Wheel robot sliding mode formation fault-tolerant control method based on event triggering

Abstract

The present specification discloses an event-triggered wheeled robot sliding mode formation fault-tolerant control method, so that the wheeled robot formation has good tracking performance when performing tasks and reduces the large occupation of system resources by communication, through the analysis of the actual situation of the wheeled robot operation, the influence of the wheeled robot formation on the external interference is considered, and the state observer is used to estimate and compensate the unknown external disturbance and bias fault, a kind of event-triggered sliding mode fault-tolerant controller is designed. At the same time, the global stability of the closed-loop system formed under the action of the controller is ensured and the Zeno phenomenon is avoided, and the simulation examples verify that the method has certain advantages.

Description

Technical Field

[0001] This invention relates to the field of wheeled robot control, specifically to the problem of trajectory tracking control in cooperative formation of wheeled robots. Background Technology

[0002] For the past two decades, distributed cooperative control of multi-robot systems has been a research hotspot, especially formation tracking control, which is a key issue in distributed cooperative control. The widespread application of wheeled robots in practical scenarios such as handling, support, and exploration, and the unique advantages of multiple robots in completing certain complex tasks compared to single robots, have made research in this field increasingly important.

[0003] However, current control methods for multi-wheeled robots mostly employ time-triggered mechanisms to acquire the state information of neighboring nodes. This means that neighboring node state updates are based on the system's sampling period, rather than the system's output state, which significantly increases the system's computational cost. Furthermore, under time-triggered control, frequent communication between wheeled robots greatly increases the system's burden and makes it vulnerable to attacks.

[0004] Meanwhile, actuator failures, uncertain dynamics, and external interference in wheeled robots can significantly impact the transient and steady-state performance of wheeled robot formation systems. In severe cases, this can lead to instability in the formation system or even collisions between robots.

[0005] Patent CN114035588A proposes an event-triggered trajectory tracking control method for wheeled robots. Its main contribution lies in inventing an event-triggered trajectory tracking control method. However, this method only studies the event-triggered trajectory tracking problem of a single robot and does not address the cooperative control problem in scenarios involving multiple wheeled robots in formation. Patent CN113703451A proposes an adaptive fault-tolerant control method for multi-wheeled robot formations. Its main contribution is designing an adaptive fault-tolerant torque control law for the formation of wheeled robots. However, this method does not study event-triggered scenarios.

[0006] Currently, due to the significant impact of actuator failures and unknown external disturbances on the control performance of nonlinear systems, many scholars both domestically and internationally have conducted in-depth research on these issues. Studies have shown that state observers can effectively estimate and compensate for unknown disturbances, while the design of sliding mode controllers can guarantee the stability and robustness of the closed-loop system globally. In the cooperative control of multi-wheeled robots, using event-triggered mechanisms to save communication resources is one of the key research focuses both domestically and internationally. It is worth noting that this method has not been reported in detail in publicly available materials and other literature based on wheeled robot formations. Summary of the Invention

[0007] This invention aims to reduce communication costs, improve system efficiency, and effectively eliminate the effects of interference. Therefore, it provides an event-triggered fault-tolerant control method for cooperative trajectory tracking of wheeled robots. To achieve this goal, this invention employs Lyapunov stability theory and designs a sliding mode trajectory tracking control law for wheeled robots based on a state observer. This method overcomes the shortcomings of existing control methods and improves the performance and reliability of the system.

[0008] Step 1: Establish the kinematic and dynamic model of the i-th wheeled robot, including fault information. The specific design process is as follows:

[0009] 1.1 The kinematic and dynamic mathematical model of the i-th wheeled robot is shown below:

[0010]

[0011] in, Indicates the first Wheeled robot For the pose vector of the wheeled robot, , These represent the coordinates of the wheeled robot along the X and Y axes, respectively. This refers to the azimuth angle of the wheeled robot. Let be the first derivative of the pose vector of the wheeled robot. , These represent the linear velocity and angular velocity of the wheeled robot, respectively. , , These are the angular velocities of the left and right wheels, respectively. It is the control torque applied to the robot's wheels, matrix , , The definitions are as follows:

