Mechanical arm MPC-SMC composite control method under uncertainty and constraint

Abstract

The invention discloses a mechanical arm MPC-SMC compound control method under uncertainty and constraint, and relates to the field of multi-degree-of-freedom mechanical arm trajectory tracking control. The method comprises the following steps: S1, enabling a mechanical arm system to track the motion of a nominal system model by using a nonsingular terminal sliding mode control method; s2, a radial basis function neural network compensator is designed, and the uncertainty of a mechanical arm system is compensated; s3, based on an inverse dynamics method, feedback linearization decoupling is carried out on the nominal system to form n SISO double-integral systems, and discretization is carried out on the n SISO double-integral systems; and S4, on the basis of the decoupled and discretized double-integral system, designing a model prediction controller, and realizing the tracking of the time-varying trajectory. According to the method, the dynamic performance and the positioning precision of a mechanical arm system influenced by uncertainty and constraint are improved; and meanwhile, the robustness of the mechanical arm to model uncertainty, load change and external disturbance is guaranteed, the system trajectory tracking precision is further guaranteed, and high-precision trajectory tracking control and input torque optimization are achieved.

Description

Technical Field

[0001] The present invention relates to the field of trajectory tracking control of a multi-degree-of-freedom robot arm, and in particular to a trajectory tracking control method of a robot arm under the influence of uncertainty and constraints. Background Art

[0002] With the increasing application of robotic arms in complex tasks such as industrial manufacturing, grinding, orbital tasks, and surgery, higher requirements are placed on the high-precision trajectory tracking control of robotic arms. The basic task of trajectory tracking control of robotic arms is to control the motion trajectory of the end effector of the robotic arm by applying control torque to the robotic arm joints to track the reference trajectory. The dynamic model of the robotic arm is a highly nonlinear multi-input and multi-output system. Although the robotic arm controller designed using an accurate dynamic model can achieve good performance, the robotic arm working in the actual environment inevitably has uncertainties and interferences, such as model parameter perturbations, load changes, time-varying friction, and unmodeled dynamics, which brings great challenges to the implementation of the robotic arm controller.

[0003] New application requirements not only place higher demands on the trajectory tracking accuracy of the robot arm, but also on the optimality of trajectory tracking control. At the same time, the robot arm is usually subject to various constraints, such as mechanical angle restrictions, joint driving force restrictions, etc. Failure to consider the constraints in the controller design may lead to unpredictable responses or even failures.

[0004] For the above requirements, model predictive control is a suitable and effective method to solve this type of problem. Model predictive control (MPC) is an optimal control strategy widely used in industrial applications. It has received widespread attention in robot control because it can effectively control complex systems and handle constraints. However, since the robot system is a multi-input and multi-output nonlinear system, compared with the classical control method, the nonlinear model predictive control algorithm will take a lot of time to solve the optimization problem online, and the linear MPC based on feedback linearization cannot handle the uncertainty of the robot system. The introduction of robust model predictive control can handle uncertainty and disturbances, but it will inevitably introduce conservatism and lead to poor control performance. Therefore, it is necessary to study a method that can handle the uncertainty and constraints of the robot while taking into account the control performance. Summary of the invention

[0005] In view of the shortcomings of the prior art, the present invention provides a MPC-SMC composite control method for a manipulator under uncertainty and constraints. The method is a hierarchical multi-loop control scheme based on the combination of model predictive control (MPC) and neural network non-singular terminal sliding mode control (NNSMC), which can handle the uncertainty and disturbance of the manipulator system, ensure the satisfaction of system constraints, and achieve high-precision trajectory tracking control and optimization of input torque.

[0006] The technical solution adopted by the present invention is as follows:

[0007] A MPC-SMC composite control method for a manipulator under uncertainty and constraints comprises the following steps:

[0008] S1. Using a non-singular terminal sliding mode control method, the manipulator system tracks the nominal system model motion; specifically:

[0009] a1. The dynamic model of the robot arm is converted into a state space model, and the uncertainty is equivalent to the lumped disturbance and the interference term is ignored to obtain the nominal system model of the robot arm.

