Mechanical arm trajectory tracking sliding mode control method and system based on predetermined sliding set

Abstract

The invention discloses a mechanical arm trajectory tracking sliding mode control method and system based on a predetermined sliding set. The method comprises the steps that a motion equation representing a mechanical arm dynamic model is established; according to the motion equation, mechanical arm tail end joint angular displacement and angular velocity are selected as state vectors, and a mechanical arm state space model is established; according to the expected tracking trajectory, a mechanical arm tracking error vector is obtained, and a mechanical arm tracking error kinetic equation is established in combination with the mechanical arm state space model; selecting a sliding variable according to the tracking error information of the mechanical arm; a self-adaptive sliding mode tracking control law is established in combination with the sliding variable, and the sliding variable of the tracking error of the tail end joint of the mechanical arm is restrained in combination with a Lyapunov function stability analysis method; according to a Lyapunov function direct method, stabilization control integration is carried out on the constrained sliding variables, it can be ensured that a mechanical arm tail end joint tracking system cannot lose a preset sliding mode, and therefore the anti-interference performance, the fault tolerance performance and other robust performance are improved.

Description

Technical Field

[0001] The present invention relates to the technical field of robot arm trajectory tracking control, and in particular to a robot arm trajectory tracking sliding mode control method and system based on a predetermined sliding set. Background Art

[0002] In today's increasingly complex and integrated industrial processes, robotic arms are an indispensable and important tool, involving many fields such as workshops, medical treatment, and detection. With the continuous enrichment and development of control theory, mature technologies have also improved the practical application of robotic arms, greatly improving efficiency and production capacity. Tracking control is the most common and basic operation process of robotic arms in industry, and it often has high requirements for various control performance indicators of the tracking process. Sliding mode control is widely used in the design of robotic arm tracking control due to its advantages such as simple design and strong robustness.

[0003] Nowadays, many effective methods have emerged for the sliding mode tracking control of robotic arms. For example, the invention patent with the patent publication number CN118181297A discloses a robotic arm tracking control method based on fixed-time fast terminal sliding mode and proportional integral. The parameter settings can be adjusted according to the actual scenario through the system convergence time, so that the convergence time is not restricted by the initial state of the system, and the singularity problem of the terminal sliding mode in the robotic arm tracking control is overcome while greatly improving the tracking rate and accuracy. The invention patent with the patent publication number CN112904728A discloses a robotic arm sliding mode tracking control method based on an improved reaching law. By effectively combining a variety of reaching law technologies, it overcomes the jitter and slow convergence problems existing in the sliding mode control of a two-joint series robotic arm, ensuring the robustness of the system. The invention patent with patent publication number CN116566259A discloses a sliding mode anti-disturbance control method for a motor-driven robotic arm. The sliding mode control method is integrated into the anti-disturbance control framework, and the extended state observer and nonlinear error feedback control law in the anti-disturbance control are optimized, thereby retaining the original anti-disturbance performance and simplifying parameter setting, thereby improving the response speed and robustness of the system.

[0004] In short, the sliding mode-based robot tracking control technology aims to improve the convergence rate and accuracy of the system and has played many important advantages. In engineering, such results are also conducive to improving production efficiency, reducing production costs and improving product quality. Especially in the context of industrial environments with a lot of interference, nonlinearity, and nonlinearity and coupling of many joints of the robot itself, sliding mode control aimed at improving tracking control performance plays a key role in the actual robot industrial process.

