Inversion sliding mode trajectory tracking control method based on double models of four-steering-wheel industrial mobile robot

Abstract

The invention discloses an inversion sliding mode trajectory tracking control method based on double models of a four-steering-wheel industrial mobile robot. The method comprises the following steps: 1) designing a control law of a kinematics controller of the four-steering-wheel industrial mobile robot; 2) based on a control law of a kinematics controller, an outer ring kinematics inversion speed controller is built by using an inversion method; 3) designing a control law of a dynamics sliding mode controller of the four-steering-wheel industrial mobile robot; 4) building an inner ring dynamic torque controller based on a sliding mode control method; and 5) realizing double-closed-loop inversion sliding mode trajectory tracking control of the four-steering-wheel industrial mobile robot by using an outer-loop kinematics inversion speed controller and an inner-loop kinematics torque controller. According to the invention, the tracking precision in the transfer process of the mobile robot is effectively improved, and the operation stability is enhanced.

Description

Technical Field

[0001] The present invention relates to the field of robot control, and specifically to an inverse sliding mode trajectory tracking control method based on a dual model of a four-steering-wheel industrial mobile robot. Background Art

[0002] The four-steering-wheel industrial mobile robot is a typical non-linear system. Under good ground conditions and low-speed driving, the kinematic characteristics of the mobile robot are prominent, so the kinematic model can be used to achieve effective trajectory tracking. However, in practical applications, when it runs on a wet or uneven road surface, non-linear changes in the lateral force of the tires may occur and even wheel slippage may be caused. Therefore, environmental interference factors and uncertainties such as system actuators will have a certain impact on its tracking performance. At the same time, considering the dynamic characteristics of the robot under heavy load conditions, relying solely on a kinematics-based controller can no longer meet the requirements of trajectory tracking accuracy. In addition, considering the problem that the model predictive controller established based on the kinematic model of the mobile robot is difficult to solve the lateral control error, and considering that the vast majority of working conditions of the research object in this paper carry large-weight workpieces, its dynamic characteristics cannot be ignored. Summary of the Invention

[0003] The purpose of the present invention is to provide an inverse sliding mode trajectory tracking control method based on a dual model of a four-steering-wheel industrial mobile robot, including the following steps:

[0004] 1) Construct a three-dimensional digital simplified model of the four-steering-wheel industrial mobile robot;

[0005] 2) Import the three-dimensional digital simplified model into ADAMS and add dynamic information to the three-dimensional digital simplified model;

[0006] 3) Design the control law of the kinematic controller of the four-steering-wheel industrial mobile robot;

[0007] 4) Based on the control law of the kinematic controller, use the inversion method to build an outer-loop kinematic inversion speed controller;

[0008] 5) Design the control law of the dynamic sliding mode controller of the four-steering-wheel industrial mobile robot;

[0009] 6) Based on the sliding mode control method, build an inner-loop dynamic torque controller;

[0010] 7) Use a non-linear extended state observer to estimate the external disturbance and feedback the external disturbance to the output of the dynamic sliding mode controller;

[0011] 8) Use the outer-loop kinematic inversion speed controller and the inner-loop dynamic torque controller to achieve the dual-loop inverse sliding mode trajectory tracking control of the four-steering-wheel industrial mobile robot.

[0012] Further, the steps of constructing a three-dimensional digital simplified model of a four-steering-wheel industrial mobile robot include:

[0013] 1.1) Based on the physical object of the four-steering-wheel industrial mobile robot, use SolidWorks to establish a three-dimensional digital model of the four-steering-wheel industrial mobile robot;

[0014] 1.2) Simplify the three-dimensional digital model of the four-steering-wheel industrial mobile robot, only retaining the vehicle body, four steering wheels and six driven wheels, so as to construct a three-dimensional digital simplified model of the four-steering-wheel industrial mobile robot.

[0015] Further, the three-dimensional digital simplified model of the four-steering-wheel industrial mobile robot includes vehicle body one, vehicle body two, vehicle body three and two telescopic arms;

[0016] Among them, one steering wheel and two driven wheels are arranged on the chassis of vehicle body one, the steering wheel and the two driven wheels are arranged in a triangular shape, and the axes of the two driven wheels coincide;

[0017] One steering wheel and two driven wheels are arranged on the chassis of vehicle body two, the steering wheel and the two driven wheels are arranged in a triangular shape, and the axes of the two driven wheels coincide; the two driven wheels of vehicle body one and the two driven wheels of vehicle body two are arranged in axial symmetry;

[0018] Steering wheels and driven wheels are respectively arranged at the four corners of the chassis of vehicle body three, the two steering wheels are arranged diagonally, the two driven wheels are arranged diagonally, and the axes of the steering wheel and the driven wheel on the same side coincide.

[0019] Further, the steps of adding dynamic information to the three-dimensional digital simplified model include:

[0020] 2.1) Set two MARKER points at the position of the vehicle body for installing the upper fixture ring as the application points of external forces;

[0021] 2.2) Apply driving pairs and rotating pairs to each steering wheel and driven wheel of the three-dimensional digital simplified model;

[0022] 2.3) Gradually apply a bearing capacity equal to the load on the reserved MARKER points, add a tension damping disc spring group to the steering wheels, and set the parameters of the tension damping disc spring group and the road surface parameters, so as to simulate the transfer process of the four-steering-wheel industrial mobile robot for workpieces;

[0023] The parameters of the tension damping disc spring group include the stiffness, tension force and damping coefficient of the tension damping disc spring group; the road surface parameters include static friction coefficient, dynamic friction coefficient, viscous boundary velocity, friction boundary velocity, friction index and tire penetration depth.

[0024] Further, the steps of designing the control law of the kinematic controller of the four-steering-wheel industrial mobile robot include:

[0025] 3.1) Construct a pose error model for the four-steering-wheel industrial mobile robot to describe the deviation between the actual pose and the reference pose of the four-steering-wheel industrial mobile robot at a certain moment during the movement process. The steps are as follows:

[0026] 3.1.1) Construct a pose error model for the four-steering-wheel industrial mobile robot in the earth coordinate system, that is:

[0027]

[0028] In the formula, x e , y e , respectively represent the relative errors in the x, y, and angular directions; represents the pose of the reference trajectory of the mobile robot in the global coordinate system; represents the pose of the actual trajectory of the mobile robot in the global coordinate system;

[0029] 3.1.2) Convert the pose error model of the four-steering-wheel industrial mobile robot in the earth coordinate system into the pose error in the vehicle body coordinate system of the mobile robot, that is:

[0030]

[0031] In the formula, e x , e y , respectively represent the relative errors in the x, y, and angular directions;

[0032] 3.1.3) Differentiate the pose error in the vehicle body coordinate system of the mobile robot to obtain:

[0033]

[0034] In the formula, respectively represent the differentials of the relative errors; represents the differential of the pose of the reference trajectory of the mobile robot in the global coordinate system; represents the differential of the pose of the actual trajectory of the mobile robot in the global coordinate system;

[0035] 3.1.4) Based on formula (3), construct a pose error model for the four-steering-wheel industrial mobile robot, that is:

[0036]

