavy-duty AGV's straight-line trajectory, and has a high-precision trajectory tracking effect. The error in the x direction is smaller than that in the y direction. The MPC control method based on the Koopman model has a certain gap with the desired trajectory, theoretically proving the feasibility of using the heavy-duty AGV Koopman model to achieve trajectory tracking control.

[0088] Figure 5 This is the control input torque during the simulation, that is, the output torques of 12 motors. This torque is the output torque at the motor end rather than after the reducer. From Figure 5 it can be known that the MPC controller based on the Koopman model can output a stable torque during the simulation.

[0089] (2) Circular trajectory

[0090] The circular desired trajectory equation is shown in Equation (3):

[0091]

[0092] The initial state of the desired trajectory is selected as:

[0093]

[0094] The initial state of the heavy-duty AGV is:

[0095]

[0096] The MPC parameters are the same as those of the straight-line reference trajectory. The simulation results of the circular trajectory are as follows Figures 6 - 8 shown. From Figures 6 - 8 it can be seen that the MPC controller based on the Koopman model can well complete the tracking control of the heavy-duty AGV's circular trajectory, and has a high-precision trajectory tracking effect. The errors in the x and y directions are periodic, and the maximum error is about 13 mm. The error in the θ direction is about -5 mm. The MPC control method based on the Koopman model has a certain gap with the desired trajectory.

[0097] From Figure 9 it can be seen that the MPC controller based on the Koopman model can output a stable torque during the simulation, and the output torque is larger than that of the straight-line trajectory.

[0098] In summary, the MPC controller based on the Koopman model shows satisfactory performance without any prior knowledge and model information, and can well complete the trajectory tracking control of heavy-load AGVs.

[0099] It should be noted that when navigating, the heavy-load AGV plans its own trajectory according to the targets issued by the dispatching software, and uses a lidar for real-time obstacle avoidance during the movement process. In actual on-site applications, the walking path of the heavy-load AGV can be satisfied by a combination of straight lines and arcs. The technical parameters of the heavy-load AGV are shown in the following table.

[0100] Technical Parameters of Heavy-Load AGV

[0101]

[0102] Before simulation and experiments, it is necessary to use training data to train the Koopman model and solve the Koopman high-dimensional linear model. Since the experimental data is in the world coordinate system, the simulation data also needs to be converted to the world coordinate system to ensure the consistency of the simulation and experimental data and the consistency of the data source of the Koopman model. At the same time, it is necessary to determine the basis function and the dimension of the dimension-raising function selected by the Koopman model; during the experiment, compare the fitting degree of the Koopman model, the Lagrangian dynamics model and the real model to further evaluate the accuracy of the Koopman model.

[0103] To more clearly illustrate the specific implementation manner of the present invention, the following provides an embodiment:

[0104] 1. Experimental Environment Setup

[0105] The experimental environment setup of the present invention includes the introduction of the experimental object, the acquisition of experimental data, the experimental constraint conditions, and the accuracy evaluation of the experimental model.

[0106] (1) Introduction of Experimental Object and Process

[0107] In the present invention, the heavy-load AGV platform consists of 6 groups of differential drive units, and each group of differential drive units consists of two drive wheels. The drive wheel motors use DC servo motors, with a single motor power of 1.5 kW, a reduction ratio of 100, and a power supply voltage of 48V. The core controller of the heavy-load AGV uses UNO-3285G, the core processor is i7-6822EQ, with 8GB of memory, and 2 CAN cards are built-in. One CAN card is connected to the driver and the motor end encoder, and the other CAN card is connected to the rotary encoders of 6 groups of differential drive units. The control system architecture is shown in the second chapter. The experimental flow chart of the heavy-load AGV is as Figure 10 shown.

[0108] (2) Input Constraint Conditions and Data Acquisition

[0109] Assume that the overloaded AGV moves normally without slipping and skidding. To select the appropriate input torque, the following strategic restrictions are imposed.

