Mpc-smc hybrid control method for robot manipulator under uncertainty and constraints

Abstract

The application discloses a kind of uncertainty and under mechanical arm MPC-SMC compound control method, it is related to the field of trajectory tracking control of multi-degree-of-freedom mechanical arm.The method comprises the following steps: S1, using non-singular terminal sliding mode control method, make mechanical arm system track nominal system model movement;S2, design radial basis neural network compensator, compensate the uncertainty of mechanical arm system;S3, based on inverse dynamics method, the nominal system feedback linearization is decoupled into n SISO double integral system and is discretized;S4, based on decoupled and the double integral system of discretization, design model predictive controller, realize the tracking of time-varying trajectory.The application improves the dynamic performance and positioning accuracy of mechanical arm system affected by uncertainty and constraint;At the same time, it guarantees the robustness of mechanical arm to model uncertainty, load variation and external disturbance, also guarantees the accuracy of system trajectory tracking, realizes high-precision trajectory tracking control and the optimization of input torque.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of multi-freedom mechanical arm trajectory tracking control, in particular to a trajectory tracking control method of a mechanical arm under the influence of uncertainty and constraints. BACKGROUND

[0002] With the increasing application of mechanical arms in industrial manufacturing, polishing, orbital tasks, surgery and other complex tasks, higher requirements are put forward for high-precision trajectory tracking control of the mechanical arms. The basic task of mechanical arm trajectory tracking control is to control the motion trajectory of the end effector of the mechanical arm by applying a control torque to the joints of the mechanical arm to track the reference trajectory. The dynamic model of the mechanical arm is a highly nonlinear multi-input multi-output system. Although the mechanical arm controller designed using the precise dynamic model can obtain good performance, the mechanical arm working in the actual environment inevitably has uncertainty and disturbance, such as model parameter perturbation, load variation, time-varying friction force and unmodeled dynamics, which brings great challenges to the implementation of the mechanical arm controller.

[0003] For new application requirements, not only is there a higher requirement for the trajectory tracking accuracy of the mechanical arm, but there is also a demand for the optimality of the trajectory tracking control. Meanwhile, the mechanical arm is usually subject to various constraints, such as mechanical angle limitation, joint driving force limitation, etc.; without considering the constraints in the controller design, unpredictable responses or even failures can be caused.

[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, and has received wide attention in robot control due to its ability to effectively control complex systems and handle constraints. However, compared with classical control methods, the nonlinear model predictive control algorithm spends a lot of time on solving the optimization problem online, and the linear MPC based on feedback linearization cannot handle the uncertainty of the mechanical arm system. The introduction of robust model predictive control can handle uncertainty and disturbance, but will inevitably introduce conservatism and cause the control performance to deteriorate. Therefore, it is necessary to study a method that can handle the uncertainty and constraints of the mechanical arm while taking into account the control performance. SUMMARY

[0005] In view of the deficiencies in the prior art, the present application provides a MPC-SMC compound control method of a mechanical arm 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 mechanical arm system and ensure the satisfaction of system constraints, achieving high-precision trajectory tracking control and optimization of input torque.

[0006] The technical solutions adopted by the present application are as follows:

[0007] A mechanical arm MPC-SMC hybrid control method under uncertainty and constraint, comprising the following steps:

[0008] S1. Use the nonsingular terminal sliding mode control method to make the mechanical arm system track the nominal system model motion; specifically:

[0009] a1. Convert the mechanical arm dynamics model into a state space model, and equivalent the uncertainty as a lumped disturbance and ignore the disturbance term, to obtain the nominal system model of the mechanical arm.

