Mechanical arm sliding mode control buffeting analysis method based on system response characteristics

Abstract

The invention discloses a mechanical arm sliding mode control buffeting analysis method based on system response characteristics. The mechanical arm sliding mode control buffeting analysis method comprises the steps that a computer periodic sampling process and the mechanical arm response characteristics are obtained; based on the mechanical arm response characteristics, a mechanical arm response model is established, and model parameters are identified; based on the model parameters, a mechanical arm sliding mode control mathematical model is established; on the basis, a sliding mode control buffeting analysis model is established. Based on the system response characteristics, the mechanical arm sliding mode control input and output relation is established, the source of sliding mode control buffeting is analyzed, and the relation between sliding mode control buffeting and the mechanical arm response characteristics, the sampling period, the control gain and mechanical arm input is established. A window function is utilized, system output is divided into steady-state output and transient-state output, and transient-state output of a previous stage serves as additional input of a subsequent stage.

Description

Technical Field

[0001] The present application belongs to the technical field of robot arm control analysis, and specifically relates to a robot arm sliding mode control chattering analysis method based on system response characteristics. Background Art

[0002] Robots have broad application prospects in the fields of medicine, agricultural production, national defense, etc. The robotic arm is the main actuator of the robot, and how to improve the control accuracy of the robotic arm is a significant issue in the field of robot control.

[0003] Robotic arm control methods are primarily categorized as those that rely on mathematical models and those that do not. Model-dependent methods involve establishing a mathematical model of the robot's input and output, and designing a feedback controller based on stability theory to achieve control. However, robotic arms are complex structures and processes, and their mathematical models are also dependent on factors such as their position and posture. These parameters vary with position and posture, and the accuracy of the model directly impacts the robot's control performance.

[0004] Sliding mode control involves designing a stable sliding surface. When the state trajectory deviates from the sliding surface, the sliding mode controller is used to quickly return to the sliding surface. In theory, sliding mode control can ensure that the state trajectory always remains on the stable sliding surface. Sliding mode control does not rely on mathematical models and quickly returns to the sliding surface when it deviates from the sliding surface, resulting in good robustness. However, in engineering, when the state trajectory reaches the sliding mode surface, it often fails to slide strictly along the sliding mode surface toward the equilibrium point. Instead, it traverses back and forth between the two sides of the sliding mode surface toward the equilibrium point, resulting in chattering. This is a major obstacle to the practical application of sliding mode control. Robot arm chattering can reduce the robot's execution accuracy, generate noise, and accelerate mechanical wear.

[0005] While sliding mode control for robotic arms has been extensively researched, the mechanisms that cause chattering are rarely explored. Sliding mode control typically involves sampling computer outputs, determining whether the current trajectory deviates from the sliding mode surface based on the output, and then applying control. However, computer-sampled signals are discrete-time signals and cannot be sampled in real time; instead, they are sampled at periodic intervals. Therefore, actual sliding mode control is discontinuous. Since robotic arms are not ideal, changes in input inevitably involve a response process. Continuous computer sampling leads to the superposition of transients in the robotic arm's output, resulting in chattering.

[0006] Existing research has rarely analyzed the chattering mechanism of manipulator sliding mode control and studied the factors affecting chattering. To address this issue, the present invention proposes a chattering analysis method for manipulator sliding mode control based on the discontinuous sampling of manipulator sliding mode control and the response characteristics of the manipulator. The relationship between chattering and the sampling period, control gain, response characteristics, and control input is established, providing a new analysis method for chattering analysis of manipulator sliding mode control. Summary of the Invention

[0007] The present application provides a chattering analysis method for sliding mode control of a robotic arm based on system response characteristics to solve the above-mentioned technical problems.

[0008] To solve the above technical problems, the present application adopts a technical solution: a chattering analysis method for sliding mode control of a robotic arm based on system response characteristics, comprising:

[0009] S1. Obtain computer cycle sampling process and robot arm response characteristics;

[0010] S2. Based on the response characteristics of the robotic arm, establish a robotic arm response model and identify model parameters;

[0011] S3. Establish a mathematical model of sliding mode control for the robotic arm based on the model parameters;

[0012] S4. Based on the mathematical model of the sliding mode control of the robot arm, obtain a sliding mode control chattering analysis model.