[0110] Strategic restrictions:

[0111] ① For the two motors of each differential drive unit, the output torque directions must be the same. If they are different, the differential drive unit of this group will rotate in place and the overloaded AGV will slip.

[0112] ② The output torques T1 and T2 of one group of differential drive units, the steering angle γ of another group of differential drive units 2 and the torques T3, T5, T7, T9, T11 of one motor in the remaining five groups of differential drive units. The instantaneous center can be determined, and the output torques and steering angles of the remaining 5 groups of differential drive units can be obtained according to the sine theorem.

[0113]

[0114]

[0115]

[0116]

[0117]

[0118] Among them, l i (i = 1, 2, 3, 4, 5, 6) represents the distance from the central axis of the differential drive unit i to the instantaneous center of rotation ICR, γ i (i = 1, 2, 3, 4, 5, 6) represents the rotation angle of the differential drive unit i, h represents the distance between the rotation centers of two adjacent differential drive units in the length direction of the overloaded AGV, and s represents half of the distance between the rotation centers of two differential drive units in the width direction of the overloaded AGV.

[0119] The randomly varying input torque during the experiment will, on the one hand, damage the motor, and on the other hand, conflict with the kinematic constraints of the overloaded AGV. Therefore, the data in the experiment is obtained by manually remotely controlling the overloaded AGV to move along a random path. There are encoders at the ends of each motor for feedback, and rotary encoders are installed at the middle positions of each differential drive unit. The speed and torque of the overloaded AGV can be read in real time in the control program. A total of 50 sets of motion trajectory data were collected during the experiment. The sampling time for each trajectory is 20 s, the sampling period ΔT = 0.05 s, and the total number of data is 20,000. During the fitting process of the Koopman high-dimensional linear model, 40 sets of trajectory data were selected, and the remaining 10 sets of trajectory data were used as test data to evaluate the approximation degree between the solved Koopman high-dimensional linear model and the Lagrangian dynamics model.

[0120] 2. Evaluation of Experimental Model Accuracy

[0121] When comparing the Koopman model with the Lagrangian dynamics model, it is necessary to ensure that the two groups of models have the same control input and the same initial state. Due to the input constraint problem, the simulation of random data cannot meet the actual engineering situation. Therefore, model comparison experiments are carried out for two cases: straight line and circle. These data are given by the data selection in the present invention. The simulation requires that the sampling period and sampling time are unified. The output trajectory is evaluated by calculating RMSE and RRMSE as evaluation indicators to measure the approximation accuracy of the Koopman high-dimensional linear model and the Lagrangian dynamics model. The method for evaluating model accuracy is as Figure 11 shown.

[0122]

[0123]

[0124] Among them, x ki represents the prediction data based on Koopman, x ti represents the real data of the heavy-duty AGV, N is the total number of experimental data acquisition points, represents the state norm of xyθ.

[0125] In order to verify the accuracy of the Koopman model, a model accuracy experiment was designed. During the experiment, 50 sets of random motion trajectory data of the heavy-duty AGV were collected. The sampling time for each motion trajectory was 30 s, and the sampling interval was 0.05 s. Among them, 40 sets of random motion trajectory data were used to construct the Koopman high-dimensional linear model, and the remaining 10 sets of random motion trajectory data were used to evaluate the accuracy of the model.

[0126] The model accuracy comparison includes three trajectories: the real trajectory, the Koopman model trajectory, and the Lagrangian model trajectory. The real trajectory is obtained by controlling the motion of the heavy-duty AGV, collecting data in real time and displaying it in the ROS rviz module, as Figure 12 shown. Figure 13 is the motor output torque diagram of the real trajectory of the heavy-duty AGV. The Koopman model trajectory is the trajectory calculated by the Koopman high-dimensional linear model. The Lagrangian model trajectory is the trajectory calculated by Lagrangian dynamics.

