Robust model predictive control method for mechanical arm trajectory tracking control

Abstract

The invention discloses a robust model prediction control method for mechanical arm trajectory tracking control, and relates to the field of multi-degree-of-freedom mechanical arm trajectory tracking control. According to the method, disturbance and parameter perturbation of a mechanical arm system are converted into lumped disturbance, a mechanical arm system error model is established, and feedback linearization is conducted on an error system based on an inverse dynamics method; based on a mechanical arm system error model, a nominal system model prediction controller is designed, a terminal area, a terminal control law and corresponding terminal cost are designed, and the feasibility of rolling interval optimization in each time step is guaranteed. According to the method, the uncertainty and disturbance of the mechanical arm system can be processed, it is guaranteed that system constraints are met, and high-precision trajectory tracking control and input torque optimization are achieved.

Description

Technical Field

[0001] The invention relates to the field of multi-degree-of-freedom robot arm trajectory tracking control, and in particular to a robust model predictive control method for robot arm trajectory tracking. Background Art

[0002] With the widespread application of robotic arms in industrial autonomous manufacturing, orbital missions, surgery and other fields, the demand for high-precision control of robotic arms is becoming increasingly strong. A basic task of robotic arm control is to control the motion of the end effector to track the desired reference trajectory by applying control inputs to the robotic arm joints. Robotic arm dynamics is a multi-input and multi-output system with high nonlinearity. Although the robotic arm controller based on the accurate model can achieve good performance, disturbances are inevitable, which leads to the performance degradation of the robotic arm. In addition, the robotic arm is usually subject to a variety of constraints, such as mechanical limitations, safety limitations, and task limitations. Failure to consider the constraints in the control design may lead to unpredictable responses or even failures.

[0003] Model predictive control (MPC) is widely used in optimal control of industrial applications because it can effectively control complex systems and handle constraints. In particular, the MPC algorithm overcomes the shortcomings of the above methods. However, in the actual application of the robot system, there are inevitable model parameter perturbations, external disturbances, inaccurate modeling, etc. At the same time, due to the influence of the working environment, the robot will inevitably be constrained. Traditional model predictive control methods have high requirements on model accuracy. Inaccurate robot models may lead to poor control performance and even controller divergence. Therefore, it is necessary to study a robust model predictive control method that can handle robot uncertainty and constraints while taking into account control performance. Summary of the invention

[0004] In view of the shortcomings of the prior art, the present invention provides a robust model predictive control method for robot arm trajectory tracking. The method can handle the uncertainty and disturbance of the robot arm system, ensure the satisfaction of system constraints, and achieve high-precision trajectory tracking control and input torque optimization.

[0005] The technical solution adopted by the present invention is as follows:

[0006] A robust model predictive control method for robot arm trajectory tracking comprises the following steps:

[0007] S1. Establish a state-space model of the manipulator under the influence of disturbance, model the constraints of the manipulator, and establish a state-space model of the tracking error.

[0008] a1. Convert the robot arm dynamics model into a state space model and equate the uncertainty to a lumped disturbance.

[0009] The dynamic model of the n-DOF rotation joint rigid manipulator system with disturbance is expressed as follows:

[0010]

[0011] 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.

[0012] make is the system state variable, As the system input variables, the dynamic model of the robotic arm system can be rewritten in state space form:

[0013]

[0014] in,

[0015] represents the lumped disturbance, including external disturbance and system parameter perturbation.

[0016] For the robotic arm system, the system parameter perturbations and model inaccuracies are always finite, and the disturbances from outside the physical system are usually also finite, so the uncertainty is bounded.

[0017] a2. Establish the constraint model of the robotic arm system;

[0018] Since the joint angle of the robot arm is subject to mechanical restrictions, and the driving torque that the joint motor of the robot arm can provide is limited; therefore, the robot arm is also subject to angle constraints, angular velocity constraints and driving torque constraints. The constraint model is:

[0019]

[0020] in, is the upper limit of the robot arm joint angle and angular velocity; is the upper limit of the driving torque of the robot arm.

