CN110000780B - Runge-Kutta periodic rhythm neural network method capable of resisting periodic noise - Google Patents

Runge-Kutta periodic rhythm neural network method capable of resisting periodic noise Download PDF

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CN110000780B
CN110000780B CN201910254767.XA CN201910254767A CN110000780B CN 110000780 B CN110000780 B CN 110000780B CN 201910254767 A CN201910254767 A CN 201910254767A CN 110000780 B CN110000780 B CN 110000780B
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张智军
何铭震
陈思远
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Foshan Shunde Zhike Intelligent Technology Co ltd
South China University of Technology SCUT
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    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B25—HAND TOOLS; PORTABLE POWER-DRIVEN TOOLS; MANIPULATORS
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    • B25—HAND TOOLS; PORTABLE POWER-DRIVEN TOOLS; MANIPULATORS
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Abstract

The invention discloses a Runge-Kutta periodic rhythm neural network method capable of resisting periodic noise, which comprises the following steps: giving a terminal task, performing inverse kinematics analysis on a mechanical arm track by adopting double-index quadratic optimization, converting a quadratic optimization scheme of weighting and indexes of minimum torque and angle deviation two-norm square of the mechanical arm into a standard quadratic programming problem, solving the Carrocon-Kuen-Tak optimization condition by using a continuous time period rhythm neural network to obtain a continuous time model, obtaining a discrete period rhythm neural network by using a Longge Kutta method, and solving by using the neural network to obtain a discrete solution of the original quadratic programming problem. And finally, transmitting the result to a mechanical arm controller to drive the mechanical arm to track the track. The discrete periodic rhythm neural network designed by the invention has the capability of inhibiting periodic noise in the network model, and can eliminate the influence of the initial error of the mechanical arm on the motion planning and successfully plan the motion of the mechanical arm.

Description

Runge-Kutta periodic rhythm neural network method capable of resisting periodic noise
Technical Field
The invention relates to the technical field of motion planning and control of mechanical arms, in particular to a Runge-Kutta periodic rhythm neural network method capable of resisting periodic noise.
Background
The redundant manipulator is a special manipulator with the number of active joints larger than the absolute motion parameters of the tail end. Due to structural particularity, the robot has the advantages of high flexibility, obstacle avoidance, singularity overcoming, fault-tolerant operation realization and the like, and is widely applied to the fields of spaceflight, medical treatment, hazardous material treatment and welding robots.
Because the inverse kinematics mapping equation of the redundant manipulator has nonlinear characteristics, it is difficult to obtain an analytical solution at the position level through the kinematics equation. One common approach is to use a quadratic programming approach, which has great flexibility. In a framework based on a quadratic programming method, a neural network solver is a common solver, and has the advantage of parallel calculation. However, the equation constraints in the existing quadratic programming-based method are directly substituted into the forward kinematics equation of the robot, and the problem of initial error and error accumulation cannot be overcome by the equation constraints. Meanwhile, if there is noise interference in the neural network model, the numerical solution may deviate from the ideal value greatly, or even there is no solution, so that the motion planning of the mechanical arm fails.
In order to solve the two problems, a method is required to be provided, which can solve the inverse kinematics problem of the mechanical arm that can overcome the initial error and the error accumulation problem in a noise environment.
Disclosure of Invention
The invention mainly aims to overcome the defects of the prior art and provide a discrete neural network method for motion planning of a mechanical arm, which can inhibit periodic noise in a model and overcome the problems of initial error and error accumulation.
The object of the present invention is achieved by the following means.
