Self-adaptive fuzzy fixed time control method based on output-limited mechanical arm

Abstract

The invention relates to the technical field of mechanical arm control, in particular to a self-adaptive fuzzy fixed time control method based on an output-limited mechanical arm, and the method comprises the following steps: S1, constructing a mechanical arm dynamic model; s2, setting constraint conditions of an output state of the mechanical arm dynamic model; s3, defining a tracking error and a virtual control law of the mechanical arm dynamic model; s4, on the basis of output state limitation and fixed time control, an actual control law is designed to serve as input of the mechanical arm dynamic model, and an adaptive law is constructed; and S5, a three-level logarithmic obstacle Lyapunov function is constructed, the three-level logarithmic obstacle Lyapunov function is combined with the virtual control law, the actual control law and the self-adaptive law step by step and substituted into the virtual control law, the actual control law and the self-adaptive law, and then it is guaranteed that the output state of the mechanical arm dynamic model does not exceed the preset safety boundary according to the actual fixed time control lemma. According to the invention, high-precision trajectory tracking can be realized on the premise of ensuring safe operation of the mechanical arm.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of mechanical arm control, and more particularly, to an adaptive fuzzy fixed-time control method based on an output-limited mechanical arm. BACKGROUND

[0002] At present, various control algorithms have been gradually applied to the control practice of mechanical arms and have achieved relatively significant application results. In order to further improve the control performance, many control algorithms take solving the inherent uncertainty and physical constraints of the mechanical arm system as the core target. For example, the prior art proposes an adaptive impedance controller using a radial basis function neural network (RBFNN), which uses RBFNN to approximate and compensate the uncertainty in the model, and introduces an adaptive robust control algorithm based on RBFNN. The algorithm significantly reduces the approximation error using robust control, and finally realizes successful trajectory tracking of the robot manipulator. However, the approximation performance of the adaptive neural network is highly dependent on the number and quality of sample data, and a large number of labeled samples are required to train the network parameters. In the scene where the working conditions of the mechanical arm are variable, it is impossible to directly model by rule inference, and it can only rely on a large number of samples to cover the fuzzy interval, resulting in low approximation efficiency and not fully considering the output state constraints existing in the operation of the mechanical arm. For example, the prior art proposes an adaptive control scheme based on fuzzy approximation, which uses a fuzzy logic system to approximate the uncertainty in the mechanical arm model, and combines a robust term to achieve high-precision trajectory tracking. However, this scheme still has deficiencies in time control performance and cannot meet the explicit requirements for convergence time in actual industrial scenarios. Although the finite time control can achieve stability within a finite time, the convergence speed is significantly affected by the initial conditions and parameter design, and the robustness is insufficient. For example, the prior art proposes a finite time scheme that can guarantee that the joint angle / velocity converges within a specified error bound within a finite time and gives the explicit relationship between the convergence time and the initial conditions. However, the finite time control has weak constraints and it is difficult to guarantee that the system reaches a stable state within a fixed time without relying on the initial conditions, which is not suitable for mechanical arm systems with uncertainty. SUMMARY

[0003] The present application aims to overcome the deficiencies of the prior art in simultaneously solving the output state limitation of the mechanical arm, complex uncertainty approximation, and precise controllability of the convergence time, and provides an adaptive fuzzy fixed-time control method based on an output-limited mechanical arm, which can ensure high-precision trajectory tracking of the mechanical arm under the premise of safe operation.

[0004] To solve the above technical problems, the technical solution adopted by the present application is: An adaptive fuzzy fixed-time control method based on an output-limited mechanical arm is provided, comprising the following steps: S1. Constructing a mechanical arm dynamics model; S2. Set the constraint condition of the output state of the mechanical arm dynamics model; S3. Define the tracking error of the mechanical arm dynamics model and the virtual control law; wherein the virtual control law is: (4) In the formula, , represents a positive adjustable control gain constant, represents the state constraint of the error , represents the first derivative of the desired tracking trajectory of the joint position in the mechanical arm dynamics model ; S4. Based on the output state limit and the fixed time control, design the actual control law as the input of the mechanical arm dynamics model, and construct the adaptive law; wherein the actual control law is: (5) In the formula, represents the symmetric inertia matrix of a single joint in the mechanical arm dynamics model, represents a positive adjustable control gain constant, represents the state constraint of the error , represents a fuzzy weight estimation vector, represents a fuzzy basis function vector; S5. Construct a three-level logarithmic barrier Lyapunov function, and combine the three-level logarithmic barrier Lyapunov function with the virtual control law, the actual control law, and the adaptive law step by step, and then according to the actual fixed time control lemma, it is guaranteed that the output state of the mechanical arm dynamics model does not exceed the preset safe boundary.

