A robot arm adaptive trajectory tracking control method

Abstract

The application discloses a mechanical arm adaptive trajectory tracking control method, which is used for solving the control problem of a mechanical arm system with time-varying state constraints, external disturbance and actuator faults, and the method steps are as follows: a preset performance function and a logarithmic barrier Lyapunov function are combined to solve the time-varying full state constraint problem of the system; secondly, a disturbance observer with boundary characteristics is designed to approximate the lumped disturbance, and an actual controller is designed by combining a backstepping method, and a compensation variable is designed to compensate for the error of a first-order filter; finally, based on the Cruz-Ortiz theory, it is proved that the system still satisfies the global fixed-time stability condition under the time-varying constraint; the application has important significance for the control problem of the mechanical arm with time-varying state constraints and actuator faults.

Description

Technical Field

[0001] This application relates to a trajectory tracking control method for a robotic arm, and more specifically, to a trajectory tracking control method for a robotic arm subject to time-varying state constraints, actuator failures, and external disturbances. Background Technology

[0002] Robotic arms play a vital role in modern industrial production, performing tasks ranging from loading and unloading machine tools to automobile manufacturing. Traditional control methods, such as PID control and computational torque control, can solve simple control problems, but they cannot maintain good dynamic performance and disturbance suppression capabilities, necessitating improvements to traditional methods or the design of new ones. While existing technologies offer many control methods for addressing state constraints and actuator failures, most focus on solving single faults or fixed constraints. Furthermore, current methods still require improvement in computational complexity and stability. Summary of the Invention

[0003] This application proposes an adaptive trajectory tracking control method for a robotic arm using a time-varying obstacle Lyapunov function. This method can achieve fixed-time stability and high-precision tracking of the target trajectory under complex working conditions where the robotic arm simultaneously faces unknown disturbances, actuator failures, and time-varying state constraints.

[0004] To solve the above-mentioned technical problems, the technical solution of this application is as follows:

[0005] 1. A method is proposed that combines a pre-defined performance function with a logarithmic barrier Lyapunov function to solve the time-varying full-state constraint problem of the system.

[0006] 2. A disturbance observer with boundary constraint characteristics is designed to approximate lumped disturbances (including external disturbances, actuator failures, and unknown nonlinear functions), which significantly improves the robustness and anti-interference capability of the system.

[0007] 3. Combining the disturbance observer and the backstepping method, the actual controller is obtained through distributed recursion, and compensation variables are designed to counteract the error introduced by the first-order filter.

[0008] 4. Based on Cruz-Ortiz theory, prove that the system still satisfies global fixed-time stability under time-varying constraints.

[0009] This application provides an adaptive trajectory tracking control method for a robotic arm, which includes the following steps:

[0010] A1. Based on actuator failures and state constraints, establish a new robotic arm system;

[0011] A2. Use a disturbance observer with boundary characteristics to estimate the lumped disturbances present in the new robotic arm system;

[0012] A3. Based on the time-varying logarithmic barrier Lyapunov function and the disturbance observer, generate an adaptive controller; and control the new robotic arm system so that the trajectory tracking error of the robotic arm converges to the set steady-state accuracy within a fixed time.

[0013] In some implementations, step A1, the step of establishing the new robotic arm system, includes:

[0014] Introducing a robotic arm system, its dynamic equations are as follows:

[0015]

[0016] System state is defined as Define control input ,in Indicates the angular displacement of the robotic arm. Indicates motor current. Indicates the input voltage. Indicates the rotor inertia of the motor. Indicates the torque coefficient. Indicates the mass of a single rod. Indicates load quality. Expressed as the length of a single rod. Represented as gravity, It is expressed as the coefficient of viscous friction. Represented as armature inductance, It is expressed as the back electromotive force coefficient. Represented as armature resistance;

[0017] Combining external interference and actuator failure, the robotic arm system (33) can be simplified as follows:

[0018]

[0019] in, Treat it as a known constant. For unknown nonlinear function terms, This is referred to as external interference;

[0020] Set the state constraints as follows:

[0021]

[0022] in, It is a nonlinear time-varying function. Represents the system's control output. This represents an actuator failure, and its form is as follows:

[0023]

[0024] in, Represents the actual input to the system. , This indicates unknown bias interference in the actuator;

[0025] The new robotic arm system can be obtained by transforming and rearranging the robotic arm system (34), and its equation is as follows:

[0026]

[0027] Among them, lumped interference .