[0012]

[0013]

[0014] in, and These represent the vehicle body mass and the mass of the wheel with the motor, respectively. These represent the moment of inertia of an object rotating around a wheel diameter with a motor, the moment of inertia of an object rotating around a wheel shaft with a motor, and the moment of inertia of an object rotating through its center of mass about a vertical axis, respectively. The distance is the distance from the center of mass to the midpoint of the line connecting the drive wheels; , Where is the damping coefficient; when or The robot is an internally damped robot; when and The robot has no internal damping; and as defined, It is a positive definite invertible matrix.

[0015] 1.2 Considering the system's bias fault and unknown disturbances, the nonlinear model introducing external disturbances when the bias fault occurs can be described as follows:

[0016]

[0017] in, Indicates external interference; Indicates a bias fault, if This indicates that the actuator is working normally. This indicates that the actuator has malfunctioned; This represents the total disturbance in the dynamics, and has , It is a positive constant; and at the same time satisfies , The upper bound of the interference is represented by a positive constant.

[0018] 1.3 After adjusting the coordinate system using the transverse function, the kinematic and dynamic models are transformed into state-space equations. Simultaneously, external disturbances to the system are considered to establish a mathematical model. The specific process of transforming the kinematic and dynamic models into state-space equations is as follows:

[0019] First, introduce the update law to be designed. and horizontal function , and To address the issue that the kinematic model of a wheeled robot is an underactuated system, the lateral functions are defined as follows:

[0020]

[0021] in and All are positive numbers, and have .

[0022] Secondly, the robot's original coordinates are transformed using a lateral function:

[0023]

[0024] in, , , , These are the transformed coordinate values. Let be the rotation matrix, represented as:

[0025]

[0026] Therefore, .

[0027] Finally, the following correspondence exists between the left and right wheel velocities and their angular and linear velocities:

[0028]

[0029] The kinematic and dynamic models are then transformed into the following state-space equations:

[0030]

[0031]

[0032]

[0033] in Indicates the radius of the wheel. It represents half the width of the car body. It is a horizontal function vector.

[0034] Step 2: Considering the dynamic equations of the faulty wheeled robot in Step 1, design a state observer to estimate the external disturbances and faults experienced by the i-th wheeled robot in its dynamic equations. The state observer design is as follows:

[0035]

[0036] in Yes The observed values, Indicates to Observations Let G be the gain coefficient of the observer to be designed. for The observation error, The observation errors of integrated interference are defined as follows: , Therefore, an adaptive disturbance law is designed as follows: ， in .

[0037] Step 3: Based on Step 2, a sliding mode controller for non-singular terminals under event-triggered conditions was designed.

[0038] The design steps for an event-triggered non-singular terminal sliding mode controller are as follows:

[0039] First, the trigger function for the i-th wheeled robot is designed as follows:

[0040]

[0041] in, It is an event triggering function. and They represent the first Second and third The timing of this event, and the instant of its triggering, is determined by the following triggering conditions: , It is a positive integer.

[0042] Secondly, the pose cooperative tracking error and velocity tracking error of the i-th wheeled robot are described as follows:

[0043]

[0044] in, and It is the robot's formation trajectory tracking error. The leader's posture vector; For the first The positional relationship between the wheeled robot and the navigator; For the virtual controller to be designed, Represents robots The set of neighboring nodes; This indicates the communication status between two wheeled robots. If the first... A wheeled robot can receive the first The state information of the wheeled robot, then Conversely, ; For leaders and the The communication status of the wheeled robot, if Then the first A wheeled robot can obtain information about the leader's status. Then the first A wheeled robot cannot know the leader's state information. Assuming all followers can know the leader's state information, the error... Represented as:

[0045]

[0046] At the same time, regarding the error Find the derivative and the error. The derivative is expressed as:

[0047]

[0048] Renewal Law Event-triggered virtual control laws and Set together as:

[0049]

[0050] in, This is a normal value. To obtain the formation tracking error under the event-triggered mechanism, and and The relationship is expressed as follows:

[0051]

[0052] in, and For the first The state of the robot at the previous trigger moment.