[0010] The dynamic model of the n-DOF rotation joint rigid manipulator system with disturbance is expressed as follows:

[0011]

[0012] in, Represent the position, velocity and acceleration of n joints respectively, is an n-dimensional vector space; represents the symmetric positive definite inertia matrix, represents the centripetal Coriolis matrix, represents the gravity vector, M0(q), G0(q) represents the nominal values ​​of the manipulator inertia matrix, centripetal Corilio matrix and gravity vector, ΔM(q), ΔG(q) represents the modeling inaccuracy and parameter perturbation of the manipulator inertia matrix, centripetal Corilio matrix and gravity vector, respectively. is an n×n matrix space, is an n-dimensional vector space; τ is the driving torque vector, τ d is the external disturbance torque vector acting on the joint.

[0013] make is the system state variable, is a 2n-dimensional vector space, As the system input variables, the dynamic model of the robotic arm system can be rewritten in state space form:

[0014]

[0015] in,

[0016] represents the lumped disturbance, including external disturbance and system parameter perturbation.

[0017] Ignoring the interference term in the robot system, the nominal system of the robot is obtained as:

[0018]

[0019] Here, the superscript ~ indicates the state of the nominal system, so, and are the state variables of the nominal system and the input variables of the nominal system respectively.

[0020] a2. Define the error dynamics equation between the manipulator system and the nominal system;

[0021] The tracking error of the actual system tracking the nominal system and the derivatives of each order are defined as The error dynamics equations of the manipulator system and the nominal system including uncertainty and external disturbance are obtained by subtracting the state space of the manipulator system from the nominal system:

[0022]

[0023] a3. Define the non-singular terminal fast sliding surface and solve the control law;

[0024] Based on the error dynamics equation between the manipulator system and the nominal system, the following non-singular fast terminal sliding surface is considered:

[0025] s=e1+k1|e1| α sign(e1)+k2|e2| β sign(e2)

[0026] Wherein, k1 and k2 are two positive constants, 1<β<2, and α>β, and sign(·) represents the sign function.

[0027] Necessary conditions for motion on the sliding surface according to the state trajectory And the equivalent control law u is designed without considering the uncertainty and disturbance expression eq ;

[0028] make d=0, substitute the error dynamics equation into the derivative equation of the non-singular fast terminal sliding surface, and obtain the equivalent control law:

[0029]

[0030] In this step, the actual manipulator system is driven to follow the nominal system motion based on the non-singular terminal sliding mode control method, so that the tracking error is 0.

[0031] S2. Design a radial basis neural network compensator to compensate for the uncertainty of the robotic arm system; specifically:

[0032] Equivalent control law u eqThe system can be kept on the sliding surface when the dynamics are known. In order to meet the sliding conditions when there are uncertainties and external disturbances, a compensation control law needs to be designed. The compensation control law is designed as follows:

[0033]

[0034] The RBF neural network approximates any nonlinear uncertain function as follows:

[0035] f = θ *T φ(z)+ε

[0036] Among them, θ * is a constant ideal weight matrix, ε is the modeling error of the RBF neural network, and the input vector φ(z)=[φ1(z),φ2(z),…φ m (z)] T ; Let the function f approximate the uncertain function That is, the compensation control law u is obtained au ;

[0037] φ c (z) is the activation function, which is a Gaussian function in the following form:

[0038]

[0039] in, and σ c are the center and width of the cth nuclear unit respectively.

[0040] Combining the equivalent control law and the compensation control law, the neural network non-singular terminal sliding mode control law is:

[0041]

[0042] The adaptive law of the RBF neural network weights is:

[0043]

[0044] Where Γ is a positive definite diagonal matrix, is an estimate of the ideal weight, k>0, and ksign(s) is used to provide robustness to the estimation error of the RBF neural network.

[0045] After the uncertainty of the robot system is compensated by the neural network compensator, a robot system model without uncertainty is obtained, so that the robot system tracks the movement of the nominal system model.