[0021] a3. Establish the state space model of the tracking error of the robot system, as follows:

[0022] In state space, the tracking error equation is defined as:

[0023] x e =xx ref

[0024] in, is the error system state variable, is the reference trajectory of the robot joint, x ref1 is the reference trajectory vector of the joint angle x1, x ref2 is the reference trajectory vector of the joint angular velocity x2; the reference trajectory x of the manipulator joint ref It is bounded, continuous, smooth, and has at least a third-order continuous derivative. Commonly used robot trajectory planning algorithms usually use cubic polynomials, quintic polynomials, or b-spline interpolation; therefore, the reference trajectory obtained by the trajectory planning algorithm can at least ensure that the reference trajectory has a third-order derivative.

[0025] Derivative the tracking error equation, and substitute the reference trajectory of the manipulator joint and the manipulator state space equation into it to obtain the manipulator tracking error state space equation:

[0026]

[0027] S2. Based on the inverse dynamics method, the state space equation of the robot tracking error is feedback linearized, and the linearized SISO double integral system is discretized; then the corresponding nominal system state equation is obtained based on the discretized constrained linear SISO double integral system; specifically:

[0028] b1. Based on the inverse dynamics method, the state space equation of the robot tracking error is linearized.

[0029] The feedback linearization method is used to linearize the nonlinear MIMO (multiple-input multiple-output) error state space equation and obtain the feedback linearization control law:

[0030]

[0031] Among them, v is the auxiliary control input variable; at this time, the feedback linearization control law u is the system input variable, so that the system input variable u and the system state variable x are in a linear relationship;

[0032] Substituting the feedback linearization control law v into the tracking error state space equation, we get:

[0033]

[0034] Where d is After applying the feedback linearization method, the original MIMO nonlinear system is simplified to n SISO decoupled systems, and each joint corresponds to a constrained linear SISO double-integral system:

[0035]

[0036] Among them, the state variable x ei =[x e1i , x e2i ] T =[x 1i -x ref1i , x 2i -x ref2i ] T , subscript i=1, 2,…, n represents the i-th joint of the robotic arm.

[0037] Rewrite the constrained linear SISO double integral system into matrix form:

[0038]

[0039] Among them, x e (t) is the error system state variable, υ(t) is the error system control input variable, w(t) is the system disturbance, A is the system state matrix, B is the system input matrix, D w is the perturbation input matrix for the system.

[0040] b2. Discretize the constrained linear SISO double integral system; specifically:

[0041] Take the sampling time series is the sampling time constant, t k+1 -t k =T, T is the sampling time, and the discretized constrained linear SISO double integration system is obtained:

[0042] x e (t k+1 )=A d x e (t k )+B d υ(t k )+D w w(t k )

[0043] in, is the discretized error system state, is the control input variable of the discretized error system, is the discretized system disturbance, A d is the discretized system state matrix, B d is the discretized input matrix, D w is the discretized perturbation input matrix.

[0044] The discretized constrained linear SISO double-integral system is subject to state constraints and input constraints:

[0045]

[0046] in, is a compact set containing the origin, is a compact set containing the origin; the system perturbation w is also bounded:

[0047]

[0048] in, is a compact set containing the origin;

[0049] Let V(t k ) represents the control input variable sequence {υ(t k ),υ(t k+1 ),…υ(t k+N-1 )}, N represents the prediction interval, W(t k ) represents the disturbance sequence {w(t k ), w(t k+1 ),…w(t k+N-1 )};; Defined at time t k The initial state variable is x e (t k ), the control input variable sequence and disturbance sequence are V(t k ) and W(t k ), the solution of the constrained linear SISO double integral system equation is

[0050] The nominal system corresponding to the constrained linear SISO double-integral system is defined as:

[0051] x e (t k+1 )=A d x e (t k )+B d υ(t k )

[0052] And, when the initial state variable is x e (t k ), the control input variable sequence is V(tk ), define Represents the nominal system state equation at time t k Time solution.

[0053] S3. Design the disturbance invariant set and the Tube set, define the optimal control problem and solve it; specifically:

[0054] c1. Design a perturbation invariant set and define the matrix Make is stable; is an uncertain system with state feedback input x e (t k+1 )=A K x e (t k )+w, the minimum perturbation invariant set, A c represents A to the power of c, ∑· represents set addition, so it satisfies:

[0055]

[0056] in, Represents Minkowski set addition.

[0057] For a discrete-time system x(t k+1 )=f(x(t k ), w), where f(·) represents the difference between x(t k ), functions related to w, the set is a perturbation invariant set or a robust positive invariant set, where for all if but

[0058] c2. Design the Tube set, define the optimal control problem and solve it; specifically:

[0059] Define the following cost function:

[0060]

[0061] in:

[0062]

[0063] Among them, 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, V f (·) is the terminal cost, and l(·) represents the stage cost.