[0127] The three-channel trajectory comparison diagram and the XY trajectory comparison diagram of the heavy-duty AGV are shown in Figures 14 and Figure 15As shown in the figure. It can be seen from the figure that the Koopman-based model is closer to the true dynamic model than the Lagrangian model, providing a theoretical basis for subsequent algorithm research based on the Koopman model. In order to quantitatively illustrate the advantages of the modeling accuracy of the Koopman model, RMSE and RRMSE are used to measure the model accuracy, and the results are shown in the following table.

[0128] Comparison table of the accuracy of the Koopman model and the Lagrange model for 10 groups of experimental data

[0129]

[0130]

[0131] The remaining 10 groups of random motion trajectory data are used to test the model accuracy, and the test indicators are RMSE and RRMSE. It can be seen from the above that both the root mean square error and the relative root mean square error of the Koopman model are better than those of the Lagrange model. The RMSE of the Koopman model is on average 24.86% higher than that of the Lagrange model, and the RRMSE of the Koopman model is on average 3.24% higher than that of the Lagrange model. This shows that the prediction performance of the Koopman model is better than that of the Lagrange model, and the Koopman model is closer to the true model.

[0132] 3. Experimental verification of the MPC controller:

[0133] An experimental test is carried out to verify the trajectory tracking performance of the Koopman-based MPC controller. The trajectory used in the experiment is the same as that in the simulation. In the experiment, the parameters used in the MPC are set as follows:

[0134] Q a = diag(6,8,8,8,8,8,8,8,8,8,8,8,8,8,8)

[0135] Q N = 100*diag(2,3,3,3,3,3,3,3,3,3,3,3,3,3,3)

[0136] R c = 1×10 -5 *diag(5,5,4,7,7,7,7,7,7,7,7,7,7,7,7)

[0137] (1) Straight-line trajectory

[0138] The experimental results of the heavy-duty AGV with the expected trajectory being a straight line are as Figures 16 - 19 shown.

[0139] From Figure 16 and Figure 17 it can be seen that, in the case of no initial error, the MPC controller based on the Koopman model can well complete the tracking control of the straight-line trajectory of the heavy-load AGV. Figure 18 It can be seen that there are still certain errors in this controller. The error in the x direction is about 15 mm, the error in the y direction is about 10 mm, and the error in the θ direction is about 0.001 rad. Figure 19 This is the motor output torque diagram of the heavy-load AGV during the experiment. It can be seen from the figure that this MPC controller can output relatively stable torque during the experiment.

[0140] (2) Circular trajectory

[0141] The experimental results of the heavy-load AGV with a circular desired trajectory are as Figures 20 - 23 shown. From Figure 20 and Figure 21 it can be seen that, in the case of no initial error, the MPC controller based on the Koopman model can well complete the tracking control of the circular trajectory of the heavy-load AGV. From Figure 22 it can be seen that there are still certain errors in this controller. The error in the x direction is about 17 mm, the error in the y direction is about 12 mm, and the error in the θ direction is about 0.002 rad. Figure 23 This is the motor output torque diagram of the heavy-load AGV during the circular trajectory experiment. It can be seen from the figure that there are some errors in the output torque of this MPC controller during the experiment, and the output torque is greatly affected by environmental disturbances.

[0142] In summary, without any prior knowledge and model information, the MPC controller based on the Koopman model can well complete the trajectory tracking experiments of two reference trajectories, straight line and circle, showing satisfactory performance. The experiment shows the effectiveness of the proposed MPC controller based on the Koopman model, and this method provides an effective solution for realizing complex robot modeling and control.

[0143] Finally, the unmentioned parts of the present invention all adopt mature products and mature technical means in the prior art.

[0144] In the description of this specification, the description with reference to terms such as "one embodiment", "some embodiments", "example", "specific example", or "some examples" means that the specific features, structures, materials, or characteristics described in connection with the embodiment or example are included in the embodiments or examples of the present invention.

[0145] Although embodiments of the present invention have been shown and described, those of ordinary skill in the art can understand that various changes, modifications, substitutions, and variations can be made to these embodiments without departing from the principles and spirit of the present invention. The scope of the present invention is defined by the claims and their equivalents.

