Trajectory tracking control method for robot arm based on hyperlocal model fixed-time cooperative control

Abstract

This invention relates to a robotic arm trajectory tracking control method based on a hyperlocal model and fixed-time cooperative control. Compared with existing technologies, it overcomes the shortcomings of cooperative control, such as dependence on accurate modeling of the controlled system and its asymptotic convergence. The invention includes the following steps: establishing a hyperlocal model of the robotic arm; estimating the n-dimensional column vectors of the hyperlocal model containing known modeling and nonlinear uncertainties; designing macrovariables and dynamic evolution equations based on fixed-time stability theory; obtaining a fixed-time cooperative control law by combining the hyperlocal model and the whale optimization algorithm; and implementing trajectory tracking control of the robotic arm according to the fixed-time cooperative control law. This invention not only generates chatter-free control signals but also ensures that the convergence time of the robotic arm tracking error is no longer constrained by the initial state, allowing the tracking error to converge to zero within a fixed maximum upper limit time.

Description

Technical Field

[0001] This invention relates to the field of robotic arm control technology, specifically to a robotic arm trajectory tracking control method based on fixed-time cooperative control using a hyperlocal model. Background Technology

[0002] In recent years, with the rapid development of modern industry, robotic arms have been widely used in industrial, medical, and logistics fields, leading to an increasing demand for rapid response and high-precision trajectory tracking control. However, due to the characteristics of robotic arms—multiple inputs and multiple outputs, strong coupling, and nonlinearity—and their susceptibility to parameter perturbations and external disturbances, accurate modeling and control of robotic arms still face technical challenges. To achieve satisfactory trajectory tracking control performance, advanced control algorithms have been proposed, such as Adaptive Control (AC), Model Predictive Control (MPC), Sliding Mode Control (SMC), and Synergetic Control (SC).

[0003] Sliding mode control is widely used in robotic arm control due to its advantages such as simple structure, fast response speed, high control accuracy, and strong robustness to perturbations of internal system parameters and external disturbances. However, traditional sliding mode control suffers from problems such as long asymptotic convergence time and control chattering. To address the long convergence time of traditional sliding mode control, terminal sliding mode control based on nonlinear sliding surface design achieves finite-time convergence while improving response speed and tracking accuracy. To solve the chattering problem in sliding mode control, researchers have proposed control methods such as boundary layer method and higher-order sliding mode control, but the former sacrifices steady-state accuracy, and the latter increases the computational burden.

[0004] The concept of cooperative control is based on modern mathematics and synergetics. Cooperative control leverages the self-organizing ability and nonlinearity of open systems far from equilibrium to guide the system state towards a predetermined manifold, thereby achieving global stability of the entire system. Similar to sliding mode control, cooperative control is well-suited for reducing the order of the controlled system and exhibits strong robustness to external disturbances. Furthermore, cooperative control designs continuous dynamic processes to force the system state to approximate the manifold, fundamentally eliminating chattering in sliding mode control through the use of completely continuous control laws. Cooperative control possesses the advantages of sliding mode control, and also avoids the chattering problem of sliding mode control, along with simple stability analysis, attracting considerable research interest.

[0005] However, traditional collaborative control relies on accurate modeling of the controlled system and has a long asymptotic convergence time, which has become a key technical problem that urgently needs to be solved. Summary of the Invention

[0006] The purpose of this invention is to address the shortcomings of existing technologies, such as the dependence of cooperative control on accurate modeling of the controlled system and its asymptotic convergence, by providing a robotic arm trajectory tracking control method based on fixed-time cooperative control using a hyperlocal model to solve the above problems.

[0007] To achieve the above objectives, the technical solution of the present invention is as follows:

[0008] A robotic arm trajectory tracking control method based on fixed-time cooperative control using a hyperlocal model includes the following steps:

[0009] Establish a hyperlocal model of the robotic arm;

[0010] Estimate the n-dimensional column vectors in the hyperlocal model of the robotic arm, which contain known modeling and nonlinear uncertainties.

[0011] Design macrovariables and dynamic evolution equations based on fixed-time stability theory;

[0012] A fixed-time cooperative control law is obtained by combining the hyperlocal model of the robotic arm and the whale optimization algorithm;

[0013] Track tracking control of the robotic arm is achieved by embedding the optimized collaborative control law into the robotic arm control system.

[0014] The process of establishing the hyperlocal model of the robotic arm includes the following steps:

[0015] A complete dynamic model of an n-DOF robotic arm is established, and its expression is as follows:

[0016]

[0017] in, It is an n-dimensional column vector, representing the displacement vector, velocity vector, and acceleration vector of the robotic arm joints, respectively; It is an n-dimensional column vector representing the input torque vector; The inertia matrix represents the robotic arm; Represents the Coriolis and centripetal force matrices; It is an n-dimensional column vector representing the gravity matrix; n-dimensional column vector Represents the disturbance torque vector;

[0018] Based on the dynamic model of the robotic arm, let x1 = q, Establish a hyperlocal model of an n-DOF robotic arm.

[0019]

[0020] Where, x1=[x 11 ,x 12 ,…,x1n ] T ,x2=[x 21 ,x 22 ,…,x 2n ] T All are n-dimensional column vectors, representing joint displacement and velocity; It is the derivative of x1, representing the joint velocity; It is the derivative of x², representing the joint acceleration; Λ r =diag{α r1 ,α r2 ,…,α rn} is a diagonal matrix, and all elements on the diagonal are positive constants; Let be an n-dimensional column vector representing the known modeling and nonlinear uncertainty components of the robotic arm;

[0021]

[0022] Among them, M -1 (q) is the inverse of the inertia matrix M(q) of the robotic arm.