[0005] The adaptive fuzzy fixed time control method of the output limited mechanical arm of the application considers the mechanical arm with output state constraints and uncertainties, and converts the state constraints into boundedness conditions of the Lyapunov function by constructing a three-level logarithmic barrier Lyapunov function. Once the boundary is approached, the control gain is automatically increased to generate a "repulsive force" to prevent the boundary from being crossed. The output constraint of the mechanical arm is directly embedded to ensure that the output state is always within the safe constraint range. The adaptive fuzzy logic system is used to efficiently approximate the uncertainty of the mechanical arm dynamics model, and the actual fixed time control law is combined to realize the precise design of the convergence time, so as to finally ensure that the mechanical arm realizes high-precision trajectory tracking under the premise of safe operation.

[0006] Further, in step S1, the mechanical arm dynamics model is: (1) wherein, denote joint position, velocity, acceleration vectors, respectively; denote symmetric positive definite inertia matrices; denote Coriolis and centrifugal matrices; denote gravity torque vector; denote Jacobian matrices; denote unknown but bounded external disturbances; denote control inputs; Let joint position, velocity be denoted by , denote, the manipulator dynamics model can be obtained as: (2) wherein, denote the input of the manipulator dynamics model, denote the output of the manipulator dynamics model.

[0007] Further, in the manipulator dynamics model: .

[0008] Further, in step S2, the constraint condition comprises: ; wherein, , denote the physical limit of the state of the manipulator dynamics model.

[0009] Further, in step S3, the tracking error comprises: (3) wherein, , , denote the desired tracking trajectory of joint position, denote the virtual control law.

[0010] Further, in step S4, the adaptive law is: (6) wherein, , , is a small normal number; wherein, denote the fuzzy weight estimation vector; denote the adaptive law control gain greater than 0; denote the state constraint of error ; denote the fuzzy basis function vector, This represents the number of basis function nodes.

[0011] Further, in step S4, for the optimal fuzzy weight vector in the adaptive law... (2) p -1) power, (2) q -1) The operations of exponentiation are defined as follows:

[0012]

[0013] In the formula, Symbolic functions: when hour, ;when hour, ;when hour, .

[0014] Further, step S5 includes the following steps: S51. Construct the first barrier Lyapunov function and find its derivative, then substitute the virtual control law into the differentiated first barrier Lyapunov function; S52. Construct a second barrier Lyapunov function based on the first barrier Lyapunov function and differentiate it. An uncertain nonlinear function is obtained in the differentiated second barrier Lyapunov function. Then, the uncertain nonlinear function is approximated using a fuzzy logic system. Then, the actual control law is substituted into the differentiated and approximated second barrier Lyapunov function. S53. Construct a third barrier Lyapunov function based on the second barrier Lyapunov function and find its derivative. Then, substitute the adaptive law into the differentiated third barrier Lyapunov function. S54. According to the actual fixed-time control lemma, it is guaranteed that all closed-loop signals of the robotic arm dynamics model are consistent and eventually bounded, the constraints of the output state of the robotic arm dynamics model are never violated, and the tracking error converges to a predefined region within a fixed time independent of the initial conditions.