[0064] xe (t k ) represents the current state of the nominal system, without uncertainty; define the optimization control problem P N (x e (t k ))as follows:

[0065]

[0066] For each j∈[0,N-1], and is a set of Tubes that satisfy stricter control, state, and terminal constraints, and is defined as follows:

[0067]

[0068] in, represents set subtraction, is a terminal constraint set, satisfying the following axioms:

[0069] A1:

[0070] A2:

[0071] has one interior point, so:

[0072]

[0073] Value Function Domain Defined by:

[0074]

[0075] Question P N (x e (t k The solution of )) provides the optimal control sequence and the corresponding optimal state sequence For each t k+j Moment, there is

[0076] Therefore, for problem P N (x e (t k )) Solve and obtain the implicit model predictive control law as:

[0077]

[0078] S4. Establish a robust model predictive control optimization problem, and use the initial state as the decision variable. Finally, solve the robust implicit model predictive control law to complete the robust model predictive control for robot trajectory tracking; specifically:

[0079] Define the robust model predictive control optimization problem The definition is as follows:

[0080]

[0081] Value Function The domain of is:

[0082]

[0083] in, is a quadratic programming problem; solve The optimal control sequence can be obtained and the corresponding optimal state sequence Among them, for each For the problem if and Paired (x e0 (t k ),υ) is feasible. Therefore, The implicit model predictive control law generated by the solution of for:

[0084]

[0085] The system status is x e (t k ), the control applied is in is the optimal control sequence v * (x e (t k ))'s first element.

[0086] The robust model predictive tracking control algorithm designed in the present invention is used for tracking control of a robot in the presence of state constraints, input constraints and uncertainties. Based on Tube's model predictive control idea, the trajectory tracking control problem in the presence of uncertainties is solved.

[0087] Beneficial effects of the present invention:

[0088] (1) The present invention transforms the disturbance and parameter perturbation of the manipulator system into lumped disturbances, establishes the manipulator system error model, and performs feedback linearization on the error system based on the inverse dynamics method. Based on the manipulator system error model, the nominal system model predictive controller is designed, and the terminal region, terminal control law and corresponding terminal cost are designed to ensure the feasibility of rolling interval optimization at each time step.

[0089] (2) The present invention is based on the nominal system model predictive control, tightens the nominal system constraints, constrains the uncertain future state to a set that satisfies the conditions, and ensures that the closed-loop tracking error system is input state stable. BRIEF DESCRIPTION OF THE DRAWINGS

[0090] Figure 1 The present invention is a flow chart of a method of an embodiment. DETAILED DESCRIPTION

[0091] The specific implementation modes of the present invention are described below to facilitate those skilled in the art to 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.

[0092] Example

[0093] like Figure 1 As shown, a robust model predictive control method for robot arm trajectory tracking includes the following steps:

[0094] S1. Establish a state-space model of the manipulator under the influence of disturbance, model the constraints of the manipulator, and establish a state-space model of the tracking error.

[0095] a1. Convert the robot arm dynamics model into a state space model and equate the uncertainty to a lumped disturbance.

[0096] The dynamic model of the n-DOF rotation joint rigid manipulator system with disturbance is expressed as follows:

[0097]

[0098] 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.

[0099] make is the system state variable, As the system input variables, the dynamic model of the robotic arm system can be rewritten in state space form:

[0100]

[0101] in,

[0102] represents the lumped disturbance, including external disturbance and system parameter perturbation.

[0103] For the robotic arm system, the system parameter perturbations and model inaccuracies are always finite, and the disturbances from outside the physical system are usually also finite, so the uncertainty is bounded.

[0104] a2. Establish the constraint model of the robotic arm system;

[0105] Since the joint angle of the robot arm is subject to mechanical restrictions, and the driving torque that the joint motor of the robot arm can provide is limited; therefore, the robot arm is also subject to angle constraints, angular velocity constraints and driving torque constraints. The constraint model is:

[0106]

[0107] in, is the upper limit of the robot arm joint angle and angular velocity; is the upper limit of the driving torque of the robot arm.