Claims

1. A simulation analysis method for heavy-duty AGV trajectory tracking model based on Koopman operator, It is characterized in that The following steps are involved: Step 1: Data preprocessing; Step 2: Rotate the matrix and transform the coordinate system to transform the body coordinate system of the heavy-loaded AGV into the world coordinate system; Step 3: Determine the basis function and dimension of the upgrade; Step 4: Simulate and verify the MPC controller, including: (1) Straight line trajectory The expected trajectory equation of the straight line is shown in formula (1): The initial state of the desired trajectory selected in the text is: q d0 =(x d (0), y d (0), θ d (0)) = (0, 0, π / 6) The initial state of the overloaded AGV is: q r0 =(x r (0), y r (0), θ r (0)) = (0, 0, 0) The MPC parameters are set to: Q a = 0.1*diag(1,1,1,1,1,1,1,1,1,1,1,1,1,1,1) R c = 1 × 10 -6 * diag(1, 1, 0.1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1), (2) (2) Circular trajectory The circular expected trajectory equation is shown in formula (3): Select the initial state of the desired trajectory as: The initial state of the heavy-load AGV is:

2. According to claim 1, a simulation analysis method for a heavy-duty AGV trajectory tracking model based on a Koopman operator, Features: In the first step, the preprocessed data includes n pieces of motion trajectory data collected during the simulation process. The sampling time for each trajectory is t, and the sampling period ΔT = 0.01 s, with a total of 500 * n data. Among them, 400 * n data from 0.8n pieces of motion trajectory data are used to solve the Koopman high-dimensional linear model matrices A, B, and C, and 100 * n data from the other 0.2n pieces of motion trajectory data are used to evaluate the approximation degree between the solved Koopman high-dimensional linear model and the true Lagrangian dynamics model; and it is defined that the original dimension of the heavy-load AGV is 30-dimensional, with the first 15 dimensions being the position information including the x coordinate, y coordinate, inclination angle θ, and the angle of the motor in the heavy-load AGV, and the last 15 dimensions being the velocity information including the x-direction velocity, y-direction velocity, body angular velocity, and the angular velocity of the motor in the heavy-load AGV. of the heavy-load AGV, and the last 15 dimensions are the x-direction velocity of the y-direction body angular velocity and the angular velocity of the motor of the heavy-load AGV.

3. According to claim 1, a simulation analysis method for a heavy-duty AGV trajectory tracking model based on a Koopman operator, Features: The coordinate system conversion method in step 2 is as follows: In the vehicle body coordinate system, the force analysis of heavy-loaded AGV is shown in formula (6): where d 1 = -scosγ 1 + hsinγ 1 - d, d 2 = -scosγ 1 + hsinγ 1 + d, d 3 = scosγ 2 + hsinγ 2 - d, d 4 = s cos γ 2 + h sin γ 2 + d, d 5 = - s cos γ 3 - d, d 6 = - s cos γ 3 + d, d 7 = s cos γ 4 - d, d 8 = s cos γ 4 + d, d 9 = -scosγ 5 -hsinγ 5 -d, d 10 = -scosγ 5 -hsinγ 5 +d, d 11 = scosγ 6 -hsinγ 6 -d, d 12 = s cos γ 6 - h sin γ 6 + d; The rotation matrix from the vehicle coordinate system to the world coordinate system is: It is concluded that: The rotation matrix C is as follows: where 4. According to claim 1, a simulation analysis method for a heavy-duty AGV trajectory tracking model based on a Koopman operator, Features: In the third step, select a thin plate spline radial basis function centered at x 0 as the basis function, that is, ψ(x) = ‖x - x 0 ‖ 2 log(‖x - x 0 ‖); x 0 is a constant vector with elements uniformly distributed in the given interval, and set the dimension of the Koopman model according to the minimum of the root mean square error and the relative root mean square error.

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