[0015] Furthermore, in step S51, the first barrier Lyapunov function is constructed. : (7) The first obstacle is the Lyapunov function. The first derivative is: (8) Substituting the virtual control law The first derivative can be obtained as follows: (9) In step S52, a second barrier Lyapunov function is constructed. : (10) The second obstacle is the Lyapunov function. The first derivative is: (11) According to formulas (2) and (3), we can obtain: (12) Combining formulas (9), (11), and (12), we can obtain: (13) in, It represents an uncertain nonlinear function; Define adaptive fuzzy weight error: (14) In the formula, This represents the optimal fuzzy weight vector matrix. Represents the fuzzy weight estimation vector matrix; By approximating the uncertain nonlinear function using a fuzzy logic system, we can obtain: (15) In the formula, Represents a fuzzy basis function vector. Represents the optimal fuzzy weight vector. Represents the input vector. Indicates the minimum approximation error; Substituting formula (15) into formula (13) yields: (16) Combining formula (16) with the actual control law, we can obtain: (17) In step S53, the third barrier Lyapunov function is constructed. : (18) in, The third obstacle is the Lyapunov function. The first derivative is: (19) Substituting the adaptive law into formula (19) yields: (20) where the vector inner product can be expanded into element-wise summation:

[0016] where , denote the th element of the adaptive fuzzy weight error and fuzzy weight estimation vector k respectively; similarly, the element-wise summation of the vector inner product can be obtained; By using and Young's inequality, we have: (21) where is a positive constant, denotes the minimum approximation error; Summing up to in inequality (21) and combining with the definition of vector norm , we have the vector inner product inequality: (22) where: , , Similarly, the calculation of , can be obtained; Substituting formulas (20), (21), (22) into formula (19), we have: (23) According to the logarithmic inequality lemma, we have: (24) By rearranging formula (24), we have: (25) where:

[0017] In step S54, according to the practical fixed-time control lemma and formula (25), it can be obtained that the dynamics model of the robot arm satisfies the practical fixed-time stability, and the error signal will converge to the compact set within the fixed time which is independent of the initial state of the dynamics model of the robot arm:

[0018] The upper bound of the convergence time is constructed as:

[0019] where the fixed-time stability is satisfied, all signals are bounded, and and ; therefore, all are also bounded, i.e., the mechanical arm dynamics model state is limited within a predefined region.

[0020] Further, the logarithmic inequality lemma is: for a given constant vector , if its components satisfy , then the following inequality for each component is true:

[0021] The actual fixed-time control lemma is: set a nonlinear system ; where is a possibly discontinuous vector field, if the nonlinear system is globally finite-time stable, and its convergence time function is bounded, i.e., there is a positive constant such that , the nonlinear system is called globally fixed-time stable; if there is a positive definite and radially unbounded function such that:

[0022] where the positive constant: , the nonlinear system is practically fixed-time stable, and the solution will converge to the following residual set within a fixed time:

[0023] and the convergence time is independent of the initial state of the nonlinear system, and the upper bound of the convergence time is:

[0024] where represents a scalar and satisfies .

[0025] Compared with the prior art, the present application has the beneficial effects that: The adaptive fuzzy fixed-time control method based on the output-restricted manipulator of the application considers the manipulator with output state constraints and uncertainties, and converts the state constraints into boundedness conditions of the Lyapunov function by constructing a three-level logarithmic barrier Lyapunov function, so that the control gain is automatically increased once the boundary is approached, and the "repulsive force" is generated to prevent the boundary from being crossed, the output constraints of the manipulator are directly embedded, and the output state is ensured to be always within the safe constraint range, the adaptive fuzzy logic system is used to efficiently approximate the dynamics model of the manipulator, the dynamics model uncertainty of the manipulator is combined with the actual fixed-time control law to realize accurate design of the convergence time, and finally the high-precision trajectory tracking of the manipulator is realized under the premise of safe operation. BRIEF DESCRIPTION OF DRAWINGS

[0026] Figure 1 The flowchart of the adaptive fuzzy fixed-time control method based on the output-restricted manipulator of the application; Figure 2 The tracking performance linear graph of joint 1 and joint 2 in the output-restricted manipulator applying the application; Figure 3 The velocity error and control input linear graph of joint 1 and joint 2 in the output-restricted manipulator applying the application; Figure 4 The fuzzy system performance linear graph of joint 1 and joint 2 in the output-restricted manipulator applying the application; Figure 5 The dynamics term estimation linear graph of joint 1 and joint 2 in the output-restricted manipulator applying the application. DETAILED DESCRIPTION

[0027] The application will be further described below in conjunction with the specific embodiments. In order to enable those skilled in the art to better understand the application scheme, the technical solutions in the embodiments of the application will be clearly and completely described below in conjunction with the drawings in the embodiments of the application. Obviously, the described embodiments are only a part of the embodiments of the application, but not all the embodiments. Based on the embodiments in the application, all other embodiments obtained by those skilled in the art without creative labor should belong to the protection scope of the application.