[0005] Although the existing sliding mode control has been well applied in the robot tracking task, there are still some problems. Considering the large number of uncertain factors in the industrial environment, such as interference, time lag, faults, etc., due to their time-varying nature and difficulty in being accurately predicted, it is difficult for the traditional sliding mode controller to always suppress their negative impact on the system, resulting in the loss of the system sliding mode and performance loss. For example, when the induced norm of the sensor measurement noise covariance matrix increases, its signal level also increases, and the gain of the originally designed sliding mode controller may not exceed the new noise signal, then the output trajectory of the end joint of the robot arm will be difficult to maintain in the ideal equilibrium state, reducing the system quality and even destroying the stability. This result damages product quality and even causes safety accidents and property losses, and is not conducive to its application in robot production workshops with high tracking accuracy requirements. Summary of the invention

[0006] The technical problem to be solved by the present invention is to solve the problem of how to avoid the performance loss and safety hazards caused by the loss of sliding mode due to insufficient control gain.

[0007] In order to solve the above technical problems, the present invention provides the following technical solutions:

[0008] A sliding mode control method for tracking a robot arm trajectory based on a predetermined sliding set, comprising:

[0009] Establish the equations of motion that characterize the dynamics model of the robotic arm;

[0010] According to the equation of motion, the angular displacement and angular velocity of the end joint of the robot arm are selected as the state vector to establish the state space model of the robot arm;

[0011] According to the expected tracking trajectory, the robot tracking error vector is obtained, and the robot tracking error dynamic equation is established in combination with the robot state space model;

[0012] According to the tracking error information of the robot arm, the sliding variable is selected;

[0013] Combined with the sliding variable, an adaptive sliding mode tracking control law is established, and the sliding variable of the tracking error of the end joint of the manipulator is constrained by combining the Lyapunov function stability analysis method;

[0014] Synthesis of stabilization control of the sliding variable after constraint based on the direct method of Lyapunov function.

[0015] In one embodiment of the present invention, the motion equation representing the dynamics model of the robot arm is the following formula:

[0016] In the formula, q, They are respectively represented by the angular displacement, angular velocity, and angular acceleration of the end joint of the robot arm. τ is represented by the n-dimensional torque input vector. D0(q) is represented by the inertia matrix. It is represented as the nominal matrix of centrifugal force and Coriolis force, G0(q) is represented as the n-dimensional gravity vector, and d is represented as the external interference vector.

[0017] In one embodiment of the present invention, the state space model of the robot arm is as follows:

[0018] Angular displacement state vector: x1(t)=q; angular velocity state vector: x(t) = col{x1(t), x2(t)}; the torque input τ is the control input vector u; then the state space model of the robot is:

[0019]

[0020] in,

[0021]

[0022] In the formula, q, They are respectively represented as the angular displacement and angular velocity of the end joint of the robot arm, and I is represented as the unit matrix. It is represented as the inertia matrix under the spatial state, C0(x1(t),x2(t)) is represented as the nominal matrix of centrifugal force and Coriolis force under the spatial state, G0(x1(t)) is represented as the gravity vector under the spatial state, d is represented as the external interference vector, f(t) is represented as the bias fault vector, t is represented as time, and col is represented as a column vector.

[0023] In one embodiment of the present invention, the desired tracking trajectory of the introduced robotic arm is:

[0024] The robot tracking error vector is: e(t) = x(t) - x d (t);

[0025] Then, the dynamic equation of the robot tracking error is:

[0026]

[0027] In the formula, x d (t) is represented by the robot arm n-dimensional target tracking trajectory vector, A d It is represented as a Hurwitz stability matrix.

[0028] In one embodiment of the present invention, the sliding variable is: Where K represents the parameter data matrix to be determined.

[0029] In one embodiment of the present invention, before constructing an adaptive sliding mode tracking control law, a sliding mode tracking controller is first constructed, including:

[0030]

[0031] Among them, u1(t), u2(t), u3(t), u4(t), and u5(t) are obtained by the following formula:

[0032]

[0033] in,

[0034]

[0035] In the formula, A represents the system parameter matrix, k b (t) is represented as a time-varying performance function, s(t) is represented as a sliding variable, It is expressed as an estimate of the upper bound of the uncertainty term Δη(x1(t),x2(t)), δ(t) is expressed as a correction function, ι is expressed as a positive constant, is expressed as an estimate of the upper bound of the actuator bias fault vector.