[0037] In the formula, v xr , v yr , ω r respectively represent the speeds of the reference trajectory of the mobile robot in the x, y, and angular directions in the global coordinate system; vx , v y , ω respectively represent the velocities of the actual trajectory of the mobile robot in the x, y, and angular directions in the global coordinate system;

[0038] 3.2) Design the control objective of the kinematic controller, that is:

[0039]

[0040] 3.3) Design the control law of the kinematic controller when the sideslip angle of the center of mass of the mobile robot is zero. The steps include:

[0041] 3.3.1) Define the Lyapunov function L1, that is:

[0042]

[0043] In the formula, the parameter k2 > 0;

[0044] 3.3.2) Simplify the pose error model of the four-wheel-steering industrial mobile robot to obtain:

[0045]

[0046] In the formula, v r is the reference trajectory speed of the mobile robot; v is the actual trajectory speed of the mobile robot;

[0047] 3.3.3) Differentiate the Lyapunov function L1 to obtain:

[0048]

[0049] 3.3.4) With as the target, design the kinematic control law v c of the kinematic controller when the sideslip angle of the center of mass is zero, that is:

[0050]

[0051] In the formula, k1, k2, k3 are positive integers;

[0052] 3.4) Design the control law of the kinematic controller when the sideslip angle of the center of mass of the mobile robot is not zero. The steps include:

[0053] 3.4.1) Design the control objective of the kinematic controller, that is:

[0054]

[0055] 3.4.2) Define the Lyapunov function L'1, that is:

[0056]

[0057] In the formula, the parameter k'2 ≥ 0;

[0058] 3.4.3) Differentiate the Lyapunov function L'1 to obtain:

[0059]

[0060] 3.4.4) With as the goal, design the kinematic control law v of the kinematic controller when the centroid side slip angle is not zero c , that is:

[0061]

[0062] In the formula, k1' > 0, k'2 > 0, k3' > 0.

[0063] Furthermore, when constructing the outer loop kinematic inversion speed controller, the saturation function u is also used to optimize the outer loop kinematic inversion speed controller;

[0064] The saturation function u is as follows:

[0065]

[0066] In the formula, λ s represents the threshold of the centroid side slip angle, λ s is greater than 0; u s represents the threshold of the centroid and speed, u s is greater than 0; u1 and u2 are the saturation functions in the cases where the centroid side slip angle is zero and not zero;

[0067] Among them, the relationship between the matrix C s and the threshold λ s is as follows:

[0068]

[0069] Furthermore, the steps for designing the dynamic sliding mode controller control law of the four-steering-wheel industrial mobile robot include:

[0070] 5.1) Construct the speed error vector v of the four-steering-wheel industrial mobile robot e , that is:

[0071]

[0072] In the formula, (v xr , v yr , ω r ) is the reference speed control quantity; (v x , v y, ω) is the actual speed control quantity; (v ex , v ey , ω e ) are the speed errors in the x, y, and rotation directions;

[0073] 5.2) Construct the model of the telescopic arm of the mobile robot for gripping the workpiece, that is:

[0074]

[0075] Among them, d1, d2, d3 are the error disturbances in the lateral, longitudinal, and rotational directions respectively; m represents the total mass of the mobile robot; I z represents the moment of inertia of the mobile robot; F Di represents the driving force of the steering wheel; P Di represents the lateral force received by each steering wheel; x i , y i respectively represent the coordinates of each steering wheel; a x , a y are the accelerations of the mobile robot in the x and y axis directions respectively;

[0076] 5.3) Select the integral sliding mode function S, that is:

[0077]

[0078] In the formula, β1 > 0, β2 > 0, β3 > 0;

[0079] 5.4) Design the sliding mode reaching law That is:

[0080]

[0081] In the formula, the inverse hyperbolic sine function x is the variable; the parameter k > 0; the parameter ε > 0;

[0082] Among them, the nonlinear function fal(x, α, δ) is as follows:

[0083]

[0084] In the formula, 0 < δ < 1, α > 0; δ is a linearly symmetric discontinuous function of the interval length of fal(x, α, δ) about the origin;

[0085] 5.5) Based on the sliding mode reaching law Design the kinematic controller, that is:

[0086]

[0087] Among them, the coefficient \(E = diag[\varepsilon_1,\varepsilon_2,\varepsilon_3]\), \(Q = diag[q_1,q_2,q_3]\); the vector vector \(\varepsilon_1\gt0,\varepsilon_2\gt0,\varepsilon_3\gt0\); \(q_1\gt0,q_2\gt0,q_3\gt0\); \(0\lt\delta_1\lt1,0\lt\delta_2\lt1,0\lt\delta_3\lt1\); \(\alpha_1\gt0,\alpha_2\gt0,\alpha_3\gt0\);

[0088] 5.6) Differentiate the integral sliding mode function to obtain:

[0089]

[0090] In the formula, is the differential of the integral sliding mode function; is the differential of the velocity error in the \(x\), \(y\) and rotation angle directions;

[0091] 5.7) Substitute the velocity error with the vector and the sliding mode reaching law into formula (23) to obtain:

[0092]

[0093] 5.8) Combine formula (17) and formula (24) to obtain the control law of the dynamic sliding mode controller, that is:

[0094]

[0095] In the formula, \(F\) xd , \(F\) yd , \(M\) zd are the lateral resultant force, longitudinal resultant force and resultant moment of the four drive wheels that need equivalent control.

[0096] Furthermore, the nonlinear state extended observer is as follows:

[0097]

[0098] In the formula, \(e_1\) and \(e_2\) are the observation errors of the state observer; \(u\) and \(d\) are the observable quantities of the mobile robot state and external disturbance respectively; \(L_1 = diag(l\) 11 , \(l\) 12 , \(l\) 13 ) \(\gt0\), \(L_2 = diag(l\) 21 , \(l\) 22 , \(l\) 23 ) \(\gt0\) are the observation gain matrices; \(z_1\) and \(z_2\) are the estimated values of the system state and disturbance respectively; represents the inverse matrix of the moment of inertia.

[0099] Furthermore, during the double-closed-loop backstepping sliding mode trajectory tracking control process, the outer-loop kinematic backstepping speed controller provides the centroid speed of the mobile robot for the inner-loop dynamic torque controller, and the inner-loop dynamic torque controller is used to control the moving trajectory of the mobile robot.

[0100] Furthermore, the method for building the outer-loop kinematic backstepping speed controller is as follows: in the MTALAB environment, through the Simulink module, the outer-loop kinematic backstepping speed controller is built based on the backstepping method.

[0101] The method for building the inner-loop dynamic torque controller is as follows: in the MTALAB environment, through the Simulink module, the inner-loop dynamic torque controller is built based on the sliding mode control method.