A Runge-Kutta periodic rhythm neural network method capable of resisting periodic noise comprises the following steps:
s1, giving a task at the tail end of the mechanical arm, and performing inverse kinematics analysis on the track of the mechanical arm by adopting a double-index quadratic optimization model;
s2, converting the quadratic optimization model in the step S1 into a quadratic programming problem;
s3, converting the quadratic programming problem in the step S2 into a solution of optimization conditions of Carlo needs, Cohn and Tak;
s4, solving the optimization condition of Carlo-Cohn-Tack in the step S3 by using a continuous time periodic rhythm neural network kinetic equation under a periodic noise environment to obtain a continuous time differential equation;
s5, discretizing the continuous time differential equation in the step S4 by using a linear one-step method to obtain a discretized periodic rhythm neural network and obtain a discretized solution of a quadratic optimization model;
and S6, transmitting the result obtained in the step S5 to the manipulator controller, and driving the redundant manipulator body to track.
Further, the quadratic optimization model of step S1 is a model with the weighted sum of the mixed torque and the squared two-norm angular offset as the optimization index on the acceleration layer, and the function of the quadratic optimization model is to perform inverse kinematics analysis on the trajectory of the mechanical arm so as to reduce the external moment and the joint angular offset applied to the body in the mechanical arm tracking process;
further, the tail end track of the mechanical arm and the joint angle of the mechanical arm satisfy the following relation:
r(t)＝f(θ(t))
where t is time, R (t) e RmThe position vector of the end effector of the redundant manipulator with time as an independent variable is represented as an m-dimensional column vector, m is the working space dimension of the redundant manipulator under a Cartesian coordinate system, RmIs an m-dimensional space, f (-) is a forward kinematics equation of the mechanical arm, is determined by the self structure of the mechanical arm and is generally a smooth nonlinear equation; the second derivative is taken over time t by the equation:
Figure BDA0002013374250000021
wherein
Figure BDA0002013374250000022
Is a jacobian matrix of the redundant manipulator,
Figure BDA0002013374250000023
the partial differential is a function of the difference between the two,
Figure BDA0002013374250000024
the acceleration vector of the end effector of the redundant manipulator is represented as an m-dimensional column vector; theta (t),
Figure BDA0002013374250000025
Respectively representing redundant manipulator in terms of timeThe joint angle vector and the joint angular acceleration vector of the variables are n-dimensional column vectors, which are abbreviated as theta and theta respectively
Figure BDA0002013374250000026
RnIs an n-dimensional space, n represents the dimension of the mechanical arm joint space.
Further, the quadratic optimization model comprises a weighted sum of a hybrid torque and an angular offset squared norm and an acceleration equation constrained to feedback control of the end effector of the mechanical arm; the formula for the weighted sum is as follows:
Figure BDA0002013374250000027
Figure BDA0002013374250000031
where t is time, σ ∈ [0,1 ]]In order to optimize the weighting of the torque in the indicator,
Figure BDA0002013374250000032
respectively represents the joint angular velocity vector of the redundant manipulator with time as independent variable, which is abbreviated as
Figure BDA0002013374250000033
RnIs an n-dimensional space, n represents the dimension of the mechanical arm joint space,
Figure BDA0002013374250000034
equivalent to an optimization index of the angular deviation on an acceleration layer, upsilon is a torque vector,
Figure BDA0002013374250000035
a two-norm representation of a vector; m (theta) is the inertia matrix of the redundant manipulator,
Figure BDA0002013374250000036
is the centrifugal or Coriolis force torque vector, g (theta) is the gravity torque vectorQuantity, alpha and beta are both positive coefficients, and theta (0) represents the initial joint angle of the mechanical arm;
the equation for the acceleration constrained to feedback control of the end effector of the robotic arm is as follows:
Figure BDA0002013374250000037
r(t)＝f(θ(t))
Figure BDA0002013374250000038
λae.g. R and lambdabThe epsilon R is respectively a feedback control coefficient,
Figure BDA0002013374250000039
representing that the speed vector of the end effector of the redundant manipulator is an m-dimensional column vector;
the quadratic optimization model (including the weighted sum and the constraint equation) is used to solve for the joint angular acceleration that minimizes the weighted sum of the joint angular offset and the torque vector two-norm.