[0036] In one embodiment of the present invention, according to the adaptive parameter Get the adaptive sliding mode tracking control law and update

[0037]

[0038] Where γ1, γ2 represent bounded and positive definite adaptive gains.

[0039] In one embodiment of the present invention, the sliding variable constraining the tracking error of the end joint of the robot arm includes:

[0040] According to the exponentially decaying Lyapunov function The signal of the sliding variable s(t) satisfies the boundary performance constraint at all times when the robot is working: ||s(t)||<k b (t), under the adaptive sliding mode tracking control law, the error vector of the robot end joint tracking error system is restricted to a preset sliding set, that is, e(t)∈{e(t)∈R n :||s(t)||<k b (t)}.

[0041] In one embodiment of the present invention, the comprehensive stabilization control of the constrained sliding variable includes:

[0042] Combined with the sliding variables, the dynamic equations of the real sliding mode are obtained, the real sliding mode dynamic system obtained by the manipulator is synthesized, and the sufficient conditions for the parameter matrix K to be satisfied when it is stabilized are obtained according to the Lyapunov direct method;

[0043] Among them, the dynamic equation of the true sliding mode is:

[0044]

[0045] The sufficient condition that the parameter matrix K satisfies during stabilization is:

[0046]

[0047] In the formula, O and Q are positive symmetric matrices, and the decision variable H is an n×n matrix. The parameter matrix K to be determined is: K = O - 1 Q.

[0048] The present invention also provides a sliding mode control system for tracking a robot arm trajectory based on a predetermined sliding set, comprising:

[0049] The motion equation module is used to establish the motion equations that characterize the dynamics model of the robot arm;

[0050] The state space module is used to select the angular displacement and angular velocity of the end joint of the robot arm as the state vector according to the motion equation and establish the state space model of the robot arm;

[0051] The error dynamic equation module is used to obtain the robot tracking error vector according to the expected tracking trajectory, and establish the robot tracking error dynamic equation in combination with the robot state space model;

[0052] A sliding variable module is used to select sliding variables according to the tracking error information of the robot arm;

[0053] The constrained sliding variable module is used to combine the sliding variables, establish the adaptive sliding mode tracking control law, and combine the Lyapunov function stability analysis method to constrain the sliding variables of the tracking error of the end joint of the manipulator;

[0054] The control synthesis module is used to synthesize the stabilization control of the sliding variable after constraint according to the direct method of Lyapunov function.

[0055] Compared with the prior art, the beneficial effect of the present invention is that the present invention can ensure that the robot arm end joint tracking system will not lose the predetermined sliding mode, thereby improving the robust performance such as anti-interference and fault tolerance. By introducing the performance function to successfully constrain the real-time boundary of the sliding variable, it will not be affected by disturbances and failures, effectively reducing the risk of the robot arm end joint deviating from the ideal working feature point during execution, and the preset performance function also introduces a certain degree of freedom to optimize the tracking performance, so the present invention can provide a new solution to the sliding mode tracking control problem of the robot arm in an unreliable industrial environment.

[0056] The present invention adopts a reduced-order sliding mode solution instead of an integral sliding mode, considering that the sliding mode model order is reduced, and the computational burden of solving the controller parameter matrix is ​​reduced. In addition, the negative effects of integral saturation and the time derivative of the integral sliding variable on the performance of the closed-loop system are avoided.

[0057] The present invention utilizes the specified robot arm tracking error system to converge to a preset sliding set, which effectively reduces the deterioration of its performance caused by faults and interferences. In traditional sliding mode control based on parameter identification, the sliding set to which the system converges is more sensitive to faults and interferences, and it is difficult to ensure good performance. BRIEF DESCRIPTION OF THE DRAWINGS

[0058] Figure 1 The present invention is a flowchart of a sliding mode control method for tracking a robot arm trajectory based on a predetermined sliding set according to an embodiment of the present invention.

