A mobile robot trajectory tracking method based on backstepping sliding mode

Abstract

The present invention belongs to the field of machine control, and relates to a wheeled mobile robot trajectory tracking control technology, and more specifically, is a mobile robot trajectory tracking method based on backstepping sliding mode. The method comprises the following steps: step 1: establishing a kinematic model and a dynamic model of a mobile robot; step 2: designing a kinematic controller of a mobile robot based on backstepping method; step 3: designing a dynamic controller of a mobile robot based on a sliding mode variable structure of a combined reaching law; step 4: obtaining the actual posture and actual speed of the mobile robot according to the kinematic model of the robot; inputting the desired posture and actual posture of the mobile robot into the backstepping kinematic controller to obtain a virtual control variable, inputting the virtual control variable and the actual speed into the dynamic controller to obtain a control torque, inputting the control torque and external disturbance into the dynamic model, and outputting acceleration, so that the actual posture of the mobile robot reaches the desired posture.

Description

Technical Field

[0001] The invention belongs to the field of machine control, and relates to a wheeled mobile robot trajectory tracking control technology, and more specifically, to a mobile robot trajectory tracking method based on backstepping sliding mode. Background Art

[0002] Wheeled mobile robots are intelligent mobile platforms that integrate multiple functions such as environmental perception, dynamic decision-making and planning, and behavior control and execution. Compared with traditional industrial robots, mobile robots have the characteristics of flexible movement, strong carrying capacity and high work efficiency. In recent years, they have been widely used in the service industry, aerospace, earthquake relief and other fields.

[0003] Robotics is an important branch in the field of automation and control, and has been widely used in industrial production and daily life. Among them, wheeled mobile robots are the most common type of mobile robots in daily life. They play an important role in industry, military, civil and scientific exploration, and can replace humans to perform information collection and processing tasks in harsh environments such as deep-sea surveys and disaster surveys. Currently, exploring the motion performance of mobile robots in complex environments is a hot topic of research.

[0004] The non-holonomic wheeled mobile robot is a typical multi-input and multi-output nonlinear system. Therefore, trajectory tracking is the basis of mobile robot motion control. The purpose is to design a reasonable control method to enable the mobile robot to quickly and stably track the pre-set curve with time as the variable function. Traditionally, the methods for solving mobile robot trajectory tracking include PID control, backstepping control, sliding mode control or adaptive control, etc. These methods have achieved good tracking effects, but when there are uncertain factors in the system or the path has inflection points or the initial error is large, its motion often cannot achieve the ideal tracking effect. In recent years, with the development of artificial intelligence, the application technology of fuzzy neural network has shown a good application prospect in the field of control, providing an effective way to solve the high nonlinearity, coupling and unmodeled uncertainty in robot control. Summary of the invention

[0005] In order to overcome the shortcomings of the prior art, the present invention provides a mobile robot trajectory tracking method based on backstepping sliding mode. First, the kinematics and dynamics modeling of the mobile robot is carried out; then the backstepping method is used to realize the trajectory tracking of the mobile robot; then the speed tracking control law is designed based on the dynamic model of the mobile robot using the sliding mode variable structure method; finally, the precise trajectory tracking of the mobile robot is realized according to the designed control law. This method can effectively weaken the chattering phenomenon of the control system, and can improve the motion performance of the mobile robot when there is an inflection point in the expected path, and improve the anti-interference ability of the mobile robot and the motion accuracy of trajectory tracking.

[0006] To achieve the above object, the present invention adopts the following technical solution:

[0007] A mobile robot trajectory tracking method based on backstepping sliding mode comprises the following steps:

[0008] Step 1: Establish the kinematic model and dynamic model of the mobile robot;

[0009] Step 2: Design the kinematic controller of the mobile robot based on the backstepping method;

[0010] Step 3: Design the dynamic controller of the mobile robot based on the sliding mode variable structure of the combined reaching law;

[0011] Step 4: According to the robot's kinematic model, the actual position and actual speed of the mobile robot are obtained; the desired position and actual position of the mobile robot are input into the backstepping kinematic controller to obtain virtual control variables, the virtual control variables and actual speed are input into the dynamic controller to obtain the control torque, the control torque and external disturbance are input into the dynamic model, and the acceleration is output so that the actual position of the mobile robot reaches the desired position.