[0010] The dynamics model of the n-degree-of-freedom revolute joint rigid mechanical arm system with disturbance is represented as follows:

[0011]

[0012] wherein, denote the position, velocity and acceleration of the n joints respectively, is an n-dimensional vector space; denotes a symmetric positive definite inertia matrix, denotes a centripetal Coriolis matrix, denotes a gravity vector, M0(q), G0(q) respectively denote the nominal values of the mechanical arm inertia matrix, the centripetal Coriolis matrix and the gravity vector, ΔM(q), ΔG(q) respectively denote the modeling inaccuracies and parameter perturbations of the mechanical arm inertia matrix, the centripetal Coriolis matrix and the gravity vector, is an n x 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] Let be the system state variable, be a 2n-dimensional vector space, be the system input variable, then the dynamics model of the mechanical arm system can be rewritten in the state space form:

[0014]

[0015] wherein,

[0016] denotes a lumped disturbance, including external disturbance and system parameter perturbation.

[0017] Ignoring the disturbance term in the mechanical arm system, the nominal system of the mechanical arm is:

[0018]

[0019] where the superscript ~ denotes the state of the nominal system, thus, and are the state variable of the nominal system and the input variable of the nominal system, respectively.

[0020] a2. Define the error dynamics equation of the robotic arm system and the nominal system;

[0021] Define the tracking error of the actual system tracking the nominal system and the derivatives of each order as Differencing the state space of the robotic arm system and the nominal system obtains the error dynamics equation of the robotic arm system and the nominal system containing uncertainties and external disturbances:

[0022]

[0023] a3. Define the nonsingular terminal fast sliding mode surface and solve the control law;

[0024] Based on the error dynamics equation of the robotic arm system and the nominal system, consider the following nonsingular fast terminal sliding mode surface:

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

[0026] where k1, k2 are two normal numbers, 1 < β < 2, and α > β, and sign(·) represents the sign function.

[0027] According to the necessary condition of the state trajectory moving on the sliding mode surface and without considering the expression of the equivalent control law u eq designed by the uncertainties and disturbances.

[0028] Let d = 0, and substitute the error dynamics equation into the derivative equation of the nonsingular fast terminal sliding mode surface to obtain the equivalent control law:

[0029]

[0030] In this step, based on the nonsingular terminal sliding mode control method, the actual robotic arm system is driven to follow the nominal system, so that the tracking error is 0.

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

[0032] the equivalent control law u eqThe system can be kept on the sliding surface in the dynamic known case. In order to meet the sliding condition when there is uncertainty and external disturbance, a compensation control law is needed to be designed, which is designed as:

[0033]

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

[0035] f = θ *T φ(z) + ε

[0036] where θ * is the 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 , and the compensation control law u au is obtained.

[0037] φ c (z) is an activation function, which is selected as a Gaussian function, and its form is as follows:

[0038]

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

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

[0041]

[0042] where the adaptive law of the RBF neural network weight is:

[0043]

[0044] where Γ is a positive definite diagonal matrix, is the estimation 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 manipulator system is compensated by the neural network compensator, a manipulator system model without uncertainty is obtained, so that the manipulator system tracks the nominal system model motion.

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

[0047] b1. The nonlinear MIMO (Multiple Input Multiple Output) nominal system model is linearized by using the inverse dynamics method of feedback linearization to obtain the feedback linearization control law:

[0048]

[0049] wherein, is an auxiliary control variable; at this time, the feedback linearization control law is the nominal system input variable so that the nominal system input variable is in a linear relationship with the nominal system state variable .

[0050] Substituting the feedback linearization control law into the nominal system of the robot arm, a constrained linear SISO double integral system is obtained:

[0051]

[0052] After applying the feedback linearization method, the original MIMO system is simplified into n constrained linear SISO double integral systems, each joint corresponding to a constrained linear SISO double integral system, wherein the state variable represents the i-th joint of the robot arm:

[0053]

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

[0055] The above equation of the constrained linear SISO double integral system is written in matrix form as:

[0056]

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

[0058] b2. The constrained linear SISO double integral system is discretized, and the sampling time sequence is taken as the sampling time constant, t k+1 -t k = T, T is the sampling time, to obtain the discretized system of the constrained linear SISO double integral system:

[0059]

[0060] where A d denotes the discretized system matrix of A d denotes the discretized input matrix of B

[0061] S4. Based on the decoupled and discretized double-integrator system, a model predictive controller (MPC) is designed to achieve the tracking of time-varying trajectories; specifically:

[0062] At time t k , the predicted state trajectory is defined as subject to the system dynamics, where N > 0 is the prediction horizon. At the state trajectory , at time t k , the predicted control input variable u

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

[0064]

[0065] where denotes the cost function, and the symbol denotes the vector two-norm weighted by the matrix W is the error of the nominal system tracking the desired trajectory, is the predicted control input variable; x req (t k+c ) is the desired trajectory; the error weight matrix is a positive semi-definite matrix, the 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 the terminal constraint set, such that

[0066] The terminal constraint set is defined as follows:

[0067]

[0068] where ρ is a positive real number, and for the desired trajectory x ref , is the equilibrium point of the system, and contains the origin as an interior point.

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

[0070]

[0071] K LQ is the control gain of an infinite horizon linear quadratic (LQ) controller with the same cost function; for the value of rho, such that there exists:

[0072]

[0073] terminal cost satisfies:

[0074]

[0075] In order to guarantee the stability of the controlled system, the matrix Pi 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 The first element of the sequence is applied in the first sampling time The control input of the nominal system of the manipulator is realized, that is, the high-precision trajectory tracking control and the optimization of the input torque are realized.

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

[0079] (1) The present application utilizes the robustness of the neural network non-singular terminal sliding mode control in the inner loop, adopts the radial basis neural network to approximate the uncertainty of the manipulator system, processes the uncertainty and external disturbance of the manipulator system, and the non-singular terminal sliding mode control guarantees that the actual manipulator follows the nominal system movement, and eliminates the uncertainty of the manipulator system.

[0080] (2) In the outer loop, based on the feedback linearization decoupling and discretization of the constrained linear SISO double integral system model, a model predictive controller (MPC) is designed, the prediction ability of the model predictive controller is utilized, the system control quantity is obtained under the condition that the controlled system satisfies the input constraint, and high-precision trajectory tracking control and optimization of the input torque are realized. BRIEF DESCRIPTION OF DRAWINGS

[0081] Figure 1 The method flowchart of the embodiment of the present application.

[0082] Figure 2 The control framework of the model predictive control-neural network non-singular terminal sliding mode control.

[0083] Figure 3 The structure diagram of the radial basis neural network compensator. Detailed Implementation

[0084] The specific embodiments of the present invention are described below to enable those skilled in the art to understand the present invention. However, it should be understood that the present invention is not limited to the scope of the specific embodiments. For those skilled in the art, various changes are obvious as long as they are within the spirit and scope of the present invention as defined and determined by the appended claims. All inventions utilizing the concept of the present invention are protected.

[0085] This embodiment presents a composite control method for a robotic arm under uncertainty and constraints, such as MPC-SMC. Figure 1 As shown, it includes the following steps:

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

[0087] a1. Transform the dynamic model of the robotic arm into a state-space model, and equate the uncertainty to a lumped disturbance and ignore the disturbance term to obtain the nominal system model of the robotic arm.

[0088] The dynamic model of a rigid robotic arm system with an n-DOF rotary joint and perturbation is represented as follows:

[0089]

[0090] in, These represent the position, velocity, and acceleration of n joints, respectively. It is an n-dimensional vector space; This represents a symmetric positive definite inertia matrix. Represents a centripetal Coriolis matrix. M0(q) represents the gravity vector. G0(q) represents the nominal values ​​of the robotic arm's inertia matrix, centripetal Coriolis matrix, and gravity vector, respectively, and ΔM(q), ΔG(q) represents the modeling inaccuracies and parameter perturbations of the robotic arm's inertia matrix, centripetal Coriolis matrix, and gravity vector, respectively. Given an n×n matrix space, Let be an n-dimensional vector space; v is the driving torque vector, τ d This is the external disturbance torque vector acting on the joint.