[0059] Figure 2 It is a schematic diagram of the real-time boundary of the sliding variable of the tracking error system of the constrained robot arm according to an embodiment of the present invention.

[0060] Figure 3 Schematic diagram of the tracking error trajectory of the end joint of the robot arm under the adaptive sliding mode tracking control law of an embodiment of the present invention.

[0061] Figure 4 This is a comparison diagram of the sliding variable boundaries under the scheme proposed in the present invention and the traditional adaptive sliding mode tracking control scheme.

[0062] Figure 5 Schematic diagram of the tracking error trajectory of the end joint of the robot under the traditional adaptive sliding mode tracking control scheme.

[0063] Figure 6 The figure is a schematic diagram of the tracking error trajectory of the end joint of the robot arm under the scheme proposed by the present invention and the traditional adaptive sliding mode tracking control scheme.

[0064] Figure 7 The present invention is a block diagram of a sliding mode control system for tracking a robot arm trajectory based on a predetermined sliding set according to an embodiment of the present invention. DETAILED DESCRIPTION

[0065] In order to facilitate those skilled in the art to understand the technical solution of the present invention, the technical solution of the present invention is further described in conjunction with the accompanying drawings of the specification.

[0066] The terms "first" and "second" are used for descriptive purposes only and should not be understood as indicating or implying relative importance or implicitly indicating the number of the indicated technical features. Thus, a feature defined as "first" or "second" may explicitly or implicitly include one or more of the features. In the description of this application, the meaning of "plurality" is two or more, unless otherwise clearly and specifically defined.

[0067] See also Figure 1 As shown, the present invention provides a sliding mode control method for tracking a robot arm trajectory based on a predetermined sliding set, comprising:

[0068] S10, establishing a motion equation to characterize the dynamics model of the robot arm.

[0069] In one embodiment of the present invention, the following constraints are imposed on the multi-DOF manipulator: all mechanical structures of the manipulator are rigid. Accordingly, the following Euler-Lagrangian dynamic equations are established for the manipulator with n degrees of freedom:

[0070]

[0071] In the formula, q, They are respectively represented by the angular displacement, angular velocity, and angular acceleration of the end joint of the robot arm, and q∈R n , R is represented by a real number, τ is represented by an n-dimensional torque input vector, D0(q) is represented by an inertia matrix, It is represented by the nominal matrix of centrifugal force and Coriolis force, G0(q) is represented by the n-dimensional gravity vector, and d is represented by the external interference vector.

[0072] S20, according to the motion equation, select the angular displacement and angular velocity of the end joint of the robot arm as the state vector, and establish the state space model of the robot arm.

[0073] In one embodiment of the present invention, the angular displacement and angular velocity of the end joint of the robotic arm are selected as the state vector:

[0074] Angular displacement state vector: x1(t)=q; angular velocity state vector: x(t) = col{x1(t), x2(t)}. The torque input τ is the control input vector u, and the robot dynamics model can be described by the following state space equation:

[0075]

[0076] in,

[0077]

[0078] Considering that the long-term use of the robot arm equipment will inevitably cause wear and tear, the resulting actuator failure. For example, the bias fault vector f(t) that causes the actuator output to shift position, the corresponding system dynamics (2) should be:

[0079]

[0080] In the formula, I represents the unit matrix, It is represented as the inertia matrix under the spatial state, C0(x1(t),x2(t)) is represented as the nominal matrix of centrifugal force and Coriolis force under the spatial state, G0(x1(t)) is represented as the gravity vector under the spatial state, d is represented as the external interference vector, f(t) is represented as the bias fault vector, t is represented as time, and col is represented as a column vector.

[0081] S30, obtaining a robot arm tracking error vector according to a desired tracking trajectory, and establishing a robot arm tracking error dynamics equation in combination with a robot arm state space model.