[0013] Furthermore, the method of step S2 includes:

[0014] S21. Build input and output convolutional model;

[0015] S22. Identify model parameters of the input-output convolution model based on the particle swarm optimization algorithm and the gradient algorithm;

[0016] S23. Based on the model parameters, construct a robotic arm response model.

[0017] Furthermore, the method of step S21 includes:

[0018] Based on formula (1), the input and output convolution model is obtained; wherein, formula (1) is:

[0019] q(t)=h(t)*τ(t)=ηt -β *τ(t) (1);

[0020] Where η and β are model parameters; q(t) represents the angular displacement of the robot arm, τ(t) represents the external torque of the robot arm, and * represents the convolution operation.

[0021] Furthermore, the method of step S3 includes:

[0022] Based on formula (2), a mathematical model of sliding mode control of the robot arm is constructed; wherein, formula (2) is:

[0023] q(t)-q(0)=h(t)*(-(q(t)-τ(t))sign(τ(t)-q(t))(u(t-kΔT)-u(t-kΔT-εΔT)))+h(t)*(τ(t)(u(t-kΔT-εΔT)-u(t-(k+1)εΔT))) (2);

[0024] Where ΔT represents the computer sampling period, ε is the duty cycle; δ(tt k ) indicates that at t k The ideal pulse at time q(0) represents the angular displacement of the robot arm at the initial time.

[0025] Furthermore, the method of step S4 includes:

[0026] S41. Before loading the load pulse, obtain the input and output time domain model;

[0027] S42. Calculate the first transient superposition output within the first sampling period based on the input-output time domain model;

[0028] S43. Based on the first transient superposition output, obtaining a second transient superposition output within a second sampling period;

[0029] S44. Based on the second transient superposition output, establish a k-th pulse load sliding mode control chattering analysis model.

[0030] Furthermore, the method of step S41 includes:

[0031] Based on formula (3), the input and output time domain model is obtained; where formula (3) is:

[0032]

[0033] Where q(t) represents the angular displacement output within the time period 0≤t<εΔT, represents the inverse Laplace transform; H(s) is the Laplace transform of h(t), and s represents the Laplace operator; Represents the output generated by the external input torque.

[0034] Furthermore, the method of step S42 includes:

[0035] Based on formulas (4)-(5), the first transient superposition output is obtained; wherein formulas (4)-(5) are:

[0036]

[0037] in, represents the transient output generated by the input in the time period 0≤t<εΔT; q(εΔT) represents the output generated by the initial state at that moment; Indicates the output produced by the current input.

[0038] Furthermore, the method of step S43 includes:

[0039] Based on formula (6), the unit step function μ(t) is introduced and the window function is constructed to separate the input before the pulse trigger; where formula (6) is:

[0040] Based on formula (6), the second transient superposition output is obtained; wherein formula (6) is:

[0041]

[0042] in, It represents the output of the transient generated in the time period 0≤t<εΔT within the time period ΔT≤t<ΔT+εΔT; It indicates the output of the transient generated in the time period εΔT≤t<ΔT within the time period ΔT≤t<ΔT+εΔT.

[0043] Furthermore, the method of step S44 includes:

[0044] Based on formula (7), the k-th sliding mode control chattering analysis model is obtained; wherein, formula (7) is:

[0045]

[0046] in:

[0047] Represents the transient caused by sliding mode control.

[0048] The beneficial effects of this application are as follows: Based on the system response characteristics, this application establishes the input-output relationship of the sliding mode control of the manipulator, analyzes the root cause of sliding mode control chattering, and establishes the relationship between sliding mode control chattering and the manipulator response characteristics, sampling period, control gain, and manipulator input. This application uses a window function to divide the system output into steady-state output and transient output, with the transient output of the previous stage serving as an additional input for the subsequent stage. BRIEF DESCRIPTION OF THE DRAWINGS

[0049] Figure 1 This is a flow chart of an embodiment of a method for analyzing chattering of a sliding mode control of a robotic arm based on system response characteristics of the present application;

[0050] Figure 2 This is a structural diagram of a sliding mode control for a manipulator according to an embodiment of a chattering analysis method for sliding mode control of a manipulator based on system response characteristics of the present application;

[0051] Figure 3 yes Figure 1 A schematic flow chart of an embodiment of step S2 in FIG.

