A method and system for trajectory tracking of a robot arm based on a predetermined sliding set

Abstract

The application discloses a kind of mechanical arm trajectory tracking sliding mode control method and system based on predetermined sliding set, method includes: the motion equation of the dynamic model of the representation mechanical arm is established;According to the motion equation, select the angular displacement and angular velocity of the end joint of mechanical arm as state vector, establish the state space model of mechanical arm;According to the desired tracking trajectory, obtain the tracking error vector of mechanical arm, combined with the state space model of mechanical arm, establish the dynamic equation of mechanical arm tracking error;According to the tracking error information of mechanical arm, select sliding variable;Combined with sliding variable, establish adaptive sliding mode tracking control law, and combine Lyapunov function stability analysis method to constrain the sliding variable of the end joint tracking error of mechanical arm;According to Lyapunov function direct method, the synthesis of the stabilizing control of the constrained sliding variable, the application can ensure that the end joint tracking system of mechanical arm does not lose predetermined sliding mode, to improve the robust performance such as anti-interference, fault tolerance.

Description

TECHNICAL FIELD

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

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

[0003] Nowadays, there are many effective methods for sliding mode tracking control of mechanical arms. For example, the invention patent with the patent publication number CN118181297A discloses a mechanical arm tracking control method based on fixed-time fast terminal sliding mode and proportional integral, which adjusts the parameter setting according to the actual scene through the system convergence time, so that the convergence time is not subject to the initial state of the system, overcomes the singularity problem of terminal sliding mode in mechanical arm tracking control, and greatly improves the tracking rate and accuracy. The invention patent with the patent publication number CN112904728A discloses a mechanical arm sliding mode tracking control method based on improved reaching law, which effectively combines various reaching law technologies to overcome the chattering and slow convergence problems in the sliding mode control of two-joint serial mechanical arms, ensuring the robustness of the system. The invention patent with the patent publication number CN116566259A discloses a mechanical arm sliding mode active disturbance rejection control method for motor direct drive, which integrates the sliding mode control method into the active disturbance rejection control framework, optimizes the extended state observer and nonlinear error feedback control law in the active disturbance rejection control, retains the original disturbance rejection performance and simplifies the parameter setting, and improves the response speed and robustness of the system.

[0004] In summary, the mechanical arm tracking control technology based on sliding mode aims to improve the system convergence rate and accuracy, which has many important advantages. Such results are also beneficial to improving production efficiency, reducing production cost and improving product quality in engineering. Especially in the industrial environment with a large amount of interference, nonlinearity, and nonlinearity and coupling of many joints of the mechanical arm, the sliding mode control aiming to improve the tracking control performance plays a key role in the actual mechanical arm industrial process.

[0005] Although the existing sliding mode control is well applied in the robot tracking task, there are still some problems. Considering a large number of uncertain factors in the industrial environment, such as disturbance, time delay, fault, etc., due to their time-varying nature and difficulty in accurate prediction, the traditional sliding mode controller is difficult to always suppress its negative impact on the system, resulting in loss of system sliding mode and performance loss. For example, when the sensor measurement noise covariance matrix induces norm increase, the signal magnitude also increases, and the gain of the originally designed sliding mode controller may not exceed the new noise signal, so the output trajectory of the robot end joint is difficult to maintain in the ideal balance state, reducing the system quality and even destroying the stability. This result damages the product quality and even causes safety accidents and property losses, and is more unfavorable for application in the robot production workshop with high tracking accuracy requirements. SUMMARY

[0006] The technical problem to be solved by the present application is to solve the problem of avoiding performance loss and safety hazards caused by sliding mode loss due to insufficient control gain for the robot

[0007] To solve the above technical problems, the present application provides the following technical solutions:

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

[0009] establishing a motion equation representing a robot dynamics model;

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

[0011] According to the expected tracking trajectory, the robot tracking error vector is obtained, and the robot state space model is combined to establish a robot tracking error dynamics equation;

[0012] According to the robot tracking error information, a 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 robot end joint tracking error is constrained by combining the Lyapunov function stability analysis method;

[0014] According to the Lyapunov function direct method, the constrained sliding variable is comprehensively controlled.