[0082] In one embodiment of the present invention, the desired tracking trajectory of the introduced robotic arm is:

[0083]

[0084] The robot tracking error vector is: e(t) = x(t) - x d (t); According to the state space model of the manipulator, the dynamic equation of the manipulator tracking error can be obtained as:

[0085]

[0086] In the formula, x d (t) is represented by the robot arm n-dimensional target tracking trajectory vector, A d is represented as a Hurwitz stability matrix, and A d ∈R n×n , R is a real number.

[0087] S40, selecting a sliding variable according to the robot arm tracking error information.

[0088] In one embodiment of the present invention, the sliding variable is selected as:

[0089]

[0090] In the formula, K represents the parameter matrix to be determined, and K∈R n×n , It is represented as a matrix of the robot tracking error vector e(t).

[0091] S50, combining the sliding variables, establish an adaptive sliding mode tracking control law, and combine the Lyapunov function stability analysis method to constrain the sliding variables of the robot end joint tracking error.

[0092] In one embodiment of the present invention, an adaptive sliding mode tracking control law is constructed, including:

[0093]

[0094] Among them, u1(t), u2(t), u3(t), u4(t), and u5(t) are obtained by the following formulas (8) to (12):

[0095]

[0096] in,

[0097]

[0098] In the formula, A represents the system parameter matrix, k b (t) is represented as a time-varying performance function, s(t) is represented as a sliding variable, It is expressed as an estimate of the upper bound of the uncertainty term Δη(x1(t),x2(t)), δ(t) is expressed as a correction function, ι is expressed as a positive constant, is expressed as an estimate of the upper bound of the actuator bias fault vector.

[0099] In this embodiment, k b (t) is a time-varying performance function, which can be selected as and e -μt It is expressed as an exponential function, μ is a positive constant, and δ(t) is a correction function to prevent the singularity caused by the controller denominator being zero and satisfy the energy constraint:

[0100] In this embodiment, Considering that Δη(x1(t), x2(t)) is a function of the bounded unknown disturbance vector d, it is itself also a bounded unknown term.

[0101] In this embodiment, when the exponential convergence barrier Lyapunov function stability analysis method constrains the sliding variable of the robot end joint tracking error, it can be updated according to formula (13): Update according to formula (14)

[0102] Among them, formulas (13) and (14) are as follows:

[0103]

[0104] Where γ1, γ2 represent bounded and positive adaptive gains, and the initial values ​​of the adaptive parameters are are all finite values.

[0105] In this embodiment, two adaptive parameter errors can be further obtained: The dynamic equation of is obtained by formula (15) and formula (16):

[0106]

[0107]

[0108] Among them, the adaptive parameter errors are According to the exponentially decaying Lyapunov function The analysis is as follows:

[0109] The energy function is selected as: Taking its derivative we get: According to the sliding variable (6), its time derivative is:

[0110]

[0111] Correspondingly, the energy function derivative can be redescribed as:

[0112]

[0113] Considering the boundary conditions of the disturbance vector and the fault vector: ||Δη(x1(t), x2(t))||≤k, It can be seen that:

[0114]

[0115] According to the basic inequality: Among them, a represents a finite-dimensional vector, which can be further deduced as follows:

[0116]

[0117] Combining the above results, applying the sliding mode controller formulas (7) to (12) and the adaptive law error dynamic formulas (14) and (15), we can finally deduce:

[0118]

[0119] According to the barrier Lyapunov function Property: When the function is bounded, the signal of the sliding variable s(t) satisfies the boundary performance constraint at all times when the robot is working: ||s(t)||<k b (t).

[0120] In summary, under the designed adaptive sliding mode tracking control law (Formulas (7) to (12)) and the adaptive law error dynamic formulas ((14), (15)), the error vector of the robot end joint tracking error system (5) can be restricted to a preset sliding set, that is:

[0121] e(t)∈{e(t)∈R n :||s(t)||<k b (t)}, (17);

[0122] The above results also show that under the designed sliding mode controller, the robot end joint tracking error system (Formula (5)) obtains a "real sliding mode" that depends on the sliding variable. At the same time, it can be seen that the real-time boundary of the system sliding variable is affected by the artificially set performance function k b (t), but is independent of the fault f(t) and the disturbance Δη(x1(t), x2(t)).