[0023] The estimation of the n-dimensional column vectors in the hyperlocal model of the robotic arm, which includes known modeling and nonlinear uncertainties, is as follows: The non-asymptotically convergent algebraic identification technique is used to estimate F... r The estimation is performed, including the following steps:

[0024] Representing a single-input, single-output, v-order nonlinear system as a hyperlocal model:

[0025] y (v) =F+αu;

[0026] Where u and y represent the system input and output respectively, v is the highest order of the system, set to 1 or 2, α is the scaling factor of the system input, so that the system output y and input u are kept on the same order of magnitude, and F represents the known modeling and nonlinear uncertainty part in the hyperlocal model, which includes all other parts except the αu term;

[0027] Assuming v = 1, performing a Laplace transform on the above equation yields:

[0028]

[0029] Where y0 represents the initial state of the system, Y is the frequency domain representation of the system output y(t) through the Laplace transform, s is the complex frequency domain variable in the Laplace transform, and U is the frequency domain representation of the input u(t) through the Laplace transform. It is an estimate of F in the frequency domain through the Laplace transform;

[0030] To eliminate the influence of the system's initial state, multiply both sides of the equation by d / ds, then:

[0031]

[0032] To avoid the noise amplification effect caused by differentiation, both sides of the above equation are multiplied by s. -2 Then, an inverse Laplace transform is performed to obtain the time-domain expression for estimating F based on algebraic identification techniques:

[0033]

[0034] Among them, T F =n F T s T represents the size of the sliding window. s The sampling period is n. F This is the length of the time window, set according to the actual system; δ represents the time variable, y(δ) is the system output, and u(δ) is the system input;

[0035] Hyperlocal model based on robotic arm:

[0036]

[0037] Where, x1=[x 11 ,x 12 ,…,x 1n ] T ,x2=[x 21 ,x 22 ,…,x 2n ] T All are n-dimensional column vectors, representing the displacement and velocity of each joint; It is the derivative of x1, representing the joint velocity; It is the derivative of x², representing the joint acceleration; It is an n-dimensional column vector representing the input torque vector;

[0038] Λ r =diag{α r1 ,α r2 ,…,α rn} is a diagonal matrix, and all elements on the diagonal are positive constants; Let be an n-dimensional column vector representing the known modeling and nonlinear uncertainties in the hyperlocal model of the robotic arm;

[0039] Combining the above algebraic identification techniques, we obtain the n-dimensional column vector F. r The estimated value for:

[0040]

[0041] The design of macrovariables and dynamic evolution equations based on fixed-time stability theory is as follows: According to fixed-time stability theory and combined with the dynamic control objectives of the robotic arm, the macrovariables are designed in a fixed-time form, and dynamic evolution equations are designed to constrain the way the macrovariables move to the invariant manifold; this includes the following steps:

[0042] definition Let the reference displacement, reference velocity, and reference acceleration be the values ​​for each joint of the robotic arm, and define them as follows:

[0043]

[0044] in, It is an n-dimensional column vector, representing the displacement vector, velocity vector, and acceleration vector of the robotic arm joints, respectively; Let each be an n-dimensional column vector, representing the desired displacement vector, desired velocity vector, and desired acceleration vector of the robotic arm joints, respectively. All are n-dimensional column vectors, representing the joint displacement error, velocity error, and acceleration error, respectively; It is the derivative of e1, representing the velocity error of the joint; It is the derivative of e1, representing the acceleration error of the joint;

[0045] Given that the goal of robotic arm dynamics control is to enable each joint of the robotic arm to quickly track a given reference trajectory, which includes position tracking and velocity tracking of each joint, the macrovariable is defined as a combination of joint displacement error and velocity error. Combining this with fixed-time stability theory, the above macrovariable is designed in a fixed-time form, resulting in:

[0046] ψ r =e2+k1e1 m / n +k2e1 n / m ,

[0047] Where, ψ r =[ψ r1 ,ψ r2 ,…,ψ rn ] T k1 and k2 are diagonal matrices, and all diagonal elements are greater than 0; m and n are both positive odd numbers, and satisfy the following conditions: Differentiating the above equation, we get:

[0048]

[0049] By designing dynamic evolution equations to constrain the movement of macrovariables to invariant manifolds, the dynamic evolution equations are designed as follows:

[0050]

[0051] Where p and q are both positive odd numbers, and satisfy... T r =diag{T r1 ,T r2 ,…,T rn} is a diagonal matrix, and all its diagonal elements are greater than 0. It is a time constant, representing the time it takes for the state variables to converge to the manifold through a dynamic process. Theoretically, T r The smaller the value of , the faster the system's dynamic response speed;

[0052] When the dynamic evolution equation is designed as the above equation, then we have right Differentiating, we get:

[0053]

[0054] Since p and q are both positive odd numbers, and satisfy... Therefore, qp is a positive even number. It is a monotonic function, therefore it satisfies the condition: Reversible and differentiable;

[0055] When ψ r When = 0, according to The following conditions were met: according to have to:

[0056]

[0057] Since p and q are both odd, then p + q is even. Therefore, for any ψ r ≠0, That is, the conditions are met:

[0058] Under the constraints of the designed dynamic evolution equation, the macrovariable ψ r It converges within a fixed time limit and remains on the invariant manifold ψ. r =0.

[0059] The fixed-time cooperative control law obtained by combining the hyperlocal model of the robotic arm and the whale optimization algorithm is as follows: Based on the designed macrovariables and dynamic evolution equations combined with the hyperlocal model, the fixed-time cooperative control law is obtained, and the key control parameters are tuned using the whale optimization algorithm; including the following steps:

[0060] Substituting the designed macrovariables into the dynamic evolution equation, we get:

[0061]

[0062] in: Each is an n-dimensional column vector, representing the joint's displacement error, velocity error, and acceleration error, respectively; T r =diag{T r1 ,T r2 ,…,T rn} is a diagonal matrix, and all elements on its diagonal are greater than 0; ψ is a time constant. r =[ψ r1 ,ψ r2 ,…,ψ rn ] T , are the designed macro variables; k1, k2 are diagonal matrices, and all diagonal elements are greater than 0; p, q are both positive odd numbers, and satisfy . Both m and n are positive odd numbers, and satisfy the following conditions:

[0063] Combining the hyperlocal model of the robotic arm and algebraic identification technology, the fixed-time cooperative control law based on the hyperlocal model of the robotic arm is derived as follows:

[0064]

[0065] in: Λ is the reference acceleration for the robotic arm joints. r =diag{α r1 ,α r2 ,…,α rm} is a diagonal matrix, and all elements on the diagonal are positive constants; The known modeling and nonlinear uncertainty F in the hyperlocal model of the robotic arm. r The estimated value;

[0066] Under the control law described above, the macrovariable converges to the invariant manifold within a fixed time upper limit, and the convergence time satisfies:

[0067]

[0068] Wherein, λ(T) r -1 ) min For diagonal matrix T r -1 The smallest eigenvalue is a positive constant.