[0052] Figure 4 yes Figure 1 A schematic flow chart of an embodiment of step S4 in FIG.

[0053] Figure 5 This is a simulation diagram of β=0.75 in Example 1 of the chattering analysis method for sliding mode control of a robotic arm based on system response characteristics of the present application;

[0054] Figure 6 This is a simulation diagram of β=0.5 in Example 1 of the chattering analysis method for sliding mode control of a robotic arm based on system response characteristics of the present application. DETAILED DESCRIPTION

[0055] In order to make the objectives, technical solutions and advantages of the present invention more clear, the present invention is further described in detail below with reference to specific embodiments.

[0056] In the following description, many specific details are set forth to facilitate a full understanding of the present invention. However, the present invention may also be implemented in other ways different from the description. Therefore, the present invention is not limited to the specific embodiments disclosed in the following specification.

[0057] See Figure 1 , Figure 1 This is a flow chart of an embodiment of a method for analyzing chattering of a sliding mode control of a robotic arm based on system response characteristics. The method includes:

[0058] S1. Obtain the computer cycle sampling process and the robot arm response characteristics.

[0059] Specifically, sliding mode control monitors the state trajectory in real time. When the state trajectory deviates from the sliding surface, it quickly reaches the sliding surface through the control law. Ideally, it is always hoped that the change in the angular displacement of the robot arm is proportional to the external torque. However, computers can only perform digital signal processing and discrete periodic sampling. At the same time, the robot arm is not an ideal rigid system. Different robots have different response characteristics h(t), and their input and output have a response process. Sliding mode control can be used Figure 2 The structure shown in the figure is studied. Due to the periodic sampling and the response characteristics of the manipulator, chattering is inevitable.

[0060] S2. Based on the response characteristics of the robotic arm, establish a robotic arm response model and identify the model parameters.

[0061] Specifically, step S2 includes:

[0062] S21. Construct an input-output convolution model.

[0063] Specifically, an integer-order differential dynamics model of the robot arm is established according to an inertia matrix of the robot arm, a Coriolis matrix, a centrifugal force matrix, a gravity torque matrix and a driving torque. According to a convolution property, an integer-order differential of a function can be regarded as a convolution of the function and an ideal impulse response differential. The integer-order differential can be regarded as an ideal impulse response model, which can only reflect local time information and cannot well describe a convolution process related to global time.

[0064] More generally, any impulse response h(t) of the robot arm is a function decaying with time and can be expressed as h(t) = ηt -β . Considering a response process of the robot arm, input and output of the robot arm can be expressed as:

[0065] q(t) = h(t) * τ(t) = ηt -β *τ(t) (1);

[0066] where η and β are model parameters, q(t) represents an angular displacement of the robot arm, τ(t) represents an external torque of the robot arm, and * represents a convolution operation.

[0067] S22. Identify model parameters of the input-output convolution model based on a particle swarm optimization algorithm and a gradient algorithm.

[0068] Specifically, a convolution operation in formula (1) is converted into an algebraic operation by using a Legendre transform, and the parameters η and β are identified by combining the particle swarm optimization algorithm and the gradient algorithm.

[0069] S23. Construct a response model of the robot arm based on the model parameters.

[0070] S3. Establish a sliding mode control mathematical model of the robot arm based on the model parameters.

[0071] Specifically, according to the structure shown in Figure 2 , the following feedback model is established:

[0072] q(t) - q(0) = h(t) * (-(q(t) - τ(t)) sign(τ(t) - q(t)) (u(t-kΔT) - u(t-kΔT-εΔT)) + h(t) * (τ(t) (u(t-kΔT-εΔT) - u(t-(k+1)εΔT))) (2);

[0073] where ΔT represents a computer sampling period, ε is a duty ratio, δ(t-t k ) represents an ideal impulse at t k , and q(0) represents an angular displacement of the robot arm at an initial time.