[0053] Then, consider the error. The non-singular terminal sliding mode is defined as:

[0054]

[0055] The derivative of the sliding surface is:

[0056]

[0057] in, and They are two constants that satisfy... and , and It is an odd number, satisfying .

[0058] Finally, the adaptive non-singular terminal sliding mode controller The design is as follows:

[0059]

[0060] in, , and It is a positive number.

[0061] Step 4: Prove the stability of the estimation errors of the wheeled robot system state, disturbances and fault terms.

[0062] In this invention, the Lyapunov function V is selected as:

[0063]

[0064] in , .

[0065] For Lyapunov functions Differentiation yields:

[0066]

[0067] Substituting the disturbance adaptive law, we have:

[0068]

[0069] According to Young's inequality The derivative changes as follows:

[0070]

[0071] Similarly, for Lyapunov functions After differentiation, we get:

[0072]

[0073] Substituting the virtual controller and update law, The derivative changes as follows:

[0074]

[0075] Based on the event triggering conditions, the sliding mode controller, and Young's inequality. The derivative changes as follows:

[0076]

[0077] in, yes The The sum of the lines is represented as , The definition is when the first The robot can obtain the first The state of the robot ,otherwise , The adjacency matrix represents the formation of wheeled robots. The first in Line 1 Column elements are Its specific form is as follows:

[0078]

[0079] According to Lyapunov ,get The derivative:

[0080]

[0081] in, express Given the identity matrix, design the gain matrix. for

[0082] , It is a non-negative definite matrix.

[0083] According to the gain matrix The design above can be transformed into:

[0084]

[0085] Therefore, there is ,in,

[0086]

[0087] Based on Lyapunov theory, it is ultimately proved that the estimation errors of the wheeled robot system's state, fault approximation terms, and disturbance terms are uniformly bounded, demonstrating that the closed-loop control system achieves precise tracking control of the wheeled robot formation trajectory. Furthermore, because the wheeled robot system's state is uniformly bounded, and Bounded, and also ,so , It is a positive integer, and it also has the following characteristics depending on the event triggering conditions: That is, during the interval between two event triggers, The change is greater than Therefore there is Therefore, this event triggering mechanism avoids the Zeno phenomenon and effectively reduces the communication burden. Attached Figure Description

[0088] Figure 1 Block diagram of a sliding mode controller for a wheeled robot formation based on event triggering and state observers.

[0089] Figure 2 A comparison of event-triggered simple sliding mode and state observer sliding mode trajectory tracking in the XOY coordinate system.

[0090] Figure 3 The tracking curves of the multi-wheeled robot on x, y, and θ are shown.

[0091] Figure 4 The tracking errors of the multi-wheeled robot in x, y, θ respectively.

[0092] Figure 5 A comparison chart of the tracking errors of event-triggered simple sliding mode and state observer-based sliding mode for multi-wheeled robots.

[0093] Figure 6 For the control input of wheeled robots

[0094] Figure 7 Event triggering graph Detailed Implementation

[0095] The present invention will be further explained in detail below with reference to the accompanying drawings and embodiments. The specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.

[0096] To enable those skilled in the art to better understand the implementation of this invention, a simulation of wheeled robot formation tracking control will be performed using Matlab software to verify its reliability. We consider a formation of three wheeled robots.

[0097] The vehicle body mass and the mass of the wheels with motors are respectively taken as... Moment of inertia The relevant parameters for other wheeled robots are set as follows: , .

[0098] The initial positions and velocity states of the three wheeled robots are shown in Table 1:

[0099]

[0100] The desired position of each wheeled robot relative to the center of the formation is represented as:

[0101]

[0102]

[0103] .

[0104] The navigator trajectory of the wheeled robot is set as follows:

[0105] .