[0012] The technical solution is further optimized, and the kinematic model of the mobile robot is:

[0013] The actual position of the mobile robot at any time is:

[0014]

[0015] In formula (1), x, y and They represent the horizontal, vertical and heading angle positions of the mobile robot respectively. The actual speed of the mobile robot at any time is:

[0016] v=[v,ω] T (26)

[0017] In formula (2), v is the linear velocity of the mobile robot, and ω is the angular velocity of the mobile robot.

[0018] When there is no sliding or side slipping between the wheels and the ground, the motion model of the mobile robot can be expressed as:

[0019]

[0020] In formula (3), A(q) is:

[0021]

[0022] Formula (4) is the Jacobian matrix, which represents the relative relationship between posture and velocity.

[0023] This technical solution is further optimized, the dynamic model of the mobile robot is:

[0024] The mass m and moment of inertia I of the mobile robot, the dynamic model can be expressed as follows:

[0025]

[0026] In formula (5), is the acceleration vector, τ and τ d are the input torque and disturbance respectively, and M is:

[0027]

[0028] The technical solution is further optimized, and the kinematic controller of the mobile robot is designed as follows:

[0029] The goal of the kinematic controller is to c Under the action of, when time t tends to infinity, the actual position of the mobile robot It also gradually converges to the desired posture and the expected speed v r =[v r ,ω r ] T , expected posture where v r and ω r are the expected linear velocity and expected angular velocity of the mobile robot, respectively, r ,y r and They are the expected lateral position, longitudinal position and heading angle position of the mobile robot respectively;

[0030] Therefore, the expected speed of the mobile robot in the global coordinate system is:

[0031]

[0032] The pose error is defined as:

[0033]

[0034] where e x , e y and are the error components of the posture error along the X direction, Y direction and heading angle direction, so the posture error vector can be obtained as:

[0035]

[0036] The posture error differential equation is obtained from formula (8) and formula (9):

[0037]

[0038] in and are the differential forms of the posture errors in the lateral, longitudinal and heading directions respectively. To keep the system stable, the Lyapunov function is constructed based on the backstepping design idea:

[0039]

[0040] Deriving equation (11) and substituting equation (10) into it, we get:

[0041]

[0042] In summary, to ensure Design a kinematics-based backstepping trajectory tracking controller:

[0043]

[0044] where v c and ω c is the output virtual control quantity, k1, k2, k3 are all positive real numbers.

[0045] The technical solution is further optimized, the dynamic controller:

[0046] The velocity tracking error of the mobile robot is defined as:

[0047]

[0048] Considering that the dynamic model of the wheeled mobile robot is a first-order nonlinear function, the proportional integral sliding surface is selected here:

[0049]

[0050] Among them, μ1 and μ2 are the integral parameters of the sliding surface, and both are greater than 0, which can improve the robustness of the system. When the system state moves to the sliding surface at S = 0:

[0051]

[0052] Where μ=diag(μ1,μ2)>0, so choosing a suitable μ can make e v tends to 0, and the derivative of formula (15) is combined with formula (5) to obtain the following:

[0053]

[0054] Let formula (17) be 0, and the equivalent control law is:

[0055]

[0056] Therefore, its vector expression is:

[0057]

[0058] However, actual mobile robot systems generally have external interference, so the switching control law τ is introduced sw , convert formula (19) into the following form:

[0059]

[0060] Where β=diag(β1,β2)>0, sgn(s)=[sgn(s1),sgn(s2)].

