Mechanical arm trajectory tracking control method based on hyper-local model fixed time cooperative control

Abstract

The invention relates to a mechanical arm trajectory tracking control method based on hyper-local model fixed time cooperative control. Compared with the prior art, the defects that cooperative control depends on accurate modeling of a controlled system and progressive convergence exists are overcome. The method comprises the following steps that a mechanical arm hyper-local model is established; estimating n-dimensional column vectors including known modeling and nonlinear uncertainty parts in the mechanical arm hyper-local model; designing a macro variable and a dynamic evolution equation based on a fixed time stability theory; a fixed time cooperative control law is obtained by combining a mechanical arm hyper-local model and a whale optimization algorithm; and trajectory tracking control over the mechanical arm is achieved according to the fixed time cooperative control law. According to the method, a buffeting-free control signal can be generated, the convergence time of the tracking error of the mechanical arm is not restrained by the initial state any more, and the tracking error can be converged to zero within the fixed maximum upper limit time.

Description

Technical Field

[0001] The present invention relates to the technical field of robotic arm control, specifically to a robotic arm trajectory tracking control method based on fixed-time cooperative control of a super-local model. Background Art

[0002] In recent years, with the rapid development of modern industry, robotic arms have been widely used in industrial, medical, logistics and other fields, and the demand for their fast response and high-precision trajectory tracking control has been increasing continuously. However, due to the characteristics of multi-input multi-output, strong coupling and nonlinearity of robotic arms, and being easily affected by parameter perturbations and external disturbances, the accurate modeling and control of robotic arms still face technical challenges. In order to obtain satisfactory trajectory tracking control performance, advanced control algorithms have been successively proposed, such as Adaptive Control (AC), Model Predict Control (MPC), Sliding Mode Control (SMC), Synergetic Control (SC), etc.

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

[0004] The idea of synergetic control is based on modern mathematics and synergetics. Synergetic control utilizes the self-organization ability of an open system far from equilibrium and the nonlinearity of the system, and realizes the global stability of the entire system by guiding the system state to converge to a predetermined manifold. Similar to sliding mode control, synergetic control is very suitable for reducing the order of the controlled system and has strong robustness to external disturbances. At the same time, synergetic control designs a continuous dynamic process to force the system state to approach the manifold, and it fundamentally eliminates the chattering of sliding mode control by using a completely continuous control law. Synergetic control has the advantages of sliding mode control and has the advantages of no chattering problem of sliding mode control and simple stability analysis, which has attracted the research interest of scholars.

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

[0006] The objective of the present invention is to solve the deficiencies of the existing technology in the dependence of cooperative control on the accurate modeling of the controlled system and its asymptotic convergence, and to provide a robotic arm trajectory tracking control method based on superlocal model fixed-time cooperative control to solve the above problems.

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

[0008] A robotic arm trajectory tracking control method based on superlocal model fixed-time cooperative control, comprising the following steps:

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

[0010] Estimate the n-dimensional column vector containing the known modeling and nonlinear uncertainty parts in the superlocal model of the robotic arm;

[0011] Design the macro variable and the dynamic evolution equation based on the fixed-time stability theory;

[0012] Combine the superlocal model of the robotic arm and the whale optimization algorithm to obtain the fixed-time cooperative control law;

[0013] Implement the trajectory tracking control of the robotic arm according to the fixed-time cooperative control law: embed the optimized cooperative control law into the robotic arm control system to achieve the trajectory tracking control of the robotic arm.

[0014] The establishment of the superlocal model of the robotic arm includes the following steps:

[0015] Establish a complete dynamic model of an n-degree-of-freedom robotic arm, and its expression is as follows:

[0016]

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

[0018] According to the dynamic model of the robotic arm, let x1 = q, Establish a superlocal model of an n-degree-of-freedom robotic arm,

[0019]

[0020] Among them, x1 = [x 11 ,x 12 ,…,x1n T , x2 = [x 21 , x 22 , …, x 2n T are all n - dimensional column vectors, representing joint displacements and velocities; is the derivative of x1, representing joint velocity; is the derivative of x2, representing joint acceleration; Λ r = diag{α r1 , α r2 , …, α rn} is a diagonal matrix, and all elements on the diagonal are positive constants; is an n - dimensional column vector, representing the known modeling and non - linear uncertainty part of the robotic arm;

[0021]

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

[0023] Estimating the n - dimensional column vector containing the known modeling and non - linear uncertainty part in the hyper - local model of the robotic arm: Estimate F r using non - asymptotic convergent algebraic identification techniques; including the following steps:

[0024] Represent a single - input single - output v - order non - linear system in the form of a hyper - local model:

[0025] y (v) = F + αu;

[0026] where, u and y represent the input and output of the system respectively, v is the highest order of the system, set to 1 or 2, α is the proportionality factor of the system input, making the system output y and input u stay in the same order of magnitude, and F represents the known modeling and non - linear uncertainty part in the hyper - local model, including all other parts except the αu term;

[0027] Assume v = 1, perform Laplace transform on the above formula, and get:

[0028]

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

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

[0031]

[0032] To avoid the noise amplification effect caused by differentiation, multiply both sides of the above equation by s -2 , and then perform the inverse Laplace transform. The time-domain expression for estimating F based on the algebraic identification technique is obtained as:

[0033]

[0034] where, T F = n F T s represents the size of the sliding window, T s is the sampling period, n F is the time window length, which is 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] Based on the super-local model of the robotic arm:

[0036]

[0037] where, x1 = [x 11 , x 12 , …, x 1n T , x2 = [x 21 , x 22 , …, x 2n T are both n-dimensional column vectors, representing the joint displacements and velocities; is the derivative of x1, representing the joint velocity; is the derivative of x2, representing the joint acceleration; 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; is an n-dimensional column vector, representing the known modeling and nonlinear uncertain parts in the super-local model of the robotic arm;

[0039] Combining the above algebraic identification technique, the estimated value r of the n-dimensional column vector F is:

[0040] ​​

[0041] The macro variable and the dynamic evolution equation designed based on the fixed-time stability theory are as follows: According to the fixed-time stability theory, combined with the dynamic control objective of the robotic arm, the macro variable is designed in the form of fixed time, and the dynamic evolution equation is designed to constrain the way the macro variable moves to the invariant manifold; the steps are as follows:

[0042] Define as the reference displacement, reference velocity, and reference acceleration of each joint of the robotic arm, and define:

[0043]

[0044] where is an n-dimensional column vector, representing the displacement vector, velocity vector, and acceleration vector of the robotic arm joints respectively; is an n-dimensional column vector, representing the desired displacement vector, desired velocity vector, and desired acceleration vector of the robotic arm joints respectively; are all n-dimensional column vectors, representing the displacement error, velocity error, and acceleration error of the joints respectively; is the derivative of e1, representing the velocity error of the joint; is the derivative of e1, representing the acceleration error of the joint;

[0045] In view of the fact that the objective of the robotic arm dynamic control is to make each joint of the robotic arm quickly track the given reference trajectory, which includes the position tracking and velocity tracking of each joint, the macro variable is defined in the form of a combination of joint displacement error and velocity error; combined with the fixed-time stability theory, the above macro variable is designed in the form of fixed time, and we get:

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

[0047] where ψ r = [ψ r1 , ψ r2 , …, ψ rn T ; k1, k2 are diagonal matrices, and the elements on the diagonal are all greater than 0; m, n are both positive odd numbers, and satisfy Taking the derivative of the above formula, we get:

[0048]

[0049] By designing the dynamic evolution equation to constrain the way the macro variable moves to the invariant manifold, the dynamic evolution equation is designed as:

[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 elements on its diagonal are greater than 0. It is the time constant, representing the time for the state variable to converge to the manifold through the dynamic process. In theory, the smaller the value of T r , the faster the dynamic response speed of the system;

[0052] When the dynamic evolution equation is designed as the above formula, then Taking the derivative of , we get:

[0053]

[0054] Since p and q are both positive odd numbers and satisfy then q - p is a positive even number. Therefore is a monotonic function. Therefore, it satisfies the conditions: It is invertible and differentiable;

[0055] When ψ r = 0, according to it is found that it satisfies the conditions: According to we get:

[0056]

[0057] Since p and q are both odd numbers, then p + q is an even number. Therefore, for any ψ r ≠ 0, that is, it satisfies the conditions:

[0058] Under the constraint of the designed dynamic evolution equation, the macro variable ψ r converges and remains on the invariant manifold ψ r = 0 within a fixed time upper limit.

[0059] The fixed-time cooperative control law obtained by combining the manipulator's superlocal model and the whale optimization algorithm is: The fixed-time cooperative control law is obtained by combining the designed macro variable and the dynamic evolution equation with the superlocal model, and the whale optimization algorithm is used to tune the key control parameters; it includes the following steps:

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

[0061]

[0062] where: They are all n-dimensional column vectors, representing the displacement error, velocity error, and acceleration error of the joint respectively; T r = diag{T r1 , T r2 , …, T rn} is a diagonal matrix, and all elements on its diagonal are greater than 0, which is the time constant; ψ r = [ψ r1 , ψ r2 , …, ψ rn T , which is the designed macro variable; k1, k2 are diagonal matrices, and all elements on the diagonal are greater than 0; p, q are both positive odd numbers, and satisfy m, n are both positive odd numbers, and satisfy

[0063] Combining the hyperlocal model of the manipulator and the algebraic identification technology, the fixed-time cooperative control law based on the hyperlocal model of the manipulator is obtained through derivation as follows:

[0064]

[0065] where: is the reference acceleration of the manipulator joint; Λ r = diag{α r1 , α r2 , …, α rm} is a diagonal matrix, and all elements on the diagonal are positive constants; is the estimated value of the known modeling and nonlinear uncertain part F r in the hyperlocal model of the manipulator;

[0066] Under the action of the above control law, the macro variable converges to the invariant manifold within a fixed-time upper bound, and the convergence time satisfies:

[0067]

[0068] where, λ(T r -1 ) min is the minimum eigenvalue of the diagonal matrix T r -1 , which is a positive constant,

[0069] After the macro variable reaches the invariant manifold ψ r = 0, the displacement error and velocity error of the manipulator both converge within a fixed-time upper bound, and the convergence time satisfies:

[0070]

[0071] ​For the proposed fixed-time cooperative control based on the hyperlocal model of the robotic arm, a fitness function is designed and trained based on the whale optimization algorithm to determine the key control parameters:

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

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

[0074]

[0075] where T end is the simulation duration, e1(t) is the displacement error of the robotic arm, λ is the penalty coefficient of the overshoot, ρ is the overshoot, and t is the time variable.