[0069] Macro variables reach invariant manifold ψ r After the value equals 0, both the displacement error and velocity error of the robotic arm converge within a fixed time upper limit, and the convergence time satisfies the following:

[0070]

[0071] For the proposed fixed-time cooperative control based on a hyperlocal model of a robotic arm, a fitness function is designed, and key control parameters are determined through training using the whale optimization algorithm.

[0072] First, set the number of search agents and the maximum number of iterations for WOA, and initialize the search range of key control parameters;

[0073] Secondly, an improved time error integral is used as the fitness function to simultaneously consider error convergence speed and overshoot suppression. The fitness function J is calculated as follows:

[0074]

[0075] Among them, T end Let e1(t) be the simulation duration, e1(t) be the displacement error of the robotic arm, λ be the penalty coefficient for overshoot, ρ be the overshoot, and t be the time variable.

[0076] Then, iterative optimization is performed;

[0077] Finally, the key control parameter T optimized through WOA will be... r Substitute k1 and k2 into the fixed-time cooperative control law, and perform a simulation test on the trajectory tracking of the robotic arm based on the obtained fixed-time cooperative control law. Determine whether the optimization convergence criterion is met. If it is met, output the optimal control parameters; otherwise, continue iterating.

[0078] The process of achieving trajectory tracking control of the robotic arm based on a fixed-time cooperative control law includes the following steps:

[0079] By using distributed high-precision encoders and torque sensors, the robot arm can acquire real-time information on the angles, angular velocities, end-effector poses, and load torques of each joint.

[0080] Calculate the deviation between the current trajectory state and the reference trajectory state;

[0081] Generate a fixed-time cooperative control law based on the hyperlocal model of the robotic arm:

[0082]

[0083] Among them: Λ r =diag{α r1 ,α r2 ,…,α rn} is a diagonal matrix, and all elements on the diagonal are positive constants; The desired acceleration of the robotic arm joint; The known modeling and nonlinear uncertainty F in the hyperlocal model of the robotic arm. r The estimated value; Both are n-dimensional column vectors, representing the joint displacement error and velocity error, respectively; T r =diag{T r1 ,T r2 ,…,T rn} is a diagonal matrix, and all elements on its diagonal are greater than 0; ψ is a time constant. r =[ψ r1 ,ψ r2 ,…,ψ rn ] T , are the designed macro variables; k1, k2 are diagonal matrices, and all diagonal elements are greater than 0; p, q are both positive odd numbers, and satisfy . Both m and n are positive odd numbers, and satisfy the following conditions: The generated τ * The signal is converted into a drive signal by the power amplifier module, and then transmitted through the heterogeneous actuator to achieve precise distribution and coordinated output of joint torque, ensuring that the end effector moves along the desired trajectory.

[0084] Beneficial effects

[0085] The robotic arm trajectory tracking control method based on fixed-time cooperative control of hyperlocal model of the present invention can not only generate a jitter-free control signal compared with the prior art, but also realize that the convergence time of the robotic arm tracking error is no longer constrained by the initial state, and the tracking error can converge to zero within a fixed maximum upper limit time.

[0086] This invention establishes a hyperlocal model of the robotic arm based on a dynamic model that takes into account uncertainties. It then uses algebraic identification techniques to estimate the known and uncertain parts of the robotic arm's dynamic model online, laying the foundation for the proposed control system to break free from the dependence on precise modeling of the robotic arm.

[0087] This invention not only solves the problem of dependence on accurate modeling of the controlled system and its asymptotic convergence in traditional cooperative control, but also improves the dynamic control performance and steady-state control accuracy of the system, and enhances the robustness of the robotic arm against external disturbances. Attached Figure Description

[0088] Figure 1 This is a sequence diagram of the method of the present invention;

[0089] Figure 2 Here is the control block diagram for an n-DOF robotic arm;

[0090] Figure 3 A comparison chart of trajectory tracking curves of a two-degree-of-freedom robotic arm under different initial states;

[0091] Figure 4 A speed comparison chart for determining controller parameters using the trial-and-error method and the WOW algorithm;

[0092] Figure 5 , Figure 6 A comparison chart showing the trajectory tracking of a two-degree-of-freedom robotic arm with parameters obtained from the trial-and-error method and the WOW algorithm. Detailed Implementation

[0093] To provide a better understanding of the structural features and effects achieved by the present invention, a detailed description is provided below, accompanied by preferred embodiments and accompanying drawings:

[0094] like Figure 1 and Figure 2 As shown, the present invention provides a robotic arm trajectory tracking control method based on hyperlocal model fixed-time cooperative control, comprising the following steps:

[0095] The first step is to establish a hyperlocal model of the robotic arm.

[0096] The model analysis of robotic arms is typically conducted from two perspectives: kinematics and dynamics. Generally, the pose of the end effector of a robotic arm is described in Cartesian space, while the joint variables of the robotic arm are defined in joint space. Kinematics primarily establishes the correspondence between the pose of the end effector in joint space and the workspace from a motion perspective. The dynamics of the robotic arm, on the other hand, involves designing the control torques required for each joint to enable the robotic arm to move along a specified trajectory.

[0097] Therefore, the control variables involved in kinematics are joint position and velocity, while the control variables involved in dynamics are joint torque. The Lagrange method is used to model the dynamics of the robotic arm.

[0098] For an n-DOF robotic arm, its Lagrange function can be expressed as:

[0099] L = KP

[0100] Where K and P represent the total kinetic energy and total potential energy of the robotic arm, respectively.

[0101] Therefore, the motion equations of each joint of the robotic arm are:

[0102]

[0103] Where, τ ri q is the input torque of the i-th joint of the robotic arm. i , It represents the angular displacement and angular velocity of the i-th joint in the generalized coordinate system.

