Adaptive fuzzy fixed-time control method for output constrained manipulators

Abstract

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, comprising the following steps: S1. constructing a mechanical arm dynamics model; S2. setting a constraint condition for the output state of the mechanical arm dynamics model; S3. defining a tracking error and a virtual control law of the mechanical arm dynamics model; S4. based on output state limitation and fixed-time control, designing an actual control law as the input of the mechanical arm dynamics model, and constructing an adaptive law; S5. constructing a three-stage logarithmic barrier Lyapunov function, and combining 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, ensuring that the output state of the mechanical arm dynamics model does not exceed the preset safety boundary. The present application can ensure high-precision trajectory tracking of the mechanical arm under the premise of safe operation.

Description

Technical Field

[0001] This invention relates to the technical field of robotic arm control, and more specifically, to an adaptive fuzzy fixed-time control method for a robotic arm with limited output. Background Technology

[0002] Currently, various control algorithms have been gradually applied to the control practice of robotic arms, achieving significant results. To further improve control performance, many control algorithms take addressing the inherent uncertainties and physical constraints of robotic arm systems as their core objective. For example, existing technologies have proposed an adaptive impedance controller using radial basis function neural networks (RBFNNs). RBFNNs are used to approximate and compensate for uncertainties in the model, and an adaptive robust control algorithm is introduced based on RBFNNs. This algorithm significantly reduces approximation errors through robust control, ultimately achieving successful trajectory tracking of the robotic arm. However, the approximation performance of adaptive neural networks is highly dependent on the quantity and quality of sample data, requiring extensive training with labeled samples to optimize network parameters. In scenarios with varying robotic arm operating conditions, direct rule-based reasoning modeling is not feasible; instead, a large number of samples must cover fuzzy regions, resulting in low approximation efficiency and insufficient consideration of output state constraints during robotic arm operation. For example, existing technologies propose adaptive control schemes based on fuzzy approximation, which use fuzzy logic systems to approximate the uncertainties in the robotic arm model and combine robust term design to achieve high-precision trajectory tracking. However, this scheme still has shortcomings in time control performance and cannot meet the explicit requirements for convergence time in actual industrial scenarios. While finite-time control can achieve stability within a finite time, its convergence speed is significantly affected by initial conditions and parameter design, resulting in insufficient robustness. Another example is a finite-time scheme proposed in existing technologies, which can guarantee that the joint angle / angular velocity converges within a specified error boundary within a finite time and provides an explicit relationship between the convergence time and initial conditions. However, finite-time control has weak constraints and is difficult to guarantee that the system reaches a stable state within a fixed time without relying on initial conditions, making it unsuitable for robotic arm systems with uncertainties. Summary of the Invention

[0003] The purpose of this invention is to overcome the shortcomings of existing technologies in simultaneously addressing the limitations of the output state of a robotic arm, the approximation of complex uncertainties, and the precise control of convergence time. It provides an adaptive fuzzy fixed-time control method for a robotic arm with limited output, which can ensure that the robotic arm achieves high-precision trajectory tracking under the premise of safe operation.

[0004] To solve the above-mentioned technical problems, the technical solution adopted by the present invention is as follows:

[0005] An adaptive fuzzy fixed-time control method for an output-constrained robotic arm is provided, comprising the following steps:

[0006] S1. Construct a dynamic model of the robotic arm;

[0007] S2. Set the constraints for the output state of the robotic arm's dynamic model;

[0008] S3. Define the tracking error and virtual control law of the robotic arm dynamics model; wherein the virtual control law is:

[0009] (4)

[0010] 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;

[0011] S4. Based on the constraints of output state and fixed-time control, design an actual control law as the input to the robotic arm's dynamics model, and construct an adaptive law; wherein, the actual control law... for:

[0012] (5)

[0013] In the formula, This represents the symmetric inertia matrix of a single joint in the robotic arm's dynamics model. This represents a positive, adjustable control gain constant. Indicates error State constraints, Represents the fuzzy weight estimation vector. Represents a fuzzy basis function vector;

[0014] S5. Construct a three-level logarithmic obstacle Lyapunov function, and combine the three-level logarithmic obstacle Lyapunov function with the virtual control law, the actual control law, and the adaptive law step by step. Then, according to the actual fixed-time control lemma, ensure that the output state of the robotic arm dynamics model does not exceed the preset safety boundary.

