A robust model predictive control method for manipulator trajectory tracking control

Abstract

The application discloses a kind of robust model prediction control methods for mechanical arm trajectory tracking control, it is related to the field of multi-degree-of-freedom mechanical arm trajectory tracking control.The method transforms the disturbance and parameter perturbation of mechanical arm system into lumped disturbance, establishes the error model of mechanical arm system, and carries out feedback linearization to error system based on inverse dynamics method;Based on the error model of mechanical arm system, design the nominal system model predictive controller, design the terminal region, terminal control law and the corresponding terminal cost, ensure the feasibility of rolling interval optimization at each time step.The method of the application can handle the uncertainty and disturbance of mechanical arm system, and ensure the satisfaction of system constraints, realize high-precision trajectory tracking control and input torque optimization.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of multi-freedom mechanical arm trajectory tracking control, in particular, to a robust model predictive control method for mechanical arm trajectory tracking. BACKGROUND

[0002] With the wide application of mechanical arms in the fields of industrial autonomous manufacturing, orbital missions, surgery, etc., the demand for high-precision control of mechanical arms is increasingly strong. A basic task of mechanical arm control is to control the motion of the end effector by applying control input to the joints of the mechanical arm to track the required reference trajectory. The dynamics of the mechanical arm is a multi-input multi-output system with high nonlinearity. Although the mechanical arm controller based on the accurate model can obtain good performance, disturbances are inevitable, which leads to the performance degradation of the mechanical arm. In addition, the mechanical arm is usually subject to various constraints, such as mechanical constraints, safety constraints and task constraints. Without considering the constraints in the control design, unpredictable responses or even failures may occur

[0003] Model predictive control (MPC) is widely used in optimal control of industrial applications due to its ability to effectively control complex systems and handle constraints. In particular, the MPC algorithm overcomes the shortcomings of the above methods. However, in the actual application process of the mechanical arm system, model parameter perturbation, external disturbance, inaccurate modeling, etc. are inevitable, and at the same time, due to the influence of the working environment, the mechanical arm will inevitably be subject to constraints. The traditional model predictive control method has high requirements for model accuracy, and an inaccurate mechanical arm model may lead to poor control performance or even controller divergence. Therefore, it is necessary to study a robust model predictive control method that can handle the uncertainty and constraints of the mechanical arm while considering the control performance. SUMMARY

[0004] In view of the deficiencies in the prior art, the present application provides a robust model predictive control method for mechanical arm trajectory tracking. This method 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.

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

[0006] A robust model predictive control method for mechanical arm trajectory tracking, comprising the following steps:

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

[0008] a1. Convert the mechanical arm dynamics model into a state space model, and equivalent the uncertainty as a lumped disturbance.

[0009] The dynamic model of a rigid robotic arm system with an n-DOF rotary joint and perturbation is represented as follows:

[0010]

[0011] 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; τ is the driving torque vector, τ d This is the external disturbance torque vector acting on the joint.

[0012] make For system state variables, 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:

[0013]

[0014] in,

[0015] This represents lumped disturbances, including external disturbances and system parameter perturbations.

[0016] For robotic arm systems, the perturbations of system parameters and the inaccuracies of the model are always finite, and disturbances from outside the physical system are also typically finite; therefore, the uncertainty is... It is bounded.

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

[0018] Because the joint angles of the robotic arm are mechanically limited, and the driving torque that the joint motors can provide is finite, the robotic arm is also subject to angle constraints, angular velocity constraints, and driving torque constraints. The constraint model is as follows:

[0019]

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

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

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

[0023] x e = x - x ref

[0024] wherein, is the error system state variable, is the joint reference trajectory of the robot arm, x ref1 is the reference trajectory vector of joint angle x1, x ref2 is the reference trajectory vector of joint angular velocity x2; the joint reference trajectory x ref of the robot arm is bounded and continuous, smooth enough, and at least has three continuous derivatives. Commonly used trajectory planning algorithms for robot arms usually use cubic polynomials, quintic polynomials or b-spline interpolation; therefore, the reference trajectory obtained through the trajectory planning algorithm can at least guarantee that the reference trajectory has three derivatives.