[0046] S3. Based on the inverse dynamics method, the nominal system feedback linearization is decoupled into n SISO (single input single output) double-integral systems; specifically:

[0047] b1. Using the inverse dynamics method of feedback linearization, the nonlinear MIMO (multiple-input multiple-output) nominal system model is linearized to obtain the feedback linearization control law:

[0048]

[0049] in, is the auxiliary control variable; at this time, the feedback linearization control law is the nominal system input variable Make the nominal system input variable With the nominal system state variable It is a linear relationship;

[0050] Feedback linearization control law Substituting into the nominal system of the robot arm, we get the constrained linear SISO double-integral system:

[0051]

[0052] After applying the feedback linearization method, the original MIMO system is simplified to n constrained linear SISO double-integral systems, each joint corresponds to a constrained linear SISO double-integral system, where the state variable Represents the i-th joint of the robotic arm:

[0053]

[0054] in, represents the auxiliary control variable of the i-th joint;

[0055] Write the above constrained linear SISO double integral system in matrix form:

[0056]

[0057] in is the state vector, is the current control variable, where is a compact set containing the origin, and is the system state matrix, is the input matrix of the system.

[0058] b2. Discretize the constrained linear SISO double-integral system and take the sampling time series is the sampling time constant, t k+1 -t k =T, T is the sampling time, and the discretized system of the constrained linear SISO double-integral system is obtained:

[0059]

[0060] Among them, A d represents the system matrix after A is discretized, B d Represents the input matrix after discretization of B.

[0061] S4. Based on the decoupled and discretized double-integral system, a model predictive controller (MPC) is designed to track the time-varying trajectory; specifically:

[0062] In t k At this moment, the predicted state trajectory is defined as Satisfy system dynamics, where N>0 is the prediction interval. Next, at t k At this moment, by using the MPC method, the predictive control input variable can be obtained by solving the finite interval optimization control problem.

[0063] The nominal system trajectory tracking control MPC problem is established as follows:

[0064]

[0065] in, represents the cost function, symbol represents the vector bi-norm of the weighted matrix W, is the error of the nominal system tracking the desired trajectory, Input variables for predictive control; x req (t k+c ) is the expected trajectory; the error weight matrix is a semi-positive definite matrix, input weight matrix is a positive definite matrix, the terminal weight matrix is a positive definite matrix; is the input constraint set, is the state constraint set, is a terminal constraint set, such that

[0066] Terminal Constraint Set The definition is as follows:

[0067]

[0068] Among them, ρ is a positive real number, for the expected trajectory x ref , is the equilibrium point of the system, and Contains the origin as an interior point.

[0069] In order to define the terminal constraint set and terminal penalty, an auxiliary control law K is introduced au :

[0070]

[0071] K LQ is the control gain of an infinite interval linear quadratic (LQ) controller with the same cost function; for values ​​of ρ, such that exist:

[0072]

[0073] Terminal cost satisfy:

[0074]

[0075] In order to ensure the stability of the controlled system, the matrix Π is the solution of the following Riccati equation:

[0076]

[0077] By solving the nominal system trajectory tracking control MPC problem, the control input variable sequence on the control interval is obtained as follows: Applies the first element of this sequence at the first sampling time It is the control input of the nominal system of the robot arm, which realizes high-precision trajectory tracking control and optimization of input torque.

[0078] The beneficial effects of the present invention are as follows:

[0079] (1) The present invention utilizes the robust characteristics of the non-singular terminal sliding mode control of the neural network in the inner loop, adopts the radial basis neural network to approximate the uncertainty of the robot system, and processes the uncertainty and external disturbance of the robot system. The non-singular terminal sliding mode control ensures that the actual robot follows the nominal system movement, eliminating the uncertainty of the robot system.

[0080] (2) In the outer loop, a model predictive controller (MPC) is designed based on the constrained linear SISO double-integral system model that is decoupled and discretized by feedback linearization. The prediction capability of the model predictive controller is utilized to obtain the system control quantity while ensuring that the controlled system meets the input constraints, thereby achieving high-precision trajectory tracking control and input torque optimization. BRIEF DESCRIPTION OF THE DRAWINGS

[0081] Figure 1 The figure is a flow chart of a method according to an embodiment of the present invention.

[0082] Figure 2 It is a control framework for model predictive control-neural network non-singular terminal sliding mode control.