[0074] S4. Based on the mathematical model of the sliding mode control of the robot arm, obtain a sliding mode control chattering analysis model.

[0075] Specifically, step S4 includes:

[0076] S41. Obtain input and output time domain models.

[0077] Specifically, when 0≤t<εΔT, the input-output time domain model is obtained by formula (8):

[0078] q(t)-q(0)=h(t)*(-(q(t)-τ(t))) (8);

[0079] Performing Laplace transform yields:

[0080]

[0081] Where Q(s), H(s), and T(s) are the Laplace transforms of q(t), h(t), and τ(t), respectively, and s represents the Laplace operator.

[0082] According to formula (3), we can get the following by performing inverse Laplace transform:

[0083]

[0084] Where q(t) represents the angular displacement output within the time period 0≤t<εΔT, represents the inverse Laplace transform; H(s) is the Laplace transform of h(t), and s represents the Laplace operator; Represents the output generated by the external input torque.

[0085] S42. Calculate a first transient superposition output within a first sampling period based on the input-output time domain model.

[0086] Specifically, when εΔT≤t<ΔT:

[0087] According to formula (2), we have:

[0088] q(t)-q(0)=h(t)*(-(q(t)-τ(t))(μ(t)-μ(t-εΔT)))+h(t)*(τ(t)(u(tεΔT)-μ(t-ΔT))) (9);

[0089] By the following formula:

[0090] q(εΔT)=q(0)+h(t)*(-(q(t)-τ(t))(μ(t)-μ(t-εΔT)))(μ(t)-μ(t-εΔT))| t＝εΔT , we can get:

[0091] q(t)-q(εΔT)=h(t)*(-(q(t)-τ(t))(μ(t)-μ(t-εΔT)))μ(t-εΔT)+h(t)*(τ(t)(u(t-kΔT-εΔT)-u(t-(k+1)ΔT))) (10);

[0092] where h(t)*(-(q(t)-τ(t))(μ(t)-μ(t-εΔT)))μ(t-εΔT) represents the transient output generated by the input within the time period εΔT≤t<ΔT.

[0093] For simplicity, it is written as:

[0094]

[0095] From this, we can see that formula (10) can be expressed as:

[0096]

[0097]

[0098] Similar to formula (3), we can get:

[0099]

[0100] in, represents the transient output generated by the input in the time period 0≤t<εΔT; q(εΔT) represents the output generated by the initial state at that moment; Indicates the output produced by the current input.

[0101] S43. Based on the first transient superposition output, obtain a second transient superposition output within a second sampling period.

[0102] Specifically, when ΔT≤t<ΔT+εΔT, similar to step S42,

[0103]

[0104] Using Laplace transform and inverse Laplace transform:

[0105]

[0106] in, It represents the output of the transient generated in the time period 0≤t<εΔT within the time period ΔT≤t<ΔT+εΔT; It indicates the output of the transient generated in the time period εΔT≤t<ΔT within the time period ΔT≤t<ΔT+εΔT.

[0107] S44. Based on the second transient superposition output, the kth times of sliding mode control chattering analysis model is established.

[0108] Specifically, based on the above process step by step recursion, we can get:

[0109]

[0110] Wherein:

[0111] represents the transient state due to sliding mode control.

[0112] It can be seen that the chattering is related to the system control system response characteristics, control period, feedback gain and other factors.

[0113] Example 1

[0114] Referring to Figure 5-6 The mechanical arm pulse load related parameters are as follows:

[0115] Let the mechanical arm impulse response be: The sampling period is 1 second, and the sliding mode control duty cycle is 0.5. When the input torque τ(t) takes different values, the simulation is performed, Figure 5 When β=0.75, τ(t) is a step function, the computer sampling period is 1 second, and the angular displacement simulation diagram of the mechanical arm with a duty cycle of 0.5 is shown in the following figure: Figure 6 When β=0.5, τ(t) is a step function, the computer sampling period is 1 second, and the angular displacement simulation diagram of the mechanical arm with a duty cycle of 0.5 is shown in the following figure.