[0108] a3. Establish the state space model of the tracking error of the robot system, as follows:

[0109] In state space, the tracking error equation is defined as:

[0110] x e =xx ref in, is the error system state variable, is the reference trajectory of the robot joint, x ref1is the reference trajectory vector of the joint angle x1, x ref2 is the reference trajectory vector of the joint angular velocity x2; the reference trajectory x of the manipulator joint ref It is bounded, continuous, smooth, and has at least a third-order continuous derivative. Commonly used robot trajectory planning algorithms usually use cubic polynomials, quintic polynomials, or b-spline interpolation; therefore, the reference trajectory obtained by the trajectory planning algorithm can at least ensure that the reference trajectory has a third-order derivative.

[0111] Derivative the tracking error equation, and substitute the reference trajectory of the manipulator joint and the manipulator state space equation into it to obtain the manipulator tracking error state space equation:

[0112]

[0113] S2. Based on the inverse dynamics method, the state space equation of the robot tracking error is feedback linearized, and the linearized SISO double integral system is discretized; then the corresponding nominal system state equation is obtained based on the discretized constrained linear SISO double integral system; specifically:

[0114] b1. Based on the inverse dynamics method, the state space equation of the robot tracking error is linearized.

[0115] The feedback linearization method is used to linearize the nonlinear MIMO (multiple-input multiple-output) error state space equation and obtain the feedback linearization control law:

[0116]

[0117] Among them, v is the auxiliary control input variable; at this time, the feedback linearization control law u is the system input variable, so that the system input variable u and the system state variable x are in a linear relationship;

[0118] Substituting the feedback linearization control law u into the tracking error state space equation, we get:

[0119]

[0120] Where d is After applying the feedback linearization method, the original MIMO nonlinear system is simplified to n SISO decoupled systems, and each joint corresponds to a constrained linear SISO double-integral system:

[0121]

[0122] Among them, the state variable x ei =[x e1i , x e2i ] T =[x 1i-x ref1i , x 2i -x ref2i ] T , subscript i=1, 2,…, n represents the i-th joint of the robotic arm.

[0123] Rewrite the constrained linear SISO double integral system into matrix form:

[0124]

[0125] Among them, x e (t) is the error system state variable, υ(t) is the error system control input variable, w(t) is the system disturbance, A is the system state matrix, B is the system input matrix, D w is the perturbation input matrix for the system.

[0126] b2. Discretize the constrained linear SISO double integral system; specifically:

[0127] Take the sampling time series is the sampling time constant, t k+1- t k =T, T is the sampling time, and the discretized constrained linear SISO double integration system is obtained:

[0128] x e (t k+1 )=A d x e (t k )+B d υ(t k )+D w w(t k )

[0129] in, is the discretized error system state, is the control input variable of the discretized error system, is the discretized system disturbance, A d is the discretized system state matrix, B d is the discretized input matrix, D w is the discretized perturbation input matrix.

[0130] The discretized constrained linear SISO double-integral system is subject to state constraints and input constraints:

[0131]

[0132] in, is a compact set containing the origin, is a compact set containing the origin; the system perturbation w is also bounded:

[0133]

[0134] in, is a compact set containing the origin;

[0135] Let V(t k ) represents the control input variable sequence {υ(t k ),υ(t k+1 ),…υ(t k+N-1 )}, N represents the prediction interval, W(t k ) represents the disturbance sequence {w(t k ), w(t k+1 ),…w(t k+N-1 )}; defined at time t k The initial state variable is x e (t k ), the control input variable sequence and disturbance sequence are V(t k ) and W(t k ), the solution of the constrained linear SISO double integral system equation is

[0136] The nominal system corresponding to the constrained linear SISO double-integral system is defined as:

[0137] x e (t k+1 )=A d x e (t k )+B d υ(t k )

[0138] And, when the initial state variable is x e (t k ), the control input variable sequence is V(t k ), define Represents the nominal system state equation at time t k Time solution.

[0139] S3. Design the disturbance invariant set and the Tube set, define the optimal control problem and solve it; specifically:

[0140] c1. Design a perturbation invariant set and define the matrix Make is stable; is an uncertain system with state feedback input x e (t k+1 )=A K x e (t k)+w, the minimum perturbation invariant set, A c represents A to the power of c, ∑· represents set addition, so it satisfies:

[0141]

[0142] in, Represents Minkowski set addition.