[0028] In some implementations, step A2 further includes:

[0029] The perturbation observer based on the boundary function is designed as follows:

[0030]

[0031] , ; It is a boundary function, and its design form is shown below:

[0032]

[0033] in, It is the smallest positive number designed;

[0034] The lumped disturbance term in the new robotic arm system can be estimated using the disturbance observer, such that: Furthermore, the estimation error can converge to 0 within a fixed time, and the upper bound of the convergence time is calculable and only affected by the set parameters.

[0035] In some implementations, step A3 includes:

[0036] A31. Design multiple virtual controllers based on the time-varying logarithmic barrier Lyapunov function;

[0037] A32. Design the adaptive controller based on the virtual controller and the disturbance observer;

[0038] A33. The adaptive controller performs trajectory tracking control on the new robotic arm system.

[0039] In some implementations, step A31 includes:

[0040] The time-varying logarithmic barrier Lyapunov function is designed as follows:

[0041]

[0042] in It is a performance function that is set. By adjusting the parameters to make It holds true at any given time. ,in , The desired trajectory is set; It is a virtual controller The output result after passing through the first-order filter;

[0043] In some implementation schemes, design This indicates the estimation error, due to Since all values ​​are close to a local minimum of 0, it can be concluded that there exist positive real numbers. Make Established;

[0044] The first virtual controller is generated by combining the time-varying logarithmic barrier Lyapunov function:

[0045]

[0046] in , , , The first-order filter is designed as follows:

[0047]

[0048] Among them, virtual controller For input, It is a defined positive real number. For the filter output, finally Represented as the compensation variable of the filter, it is designed as follows:

[0049] .

[0050] In some implementations, a second virtual controller is generated by combining the time-varying logarithmic barrier Lyapunov function:

[0051]

[0052] in, , Corresponding The compensation variables are designed as follows:

[0053] .

[0054] In some implementations, step A32 includes:

[0055] Design the adaptive controller based on the virtual controller and the disturbance observer:

[0056]

[0057] in The corresponding performance function is designed as follows: .

[0058] Beneficial effects

[0059] 1. Even under complex operating conditions such as time-varying state constraints, actuator failures, and external disturbances, the system still achieves global fixed-time stability. The upper bound of the convergence time depends only on the preset parameters, which improves the applicability and stability of the control algorithm.

[0060] 2. The control method uses only one adaptive variable that is updated in real time, which simplifies the calculation process and reduces the computational load, making it suitable for use in real-time control scenarios.

[0061] 3. By designing a preset performance function and a disturbance observer, the tracking error is strictly constrained within a preset range, which significantly enhances the dynamic response speed and anti-interference capability of the system. Attached Figure Description

[0062] Figure 1 This is a flowchart of the adaptive control method designed in this application;

[0063] Figure 2 yes Figure 1 The specific flowchart described in step one;

[0064] Figure 3 yes Figure 1 The specific flowchart described in step two;

[0065] Figure 4 yes Figure 1 The specific flowchart described in step three;

[0066] Figure 5 The system demonstrates the tracking effect of the robotic arm's angular displacement against a set reference trajectory, and intuitively shows the system's key performance indicators such as convergence speed and tracking accuracy.

[0067] Figure 6These are the trajectory of the robotic arm's angular velocity and the fluctuation curve of the motor current over time, respectively. Both satisfy the time-varying constraint at any given moment.

[0068] Figure 7 The figure shows the trajectory curves of the tracking errors of each state variable, which verifies that the method in this application has significant advantages in convergence speed and control accuracy.

[0069] Figure 8 The curve of the input voltage is shown, and its amplitude is affected by the failure of the actuator but still remains bounded. Detailed Implementation

[0070] The present application will now be described in detail with reference to the accompanying drawings and embodiments. Various examples are provided by way of explanation and not by way of limitation. In fact, those skilled in the art will recognize that modifications and variations can be made to the present application without departing from the scope or spirit thereof. For example, a feature shown or described as part of one embodiment may be used in another embodiment to produce yet another embodiment. Therefore, it is desirable that the present application encompass such modifications and variations that fall within the scope of the appended claims and their equivalents.