[0028] Those skilled in the art should understand that the embodiments of the application can be provided as a method, a system, or a computer program product. Therefore, the application can adopt a complete hardware embodiment, a complete software embodiment, or an embodiment combining software and hardware aspects.

[0029] It should be noted that the terms "comprising" and "having" and any variations thereof in the specification and in the claims are intended to cover a non-exclusive inclusion, for example, a process, method, system, product or apparatus that comprises a list of steps or units not merely those stated explicitly in the specification, but can include other steps or units not expressly listed or inherent to such process, method, product or apparatus.

[0030] It should be noted that the embodiments in the present application and the features in the embodiments can be combined with each other without conflict. The present application will be described in detail below with reference to the accompanying drawings and in combination with the embodiments.

[0031] Embodiment one As Figure 1 shown is the first embodiment of the adaptive fuzzy fixed-time control method based on the output limited manipulator, comprising the following steps: S1. Constructing a manipulator dynamics model; S2. Setting the constraint condition of the output state of the manipulator dynamics model; S3. Defining the tracking error of the manipulator dynamics model and the virtual control law; wherein the virtual control law is: (4) In the formula, , represents a positive adjustable control gain constant, represents the state constraint of the error , represents the first derivative of the desired tracking trajectory of the joint position in the manipulator dynamics model ; S4. Based on the output state limited and fixed-time control, designing the actual control law as the input of the manipulator dynamics model, and constructing the adaptive law; wherein the actual control law is: (5) In the formula, represents the symmetric inertia matrix of a single joint in the manipulator dynamics model, represents a positive adjustable control gain constant, represents the state constraint of the error , represents the fuzzy weight estimation vector, represents the fuzzy basis function vector; S5. Construct a three-stage logarithmic barrier Lyapunov function, and combine the three-stage logarithmic barrier Lyapunov function with the virtual control law, the actual control law and the adaptive law step by step, and then according to the actual fixed-time control lemma, the output state of the robot arm dynamics model is ensured not to exceed the preset safety boundary.

[0032] The adaptive fuzzy fixed-time control method based on the output limited robot arm of the application considers the robot arm with output state constraints and uncertainties, and converts the state constraints into boundedness conditions of the Lyapunov function by constructing a three-stage logarithmic barrier Lyapunov function. Once the boundary is approached, the control gain is automatically increased to generate a "repulsive force" to prevent the boundary from being crossed. The output constraints of the robot arm are directly embedded to ensure that the output state is always within the safe constraint range. The adaptive fuzzy logic system is used to efficiently approximate the robot arm dynamics model and the uncertainty of the robot arm dynamics model, and the actual fixed-time control law is combined to realize accurate design of the convergence time, so as to ultimately ensure high-precision trajectory tracking of the robot arm under the premise of safe operation.

[0033] Embodiment two This embodiment is a second embodiment of the adaptive fuzzy fixed-time control method based on the output limited robot arm. This embodiment is similar to embodiment one, except that in step S1, the robot arm dynamics model is: (1) In the formula, respectively represent joint position, velocity and acceleration vectors; represents a symmetric positive definite inertia matrix; represents a Coriolis and centrifugal matrix; represents a gravity torque vector; represents a Jacobian matrix; represents an unknown but bounded external disturbance; represents a control input; Let the joint position and velocity use , respectively, and the robot arm dynamics model can be obtained as: (2) In the formula, represents the input of the robot arm dynamics model, represents the output of the robot arm dynamics model.

[0034] Specifically, in the robot arm dynamics model: .

[0035] Specifically, in step S2, the constraint conditions include: ; wherein, , is a pre-defined column vector with elements being real numbers, representing physical limits of the state of the robot dynamics model.