[0143] For a discrete-time system x(t k+1 )=f(x(t k ), w), where f(·) represents the difference between x(t k ), functions related to w, the set is a perturbation invariant set or a robust positive invariant set, where for all if but

[0144] c2. Design the Tube set, define the optimal control problem and solve it; specifically:

[0145] Define the following cost function:

[0146]

[0147] in:

[0148]

[0149] Among them, 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, V f (·) is the terminal cost, and l(·) represents the stage cost.

[0150] x e (t k ) represents the current state of the nominal system, without uncertainty; define the optimization control problem P N (x e (t k ))as follows:

[0151]

[0152] For each j∈[0,N-1], and is a set of Tubes that satisfy stricter control, state, and terminal constraints, and is defined as follows:

[0153]

[0154] in, represents set subtraction, is a terminal constraint set, satisfying the following axioms:

[0155] A1:

[0156] A2:

[0157] has one interior point, so:

[0158]

[0159] Value Function Domain Defined by:

[0160]

[0161] Question P N (x e (t k The solution of )) provides the optimal control sequence and the corresponding optimal state sequence For each t k+j Moment, there is

[0162] Therefore, for problem P N (x e (t k )) Solve and obtain the implicit model predictive control law as:

[0163]

[0164] S4. Establish a robust model predictive control optimization problem, and use the initial state as the decision variable. Finally, solve the robust implicit model predictive control law to complete the robust model predictive control for robot trajectory tracking; specifically:

[0165] Define the robust model predictive control optimization problem The definition is as follows:

[0166]

[0167] Value Function The domain of is:

[0168]

[0169] in, is a quadratic programming problem; solve The optimal control sequence can be obtained and the corresponding optimal state sequence Among them, for each For the problem if and Paired (x e0 (t k ),υ) is feasible. Therefore, The implicit model predictive control law generated by the solution of for:

[0170]

[0171] The system status is x e (t k ), the control applied is in is the optimal control sequence v * (x e (t k ))'s first element.

[0172] In summary, the present invention designs a robust model predictive control method based on the Tube method, which is particularly suitable for processing additional disturbances. In order to improve the practicality of the algorithm, the present invention sacrifices some optimality to a certain extent to simplify the algorithm. Similar control objectives are achieved by solving the open-loop optimization problem with the same computational complexity as the traditional MPC online. A finite-dimensional parameterization including an open-loop control sequence and a simple local feedback controller is used to replace the decision variables, thereby simplifying the model. The nominal system constraints are tightened, the uncertain future state is constrained to a set that satisfies the conditions, and the closed-loop tracking error system is input-state stable.

Claims

1. A robust model predictive control method for robot arm trajectory tracking, characterized in that: The steps include: S1. Establish a state space model of the manipulator under the influence of disturbance, model the constraints of the manipulator, and establish a state space model of the tracking error; S2. Based on the inverse dynamics method, the state space equation of the robot tracking error is feedback linearized, and the linearized SISO double integral system is discretized; then the corresponding nominal system state equation is obtained based on the discretized constrained linear SISO double integral system; S3. Design the disturbance invariant set and Tube set, define the optimal control problem and solve it; S4. Establish a robust model predictive control optimization problem, and use the initial state as the decision variable. Finally, the robust implicit model predictive control law is obtained to complete the robust model predictive control for robot trajectory tracking.

2. A robust model predictive control method for robot arm trajectory tracking according to claim 1, characterized in that: Step S1 specifically includes: a1. Convert the robot arm dynamics model into a state space model and equate the uncertainty to a lumped disturbance; 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, As the system input variable, 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; a2. Establish the constraint model of the robotic arm system; The robot arm is subject to angle constraints, angular velocity constraints and driving torque constraints. The constraint model is: in, is the upper limit of the robot arm joint angle and angular velocity; is the upper limit of the driving torque of the robot arm; a3. Establish the state space model of the tracking error of the robot system, as follows: In state space, the tracking error equation is defined as: x e =x-x ref in, is the error system state variable, is the reference trajectory of the robot joint, x ref1 is the reference trajectory vector of the joint angle x1, x ref2 is the reference trajectory vector of the joint angular velocity x2; the reference trajectory x of the manipulator joint ref It is bounded, continuous, smooth, and has at least three-order continuous derivatives. Derivative the tracking error equation, and substitute the reference trajectory of the manipulator joint and the manipulator state space equation into it to obtain the manipulator tracking error state space equation:

3. A robust model predictive control method for robot arm trajectory tracking according to claim 2, characterized in that: Step S2 specifically includes: b1. Based on the inverse dynamics method, the state space equation of the robot tracking error is linearized; The feedback linearization method is used to linearize the nonlinear MIMO error state space equation and obtain the feedback linearization control law: Among them, v is the auxiliary control input variable; at this time, the feedback linearization control law u is the system input variable, so that the system input variable u and the system state variable x are in a linear relationship; Substituting the feedback linearization control law u into the tracking error state space equation, we get: Where d is Each joint corresponds to a constrained linear SISO double-integral system: Among them, the state variable x ei =[x e1i , x e2i ] T =[x 1i -x ref1i , x 2i -x ref2i ] T , subscript i = 1, 2, ..., n represents the i-th joint of the robot arm; Rewrite the constrained linear SISO double integral system into matrix form: Among them, x e (t) is the error system state variable, v(t) is the error system control input variable, w(t) is the system disturbance, A is the system state matrix, B is the system input matrix, D w is the disturbance input matrix of the system; b2. Discretize the constrained linear SISO double integral system; specifically: Take the sampling time series t k , is the sampling time constant, t k+1 -t k =T, T is the sampling time, and the discretized constrained linear SISO double integration system is obtained: x e (t k+1) =A d x e (t k )+B d υ(t k )+D w w(t k ) in, is the discretized error system state, is the control input variable of the discretized error system, is the discretized system disturbance, A d is the discretized system state matrix, B d is the discretized input matrix, D w is the discretized disturbance input matrix; The discretized constrained linear SISO double-integral system is subject to state constraints and input constraints: in, is a compact set containing the origin, is a compact set containing the origin; the system perturbation w is also bounded: in, is a compact set containing the origin; Defined at time t k The initial state variable is x e (t k ), control input variable sequence V(t k ) is {v(t k ), v(t k+1 ),…v(t k+N-1) }, the disturbance sequence W(t k ) is {w(t k ), w(t k+1 ),…w(t k+N-1 )}, the solution of the constrained linear SISO double integral system equation is b3. Define the nominal system state equation corresponding to the discretized constrained linear SISO double-integral system as: x e (t k+1 )=A d x e (t k )+B d υ(t k ) And, when the initial state variable is x e (t k ), the control input variable sequence is V(t k ), define Represents the nominal system state equation at time t k Time solution.

4. A robust model predictive control method for robot arm trajectory tracking according to claim 3, characterized in that: Step S3 specifically includes: c1. Design a perturbation invariant set and define the matrix Make is stable; is an uncertain system with state feedback input x e (t k+1 )=A K x e (t k )+w, the minimum perturbation invariant set, A c represents A to the power of c, ∑· represents set addition, so it satisfies: in, represents Minkowski set addition; For a discrete-time system x(t k+1 )=f(x(t k ), w), where f(·) represents the difference between x(t k ), functions related to w, the set is a perturbation invariant set or a robust positive invariant set, where for all if but c2. Design the Tube set, define the optimal control problem and solve it; specifically: Define the following cost function: in: Among them, 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, V f (·) is the terminal cost, l(·) represents the stage cost; x e (t k ) represents the current state of the nominal system, without uncertainty; define the optimization control problem P N (x e (t k ))as follows: For each j∈[0,N-1], and is a set of Tubes that satisfy stricter control, state, and terminal constraints, and is defined as follows: in, represents set subtraction, is a terminal constraint set, satisfying the following axioms: has one interior point, so: Value Function Domain Defined by: Question P N (x e (t k The solution of )) provides the optimal control sequence and the corresponding optimal state sequence For each t k+j Moment, there is Therefore, for problem P N (x e (t k )) Solve and obtain the implicit model predictive control law as:

5. A robust model predictive control method for robot arm trajectory tracking according to claim 4, characterized in that: Step S4 specifically includes: Define the robust model predictive control optimization problem Value Function The domain of is: in, is a quadratic programming problem; solve The optimal control sequence can be obtained and the corresponding optimal state sequence Among them, for each For the problem if and Paired (x e0 (t k ), v) is feasible; therefore, The implicit model predictive control law generated by the solution of for: The system status is x e (t k ), the control applied is in is the optimal control sequence v * (x e (t k ))'s first element.

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