[0083] Figure 3 This is the structural diagram of the radial basis neural network compensator. DETAILED DESCRIPTION

[0084] The specific implementation modes of the present invention are described below so that those skilled in the art can understand the present invention. However, it should be clear that the present invention is not limited to the scope of the specific implementation modes. For those of ordinary skill in the art, as long as various changes are within the spirit and scope of the present invention as defined and determined by the attached claims, these changes are obvious, and all inventions and creations utilizing the concept of the present invention are protected.

[0085] A MPC-SMC composite control method for a manipulator under uncertainty and constraints in this embodiment is as follows: Figure 1 As shown, the following steps are included:

[0086] S1. Using a non-singular terminal sliding mode control method, the manipulator system tracks the nominal system model motion; specifically:

[0087] a1. The dynamic model of the robot arm is converted into a state space model, and the uncertainty is equivalent to the lumped disturbance and the interference term is ignored to obtain the nominal system model of the robot arm.

[0088] The dynamic model of the n-DOF rotation joint rigid manipulator system with disturbance is expressed as follows:

[0089]

[0090] in, Represent the position, velocity and acceleration of n joints respectively, is an n-dimensional vector space; represents the symmetric positive definite inertia matrix, represents the centripetal Coriolis matrix, represents the gravity vector, M0(q), G0(q) represents the nominal values ​​of the manipulator inertia matrix, centripetal Corilio matrix and gravity vector, ΔM(q), ΔG(q) represents the modeling inaccuracy and parameter perturbation of the manipulator inertia matrix, centripetal Corilio matrix and gravity vector, respectively. is an n×n matrix space, is an n-dimensional vector space; v is the driving torque vector, τ d is the external disturbance torque vector acting on the joint.

[0091] make is the system state variable, is a 2n-dimensional vector space, As the system input variables, the dynamic model of the robotic arm system can be rewritten in state space form:

[0092]

[0093] in,

[0094] represents the lumped disturbance, including external disturbance and system parameter perturbation.

[0095] Ignoring the interference term in the robot system, the nominal system of the robot is obtained as:

[0096]

[0097] Here, the superscript ~ indicates the state of the nominal system, so, and are the state variables of the nominal system and the input variables of the nominal system respectively.

[0098] a2. Define the error dynamics equation between the manipulator system and the nominal system;

[0099] The tracking error of the actual system tracking the nominal system and the derivatives of each order are defined as The error dynamics equations of the manipulator system and the nominal system including uncertainty and external disturbance are obtained by subtracting the state space of the manipulator system from the nominal system:

[0100]

[0101] a3. Define the non-singular terminal fast sliding surface and solve the control law;

[0102] Based on the error dynamics equation between the manipulator system and the nominal system, the following non-singular fast terminal sliding surface is considered:

[0103] s=e1+k1|e1| α sign(e1)+k2|e2| β sign(e2)

[0104] Wherein, k1 and k2 are two positive constants, 1<β<2, and α>β, and sign(·) represents the sign function.

[0105] Necessary conditions for motion on the sliding surface according to the state trajectory And the equivalent control law u is designed without considering the uncertainty and disturbance expression eq ;

[0106] make d=0, substitute the error dynamics equation into the derivative equation of the non-singular fast terminal sliding surface, and obtain the equivalent control law:

[0107]

[0108] In this step, the actual manipulator system is driven to follow the nominal system motion based on the non-singular terminal sliding mode control method, so that the tracking error is 0.

[0109] S2. Design as Figure 3 The radial basis neural network compensator shown compensates for the uncertainty of the robotic arm system; specifically:

[0110] Equivalent control law u eq The system can be kept on the sliding surface when the dynamics are known. In order to meet the sliding conditions when there are uncertainties and external disturbances, a compensation control law needs to be designed. The compensation control law is designed as follows:

[0111]

[0112] The RBF neural network approximates any nonlinear uncertain function as follows:

[0113] f = θ *T φ(z)+ε

[0114] Among them, θ * is a constant ideal weight matrix, ε is the modeling error of the RBF neural network, and the input vector φ(z)=[φ1(z),φ2(z),…φ m (z)] T ; Let the function f approximate the uncertain function That is, the compensation control law u is obtained au ;

[0115] φ c (z) is the activation function, which is a Gaussian function in the following form:

[0116]

[0117] in, and σ c are the center and width of the cth nuclear unit respectively.