[0071] In the description of this application, the terms "longitudinal," "lateral," "upper," "lower," "front," "rear," "left," "right," "vertical," "horizontal," "top," and "bottom," etc., indicate the orientation or positional relationship based on the orientation or positional relationship shown in the accompanying drawings. They are used only for the convenience of describing this application and do not require that this application be constructed and operated in a specific orientation, and therefore should not be construed as limiting this application. The terms "connected," "linked," and "set up" used in this application should be interpreted broadly. For example, they can refer to fixed connections or detachable connections; direct connections or indirect connections through intermediate components; wired connections, radio connections, or wireless communication signal connections. Those skilled in the art can understand the specific meaning of the above terms according to the specific circumstances.

[0072] The accompanying drawings illustrate one or more examples of this application. The detailed description uses numerical and alphabetic designations to refer to features in the drawings. Similar or analogous designations in the drawings and description have been used to refer to similar or analogous parts of this application. As used herein, the terms “first,” “second,” and “third,” etc., are used interchangeably to distinguish one component from another and are not intended to indicate the location or importance of individual components.

[0073] The implementation process of this application will be further described in detail below with reference to the accompanying drawings. Figure 1 The design flowchart of the adaptive control method in this application includes the following steps:

[0074] Step 1: Combine the preset performance function as a logarithmic barrier Lyapunov function.

[0075] Step 2: Design a disturbance observer with boundary characteristics, and then combine it with the backstepping method to obtain the actual controller. Design compensation variables to offset the error introduced by the first-order filter.

[0076] Step 3: Prove that the system satisfies global fixed-time stability under time-varying constraints.

[0077] according to Figure 2 As can be seen, the specific steps of step 1 are as follows:

[0078] Step 1.1: Describes the robotic arm system, actuator failures, and state constraint issues.

[0079] For a class of single-link flexible robotic arm systems with external disturbances, state constraints, and actuator failures, the system state is defined as... Define control input ,in Indicates the angular displacement of the robotic arm. Indicates motor current. The dynamic equations of the robotic arm system, representing the input voltage, are as follows:

[0080]

[0081] in, Indicates the rotor inertia of the motor. Indicates the torque coefficient. Indicates the mass of a single rod. Indicates load quality. Expressed as the length of a single rod. Represented as gravity, It is expressed as the coefficient of viscous friction. Represented as armature inductance, It is expressed as the back electromotive force coefficient. It is represented as armature resistance.

[0082] Finally, assuming that the robotic arm system has external interference and actuator failure, simplifying (1) yields:

[0083]

[0084] in Treat it as a known constant. For unknown nonlinear function terms, This is represented as an external disturbance, and the state constraints are set as follows:

[0085]

[0086] in, It is a nonlinear time-varying function. Represents the system's control output. This represents an actuator failure, and its failure modes are as follows:

[0087]

[0088] in Represents the actual input to the system. The actuator completely failed at that time. The actuator was intact at the time. This application investigates the case where the actuator partially fails. Suppose it is an unknown and slowly time-varying real number. This indicates an additive unknown disturbance to the actuator.

[0089] By transforming and rearranging (2), we can obtain:

[0090]

[0091] Among them, centralized interference .

[0092] The coordinate transformation of the system is as follows:

[0093]

[0094] in , The desired trajectory is set. It is a virtual controller The output result after passing through a first-order filter.

[0095] Step 1.2: Set the preset performance function as the boundary of the barrier Lyapunov function.

[0096] To satisfy the time-varying full-state constraints, a logarithmic barrier Lyapunov function is introduced, with the following form:

[0097]

[0098] in It is a performance function that is set. t represents time, and the parameter is adjusted to make... It holds true at any given time. In the set It is continuously differentiable. Equation (7) defines the systematic error. Limited to time-varying performance functions middle, They decided respectively The initial upper bound and the upper bound after stabilization.

[0099] according to Figure 3 Therefore, the specific steps of step 2 are as follows:

[0100] Step 2.1: Design boundary functions to construct the perturbation observer

[0101] Based on the system model, the perturbation observer based on the boundary function is designed as follows:

[0102] , , It is a boundary function, and its design form is shown below:

[0103]

[0104] in It is the smallest positive number designed.