[0015] In an embodiment of the present application, the motion equation representing the robot dynamics model is as follows:

[0016] In the formula, q, respectively represent the angular displacement, angular velocity and angular acceleration of the joint at the end of the mechanical arm, τ represents the n-dimensional torque input vector, D0(q) represents the inertia matrix, represents the nominal matrix of centrifugal force and Coriolis force, G0(q) represents the n-dimensional gravity vector, and d represents the external disturbance vector.

[0017] In an embodiment of the present application, the state space model of the mechanical arm is as follows:

[0018] The angular displacement state vector is x1(t)=q, and the angular velocity state vector is x2(t)= x(t)=col{x1(t),x2(t)}; the torque input τ is the control input vector u; and the state space model of the mechanical arm is as follows:

[0019]

[0020] wherein,

[0021]

[0022] In the formula, q, respectively represent the angular displacement and angular velocity of the joint at the end of the mechanical arm, I represents the unit matrix, represents the inertia matrix in the space state, C0(x1(t),x2(t)) represents the nominal matrix of centrifugal force and Coriolis force in the space state, G0(x1(t)) represents the gravity vector in the space state, d represents the external disturbance vector, f(t) represents the bias fault vector, t represents time, and col represents the column vector.

[0023] In an embodiment of the present application, the desired tracking trajectory of the mechanical arm is as follows:

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

[0025] The mechanical arm tracking error dynamics equation is as follows:

[0026]

[0027] In the formula, x d (t) represents the n-dimensional target tracking trajectory vector of the mechanical arm, A d represents the Hurwitz stable matrix.

[0028] In an embodiment of the present application, the sliding variable is as follows: In the formula, K represents a to-be-determined parameter matrix.

[0029] In an embodiment of the present application, before constructing the adaptive sliding mode tracking control law, a sliding mode tracking controller is constructed, comprising:

[0030]

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

[0032]

[0033] wherein,

[0034]

[0035] wherein, A represents a system parameter matrix, k b (t) represents a time-varying performance function, s(t) represents a sliding variable, represents an estimation of the upper bound of the uncertain term Δη(x1(t), x2(t)), δ(t) represents a correction function, ι represents a normal number, represents an estimation of the upper bound of the actuator bias fault vector.

[0036] In an embodiment of the present application, the adaptive parameter is obtained according to the adaptive sliding mode tracking control law, and the

[0037]

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

[0039] In an embodiment of the present application, the sliding variable that restricts the end joint tracking error of the manipulator comprises:

[0040] According to the exponential decay Lyapunov function the signal of the sliding variable s(t) satisfies the boundary performance constraint: ||s(t)||<k b (t) at all times of the operation of the manipulator, under the adaptive sliding mode tracking control law, the error vector of the end joint tracking error system of the manipulator is limited in a preset sliding set, i.e. e(t)∈{e(t)∈R n :||s(t)||<k b (t)}.

[0041] In an embodiment of the present application, the synthesis of the stabilized control of the restricted sliding variable comprises:

[0042] The dynamic equation of the real sliding mode is obtained in combination with the sliding variable, the real sliding mode dynamic system of the obtained mechanical arm is synthesized, and a sufficient condition for the parameter matrix K to satisfy when stabilizing is obtained according to the Lyapunov direct method.

[0043] The dynamic equation of the real sliding mode is:

[0044]

[0045] The sufficient condition for the parameter matrix K to satisfy when stabilizing is:

[0046]

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

[0048] The application also provides a mechanical arm trajectory tracking sliding mode control system based on a predetermined sliding set, which comprises:

[0049] A motion equation module is configured to establish a motion equation representing a dynamic model of the mechanical arm.

[0050] A state space module is configured to select joint angular displacement and angular velocity of the mechanical arm as a state vector according to the motion equation, and establish a state space model of the mechanical arm.

[0051] An error dynamic equation module is configured to obtain a tracking error vector of the mechanical arm according to an expected tracking trajectory, and establish a dynamic equation of the tracking error of the mechanical arm in combination with the state space model of the mechanical arm.

[0052] A sliding variable module is configured to select a sliding variable according to the tracking error information of the mechanical arm.