[0025] Derive the tracking error equation and substitute the joint reference trajectory of the robot arm and the state space equation of the robot arm to obtain the state space equation of the robot arm tracking error:

[0026]

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

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

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

[0030]

[0031] wherein v is an 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] Substitute the feedback linearization control law v into the tracking error state space equation to obtain:

[0033]

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

[0035]

[0036] where 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.

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

[0038]

[0039] where 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 state matrix of the system, B is the input matrix of the system, D w is the disturbance input matrix of the system.

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

[0041] Take the sampling time sequence is the sampling time constant, t k+1 -t k = T, T is the sampling time, to get the discretized constrained linear SISO double-integrator system:

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

[0043] where x d is the discretized error system state, for the discretized error system, for the discretized system disturbance, A d for the discretized system state matrix, B d for the discretized input matrix, D w for the discretized disturbance input matrix.

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

[0045]

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

[0047]

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

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

[0050] The nominal system corresponding to the constrained linear SISO double-integrator 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 ) is defined as denotes the solution of the nominal system state equation at time t k .

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

[0054] c1. Design disturbance invariant set, define matrix such that is stable; let be the minimal disturbance invariant set of the uncertain system x e (t k+1 ) = A K x e (t k ) + w with A c denoting the c-th power of A, and denoting set addition, thus satisfying:

[0055]

[0056] where denotes Minkowski set addition.

[0057] For a discrete-time system x(t k+1 ) = f(x(t k ), w) with f(·) denoting a function of x(t k ) and w, the set is a disturbance invariant set or robust positive invariant set, where for all if then

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

[0059] Define the cost function as follows:

[0060]

[0061] where:

[0062]

[0063] where 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, and V f (·) is the terminal cost, and l(·) denotes the stage cost.

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

[0065]

[0066] For each j e [0, N - 1], while is the Tube set of control sequences that satisfy more stringent control, state, and terminal constraints, defined as follows:

[0067]

[0068] where, denotes set subtraction, is the terminal constraint set, satisfying the following axioms:

[0069] A1:

[0070] A2:

[0071] There is an interior point, thus:

[0072]

[0073] The definition domain of the value function is defined by:

[0074]

[0075] The solution of the problem P N (x e (t k )) provides the optimal control sequence and the corresponding optimal state sequence where for each time instant t k+j , there is

[0076] Therefore, solving the problem P N (x e (t k )) gives the implicit model predictive control law as:

[0077]

[0078] ​S4. A robust model predictive control optimization problem is established, and the initial state is taken as a decision variable, and finally a robust implicit model predictive control law is solved to complete the robust model predictive control for trajectory tracking of the mechanical arm; specifically:

[0079] A robust model predictive control optimization problem is defined is defined as follows:

[0080]

[0081] The definition domain of the value function is:

[0082]

[0083] wherein, is a quadratic programming problem; solving can obtain the optimal control sequence and the corresponding optimal state sequence wherein, for each for the problem if and the pair (x e0 (t k ), υ) is feasible. Therefore, the implicit model predictive control law produced by the solution of is:

[0084]

[0085] When the system state is x e (t k ), the applied control is wherein is the first element of the optimal control sequence υ * (x e (t k )).

[0086] The robust model predictive tracking control algorithm designed in the application is used for tracking control of a robot under the condition of existing state constraints and input constraints and uncertainty, and solves the trajectory tracking control problem under the condition of existing uncertainty based on the Tube model predictive control idea.

[0087] The application has the beneficial effects that:

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

[0089] (2) The present application is based on the nominal system model predictive control, the nominal system constraint is tightened, the uncertain future state constraint is confined in the set satisfying the condition, and the closed-loop tracking error system is input-state stable. BRIEF DESCRIPTION OF DRAWINGS

[0090] Figure 1 A method flowchart for the embodiment. DETAILED DESCRIPTION

[0091] The specific embodiments of the present application are described below to facilitate the understanding of the present application by those skilled in the art, but it should be clear that the present application is not limited to the scope of the specific embodiments, and for those skilled in the art, it is obvious that various changes are within the spirit and scope of the present application defined and determined by the appended claims, and all the inventions utilizing the concept of the present application are within the scope of protection.