[0123] Furthermore, combined with the sliding variable (Formula (6)), we know that e2(t) = Ke1(t) - s(t), and the tracking error system dynamics (Formula 5) can be re-described as:

[0124]

[0125] In the formula, Correspondingly, the dynamic equation of the true sliding mode of the tracking error system (Formula (18)) can be obtained as:

[0126]

[0127] S60, synthesis of stabilization control of sliding variables after constraints according to the direct method of Lyapunov functions.

[0128] In one embodiment of the present invention, the sliding mode dynamic system (19) obtained by the robot arm is synthesized, and the sufficient condition that the parameter matrix K satisfies when the robot arm is stabilized can be obtained according to the Lyapunov direct method:

[0129]

[0130] In the formula, O and Q are positive symmetric matrices, and the decision variable H is an n×n matrix. The parameter matrix K to be determined is: K = O - 1 Q.

[0131] Specific example

[0132] The design steps of sliding mode control for robot trajectory tracking under the influence of external interference and actuator offset failure include:

[0133] The parameters in the motion equation of the robot dynamics model are set as follows:

[0134] In step S10, the parameters of the motion equation representing the dynamics model of the robot arm are set as follows:

[0135]

[0136] The physical meaning of each parameter is: m1, m2, l1, l2 represent the mass and length of the two joints of the robot arm, and the angular displacement and angular velocity vectors of the end joint of the robot arm are: q = [q 11 q 12 ] T ,

[0137] According to step S20, the angular displacement and angular velocity of the end joint of the robot arm are selected as the state vector: The torque input is the control input vector u, and the robot dynamics model can be described by the following state space equation

[0138]

[0139] Furthermore, the expected tracking trajectory is introduced: Coefficient A d is the following Hurwitz matrix:

[0140]

[0141] Among them, x d (t) is the 4-dimensional target trajectory vector of the robot arm, is its first-order derivative.

[0142] In step S30, let the robot tracking error vector be e(t)=x(t)-x d (t), according to the state space model of the manipulator, the dynamic equation of the tracking error is:

[0143]

[0144] Combined with the robot tracking error vector, according to S40, the sliding variable is selected:

[0145]

[0146] The parameter matrix K can be obtained by solving formula (19) using the linear matrix inequality toolbox of Matlab, and the solution is:

[0147]

[0148] The sliding mode tracking controller is expressed as:

[0149]

[0150] u1(t), u2(t), u3(t), u4(t), u5(t) are obtained by formulas (8)-(12):

[0151]

[0152] In the formula

[0153]

[0154] The performance function is set as μ=2. ι=0.5. δ(t)=0.5e -0.2t is a correction function. Adaptive parameters Obtained by the adaptive update law:

[0155]

[0156]

[0157] Furthermore, two adaptive parameter errors can be obtained: The dynamic equation of is obtained by formula (15) and formula (16):

[0158]

[0159] Among them, γ1, γ2 are bounded and positive adaptive gains, let γ1 = 0.65, γ2 = 0.8, the initial value of the adaptive parameter Set to zero, the adaptive parameter errors are

[0160] Simulation experiment

[0161] The specific embodiment method is applied to the simulation experiment, and the performance is compared with the traditional adaptive sliding mode tracking control. The external disturbance signal of the manipulator system implemented by the present invention is d(t)=col{-2.6sin(2t,-1.8cos(13.6t)}, and the actuator offset fault is:

[0162] f(t)=col{f1(t),f2(t)},

[0163]