[0015] The present invention provides an adaptive fuzzy fixed-time control method for output-constrained robotic arms. Considering the output state constraints and uncertainties of the robotic arm, this method constructs a three-level logarithmic barrier Lyapunov function to transform the state constraints into boundedness conditions of the Lyapunov function. Once the boundary is approached, the control gain automatically increases, generating a "repulsive force" to prevent overshooting. This directly embeds the output constraints of the robotic arm, ensuring that the output state remains within the safe constraint range. An adaptive fuzzy logic system is used to efficiently approximate the uncertainty of the robotic arm's dynamics model. Combined with a practical fixed-time control law, the convergence time is precisely designed, ultimately ensuring that the robotic arm achieves high-precision trajectory tracking while operating safely.

[0016] Further, in step S1, the dynamic model of the robotic arm is:

[0017] (1)

[0018] In the formula, These represent the joint position, velocity, and acceleration vectors, respectively. Represents a symmetric positive definite inertia matrix; Represent the Coriolis and centrifugal matrices; Represents the gravitational torque vector; Represents the Jacobian matrix; Represents an unknown but bounded external disturbance; Indicates control input;

[0019] Use joint position and speed separately , The dynamic model of the robotic arm can be derived as follows:

[0020] (2)

[0021] In the formula, This represents the input to the robotic arm's dynamics model. This represents the output of the robotic arm dynamics model.

[0022] Furthermore, in the aforementioned robotic arm dynamics model:

[0023] .

[0024] Further, in step S2, the constraints include:

[0025] ;

[0026] In the formula, , , which represents the physical constraints on the state of the robotic arm's dynamic model.

[0027] Further, in step S3, the tracking error includes:

[0028] (3)

[0029] In the formula, , , The expected tracking trajectory representing the joint position. This represents a virtual control law.

[0030] Further, in step S4, the adaptive law is:

[0031] (6)

[0032] 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.

[0033] 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:

[0034]

[0035]

[0036] In the formula, Symbolic functions:

[0037] when hour, ;when hour, ;when hour, .

[0038] Further, step S5 includes the following steps:

[0039] S51. Construct the first barrier Lyapunov function and find its derivative, then substitute the virtual control law into the differentiated first barrier Lyapunov function;

[0040] 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.

[0041] 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.

[0042] 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.

[0043] Furthermore, in step S51, the first barrier Lyapunov function is constructed. :

[0044] (7)

[0045] The first obstacle is the Lyapunov function. The first derivative is:

[0046] (8)

[0047] Substituting the virtual control law into The first derivative can be obtained as follows:

[0048] (9)

[0049] In step S52, a second barrier Lyapunov function is constructed. :

[0050] (10)

[0051] The second obstacle is the Lyapunov function. The first derivative is:

[0052] (11)

[0053] According to formulas (2) and (3), we can obtain:

[0054] (12)

[0055] Combining formulas (9), (11), and (12), we can obtain:

[0056] (13)

[0057] in, It represents an uncertain nonlinear function;

[0058] Define adaptive fuzzy weight error:

[0059] (14)

[0060] In the formula, This represents the optimal fuzzy weight vector matrix. Represents the fuzzy weight estimation vector matrix;

[0061] By approximating the uncertain nonlinear function using a fuzzy logic system, we can obtain:

[0062] (15)

[0063] In the formula, Represents a fuzzy basis function vector. Represents the optimal fuzzy weight vector. Represents the input vector. Indicates the minimum approximation error;

[0064] Substituting formula (15) into formula (13) yields:

[0065] (16)

[0066] Combining formula (16) with the actual control law, we can obtain:

[0067] (17)

[0068] In step S53, the third barrier Lyapunov function is constructed. :

[0069] (18)

[0070] in, The third obstacle is the Lyapunov function. The first derivative is:

[0071] (19)

[0072] Substituting the adaptive law into formula (19), we get:

[0073] (20)

[0074] Where, vector dot product It can be expanded into element-wise summation:

[0075]

[0076] 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;

[0077] use And from Young's inequality, we can obtain:

[0078] (twenty one)

[0079] in, For positive integers, Indicates the minimum approximation error;