Further, the conversion manner of step S2 is as follows:
xT(t)Q(t)x(t)/2+μT(t)x(t)
Jx(t)＝y(t)
Figure BDA00020133742500000310
Q(t)＝(1-σ)I+σMT(θ)M(θ)
Figure BDA00020133742500000311
Figure BDA00020133742500000312
wherein xT(t)Q(t)x(t)/2+μT(t) x (t) is expressed as a performance index function, Jx (t) y (t) is expressed as a constraint function borne by the performance index function, and x (t) is expressed as a joint angular acceleration vector to be solved
Figure BDA0002013374250000046
T is matrix transpose, I is belonged to Rn×nIs an identity matrix, Rn×nThe method comprises the steps of representing an n multiplied by n dimensional space, representing the dimension of a mechanical arm joint space by n, respectively representing J (theta) and theta (t) by J (theta), and representing a rotational inertia matrix of the mechanical arm by M (theta).
Step S2 converts the quadratic optimization model into a standard quadratic programming problem form.
Further, step S3 is to perform transformation by using lagrangian equation, where the transformation process is to combine the constraint function and the performance indicator function to obtain the following lagrangian function L (x) (t), λ (t), t is:
Figure BDA0002013374250000041
where λ (t) is the Lagrangian multiplier vector, λ (t) is ∈ RmAccording to the carlo-kuen-tower optimization conditions:
Figure BDA0002013374250000042
wherein the content of the first and second substances,
Figure BDA0002013374250000043
is a partial differential sign. The system of partial derivative equations obtained according to the optimization conditions of Carlo-Cohen-Tak is written in a matrix form:
A(t)X(t)＝B(t)
wherein the content of the first and second substances,
Figure BDA0002013374250000044
while assuming A (t) as a nonsingular matrix, R(n+m)×(n+m)A space representing (n + m) × (n + m);
further, the solving process of step S4 is as follows, and the error equation is set as follows:
∈(t)＝A(t)X(t)-B(t)
the error vector e (t) represents the difference value of A (t) X (t) and B (t), in order to obtain the X (t) vector to be solved, the periodic rhythm neural network is used to lead the e (t) to trend to a zero vector, and the kinetic equation of the periodic rhythm neural network is as follows:
Figure BDA0002013374250000045
φ(t)＝φ(t-T)+ρ∈(t)
wherein the content of the first and second substances,
Figure BDA0002013374250000051
denotes the derivative of ∈ (t) with respect to time, γ is a positive real number, φ (t) and ω (t) are the compensation matrix and the noise vector, respectively, ω (t) ∈ R(n+m)ρ is a positive real number; the kinetic equation enables the error to be converged at a rate of over exponential (meaning that the convergence form is an exponential function form), the convergence rate is determined by gamma and rho, and the error can be prompted to be converged from any initial error e0The convergence is 0 along with the time, namely the track tracking of the robot can eliminate the disturbance and the error suffered in the control process, and a continuous time differential equation is obtained by combining an error equation and a kinetic equation:
Figure BDA0002013374250000052
wherein A, B and X are respectively the abbreviations of A (t), B (t) and X (t),
Figure BDA0002013374250000053
is divided into A, B, the first derivative of X with respect to time, A-1Is the inverse of the matrix of a,
Figure BDA0002013374250000054
can be considered the slope or rate of change of X. .
Further, the linear one-step method in step S5 is a lattice tata method, and after discretizing the continuous time differential equation in step S4, the following discretized periodic rhythm neural network is obtained:
Figure BDA0002013374250000055
wherein k is an index value, k is a non-negative integer, τ is a step length, and t is k × τ; xkDenotes the value of X (t) at time t ═ k τ, and the value X at the next timek+1From the value X of the current momentkPlus a step τ and an estimated slope (K)1+2K2+2K3+K4) The product of/6; k1Represents the slope of X
Figure BDA0002013374250000056
Taking a value at the time t; k2Representing the time period t, t + tau]The slope of the midpoint is taken, wherein X is the slope K by the Euler method1To determine the value of (t + tau/2), i.e. X ═ Xk+τK1/2；K3Also represents the time period t, t + tau]The slope of the midpoint is taken, but X is taken as the slope K by the Euler method2To determine the value of (t + tau/2), i.e. X ═ Xk+τK2/2；K4Represents the slope of X
Figure BDA0002013374250000057
Taking the value at time t + T, and taking the value X as K3Determining; the value of X at the initial moment is X0Namely, X (0) is a zero vector.