[0118] Combining the equivalent control law and the compensation control law, the neural network non-singular terminal sliding mode control law is:

[0119]

[0120] The adaptive law of the RBF neural network weights is:

[0121]

[0122] Where Γ is a positive definite diagonal matrix, is an estimate of the ideal weight, k>0, and ksign(s) is used to provide robustness to the estimation error of the RBF neural network.

[0123] After the uncertainty of the robot system is compensated by the neural network compensator, a robot system model without uncertainty is obtained, so that the robot system tracks the movement of the nominal system model.

[0124] S3. Based on the inverse dynamics method, the nominal system feedback linearization is decoupled into n SISO (single input single output) double-integral systems; specifically:

[0125] b1. Using the inverse dynamics method of feedback linearization, the nonlinear MIMO (multiple-input multiple-output) nominal system model is linearized to obtain the feedback linearization control law:

[0126]

[0127] in, is the auxiliary control variable; at this time, the feedback linearization control law is the nominal system input variable Make the nominal system input variable With the nominal system state variable It is a linear relationship;

[0128] Feedback linearization control law Substituting into the nominal system of the robot arm, we get the constrained linear SISO double-integral system:

[0129]

[0130] After applying the feedback linearization method, the original MIMO system is simplified to n constrained linear SISO double-integral systems, each joint corresponds to a constrained linear SISO double-integral system, where the state variable Represents the i-th joint of the robotic arm:

[0131]

[0132] in, represents the auxiliary control variable of the i-th joint;

[0133] Write the above constrained linear SISO double integral system in matrix form:

[0134]

[0135] in is the state vector, is the current control variable, where is a compact set containing the origin, and is the system state matrix, is the input matrix of the system.

[0136] b2. Discretize the constrained linear SISO double-integral system and take the sampling time series is the sampling time constant, t k+1 -t k =T, T is the sampling time, and the discretized system of the constrained linear SISO double-integral system is obtained:

[0137]

[0138] Among them, A d represents the system matrix after A is discretized, B d Represents the input matrix after discretization of B.

[0139] S4. Based on the decoupled and discretized double-integral system, a model predictive controller (MPC) is designed to track the time-varying trajectory; specifically:

[0140] In t k At this moment, the predicted state trajectory is defined as Satisfy system dynamics, where N>0 is the prediction interval. Next, at t k At this moment, by using the MPC method, the predictive control input variable can be obtained by solving the finite interval optimization control problem.

[0141] The nominal system trajectory tracking control MPC problem is established as follows:

[0142]

[0143] in, represents the cost function, symbol represents the vector bi-norm of the weighted matrix W, is the error of the nominal system tracking the desired trajectory, Input variables for predictive control; x ref (t k+c ) is the expected trajectory; the error weight matrix is a semi-positive definite matrix, input weight matrix is a positive definite matrix, the terminal weight matrix is a positive definite matrix; is the input constraint set, is the state constraint set, is a terminal constraint set, such that

[0144] Terminal Constraint Set The definition is as follows:

[0145]

[0146] Among them, ρ is a positive real number, for the expected trajectory x ref , is the equilibrium point of the system, and Contains the origin as an interior point.

[0147] In order to define the terminal constraint set and terminal penalty, an auxiliary control law k is introduced au :

[0148]

[0149] K LQ is the control gain of an infinite interval linear quadratic (LQ) controller with the same cost function; for values ​​of ρ, such that exist:

[0150]

[0151] Terminal cost satisfy:

[0152]

[0153] In order to ensure the stability of the controlled system, the matrix ∏ is the solution of the following Riccati equation:

[0154]

[0155] By solving the nominal system trajectory tracking control MPC problem, the control input variable sequence on the control interval is obtained as follows: Applies the first element of this sequence at the first sampling time It is the control input of the nominal system of the robot arm, which realizes high-precision trajectory tracking control and optimization of input torque.