[0106] When the wheeled robot formation operates under the proposed controller, the corresponding weighted adjacency matrix is ​​designed as follows:

[0107]

[0108] Simultaneously, the three wheeled robots were positioned relative to the center of the formation according to the predetermined formation and began coordinated operation under control input; provided that the closed-loop system is stable and bounded, the constant parameters in the controller were set as follows:

[0109] k1 = 1.5, γ1 = 1.6, γ f=1.8, a=b=1, p=5, q=3, σ i1 =0.001, σ i2 =0.001; the threshold for event triggering is set to... =0.1.

[0110] External disturbances can be simulated as:

[0111] .

[0112] The actuator fault design is as follows: when t≥6, bias fault occurs. It can be represented as:

[0113] .

[0114] The results show that, compared with the time-triggered mechanism, the event-triggered sliding mode control mechanism in this paper does not require continuous communication and achieves trajectory tracking of the formation. Furthermore, the addition of a state observer for bias faults and external disturbances significantly improves the compensation effect. Through the comparison of the above simulation results, the effectiveness and feasibility of the non-singular terminal sliding mode control strategy based on the state observer under event triggering proposed in this invention are verified, which meets the expectations.

[0115] Finally, it should be noted that this invention does not explain in detail the common knowledge recognized by those skilled in the art. The above description is only a specific embodiment of this invention and is not intended to limit this invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of this invention should be included within the protection scope of this invention.

Claims

1. A fault-tolerant control method for sliding mode formation of a wheeled robot based on event triggering, characterized in that, It consists of the following steps: Step 1, establish the first Kinematic and dynamic models of a wheeled robot; Step 2: Based on Step 1, design a dynamic state observer to estimate the compensation bias fault and unknown external disturbances; Step 3: Based on Step 2, a sliding mode controller for non-singular terminals under event-triggered conditions was designed. Step 4: Verify the stability of the closed-loop control of the wheeled robot formation system and confirm that the designed event triggering mechanism has eliminated Zeno behavior; Step 1 contains the fault information. The process of establishing the kinematics and dynamics model of the wheeled robot is as follows: First, the kinematic and dynamic mathematical model of the wheeled robot is shown below: ,in, Indicates the first A wheeled robot, For the pose vector of the wheeled robot, , These represent the coordinates of the wheeled robot along the X and Y axes, respectively. The azimuth angle of the wheeled robot. Let be the first derivative of the pose vector of the wheeled robot. , These represent the linear velocity and angular velocity of the wheeled robot, respectively. , These are the angular velocities of the left and right wheels, respectively. , It is the control torque applied to the robot's wheels, matrix , , The definitions are as follows: ,in, and These represent the vehicle body mass and the mass of the wheel with the motor, respectively. These represent the moment of inertia of an object rotating around a wheel diameter with a motor, the moment of inertia of an object rotating around a wheel shaft with a motor, and the moment of inertia of an object rotating through its center of mass about a vertical axis, respectively. The distance is the distance from the center of mass to the midpoint of the line connecting the drive wheels; , Where is the damping coefficient; when or The robot is an internally damped robot; when and The robot has no internal damping; and as defined, It is a positive definite invertible matrix; Then, we introduce the horizontal function. , , and the update law of the design This addresses the problem that the kinematic model of a wheeled robot is an underactuated system. First, define the horizontal functions as follows: ,in and All are positive numbers, and have ; Secondly, the robot's original coordinates are transformed using the lateral function through the following equation: ,in, , , , These are the transformed coordinate values ​​and orientation angles, respectively. Let be the rotation matrix, represented as: Therefore, ; The following correspondence exists between the left and right wheel velocities and their angular and linear velocities: The kinematic equations of the original model are then transformed into the following form: ,in, Indicates the radius of the wheel. It represents half the width of the car body. It is a horizontal function vector. It is a non-singular matrix. ; Finally, considering both the system's bias fault and unknown disturbances, the dynamic model with bias faults and external disturbances can be described as follows: ,in, Indicates external interference. Indicates a bias fault, if This indicates that the actuator is working normally. This indicates that the actuator has malfunctioned; This represents the total disturbance in the dynamics, and has , It is a positive constant; and at the same time, it has , , , The upper bound of the interference is represented by a positive constant.