[0102] The technical effect of the present invention is beyond doubt. Based on the kinematic and dynamic double models of the four-steering-wheel industrial mobile robot, the present invention proposes a backstepping sliding mode trajectory tracking control method, which uses an extended state observer to observe the uncertain disturbance and compensate it into the controller, reducing the influence of the uncertain disturbance on the trajectory tracking accuracy, effectively improving the tracking accuracy during the transfer process of the mobile robot, and enhancing the operation stability at the same time. Description of the Drawings

[0103] Figure 1 is the chassis configuration schematic diagram of the physical model of the present invention;

[0104] Figure 2 is the three-dimensional model layout schematic diagram of the physical object of the present invention;

[0105] Figure 3 is the virtual prototype schematic diagram of the physical model of the present invention

[0106] Figure 4 is the trajectory tracking pose error schematic diagram of the present invention;

[0107] Figure 5 is the schematic diagram of the double-model trajectory tracking controller principle of the present invention;

[0108] Figure 6 is the schematic diagram of the kinematic trajectory tracking controller principle of the present invention;

[0109] Figure 7 is the Simulink model of the double-model trajectory tracking controller of the present invention;

[0110] Figure 8 is the trajectory tracking effect diagram of the kinematic backstepping speed controller of the present invention;

[0111] Figure 9 is the comparison diagram of the kinematic backstepping speed controller trajectory tracking saturation function optimization of the present invention;

[0112] Figure 10 This is the trajectory tracking effect diagram of the double closed-loop trajectory tracking controller of the present invention. Detailed implementation manners

[0113] The present invention will be further described below in conjunction with embodiments, but it should not be understood that the above-mentioned subject scope of the present invention is limited to the following embodiments. Without departing from the above technical idea of the present invention, various substitutions and changes made according to ordinary technical knowledge and customary means in the art shall be included within the protection scope of the present invention.

[0114] Embodiment 1:

[0115] Refer to Figures 1 to 10 , the inversion sliding mode trajectory tracking control method based on the double models of a four-steering-wheel industrial mobile robot includes the following steps:

[0116] 1) Construct a three-dimensional digital simplified model of the four-steering-wheel industrial mobile robot;

[0117] 2) Import the three-dimensional digital simplified model into ADAMS and add dynamic information to the three-dimensional digital simplified model;

[0118] 3) Design the control law of the kinematic controller of the four-steering-wheel industrial mobile robot;

[0119] 4) Based on the control law of the kinematic controller, use the inversion method to build an outer-loop kinematic inversion speed controller;

[0120] 5) Design the control law of the dynamic sliding mode controller of the four-steering-wheel industrial mobile robot;

[0121] 6) Based on the sliding mode control method, build an inner-loop dynamic torque controller;

[0122] 7) Use a nonlinear extended state observer to estimate external disturbances and feedback the external disturbances into the output of the dynamic sliding mode controller;

[0123] 8) Use the outer-loop kinematic inversion speed controller and the inner-loop dynamic torque controller to realize the double closed-loop inversion sliding mode trajectory tracking control of the four-steering-wheel industrial mobile robot.

[0124] Embodiment 2:

[0125] The inversion sliding mode trajectory tracking control method based on the double models of a four-steering-wheel industrial mobile robot has the same technical content as that in Embodiment 1. Further, the steps of constructing the three-dimensional digital simplified model of the four-steering-wheel industrial mobile robot include:

[0126] 1.1) Based on the physical object of the four-steering-wheel industrial mobile robot, use SolidWorks to establish a three-dimensional digital model of the four-steering-wheel industrial mobile robot;

[0127] 1.2) Simplify the 3D digital model of the four-steering-wheel industrial mobile robot, and only retain the vehicle body, four steering wheels and six driven wheels, so as to construct a 3D digital simplified model of the four-steering-wheel industrial mobile robot.

[0128] Embodiment 3:

[0129] Based on the inversion sliding mode trajectory tracking control method of the double model of the four-steering-wheel industrial mobile robot, the technical content is the same as any one of Embodiments 1-2. Further, the 3D digital simplified model of the four-steering-wheel industrial mobile robot includes Vehicle Body 1, Vehicle Body 2, Vehicle Body 3 and two telescopic arms;

[0130] Among them, one steering wheel and two driven wheels are arranged on the chassis of Vehicle Body 1. The steering wheel and the two driven wheels are arranged in a triangular shape, and the axes of the two driven wheels coincide;

[0131] One steering wheel and two driven wheels are arranged on the chassis of Vehicle Body 2. The steering wheel and the two driven wheels are arranged in a triangular shape, and the axes of the two driven wheels coincide; The two driven wheels of Vehicle Body 1 and the two driven wheels of Vehicle Body 2 are arranged in an axisymmetric manner;

[0132] Steering wheels and driven wheels are respectively arranged at the four corners of the chassis of Vehicle Body 3. The two steering wheels are arranged diagonally, and the two driven wheels are arranged diagonally. The axes of the steering wheel and the driven wheel on the same side coincide.

[0133] Embodiment 4:

[0134] Based on the inversion sliding mode trajectory tracking control method of the double model of the four-steering-wheel industrial mobile robot, the technical content is the same as any one of Embodiments 1-3. Further, the steps of adding dynamic information to the 3D digital simplified model include:

[0135] 2.1) Set two MARKER points at the position of the vehicle body for installing the upper fixture ring as the application points of external forces;

[0136] 2.2) Apply driving pairs and rotating pairs to each steering wheel and driven wheel of the 3D digital simplified model;

[0137] 2.3) Gradually apply a bearing capacity equal to the load on the reserved MARKER points, add a tension damping disc spring group to the steering wheels, and set the parameters of the tension damping disc spring group and the road surface parameters, so as to simulate the process of the four-steering-wheel industrial mobile robot transporting workpieces;

[0138] The parameters of the tension damping disc spring group include the stiffness, tension force and damping coefficient of the tension damping disc spring group; the road surface parameters include static friction coefficient, dynamic friction coefficient, viscous boundary velocity, friction boundary velocity, friction index and tire penetration depth.

[0139] Embodiment 5:

[0140] Inverse Sliding Mode Trajectory Tracking Control Method Based on Dual Models of Four-Wheel Industrial Mobile Robot, the technical content is the same as any one of Embodiments 1-4. Further, the steps of designing the control law of the kinematic controller of the four-wheel industrial mobile robot include:

[0141] 3.1) Construct a pose error model of the four-wheel industrial mobile robot to describe the deviation between the actual pose and the reference pose at a certain moment during the movement of the four-wheel industrial mobile robot. The steps include:

[0142] 3.1.1) Construct a pose error model of the four-wheel industrial mobile robot in the earth coordinate system, that is:

[0143]

[0144] In the formula, x e , y e , respectively represent the relative errors in the x, y, and angular directions; represents the pose of the reference trajectory of the mobile robot in the global coordinate system; represents the pose of the actual trajectory of the mobile robot in the global coordinate system;

[0145] 3.1.2) Convert the pose error model of the four-wheel industrial mobile robot in the earth coordinate system into the pose error in the body coordinate system of the mobile robot, that is:

[0146]

[0147] In the formula, e x , e y , respectively represent the relative errors in the x, y, and angular directions;

[0148] 3.1.3) Differentiate the pose error in the body coordinate system of the mobile robot to obtain:

[0149]

[0150] In the formula, respectively represent the differentials of the relative errors; represents the differential of the pose of the reference trajectory of the mobile robot in the global coordinate system; represents the differential of the pose of the actual trajectory of the mobile robot in the global coordinate system;