[0091] make For system state variables, Given a 2n-dimensional vector space, If we take the system input variables as input variables, the dynamic model of the robotic arm system can be rewritten in state-space form:

[0092]

[0093] where,

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

[0095] Neglecting the disturbance term in the manipulator system, the nominal system of the manipulator is obtained as:

[0096]

[0097] where the superscript ~ denotes the state of the nominal system, thus, and are the state variable of the nominal system and the input variable of the nominal system, respectively.

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

[0099] Define the tracking error of the actual system tracking the nominal system and the derivatives of each order as Differencing the state space of the manipulator system and the nominal system, the error dynamics equation of the manipulator system and the nominal system containing uncertainty and external disturbance is obtained as:

[0100]

[0101] a3. Define the nonsingular terminal fast sliding mode surface and solve the control law;

[0102] Based on the error dynamics equation of the manipulator system and the nominal system, consider the following nonsingular fast terminal sliding mode surface:

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

[0104] where k1, k2 are two normal numbers, 1 < β < 2, and α > β, and sign(·) denotes the sign function.

[0105] According to the necessary condition of the state trajectory moving on the sliding mode surface and without considering the expression of the equivalent control law u eq designed without considering the expression of the uncertainty and disturbance.

[0106] Let d = 0, and substitute the error dynamics equation into the derivative equation of the nonsingular fast terminal sliding mode surface to obtain the equivalent control law:

[0107]

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

[0109] S2. Design a radial basis function neural network compensator as shown in Figure 3 , to compensate for the uncertainty of the mechanical arm system; specifically:

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

[0111]

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

[0113] f = θ *T φ(z) + ε

[0114] where 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 , and the compensation control law u au is obtained.

[0115] φ c (z) is an activation function, which is selected as a Gaussian function, and has the following form:

[0116]

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

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

[0119]

[0120] where the adaptive law of the RBF neural network weight is:

[0121]

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

[0123] After the compensation of the uncertainty of the manipulator system by the neural network compensator, a manipulator system model without uncertainty is obtained, so that the manipulator system tracks the motion 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 (multi-input multi-output) nominal system model is linearized to obtain the feedback linearization control law:

[0126]

[0127] wherein, is an auxiliary control variable; at this time, the feedback linearization control law is the input variable of the nominal system so that the input variable of the nominal system and the state variable of the nominal system are in a linear relationship;

[0128] Substitute the feedback linearization control law into the nominal system of the manipulator to obtain a constrained linear SISO double integral system:

[0129]

[0130] After applying the feedback linearization method, the original MIMO system is simplified into n constrained linear SISO double integral systems, each joint corresponding to a constrained linear SISO double integral system, wherein the state variable represents the i-th joint of the manipulator:

[0131]

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

[0133] The above equation of the constrained linear SISO double integral system is written in matrix form as:

[0134]

[0135] wherein is a state vector, is the current control variable, wherein 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-integrator system with sampling time sequence is the sampling time constant, t k+1 -t k = T, T is the sampling time, to get the discrete system of the constrained linear SISO double-integrator system:

[0137]

[0138] where, A d represents the system matrix of A after discretization, B d represents the input matrix of B after discretization.

[0139] S4. Based on the decoupled and discretized double-integrator system, design a model predictive controller (MPC) to realize the tracking of time-varying trajectory; specifically:

[0140] At t k , define the predicted state trajectory as satisfies the system dynamics, where N > 0 is the prediction interval. At the state trajectory , at t k , use the MPC method to solve the finite interval optimization control problem, that is, the predicted control input variable

[0141] Establish the nominal system trajectory tracking control MPC problem as follows:

[0142]

[0143] where, represents the cost function, and the symbol represents the vector two-norm weighted by the matrix W, is the error of the nominal system tracking the expected trajectory, is the predicted control input variable; x ref (t k+c ) is the expected trajectory; the error weight matrix is a positive semi-definite matrix, the 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 the terminal constraint set, so that

[0144] The terminal constraint set is defined as follows:

[0145]

[0146] where ρ is a positive real number, for the desired trajectory x ref , is the equilibrium point of the system, and contains the origin as an interior point.