[0061] (3) In the early stage of sliding mode motion, in order to ensure that the system state quickly approaches the sliding mode surface to weaken chattering, the following exponential approach law is selected:

[0062]

[0063] Its control law is:

[0064]

[0065] Among them, η=diag(η1,η2)>0, q=diag(q1,q2)>0. The above η, q and ε are all adjustable parameters of the reaching law. By reasonably adjusting these parameters, the quality of the arrival process can be guaranteed and the chattering phenomenon in the variable structure system can be reduced;

[0066] (4) In the late stage and stable stage of sliding mode motion, in order to weaken the chattering phenomenon in the stable state, the following speed reaching law is selected here:

[0067]

[0068] Its control law is:

[0069]

[0070] Where ε=diag(ε1,ε2)>0, vs(S,ε)=[ε1|e v |sgn(s1),-ε2|e ω |sgn(s2)] T .

[0071] Different from the prior art, the above technical solution has the following beneficial effects:

[0072] The present invention can realize motion control of a mobile robot, effectively weaken the chattering phenomenon existing in the system, improve the trajectory tracking and anti-interference capabilities of the mobile robot when there are inflection points in the expected trajectory, and improve the motion accuracy of the trajectory tracking of the mobile robot. BRIEF DESCRIPTION OF THE DRAWINGS

[0073] Figure 1 is a schematic diagram of a mobile robot model of the present invention;

[0074] Figure 2 It is a system structure block diagram of the present invention;

[0075] Figure 3 It is a diagram showing the tracking effect of a circular and a broken line trajectory according to an embodiment of the present invention. DETAILED DESCRIPTION

[0076] In order to explain the technical content, structural features, achieved objectives and effects of the technical solution in detail, the following is a detailed description in conjunction with specific embodiments and accompanying drawings.

[0077] The present invention discloses a mobile robot trajectory tracking control method based on a backstepping sliding mode method. A mobile robot is a typical nonlinear system with strongly coupled, time-varying and nonlinear dynamic characteristics. It is usually inevitable to have uncertain factors such as parameter errors, unmodeled dynamics, observation noise and uncertain external interference. Therefore, based on the mobile robot, its kinematic controller and dynamic controller are designed separately, which will effectively solve these problems and improve the motion accuracy of its trajectory tracking.

[0078] The wheeled mobile robot is driven by differential speed, has a simple structure and is flexible in steering. The present invention is directed to a differentially driven mobile robot, and solves the following technical problems: when there is an inflection point in the motion path of the mobile robot or the initial error is relatively large, the resulting speed jump will cause the final trajectory tracking effect of the mobile robot to deteriorate. In addition, since the wheeled mobile robot system is a typical nonlinear system with strong coupling, time-varying and nonlinear dynamic characteristics, there are usually unavoidable uncertainties such as parameter errors, unmodeled dynamics, observation noise and uncertain external interference, which will also have an adverse effect on the trajectory tracking performance of the mobile robot. Therefore, in the above state, how to control the motion state of the mobile robot so that it can better track a predetermined path that changes with time, so as to move stably according to the expected state.

[0079] The mobile robot trajectory tracking control method based on the backstepping sliding mode method includes the following steps:

[0080] Step 1: Kinematic and dynamic modeling of the mobile robot;

[0081] See also Figure 1The figure shows a simplified model of the mobile robot. The global coordinate system XOY is a Cartesian inertial coordinate system that allows the mobile robot to move in it and is used to represent environmental information. The center of mass of the mobile robot and the center of the drive wheel axis coincide at point P, with a mass of m and a moment of inertia of I. The forward linear velocity of the center of mass of the mobile robot is v, the angular velocity of the center of mass is ω, and the angle between the direction of movement of the mobile robot and the X-axis is is the heading angle, so a physical analysis is performed on the center of mass P of the mobile robot.