[0076] Then, perform iterative optimization;

[0077] Finally, substitute the key control parameters T r , k1, and k2 optimized by the WOA into the fixed-time cooperative control law, and conduct a robotic arm trajectory tracking simulation test based on the obtained fixed-time cooperative control law to determine whether the optimization convergence criterion is satisfied. If it is satisfied, output the optimal control parameters; otherwise, continue the iteration.

[0078] The implementation of the trajectory tracking control of the robotic arm according to the fixed-time cooperative control law includes the following steps:

[0079] Through the distributed high-precision encoders and torque sensors, the joint angles, angular velocities, end poses, and load torques of the robotic arm are obtained in real time;

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

[0084] Beneficial effects

[0085] The robotic arm trajectory tracking control method based on the superlocal model fixed-time cooperative control of the present invention can not only generate a control signal without chattering compared with the prior art, but also realizes that the convergence time of the tracking error of the robotic arm is no longer restricted by the initial state, and the tracking error can converge to zero within a fixed maximum upper limit time.

[0086] Based on the dynamic model of the robotic arm considering uncertainties, the present invention establishes a superlocal model of the robotic arm, and uses algebraic identification technology to online estimate the known part and the uncertain part in the dynamic model of the robotic arm, laying a foundation for the proposed control to get rid of the dependence on the precise modeling of the robotic arm.

[0087] The present invention not only solves the dependence of traditional cooperative control on the accurate modeling of the controlled system and its asymptotic convergence problem, 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. Description of the drawings

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

[0089] Figure 2 is the control block diagram of the n-degree-of-freedom robotic arm;

[0090] Figure 3 is the comparison diagram of the trajectory tracking curves of the two-degree-of-freedom robotic arm under different initial states;

[0091] Figure 4 is the speed comparison diagram of determining the controller parameters by the trial-and-error method and the WOW algorithm; ​

[0092] Figure 5 , Figure 6 It is a comparison chart of the trajectory tracking of a two-degree-of-freedom robotic arm with parameters obtained by the trial-and-error method and the WOW algorithm. Specific implementation mode

[0093] To have a further understanding and recognition of the structural features and achieved effects of the present invention, the following is a detailed description with preferred embodiments and accompanying drawings:

[0094] As Figure 1 and Figure 2 shown, a robotic arm trajectory tracking control method based on super-local model fixed-time cooperative control according to the present invention includes the following steps:

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

[0096] The model analysis of the robotic arm is usually carried out from two perspectives: kinematics and dynamics. Generally, the pose of the end effector of the robotic arm is described in the Cartesian space, while the joint variables of the robotic arm are defined in the joint space. Kinematics mainly establishes the correspondence between the pose of the end effector of the robotic arm in the joint space and the working space from the perspective of motion. The dynamics of the robotic arm realizes the movement of the robotic arm along the specified trajectory by designing the control torque required for each joint.

[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-degree-of-freedom robotic arm, its Lagrange function can be expressed as:

[0099] L = K - P

[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 is the input torque of the i-th joint of the robotic arm, and q i , is the angular displacement and angular velocity of the i-th joint in the generalized coordinates.

[0104] Combined with the Lagrange function of the robotic arm, the above formula is rewritten as:

[0105]

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

[0107]

[0108] where, are n-dimensional column vectors, representing the displacement vector, velocity vector, and acceleration vector of the robotic arm joints respectively; is an n-dimensional column vector, representing the input torque vector; represents the inertia matrix of the robotic arm; represents the Coriolis and centripetal force matrix;

[0109] is an n-dimensional column vector, representing the gravity matrix. Considering the influence of uncertain factors such as non-linear friction and unknown loads in the robotic arm system, the n-dimensional column vector is used to represent the disturbance torque vector. In addition to the robotic arm body, the robotic arm also includes actuators and reducers. The above dynamic model of the robotic arm body alone cannot describe the complete dynamic model of the robotic arm. Therefore, the mathematical model of the joint module is combined with the dynamic model of the robotic arm body to establish a complete robotic arm model. Based on the modeling of the robotic arm joint module, the robotic arm body is regarded as the load of the joint module, then there is:

[0110]

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

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

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

[0114]

[0115] where,

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

[0117]

[0118] where, are n-dimensional column vectors, representing the displacement vector, velocity vector, and acceleration vector of the robotic arm joints respectively; is an n-dimensional column vector, representing the input torque vector; represents the inertia matrix of the robotic arm; represents the Coriolis and centripetal force matrix; is an n-dimensional column vector, representing the gravity matrix; n-dimensional column vector represents the disturbance torque vector.