[0036] In particular, in step S3, the tracking error comprises: (3) wherein, , , represents a desired tracking trajectory of the joint position, represents a virtual control law.

[0037] In particular, in step S4, the adaptive law is: (6) wherein, , , is a small real number; wherein, represents a fuzzy weight estimation vector; represents an adaptive law control gain greater than 0; represents a state constraint of the error ; represents a fuzzy basis function vector, represents a number of basis function nodes; wherein, for the optimal fuzzy weight vector , the operations of the (2 p -1)th power and the (2 q -1)th power of the error are respectively defined as:

[0038]

[0039] wherein, represents a sign function: when , ; when , ; when , .

[0040] In particular, step S5 comprises the following steps: S51. constructing a first barrier Lyapunov function and taking derivative, and then substituting the virtual control law into the first barrier Lyapunov function after taking derivative; S52. Construct a second barrier Lyapunov function based on the first barrier Lyapunov function and derive, an uncertain nonlinear function is obtained in the derived second barrier Lyapunov function; then, the uncertain nonlinear function is approximated by using a fuzzy logic system; then, the actual control law is substituted into the derived and approximated second barrier Lyapunov function; S53. Construct a third barrier Lyapunov function based on the second barrier Lyapunov function and derive, then substitute the adaptive law into the derived third barrier Lyapunov function; S54. According to the actual fixed-time control lemma, it is ensured that all closed-loop signals of the robotic arm dynamics model are uniformly ultimately bounded, the constraint condition of the output state of the robotic arm dynamics model is not violated at all times, and the tracking error converges to a predefined region within a fixed time that is independent of the initial condition.

[0041] Embodiment Three This embodiment is a third embodiment of the adaptive fuzzy fixed-time control method based on an output-limited robotic arm, which is similar to Embodiment One or Two, except that in step S51, a first barrier Lyapunov function is constructed in this embodiment. (7) The first derivative of the first barrier Lyapunov function is: (8) Substituting the virtual control law (4) into the first derivative of can be obtained: (9) In step S52, a second barrier Lyapunov function is constructed: (10) The first derivative of the second barrier Lyapunov function is: (11) According to formulas (2) and (3), we have: (12) Combining formulas (9), (11), and (12), we have: (13) where represents an uncertain nonlinear function; Define the adaptive fuzzy weight error as: (14) In the formula, This represents the optimal fuzzy weight vector matrix. Represents the fuzzy weight estimation vector matrix; By approximating an uncertain nonlinear function using a fuzzy logic system, we can obtain: (15) In the formula, Represents a fuzzy basis function vector. Represents the optimal fuzzy weight vector. Represents the input vector. Indicates the minimum approximation error; Substituting formula (15) into formula (13) yields: (16) Combining formula (16) with the actual control law (5), we can obtain: (17) In step S53, the third barrier Lyapunov function is constructed. : (18) in, The third obstacle is the Lyapunov function. The first derivative is: (19) Substituting the adaptive law (6) into formula (19), we get: (20) Where, vector dot product It can be expanded into element-wise summation:

[0042] In the formula, , These represent the adaptive fuzzy weighting errors, respectively. Fuzzy weight estimation vector The k Each element; similarly, the vector dot product can be obtained. element-wise summation; use Introducing Young's inequality, we obtain the element-by-element inequalities: (twenty one) in, For positive integers, Indicates the minimum approximation error; In inequality (21), for the first two inequalities arrive Summing up and combining the definition of vector norm The vector inner product inequality can be obtained: (22) Wherein: , , Similarly, the calculation of , ; Substitute formulas (20), (21), and (22) into formula (19) to obtain: (23) According to the logarithmic inequality lemma and formula (23), we have: (24) Rearranging formula (24) gives: (25) Wherein: ; In step S54, according to the actual fixed time control lemma and formula (25), it can be obtained that the mechanical arm dynamics model satisfies the practical fixed time stability, and the error signal will converge to the compact set within the fixed time independent of the initial state of the mechanical arm dynamics model:

[0043] The upper bound of the display convergence time is constructed: ; According to the above proof, it can be obtained that the fixed time stability is satisfied, all signals are bounded, and , and ; therefore, For all , it is also bounded, that is, the state of the mechanical arm dynamics model is limited in the predefined area.