The discretization periodic rhythm neural network is iterated, and the solution X of the moment t ═ k τ is arbitrarily indexedkAnd optimizing the discrete solution of the model for the quadratic form at the index moment.
Compared with the prior art, the invention has the following advantages and beneficial effects:
the invention has the capability of inhibiting noise in the neural network model by designing the discretized periodic rhythm neural network, can eliminate the influence of the initial error of the mechanical arm on the motion planning, and successfully plans the mechanical arm to obtain the motion trail.
Drawings
FIG. 1 is a schematic flow chart of a design concept for solving a Runge-Kutta periodic rhythm neural network for periodic noise resistance;
FIG. 2 is a schematic flow chart of an embodiment;
FIG. 3 is a trace diagram of the hexalobal flower-shaped end task in the example;
FIG. 4 is a graph of noise vectors in an embodiment;
fig. 5 is a schematic diagram of the position error in the cartesian coordinate system of the actual trajectory and the ideal trajectory in the tracking process of the mechanical arm according to the embodiment.
Detailed Description
The following describes the embodiments of the present invention in further detail with reference to the examples and the drawings, but the embodiments of the present invention are not limited thereto.
A method of the longkutta periodic rhythm neural network capable of resisting periodic noise as shown in fig. 1 and 2, comprising the steps of:
s1, giving a terminal task, wherein a quadratic optimization model takes the weighted sum of a mixed torque and the quadratic norm of angular deviation as an optimization index on an acceleration layer, the quadratic optimization model is used for reducing the external moment and the joint angular deviation of a body in the mechanical arm tracking process, and performing inverse kinematics analysis on the mechanical arm track, and the weighted sum formula is as follows:
Figure BDA0002013374250000061
Figure BDA0002013374250000062
where t is time, σ is 0.8, θ (t) is the weight of torque in the optimization index,
Figure BDA0002013374250000063
Respectively representing the joint angle vector, the joint angular velocity vector and the joint angular acceleration vector of the redundant manipulator with time as independent variables as six-dimensional column vectors which are respectively abbreviated as theta, DEG, and DEG, in the form of a linear vector,
Figure BDA0002013374250000071
And
Figure BDA0002013374250000072
R6is a six-dimensional space, the dimension of the mechanical arm joint space is six,
Figure BDA0002013374250000073
Figure BDA0002013374250000074
equivalent to an optimization index of the angular deviation on an acceleration layer, upsilon is a torque vector,
Figure BDA0002013374250000075
a two-norm representation of a vector; m (theta) is the inertia matrix of the redundant manipulator,
Figure BDA0002013374250000076
the vector is a centrifugal force or a coriolis force torque vector, g (θ) is a gravity torque vector, α is 100, β is 2500 is a coefficient, and θ (0) represents an initial joint angle of the robot arm;
the equation for the acceleration constrained to feedback control of the end effector of the robotic arm is as follows:
Figure BDA0002013374250000077
r(t)＝f(θ(t))
Figure BDA0002013374250000078
wherein λ isa＝λb20 are feedback control coefficients respectively; wherein t is time, r (t),
Figure BDA0002013374250000079
respectively representing time-independent position vectors of an end effector of a redundant manipulatorThe velocity vector and the acceleration vector are three-dimensional column vectors, the working space dimension of the redundant manipulator under a Cartesian coordinate system is three, R3The method is a three-dimensional space, wherein f (-) is a forward kinematics equation of the mechanical arm, is determined by the self structure of the mechanical arm and is generally a smooth nonlinear equation; wherein
Figure BDA00020133742500000710
Is a jacobian matrix of the redundant manipulator,
Figure BDA00020133742500000711
is a partial differential sign;
s2, converting the quadratic optimization model in the step S1 into a standard quadratic programming problem;
the transformation mode is as follows:
xT(t)Q(t)x(t)/2+μT(t)x(t)
Jx(t)＝y(t)
Figure BDA00020133742500000714
Q(t)＝(1-σ)I+σMT(θ)M(θ)
Figure BDA00020133742500000712
Figure BDA00020133742500000713
wherein xT(t)Q(t)x(t)/2+μT(t) x (t) is expressed as a performance index function, Jx (t) y (t) is expressed as a constraint function borne by the performance index function, and x (t) is expressed as a joint angular acceleration vector to be solved
Figure BDA0002013374250000084
T is matrix transpose, I is belonged to R6×6Is an identity matrix, R6×6Representing a 6 x 6 dimensional space, the dimensions of the robotic arm joint space are six,j and theta respectively represent J (theta) and theta (t), M (theta) represents a rotational inertia matrix of the mechanical arm, Q (t) represents that a Hessian matrix is a semi-positive definite matrix, and mu (t) represents a quadratic programming primary term coefficient vector.