[0151] 3.1.4) Based on formula (3), construct a pose error model of the four-wheel industrial mobile robot, that is:

[0152]

[0153] wherein, v xr , v yr , ω r respectively represent the speeds of the mobile robot's reference trajectory in the x, y, and angular directions in the global coordinate system; v x , v y , ω respectively represent the speeds of the mobile robot's actual trajectory in the x, y, and angular directions in the global coordinate system;

[0154] 3.2) Design the control objective of the kinematic controller, that is:

[0155]

[0156] 3.3) Design the control law of the kinematic controller when the sideslip angle of the center of mass of the mobile robot is zero. The steps include:

[0157] 3.3.1) Define the Lyapunov function L1, that is:

[0158]

[0159] wherein, the parameter k2 > 0;

[0160] 3.3.2) Simplify the pose error model of the four-wheel-steering industrial mobile robot to obtain:

[0161]

[0162] wherein, v r is the speed of the mobile robot's reference trajectory; v is the speed of the mobile robot's actual trajectory;

[0163] 3.3.3) Differentiate the Lyapunov function L1 to obtain:

[0164]

[0165] 3.3.4) With as the target, design the kinematic control law v c of the kinematic controller when the sideslip angle of the center of mass is zero, that is:

[0166]

[0167] wherein, k1, k2, k3 are positive integers;

[0168] 3.4) Design the control law of the kinematic controller when the sideslip angle of the center of mass of the mobile robot is not zero. The steps include:

[0169] 3.4.1) Design the control objective of the kinematic controller, that is:

[0170]

[0171] 3.4.2) Define the Lyapunov function \(L'_1\), that is:

[0172]

[0173] where the parameter \(k'_2\geq0\);

[0174] 3.4.3) Take the derivative of the Lyapunov function \(L'_1\) to obtain:

[0175]

[0176] 3.4.4) With as the goal, design the kinematic control law \(v\) of the kinematic controller when the sideslip angle of the center of mass is non-zero c , that is:

[0177]

[0178] where \(k_1'\gt0\), \(k'_2\gt0\), \(k_3'\gt0\).

[0179] Example 6:

[0180] Based on the inversion sliding mode trajectory tracking control method of a four-steering-wheel industrial mobile robot with a dual model, the technical content is the same as any one of Examples 1-5. Further, when constructing the outer-loop kinematic inversion speed controller, the saturation function \(u\) is also used to optimize the outer-loop kinematic inversion speed controller;

[0181] The output of the speed control amount is proportional to the error value. When the error value is extremely large, the high-speed control output may cause a large torque output that exceeds the actual working ability of the motor. In addition, due to the gear train layout of the mobile robot and the chassis space limitation, the steering wheel cannot achieve a 360° rotation. Therefore, it is also necessary to limit the angle of the steering wheel.

[0182] Therefore, the saturation function is introduced to realize controller optimization, and the control law is restricted from two aspects of speed and deflection angle (i.e., angular velocity).

[0183] The saturation function \(u\) is as follows:

[0184]

[0185] where \(\lambda\) s represents the threshold value of the sideslip angle of the center of mass, \(\lambda\) s is greater than 0; \(u\) s represents the threshold value of the center of mass and speed, \(u\) s is greater than 0; \(u_1\), \(u_2\) are the saturation functions in the cases where the sideslip angle of the center of mass is zero and non-zero;

[0186] Among them, matrix C s and threshold λ s have the following relationship:

[0187]

[0188] Embodiment 7:

[0189] An inversion sliding mode trajectory tracking control method based on a dual model of a four-steering-wheel industrial mobile robot. The technical content is the same as any one of Embodiments 1-6. Further, the steps of designing the control law of the dynamic sliding mode controller for the four-steering-wheel industrial mobile robot include:

[0190] 5.1) Construct the speed error vector v of the four-steering-wheel industrial mobile robot e , that is:

[0191]

[0192] In the formula, (v xr , v yr , ω r ) is the reference speed control quantity; (v x , v y , ω) is the actual speed control quantity; (v ex , v ey , ω e ) are the speed errors in the x, y, and rotation directions;

[0193] 5.2) Construct the model of the telescopic arm of the mobile robot holding the workpiece, that is:

[0194]

[0195] Among them, d1, d2, and d3 are the error disturbances in the lateral, longitudinal, and rotational directions respectively; m represents the total mass of the mobile robot; I z represents the moment of inertia of the mobile robot; F Di represents the driving force of the steering wheel; P Di represents the lateral force received by each steering wheel; x i , y i respectively represent the coordinates of each steering wheel; a x , a y are the accelerations of the mobile robot in the x and y axis directions respectively;

[0196] 5.3) Select the integral sliding mode function S, that is:

[0197]

[0198] In the formula, β1 > 0, β2 > 0, β3 > 0;

[0199] 5.4) Design of sliding mode reaching law That is:

[0200]

[0201] In the formula, the inverse hyperbolic sine function x is a variable; parameter k > 0; parameter ε > 0;

[0202] Among them, the nonlinear function fal(x, α, δ) is as follows:

[0203]

[0204] In the formula, 0 < δ < 1, α > 0; δ is a linear symmetric discontinuous function of the interval length contained in fal(x, α, δ) with respect to the origin;

[0205] 5.5) Based on the sliding mode reaching law Design the kinematic controller, that is:

[0206]

[0207] Among them, the coefficient E = diag[ε1, ε2, ε3], Q = diag[q1, q2, q3]; vector vector ε1 > 0, ε2 > 0, ε3 > 0; q1 > 0, q2 > 0, q3 > 0; 0 < δ1 < 1, 0 < δ2 < 1, 0 < δ3 < 1; α1 > 0, α2 > 0, α3 > 0;

[0208] 5.6) Differentiate the integral sliding mode function to obtain:

[0209]

[0210] In the formula, is the differential of the integral sliding mode function; is the differential of the velocity error in the x, y, and rotation angle directions;

[0211] 5.7) Substitute the velocity error with the vector and the sliding mode reaching law into formula (23) to obtain:

[0212]

[0213] 5.8) Combine formula (17) and formula (24) to obtain the control law of the dynamic sliding mode controller, that is:

[0214]

[0215] In the formula, F xd 、F yd 、M zdThe lateral resultant force, longitudinal resultant force, and resultant moment of the four steering wheels to be equivalently controlled.

[0216] Embodiment 8:

[0217] The inversion sliding mode trajectory tracking control method based on the dual models of the four-wheel industrial mobile robot has the same technical content as any one of Embodiments 1-7. Further, the nonlinear state expansion observer is as follows:

[0218]

[0219] In the formula, e1 and e2 are the observation errors of the state observer respectively; u and d are the observable quantities of the state variables of the mobile robot and external disturbances respectively; L1 = diag(l 11 , l 12 , l 13 ) > 0, L2 = diag(l 21 , l 22 , l 23 ) > 0 are the observation gain matrices; z1 and z2 are the estimated values of the system state variables and disturbances respectively; Denotes the inverse matrix of the moment of inertia.