[0104] Combining the robotic arm's Lagrange function, the above equation can be rewritten as:

[0105]

[0106] Based on the above equation, the dynamic model of the n-degree-of-freedom robotic arm can be further obtained as follows:

[0107]

[0108] in, It is an n-dimensional column vector, representing the displacement vector, velocity vector, and acceleration vector of the robotic arm joints, respectively; It is an n-dimensional column vector representing the input torque vector; The inertia matrix represents the robotic arm; Represents the Coriolis and centripetal force matrices;

[0109] It is an n-dimensional column vector representing the gravity matrix. Considering the uncertainties in the robotic arm system, such as nonlinear friction and unknown loads, an n-dimensional column vector is used. This represents the disturbance torque vector. Besides the robotic arm itself, the robotic arm also includes actuators and reducers. The above-mentioned robotic arm body dynamics model alone cannot describe the complete dynamics of the robotic arm. Therefore, the mathematical model of the joint modules is combined with the robotic arm body dynamics model to establish a complete robotic arm model. Based on the modeling of the robotic arm joint modules, and considering the robotic arm body as the load of the joint modules, we have:

[0110]

[0111] in, All are diagonal matrices, representing the moment of inertia and the coefficient of viscous friction, respectively;

[0112] It is an n-dimensional column vector representing the electromagnetic torque of the permanent magnet synchronous motor; All are diagonal matrices, representing the reduction ratio and transmission efficiency of the harmonic reducer, respectively.

[0113] Therefore, the complete dynamic model of the n-degree-of-freedom robotic arm can be obtained as follows:

[0114]

[0115] in,

[0116] (1) Establish the complete dynamic model of the n-degree-of-freedom robotic arm, and its expression is as follows:

[0117]

[0118] in, It is an n-dimensional column vector, representing the displacement vector, velocity vector, and acceleration vector of the robotic arm joints, respectively; It is an n-dimensional column vector representing the input torque vector; The inertia matrix represents the robotic arm; Represents the Coriolis and centripetal force matrices; It is an n-dimensional column vector representing the gravity matrix; n-dimensional column vector This represents the disturbance torque vector.

[0119] (2) Based on the dynamic model of the robotic arm, let x1 = q, Establish a hyperlocal model of an n-DOF robotic arm.

[0120]

[0121] Where, x1=[x 11 ,x 12 ,…,x 1n ] T ,x2=[x 21 ,x 22 ,…,x 2n ] T All are n-dimensional column vectors, representing joint displacement and velocity; It is the derivative of x1, representing the joint velocity; It is the derivative of x², representing the joint acceleration; Λ r =diag{α r1 ,α r2 ,…,α rn} is a diagonal matrix, and all elements on the diagonal are positive constants; Let be an n-dimensional column vector representing the known modeling and nonlinear uncertainty components of the robotic arm;

[0122]

[0123] Among them, M -1 (q) is the inverse of the inertia matrix M(q) of the robotic arm.

[0124] The second step involves estimating the n-dimensional column vectors in the hyperlocal model of the robotic arm, which contain known modeling and nonlinear uncertainties. This estimation is performed using a non-asymptotically convergent algebraic identification techniques to evaluate F. r To make an estimate, the following steps are included:

[0125] (1) Represent a single-input, single-output v-order nonlinear system in the form of a hyperlocal model:

[0126] y (v) =F+αu;

[0127] Where u and y represent the system input and output respectively, v is the highest order of the system, set to 1 or 2, α is the scaling factor of the system input, used to keep the system output y and input u on the same order of magnitude, and F represents the sum of system uncertainty and nonlinearity, which includes all system information except for the αu term;

[0128] Assuming v = 1, performing a Laplace transform on the above equation yields:

[0129]

[0130] Where y0 represents the initial state of the system, Y is the frequency domain representation of the system output y(t) through the Laplace transform, s is the complex frequency domain variable in the Laplace transform, and U is the frequency domain representation of the input u(t) through the Laplace transform. It is an estimate of F in the frequency domain through the Laplace transform;

[0131] To eliminate the influence of the system's initial state, multiply both sides of the equation by d / ds, then:

[0132]

[0133] To avoid the adverse effects of differentiation, both sides of the above equation are multiplied by s. -2 Then, an inverse Laplace transform is performed to obtain the time-domain representation of the algebraic identification technique:

[0134]

[0135] Among them, T F =n F T s T represents the size of the sliding window. s The sampling period is n. F This is the length of the time window, set according to the actual system; δ represents the time variable, y(δ) is the system output, and u(δ) is the system input;

[0136] (2) Hyperlocal model based on robotic arm:

[0137]

[0138] Where, x1=[x 11 ,x 12 ,…,x 1n ] T ,x2=[x 21 ,x 22 ,…,x 2m ] T All are n-dimensional column vectors, representing the displacement and velocity of each joint; It is the derivative of x1, representing the joint velocity; It is the derivative of x², representing the joint acceleration; It is an n-dimensional column vector representing the input torque vector;

[0139] Λ r =diag{α r1 ,α r2 ,…,α rn} is a diagonal matrix, and all elements on the diagonal are positive constants;

[0140] Let be an n-dimensional column vector representing the known modeling and nonlinear uncertainty components of the robotic arm;

[0141] Combining the above algebraic identification techniques, we obtain the n-dimensional column vector F. r The estimated value for:

[0142]

[0143] The third step involves designing macrovariables and dynamic evolution equations based on fixed-time stability theory. According to fixed-time stability theory and combined with the robotic arm's dynamic control objectives, the macrovariables are designed in a fixed-time form, and dynamic evolution equations are designed to constrain the way the macrovariables move to the invariant manifold.

[0144] The core idea of ​​cooperative control is the principle of directional self-organization. Specifically, it refers to the process of achieving the system's control objective and guiding the system to an equilibrium state by forming cooperative control that maintains the dynamic invariance of the system's internal or external characteristics, enabling the interaction and feedback between the system's state variables. The cooperative control evolution process is divided into two stages: the first stage is the process of the system's state variables reaching the invariant manifold, in which the system's motion depends on the design of the dynamic evolution equations; the second stage is the process of the system moving to an equilibrium state after reaching the invariant manifold. The invariant manifold, defined as the constraints in the system's state space, determines the motion of this process, and the form of the invariant manifold is designed according to the final control objective.