[0053] A constrained sliding variable module is configured to establish an adaptive sliding mode tracking control law in combination with the sliding variable, and constrain the sliding variable of the tracking error of the joint of the mechanical arm in combination with a Lyapunov function stability analysis method.

[0054] A control synthesis module is configured to perform stabilizing control synthesis on the constrained sliding variable according to the Lyapunov function direct method.

[0055] Compared with the prior art, the application has the beneficial effects that the application can ensure that the end joint tracking system of the mechanical arm does not lose the predetermined sliding mode, thereby improving the robust performance such as anti-interference and fault tolerance. By introducing the performance function, the real-time boundary of the sliding variable is successfully constrained, so that it is not affected by the disturbance and the fault, and the risk of deviating from the ideal working characteristic point of the end joint of the mechanical arm during execution is effectively reduced. The preset performance function also introduces a certain degree of freedom to optimize the tracking performance, so that the application can provide a new solution for the sliding mode tracking control problem of the mechanical arm in an unreliable industrial environment.

[0056] The application adopts the reduced-order sliding mode scheme instead of the integral sliding mode, and the calculation burden of solving the controller parameter matrix is reduced due to the reduced order of the sliding mode model. In addition, the integral saturation and the negative effect of the time derivative of the integral sliding variable on the performance of the closed-loop system are avoided.

[0057] The application utilizes the fact that the tracking error system of the designated mechanical arm converges in the preset sliding set, effectively reduces the deterioration of the performance caused by the fault and the disturbance, and the sliding set to which the system converges in the traditional sliding mode control based on parameter identification is sensitive to the fault and the disturbance, and it is difficult to ensure good performance. BRIEF DESCRIPTION OF DRAWINGS

[0058] Figure 1 A flow chart of a mechanical arm trajectory tracking sliding mode control method based on a predetermined sliding set according to an embodiment of the application.

[0059] Figure 2 A schematic diagram of a sliding variable real-time boundary of a mechanical arm tracking error system according to an embodiment of the application.

[0060] Figure 3 A mechanical arm end joint tracking error trajectory schematic diagram under an adaptive sliding mode tracking control law according to an embodiment of the application.

[0061] Figure 4 A sliding variable boundary comparison diagram between the scheme proposed by the application and the traditional adaptive sliding mode tracking control scheme.

[0062] Figure 5 A mechanical arm end joint tracking error trajectory schematic diagram under the traditional adaptive sliding mode tracking control scheme.

[0063] Figure 6 A mechanical arm end joint tracking error trajectory schematic diagram under the traditional adaptive sliding mode tracking control scheme.

[0064] Figure 7 A mechanical arm trajectory tracking sliding mode control system block diagram based on a predetermined sliding set according to an embodiment of the application. DETAILED DESCRIPTION

[0065] For the person skilled in the art to understand the technical scheme of the present application, the technical scheme of the present application will be further described in conjunction with the drawings of the specification.

[0066] The terms "first", "second", "third", etc. are used only for the purpose of description and should not be understood as indicating or implying relative importance or a specific number of the technical features indicated. Therefore, the features defined with "first", "second", etc. can explicitly or implicitly include one or more of the features. In the description of the present application, the meaning of "a plurality of" is two or more, unless otherwise specifically limited.

[0067] Please refer to Figure 1 As shown in the drawings, the present application provides a mechanical arm trajectory tracking sliding mode control method based on a predetermined sliding set, comprising:

[0068] S10, a motion equation representing a mechanical arm dynamics model is established.

[0069] In an embodiment of the present application, the following constraints are made to the multi-degree-of-freedom mechanical arm: all mechanical structures of the mechanical arm are rigid. Correspondingly, the following Euler-Lagrange dynamics equation is established for a mechanical arm with n degrees of freedom:

[0070]

[0071] In the formula, q、 respectively represent three vectors of joint angular displacement, angular velocity and angular acceleration of the end of the mechanical arm, and q R n , R represents a real number, τ represents an n-dimensional torque input vector, D0(q) represents an inertia matrix, represents a centrifugal force and Coriolis force nominal matrix, G0(q) represents an n-dimensional gravity vector, and d represents an external disturbance vector.

[0072] S20, according to the motion equation, the joint angular displacement and angular velocity of the end of the mechanical arm are selected as the state vector, and the mechanical arm state space model is established.