S3, converting the quadratic programming problem of the mechanical arm in the step S2 into solution of the optimization condition of Carlo needs-Kuen-Tack, specifically, converting by using a Lagrange equation, wherein the conversion process is as follows:
combining the constraint function and the performance indicator function to obtain a Lagrangian function L (x (t), λ (t), t) as: l (x (t), λ (t), t) ═ xT(t)Q(t)x(t)/2+μT(t)x(t)+λT(t)(Jx(t)-y(t))
Wherein λ (t) is a Lagrange multiplier vector, λ (t) is ∈ RmAccording to the carlo-kuen-tower optimization conditions:
Figure BDA0002013374250000081
writing the partial derivative equation in matrix form:
A(t)X(t)＝B(t)
wherein the content of the first and second substances,
Figure BDA0002013374250000082
while assuming A (t) as a nonsingular matrix, R9×9Representing a 9 x 9 dimensional space, and x (t) representing a combined vector of the angular acceleration vector of the mechanical arm joint to be solved and a lagrange multiplier vector.
S4, under the periodic noise environment, solving the optimization condition of Carlo-Couin-Tack in the step S3 by using a continuous time periodic rhythm neural network to obtain a continuous time differential equation; the solution process is as follows, and the error equation is set as follows:
∈(t)＝A(t)X(t)-B(t)
the error vector e (t) represents the difference value of A (t) X (t) and B (t), in order to obtain the X (t) vector to be solved, a periodic rhythm neural network dynamic equation is constructed by utilizing the periodic rhythm neural network to lead the e (t) to trend to a zero vector:
Figure BDA0002013374250000083
φ(t)＝φ(t-T)+ρ∈(t)
where φ (t) and ω (t) are the compensation matrix and the noise vector, respectively, and ω (t) is ∈ R9γ is 1, ρ is 50; the kinetic equation enables the error to be converged at a rate exceeding the exponential, the convergence rate is determined by gamma and rho, and the error can be prompted to be from any initial error e0Convergence to 0 over time means that the trajectory tracking of the robot can eliminate the disturbances and errors suffered in the control process. The ith element in the noise vector is ωi＝p*ζi*fT(10πt)+q*Ψ(t)；p＝200，q＝20，ζiRandom numbers distributed in a Gaussian distribution in nine dimensions, fT(10 π t) denotes sin (10 π t) or coe (10 π t), Ψ (t) is a random number that is distributed in a Gaussian at different times.
The nine sub-graphs of fig. 4 are added to a vector equation a (t) x (t) -b (t) to be solved respectively, nine dimensions are used as noise interference, wherein the noise vector is a vector taking time as an independent variable, the dimension is nine dimensions, the nine sub-graphs are graphs of the nine dimensions in the noise vector respectively, the abscissa of the graph is time, and the ordinate is noise size.
And (3) simultaneous error equations and kinetic equations obtain a continuous time differential equation:
Figure BDA0002013374250000091
wherein A, B and X are respectively the abbreviations of A (t), B (t) and X (t),
Figure BDA0002013374250000092
is divided into A, B, the first derivative of X with respect to time, A-1Is the inverse matrix of a.