[0220] Embodiment 9:

[0221] The inversion sliding mode trajectory tracking control method based on the dual models of the four-wheel industrial mobile robot has the same technical content as any one of Embodiments 1-8. Further, in the process of dual-loop inversion sliding mode trajectory tracking control, the outer-loop kinematic inversion speed controller provides the centroid speed of the mobile robot for the inner-loop dynamic torque controller, and the inner-loop dynamic torque controller is used to control the moving trajectory of the mobile robot.

[0222] Embodiment 10:

[0223] The inversion sliding mode trajectory tracking control method based on the dual models of the four-wheel industrial mobile robot has the same technical content as any one of Embodiments 1-9. Further, the method for building the outer-loop kinematic inversion speed controller is: based on the inversion method, build the outer-loop kinematic inversion speed controller through the Simulink module in the MTALAB environment.

[0224] The method for building the inner-loop dynamic torque controller is: based on the sliding mode control method, build the inner-loop dynamic torque controller through the Simulink module in the MTALAB environment.

[0225] Embodiment 11:

[0226] The steps of the inversion sliding mode trajectory tracking control method based on the dual models of the four-wheel industrial mobile robot are as follows:

[0227] Step 1: Precisely establish a 3D digital model of the physical model of the mobile robot in SolidWorks.

[0228] Step 2: Simplify the 3D model and import it into ADAMS, and perform operations such as adding constraints to verify the kinematic and dynamic models of the mobile robot.

[0229] Step 3: According to the idea of the inversion method, design the control law of its kinematic controller on the premise of considering whether the sideslip angle of the center of mass of the mobile robot is zero.

[0230] Step 4: In the MTALAB environment, through the Simulink module, build an outer-loop kinematic inversion speed controller based on the inversion method, and introduce a saturation function for optimization to ensure tracking accuracy and real-time performance.

[0231] Step 5: According to the idea of the sliding mode control method, design the control law of the dynamic controller of the mobile robot.

[0232] Step 6: In the MTALAB environment, through the Simulink module, build an inner-loop dynamic torque controller based on the sliding mode control method. Considering the influence of the external environment and its own parameters on the overall tracking accuracy, use a state observer to estimate the external disturbance and provide real-time feedback to ensure the stability of tracking.

[0233] Step 7: Finally, realize the double-loop inversion sliding mode trajectory tracking control based on the kinematic and dynamic double models of the mobile robot.

[0234] In Step 1, it is necessary to establish an accurate 1:1 3D model according to the physical dimensions of the mobile robot to facilitate the virtual simulation verification of the trajectory tracking control algorithm in the subsequent steps and obtain the trajectory tracking comparison curve.

[0235] In Step 2, in order to avoid the difficulty of solving caused by the over-complexity of the system, it is necessary to simplify the complete 3D model and import it into ADAMS to add constraints, forces, parameters, etc., and then verify the correctness of the kinematic and dynamic models by writing scripts.

[0236] In Step 3, based on the Lyapunov stability theorem, the nonlinear system is decomposed into low-order subsystems through virtual control variables, and a Lyapunov function is constructed to derive the adaptive law of the control parameters and the control law of the controller, realizing the feedback control of the entire system and strengthening the stability and real-time performance of the system.

[0237] In step 4, the Simulink module in MATLAB is used to verify the cases where the sideslip angle of the center of mass of the mobile robot is zero and non-zero respectively, and the optimized curve of the saturation function is added for comparison, so as to obtain the verification result of the overall trajectory tracking effect of the kinematic inversion speed controller of the four-steering-wheel industrial mobile robot.

[0238] In step 5, the system torque output is adjusted according to the virtual speed control input of the outer-loop kinematic controller. The sliding mode surface of the system is designed based on the sliding-mode control method (SMC). Under the control of the controller, the system state converges from outside the sliding mode surface to the sliding mode surface and gradually reaches the origin of the system.

[0239] In step 6, the Simulink module in MATLAB is used to build the ordinary inversion sliding mode controller and the inversion sliding mode controller with a disturbance observer for the mobile robot respectively, and relevant effects are compared after simulation.

[0240] Example 12:

[0241] The inversion sliding mode trajectory tracking control method based on the dual model of the four-steering-wheel industrial mobile robot is as follows:

[0242] As Figure 1 shown, the chassis configuration of the four-steering-wheel industrial mobile robot adopts the form of "four steering wheels - six driven wheels", and other components such as the electric control box are installed in the gaps; the overall telescopic function schematic diagram of the mobile robot is as Figure 2 shown, which is composed of vehicle body one, two, three and two telescopic arms respectively, so that the mobile robot can freely expand and contract to meet the transfer requirements when transferring workpieces of different lengths. As Figure 3 shown, the four-steering-wheel industrial mobile robot itself is a system with a complex structure and numerous parts. Therefore, it is very difficult to directly build a virtual prototype model of the whole vehicle and then simulate and solve it. Therefore, the mechanical structure is simplified, and then the file in SolidWorks is imported into ADAMS. The virtual prototype model established is as Figure 3 shown. Irrelevant structures such as the wire reel, electric control box, and electronic fence are removed, and only the whole vehicle body, steering wheels and turning wheels are retained. At the same time, two MARKER points are set at the installation positions of the upper fixture ring and the body, which are used as the positions for applying external forces. Driving pairs and rotating pairs are applied to each steering wheel and turning wheel, and the bearing capacity equal to the load is gradually applied at the reserved MARKER points; tensioning shock-absorbing disc springs are added to the steering wheels, and parameters such as the stiffness and tension of the disc spring group are set; the road surface parameters are set. The specific parameter values are shown in Table 1.

[0243] Table 1 Key simulation parameters of mobile robot virtual prototype

[0244]

[0245] The specific principle is as follows:

[0246] Considering that the pose error of the mobile robot during trajectory tracking comes from multiple aspects, it is necessary to establish a pose error model to describe the deviation between the actual pose and the reference pose of the mobile robot at a certain moment during the movement process. Through the pose error model, the absolute error in the global coordinate system XOY can be converted into the relative error in the vehicle coordinate system xoy, which is more convenient for the subsequent design of the trajectory tracking controller.

[0247] Taking the centroid of the mobile robot as the reference point, its pose error model is as Figure 4 shown. The reference trajectory pose of the mobile robot is represented by , where x r , y r , respectively represent the poses of the reference trajectory in the x, y, and angular directions in the global coordinate system; the actual trajectory pose is represented by , where x, y, respectively represent the poses of the actual vehicle trajectory in the x, y, and angular directions in the global coordinate system; the reference speed control quantity is v r = [v xr , v yr , ω r T , where v xr , v yr , ω r respectively represent the speeds of the reference trajectory in the x, y, and angular directions in the global coordinate system; the actual speed control quantity is represented by v = [v x , v y , ω] T , where v x , v y , ω respectively represent the speeds of the actual trajectory in the x, y, and angular directions in the global coordinate system;. According to the kinematic model, the pose error model in the global coordinate system during the trajectory tracking process can be expressed as:

[0248]

[0249] In Equation (1), x e , y e , ​respectively represent the relative errors in the x, y, and angular directions.

[0250] Converting to the pose error in the body coordinate system of the mobile robot can be expressed as:

[0251]

[0252] In Equation (2), e x , e y , respectively represent the relative errors in the x, y, and angular directions.