[0119] (2) According to the dynamic model of the robotic arm, let x1 = q, establish a hyperlocal model of the n-degree-of-freedom robotic arm,

[0120]

[0121] where, x1 = [x 11 , x 12 , …, x 1n T , x2 = [x 21 , x 22 , …, x 2n T are both n-dimensional column vectors, representing joint displacements and velocities; is the derivative of x1, representing the joint velocity; 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; is an n-dimensional column vector, representing the known modeling and nonlinear uncertainty part of the robotic arm;

[0122]

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

[0124] In the second step, estimate the n-dimensional column vector containing the known modeling and nonlinear uncertainty part in the hyperlocal model of the robotic arm. Use non-asymptotic convergent algebraic identification technology to estimate F r ; including the following steps:

[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] ​​Among them, u and y represent the input and output of the system respectively, v is the highest order of the system, set to 1 or 2, α is the scale factor of the system input, used to keep the system output y and the input u at the same order of magnitude, F represents the sum of system uncertainty and nonlinearity, and contains all system information except the αu term;

[0128] Assume v = 1, perform Laplace transform on the above formula, and get:

[0129]

[0130] Among them, the y0 term is the initial state of the system, Y is the representation of the system output y(t) in the frequency domain through Laplace transform, s is the complex frequency domain variable in Laplace transform, U is the representation of the input u(t) in the frequency domain through Laplace transform, is the estimate of F in the frequency domain through Laplace transform;

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

[0132]

[0133] To avoid the adverse effects brought by differentiation, multiply both sides of the above formula by s -2 , and then perform inverse Laplace transform, and the expression form of the algebraic identification technology in the time domain is obtained as:

[0134]

[0135] Among them, T F = n F T s represents the size of the sliding window, T s is the sampling period, n F is the time window length, 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 the robotic arm:

[0137]

[0138] Among them, x1 = [x 11 , x 12 , …, x 1n T , x2 = [x 21 , x 22 , …, x 2m T are all n-dimensional column vectors, representing the displacements and velocities of each joint; is the derivative of x1, representing the joint velocity;​​ is the derivative of x2, representing joint acceleration; 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] is an n-dimensional column vector, representing the known modeling and nonlinear uncertainty part of the robotic arm;

[0141] Combining the above algebraic identification techniques, the estimated value of the n-dimensional column vector F r is obtained as follows: is:

[0142]

[0143] In the third step, macro variables and dynamic evolution equations are designed based on the fixed-time stability theory. According to the fixed-time stability theory, combined with the dynamic control objective 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.

[0144] The core idea of cooperative control is the principle of directed self-organization, which specifically refers to the process of achieving the control objective of the system and guiding the system to reach the equilibrium state by forming cooperative control that maintains the dynamic invariant characteristics inside or outside the system, enabling the interaction and feedback between the system state variables; the cooperative control evolution process is divided into two stages: the first stage is the process in which the system state variables reach the invariant manifold, and the motion of the system in this process depends on the design of the dynamic evolution equations; the second stage is the process in which the system moves to the equilibrium state after reaching the invariant manifold, and the invariant manifold, which is defined as the constraint condition in the system state space, determines the motion of this process, and the form of the invariant manifold is designed according to the final control objective.

[0145] Writing the nth-order nonlinear system in the form of a differential equation, then:

[0146]

[0147] where x = [x1, …, x n T represents the system state variables; is the derivative of x;

[0148] f i , g n are smooth nonlinear functions describing the system dynamics; u represents the system input; t is time; f n ​The dynamic function representing the nth state variable depends on the state vector x of the system;

[0149] According to cooperative control, the control input u is obtained to ensure that the system moves from an arbitrary initial state to the invariant manifold and then to the equilibrium state. According to the control objective and the settling time, the macro variable is expressed as:

[0150] M1 = {x: ψ = ψ(x,t) = 0},

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

[0152] The macro variable is designed in the form of a linear combination of system state variables. By this design form of the macro variable, the control objective of the system is transformed into making the system converge to the invariant manifold, i.e., ψ = 0; Once the manifold design is completed, it is equivalent to adding new constraint conditions in the state space, achieving system order reduction;

[0153] Set the designed dynamic evolution equation to ensure that the macro variable converges to the 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 for the state variable to converge to the manifold through the dynamic process, affecting the convergence speed of the system state. The smaller the value of T, the faster the system dynamic response speed, but its value is restricted by the system stability requirement. An overly small value of T leads to system instability;

[0156] Set is a differentiable function of ψ and must satisfy the following conditions:

[0157] is invertible and differentiable,

[0158]

[0159] Based on the 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; is the derivative of the function y; y0 represents the initial condition of the system at t = 0; Both a and b are greater than 0; Both m and n are positive odd numbers and satisfy

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

[0163]

[0164] Designing the macro variable and the dynamic evolution equation based on the fixed-time stability theory includes the following steps:

[0165] (1) Define as the reference displacement, reference velocity, and reference acceleration of each joint of the robotic arm, and define:

[0166]

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

[0168] (2) Given that the goal of the robotic arm dynamics control is to make each joint of the robotic arm quickly track the given reference trajectory, which includes the position tracking and velocity tracking of each joint. Therefore, the macro variable is defined in the form of 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 onto the invariant manifold ψ = 0, the convergence speed of the state error will decrease as the state error decreases, and finally approach 0 infinitely, that is, the system state can achieve asymptotic convergence, but not finite-time convergence, and the convergence time is also related to the initial state of the system. However, this design of the finite-time macro variable whose convergence time is related to the initial state of the system makes the dynamic performance analysis of the system more complex. Therefore, it is urgent to develop a macro variable with fixed-time convergence that is independent of the initial state of the system.