S5, discretizing the continuous time differential equation in the step S4 by using a linear one-step method to obtain a solution of a quadratic programming problem; the linear one-step method is a Runge Kutta method, and after the continuous time differential equation in the step S4 is discretized, the following discretized periodic rhythm neural network is obtained:
Figure BDA0002013374250000093
wherein k is an index value, k is a non-negative integer, τ is 0.001, and t is k × τ; xkDenotes the value of X (t) at time t ═ k τ, and the value X at the next timek+1From the value X of the current momentkPlus a step τ and an estimated slope (K)1+2K2+2K3+K4) The product of/6; k1Represents the slope of X
Figure BDA0002013374250000094
Taking a value at the time t; k2Representing the time period t, t + tau]The slope of the midpoint is taken, wherein X is the slope K by the Euler method1To determine the value of (t + tau/2), i.e. X ═ Xk+τK1/2；K3Also represents the time period t, t + tau]The slope of the midpoint is taken, but X is taken as the slope K by the Euler method2To determine the value of (t + tau/2), i.e. X ═ Xk+τK2/2；K4Represents the slope of X
Figure BDA00020133742500001013
Taking the value at time t + T, and taking the value X as K3Determining; the value of X at the initial moment is X0I.e., X (0) is a zero vector, and the discretized periodic rhythm neural network knows that the discretized equation can be self-initiated.
The discretization periodic rhythm neural network is iterated circularly until t is equal to N, wherein N is the time of one motion tracking, and the time N for completing one motion tracking is set to 8 seconds in the embodiment. Solution X at arbitrary index time t ═ k τkThe process is shown in the loop portion of FIG. 2 for a discrete solution to the performance indicator function at that index time.
Figure BDA0002013374250000101
Figure BDA0002013374250000102
Figure BDA0002013374250000103
rz＝0
Figure BDA0002013374250000104
Figure BDA0002013374250000105
Figure BDA0002013374250000106
Figure BDA0002013374250000107
Figure BDA0002013374250000108
Figure BDA0002013374250000109
Wherein the content of the first and second substances,
Figure BDA00020133742500001010
expressing the relation of angle and time under a polar coordinate system, and giving out an expression of the task space position of which the tail end track is in a hexalobal flower shape
Figure BDA00020133742500001011
And the parameter expressions of the speed and the acceleration to the time are respectively
Figure BDA00020133742500001012
Assuming that the angular acceleration at each step is a constant value, calculating the joint angular velocity and the joint angle at each step, and finally obtaining a six-leaf flower-shaped tail end task trajectory diagram shown in FIG. 3; fig. 5 shows the position error in the cartesian coordinate system of the actual trajectory and the ideal trajectory as a function of time during the robot arm tracking process, in this example, the abscissa is time (in seconds) and the ordinate is error (in meters).
And S6, transmitting the result obtained in the step S5 to the manipulator controller, and driving the redundant manipulator body to track. And finally, transmitting the joint angle obtained by solving through the discretization periodic rhythm neural network solver to a manipulator controller, further controlling the redundant manipulator body, realizing the track tracking function of the end effector and realizing the method of the embodiment.
The above embodiments are preferred embodiments of the present invention, but the present invention is not limited to the above embodiments, and any other changes, modifications, substitutions, combinations, and simplifications which do not depart from the spirit and principle of the present invention should be construed as equivalents thereof, and all such changes, modifications, substitutions, combinations, and simplifications are intended to be included in the scope of the present invention.