[0253] Expanding Equation 2 and taking the differential gives the system error differential equation as:

[0254]

[0255] In Equation (3), respectively represent the differentials of the relative errors.

[0256] The pose error vector of the mobile robot can be expressed as:

[0257]

[0258] It can be seen from Equation 4 that the control quantity u of the mobile robot = [v x , v y , ω] T and the error quantity The relationship between them. If the mobile robot is to perform trajectory tracking motion along the reference trajectory, the pose error of the whole vehicle can be measured in real time by on-vehicle sensors, and then the pose error of the mobile robot can be adjusted to converge through the control quantity u, so as to achieve high-precision trajectory tracking control.

[0259] According to the backstepping method idea, a kinematic controller needs to be designed so that the pose error vector shown in Equation 4 satisfies the equation:

[0260]

[0261] The sideslip angle of the center of mass is zero

[0262] When the sideslip angle of the center of mass of the mobile robot is zero, that is, when the angle between the velocity direction and the x-axis direction of the body coordinate system is zero, only the control quantity u = [v, ω] T needs to be designed for the control law, and the following Lyapunov function is defined:

[0263]

[0264] It can be seen from Equation 6 that L1 ≥ 0. When the pose error is 0, L1 = 0; when the pose error is not 0, L1 > 0. When the sideslip angle is zero, the reference control quantity and the actual control quantity have vy = v yr = 0, v x = v, v xr = v r , so after simplifying Equation 4, we get:

[0265]

[0266] Differentiating Equation 6 gives:

[0267]

[0268] To make The kinematic control law for a kinematic controller with a zero centroidal side slip angle can be obtained as:

[0269]

[0270] In Equation 9, k1, k2, and k3 are all positive integers and are the setting parameters of the kinematic controller.

[0271] (1) The centroidal side slip angle is not zero

[0272] When the centroidal side slip angle of the mobile robot is not zero, that is, there is an angle between the velocity direction and the x-axis direction of the vehicle body coordinate system. In addition to satisfying Equation 5, the following equation also needs to be satisfied:

[0273]

[0274] So at this time, it is necessary to design a control law for the control quantity u = [v x , v y , ω] T Define the following Lyapunov function:

[0275]

[0276] Combining Equation 4 and differentiating Equation 11 gives:

[0277]

[0278] Similarly, to make The kinematic control law for a kinematic controller with a non-zero centroidal side slip angle can be obtained as:

[0279]

[0280] Similarly, k1' > 0, k'2 > 0, k3' > 0, which are the setting parameters of the kinematic controller. The main function of the kinematic controller is to provide the centroid velocity of the mobile robot for the dynamic controller.

[0281] Controller optimization

[0282] After obtaining the kinematic control laws in two cases, the desired speeds and angles of each steering wheel can be obtained. It can be observed that the output of the speed control amount is proportional to the error value. When the error value is extremely large, the high-speed control output may result in a large torque output that exceeds the actual working ability of the motor. In addition, due to the gear train layout of the mobile robot and the chassis space limitation, the steering wheel cannot achieve a 360° rotation. Therefore, it is also necessary to limit the angle of the steering wheel.

[0283] In summary, it is necessary to optimize the kinematic controller. That is, the control law is restricted from two aspects: speed and deflection angle (i.e., angular velocity). The present invention introduces a saturation function to achieve controller optimization, as shown in the following formula:

[0284]

[0285] In Equation 14, λ s represents the threshold value of the centroid side slip angle, and its value is greater than 0; u s represents the threshold value of the centroid and speed, and its value is also greater than 0. The relationship between C s and λ s can be expressed by Equation 15.

[0286]

[0287] The speed error of the four-wheel industrial mobile robot is represented by the vector v e . The reference speed control amount v r = [v xr , v yr , ω r T , and the actual speed control amount is v = [v x , v y , ω] T . Therefore, v e can be expressed as:

[0288]

[0289] According to the rotational dynamics model of the mobile robot, the simplified controlled object model is as shown in the following formula:

[0290]

[0291] Where d1, d2, and d3 are the error disturbances in the lateral, longitudinal, and rotational directions respectively.

[0292] The integral sliding mode function is selected as:

[0293]

[0294] ​where β1 > 0, β2 > 0, β3 > 0. When S = 0, the system is asymptotically stable.

[0295] After the system reaches the sliding surface, it may move along the sliding surface with high-frequency up and down crossing the sliding surface, resulting in high-frequency chattering phenomenon, which may damage the actuator. Therefore, an appropriate reaching law needs to be selected to improve the dynamic performance of the reaching motion. The typical sliding mode reaching laws are shown in Table 2.

[0296] Table 2 Typical sliding mode approach law

[0297]

[0298] The present invention takes into account reducing the chattering during the reaching process and quickly approaching the switching surface, and designs a new reaching law using the special power function fal(x, α, δ) as shown in Equation 19:

[0299]

[0300] where the non-linear function fal(x, α, δ) is:

[0301]

[0302] where: 0 < δ < 1, α > 0. δ is a linearly symmetric discontinuous function of the interval length contained in fal(x, α, δ) about the origin. When s approaches 0, the reaching speed of the system will increase, enabling the system to reach the sliding surface faster within a finite time. At the same time, fal(x, α, δ) is smooth and continuous near the sliding surface, and the high-frequency chattering phenomenon of the control input is effectively weakened.

[0303] The inverse hyperbolic sine function is:

[0304]

[0305] Therefore, according to the above reaching law, the system kinematic controller can be designed as:

[0306]

[0307] where the coefficient E = diag[ε1, ε2, ε3], Q = diag[q1, q2, q3].

[0308] In addition:

[0309]

[0310] Finally, the sliding mode control law is designed. Differentiating the sliding surface shown in Equation 18 gives:

[0311]

[0312] Substituting Equations 16 and 19 into the above equation gives:

[0313]

[0314] By simultaneously considering Equation 17 and Equation 26, the final output of the kinetic sliding mode controller is obtained as:

[0315]

[0316] It can be seen from Equation 27 that there are uncertain disturbance terms in the output of the torque sliding mode controller. To reduce its impact on the control accuracy of the controller, a disturbance observer needs to be designed to observe and estimate it and feedback it to the sliding mode controller.

[0317] For the uncertain disturbance terms of the four-steering-wheel industrial mobile robot system, the use of a gain matrix for compensation can be selected, but this will reduce the control accuracy of the controller. Therefore, the present invention will introduce a Nonlinear Expansion State Observer (NESO) to estimate the system disturbance and feedback the disturbance estimation value to the output of the kinetic torque controller, thereby effectively reducing the disturbance impact and improving the trajectory tracking control accuracy of the controller.

[0318] By transforming Equation 17, we get:

[0319]

[0320] Rewrite System 28 as:

[0321]

[0322] where a = [v y ω, -v x ω, 0] T , u = [F x , F y , M z T , d = [d1, d2, d3] T ,

[0323] Since System 29 is a non-linear coupling system, according to the design theory of the second-order state observer, the following non-linear state expansion observer is designed:

[0324]

[0325] In Equation 30, e1 and e2 are the observation errors of the state observer; u and d are the observable quantities of the mobile robot state and external disturbance, respectively; L1 = diag(l 11 , l 12 , l 13 ) > 0, and L2 = diag(l 21 , l 22 , l 23 ) > 0 are the observation gain matrices; z1 and z2 are the estimated values of the system state and disturbance, respectively. In summary, as long as appropriate gain matrices and other parameters are selected, the estimation of the total disturbance can be achieved, and the observation error can converge to a very small range near the origin.