[0164] Figure 2 It shows that the method of the present invention can ensure that the robot end joint tracking error system obtains a preset sliding set: e(t)∈{e(t)∈R 4:||s(t)||<k b (t)}, in this context, the robot joint tracking error signal can always maintain the sliding mode, and the real-time boundary of the sliding variable is constrained by a given exponential decay performance function, corresponding to Figure 3 The error trajectory curve also shows that the steady-state error of the tracking system is ideal and the tracking performance is excellent even under faults and interference. This result shows that the design of this method can ensure that the robot arm can track the target trajectory in real time. Figure 4 and Figure 5 The figure shows the Euclidean norm of the robot arm sliding variable and the error signal trajectory curve under the traditional adaptive sliding mode control scheme. Due to the lack of performance constraints on the real-time boundary of the sliding variable, it is more sensitive to faults and interferences, and the corresponding error signal also has large overshoot and steady-state errors. In order to further highlight the superiority of the present invention in the performance of the robot arm tracking error system, Figure 6 The integral comparison curve of the absolute error is given, and it can be seen that the tracking error under the proposed method is smaller and more stable. Therefore, the proposed method has good robust performance and provides a new solution to problems such as industrial robot arm tracking control.

[0165] It is obvious to those skilled in the art that the present invention is not limited to the details of the exemplary embodiments described above, and that the present invention can be implemented in other specific forms without departing from the spirit or essential features of the present invention. Therefore, the embodiments should be regarded as exemplary and non-limiting from any point of view, and the scope of the present invention is defined by the appended claims rather than the above description, and it is intended that all changes falling within the meaning and scope of the equivalent elements of the claims are included in the present invention, and any reference numerals in the claims should not be regarded as limiting the claims involved.

[0166] The above-described embodiments merely represent implementation methods of the invention. The protection scope of the present invention is not limited to the above-described embodiments. For those skilled in the art, several modifications and improvements may be made without departing from the concept of the present invention, which all belong to the protection scope of the present invention.

Claims

1. A sliding mode control method for tracking the trajectory of a robot arm based on a predetermined sliding set, characterized in that: include: Establish the equations of motion that characterize the dynamics model of the robotic arm; According to the equation of motion, the angular displacement and angular velocity of the end joint of the robot arm are selected as the state vector to establish the state space model of the robot arm; According to the expected tracking trajectory, the robot tracking error vector is obtained, and the robot tracking error dynamic equation is established in combination with the robot state space model; According to the tracking error information of the robot arm, the sliding variable is selected; Combined with the sliding variable, an adaptive sliding mode tracking control law is established, and the sliding variable of the tracking error of the end joint of the manipulator is constrained by combining the Lyapunov function stability analysis method; Synthesis of stabilization control of the sliding variable after constraint based on the direct method of Lyapunov function.

2. The sliding mode control method for tracking the trajectory of a manipulator based on a predetermined sliding set according to claim 1, characterized in that: The motion equation that characterizes the dynamics model of the robotic arm is as follows: In the formula, q, They are respectively represented by the angular displacement, angular velocity, and angular acceleration of the end joint of the robot arm. τ is represented by the n-dimensional torque input vector. D0(q) is represented by the inertia matrix. It is represented by the nominal matrix of centrifugal force and Coriolis force, G0(q) is represented by the n-dimensional gravity vector, and d is represented by the external interference vector.

3. The sliding mode control method for tracking the trajectory of a manipulator based on a predetermined sliding set according to claim 1, characterized in that: The state space model of the robot arm is as follows: Angular displacement state vector: x1(t)=q; angular velocity state vector: x(t) = col{x1(t), x2(t)}; the torque input τ is the control input vector u; then the state space model of the manipulator is: in, In the formula, q, They are respectively represented as the angular displacement and angular velocity of the end joint of the robot arm, and I is represented as the unit matrix. It is represented as the inertia matrix under the spatial state, C0(x1(t),x2(t)) is represented as the nominal matrix of centrifugal force and Coriolis force under the spatial state, G0(x1(t)) is represented as the gravity vector under the spatial state, d is represented as the external interference vector, f(t) is represented as the bias fault vector, t is represented as time, and col is represented as a column vector.