[0080] In inequality (21) arrive Summation, and combining it with the definition of vector norm. We can obtain the vector dot product inequality:

[0081] (twenty two)

[0082] in:

[0083] , ,

[0084] Similarly, we can obtain , Calculation;

[0085] Substituting formulas (20), (21), and (22) into formula (19), we get:

[0086] (twenty three)

[0087] According to the lemma of the logarithmic inequality, we can obtain:

[0088] (twenty four)

[0089] Rearranging formula (24) yields:

[0090] (25)

[0091] in:

[0092]

[0093] In step S54, according to the practical fixed-time control lemma and formula (25), the robotic arm dynamics model satisfies practical fixed-time stability, and the error signal will converge to a compact set within a fixed time independent of the initial state of the robotic arm dynamics model:

[0094]

[0095] Construct an upper bound for the explicit convergence time:

[0096]

[0097] Among them, it satisfies the following conditions: fixed-time stability, all signals are bounded, and... ,as well as ;therefore, For all It is also bounded, meaning that the state of the robotic arm's dynamic model is restricted to a predefined region.

[0098] Furthermore, the lemma of the logarithmic inequality states that for a given constant vector... If its components satisfy Then the following inequality holds for each component:

[0099]

[0100] The actual fixed-time control lemma is as follows:

[0101] Setting up a nonlinear system ;

[0102] in, Let be a potentially discontinuous vector field, if the nonlinear system is globally finite-time stable, and its convergence time function is... It is bounded, meaning there exists a positive constant. , making If the nonlinear system is stable, then it is called a globally fixed-time stable system.

[0103] If there exists a positive definite and radially unbounded function , so that:

[0104]

[0105] Among them, the positive constants are: Then the nonlinear system is practically fixed-time stable, and the solution will converge to the following residual set in a fixed time:

[0106]

[0107] Furthermore, the convergence time is independent of the initial state of the nonlinear system, and the upper bound of the convergence time is:

[0108]

[0109] in, Describe a scalar and satisfy .

[0110] Compared with the prior art, the beneficial effects of the present invention are:

[0111] The present invention provides an adaptive fuzzy fixed-time control method for output-constrained robotic arms. Considering the output state constraints and uncertainties of the robotic arm, this method constructs a three-level logarithmic barrier Lyapunov function to transform the state constraints into boundedness conditions of the Lyapunov function. Once the boundary is approached, the control gain automatically increases, generating a "repulsive force" to prevent overshooting. This directly embeds the output constraints of the robotic arm, ensuring that the output state remains within the safe constraint range. An adaptive fuzzy logic system is used to efficiently approximate the uncertainty of the robotic arm's dynamics model. Combined with a practical fixed-time control law, the convergence time is precisely designed, ultimately ensuring that the robotic arm achieves high-precision trajectory tracking while operating safely. Attached Figure Description

[0112] Figure 1 This is a flowchart of the adaptive fuzzy fixed-time control method based on an output-constrained robotic arm according to the present invention;

[0113] Figure 2 Linear graph of the tracking performance of joints 1 and 2 in the output-limited robotic arm for applying the present invention;

[0114] Figure 3 Linear graph of speed error and control input for joints 1 and 2 in the output-limited robotic arm of the present invention;

[0115] Figure 4 A fuzzy system performance linear graph of joints 1 and 2 in an output-constrained robotic arm for applying the present invention;

[0116] Figure 5 Linear graphs are estimated for the dynamics of joints 1 and 2 in the output-constrained robotic arm using the present invention. Detailed Implementation

[0117] The present invention will be further described below with reference to specific embodiments. To enable those skilled in the art to better understand the present invention, the technical solutions of the embodiments will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are merely some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort should fall within the scope of protection of the present invention.

[0118] Those skilled in the art will understand that embodiments of the present invention can be provided as methods, systems, or computer program products. Therefore, the present invention can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects.

[0119] It should be noted that the terms "comprising" and "having" and any variations thereof in the specification, claims and accompanying drawings of this invention are intended to cover non-exclusive inclusion. For example, a process, method, system, product or device that includes a series of steps or units is not necessarily limited to those steps or units that are explicitly listed, but may include other steps or units that are not explicitly listed or that are inherent to such processes, methods, products or devices.