[0326] For the four-steering-wheel industrial mobile robot system, under the combined action of the inverse kinematics velocity controller and the sliding-mode dynamic torque controller, appropriate control parameters need to be selected to make the closed-loop system reach an asymptotically stable state. To verify the stability of the dual-model controller, for the wheeled mobile robot system, the Lyapunov function is constructed as follows:

[0327] V L = V1 + V2 (31)

[0328] Where:

[0329]

[0330] Taking the derivatives of Equation 32 and Equation 33 gives:

[0331]

[0332] As known from the foregoing It can also be known by observing Equation 35 So Therefore, V = V1 + V2 ≥ 0, which can prove that the system is asymptotically stable.

[0333] According to the actual working condition requirements of the four-steering-wheel industrial mobile robot, a chassis layout method of "four-steering-wheel - six-steering-wheel" is proposed, and the overall structure design of three car bodies and two connecting arms is completed. According to the characteristics of the four-steering-wheel industrial mobile robot that needs to perform multi-motion mode switching during the transfer process, its kinematic model is established; according to the characteristics of the mobile robot with heavy load and prominent mechanical properties, its dynamic model is established, and the correctness of the model under three main working conditions of straight-line motion, self-rotation, and curve motion is verified through virtual prototypes. The correctness and accuracy of the dual models are verified, laying a theoretical foundation for the subsequent research on trajectory tracking control methods.

[0334] The double - closed - loop inversion sliding - mode trajectory tracking control strategy consists of an outer - loop kinematic inversion tracking controller and an inner - loop dynamic sliding - mode tracking controller. The outer loop controls the speed during the tracking process of the mobile robot based on the inversion control principle and introduces a saturation function for optimization, outputting the speed control quantity to the inner loop. The inner loop controls the vehicle's mechanical characteristics during the tracking process of the mobile robot based on the sliding - mode control principle and introduces a disturbance observer to estimate external disturbances and system uncertainties, ultimately achieving double - closed - loop control. The simulation results prove that the stability and robustness of this controller meet the actual transfer requirements.

[0335] The four - omnidirectional - wheel industrial mobile robot that can carry heavy loads has excellent mobility, stability, and load - bearing capacity, and is suitable for transfer scenarios with a narrow working space. Studying the trajectory tracking control method for the four - omnidirectional - wheel industrial mobile robot is beneficial to improving its transfer efficiency and accuracy, making it more adaptable to complex and changeable working environments. Based on the existing motion control theory of the four - omnidirectional - wheel industrial mobile robot, the present invention proposes a reasonable and superior trajectory tracking control method, which can better meet the actual transfer requirements of materials.

Claims

1. An inversion sliding mode trajectory tracking control method based on a dual model of a four-steering-wheel industrial mobile robot, characterized in that, It includes the following steps: 1) Construct a three-dimensional digital simplified model of the four-steering-wheel industrial mobile robot. 2) Import the three-dimensional digital simplified model into ADAMS and add dynamic information to the three-dimensional digital simplified model; 3) Design the control law of the kinematic controller of the four-steering-wheel industrial mobile robot; 4) Based on the control law of the kinematic controller, use the inversion method to build an outer-loop kinematic inversion speed controller; 5) Design the control law of the dynamic sliding mode controller of the four-steering-wheel industrial mobile robot; 6) Based on the sliding mode control method, build an inner-loop dynamic torque controller; 7) Use a nonlinear extended state observer to estimate external disturbances and feedback the external disturbances to the output of the dynamic sliding mode controller; 8) Use the outer-loop kinematic inversion speed controller and the inner-loop dynamic torque controller to achieve the double-loop inversion sliding mode trajectory tracking control of the four-steering-wheel industrial mobile robot.

2. The inversion sliding mode trajectory tracking control method based on the dual models of a four-steering-wheel industrial mobile robot according to claim 1, wherein The steps of constructing the three-dimensional digital simplified model of the four-steering-wheel industrial mobile robot include: 1) Based on the physical object of the four-steering-wheel industrial mobile robot, use SolidWorks to establish a three-dimensional digital model of the four-steering-wheel industrial mobile robot; 2) Simplify the three-dimensional digital model of the four-steering-wheel industrial mobile robot, only retaining the vehicle body, four steering wheels and six driven wheels, so as to construct a three-dimensional digital simplified model of the four-steering-wheel industrial mobile robot.

3. The inversion sliding mode trajectory tracking control method based on the double model of the four-steering-wheel industrial mobile robot according to claim 2, wherein The three-dimensional digital simplified model of the four-steering-wheel industrial mobile robot includes vehicle body 1, vehicle body 2, vehicle body 3 and two telescopic arms; Among them, one steering wheel and two driven wheels are arranged on the chassis of vehicle body 1. The steering wheel and the two driven wheels are arranged in a triangular shape, and the axes of the two driven wheels coincide; One steering wheel and two driven wheels are arranged on the chassis of vehicle body 2. The steering wheel and the two driven wheels are arranged in a triangular shape, and the axes of the two driven wheels coincide; The two driven wheels of vehicle body 1 and the two driven wheels of vehicle body 2 are arranged axially symmetrically; Steering wheels and driven wheels are arranged at the four corners of the chassis of vehicle body 3 respectively. Two steering wheels are arranged diagonally, and two driven wheels are arranged diagonally. The axes of the steering wheel and the driven wheel on the same side coincide.

4. The inversion sliding mode trajectory tracking control method based on the dual models of a four-steering-wheel industrial mobile robot according to claim 1, characterized in that, The steps of adding dynamic information to the three-dimensional digital simplified model include: 1) Set two MARKER points at the position of the vehicle body for installing the upper fixture ring as the application points of external forces; 2) Apply driving pairs and rotating pairs to each steering wheel and driven wheel of the three-dimensional digital simplified model; 3) Gradually apply a bearing capacity equal to the load on the reserved MARKER points, add a tension damping disc spring group to the steering wheels, and set the parameters of the tension damping disc spring group and the road surface parameters, so as to simulate the transfer process of the four-steering-wheel industrial mobile robot for workpieces; The parameters of the tension damping disc spring group include the stiffness, tension force and damping coefficient of the tension damping disc spring group; The road surface parameters include static friction coefficient, dynamic friction coefficient, viscous boundary velocity, friction boundary velocity, friction index, and tire penetration depth.