[0156] like Figure 2 As shown, in this embodiment, the method adopts a model predictive control-neural network non-singular terminal sliding mode control robust multi-loop hierarchical control method; on the inner loop, the robust characteristics of the non-singular terminal sliding mode control are utilized to drive the actual robotic arm system to follow the movement of the nominal system, and the radial basis neural network is utilized to compensate for the uncertainty of the robotic arm system, thereby eliminating the uncertainty of the robotic arm system; on the outer loop, the nominal system model is feedback linearized based on the inverse dynamics method, decoupled into n single-input and single-output dual-integral systems, and the predictive capability of the model predictive control is utilized to achieve high-precision tracking of the time-varying trajectory.

[0157] In summary, through the above design, the present invention improves the dynamic performance and positioning accuracy of the robotic arm system affected by uncertainty and constraints, and reduces the steady-state tracking error; at the same time, the present invention combines the robustness of sliding mode control with the predictive ability of model predictive control, which not only ensures the robustness of the robotic arm to model uncertainty, load changes and external disturbances, but also ensures the accuracy of system trajectory tracking, and at the same time realizes high-precision trajectory tracking control and optimization of input torque.

Claims

1. A MPC-SMC composite control method for a manipulator under uncertainty and constraints, characterized in that: The steps include: S1. Using the non-singular terminal sliding mode control method, the manipulator system tracks the nominal system model motion; S2. Design a radial basis neural network compensator to compensate for the uncertainty of the robotic arm system; S3. Based on the inverse dynamics method, the nominal system feedback linearization is decoupled into n SISO double-integral systems and discretized; S4. Based on the decoupled and discretized double-integral system, a model predictive controller is designed to track the time-varying trajectory.

2. The MPC-SMC composite control method for a manipulator under uncertainty and constraints as claimed in claim 1, characterized in that: In step S1, a non-singular terminal sliding mode control method is used to make the manipulator system track the motion of the nominal system model, including the following steps: a1. The robot dynamics model is transformed into a state space model, and the uncertainty is equivalent to a lumped disturbance and the interference term is ignored to obtain the nominal system model of the robot; The dynamic model of the n-DOF rotation joint rigid manipulator system with disturbance is expressed as follows: in, Represent the position, velocity and acceleration of n joints respectively, is an n-dimensional vector space; represents the symmetric positive definite inertia matrix, represents the centripetal Coriolis matrix, represents the gravity vector, M0(q), G0(q) represents the nominal values ​​of the manipulator inertia matrix, centripetal Corilio matrix and gravity vector, ΔM(q), ΔG(q) represents the modeling inaccuracy and parameter perturbation of the manipulator inertia matrix, centripetal Corilio matrix and gravity vector, respectively. is an n×n matrix space, is an n-dimensional vector space; τ is the driving torque vector, τ d is the external disturbance torque vector acting on the joint; make is the system state variable, is a 2n-dimensional vector space, As the system input variables, the dynamic model of the robotic arm system is rewritten in state space form: in, represents the lumped disturbance, including external disturbance and system parameter perturbation; Ignoring the interference term in the robot system, the nominal system of the robot is obtained as: Here, the superscript ~ indicates the state of the nominal system, so, and are the state variables of the nominal system and the input variables of the nominal system respectively; a2. Define the error dynamics equation between the manipulator system and the nominal system; The tracking error of the actual system tracking the nominal system and the derivatives of each order are defined as The error dynamics equations of the manipulator system and the nominal system containing uncertainty and external disturbances are obtained by subtracting the state space of the manipulator system from the nominal system: a3. Define the non-singular terminal fast sliding surface and solve the control law; Based on the error dynamics equation between the manipulator system and the nominal system, a non-singular fast terminal sliding surface is defined: s=e1+k1|e1| α sign(e1)+k2|e2| β sign(e2) Wherein, k1 and k2 are two positive constants, 1<β<2, and α>β, and sign(·) represents the sign function; make d=0, substitute the error dynamics equation into the derivative equation of the non-singular fast terminal sliding surface, and obtain the equivalent control law: In this step, the actual manipulator system is driven to follow the nominal system motion based on the non-singular terminal sliding mode control method, so that the tracking error is 0.