[0120] It should be noted that, unless otherwise specified, the embodiments and features described in the present invention can be combined with each other. The present invention will now be described in detail with reference to the accompanying drawings and embodiments.

[0121] Example 1

[0122] like Figure 1 The first embodiment of the adaptive fuzzy fixed-time control method for a robot arm with limited output, as shown in the present invention, includes the following steps:

[0123] S1. Construct a dynamic model of the robotic arm;

[0124] S2. Set the constraints for the output state of the robotic arm's dynamic model;

[0125] S3. Define the tracking error and virtual control law for the robotic arm's dynamics model; where the virtual control law is:

[0126] (4)

[0127] 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;

[0128] S4. Based on output state constraints and fixed-time control, design an actual control law as the input to the robotic arm's dynamics model, and construct an adaptive law; wherein, the actual control law... for:

[0129] (5)

[0130] In the formula, This represents the symmetric inertia matrix of a single joint in the dynamic model of a robotic arm. This represents a positive, adjustable control gain constant. Indicates error State constraints, Represents the fuzzy weight estimation vector. Represents a fuzzy basis function vector;

[0131] 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. Then, according to the actual fixed-time control lemma, ensure that the output state of the robotic arm dynamics model does not exceed the preset safety boundary.

[0132] The present invention provides an adaptive fuzzy fixed-time control method for output-constrained robotic arms. Considering the output state constraints and uncertainties of the robotic arm, this method constructs a three-level logarithmic barrier Lyapunov function to transform the state constraints into boundedness conditions of the Lyapunov function. Once the boundary is approached, the control gain automatically increases, generating a "repulsive force" to prevent overshooting. This directly embeds the output constraints of the robotic arm, ensuring that the output state remains within the safe constraint range. An adaptive fuzzy logic system is used to efficiently approximate the uncertainty of the robotic arm's dynamics model. Combined with a practical fixed-time control law, the convergence time is precisely designed, ultimately ensuring that the robotic arm achieves high-precision trajectory tracking while operating safely.

[0133] Example 2

[0134] This embodiment is the second embodiment of the adaptive fuzzy fixed-time control method based on an output-constrained robotic arm. This embodiment is similar to the first embodiment, except that in step S1, the robotic arm dynamics model is as follows:

[0135] (1)

[0136] In the formula, These represent the joint position, velocity, and acceleration vectors, respectively. Represents a symmetric positive definite inertia matrix; Represent the Coriolis and centrifugal matrices; Represents the gravitational torque vector; Represents the Jacobian matrix; Represents an unknown but bounded external disturbance; Indicates control input;

[0137] Use joint position and speed separately , The dynamic model of the robotic arm can be derived as follows:

[0138] (2)

[0139] In the formula, This represents the input to the robotic arm's dynamics model. This represents the output of the robotic arm's dynamics model.

[0140] Specifically, in the robotic arm dynamics model:

[0141] .

[0142] Specifically, in step S2, the constraints include:

[0143] ;

[0144] In the formula, , is a predefined column vector, where the elements are positive integers representing the physical constraints of the robotic arm's dynamics model state.

[0145] Specifically, in step S3, the tracking error includes:

[0146] (3)

[0147] In the formula, , , The expected tracking trajectory representing the joint position. This represents a virtual control law.

[0148] Specifically, in step S4, the adaptive law is:

[0149] (6)

[0150] 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. Indicates the number of basis function nodes;

[0151] Among them, for the optimal fuzzy weight vector (2) p -1) power, (2) q -1) The operations of exponentiation are defined as follows:

[0152]

[0153]

[0154] In the formula, Symbolic functions:

[0155] when hour, ;when hour, ;when hour, .

[0156] Specifically, step S5 includes the following steps:

[0157] S51. Construct the first barrier Lyapunov function and find its derivative, then substitute the virtual control law into the differentiated first barrier Lyapunov function;

[0158] 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.

[0159] 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.

[0160] S54. Based on 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 the predefined region within a fixed time independent of the initial conditions.