[0105] Constructing auxiliary variables Suppose at a certain moment The Lyapunov function is chosen as follows:

[0106]

[0107] Taking its derivative with respect to time, we get:

[0108]

[0109] And because Therefore, we can know Therefore, it is true, and thus we can further conclude that:

[0110]

[0111] in, , According to the fixed-time stability theorem, the perturbation observer can converge to a fixed time. Within its smallest neighborhood. And with an upper bound on the convergence time. It can be calculated as:

[0112]

[0113] Therefore, if the observer is used to observe the lumped disturbance components of the system, including external disturbances, actuator failures, and unknown nonlinear functions, then the observer error will be significant when the upper bound of the lumped disturbance is unknown. A minimal neighborhood that converges to 0 in a fixed time. Within the range. The upper bound of the convergence time is computable and is only affected by the set parameters.

[0114] When observer error converged to When the range is defined, we can obtain the following:

[0115]

[0116] Will Substituting into the robotic arm model and simplifying, we get:

[0117]

[0118] in This represents the error of lumped disturbance and boundary function, and because... Since all values ​​are close to a local minimum of 0, it can be concluded that there exist positive real numbers. Make Established.

[0119] Step 2.2: Design virtual controllers and compensation variables step by step using a perturbation observer and backstepping method.

[0120] To design the virtual controller for the first step, a first-order filter is first applied:

[0121]

[0122] Among them, virtual controller For input, It is a defined positive real number. This is the filter output.

[0123] According to equation (6), the error Taking the time derivative and substituting it into the model, we get:

[0124]

[0125] To compensate for the error introduced by the first-order filter ,right Introducing compensation variables Its derivative form is shown below:

[0126]

[0127] in It is a positive real number.

[0128] definition ,right Differentiating, we get:

[0129]

[0130] Choose the Lyapunov function and will In Replace with And by taking the derivative, we get:

[0131]

[0132] in , , noticed Yes Compensation, through a compensation mechanism to drive error This approach brings the system closer to the origin's neighborhood, further improving the system's control accuracy. Replace with It cancels out the filtering error. The initial value is 0 and the system will tend to the smallest neighborhood of the origin after it stabilizes.

[0133] Design of virtual controller for:

[0134]

[0135] in Bring the virtual controller into the Lyapunov function Zhongde:

[0136]

[0137] Design the virtual controller in step 2, and define... Construct the compensation variables in step 2 for:

[0138]

[0139] Choose the Lyapunov function for:

[0140]

[0141] in The virtual controller designed for:

[0142]

[0143] in Substitute it into the Lyapunov function And by taking the derivative, we get:

[0144]

[0145] in .

[0146] Step 2.3: Obtain the adaptive controller by back-engineering from the virtual controller.

[0147] definition ,because There is no filter error in the input, so there is no need to compensate for it, and therefore no compensation variable is required.

[0148] Choose the Lyapunov function for

[0149]

[0150] in Design an adaptive controller for:

[0151]

[0152] in Substituting the adaptive controller into the designed Lyapunov function and taking the derivative, we obtain:

[0153]

[0154] in ,

[0155] .

[0156] according to Figure 4 Therefore, the specific steps of step 3 are as follows:

[0157] Step 3.1: Substitute the adaptive controller u into the time-varying barrier Lyapunov function V designed in Step 1 and take the derivative.

[0158] Choose one of the Lyapunov functions shown below. By selecting appropriate parameters, It always holds true.

[0159] By citing the theories of Cruz-Ortiz scholars, for arbitrary constants The following inequalities hold:

[0160]

[0161] set up Substituting this into the derivative of the Lyapunov function V, we get:

[0162]

[0163] Step 3.2: According to the theory of Cruz-Ortiz, the system is stable within a fixed time under state constraints.