3. The MPC-SMC composite control method for a manipulator under uncertainty and constraints as claimed in claim 2, characterized in that: In step S2, a radial basis function neural network compensator is designed to compensate for the uncertainty of the robot system, including the following steps: The designed compensation control law is: The RBF neural network approximates any nonlinear uncertain function as follows: f=θ *T φ(z)+ε Among them, θ * is a constant ideal weight matrix, ε is the modeling error of the RBF neural network, and the input vector φ(z)=[φ1(z),φ2(z),…φ m (z)] T ; Let the function f approximate the uncertain function That is, the compensation control law u is obtained au ; φ c (z) is the activation function, which is a Gaussian function in the following form: in, and σ c are the center and width of the cth nuclear unit, respectively; Combining the equivalent control law and the compensation control law, the neural network non-singular terminal sliding mode control law is obtained: Among them, the adaptive law of RBF neural network weights is: Where Γ is a positive definite diagonal matrix, is the estimate of the ideal weight, k>0, ksign(s) is used to provide robustness to the estimation error of the RBF neural network; After the uncertainty of the robot system is compensated by the neural network compensator, a robot system model without uncertainty is obtained, so that the robot system tracks the movement of the nominal system model.

4. The MPC-SMC composite control method for a manipulator under uncertainty and constraints as claimed in claim 3, characterized in that: In step S3, based on the inverse dynamics method, the nominal system feedback linearization is decoupled into n SISO double-integral systems and discretized, including the following steps: b1. Using the inverse dynamics method of feedback linearization, the nonlinear MIMO nominal system model is linearized to obtain the feedback linearization control law: in, is the auxiliary control variable; at this time, the feedback linearization control law Enter variables for the nominal system Make the nominal system input variable With the nominal system state variable It is a linear relationship; Feedback linearization control law Substituting into the nominal system of the robot arm, we get the constrained linear SISO double-integral system: After applying the feedback linearization method, the nonlinear MIMO nominal system is simplified to n constrained linear SISO double-integral systems, one for each joint, where the state variables Represents the i-th joint of the robotic arm: in, represents the auxiliary control variable of the i-th joint; Write the constrained linear SISO double-integral system in matrix form: in is the state vector, is the current control variable, where is a compact set containing the origin, and is the system state matrix, is the input matrix of the system; b2. Discretize the constrained linear SISO double-integral system and take the sampling time series is the sampling time constant, t k+1 -t k =T, T is the sampling time, and the discretized system of the constrained linear SISO double-integral system is obtained: Among them, A d represents the system matrix after discretization of A, B d Represents the input matrix after discretization of B.

5. The MPC-SMC composite control method for a manipulator under uncertainty and constraints as claimed in claim 4, characterized in that: In step S4, based on the decoupled and discretized double-integral system, a model predictive controller is designed to track the time-varying trajectory, including the following steps: In t k At this moment, the predicted state trajectory is defined as Satisfy system dynamics, where N>0 is the prediction interval; The nominal system trajectory tracking control MPC problem is established as follows: in, represents the cost function, symbol represents the vector bi-norm of the weighted matrix W, is the error of the nominal system tracking the desired trajectory, Input variables for predictive control; x ref (t k+c ) is the expected trajectory; the error weight matrix is a semi-positive definite matrix, input weight matrix is a positive definite matrix, the terminal weight matrix is a positive definite matrix; is the input constraint set, is the state constraint set, is a terminal constraint set, such that Terminal Constraint Set The definition is as follows: Among them, ρ is a positive real number, for the expected trajectory x ref , is the equilibrium point of the system, and Contains the origin as an interior point; In order to define the terminal constraint set and terminal penalty, an auxiliary control law k is introduced au : K LQ is the control gain of the infinite interval linear quadratic controller with the same cost function; for values ​​ρ, such that exist: Terminal price satisfy: In order to ensure the stability of the controlled system, the matrix Π is the solution of the following Riccati equation: By solving the nominal system trajectory tracking control MPC problem, the control input variable sequence on the control interval is obtained as follows: Applies the first element of the control input variable sequence at the first sampling time It is the control input of the nominal system of the robot arm to achieve high-precision trajectory tracking control and optimization of input torque.

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