[0092] EMBODIMENT

[0093] As shown in Figure 1 A robust model predictive control method for mechanical arm trajectory tracking includes the following steps:

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

[0095] a1. Convert the mechanical arm dynamics model into a state space model, and equivalent the uncertainty as a lumped disturbance.

[0096] The dynamics model of an n-degree-of-freedom revolute joint rigid mechanical arm system with disturbance is represented as follows:

[0097]

[0098] wherein, q, q and q represent the position, velocity and acceleration of the n joints, respectively, is an n-dimensional vector space; represents a symmetric positive definite inertia matrix, represents a centripetal Coriolis matrix, represents a gravity vector, M0(q), G0(q) represents the nominal values of the inertia matrix, the centripetal Coriolis matrix and the gravity vector of the manipulator, respectively, ΔM(q), ΔG(q) represents the modeling inaccuracies and parameter perturbations of the inertia matrix, the centripetal Coriolis matrix and the gravity vector of the manipulator, respectively, is the n x n matrix space, is the n-dimensional vector space; τ is the driving torque vector, τ d is the external disturbance torque vector acting on the joint.

[0099] Let be the system state variable, be the system input variable, then the dynamics model of the manipulator system can be rewritten in the state space form:

[0100]

[0101] where,

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

[0103] For the manipulator system, the system parameter perturbation and the model inaccuracies are always limited, and the disturbance from the outside of the physical system is usually limited, so the uncertainty is bounded.

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

[0105] Since the joint angle of the manipulator will be subject to mechanical constraints, and the driving torque that the joint motor of the manipulator can provide is limited; therefore, the manipulator will also be subject to angle constraints, angular velocity constraints and driving torque constraints, and the constraint model is:

[0106]

[0107] where, is the upper limit of the joint angle and angular velocity of the manipulator; is the upper limit of the driving torque of the manipulator.

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

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

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

[0111] The derivative of the tracking error equation is taken, and the robotic arm joint reference trajectory and the robotic arm state space equation are substituted into it to obtain the robotic arm tracking error state space equation:

[0112]

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

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

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

[0116]

[0117] where v is an 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] The feedback linearization control law u is substituted into the tracking error state space equation to obtain:

[0119]

[0120] where d is After applying the feedback linearization method, the original MIMO nonlinear system is simplified into n SISO decoupled systems, each joint corresponding to a constrained linear SISO double-integrator system:

[0121]

[0122] where the state variable x ei = [x e1i , x e2i ] T = [x 1ix ref1i , x 2i x ref2i ] T , where subscript i = 1, 2, …, n denotes the i-th joint of the robot arm.

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

[0124]

[0125] where 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 state matrix of the system, B is the input matrix of the system, D w is the disturbance input matrix of the system.

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

[0127] Take the sampling time sequence as the sampling time constant, t k+1- t k = T, T is the sampling time, to obtain the discretized constrained linear SISO double-integrator system:

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

[0129] where x is the discretized error system state, is the discretized error system control input variable, 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.

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

[0131]

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

[0133]

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

[0135] Let V(t k ) denote the control input variable sequence {υ(t k ), υ(t k+1 ),…υ(t k+N-1 )}, N denote the prediction horizon, W(t k ) denote the disturbance sequence {w(t k ), w(t k+1 ),…w(t k+N-1 )}; define the initial state variable at time t k as x e (t k ), the control input variable sequence and the disturbance sequence as V(t k ) and W(t k ) respectively, the solution of the constrained linear SISO double-integrator system equation is

[0136] Define the nominal system corresponding to the constrained linear SISO double-integrator system 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 ) and the control input variable sequence is V(t k ), define denotes the solution of the nominal system state equation at time t k .

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

[0140] c1. Design the disturbance invariant set, define the matrix such that is stable; let be the uncertain system with state feedback input x e (t k+1 ) = A K x e (t k) + w, A c denotes the c-th power of A, ∑· denotes set addition, thus satisfying:

[0141]

[0142] wherein, denotes Minkowski set addition.