[0145] If we write the nth-order nonlinear system in the form of differential equations, then:

[0146]

[0147] Where x = [x1, ..., x n ] T Represents system state variables; It is the derivative of x;

[0148] f i g n A smooth nonlinear function describing the system dynamics; u represents the system input; t is time; f nThe dynamic function representing the nth state variable depends on the system's state vector x;

[0149] Based on cooperative control, the control input u is obtained to ensure that the system moves from any initial state to an invariant manifold and then to an equilibrium state. The macro variables are represented as follows, according to the control objective and the settling time:

[0150] M1 = {x:ψ = ψ(x,t) = 0},

[0151] Where M1 is the set defining the system equilibrium state, ψ is the designed macro variable, and ψ(x,t) is the function describing the system state;

[0152] By designing macro variables as linear combinations of system state variables, the control objective of the system is transformed into making the system converge to an invariant manifold, i.e., ψ = 0. Once the manifold design is completed, it is equivalent to adding new constraints in the state space, thus achieving system order reduction.

[0153] Define a dynamic evolution equation to ensure that the macrovariables converge to an invariant manifold; the design form of the dynamic evolution equation is expressed as:

[0154]

[0155] Where T is the time parameter to be designed, representing the time it takes for the state variable to converge to the manifold through the dynamic process, which affects the convergence speed of the system state. The smaller the value of T, the faster the dynamic response speed of the system. However, its value is constrained by the system stability requirements. Too small a value of T will lead to system instability.

[0156] set up It is a differentiable function of ψ, and must satisfy the following conditions:

[0157] Reversible and differentiable

[0158]

[0159] Fixed-time stability theory, consider the following differential equation

[0160]

[0161] Where y is a function of time t, describing the state or response of the system at any time; y is the derivative of the function y; y0 represents the initial conditions of the system at t=0; a and b are both greater than 0; m and n are both positive odd numbers and satisfy the following conditions:

[0162] Assuming the convergence time of the differential equation to the origin is T(y0), then y will converge within a fixed upper bound time T. max(y) converges to the equilibrium point, i.e., the origin, and the convergence time is:

[0163]

[0164] The design of macrovariables and dynamic evolution equations based on fixed-time stability theory includes the following steps:

[0165] (1) Definition Let the reference displacement, reference velocity, and reference acceleration be the values ​​for each joint of the robotic arm, and define them as follows:

[0166]

[0167] in, It is an n-dimensional column vector, representing the displacement vector, velocity vector, and acceleration vector of the robotic arm joints, respectively; Let each be an n-dimensional column vector, representing the desired displacement vector, desired velocity vector, and desired acceleration vector of the robotic arm joints, respectively. All are n-dimensional column vectors, representing the joint displacement error, velocity error, and acceleration error, respectively; It is the derivative of e1, representing the velocity error of the joint; It is the derivative of e², representing the acceleration error of the joint.

[0168] (2) Given that the goal of robotic arm dynamics control is to enable each joint of the robotic arm to quickly track a given reference trajectory, which includes position tracking and velocity tracking of each joint, the macro variable is defined as a combination of joint displacement error and velocity error:

[0169]

[0170] Where: k is a diagonal matrix, and all elements on the diagonal are greater than 0;

[0171] When the system state moves to the invariant manifold ψ = 0, the convergence rate of the state error decreases as the state error decreases, eventually approaching zero. This means the system state can achieve asymptotic convergence, but it is not finite-time convergence, and the convergence time is still related to the initial state. However, this finite-time macrovariable design, where the convergence time of the system state depends on the initial state, makes the dynamic performance analysis of the system quite complex. Therefore, there is an urgent need to develop a fixed-time convergent macrovariable that is independent of the initial state of the system.

[0172] Therefore, combining the fixed-time stability theory, and designing the above macrovariables in a fixed-time form, we get:

[0173] ψ r =e2+k1e1 m / n +k2e1 n / m ,

[0174] Where, ψ r =[ψ r1 ,ψ r2 ,…,ψ rn ] T k1 and k2 are diagonal matrices, and all diagonal elements are greater than 0; m and n are both positive odd numbers, and satisfy the following conditions: Differentiating the above equation, we get:

[0175]

[0176] (3) After completing the macrovariable design, the motion mode of the robotic arm's state variables from the initial state to the invariant manifold to the equilibrium point is determined. However, the motion mode of the macrovariables from the initial state to the invariant manifold is still undefined. Therefore, it is necessary to constrain the motion mode of the macrovariables to the invariant manifold by designing dynamic evolution equations. The dynamic evolution equations are designed as follows:

[0177]

[0178] Where p and q are both positive odd numbers, and satisfy... T r =diag{T r1 ,T r2 ,…,T rn} is a diagonal matrix, and all its diagonal elements are greater than 0. It is a time constant, representing the time it takes for the state variables to converge to the manifold through a dynamic process. Theoretically, T r The smaller the value of , the faster the system's dynamic response speed;

[0179] When the dynamic evolution equation is designed as the above equation, then we have First of all, Differentiating, we get:

[0180]

[0181] Since p and q are both positive odd numbers, and satisfy... Therefore, qp is a positive even number. It is a monotonic function, therefore it satisfies the condition: Reversible and differentiable;

[0182] When ψ r When = 0, according to The following conditions were met: according to have to:

[0183]

[0184] Since p and Q are both odd, then p+q is even. Therefore, for any ψr ≠0, That is, the conditions are met:

[0185] Under the constraints of the designed dynamic evolution equation, the macrovariable ψ r It converges within a fixed time limit and remains on the invariant manifold ψ. r =0.