[0073] In an embodiment of the present application, the joint angular displacement and angular velocity of the end of the mechanical arm are selected as the state vector:

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

[0075]

[0076] wherein,

[0077]

[0078] Considering that the mechanical arm device is inevitably worn out after long-term use, thus causing actuator failure. For example, the bias failure vector f(t) that causes the position offset of the actuator output, and the corresponding system dynamics (2) should be:

[0079]

[0080] In the formula, I represents a unit matrix, represents the inertia matrix in the space state, C0(x1(t), x2(t)) represents the nominal matrix of centrifugal force and coriolis force in the space state, G0(x1(t)) represents the gravity vector in the space state, d represents the external disturbance vector, f(t) represents the bias failure vector, t represents time, and col represents a column vector.

[0081] S30, according to the desired tracking trajectory, the tracking error vector of the mechanical arm is obtained, and the mechanical arm tracking error dynamics equation is established combined with the mechanical arm state space model.

[0082] In an embodiment of the present application, the desired tracking trajectory of the mechanical arm is:

[0083]

[0084] The tracking error vector of the mechanical arm is e(t) = x(t)-x d (t) ; according to the mechanical arm state space model, the mechanical arm tracking error dynamics equation is:

[0085]

[0086] In the formula, x d (t) represents the n-dimensional target tracking trajectory vector of the mechanical arm, A d represents the Hurwitz stable matrix, and A d ∈R n×n , R is a real number.

[0087] S40, according to the tracking error information of the mechanical arm, a sliding variable is selected.

[0088] In an embodiment of the present application, the selected sliding variable is:

[0089]

[0090] In the formula, K represents a to-be-determined parameter matrix, and K ∈ R n×n , represents the matrix of the tracking error vector e(t) of the mechanical arm.

[0091] S50, combined with the sliding variable, the adaptive sliding mode tracking control law is established, and the Lyapunov function stability analysis method is combined to constrain the sliding variable of the tracking error of the end joint of the robot arm.

[0092] In an embodiment of the application, an adaptive sliding mode tracking control law is constructed, comprising:

[0093]

[0094] Wherein, u1(t), u2(t), u3(t), u4(t), u5(t) are obtained by the following formulas (8)~(12):

[0095]

[0096]

[0097]

[0098] In the formula, A represents a system parameter matrix, k b (t) is a time-varying performance function, s(t) is a sliding variable, is an estimate of the upper bound of the uncertain term Δη(x1(t), x2(t)), δ(t) is a correction function, ι is a normal number, is 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 is an exponential function, and μ is a normal number, and δ(t) is a correction function to prevent singularity caused by zero denominator of the controller, which satisfies the energy constraint condition:

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

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

[0102] Wherein, formula (13), (14) are as follows:

[0103] ​

[0104] where γ1, γ2 are bounded and positive adaptive gains, and the initial values of adaptive parameters are finite values.

[0105] In this example, the dynamic equations of two adaptive parameter errors are obtained from equations (15) and (16):

[0106]

[0107]

[0108] where the adaptive parameter errors are According to the Lyapunov function with exponential decay the analysis is as follows,

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

[0110]

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

[0112]

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

[0114]

[0115] According to the basic inequality: where a is a finite-dimensional vector, and further it can be derived that:

[0116]

[0117] Combining the above results, applying the sliding mode controller equations (7)-(12) and the adaptive law error dynamic equations (14), (15), the final result can be obtained as:

[0118]

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

[0120] In summary, under the designed adaptive sliding mode tracking control law (equations (7)-(12)) and the error dynamic equations of adaptive law ((14), (15)), the error vector of the manipulator end joint tracking error system (5) can be limited in 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 manipulator end joint tracking error system (equation (5)) obtains a "real sliding mode" depending on the sliding variable. At the same time, it is known that the real-time boundary of the system sliding variable is constrained by the artificially set performance function k b (t) and does not depend on the fault f(t) and the disturbance Δη(x1(t), x2(t)).

[0123] Further, combined with the sliding variable (equation (6)), e2(t) = Ke1(t)-s(t), and the tracking error system dynamic (equation five) can be described as:

[0124]

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

[0126]

[0127] S60, according to the Lyapunov function direct method, the constrained sliding variable is stabilized and controlled.