5. The inversion sliding mode trajectory tracking control method based on the dual models of a four-steering-wheel industrial mobile robot according to claim 1, characterized in that The steps of designing the control law of the kinematic controller of the four-steering-wheel industrial mobile robot include: 1) Construct a pose error model of the four-steering-wheel industrial mobile robot to describe the deviation between the actual pose and the reference pose at a certain moment during the movement of the four-steering-wheel industrial mobile robot. The steps include: 1.1) Construct the pose error model of the four-steering-wheel industrial mobile robot in the earth coordinate system, i.e.: where x e , y e , respectively represent the relative errors in the x, y, and angular directions; represents the pose of the mobile robot's reference trajectory in the global coordinate system; represents the pose of the mobile robot's actual trajectory in the global coordinate system; 1.2) Convert the pose error model of the four-steering-wheel industrial mobile robot in the earth coordinate system into the pose error in the vehicle body coordinate system of the mobile robot, i.e.: where, e x , e y , respectively represent the relative errors in the x, y, and angular directions; 1.3) Differentiate the pose error in the vehicle body coordinate system of the mobile robot to obtain: wherein, respectively represent the differentials of the relative error; represents the differential of the pose of the reference trajectory of the mobile robot in the global coordinate system; represents the differential of the pose of the actual trajectory of the mobile robot in the global coordinate system; 1.4) Based on formula (3), construct the pose error model of the four-steering-wheel industrial mobile robot, i.e.: where, v xr , v yr , ω r respectively represent the velocities of the reference trajectory of the mobile robot in the x, y, and angular directions in the global coordinate system; v x , v y , ω respectively represent the velocities of the actual trajectory of the mobile robot in the x, y, and angular directions in the global coordinate system; 2) Design the control objective of the kinematic controller, i.e.: 3) Design the control law of the kinematic controller when the sideslip angle of the center of mass of the mobile robot is zero. The steps include: 3.1) Define the Lyapunov function L1, i.e.: where the parameter k2 > 0; 3.2) Simplify the pose error model of the four-steering-wheel industrial mobile robot to obtain: where, v r is the reference trajectory speed of the mobile robot; v is the actual trajectory speed of the mobile robot; 3.3) Differentiate the Lyapunov function L1 to obtain: 3.4) With as the target, design the kinematic control law v of the kinematic controller when the centroid side slip angle is zero c , that is: where k1, k2, k3 are positive integers; 4) Design the control law of the kinematic controller when the sideslip angle of the center of mass of the mobile robot is not zero. The steps include: 4.1) Design the control objective of the kinematic controller, i.e.: 4.2) Define the Lyapunov function L'1, i.e.: where the parameter k'2 ≥ 0; 4.3) Differentiate the Lyapunov function L'1 to obtain: 4.4) With as the target, design the kinematic control law v of the kinematic controller when the centroid side slip angle is not zero c , that is: where k1' > 0, k'2 > 0, k3' > 0.

6. The inversion sliding mode trajectory tracking control method based on the dual models of a four-steering-wheel industrial mobile robot according to claim 1, wherein When constructing the outer-loop kinematic inversion speed controller, the saturation function u is also used to optimize the outer-loop kinematic inversion speed controller; The saturation function u is as follows: where λ s represents the threshold value of the centroid side slip angle, and λ s is greater than 0; u s represents the threshold value of the centroid and velocity, and u s is greater than 0; u1 and u2 are saturation functions in the cases where the centroid side slip angle is zero and non-zero respectively; Among them, matrix C s and threshold λ s are related as follows:

7. The inversion sliding mode trajectory tracking control method based on the dual models of a four-steering-wheel industrial mobile robot according to claim 1, wherein The steps for designing the control law of the dynamic sliding mode controller of the four-steering-wheel industrial mobile robot include: 1) Construct the speed error vector v of the four-steering-wheel industrial mobile robot e , namely: where (v xr , v yr , ω r ) is the reference speed control quantity; (v x , v y , ω) is the actual speed control quantity; (v ex , v ey , ω e ) is the speed error in the x, y, and rotation angle directions; 2) Construct the model of the telescopic arm of the mobile robot holding the workpiece, i.e.: Among them, d1, d2, and d3 are error disturbances in the horizontal, vertical, and rotational directions respectively; m represents the total mass of the mobile robot; I z represents the moment of inertia of the mobile robot; F Di represents the driving force of the steering wheel; P Di represents the lateral force received by each steering wheel; x i , y i respectively represent the coordinates of each steering wheel; a x , a y are the accelerations of the mobile robot in the x and y axis directions respectively; 3) Select the integral sliding mode function S, i.e.: where β1 > 0, β2 > 0, β3 > 0; 4) Design the sliding mode reaching law That is: wherein, the inverse hyperbolic sine function x is a variable; parameter k > 0; parameter ε > 0; where the nonlinear function fal(x, α, δ) is as follows: where 0 < δ < 1, α > 0; δ is a linear symmetric discontinuous function of the interval length contained in fal(x, α, δ) with respect to the origin; 5) Based on the sliding mode reaching law Design a kinematic controller, namely: Among them, the coefficient \(E = diag[\varepsilon_1,\varepsilon_2,\varepsilon_3]\), \(Q = diag[q_1,q_2,q_3]\); the vector vector \(\varepsilon_1\gt0,\varepsilon_2\gt0,\varepsilon_3\gt0\); \(q_1\gt0,q_2\gt0,q_3\gt0\); \(0\lt\delta_1\lt1,0\lt\delta_2\lt1,0\lt\delta_3\lt1\); \(\alpha_1\gt0,\alpha_2\gt0,\alpha_3\gt0\); 6) Differentiate the integral sliding mode function to obtain: In the formula, is the differential of the integral sliding mode function; is the differential of the velocity errors in the x, y, and rotation directions; 7) Substitute the velocity error, vector, and sliding mode reaching law into formula (23) to obtain: 8) Combine formula (17) and formula (24) to obtain the control law of the dynamic sliding mode controller, i.e.: where F xd , F yd , M zd are the lateral resultant force, longitudinal resultant force and resultant moment of the four steering wheels that need to be equivalently controlled.

8. The inversion sliding mode trajectory tracking control method based on the dual models of a four-steering-wheel industrial mobile robot according to claim 1, characterized in that The nonlinear state extended observer is as follows: where e1 and e2 are the observation errors of the state observer; u and d are the observable quantities of the mobile robot state and external disturbance respectively; L1 = diag(l 11 , l 12 , l 13 ) > 0, L2 = diag(l 21 , l 22 , l 23 ) > 0 are the observation gain matrices; z1 and z2 are the estimated values of the system state and disturbance respectively; represents the inverse matrix of the moment of inertia.

9. The inversion sliding mode trajectory tracking control method based on the dual models of a four-steering-wheel industrial mobile robot according to claim 1, characterized in that In the process of double-loop inversion sliding mode trajectory tracking control, the outer-loop kinematic inversion speed controller provides the centroid velocity of the mobile robot for the inner-loop dynamic torque controller, and the inner-loop dynamic torque controller is used to control the moving trajectory of the mobile robot.

10. The inversion sliding mode trajectory tracking control method based on the dual models of a four-steering-wheel industrial mobile robot according to claim 1, wherein, The method for building the outer-loop kinematic inversion speed controller is: Based on the inversion method, build the outer-loop kinematic inversion speed controller through the Simulink module in the MTALAB environment. The method for building the inner-loop dynamic torque controller is: Based on the sliding mode control method, build the inner-loop dynamic torque controller through the Simulink module in the MTALAB environment.

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