[0186] The fourth step involves deriving a fixed-time cooperative control law by combining the robotic arm's hyperlocal model and the Whale Optimization Algorithm (WOA). The fixed-time cooperative control law is obtained by combining the designed macrovariables and dynamic evolution equations with the hyperlocal model, and then using the Whale Optimization Algorithm to tune the controller parameters. This includes the following steps:

[0187] (1) Substituting the designed macrovariables into the dynamic evolution equation, we have:

[0188]

[0189] in: Each is an n-dimensional column vector, representing the joint's displacement error, velocity error, and acceleration error, respectively; T r =diag{T r1 ,T r2 ,…,T rn} is a diagonal matrix, and all elements on its diagonal are greater than 0; ψ is a time constant. r =[ψ r1 ,ψ r2 ,…,ψ rn ] R , are the designed macro variables; k1, k2 are diagonal matrices, and all diagonal elements are greater than 0; p, q are both positive odd numbers, and satisfy . Both m and n are positive odd numbers, and satisfy the following conditions:

[0190] Combining the hyperlocal model of the robotic arm and algebraic identification technology, the fixed-time cooperative control law based on the hyperlocal model of the robotic arm is derived as follows:

[0191]

[0192] in: Λ is the reference acceleration for the robotic arm joints. r =diag{α r1 ,α r2 ,…,α rn} is a diagonal matrix, and all elements on the diagonal are positive constants; The known modeling of the robotic arm and the nonlinear uncertainty part F r The estimated value;

[0193] Under the control law described above, the macrovariable converges to the invariant manifold within a fixed time upper limit, and the convergence time satisfies:

[0194]

[0195] Wherein, λ(T) r -1 ) min For diagonal matrix T r -1 The smallest eigenvalue is a positive constant.

[0196] Macro variables reach invariant manifold ψ r After the value equals 0, both the displacement error and velocity error of the robotic arm converge within a fixed time upper limit, and the convergence time satisfies the following:

[0197]

[0198] (2) For the parameters of the proposed fixed-time collaborative controller based on the hyperlocal model of the robotic arm, a fitness function is designed and determined by training based on the whale optimization algorithm.

[0199] Time parameter T r T is a crucial parameter affecting the convergence speed of macro variables, and its value directly determines the dynamic performance of the robotic arm. r The smaller the value of T, the faster the convergence speed of the macro variable; however, an excessively small T... r Choosing the wrong value can cause the robotic arm to become unstable. Therefore, it is necessary to minimize the time parameter T while ensuring the stable operation of the robotic arm. r The values ​​of parameters k1 and k2 primarily affect the convergence speed of the robot arm's state variables after the macro variables reach the invariant manifold. Therefore, determining T... r k1 and k2 are key parameters of the controller.

[0200] First, set the number of search agents and the maximum number of iterations for the WOA algorithm, and initialize the search range of the key parameters of the controller.

[0201] Secondly, the improved integral of time error (IITAE) is used as the fitness function to simultaneously consider error convergence speed and overshoot suppression. The fitness calculation formula is as follows:

[0202]

[0203] Among them, T endLet e1(t) be the simulation duration, e1(t) be the displacement error of the robotic arm, λ be the penalty coefficient for overshoot, and ρ be the overshoot.

[0204] Then, iterative optimization is performed. The most significant feature of the WOA algorithm compared to other metaheuristic algorithms is its simulation of hunting behavior. Its predatory behavior includes three phases: surrounding the prey, bubble net attack, and exploration phase.

[0205] Encircle the prey: Adjust the position of the search agent using the current optimal solution to gradually converge it.

[0206]

[0207] in, The distance between the current search agent and the optimal agent position is represented by t, where t represents the current iteration number. This represents the position of the current optimal agent, i.e., the optimal solution. This is the coefficient vector.

[0208] The exploitation phase of a bubble web attack: Based on a shrinking encirclement mechanism and a logarithmic spiral search strategy, different search methods are selected with a 50% probability.

[0209] Exploration phase of random prey search: Randomly select a reference agent to adjust the position of the search agent globally.

[0210] Finally, the key control parameter T optimized through WOA will be... r Substitute k1 and k2 into the fixed-time cooperative control law, and perform a simulation test on the trajectory tracking of the robotic arm based on the obtained fixed-time cooperative control law. If the condition is met, output the optimal control parameters; otherwise, continue iterating.

[0211] The fifth step is to implement trajectory tracking control of the robotic arm based on the fixed-time cooperative control law: the optimized cooperative control law is embedded into the robotic arm control system to realize trajectory tracking control of the robotic arm.

[0212] (1) By using distributed high-precision encoders and torque sensors, the angle, angular velocity, end pose and load torque information of each joint of the robotic arm can be obtained in real time.

[0213] (2) Calculate the deviation between the current trajectory state and the reference trajectory state.

[0214] (3) Substitute the real-time status information into the design's control law prototype:

[0215]

[0216] Among them: Λ r =diag{α r1 ,α r2 ,…,α rn} is a diagonal matrix, and all elements on the diagonal are positive constants; The desired acceleration of the robotic arm joint; It is an estimate of the known modeling of the robotic arm and the nonlinear uncertainty F; Both are n-dimensional column vectors, representing the joint displacement error and velocity error, respectively; T r =diag{T r1 ,T r2 ,…,T rn} is a diagonal matrix, and all elements on its diagonal are greater than 0; ψ is a time constant. r =[ψ r1 ,ψ r2 ,…,ψ rn ] T , are the designed macro variables; k1, k2 are diagonal matrices, and all diagonal elements are greater than 0; p, q are both positive odd numbers, and satisfy . Both m and n are positive odd numbers, and satisfy the following conditions:

[0217] The calculated control command τ * The signal is converted into a drive signal by a power amplifier module, and then used by heterogeneous actuators such as harmonic deceleration servo motors to achieve precise distribution and coordinated output of joint torque, ensuring that the end effector moves along the desired trajectory.

[0218] The experimental environment used in this invention is as follows: Taking a two-degree-of-freedom (DOF) robotic arm as an example, a simulation model of the two-DOF robotic arm is established to verify the effectiveness of the proposed fixed-time cooperative control based on the hyperlocal model of the robotic arm. In the simulation study, the sampling step size of the robotic arm dynamics control is set to 1 ms, and the total simulation time is set to 20 s. The two-DOF robotic arm dynamics model used in the simulation is as follows:

[0219]

[0220] in, The inertia matrix represents the robotic arm; Represents the Coriolis and centripetal force matrices; It is an n-dimensional column vector representing the gravity matrix, and q1 and q2 represent the angular displacements of joint 1 and joint 2 of the robotic arm, respectively. This represents the angular velocities of joints 1 and 2 of the robotic arm. a4=(m1l c1 +m2l1)g,a5=m2l c2 g. Where m1 and m2 are the masses of connecting rod 1 and connecting rod 2, respectively, and l1 and l2 are the lengths of connecting rod 1 and connecting rod 2, respectively. c1 ,l c2I1 and I2 are the distances from the center of mass of link 1 to joint 1 and from the center of mass of link 2 to joint 2, respectively; I1 and I2 are the moments of inertia of link 1 and link 2, respectively; and g is the acceleration due to gravity.