[0161] Example 3

[0162] This embodiment is the third embodiment of the adaptive fuzzy fixed-time control method based on an output-constrained robotic arm. This embodiment is similar to Embodiment 1 or 2, except that in this embodiment, in step S51, the first obstacle Lyapunov function is constructed. :

[0163] (7)

[0164] The first obstacle is the Lyapunov function. The first derivative is:

[0165] (8)

[0166] Substituting the virtual control law (4) into The first derivative can be obtained as follows:

[0167] (9)

[0168] In step S52, a second barrier Lyapunov function is constructed. :

[0169] (10)

[0170] The second obstacle is the Lyapunov function. The first derivative is:

[0171] (11)

[0172] According to formulas (2) and (3), we can obtain:

[0173] (12)

[0174] Combining formulas (9), (11), and (12), we can obtain:

[0175] (13)

[0176] in, It represents an uncertain nonlinear function;

[0177] Define adaptive fuzzy weight error:

[0178] (14)

[0179] In the formula, This represents the optimal fuzzy weight vector matrix. Represents the fuzzy weight estimation vector matrix;

[0180] By approximating an uncertain nonlinear function using a fuzzy logic system, we can obtain:

[0181] (15)

[0182] In the formula, Represents a fuzzy basis function vector. Represents the optimal fuzzy weight vector. Represents the input vector. Indicates the minimum approximation error;

[0183] Substituting formula (15) into formula (13) yields:

[0184] (16)

[0185] Combining formula (16) with the actual control law (5), we can obtain:

[0186] (17)

[0187] In step S53, the third barrier Lyapunov function is constructed. :

[0188] (18)

[0189] in, The third obstacle is the Lyapunov function. The first derivative is:

[0190] (19)

[0191] Substituting the adaptive law (6) into formula (19), we get:

[0192] (20)

[0193] Where, vector dot product It can be expanded into element-wise summation:

[0194]

[0195] 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;

[0196] use Introducing Young's inequality, we obtain the element-by-element inequalities:

[0197] (twenty one)

[0198] in, For positive integers, Indicates the minimum approximation error;

[0199] In inequality (21), for the first two inequalities arrive Summation, and combining it with the definition of vector norm. We can obtain the vector dot product inequality:

[0200] (twenty two)

[0201] in:

[0202] , ,

[0203] Similarly, we can obtain , Calculation;

[0204] Substituting formulas (20), (21), and (22) into formula (19), we get:

[0205] (twenty three)

[0206] According to the lemma of logarithmic inequality and formula (23), we can obtain:

[0207] (twenty four)

[0208] Rearranging formula (24) yields:

[0209] (25)

[0210] in:

[0211] ;

[0212] In step S54, according to the practical fixed-time control lemma and formula (25), the robotic arm dynamics model satisfies practical fixed-time stability, and the error signal will converge to a compact set within a fixed time independent of the initial state of the robotic arm dynamics model:

[0213]

[0214] Construct an upper bound for the explicit convergence time:

[0215] ;

[0216] Based on the above proof, it can be concluded that the signal satisfies the following conditions: fixed-time stability, all signals are bounded, and... ,as well as ;therefore, For all It is also bounded, meaning that the state of the robotic arm's dynamic model is restricted to a predefined region.

[0217] Specifically, in step S53, the logarithmic inequality lemma states: for a given constant vector... If its components satisfy Then the following inequality holds for each component: .

[0218] Specifically, in step S54, the actual fixed-time control lemma is:

[0219] Setting up a nonlinear system ;

[0220] in, Given a potentially discontinuous vector field, if the nonlinear system is globally finite-time stable and its convergence time function is... It is bounded, meaning there exists a positive constant. , making If the nonlinear system is stable, then it is called a globally fixed-time stable system.

[0221] If there exists a positive definite and radially unbounded function , so that:

[0222]

[0223] Among them, the positive constants are: Then the nonlinear system is practically fixed-time stable, and the solution will converge to the following residual set in a fixed time:

[0224]

[0225] Furthermore, the convergence time is independent of the initial state of the nonlinear system, and the upper bound of the convergence time is:

[0226]

[0227] in, Describe a scalar and satisfy .