[0172] Therefore, combining the fixed-time stability theory, the above macro variable is designed in the form of fixed time, and we get:

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

[0174] Among them, ψ r =[ψ r1 , ψ r2 ,…, ψ rn T ; k1, k2 are diagonal matrices, and the elements on the diagonal are all greater than 0; m, n are both positive odd numbers, and satisfy Taking the derivative of the above formula, then:

[0175]

[0176] (3) After completing the macro variable design, the motion mode of the manipulator state variable during the stage of moving from the arrival at the invariant manifold to the equilibrium point is determined. However, the motion mode of the macro variable from the initial state to the invariant manifold has not been defined yet. Therefore, it is necessary to constrain the motion mode of the macro variable to the invariant manifold by designing the dynamic evolution equation, and the dynamic evolution equation is designed as:

[0177]

[0178] Among them, p, q are both positive odd numbers, and satisfy T r =diag{T r1 , T r2 ,…, T rn} is a diagonal matrix, and each element on its diagonal is greater than 0. It is a time constant, indicating the time for the state variable to converge to the manifold through the dynamic process. Theoretically, the smaller the value of T r , the faster the system dynamic response speed;

[0179] When the dynamic evolution equation is designed as the above formula, then there is First, take the derivative of , and get:

[0180]

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

[0182] When ψ r =0, according to it is found that it satisfies the condition: According to get:

[0183]

[0184] Since p, Q are both odd numbers, then p + q is an even number. Therefore, for any ψ​r ≠0, i.e., satisfying the condition:

[0185] Under the constraint of the designed dynamic evolution equation, the macro variable ψ r converges and remains on the invariant manifold ψ r = 0 within a fixed time upper bound.

[0186] Step 4: Combine the manipulator's superlocal model and the whale optimization algorithm to obtain a fixed-time cooperative control law. The fixed-time cooperative control law obtained by combining the manipulator's superlocal model and the whale optimization algorithm (WOA) is as follows: The fixed-time cooperative control law is obtained by combining the designed macro variable and the dynamic evolution equation with the superlocal model, and the whale optimization algorithm is used to tune the controller parameters. It includes the following steps:

[0187] (1) Substitute the designed macro variable into the dynamic evolution equation, then we have:

[0188]

[0189] where: are all n-dimensional column vectors, representing the displacement error, velocity error, and acceleration error of the joints respectively; T r = diag{T r1 , T r2 , …, T rn} is a diagonal matrix, and all elements on its diagonal are greater than 0, which are time constants; ψ r = [ψ r1 , ψ r2 , …, ψ rn R , is the designed macro variable; k1, k2 are diagonal matrices, and the elements on the diagonal are all greater than 0; p, q are both positive odd numbers, and satisfy m, n are both positive odd numbers, and satisfy

[0190] Combining the manipulator's superlocal model and algebraic identification technology, the fixed-time cooperative control law based on the manipulator's superlocal model is obtained through derivation as:

[0191]

[0192] where: is the reference acceleration of the manipulator joint; Λ r = diag{α r1 , α r2 , …, α rn} is a diagonal matrix, and all elements on the diagonal are positive constants;​ is the known modeling and non - linear uncertainty part F of the robotic arm r the estimated value;

[0193] Under the action of the above control law, the macro - variable converges to the invariant manifold within a fixed time upper - bound, and the convergence time satisfies:

[0194]

[0195] where, λ(T r -1 ) min is the minimum eigenvalue of the diagonal matrix T r -1 and is a positive constant,

[0196] After the macro - variable reaches the invariant manifold ψ r = 0, both the displacement error and velocity error of the robotic arm converge within a fixed time upper - bound, and the convergence time satisfies:

[0197]

[0198] (2) Design a fitness function for the fixed - time cooperative controller parameters proposed based on the hyper - local model of the robotic arm, and determine them through training based on the whale optimization algorithm.

[0199] The time parameter T r is an important parameter affecting the convergence speed of the macro - variable, and its value directly determines the dynamic performance of the robotic arm. The smaller the value of T r , the faster the convergence speed of the macro - variable. However, too small a value of T r will cause the robotic arm to become unstable. Therefore, it is necessary to minimize the time parameter T r while ensuring the stable operation of the robotic arm. The values of the parameters k1 and k2 mainly affect the convergence speed of the robotic arm state variables after the macro - variable reaches the invariant manifold. Therefore, determine T r , k1, k2 as the key parameters of the controller.

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

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

[0202]

[0203] where, T endis the simulation time, e1(t) is the displacement error of the robot, λ is the penalty coefficient of overshoot, and ρ is the overshoot.

[0204] Then, iterative optimization is performed. Compared with other metaheuristic algorithms, the most notable feature of the WOA algorithm is its simulated hunting behavior. There are three types of predation behaviors: surrounding prey, bubble net attack, and exploration phase.

[0205] Surround the prey: Use the current optimal solution to adjust the search agent position so that it gradually converges

[0206]

[0207] in, represents the distance between the current search agent and the optimal agent position, t represents the current iteration number, represents the current optimal agent, that is, the position of the optimal solution. is the coefficient vector.

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

[0209] Exploration phase of random search for prey: A reference agent is randomly selected to make the search agent adjust its position globally.