[0082] Step 1.1 Establish the kinematic model of the mobile robot

[0083] The actual position of the mobile robot at any time is:

[0084]

[0085] In formula (1), x, y and They represent the horizontal, vertical and heading angle positions of the mobile robot respectively. The actual speed of the mobile robot at any time is:

[0086] V=[v,ω] T (50)

[0087] In formula (2), v is the linear velocity of the mobile robot, and ω is the angular velocity of the mobile robot.

[0088] When there is no sliding or side slipping between the wheels and the ground, the motion model of the mobile robot can be expressed as:

[0089]

[0090] In formula (3), A(q) is:

[0091]

[0092] Formula (4) is the Jacobian matrix, which represents the relative relationship between the actual posture and the actual velocity.

[0093] Step 1.2 Establish the dynamic model of the mobile robot

[0094] Considering the mass m and moment of inertia I of the mobile robot, the dynamic model can be expressed as follows:

[0095]

[0096] In formula (5), is the acceleration vector, τ and τ d are the control torque and disturbance respectively, and the mass and moment of inertia matrix M of the mobile robot is:

[0097]

[0098] Step 2: Design a kinematic controller for a mobile robot based on backstepping

[0099] like Figure 2 In the process of the mobile robot's motion, the reference speed is used as input. After its kinematic model, that is, formula (3), the actual motion posture error of the mobile robot is obtained. The backstepping method obtains the feedback controller by recursively constructing the Lyapunov function of the closed-loop system. The control law is selected so that the derivative of the Lyapunov function along the closed-loop system trajectory has certain performance, ensuring the boundedness of the closed-loop system trajectory and convergence to the equilibrium point. The selected control law is the solution to the trajectory tracking problem. This method can decompose the complex nonlinear system into subsystems that do not exceed the system order, and then design partial Lyapunov functions and intermediate virtual control quantities for each subsystem, and "step back" to the entire system, and integrate them to complete the design of the entire control law. The basic design method starts from the kernel of a high-order system. Design a virtual control law to ensure certain performance of the kernel system, such as stability; then gradually modify the algorithm of the obtained virtual control law, but the established performance should be guaranteed; and then design a real stabilization controller to achieve global regulation or tracking of the system, so that the system can achieve the desired performance indicators. The backstepping control method is applicable to uncertain nonlinear systems that can be linearized or have strict parameter feedback. In terms of designing robust or adaptive controllers for uncertain systems, especially when the disturbance or uncertainty does not meet the matching conditions, the backstepping method has obvious advantages. Therefore, a kinematic controller for a mobile robot is designed based on this method.

[0100] The goal of the kinematic controller is to c Under the action of, when time t tends to infinity, the actual position of the mobile robot It also gradually converges to the desired posture and the expected speed v r =[v r ,ω r ] T , expected posture where v r and ω r are the expected linear velocity and expected angular velocity of the mobile robot, respectively, r ,y r and They are the expected lateral posture, longitudinal posture and heading angle posture of the mobile robot respectively.

[0101] Therefore, the expected speed of the mobile robot in the global coordinate system is:

[0102]

[0103] The pose error is defined as:

[0104]

[0105] where e x , e y and are the error components of the posture error along the X direction, Y direction and heading angle direction, so the posture error vector can be obtained as:

[0106]

[0107] The posture error differential equation is obtained from formula (8) and formula (9):

[0108]

[0109] in and are the differential forms of the posture errors in the lateral, longitudinal and heading directions respectively. To keep the system stable, the Lyapunov function is constructed based on the backstepping design idea:

[0110]

[0111] Deriving equation (11) and substituting equation (10) into it, we get:

[0112]

[0113] In summary, to ensure Design a kinematics-based backstepping trajectory tracking controller:

[0114]

[0115] where v c and ω c is the output virtual control quantity, k1, k2, k3 are all positive real numbers. c Under the action of , when time t tends to infinity, the actual posture q also converges to the expected posture q r It can be seen that when the expected trajectory has an inflection point or the initial tracking error is large, it will lead to v c To produce a sudden change, the mobile robot needs to have a large torque, but this is difficult to achieve in the actual control of the mobile robot. Therefore, in the actual trajectory tracking control process, the speed tracking vv r is q→q r Based on the premise that only designing a kinematic controller cannot guarantee high-quality velocity tracking, it is necessary to design a dynamic controller.