[0143] For a discrete-time system x(t k+1 ) = f(x(t k ), w), wherein f(·) denotes a function depending on x(t k ), w, the set is a disturbance invariant set or a robust positive invariant set, wherein for all if then

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

[0145] define the following cost function:

[0146]

[0147] wherein:

[0148]

[0149] wherein 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, and V f (·) is the terminal cost, and l(·) denotes the stage cost.

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

[0151]

[0152] for each j e [0, N - 1], and is a Tube set of control sequences satisfying more stringent control, state and terminal constraints, defined as follows:

[0153]

[0154] where, denotes set subtraction, is the terminal constraint set, satisfying the following axiom:

[0155] A1:

[0156] A2:

[0157] There is an interior point, so:

[0158]

[0159] Value function Domain of is defined by:

[0160]

[0161] Problem P N (x e (t k )) provides the optimal control sequence and the corresponding optimal state sequence where for each time instant t k+j , there is

[0162] Therefore, solving problem P N (x e (t k )) gives the implicit model predictive control law as:

[0163]

[0164] S4. Establish a robust model predictive control optimization problem, take the initial state as the decision variable, and finally solve to obtain the robust implicit model predictive control law, complete the robust model predictive control for the trajectory tracking of the robot arm; Specifically:

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

[0166]

[0167] The domain of the value function is:

[0168]

[0169] where, is a quadratic programming problem; solving can obtain the optimal control sequence and the corresponding optimal state sequence where, for each For the problem If and The pair (x e0 (t k ), u) is feasible. Thus, the implicit model predictive control law resulting from the solution of is given by

[0170]

[0171] The control applied when the system state is x e (t k ) is where is the first element of the optimal control sequence u * (x e (t k )).

[0172] In summary, the present invention designs a robust model predictive control method based on Tube method, which is particularly suitable for handling additional disturbances. In order to improve the practicability of the algorithm, the present invention sacrifices some optimality to some extent to simplify the algorithm. By solving an open-loop optimization problem with the same computational complexity as traditional MPC online, a similar control objective is achieved. A finite-dimensional parameterization containing 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 constraints are constrained in the set satisfying the condition, and the closed-loop tracking error system is guaranteed to be input-state stable.​

Claims

1. A robust model predictive control method for trajectory tracking of a robotic arm, characterized in that, Comprising the following steps: S1. Establishing a mechanical arm state space model under the influence of disturbance, modeling the constraints of the mechanical arm, and establishing a state space model of tracking error; S2. Based on the inverse dynamics method, the tracking error state space equation of the mechanical arm is feedback linearized, and the linearized SISO double integral system is discretized; then based on the discretized constraint linear SISO double integral system, the corresponding nominal system state equation is obtained; S3. Designing disturbance invariant set and Tube set, defining optimal control problem and solving; S4. Establishing a robust model predictive control optimization problem, taking the initial state as the decision variable, and finally solving to obtain a robust implicit model predictive control law, completing the robust model predictive control for mechanical arm trajectory tracking; Step S1 specifically includes: a1. Convert the mechanical arm dynamics model into a state space model, and equivalent the uncertainty as a lumped disturbance; The dynamics model of an n-degree-of-freedom revolute joint rigid mechanical arm system with disturbance is represented as follows: wherein, denote the position, velocity and acceleration of 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) denote the nominal values of the robot inertia matrix, the centripetal Coriolis matrix and the gravity vector, respectively, ΔM(q), ΔG(q) denote the modeling inaccuracies and parameter perturbations of the robot inertia matrix, the centripetal Coriolis matrix and the gravity vector, respectively, 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 joints; Let be the system state variable, be the system input variable, then the dynamics model of the manipulator system can be rewritten in state space form as wherein, represents lumped disturbances, including external disturbances and system parameter perturbations; a2. Establish the constraint model of the mechanical arm system; The mechanical arm is subject to angle constraint, angular velocity constraint and driving torque constraint, and the constraint model is: wherein, is an upper limit for the joint angle and angular velocity of the robot arm; is an upper limit for the driving torque of the robot arm; a3. Establish the state space model of the tracking error of the mechanical arm system, specifically as follows: In the state space, the tracking error equation is defined as: x e = x - x ref wherein, is the error system state variable, is the joint reference trajectory of the robot arm, x ref1 is the reference trajectory vector for joint angle x1, x ref2 is the joint angular velocity x2 reference trajectory vector; the joint reference trajectory of the robot arm x ref is bounded and continuous, sufficiently smooth with at least third order continuous derivatives; Derive the tracking error equation, and substitute the mechanical arm joint reference trajectory and the mechanical arm state space equation to obtain the mechanical arm tracking error state space equation:

2. The robust model predictive control method for trajectory tracking of a robotic arm of claim 1, wherein, Step S2 specifically includes: b1. Based on the inverse dynamics method, the tracking error state space equation of the mechanical arm is linearized; Using feedback linearization method, the nonlinear MIMO error state space equation is linearized to obtain the feedback linearization control law: Where 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 linear relationship; Substitute the feedback linearization control law u into the tracking error state space equation to obtain: wherein d is Each joint corresponds to a constraint linear SISO double integral system: where the state variable x ei = [x e1i , x e2i ] T = [x 1i - x ref1i , x 2i - x ref2i ] T , the subscript i = 1, 2, …, n represents the i-th joint of the robot arm; Rewrite the constraint linear SISO double integral system into matrix form: wherein 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 state matrix of the system, B is the input matrix of the system, D w is the disturbance input matrix of the system; b2. Discretize the constraint linear SISO double integral system; Specifically: Taking the sampling time sequence for the sampling time constant, t k+1 -t k = T, T being the sampling time, resulting in the discretized constrained linear SISO double-integrator system: x e (t k+1 )=A d x e (t k )+B d v(t k )+D w w(t k ) wherein, is the discretized error system state, is the discretized error system control input variable, 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 constraint linear SISO double integral system is subject to state constraints and input constraints: wherein is a compact set containing the origin, is a compact set containing the origin; the system disturbance w is also bounded: wherein is a compact set containing the origin; The initial state variable at time t k is defined as x e (t k ), the 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 )}, and the solution of the constrained linear SISO double-integrator system equation is b3. Define the nominal system state equation corresponding to the discretized constraint linear SISO double integral system as: x e (t k+1 )=A d x e (t k )+B d v(t k ) And, when the initial state variable is x e (t k ), the control input variable sequence is V(t k ), define the solution of the nominal system state equation at time t k .

3. The robust model predictive control method for trajectory tracking of a robotic arm of claim 2, wherein, Step S3 specifically includes: c1. Design a disturbance invariant set, define matrix such that is stable; let be an uncertain system with state feedback input x e (t k+1 ) = A K x e (t k ) + w c denote the c-th power of A, ∑· denote set addition, thus satisfying: wherein denotes Minkowski set addition; For discrete-time systems x(t k+1 ) = f(x(t k ), w), where f(·) denotes a function related to x(t k ), w, the set is a disturbance invariant set or a robust positive invariant set, where for all if then c2. Design Tube set, define optimal control problem and solve; Specifically: Define the following cost function: Where: where the error weight matrix is positive semi-definite, the input weight matrix is positive definite, the terminal weight matrix is positive definite, and V f is the terminal cost, and l(·) denotes the stage cost. x e (tk) represents the nominal system current state without uncertainty; define the optimization control problem P N (x e (t k )) as follows: for each j ∈ [0, N - 1], And is the set of Tubes of control sequences that satisfy the more restrictive control, state and terminal constraints, defined as follows: wherein denotes set subtraction, is a terminal constraint set, satisfying the following axiom: There is an interior point, so: Value function Domain of definition is defined by the equation Problem P N (x e (t k )) provides the optimal control sequence and the corresponding optimal state sequence where for each t k+j time instant, there is Thus, the problem P N (x e (t k )) is solved to obtain an implicit model predictive control law:

4. The robust model predictive control method for trajectory tracking of a robotic arm of claim 3, wherein, Step S4 specifically includes: Defining robust model predictive control optimization problems Value function is defined on the domain: in, This is a quadratic programming problem; find the solution. The optimal control sequence can be obtained. and the corresponding optimal state sequence Among them, for each Regarding the question if and Pairs (x e0 (t k (),v) is feasible; therefore, by The implicit model predictive control law generated by the solution for: The system state is x e (t k ) the control applied is where is the first element of the optimal control sequence v * (x e (t k ))

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