[0128] In an embodiment of the present application, the sliding mode dynamic system (19) of the manipulator is synthesized, and according to the Lyapunov direct method, the sufficient condition for the parameter matrix K to be stabilized is:

[0129]

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

[0131] Specific example

[0132] The steps of designing the manipulator trajectory tracking sliding mode control under the influence of external disturbance and actuator offset fault include:

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

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

[0135]

[0136] wherein the physical meanings of the parameters are as follows: m1, m2, l1, and l2 represent the masses and lengths of the two connecting joints of the robot, and the angular displacement and angular velocity vectors of the end joint of the robot are respectively: 11 q 12 ] T ,

[0137] In step S20, the angular displacement and angular velocity of the end joint of the robot 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] Further, the expected tracking trajectory is introduced: The coefficient A d is the following Hurwitz matrix:

[0140]

[0141] wherein x d (t) is a 4-dimensional robot target trajectory vector, and its first-order derivative.

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

[0143]

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

[0145]

[0146] In the formula, 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 represented as:

[0149]

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

[0151]

[0152] In the formula

[0153]

[0154] In the formula, the performance function is set as μ=2, ι=0.5, δ(t)=0.5e -0.2t is a correction function, and the adaptive parameter is obtained by an adaptive update law:

[0155]

[0156]

[0157] Further, the dynamic equations of two adaptive parameter errors are obtained by formula (15) and formula (16)

[0158]

[0159] wherein γ1, γ2 are bounded and positive adaptive gains, let γ1=0.65, γ2=0.8, and the initial value of the adaptive parameter is set to zero, and the adaptive parameter errors are respectively

[0160] Simulation experiment

[0161] The specific embodiment method is applied to a simulation experiment, and the performance is compared with that of a traditional adaptive sliding mode tracking control. The mechanical arm system implemented by the application is subjected to an external disturbance signal 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 is shown that the method of the application can ensure that the end joint tracking error system of the mechanical arm obtains a preset sliding set: e(t)∈{e(t)∈R 4||s(t)|| < k b In this context, the joint tracking error signal of the manipulator can always maintain the sliding mode, the sliding variable is real-time bounded by the performance function with given exponential decay, and the corresponding Figure 3 The error trajectory curve also shows that the tracking system is ideal in steady-state error and superior in tracking performance even under faults and disturbances. This result can illustrate that the design of the method can ensure that the manipulator real-time tracks the target trajectory. Figure 4 and Figure 5 The Euclidean norm of the sliding variable and the error signal trajectory curve of the manipulator under the traditional adaptive sliding mode control scheme are shown, and due to the lack of performance constraints on the real-time boundary of the sliding variable, it is more sensitive under faults and disturbances, and the corresponding error signal also has a large overshoot and steady-state error. To further highlight the superiority of the present application in the performance of the manipulator 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 robustness, and provides a new solution for the problem of industrial manipulator tracking control.

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

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

Claims

1. A method for trajectory tracking of a robot manipulator based on a predetermined sliding set, characterized in that, Comprise: establishing a motion equation representing a dynamic model of a robot arm; selecting joint angular displacement and angular velocity of the robot arm as state vectors according to the motion equation, and establishing a state space model of the robot arm; obtaining a tracking error vector of the robot arm according to a desired tracking trajectory, and combining the state space model of the robot arm to establish a dynamic equation of tracking error of the robot arm; selecting a sliding variable according to tracking error information of the robot arm; combining the sliding variable to establish an adaptive sliding mode tracking control law, and combining Lyapunov function stability analysis method to constrain the sliding variable of the tracking error of the robot arm; comprehensive stabilization control of the constrained sliding variable according to Lyapunov function direct method. 2.The method of trajectory tracking sliding mode control for robot manipulators based on a predetermined sliding set, according to claim 1, characterized in that, The motion equation representing the dynamic model of the robot arm is as follows: ； In the formula, , , respectively represent the joint rotation angle displacement, angular velocity and angular acceleration of the end of the mechanical arm as three vectors, represent the torque input vector of dimension , represent the inertia matrix, represent the nominal matrix of centrifugal force and Coriolis force, represent the dimensional gravity vector, represent the external disturbance vector. 3.The method of trajectory tracking sliding mode control for robot manipulators 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: ; Angular velocity state vector: ; ; torque input is the control input vector ; the manipulator state space model is: ； wherein, ； ； ； In the formula, , are expressed as the joint rotation angle displacement and angular velocity of the end of the robot arm, respectively, is expressed as a unit matrix, is expressed as an inertia matrix in the space state, is expressed as a nominal matrix of centrifugal force and Coriolis force in the space state, is expressed as a gravity vector in the space state, is expressed as an external disturbance vector, is expressed as a bias fault vector, is expressed as time, is expressed as a column vector.