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

[0148]

[0149] K LQ is the control gain of an infinite-horizon linear-quadratic (LQ) controller with the same cost function; for the value of ρ, such that exists:

[0150]

[0151] The terminal cost satisfies:

[0152]

[0153] To guarantee 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 sequence of control input variables on the control interval is obtained as The first element of the sequence is applied at the first sampling time is the control input of the nominal system of the manipulator, that is, high-precision trajectory tracking control and optimization of the input torque are achieved.

[0156] As shown in Figure 2 , 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 robustness of the non-singular terminal sliding mode control is used to drive the actual manipulator system to follow the nominal system movement, and the radial basis neural network is used to compensate for the uncertainty of the manipulator system, thereby eliminating the uncertainty of the manipulator system; on the outer loop, the nominal system model is feedback linearized based on the inverse dynamics method, decoupled into n single-input single-output double-integral systems, and the predictive ability of the model predictive control is used to achieve high-precision tracking of the time-varying trajectory.

[0157] To sum up, the application improves the dynamic performance and positioning accuracy of the mechanical arm system affected by uncertainty and constraints, and reduces the steady-state tracking error through the above design; meanwhile, the application combines the robustness of the sliding mode control and the prediction ability of the model predictive control, which not only guarantees the robustness of the mechanical arm to model uncertainty, load change and external disturbance, but also guarantees the accuracy of system trajectory tracking, and realizes high-precision trajectory tracking control and input torque optimization.