[0044] Specifically, in step S53, the logarithmic inequality lemma is: for a given constant vector , if its components satisfy , then the following inequality for each component is true: .

[0045] Specifically, in step S54, the actual fixed time control lemma is: Set the nonlinear system ; Wherein, For a possibly discontinuous vector field, if a nonlinear system is globally finite time stable and its convergence time function is bounded, i.e., there exists a positive constant such that the nonlinear system is called globally fixed time stable. If there exists a positive definite and radially unbounded function such that:

[0046] where the positive constant: the nonlinear system is practically fixed time stable, and the solution will converge to the following residual set within a fixed time:

[0047] and the convergence time is independent of the initial state of the nonlinear system, and the upper bound of the convergence time is:

[0048] where is a scalar and satisfies .

[0049] The application is based on the adaptive fuzzy fixed time control method of the output limited mechanical arm, introduces three-stage logarithmic barrier Lyapunov function to realize strict constraint of the output state, converts the state constraint into the boundedness condition of the Lyapunov function, automatically increases the control gain once approaching the boundary, generates "repulsive force" to prevent border crossing, directly embeds the output constraint, and theoretically guarantees that the output of the mechanical arm dynamics model does not exceed the boundary all the time. In addition, the application also uses an adaptive fuzzy logic system to realize efficient approximation of unknown nonlinear dynamics, does not need offline training, can estimate the system uncertainty in real time, such as friction, load disturbance and modeling error; the universal approximation characteristic of the fuzzy logic system is suitable for engineering implementation; the adaptive law design ensures that the approximation error is bounded, and the robustness of the system is enhanced. Furthermore, the application applies the actual fixed time control lemma to realize preset convergence time and initial value independence, specifically, the application constructs an explicit convergence time upper bound, wherein the control parameter can be directly used for setting the convergence time, realizes "time programmable", and the convergence time is independent of the initial state of the mechanical arm dynamics model, which is suitable for repeated tasks and different initial conditions of industrial scenes. The application guarantees state safety through three-stage logarithmic barrier Lyapunov function, provides stable operation space for fuzzy approximation and fixed time control; fuzzy approximation compensates for uncertainty, can reduce the interference of the mechanical arm dynamics model error on the fixed time control law, and improves the convergence accuracy; the fixed time control guarantees fast response, can make the mechanical arm dynamics model converge quickly within the safety constraint, and avoids constraint violation caused by slow response.

[0050] As shown in Figure 2 , the actual trajectories of joint 1 and joint 2 are very close to the expected trajectories, the position error is always within the constraint range, and it can be stabilized in a very short time, which meets the fixed-time control, indicating that the present application is effective. As shown in Figure 3 , the velocity error of joint 1 and joint 2 is controlled within the constraint boundary, and the control input is reasonable, indicating that the present application is feasible in practical application. As shown in Figure 4 , the change of the weight norm of the fuzzy system shows that the fuzzy logic system can effectively approximate the unknown nonlinear dynamics, and the comparison of the position error before and after the disturbance shows that the present application has good robustness. As shown in Figure 5 , the fuzzy estimation can well approximate the actual dynamic term, which can verify the approximation ability of the fuzzy logic system.

[0051] The dynamic performance of the present application is significantly improved, which can meet the demand of high-precision scenes such as semiconductor manufacturing and medical treatment; the stability and robustness of the present application are enhanced, the global signal is bounded, and there is no risk of divergence, and all signals in the closed-loop system are uniformly ultimately bounded through the three-level logarithmic barrier Lyapunov function, that is, even if the inertia matrix changes by 20% or sudden disturbances such as external force impact are faced, stability can still be maintained; parameter tuning is more convenient, which can reduce the engineering threshold and clearly associate the control gain with the convergence time; the number of hidden layer neurons of the fuzzy logic system can be directly determined by the number of fuzzy rules, avoiding the contradiction between "excessive neurons increasing consumption time" and "insufficient neurons reducing precision" of the adaptive neural network. The engineering practicability of the present application is greatly improved, and the fuzzy weight can be adjusted in real time through the adaptive law, which can adapt to scenes such as load switching and working condition change, reducing the number of on-site debugging and maintenance cycle. The present application can also be applied to the field of high-precision manufacturing, and is suitable for scenes such as semiconductor wafer handling and automobile welding mechanical arm, which can avoid product scrapping or equipment damage caused by state over-limit.