[0221] The parameters of the two-degree-of-freedom robotic arm are shown in Table 1:

[0222] Table 1. Parameters of a Two-DOF Robotic Arm

[0223]

[0224] The controller parameters are shown in Table 2:

[0225] Table 2 Parameters of Fixed-Time Coordination Controller

[0226]

[0227] Set the robotic arm reference trajectory as follows:

[0228] To verify that the convergence time of the ULM-FTSC robotic arm is fixed, three different initial states were selected for simulation studies. Initial state 1 is: q1(0)=[0.5,0.5] T The initial state 2 is: q2(0) = [0,0] T The initial state 3 is:

[0229] q3(0) = [-0.5, -0.5] T Furthermore, the initial velocity of each joint is 0 in all three states. Considering the influence of unknown load disturbances, a disturbance torque τ is applied to the joints of the robotic arm. d = [5+sin(t), 5+sin(t)] T (N·m).

[0230] from Figure 3 It can be seen that for the proposed ULM-FTSC two-DOF manipulator, the convergence time of the displacement error of the two joints of the manipulator under three different initial states remains at around 1.6s, and the convergence time of the displacement error does not increase significantly with the increase of the initial position error. Therefore, Figure 3 The simulation results shown confirm that the proposed ULM-FTSC has a fixed-time convergence characteristic, and the convergence time is independent of the initial state.

[0231] Figure 4 , Figure 5 , Figure 6 The simulation comparison of the control performance of a two-degree-of-freedom robotic arm is based on controller parameter values ​​determined by trial and error and the WOA algorithm. From... Figure 4 It can be seen that the WOA algorithm has a fast convergence speed and can quickly determine the key parameters of the controller. From Figure 5 , Figure 6 It can be seen that, compared with the trial-and-error method for determining controller parameters, the key controller parameters determined based on the WOA algorithm can reduce the dynamic adjustment time of the robotic arm and improve its control accuracy, thereby enabling each joint of the robotic arm to quickly and accurately track the reference trajectory.

[0232] The foregoing has shown and described the basic principles, main features, and advantages of the present invention. Those skilled in the art should understand that the present invention is not limited to the above embodiments. The embodiments and descriptions in the specification are merely principles of the invention. Various changes and modifications can be made to the invention without departing from its spirit and scope, and all such changes and modifications fall within the scope of the claimed invention. The scope of protection claimed by the appended claims and their equivalents is defined.

Claims

1. A robotic arm trajectory tracking control method based on fixed-time cooperative control using a hyperlocal model, characterized in that, Includes the following steps: 11) Establish a hyperlocal model of the robotic arm; 12) For the hyperlocal model of the robotic arm, which contains known modeling and nonlinear uncertainties. Estimate the column vectors in dimensionality; 13) Design macrovariables and dynamic evolution equations based on fixed-time stability theory; 14) A fixed-time cooperative control law is obtained by combining the hyperlocal model of the robotic arm and the whale optimization algorithm; The fixed-time cooperative control law obtained by combining the hyperlocal model of the robotic arm and the whale optimization algorithm is as follows: Based on the designed macrovariables and dynamic evolution equations combined with the hyperlocal model, the fixed-time cooperative control law is obtained, and the key control parameters are tuned using the whale optimization algorithm; including the following steps: 141) Substituting the designed macrovariables into the dynamic evolution equation, we have: ， in: All The column vectors represent the displacement error, velocity error, and acceleration error of the joint, respectively. It is a diagonal matrix, and all elements on its diagonal are greater than 0, which is a time constant; , is the designed macro variable; It is a diagonal matrix, and all elements on the diagonal are greater than 0; All are positive odd numbers, and satisfy the following conditions: ; All are positive odd numbers, and satisfy the following conditions: ; Combining the hyperlocal model of the robotic arm and algebraic identification technology, the fixed-time cooperative control law based on the hyperlocal model of the robotic arm is derived as follows: ， in: This is the reference acceleration for the robotic arm joints; It is a diagonal matrix, and all elements on the diagonal are positive constants; The known modeling and nonlinear uncertainties in the hyperlocal model of the robotic arm. The estimated value; Under the control law described above, the macrovariable converges to the invariant manifold within a fixed time upper limit, and the convergence time satisfies: ， in, diagonal array The smallest eigenvalue is a positive constant. , ; Macro variables reach invariant manifolds Subsequently, the displacement and velocity errors of the robotic arm converge within a fixed time limit, and the convergence time satisfies the following: ； 142) For the proposed fixed-time cooperative control based on the hyperlocal model of the robotic arm, a fitness function is designed, and key control parameters are determined by training using the whale optimization algorithm: First, set the number of search agents and the maximum number of iterations for WOA, and initialize the search range of key control parameters; Secondly, an improved time error integral is adopted as the fitness function to simultaneously consider error convergence speed and overshoot suppression. The calculation formula is as follows: ， in, For simulation duration, This is the displacement error of the robotic arm. The penalty coefficient for overshoot. For overshoot, It is a time variable. Then, iterative optimization is performed; Finally, the key control parameters optimized through WOA will be... Substitute the fixed-time cooperative control law, and perform a simulation test on the trajectory tracking of the robotic arm based on the obtained fixed-time cooperative control law. Determine whether the optimization convergence criterion is met. If it is met, output the optimal control parameters; otherwise, continue iterating. 15) Implement trajectory tracking control of the robotic arm based on the fixed-time cooperative control law: embed the optimized cooperative control law into the robotic arm control system to realize trajectory tracking control of the robotic arm.

2. The robotic arm trajectory tracking control method based on fixed-time cooperative control using a hyperlocal model according to claim 1, characterized in that, The process of establishing the hyperlocal model of the robotic arm includes the following steps: 21) Establish a complete dynamic model of the n-degree-of-freedom robotic arm, the expression of which is as follows: ， in, for The column vectors represent the displacement vector, velocity vector, and acceleration vector of the robotic arm joints, respectively. for A column vector representing the input torque vector; The inertia matrix represents the robotic arm; Represents the Coriolis and centripetal force matrices; yes A column vector representing the gravity matrix; 3D column vector Represents the disturbance torque vector; 22) Based on the dynamic model of the robotic arm, let Establish a hyperlocal model of an n-DOF robotic arm. ， in, All A column vector representing joint displacement and velocity; yes The derivative of represents the joint velocity; yes The derivative of represents the joint acceleration; It is a diagonal matrix, and all elements on the diagonal are positive constants; for A column vector representing the known modeling and nonlinear uncertainties of the robotic arm; ， in, It is the inertia matrix of the robotic arm. Inverse matrix.