[0116] The application proposes a mechanical arm sliding mode control chattering analysis method based on system response characteristics. The method establishes the input-output relationship of the mechanical arm sliding mode control based on the system response characteristics, analyzes the root cause of the sliding mode control chattering, and establishes the relationship between the sliding mode control chattering and the mechanical arm response characteristics, sampling period, control gain and mechanical arm input. The application uses window function to divide the system output into steady state output and transient state output, and the transient state output of the previous stage is used as the additional input of the subsequent stage, which provides a basis for mechanical arm sliding mode control parameter selection.

[0117] The above only describes the embodiments of the application, and does not limit the patent scope of the application. Any equivalent structure or equivalent process transformation using the contents of the application specification and drawings, or direct or indirect application in other related technical fields, are also included in the patent protection scope of the application.

Claims

1. A chattering analysis method for sliding mode control of a robotic arm based on system response characteristics, characterized in that: include: S1. Obtain computer cycle sampling process and robot arm response characteristics; S2. Based on the response characteristics of the manipulator, establish a manipulator response model and identify model parameters; S3. Based on the model parameters, a mathematical model of sliding mode control of the robotic arm is established; S4. Based on the mechanical arm sliding mode control mathematical model, obtain a sliding mode control chattering analysis model.

2. The method according to claim 1, characterized in that The method of step S2 comprises: S21. Build input and output convolutional model; S22. Identify the model parameters of the input-output convolution model based on the particle swarm optimization algorithm and the gradient algorithm; S23. Based on the model parameters, construct the robotic arm response model.

3. The method according to claim 2, characterized in that The method of step S21 includes: Based on formula (1), the input and output convolution model is obtained; wherein, the formula (1) is: q(t)=h(t)*τ(t)=ηt -β *τ(t) (1); Where η and β are model parameters; q(t) represents the angular displacement of the robot arm, τ(t) represents the external torque of the robot arm, and * represents the convolution operation.

4. The method according to claim 3, characterized in that The method of step S3 comprises: Based on formula (2), the mathematical model of the sliding mode control of the manipulator is constructed; wherein, the formula (2) is: q(t)-q(0)=h(t)*(-(q(t)-τ(t))sign(τ(t)-q(t))(u(t-kΔT)-u(t-kΔT-εΔT)))+h(t)*(τ(t)(u(t-kΔT-εΔT)-u(t-(k+1)εΔT))) (2); Where ΔT represents the computer sampling period, ε is the duty cycle; δ(tt k ) indicates that at t k The ideal pulse at time q(0) represents the angular displacement of the robot arm at the initial time.

5. The method according to claim 4, characterized in that The method of step S4 comprises: S41. Obtain input and output time domain models; S42. Calculate the first transient superposition output within the first sampling period based on the input-output time domain model; S43. Based on the first transient superposition output, obtaining a second transient superposition output within a second sampling period; S44. Based on the second transient superposition output, establish a k-th sliding mode control chattering analysis model.

6. The method according to claim 5, characterized in that The method of step S41 includes: Based on formula (3), the input-output time domain model is obtained; wherein, the formula (3) is: Where q(t) represents the angular displacement output within the time period 0≤t<εΔT, represents the inverse Laplace transform; H(s) is the Laplace transform of h(t), and s represents the Laplace operator; Represents the output generated by the external input torque.

7. The method according to claim 6, characterized in that The method of step S42 includes: Based on formulas (4)-(5), the first transient superposition output is obtained; wherein, the formulas (4)-(5) are: in, represents the transient output generated by the input in the time period 0≤t<εΔT; q(εΔT) represents the output generated by the initial state at that moment; Indicates the output produced by the current input.

8. The method according to claim 7, characterized in that The method of step S43 includes: Based on formula (6), the second transient superposition output is obtained; wherein, the formula (6) is: in, It represents the output of the transient generated in the time period 0≤t<εΔT within the time period ΔT≤t<ΔT+εΔT; It represents the output of the transient generated in the time period εΔT≤t<ΔT during the time period ΔT≤t<ΔT+εΔT.

9. The method according to claim 8, characterized in that The method of step S44 includes: Based on formula (7), the k-th sliding mode control chattering analysis model is obtained; wherein, the formula (7) is: in: Represents the transient caused by sliding mode control.

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