4. The method of trajectory tracking sliding mode control for robot manipulators based on a predetermined sliding set according to claim 3, characterized in that, The desired tracking trajectory of the robot arm is introduced as: ; The mechanical arm tracking error vector is: ; The dynamic equation of tracking error of the robot arm is as follows: ； In the formula, Represented as a robotic arm 3D target tracking trajectory vector, It is represented as the Herwitz stability matrix.

5. The method of trajectory tracking sliding mode control for robot manipulators based on a predetermined sliding set according to claim 4, characterized in that, The sliding variable is: ; in which, is represented as a matrix of parameters to be determined.

6. The method of trajectory tracking sliding mode control for robot manipulators based on a predetermined sliding set according to claim 5, wherein, Before constructing the adaptive sliding mode tracking control law, a sliding mode tracking controller is constructed, including: ； wherein , , , , is obtained by the following equation: ； wherein ; ； ； ； ； wherein is represented as a system parameter matrix, is represented as a time-varying performance function, is represented as a sliding variable, is represented as an estimate of an upper bound on the uncertainty term is represented as a correction function, is represented as a positive constant, is represented as an estimate of an upper bound on the actuator bias fault vector.

7. The method of trajectory tracking sliding mode control for robot manipulators based on a predetermined sliding set according to claim 6, characterized in that, According to the adaptive parameters , , an adaptive sliding mode tracking control law is obtained, and the , : ； ； wherein is expressed as a bounded and positive definite adaptive gain.

8. The method of trajectory tracking sliding mode control for robot manipulators based on a predetermined sliding set according to claim 7, characterized in that, constraining the sliding variable of the tracking error of the robot arm, including: According to an exponentially decaying lyapunov function , the signal of the sliding variable satisfies the boundary performance constraint at all times of the manipulator operation: Under the adaptive sliding mode tracking control law, the error vector of the manipulator end joint tracking error system is limited in a preset sliding set, that is .

9. The method of trajectory tracking sliding mode control for robot manipulators based on a predetermined sliding set according to claim 7, wherein, comprehensive stabilization control of the constrained sliding variable, including: In combination with the sliding variable, the dynamic equation of the real sliding mode is obtained, the real sliding mode dynamic system of the robot arm is synthesized, and the parameter matrix is obtained by the Lyapunov direct method sufficient conditions wherein, the dynamic equation of the real sliding mode is as follows: ； Steady time parameter matrix A sufficient condition for satisfaction is that: ； wherein , is a positive definite symmetric matrix, the decision variable is a matrix; the parameter matrix to be solved is: .

10. A mechanical arm trajectory tracking sliding mode control system based on a predetermined sliding set, characterized by, Comprise: a motion equation module for establishing a motion equation representing a dynamic model of a robot arm; a state space module for selecting joint angular displacement and angular velocity of the robot arm as state vectors according to the motion equation, and establishing a state space model of the robot arm; an error dynamic equation module for obtaining a tracking error vector of the robot arm according to a desired tracking trajectory, and combining the state space model of the robot arm to establish a dynamic equation of tracking error of the robot arm; a sliding variable module for selecting a sliding variable according to tracking error information of the robot arm; a constrained sliding variable module for combining the sliding variable to establish an adaptive sliding mode tracking control law, and combining Lyapunov function stability analysis method to constrain the sliding variable of the tracking error of the robot arm; a control comprehensive module for comprehensive stabilization control of the constrained sliding variable according to Lyapunov function direct method.

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