2. The event-triggered sliding mode formation fault-tolerant control method for wheeled robots according to claim 1, characterized in that, The design process of the state observer in the dynamic equations is as follows: The state observer is designed to compensate for disturbances and faults affecting the dynamics of the i-th wheeled robot. Considering the dynamic equations of the faulty wheeled robot in step 2, the state observer can be designed as follows: ,in, Yes The observed values, Indicates to Observations Let G be the gain coefficient of the observer to be designed. for The observation error, The observation errors of integrated interference are defined as follows: , Therefore, an adaptive disturbance law is designed as follows: ,in .

3. The event-triggered sliding mode formation fault-tolerant control method for wheeled robots according to claim 1, characterized in that, The design steps for the event-triggered adaptive non-singular terminal sliding mode controller in step 3 are as follows: First, the trigger function for the i-th wheeled robot is designed as follows: ,in, It is an event triggering function. and They represent the first Second and third The timing of this event, and the instant of its triggering, is determined by the following triggering conditions: , It is a positive integer; Secondly, the pose cooperative tracking error and velocity tracking error of the i-th wheeled robot are described as follows: ,in, and It is the robot's formation trajectory tracking error. The leader's posture vector; For the first The positional relationship between the wheeled robot and the navigator; For the virtual controller to be designed, Represents robots The set of neighboring nodes; This indicates the communication status between two wheeled robots. If the first... A wheeled robot can receive the first The state information of the wheeled robot, then Conversely, ; For leaders and the The communication status of the wheeled robot, if Then the first A wheeled robot can obtain information about the leader's status. Then the first A wheeled robot cannot know the leader's state information. Assuming all followers can know the leader's state information, the error... Represented as: ； At the same time, regarding the error Find the derivative and the error. The derivative is expressed as: Renewal Law Event-triggered virtual control laws and Set together as: ,in, This is a normal value. To obtain the formation tracking error under the event-triggered mechanism, and and The relationship is expressed as follows: ,in, and For the first The state of the robot at the previous trigger moment; Then, consider the error. The non-singular terminal sliding mode is defined as: The derivative of the sliding surface is: ,in, and They are two constants that satisfy... and , and It is an odd number, satisfying ; Finally, the adaptive non-singular terminal sliding mode controller The design is as follows: ,in, , and It is a positive number.

4. The event-triggered sliding mode formation fault-tolerant control method for wheeled robots according to claim 3, characterized in that, Step 4 requires proving the stability of the estimation errors for the wheeled robot system's state, disturbances, and fault terms: Choosing Lyapunov functions for: ,in ， ； For Lyapunov functions Differentiation yields: Substituting the disturbance adaptive law, we have: According to Young's inequality The derivative changes as follows: Similarly, for Lyapunov functions After differentiation, we get: Substituting the virtual controller and update law, The derivative changes as follows: Based on the event triggering conditions, the sliding mode controller, and Young's inequality. The derivative changes as follows: in, yes The The sum of the lines is represented as , The definition is when the first The robot can obtain the first The state of the robot ,otherwise , The adjacency matrix represents the formation of wheeled robots. The first in Line 1 Column elements are Its specific form is as follows: According to Lyapunov ,get The derivative is: ,in, express Given the identity matrix, design the gain matrix. for , It is a non-negative definite matrix; According to the gain matrix The design above can be transformed into: Therefore, there is ,in, Based on Lyapunov theory, it is finally proved that the estimation errors of the wheeled robot system's state, fault approximation terms, and disturbance terms are uniformly bounded, demonstrating that the closed-loop control system achieves precise tracking control of the wheeled robot formation trajectory. Furthermore, because the wheeled robot system's state is uniformly bounded, and Bounded, and also ,so , It is a positive integer, and it also has the following characteristics depending on the event triggering conditions: That is, during the interval between two event triggers, The change is greater than Therefore there is .

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