3. The robotic arm trajectory tracking control method based on fixed-time cooperative control using a hyperlocal model according to claim 1, characterized in that, For the hyperlocal model of the robotic arm, which contains known modeling and nonlinear uncertainties. The column vector is estimated as follows: using a non-asymptotically convergent algebraic identification technique... Make an estimate; Includes the following steps: 31) Convert a single-input single-output... A first-order nonlinear system can be represented in the form of a hyperlocal model: ； in, and These represent the system's input and output, respectively. The highest order of the system is set to 1 or 2. The scaling factor is the input to the system, which makes the system output... and input Keep at the same order of magnitude This represents the known modeling and nonlinear uncertainties in the hyperlocal model, including, except for All other parts outside the item; Assumption Applying the Laplace transform to the above equation, we get: ， in, The term represents the initial state of the system. It is the system output. Represented in the frequency domain by the Laplace transform, These are complex frequency domain variables in the Laplace transform. It is input Represented in the frequency domain by the Laplace transform, Yes Estimation in the frequency domain via Laplace transform; To eliminate the influence of the system's initial state, multiply both sides of the equation by... ,but: ， To avoid the noise amplification effect caused by differentiation, both sides of the above equation are multiplied by . Then, an inverse Laplace transform is performed to obtain the time-domain expression for estimating F based on algebraic identification techniques: ， in, Indicates the size of the sliding window. The sampling period is This refers to the length of the time window, which is set according to the actual system. Represents a time variable. For system output, Input for the system; 32) Hyperlocal model based on robotic arm: ， in, All A column vector representing the displacement and velocity of each joint; yes The derivative of represents the joint velocity; yes The derivative of represents the joint acceleration; for A column vector representing the input torque vector; It is a diagonal matrix, and all elements on the diagonal are positive constants; for The column vector represents the known modeling and nonlinear uncertainties in the hyperlocal model of the robotic arm; Combining the above algebraic identification techniques, we obtain 3D column vector The estimated value for: 。 4. The robotic arm trajectory tracking control method based on fixed-time cooperative control using a hyperlocal model according to claim 1, characterized in that, The design of macro variables and dynamic evolution equations based on fixed-time stability theory is as follows: According to fixed-time stability theory and combined with the dynamic control target of the robotic arm, the macro variables are designed in a fixed-time form, and dynamic evolution equations are designed to constrain the way the macro variables move to the invariant manifold. Includes the following steps: 41) Definition Let the reference displacement, reference velocity, and reference acceleration be the values ​​for each joint of the robotic arm, and define them as follows: ， in, for The column vectors represent the displacement vector, velocity vector, and acceleration vector of the robotic arm joints, respectively. for The column vectors represent the desired displacement vector, desired velocity vector, and desired acceleration vector of the robotic arm joints, respectively. All The column vectors represent the displacement error, velocity error, and acceleration error of the joint, respectively. yes The derivative of represents the velocity error of the joint; yes The derivative of represents the acceleration error of the joint; 42) Given that the goal of robotic arm dynamics control is to enable each joint of the robotic arm to quickly track a given reference trajectory, which includes position tracking and velocity tracking of each joint, the macrovariable is defined as a combination of joint displacement error and velocity error; combining the fixed-time stability theory, the above macrovariable is designed in a fixed-time form, resulting in: ， in, ; It is a diagonal matrix, and all elements on the diagonal are greater than 0; All are positive odd numbers, and satisfy the following conditions: Differentiating the above equation, we get: ； 43) By designing the dynamic evolution equations, the way macrovariables move to the invariant manifold is constrained. The dynamic evolution equations are designed as follows: ， in, All are positive odd numbers, and satisfy the following conditions: ; It is a diagonal matrix with all diagonal elements greater than 0, and is a time constant representing the time it takes for the state variables to converge to the manifold through a dynamic process. Theoretically, The smaller the value of , the faster the system's dynamic response speed; When the dynamic evolution equation is designed as the above equation, then we have ,right Differentiating, we get: ， because All are positive odd numbers, and satisfy the following conditions: ,but It is a positive even number, therefore , It is a monotonic function, therefore it satisfies the condition: Reversible and differentiable; when At that time, according to The following conditions were met: , according to have to: ， because If all are odd numbers, then Since it is even, therefore for any... , That is, the conditions are met: ; Under the constraints of the designed dynamic evolution equation, macro variables It converges within a fixed time limit and remains on an invariant manifold. superior.

5. The robotic arm trajectory tracking control method based on fixed-time cooperative control using a hyperlocal model according to claim 1, characterized in that, The process of achieving trajectory tracking control of the robotic arm based on a fixed-time cooperative control law includes the following steps: 51) Real-time information on the angle, angular velocity, end effector pose, and load torque of each joint of the robotic arm is obtained through distributed high-precision encoders and torque sensors. 52) Calculate the deviation between the current trajectory state and the reference trajectory state; 53) Generate a fixed-time cooperative control law based on the hyperlocal model of the robotic arm: ， in: It is a diagonal matrix, and all elements on the diagonal are positive constants; The desired acceleration of the robotic arm joint; The known modeling and nonlinear uncertainties in the hyperlocal model of the robotic arm. The estimated value; All The column vectors represent the displacement error and velocity error of the joint, respectively. It is a diagonal matrix, and all elements on its diagonal are greater than 0, which is a time constant; , is the designed macro variable; It is a diagonal matrix, and all elements on the diagonal are greater than 0; All are positive odd numbers, and satisfy the following conditions: ; All are positive odd numbers, and satisfy the following conditions: ; The generated The signal is converted into a drive signal by the power amplifier module, and then transmitted through the heterogeneous actuator to achieve precise distribution and coordinated output of joint torque, ensuring that the end effector moves along the desired trajectory.

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