[0164] It can be known It is stable over a fixed time, and the upper bound of the total convergence time of the system can be calculated. for:

[0165]

[0166] Example: The effectiveness of the proposed control method was verified by simulating the robotic arm model (1) using MATLAB simulation software. The experiment is as follows:

[0167] set up Meanwhile, the state constraint condition is set as follows:

[0168] , , , The reference signal trajectory is set as follows: The corresponding control parameters are: The initial state of the system is , , This embodiment combines an evolutionary simulation with a robotic arm model that incorporates time-varying state constraints, actuator failures, and external disturbances, and performs experimental verification. As can be seen from simulation image 5, the output... For reference signal Rapid and accurate tracking, combined with Figure 6 It can be seen that the system satisfies Full-state time-varying constraints, Figure 7 These are the corresponding errors for each system state. As shown in the graph, all errors are within the specified performance function. , Figure 8 It is the actual controller input voltage of the system. Experiments show that this method is effective. Within seconds, the tracking error was reduced. convergence to The magnitude is less than that, and the overshoot is less than that. The proposed method significantly improves the system's robustness to actuator failures and external disturbances.

[0169] The above descriptions are merely examples of this application and are not intended to limit this application. Various modifications and variations can be made to this application by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of this application should be included within the scope of protection of this application.

Claims

1. A method for adaptive trajectory tracking control of a robotic arm, characterized in that, Includes the following steps: A1. Based on actuator failures and state constraints, establish a new robotic arm system; A2. Use a disturbance observer with boundary characteristics to estimate the lumped disturbances present in the new robotic arm system; A3. Based on the time-varying logarithmic barrier Lyapunov function and the disturbance observer, generate an adaptive controller; and control the new robotic arm system so that the trajectory tracking error of the robotic arm converges to the set steady-state accuracy within a fixed time. The time-varying logarithmic barrier Lyapunov function is designed as follows: in It is a performance function that is set. By adjusting the parameters to make It holds true at any given time, where, They decided respectively The initial upper bound and the upper bound after stabilization; ,in, It is a systematic error. , For the set desired trajectory, It is a virtual controller The output result after passing through the first-order filter; the system state is defined as follows: ,in Indicates the angular displacement of the robotic arm. Indicates the motor current; design This indicates the estimation error, due to Since all values ​​are close to a local minimum of 0, it can be concluded that there exist positive real numbers. Make Established; The first virtual controller is generated by combining the time-varying logarithmic barrier Lyapunov function: in , , , The first-order filter is designed as follows: Among them, virtual controller For input, It is a defined positive real number. For the filter output, finally Represented as the compensation variable of the filter, it is designed as follows: ， in It is a positive real number that is defined. A second virtual controller is generated by combining the time-varying logarithmic barrier Lyapunov function: Among them, the definition , , , Corresponding The compensation variables are designed as follows: ； Design the adaptive controller based on the virtual controller and the disturbance observer: in The corresponding performance function is designed as follows: .

2. The control method according to claim 1, characterized in that, In step A1, the steps for establishing the new robotic arm system include: Introducing a robotic arm system, its dynamic equations are as follows: System state is defined as Define control input ,in Indicates the angular displacement of the robotic arm. Indicates motor current. Indicates the input voltage. Indicates the rotor inertia of the motor. Indicates the torque coefficient. Indicates the mass of a single rod. Indicates load quality. Expressed as the length of a single rod. Represented as gravity, It is expressed as the coefficient of viscous friction. Represented as armature inductance, It is expressed as the back electromotive force coefficient. Represented as armature resistance; By considering external interference and actuator malfunction, the robotic arm system can be simplified as follows: in, Treat it as a known constant. For unknown nonlinear function terms, This is referred to as external interference; Set the state constraints as follows: in, It is a nonlinear time-varying function. Represents the system's control output. This represents an actuator failure, and its form is as follows: in, Represents the actual input to the system. , This indicates unknown bias interference in the actuator; By transforming and rearranging the aforementioned robotic arm system, we can obtain the new robotic arm system, whose equation is as follows: Among them, lumped interference .

3. The control method according to claim 2, characterized in that, Step A2 further includes: The perturbation observer based on the boundary function is designed as follows: , ; It is a boundary function, and its design form is shown below: in, It is the smallest positive number designed; The lumped disturbance term in the new robotic arm system can be estimated using the disturbance observer, such that: Furthermore, the estimation error can converge to 0 within a fixed time, and the upper bound of the convergence time is calculable and only affected by the set parameters.

4. The control method according to claim 3, characterized in that, Step A3 includes: A31. Design multiple virtual controllers based on the time-varying logarithmic barrier Lyapunov function; A32. Design the adaptive controller based on the virtual controller and the disturbance observer; A33. The adaptive controller performs trajectory tracking control on the new robotic arm system.

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