[0210] Finally, the key control parameter T optimized by WOA is r , k1, k2 are brought into the fixed-time collaborative control law, and the robot trajectory tracking simulation test is performed based on the obtained fixed-time collaborative control law. If it is satisfied, the optimal control parameters are output, otherwise the iteration continues.

[0211] The fifth step is to realize the trajectory tracking control of the robot arm according to the fixed-time collaborative control law: embed the optimized collaborative control law into the robot arm control system to realize the trajectory tracking control of the robot arm.

[0212] (1) Through distributed high-precision encoders and torque sensors, the robot arm’s joint angles, angular velocities, end-positions, and load torque information can be acquired 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 designed control law prototype:

[0215]

[0216] Where: r =diag{α r1 ,α r2 ,…,α rn} is a diagonal matrix, and all elements on the diagonal are positive constants; is the desired acceleration of the robotic arm joint; is the estimated value of the known modeling and non-linear uncertainty F of the robotic arm; are all n-dimensional column vectors, representing the displacement error and velocity error of the joint respectively; T r = diag{T r1 , T r2 , …, T rn} is a diagonal matrix, and all elements on its diagonal are greater than 0, which is the time constant; ψ r = [ψ r1 , ψ r2 , …, ψ rn T , which is the designed macro variable; k1, k2 are diagonal matrices, and all elements on the diagonal are greater than 0; p, q are both positive odd numbers and satisfy m, n are both positive odd numbers and satisfy

[0217] The calculated control command τ * is converted into a drive signal through the power amplifier module, and through heterogeneous actuators such as harmonic reduction servo motors, precise distribution and coordinated output of joint torques are realized to ensure that the end effector moves along the desired trajectory.

[0218] The experimental environment adopted in the present invention is as follows: taking a two-degree-of-freedom robotic arm as an example, a simulation model of the two-degree-of-freedom 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 of the robotic arm dynamic control is set to 1 ms, the total simulation duration is set to 20 s, and the dynamic model of the two-degree-of-freedom robotic arm used in the simulation is:

[0219]

[0220] Among them, represents the inertia matrix of the robotic arm; represents the Coriolis and centripetal force matrix; is an n-dimensional column vector, representing the gravity matrix, q1, q2 respectively represent the angular displacements of joint 1 and joint 2 of the robotic arm, represents the angular velocities of joint 1 and joint 2 of the robotic arm, a4 = (m1l c1 + m2l1)g, a5 = m2l c2 g. Among them, m1, m2 are the masses of link 1 and link 2 respectively, l1, l2 are the lengths of link 1 and link 2 respectively, l c1 , l c2 ​are the distances from the centroid of link 1 to joint 1 and from the centroid of link 2 to joint 2, \(I_1\) and \(I_2\) 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 manipulator are shown in Table 1:

[0222] Table 1 Parameters of the two-degree-of-freedom manipulator

[0223]

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

[0225] Table 2 Parameters of the fixed-time cooperative controller

[0226]

[0227] Set the reference trajectory of the manipulator as:

[0228] To verify that the convergence time of the manipulator with ULM-FTSC is fixed, three different initial states are selected for simulation research. Among them, the initial state 1 is: \(q_1(0)=[0.5, 0.5]\) T , the initial state 2 is: \(q_2(0)=[0, 0]\) T , and the initial state 3 is:

[0229] \(q_3(0)=[-0.5, -0.5]\) T , and the initial velocities of each joint in the three states are all 0. At the same time, considering the influence of unknown load disturbances, a disturbance torque \(\tau\) d \(=[5 + \sin(t), 5 + \sin(t)]\) T (N·m) is applied at the joints of the manipulator.

[0230] From Figure 3 it can be seen that for the two-degree-of-freedom manipulator with the proposed ULM-FTSC, the convergence times of the displacement errors of the two joints of the manipulator under the three different initial states are all maintained at about 1.6 s, and with the increase of the initial position error, the convergence time of its displacement error does not increase significantly. Therefore, Figure 3 the simulation results shown confirm that the proposed ULM-FTSC has the property of fixed-time convergence, and the convergence time is independent of the initial state.

[0231] Figure 4 、 Figure 5 、 Figure 6 are the simulation comparisons of the control performance of the two-degree-of-freedom manipulator with the controller parameter values determined by the trial-and-error method and the WOA algorithm. From Figure 4 it can be seen that the WOA algorithm has a faster convergence speed and can quickly determine the key parameters of the controller. FromFigure 5 , Figure 6 It can be seen that, compared with the determination of controller parameters by the trial-and-error method, 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 the joints of the robotic arm to quickly and accurately track the reference trajectory.

[0232] The above shows and describes 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 by the above embodiments. What is described in the above embodiments and the specification is only the principle of the present invention. Without departing from the spirit and scope of the present invention, the present invention will have various changes and improvements, and these changes and improvements all fall within the scope of the present invention claimed. The scope of protection required by the present invention is defined by the appended claims and their equivalents.