4. The sliding mode control method for tracking the trajectory of a manipulator based on a predetermined sliding set according to claim 3, characterized in that: The expected tracking trajectory of the introduced robotic arm is: The robot tracking error vector is: e(t) = x(t) - x d (t); Then, the dynamic equation of the robot tracking error is: In the formula, x d (t) is represented by the robot arm n-dimensional target tracking trajectory vector, A d It is represented as a Hurwitz stability matrix.

5. The sliding mode control method for tracking the trajectory of a manipulator based on a predetermined sliding set according to claim 4, characterized in that: The sliding variables are: Where K represents the parameter data matrix to be determined.

6. The sliding mode control method for tracking the trajectory of a manipulator based on a predetermined sliding set according to claim 5, characterized in that: Before building the adaptive sliding mode tracking control law, first build a sliding mode tracking controller, including: Among them, u1(t), u2(t), u3(t), u4(t), and u5(t) are obtained by the following formula: in, In the formula, A represents the system parameter matrix, k b (t) is represented as a time-varying performance function, s(t) is represented as a sliding variable, It is expressed as an estimate of the upper bound of the uncertainty term Δη(x1(t),x2(t)), δ(t) is expressed as a correction function, ι is expressed as a positive constant, is expressed as an estimate of the upper bound of the actuator bias fault vector.

7. The sliding mode control method for tracking the trajectory of a manipulator based on a predetermined sliding set according to claim 6, characterized in that: According to the adaptive parameters Get the adaptive sliding mode tracking control law and update Where γ1, γ2 represent bounded and positive definite adaptive gains.

8. The sliding mode control method for tracking the trajectory of a manipulator based on a predetermined sliding set according to claim 7, characterized in that: The sliding variables that constrain the tracking error of the end joint of the robot arm include: According to the exponentially decaying Lyapunov function The signal of the sliding variable s(t) satisfies the boundary performance constraint at all times when the robot is working: ||s(t)||<k b (t), under the adaptive sliding mode tracking control law, the error vector of the robot end joint tracking error system is restricted to a preset sliding set, that is, e(t)∈{e(t)∈R n :||s(t)|| <k b (t)}.

9. The sliding mode control method for tracking the trajectory of a manipulator based on a predetermined sliding set according to claim 7, characterized in that: Comprehensive stabilization control of the constrained sliding variables, including: Combined with the sliding variables, the dynamic equations of the real sliding mode are obtained, the real sliding mode dynamic system obtained by the manipulator is synthesized, and the sufficient conditions for the parameter matrix K to be satisfied when it is stabilized are obtained according to the Lyapunov direct method; Among them, the dynamic equation of the true sliding mode is: The sufficient condition that the parameter matrix K satisfies during stabilization is: In the formula, O and Q are positive symmetric matrices, and the decision variable H is an n×n matrix. The parameter matrix K to be determined is: K = O- 1 Q.

10. A sliding mode control system for tracking the trajectory of a manipulator based on a predetermined sliding set, characterized in that: include: The motion equation module is used to establish the motion equations that characterize the dynamics model of the robot arm; The state space module is used to select the angular displacement and angular velocity of the end joint of the robot arm as the state vector according to the motion equation and establish the state space model of the robot arm; The error dynamic equation module is used to obtain the robot tracking error vector according to the expected tracking trajectory, and establish the robot tracking error dynamic equation in combination with the robot state space model; A sliding variable module is used to select sliding variables according to the tracking error information of the robot arm; The constrained sliding variable module is used to combine the sliding variables, establish the adaptive sliding mode tracking control law, and combine the Lyapunov function stability analysis method to constrain the sliding variables of the tracking error of the end joint of the manipulator; The control synthesis module is used to synthesize the stabilization control of the sliding variable after constraint according to the direct method of Lyapunov function.

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