[0116] Step 3: Sliding mode variable structure control method based on combined reaching law.

[0117] The sliding mode variable structure control method is essentially a special type of nonlinear control. Its nonlinearity is manifested as control discontinuity. The difference between this control method and other control methods is that the "structure" of the system is not fixed, but can be adjusted according to the current state of the system such as deviation and its various orders in the dynamic process. Derivative Purposefully changing so that the system follows the predetermined Sliding Mode The advantage of sliding mode control is that it can overcome the uncertainty of the system and has strong resistance to disturbances and unmodeled dynamics. robustness , especially for nonlinear systems, it has good control effects. algorithm Simple, fast response, and resistant to external noise and parameters Perturbation It is robust, so it is widely used in related research on mobile robots. But at the same time, the disadvantage of this method is that when the state trajectory reaches the sliding surface, it is difficult to strictly slide along the sliding surface toward the equilibrium point, but it crosses back and forth on both sides of the sliding surface, resulting in vibration, that is, the jitter problem. Therefore, sliding mode variable structure control does not need to establish an accurate model of the controlled object, and is insensitive to external bounded interference and parameter changes. In recent years, it has gradually been applied to the motion control of mobile robots. However, sliding mode variable structure control will inevitably produce jitter, which may cause system instability and lead to failure of trajectory tracking in severe cases. The present invention proposes a sliding mode control method based on the combined convergence law based on the sliding mode variable structure, which can effectively weaken the jitter phenomenon of the control system and improve the stability of the control system and the trajectory tracking performance. Figure 2 The sliding mode control uses the virtual output variable v of the backstepping control method in step 2 c and the actual velocity v of the mobile robot as input, the purpose is to make v converge to v c , so that the system can achieve the stability of velocity tracking and ensure the accuracy of system trajectory tracking. Therefore, it is necessary to design a dynamic controller based on sliding mode variable structure.

[0118] First, the velocity tracking error of the mobile robot is defined as:

[0119]

[0120] Considering that the dynamic model of the wheeled mobile robot is a first-order nonlinear function, the proportional integral sliding surface is selected here:

[0121]

[0122] Among them, μ1 and μ2 are the integral parameters of the sliding surface, and both are greater than 0, which can improve the robustness of the system. When the system state moves to the sliding surface at S = 0:

[0123]

[0124] Where μ=diag(μ1,μ2)>0, so choosing a suitable μ can make e v Tends to 0. Taking the derivative of formula (15) and combining it with formula (5), we get the following:

[0125]

[0126] Let formula (17) be 0, and the equivalent control law is:

[0127]

[0128] Therefore, its vector expression is:

[0129]

[0130] However, actual mobile robot systems generally have external interference. In order to reduce the influence of interference in the system, the switching control law τ is introduced. sw , convert formula (19) into the following form:

[0131]

[0132] The adjustable parameters of the system are β=diag(β1,β2)>0, sgn(s)=[sgn(s1),sgn(s2)].

[0133] However, the discontinuous term sgn(s) in the control law τ will cause the system to switch back and forth between different control logics, thus causing system chattering. Therefore, the combined approaching law is introduced into the tracking problem of the mobile robot, and the chattering problem in the sliding mode motion is weakened by replacing the sign function sgn(s), and the rapidity and stability of the tracking process are guaranteed, that is, the exponential approaching law is used in the early stage of the sliding mode motion to ensure that the system state can quickly approach the sliding mode surface and weaken the chattering; the variable speed approaching law is used in the late stage and stable stage of the sliding mode motion to effectively reduce the chattering in the steady state, ensuring good steady-state performance. The specific implementation process includes the following two stages:

[0134] (5) In the early stage of sliding mode motion, in order to ensure that the system state quickly approaches the sliding mode surface to weaken chattering, the following exponential approach law is selected:

[0135]

[0136] Its control law is:

[0137]

[0138] Among them, η=diag(η1,η2)>0, q=diag(q1,q2)>0. The above η, q and ε are all adjustable parameters of the reaching law. By reasonably adjusting these parameters, the quality of the arrival process can be guaranteed and the chattering phenomenon in the variable structure system can be reduced.