Claims

1. A composite control method for a robotic arm under uncertainty and constraints, characterized in that, Includes the following steps: S1. Using a non-singular terminal sliding mode control method, the robotic arm system tracks the motion of the nominal system model; S2. Design a radial basis function neural network compensator to compensate for the uncertainties in 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 dual integral system, design a model predictive controller to track time-varying trajectories; In step S1, the non-singular terminal sliding mode control method is used to enable the robotic arm system to track the motion of the nominal system model, including the following steps: a1. Transform the dynamic model of the robotic arm into a state-space model, and convert the uncertainty into a lumped disturbance and ignore the disturbance term to obtain the nominal system model of the robotic arm; The dynamic model of a rigid robotic arm system with an n-DOF rotary joint and perturbation is represented as follows: in, These represent the position, velocity, and acceleration of n joints, respectively. It is an n-dimensional vector space; This represents a symmetric positive definite inertia matrix. Represents a centripetal Coriolis matrix. Represents the gravity vector. These represent the nominal values ​​of the robotic arm's inertia matrix, centripetal Coriolis matrix, and gravity vector, respectively. These represent the modeling inaccuracies and parameter perturbations of the robotic arm's inertia matrix, centripetal Coriolis matrix, and gravity vector, respectively. for The matrix space, It is an n-dimensional vector space; For the driving torque vector, This is the vector of external disturbance torque acting on the joint; make For system state variables, Given a 2n-dimensional vector space, If we take the system input variables as input variables, the dynamic model of the robotic arm system can be rewritten in state-space form: in, , This represents lumped disturbances, including external disturbances and system parameter perturbations; Ignoring interference terms in the robotic arm system, the nominal system of the robotic arm is obtained as follows: Here, the superscript ~ indicates the nominal system state, therefore... and These are the state variables and input variables of the nominal system, respectively. a2. Define the error dynamics equations for the robotic arm system and the nominal system; Define the tracking error of the actual system tracking the nominal system and its derivatives as follows: , , Subtracting the state space of the robotic arm system from that of the nominal system yields the error dynamic equations between the robotic arm system and the nominal system, which include uncertainties and external disturbances: a3. Define a fast sliding surface with non-singular terminals and solve for the control law; Based on the error dynamics equations of the robotic arm system and the nominal system, a non-singular fast end-effector sliding surface is defined: in, , Two positive numbers, ,and , Represents a symbolic function; make , Substituting the error dynamics equation into the derivative equation of the non-singular fast terminal sliding surface, we obtain the equivalent control law: In this step, the actual robotic arm 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. In step S2, a radial basis function neural network compensator is designed to compensate for the uncertainties of the robotic arm system, including the following steps: The design compensation control law is as follows: RBF neural networks approximate arbitrary nonlinear uncertain functions as follows: in, It is a constant ideal weight matrix. The modeling error of the RBF neural network is the input vector. , Let the function Approximating the uncertain function That is, the compensation control law was obtained. ; For the activation function, we choose the Gaussian function, which has the following form: in, and The first The center and width of each core unit; Combining the equivalent control law and the compensation control law, the sliding mode control law for the non-singular terminal of the neural network is obtained as follows: The adaptive law for the weights of the RBF neural network is as follows: in, It is a positive definite diagonal matrix. For the estimation of ideal weights, , Used to provide robustness to the estimation error of RBF neural networks; After the uncertainty of the robotic arm system is compensated by the neural network compensator, a robotic arm system model without uncertainty is obtained, which enables the robotic arm system to track the motion of the nominal system model. 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 linearized control law: in, As an auxiliary control variable; at this time, the feedback linearized control law Input variables for the nominal system This makes the nominal system input variables With nominal system state variables It is a linear relationship; Linearize the feedback control law Substituting the nominal system of the robotic arm, we obtain 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, with each joint corresponding to a constrained linear SISO double integral system, where the state variables... , Indicates the first robotic arm One joint: in, i Indicates the first Auxiliary control variables for each joint; The constrained linear SISO double integral system is written in matrix form: in For state vectors, It is the current control variable, where It is a compact set containing the origin, and The system state matrix, The input matrix of the system; b2. Discretize the constrained linear SISO double integral system and take the sampling time series. The sampling time constant, , Given the sampling time, we obtain the discretized system of the constrained linear SISO double integral system: Among them, A d Let A represent the system matrix after discretization, and B represent the system matrix after discretization. d This represents the input matrix after B is discretized.

2. The MPC-SMC composite control method for a robotic arm under uncertainty and constraints as described in claim 1, characterized in that, In step S4, based on the decoupled and discretized dual-integral system, a model predictive controller is designed to track the time-varying trajectory, including the following steps: exist At time t, the predicted state trajectory is defined as Satisfy system dynamics, where For the prediction interval; The nominal system trajectory tracking control MPC problem is established as follows: in, Represents the cost function, symbol Representation matrix Weighted vector L2 norm, The error of the nominal system in tracking the desired trajectory, For predicting control input variables; The desired trajectory; error weight matrix Given a positive semi-definite matrix, the input weight matrix is... The terminal weight matrix is ​​a positive definite matrix. It is a positive definite matrix; For the input constraint set, It is a set of state constraints. For the terminal constraint set, such that ; Terminal constraint set The definition is as follows: in, For positive real numbers, the expected trajectory , The equilibrium point of the system, and Including the origin as an interior point; To define the terminal constraint set and terminal penalty, an auxiliary control law is introduced. : It is the control gain of an infinite-interval linear quadratic controller with the same cost function; for the value , making exist: Terminal cost satisfy: To ensure the stability of the controlled system, the matrix Here are the solutions to the following Riccati equation: By solving the nominal system trajectory tracking control MPC problem, the sequence of control input variables over the control interval is obtained as follows: Apply the first element of the sequence of control input variables during the first sampling time. It serves as the control input for the nominal system of the robotic arm, enabling high-precision trajectory tracking control and optimization of input torque.

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