[0228] This invention is based on an adaptive fuzzy fixed-time control method for output-constrained robotic arms. It introduces a third-order logarithmic barrier Lyapunov function to achieve strict constraints on the output state, transforming the state constraints into boundedness conditions of the Lyapunov function. Once the boundary is approached, the control gain automatically increases, generating a "repulsive force" to prevent exceeding the boundary, directly embedding the output constraint and theoretically ensuring that the output of the robotic arm's dynamic model never exceeds the boundary. Furthermore, this invention employs an adaptive fuzzy logic system to achieve efficient approximation of unknown nonlinear dynamics without offline training, enabling real-time estimation of system uncertainties such as friction, load disturbances, and modeling errors. Utilizing the universal approximation characteristics of fuzzy logic systems, it is suitable for engineering implementation. The adaptive law design ensures bounded approximation errors, enhancing system robustness. Moreover, this invention applies the practical fixed-time control lemma to achieve a preset convergence time independent of the initial value. Specifically, this invention constructs an explicit upper bound on the convergence time, where the control parameters... It can be directly used to set the convergence time, achieving "time programmability," and the convergence time is independent of the initial state of the robotic arm dynamics model, making it suitable for repetitive tasks and industrial scenarios with different initial conditions. This invention ensures state safety through a three-level logarithmic barrier Lyapunov function, providing a stable operating space for fuzzy approximation and fixed-time control; fuzzy approximation compensates for uncertainties, reducing the interference of robotic arm dynamics model errors on the fixed-time control law and improving convergence accuracy; fixed-time control ensures rapid response, enabling the robotic arm dynamics model to converge quickly within safety constraints, avoiding constraint violations caused by slow response.

[0229] like Figure 2 As shown, the actual trajectories of joints 1 and 2 are very close to the desired trajectories, the positional errors are always within the constraints, and they can stabilize in a very short time, conforming to fixed-time control, indicating that the invention is effective. Figure 3 As shown, the speed error between joint 1 and joint 2 is controlled within the constraint boundaries, and the control input is reasonable, indicating that the present invention is feasible in practical applications. Figure 4 As shown, the change in the weight norm of the fuzzy system indicates that the fuzzy logic system can effectively approximate unknown nonlinear dynamics. The comparison of position errors before and after the perturbation shows that this invention has good robustness. Figure 5 As shown, fuzzy estimation can approximate the actual dynamic terms well, and can verify the approximation capability of fuzzy logic systems.

[0230] This invention significantly improves dynamic performance, meeting the high-precision requirements of semiconductor manufacturing, medical applications, and other fields. Its stability and robustness are enhanced; the global signal is bounded, eliminating divergence risk. Through rigorous proof using a third-order logarithmic barrier Lyapunov function, it is demonstrated that all signals in the closed-loop system are uniformly bounded, maintaining stability even under parameter perturbations such as a 20% change in the inertial matrix or sudden disturbances such as external impacts. Parameter tuning is simpler, lowering the engineering threshold, and the correlation between control gain and convergence time is clear. The number of hidden layer neurons in the fuzzy logic system can be directly determined by the number of fuzzy rules, avoiding the contradiction of "too many neurons increasing time consumption, too few neurons reducing accuracy" in adaptive neural networks. The invention greatly improves engineering practicality. By adjusting fuzzy weights in real time using adaptive laws, it can adapt to scenarios such as load switching and changing operating conditions, reducing the number of on-site debugging sessions and maintenance cycles. This invention can also be applied to high-precision manufacturing, suitable for semiconductor wafer handling, automotive welding robotic arms, and other scenarios, preventing product scrapping or equipment damage due to exceeding limits.

[0231] In the specific implementation of the above embodiments, the technical features can be combined in any non-contradictory way. For the sake of brevity, not all possible combinations of the above technical features are described. However, as long as the combination of these technical features is not contradictory, it should be considered to be within the scope of this specification.

[0232] Obviously, the above embodiments of the present invention are merely examples for clearly illustrating the present invention, and are not intended to limit the implementation of the present invention. Those skilled in the art can make other variations or modifications based on the above description. It is neither necessary nor possible to exhaustively describe all embodiments here. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the scope of protection of the claims of the present invention.