[0139] (6) In the late stage and stable stage of sliding mode motion, in order to weaken the chattering phenomenon in the stable state, the following speed reaching law is selected here:

[0140]

[0141] Its control law is:

[0142]

[0143] Where ε=diag(ε1,ε2)>0, vs(S,ε)=[ε1|e v |sgn(s1),-ε2|e ω |sgn(s2)] T .

[0144] Then, a stability analysis is performed and the effectiveness of the proposed dynamic controller is proved by using the Lyapunov function:

[0145]

[0146] Differentiating formula (25) and combining it with formula (21) yields:

[0147]

[0148] By taking the derivative of formula (25) and combining it with formula (23), we can obtain:

[0149]

[0150] Available Therefore, the dynamic control system is stable and can ensure that the actual speed v converges quickly and stably to the virtual control speed v c .

[0151] Steps 2 and 3 complete the design of the kinematic and dynamic controllers for the mobile robot, which are suitable for typical complex systems with strong coupling, time variation and nonlinearity such as wheeled mobile robot systems, and effectively reduce the jitter problem in trajectory tracking motion.

[0152] Step 4: According to the above steps and methods, the designed control system is combined to achieve accurate trajectory tracking of the mobile robot. The final control system block diagram is as follows Figure 2 shown.

[0153] According to this control method, the trajectory tracking effect of the mobile robot is as follows: Figure 3 shown.

[0154] It can be seen from the final algorithm detection results that the method for mobile robot trajectory tracking proposed in the present invention can achieve accurate trajectory tracking for a mobile robot with uncertain factors under the condition that there is an inflection point in the trajectory. The tracking effect is good, and the jitter phenomenon of the system is effectively weakened, thereby ensuring the stability and speed of trajectory tracking and realizing the motion control of the mobile robot.

[0155] It should be noted that, in this article, relational terms such as first and second, etc. are only used to distinguish one entity or operation from another entity or operation, and do not necessarily require or imply any such actual relationship or order between these entities or operations. Moreover, the terms "include", "comprise" or any other variants thereof are intended to cover non-exclusive inclusion, so that the process, method, article or terminal device including a series of elements includes not only those elements, but also other elements not explicitly listed, or also includes elements inherent to such process, method, article or terminal device. In the absence of further restrictions, the elements defined by the sentence "include..." or "comprise..." do not exclude the existence of other elements in the process, method, article or terminal device including the elements. In addition, in this article, "greater than", "less than", "exceed" and the like are understood to exclude the number itself; "above", "below", "within" and the like are understood to include the number itself.

[0156] Although the above embodiments have been described, once those skilled in the art know the basic creative concepts, they can make additional changes and modifications to these embodiments. Therefore, the above description is only an embodiment of the present invention and does not limit the patent protection scope of the present invention. Any equivalent structure or equivalent process transformation made by using the contents of the specification and drawings of the present invention, or directly or indirectly used in other related technical fields, are also included in the patent protection scope of the present invention.