[0052] In the specific content of the above specific embodiments, each technical feature can be combined arbitrarily without contradiction, and in order to make the description concise, not all possible combinations of the above technical features are described, but as long as the combination of these technical features does not exist contradiction, it should be considered as the scope of the present application.

[0053] Obviously, the above embodiments of the present application are only examples for clearly illustrating the present application, and are not intended to limit the embodiments of the present application. Based on the above description, other different forms of changes or variations can be made by those skilled in the art. Here, all embodiments are not required to be exhausted. Any modification, equivalent replacement and improvement made within the spirit and principle of the present application shall be included in the protection scope of the claims of the present application.

Claims

1. An adaptive fuzzy fixed-time control method based on output constrained manipulator, characterized in that, Comprising the steps of: S1. constructing a mechanical arm dynamics model; S2. setting a constraint condition of an output state of the mechanical arm dynamics model; S3. defining a tracking error and a virtual control law of the mechanical arm dynamics model; wherein the virtual control law is: （4） In the formula, , This represents a positive, adjustable control gain constant. Indicates error State constraints, This represents the desired tracking trajectory of the joint positions in the robotic arm's dynamics model. The first derivative; S4. Design an actual control law as an input of the mechanical arm dynamics model based on the output state constraint and the fixed time control, and construct an adaptive law; wherein the actual control law is: （5） wherein denotes a symmetric inertia matrix of a single joint in the robot dynamics model, denotes a positive adjustable control gain constant, denotes an error of a state constraint, denotes a fuzzy weight estimation vector, denotes a fuzzy basis function vector; S5. constructing a three-level logarithmic barrier Lyapunov function, and combining the three-level logarithmic barrier Lyapunov function with the virtual control law, the actual control law and the adaptive law step by step, and then guaranteeing that the output state of the mechanical arm dynamics model does not exceed a preset safety boundary according to an actual fixed-time control lemma.

2. The adaptive fuzzy fixed-time control method based on output constrained manipulator according to claim 1, wherein, In step S1, the mechanical arm dynamics model is: （1） wherein denote joint position, velocity, acceleration vectors, respectively; denote symmetric positive definite inertia matrices; denote Coriolis and centrifugal matrices; denote gravity torque vectors; denote Jacobian matrices; denote unknown but bounded external disturbances; denote control inputs; Let joint position, velocity respectively use , represent, the mechanical arm dynamics model can get: （2） wherein represents an input to the manipulator dynamics model, represents an output of the manipulator dynamics model.

3. The adaptive fuzzy fixed-time control method based on output-constrained manipulators according to claim 2, wherein, In the mechanical arm dynamics model: 。 4. The adaptive fuzzy fixed-time control method based on output-constrained manipulators according to claim 3, wherein, In step S2, the constraint condition comprises: wherein , , represents the physical limits of the state of the robot arm dynamics model.

5. The adaptive fuzzy fixed-time control method based on output-constrained manipulators according to claim 4, wherein, In step S3, the tracking error comprises: （3） wherein , , represents a desired tracking trajectory of the joint position, represents a virtual control law.

6. The adaptive fuzzy fixed-time control method based on output-constrained manipulators according to claim 5, wherein In step S4, the adaptive law is: （6） in, , , For small positive constants; in the formula, Represents the fuzzy weight estimation vector; This represents the adaptive law control gain that is greater than 0. Indicates error State constraints; Represents a fuzzy basis function vector. This represents the number of basis function nodes.

7. The adaptive fuzzy fixed-time control method based on output-constrained manipulators according to claim 6, wherein In step S4, for the optimal fuzzy weight vector in the adaptive law (2) p -1) power, (2) q -1) The operations of exponentiation are defined as follows: In the formulae, denotes the sign function: When , ; when , ; when , .