Claims

1. A robot arm trajectory tracking control method based on hyperlocal model fixed-time cooperative control, characterized in that: The following steps are involved: 11) Establish a hyperlocal model of the robotic arm; 12) Estimate the n-dimensional column vector containing known modeling and nonlinear uncertainty parts in the hyperlocal model of the manipulator; 13) Design macro variables and dynamic evolution equations based on fixed time stability theory; 14) Combining the robot arm hyperlocal model and the whale optimization algorithm to obtain the fixed-time collaborative control law; 15) Realize trajectory tracking control of the robot arm according to the fixed-time collaborative control law: embed the optimized collaborative control law into the robot arm control system to realize the trajectory tracking control of the robot arm.

2. The robot arm trajectory tracking control method based on super local model fixed time cooperative control according to claim 1 is characterized in that: The establishment of the robotic arm hyperlocal model comprises the following steps: 21) Establish a complete dynamic model of the n-DOF manipulator, and its expression is as follows: in, is an n-dimensional column vector, representing the displacement vector, velocity vector and acceleration vector of the robot joint respectively; is an n-dimensional column vector, representing the input moment vector; Represents the inertia matrix of the robot; represents the Coriolis and centripetal force matrices; is an n-dimensional column vector, representing the gravity matrix; represents the disturbance torque vector; 22) According to the dynamic model of the robot arm, let x1 = q, Establish a hyperlocal model of an n-DOF manipulator, Where x1=[x 11 ,x 12 ,…,x 1n ] T ,x2=[x 21 ,x 22 ,…,x 2n ] T Both are n-dimensional column vectors, representing joint displacement and velocity; is the derivative of x1, representing the joint velocity; 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 numbers; is an n-dimensional column vector, representing the known modeling and nonlinear uncertainty parts of the robot; Among them, M -1 (q) is the inverse matrix of the inertia matrix M(q) of the robot.

3. The robot arm trajectory tracking control method based on super local model fixed time cooperative control according to claim 1 is characterized in that: The n-dimensional column vector of the manipulator hyperlocal model containing known modeling and nonlinear uncertainties is estimated as follows: r Make an estimate; this includes the following steps: 31) A single-input single-output v-order nonlinear system is represented as a hyperlocal model: y (v) =F+αu; Where u and y represent the input and output of the system respectively, ν is the highest order of the system, which is set to 1 or 2, α is the scaling factor of the system input, which keeps the system output y and input u at 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; Assuming v = 1, perform Laplace transform on the above formula and we get: Among them, the term y0 is the initial state of the system, Y is the representation of the system output y(t) in the frequency domain through Laplace transform, s is the complex frequency domain variable in Laplace transform, and U is the representation of the input u(t) in the frequency domain through Laplace transform. is the estimate of F in the frequency domain through Laplace transform; To eliminate the influence of the initial state of the system, multiply both sides of the equation by d / ds, then: To avoid the noise amplification caused by differentiation, multiply both sides of the above equation by s -2 , and then perform inverse Laplace transform to obtain the time domain expression of F estimation based on algebraic identification technology: Among them, T F =n F T s represents the size of the sliding window, T s is the sampling period, n F is the time window length, which is set according to the actual system; δ represents the time variable, y(δ) is the system output, and u(δ) is the system input; 32) Hyperlocal model based on robotic arm: Where x1=[x 11 ,x 12 ,…,x 1n ] T ,x2=[x 21 ,x 22 ,…,x 2n ] T They are all n-dimensional column vectors, representing the displacement and velocity of each joint; is the derivative of x1, representing the joint velocity; is the derivative of x2, representing the joint acceleration; is an n-dimensional column vector, representing the input torque vector; r =diag{α r1 ,α r2 ,…,α rn } is a diagonal matrix, and all elements on the diagonal are positive numbers; is an n-dimensional column vector, representing the known modeling and nonlinear uncertain parts in the hyperlocal model of the robot arm; Combining the above algebraic identification techniques, we get the n-dimensional column vector F r Estimated value of for:

4. The robot arm trajectory tracking control method based on super local model fixed time cooperative control according to claim 1 is characterized in that: The method of designing macro variables and dynamic evolution equations based on the fixed time stability theory is as follows: according to the fixed time stability theory and in combination with the robot arm dynamics control target, the macro variables are designed in a fixed time form, and the dynamic evolution equations are designed to constrain the way the macro variables move to the invariant manifold; The following steps are involved: 41) Definition are the reference displacement, reference velocity and reference acceleration of each joint of the robot arm, and are defined as: in, is an n-dimensional column vector, representing the displacement vector, velocity vector and acceleration vector of the robot joint respectively; is an n-dimensional column vector, which represents the expected displacement vector, expected velocity vector and expected acceleration vector of the robot joint respectively; are all n-dimensional column vectors, representing the displacement error, velocity error, and acceleration error of the joint respectively; is the derivative of e1, representing the velocity error of the joint; is the derivative of e2, representing the acceleration error of the joint; 42) Considering that the goal of the robot arm dynamics control is to make each joint of the robot arm quickly track the given reference trajectory, which includes the position tracking and velocity tracking of each joint, the macro variable is defined as the combination of joint displacement error and velocity error; combined with the fixed time stability theory, the above macro variables are designed as fixed time form, and we get: ψ r s2+k1e1 m / n +k2e1 n / m , Among them, ψ r =[ψ r1 ,ψ r2 ,…,ψ rn ] T ; k1, k2 are diagonal matrices, and the elements on the diagonal are all greater than 0; m, n are both positive odd numbers, and satisfy Taking the derivative of the above formula, we get: 43) By designing the dynamic evolution equation, the way the macro variables move to the invariant manifold is constrained, and the dynamic evolution equation is designed as: Among them, 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 elements on its diagonal are greater than 0. It is a time constant, which indicates the time it takes for the state variable to converge to the manifold through the dynamic process. Theoretically, T r The smaller the value of is, the faster the system dynamic response speed is; When the dynamic evolution equation is designed as the above formula, we have right Taking the derivative, we get: Since p and q are both positive odd numbers and satisfy Then qp is a positive even number, so is a monotonic function, so it satisfies the condition: Reversible and differentiable; When r = 0, according to Found that the conditions are met: according to have to: Since p and Q are both odd numbers, p+q is an even number, so for any ψ r ≠0, That is, the conditions are met: Under the constraints of the designed dynamic evolution equation, the macro variable ψ r Converges within a fixed upper bound on time and stays on the invariant manifold ψ r =0 up.