Claims

1. An adaptive fuzzy fixed-time control method for an output-constrained robotic arm, characterized in that, Includes the following steps: S1. Construct a dynamic model of the robotic arm; wherein, the dynamic model of the robotic arm is: （1） In the formula, These represent the joint position, velocity, and acceleration vectors, respectively. Represents a symmetric positive definite inertia matrix; Represent the Coriolis and centrifugal matrices; Represents the gravitational torque vector; Represents the Jacobian matrix; Represents an unknown but bounded external disturbance; Indicates control input; Use joint position and speed separately , The dynamic model of the robotic arm can be obtained as follows: （2） In the formula, This represents the input to the robotic arm's dynamics model. This represents the output of the robotic arm dynamics model; In the robotic arm dynamics model: S2. Set the constraints for the output state of the robotic arm's dynamic model; S3. Define the tracking error and virtual control law of the robotic 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. Based on the constraints of output state and fixed-time control, design an actual control law as the input to the robotic arm's dynamics model, and construct an adaptive law; wherein, the actual control law... for: （5） In the formula, This represents the symmetric inertia matrix of a single joint in the robotic arm's dynamics model. This represents a positive, adjustable control gain constant. Indicates error State constraints, Represents the fuzzy weight estimation vector. Represents a fuzzy basis function vector; S5. Construct a three-level logarithmic obstacle Lyapunov function, and combine the three-level logarithmic obstacle Lyapunov function with the virtual control law, the actual control law, and the adaptive law step by step. Then, according to the actual fixed-time control lemma, ensure that the output state of the robotic arm dynamics model does not exceed the preset safety boundary.

2. The adaptive fuzzy fixed-time control method based on an output-constrained robotic arm according to claim 1, characterized in that, In step S2, the constraints include: In the formula, , , which represents the physical constraints on the state of the robotic arm's dynamic model.

3. The adaptive fuzzy fixed-time control method based on an output-constrained robotic arm according to claim 2, characterized in that, In step S3, the tracking error includes: （3） In the formula, , , The expected tracking trajectory representing the joint position. This represents a virtual control law.

4. The adaptive fuzzy fixed-time control method based on an output-constrained robotic arm according to claim 3, characterized in that, 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.

5. The adaptive fuzzy fixed-time control method based on an output-constrained robotic arm according to claim 4, characterized in that, 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 formula, Symbolic functions: when hour, ;when hour, ;when hour, .

6. The adaptive fuzzy fixed-time control method based on an output-constrained robotic arm according to claim 5, characterized in that, 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.

7. The adaptive fuzzy fixed-time control method based on an output-constrained robotic arm according to claim 6, characterized in that, 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 into 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), we get: （20） Where, vector dot product It can be expanded into element-wise summation: 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 And from Young's inequality, we can obtain: （21） in, For positive integers, Indicates the minimum approximation error; In inequality (21) arrive Summation, and combining it with the definition of vector norm. We can obtain the vector dot product inequality: （22） in: , Similarly, we can obtain , Calculation; Substituting formulas (20), (21), and (22) into formula (19), we get: （23） According to the lemma of the logarithmic inequality, we can obtain: （24） Rearranging formula (24) yields: （25） in: In step S54, according to the practical fixed-time control lemma and formula (25), the robotic arm dynamics model satisfies practical fixed-time stability, and the error signal will converge to a compact set within a fixed time independent of the initial state of the robotic arm dynamics model: Construct an upper bound for the explicit convergence time: Among them, it satisfies the following conditions: fixed-time stability, all signals are bounded, and... ,as well as ;therefore, For all It is also bounded, meaning that the state of the robotic arm's dynamic model is restricted to a predefined region.

8. The adaptive fuzzy fixed-time control method based on an output-constrained robotic arm according to claim 7, characterized in that, The lemma of the logarithmic inequality states that for a given constant vector... If its components satisfy Then the following inequality holds for each component: The actual fixed-time control lemma is as follows: Setting up a nonlinear system ; in, Let be a potentially discontinuous vector field, if the nonlinear system is globally finite-time stable, and its convergence time function is... It is bounded, meaning there exists a positive constant. , making If the nonlinear system is stable, then it is called a globally fixed-time stable system. If there exists a positive definite and radially unbounded function , so that: Among them, the positive constants are: Then the nonlinear system is practically fixed-time stable, and the solution will converge to the following residual set in a fixed time: Furthermore, the convergence time is independent of the initial state of the nonlinear system, and the upper bound of the convergence time is: in, Describe a scalar and satisfy .

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