Claims

1. A mobile robot trajectory tracking method based on backstepping sliding mode, characterized in that: The steps include: Step 1: Establish the kinematic model and dynamic model of the mobile robot; Step 2: Design the kinematic controller of the mobile robot based on the backstepping method; Step 3: Design the dynamic controller of the mobile robot based on the sliding mode variable structure of the combined reaching law; Step 4: According to the kinematic model of the robot, the actual position and actual speed of the mobile robot are obtained; The desired posture and actual posture of the mobile robot are input into the backstepping kinematic controller to obtain virtual control variables, the virtual control variables and actual speed are input into the dynamic controller to obtain control torque, the control torque and external disturbance are input into the dynamic model, and the acceleration is output so that the actual posture of the mobile robot reaches the desired posture; Kinematic model of the mobile robot: The actual position of the mobile robot at any time is: In formula (1), x, y and They represent the lateral, longitudinal and heading angle positions of the mobile robot respectively. The actual speed of the mobile robot at any time is: v=[v,ω] T (2) In formula (2), v is the linear velocity of the mobile robot, and ω is the angular velocity of the mobile robot; When there is no sliding or side slipping between the wheels and the ground, the motion model of the mobile robot can be expressed as: In formula (3), A(q) is: Formula (4) is the Jacobian matrix, which represents the relative relationship between posture and velocity; The dynamic model of the mobile robot: The mass m and moment of inertia I of the mobile robot, the dynamic model is expressed as follows: In formula (5), is the acceleration vector, τ and τ d are the input torque and disturbance respectively, and M is: The kinematic controller of the mobile robot is designed as follows: The goal of the kinematic controller is to c Under the action of, when time t tends to infinity, the actual position of the mobile robot It also gradually converges to the desired posture and the expected speed v r =[v r ,ω r ] T , expected posture where v r and ω r are the expected linear velocity and expected angular velocity of the mobile robot, respectively, r ,y r and They are the expected lateral position, longitudinal position and heading angle position of the mobile robot respectively; Therefore, the expected speed of the mobile robot in the global coordinate system is: The pose error is defined as: where e x , e y and are the error components of the posture error along the X direction, Y direction and heading angle direction, so the posture error vector is: The posture error differential equation is obtained from formula (8) and formula (9): in and are the differential forms of the posture errors in the lateral, longitudinal and heading directions respectively. To keep the system stable, the Lyapunov function is constructed based on the backstepping design idea: Deriving equation (11) and substituting equation (10) into it, we get: In summary, to ensure Design a kinematics-based backstepping trajectory tracking controller: where v c and ω c is the output virtual control quantity, k1, k2, k3 are all positive real numbers.

2. The mobile robot trajectory tracking method based on backstepping sliding mode according to claim 1, characterized in that: The dynamics controller: The velocity tracking error of the mobile robot is defined as: Considering that the dynamic model of the wheeled mobile robot is a first-order nonlinear function, the proportional integral sliding surface is selected here: Among them, μ1 and μ2 are the integral parameters of the sliding surface, and both are greater than 0, which can improve the robustness of the system. When the system state moves to the sliding surface at S = 0: Where μ = diag (μ1, μ2) > 0, so choosing a suitable μ can make e v tends to 0, and the derivative of formula (15) is combined with formula (5) to obtain the following: Let formula (17) be 0, and the equivalent control law is: Therefore, its vector expression is: However, actual mobile robot systems generally have external interference, so the switching control law τ is introduced sw , convert formula (19) into the following form: Where β = diag (β1, β2) > 0, sgn (s) = [sgn (s1), sgn (s2)]; (1) In the early stage of sliding mode motion, in order to ensure that the system state quickly approaches the sliding mode surface to weaken chattering, the following exponential approach law is selected: Its control law is: Among them, η=diag(η1,η2)>0, q=diag(q1,q2)>0; the above η, q and ε are all adjustable parameters of the reaching law. By reasonably adjusting these parameters, the quality of the arrival process can be guaranteed and the chattering phenomenon in the variable structure system can be reduced; (2) In the late stage and stable stage of sliding mode motion, in order to weaken the chattering phenomenon in the stable state, the following speed reaching law is selected here: Its control law is: Where ε=diag(ε1,ε2)>0, vs(S,ε)=[ε1|e v |sgn(s1),-ε2|e ω |sgn(s2)] T 。

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