5. The robot arm trajectory tracking control method based on super local model fixed time cooperative control according to claim 1 is characterized in that: The fixed-time collaborative control law obtained by combining the manipulator hyperlocal model and the whale optimization algorithm is: the fixed-time collaborative control law is obtained by combining the hyperlocal model with the designed macro variables and dynamic evolution equations, and the key control parameters are adjusted by using the whale optimization algorithm; including the following steps: 51) Substituting the designed macro variables into the dynamic evolution equation, we have: in: are all n-dimensional column vectors, representing the displacement error, velocity error, and acceleration error of the joint respectively; T r =diag{T r1 ,T r2 ,…,T rn } is a diagonal matrix, and all elements on its diagonal are greater than 0, which is a time constant; ψ r =[ψ r1 ,ψ r2 ,…,ψ rn ] T , is the designed macro variable; k1, k2 are diagonal matrices, and the elements on the diagonal are all greater than 0; p, q are both positive odd numbers, and satisfy m, n are both positive odd numbers and satisfy Combining the robot arm hyperlocal model and algebraic identification technology, the fixed-time cooperative control law based on the robot arm hyperlocal model is derived as follows: in: is the reference acceleration of the robot joint; Λ r =diag{α r1 ,α r2 ,…,α rn } is a diagonal matrix, and all elements on the diagonal are positive numbers; is the known modeling and nonlinear uncertain part F in the hyperlocal model of the robot arm r An estimated value of Under the control law above, the macro variables converge to the invariant manifold within a fixed time limit, and the convergence time satisfies: Among them, λ(T r -1 ) min is a diagonal matrix T r -1 The smallest eigenvalue of is a positive constant, The macro variables reach the invariant manifold ψ r = 0, the displacement error and velocity error of the robot arm converge within a fixed time limit, and the convergence time satisfies: 52) For the proposed fixed-time collaborative control based on the super-local model of the manipulator, a fitness function is designed and the key control parameters are determined by training based on the whale optimization algorithm: First, the number of search agents and the maximum number of iterations of WOA are set, and the search range of key control parameters is initialized; Secondly, the improved time error integral is used as the fitness function to take into account both the error convergence speed and overshoot suppression. The fitness function J is calculated as follows: Among them, T end is the simulation duration, e1(t) is the displacement error of the robot, λ is the penalty coefficient of overshoot, ρ is the overshoot, and t is the time variable. Then, iterative optimization is performed; Finally, the key control parameter T optimized by WOA is r , k1, k2 are introduced into the fixed-time collaborative control law, and the robot trajectory tracking simulation test is performed based on the obtained fixed-time collaborative control law to determine whether the optimization convergence standard is met. If so, the optimal control parameters are output, otherwise the iteration continues.

6. The robot arm trajectory tracking control method based on super local model fixed time cooperative control according to claim 1 is characterized in that: The method of implementing trajectory tracking control of the robot arm according to the fixed time cooperative control law comprises the following steps: 61) Through distributed high-precision encoders and torque sensors, the angle, angular velocity, end position and load torque information of each joint of the robot arm can be obtained in real time; 62) Calculate the deviation between the current trajectory state and the reference trajectory state; 63) Generate a fixed-time collaborative control law based on the hyperlocal model of the manipulator: Where: r =diag{α r1 ,α r2 ,…,α rn } is a diagonal matrix, and all elements on the diagonal are positive numbers; is the expected acceleration of the robot joint; is the known modeling and nonlinear uncertain part F in the hyperlocal model of the robot arm r An estimated value of are all n-dimensional column vectors, representing the displacement error and velocity error of the joint respectively; T r =diag{T r1 ,T r2 ,…,T rn } is a diagonal matrix, and all elements on its diagonal are greater than 0, which is a time constant; ψ r =[ψ r1 ,ψ r2 ,…,ψ rn ] T , is the designed macro variable; k1, k2 are diagonal matrices, and the elements on the diagonal are all greater than 0; p, q are both positive odd numbers, and satisfy m, n are both positive odd numbers and satisfy The generated τ * It is converted into a driving signal by the power amplifier module, and through heterogeneous actuators, precise distribution and coordinated output of joint torque are achieved to ensure that the end effector moves along the desired trajectory.

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