8. The adaptive fuzzy fixed-time control method based on output-constrained manipulators according to claim 7, wherein, Step S5 comprises the steps of: S51. constructing a first barrier Lyapunov function and deriving, and then substituting the virtual control law into the derived first barrier Lyapunov function; S52. constructing a second barrier Lyapunov function based on the first barrier Lyapunov function and deriving, obtaining an uncertain nonlinear function in the derived second barrier Lyapunov function; then, using a fuzzy logic system to approximate the uncertain nonlinear function; and then substituting the actual control law into the second barrier Lyapunov function after the approximation processing; S53. constructing a third barrier Lyapunov function based on the second barrier Lyapunov function and deriving, and then substituting the adaptive law into the derived third barrier Lyapunov function; S54. according to the actual fixed-time control lemma, guaranteeing that all closed-loop signals of the mechanical arm dynamics model are uniformly ultimately bounded, the constraint condition of the output state of the mechanical arm dynamics model is always not violated, and the tracking error converges to a predefined region within a fixed time that is independent of the initial condition.

9. The adaptive fuzzy fixed-time control method based on output-constrained manipulators according to claim 8, wherein, In step S51, a first barrier Lyapunov function is constructed : （7） The first obstacle Lyapunov function The first derivative of the obstacle Lyapunov function （8） Substituting the virtual control law into the first derivative gives: （9） In step S52, a second barrier Lyapunov function is constructed : （10） The first derivative of the second obstacle Lyapunov function is: （11） According to formulas (2) and (3), we have: （12） Combining formulas (9), (11) and (12), we have: （13） wherein represents an uncertain non-linear function; Define the adaptive fuzzy weight error as: （14） wherein denotes the optimal fuzzy weight vector matrix, denotes the fuzzy weight estimation vector matrix; Using a fuzzy logic system to approximate the uncertain nonlinear function, we have: （15） wherein denotes the fuzzy basis function vector, denotes the optimal fuzzy weight vector, denotes the input vector, denotes the minimum approximation error; Substituting formula (15) into formula (13), we have: （16） Combining the actual control law with formula (16), we have: （17） In step S53, a third barrier Lyapunov function is constructed : （18） wherein ; the third obstacle Lyapunov function first derivative of the third obstacle Lyapunov function （19） Substituting the adaptive law into formula (19), we have: （20） where the vector inner product which can be expanded to an element-wise sum: In the formula, , These represent the adaptive fuzzy weighting errors, respectively. Fuzzy weight estimation vector The k Each element; similarly, the vector dot product can be obtained. element-wise summation; By using and the Young inequality we obtain （21） wherein is a normal number, denotes the minimum approximation error; In inequality (21) for to summing and combining with the definition of the vector norm the vector inner product inequality is obtained: （22） Wherein: , By analogy , calculations; Substituting formulas (20), (21) and (22) into formula (19), we have: （23） According to the logarithmic inequality lemma, we have: （24） After formula (24) is arranged, we have: （25） Wherein: In step S54, according to the actual fixed-time control lemma and formula (25), the mechanical arm dynamics model satisfies the practical fixed-time stability, and the error signal converges to a compact set within a fixed time that is independent of the initial state of the mechanical arm dynamics model: Construct a display convergence time upper bound: where the fixed-time stability is satisfied, all signals are bounded, and and ; therefore, all are also bounded, i.e., the manipulator dynamics model state is limited within a predefined region.

10. The adaptive fuzzy fixed-time control method based on output-constrained manipulators according to claim 9, wherein, The logarithmic inequality lemma is: for a given constant vector , if its components satisfy , then the following inequality holds for each component: The actual fixed-time control lemma is: Setting up a nonlinear system ; wherein, is a possibly discontinuous vector field, if the nonlinear system is globally finite-time stable and its convergence time function is bounded, i.e. there exists a positive number such that then the nonlinear system is said to be globally fixed-time stable; If there exists a positive definite and radially unbounded function such that: where the positive constant: Then the nonlinear system is practically fixed-time stable, and the solution will converge to the following residual set within a fixed time: And the convergence time is independent of the initial state of the nonlinear system, and the convergence time upper bound is: wherein represents a scalar and satisfies .

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