Claims (7)

1. A Runge-Kutta periodic rhythm neural network method capable of resisting periodic noise is characterized by comprising the following steps:
s1, giving a task at the tail end of the mechanical arm, and analyzing the track of the mechanical arm by adopting a double-index quadratic optimization model;
s2, converting the quadratic optimization model in the step S1 into a quadratic programming problem;
s3, converting the quadratic programming problem in the step S2 into a solution of optimization conditions of Carlo needs, Cohn and Tak;
s4, solving the optimization condition of Carlo-Cohn-Tack in the step S3 by using a continuous time periodic rhythm neural network kinetic equation under a periodic noise environment to obtain a continuous time differential equation;
s5, discretizing the continuous time differential equation in the step S4 by using a linear one-step method to obtain a discretized periodic rhythm neural network and obtain a discretized solution of a quadratic optimization model;
s6, transmitting the result obtained in the step S5 to a manipulator controller, and driving the redundant manipulator body to track;
the quadratic optimization model comprises a weighted sum formula of a mixed torque and an angular offset two-norm square and an acceleration equation constrained to the end effector of the mechanical arm and controlled by feedback; the formula of the weighted sum is as follows:
Figure FDA0003201525270000011
Figure FDA0003201525270000012
where t is time, σ ∈ [0,1 ]]In order to optimize the weighting of the torque in the indicator,
Figure FDA0003201525270000013
the angular velocity vector of the joint which represents the time of the redundant manipulator as the independent variable is abbreviated as
Figure FDA0003201525270000014
RnIs an n-dimensional space, n represents the dimension of the mechanical arm joint space,
Figure FDA0003201525270000015
Figure FDA0003201525270000016
equivalent to an optimization index of the angular deviation on an acceleration layer, upsilon is a torque vector,
Figure FDA0003201525270000017
a two-norm representation of a vector; m (theta) is the inertia matrix of the redundant manipulator,
Figure FDA0003201525270000018
the vector is a centrifugal force or a Coriolis force torque vector, g (theta) is a gravity torque vector, alpha and beta are positive coefficients, and theta (0) represents an initial joint angle of the mechanical arm;
the equation for the feedback-controlled acceleration of the end effector constrained to the robotic arm is as follows:
Figure FDA0003201525270000019
r(t)＝f(θ(t))
Figure FDA0003201525270000021
wherein λaE.g. R and lambdabThe epsilon R is respectively a feedback control coefficient,
Figure FDA0003201525270000022
representing the velocity vector of the end effector of the redundant manipulator as an m-dimensional column vector, f (theta): Rn→RmRepresents a joint angle θ ∈ R from the robot armnThe coordinate R ∈ R to the end effector of the robotic armmThe mapping relationship between them.
2. The method according to claim 1, wherein the quadratic optimization model of step S1 is a model with a weighted sum of two-norm squares of hybrid torque and angular deviation as an optimization index on an acceleration layer, and is used for performing inverse kinematics analysis on the trajectory of the mechanical arm to reduce the external moment and the joint angular deviation applied to the body during the mechanical arm tracking process.
3. The method of claim 1, wherein the trajectory of the tip of the robotic arm and the joint angle of the robotic arm satisfy the following relationship:
r(t)＝f(θ(t))
where t is time, R (t) e RmThe position vector of the end effector of the redundant manipulator with time as an independent variable is represented as an m-dimensional column vector, m is the working space dimension of the redundant manipulator under a Cartesian coordinate system, RmIs m-dimensional space; f (-) is a forward kinematics equation of the mechanical arm, is determined by the self structure of the mechanical arm and is a smooth nonlinear equation; the second derivative is taken over time t by the equation:
Figure FDA0003201525270000023
wherein
Figure FDA0003201525270000024
Is a jacobian matrix of the redundant manipulator,
Figure FDA0003201525270000025
the partial differential is a function of the difference between the two,
Figure FDA0003201525270000026
the acceleration vector of the end effector of the redundant manipulator is represented as an m-dimensional column vector; theta (t),
Figure FDA0003201525270000027
Respectively representing a joint angle vector and a joint angle acceleration vector of the redundant manipulator with time as an independent variable, wherein the joint angle vector and the joint angle acceleration vector are n-dimensional column vectors and are respectively abbreviated as theta and theta
Figure FDA0003201525270000028
RnIs an n-dimensional space, n represents the dimension of the mechanical arm joint space.
4. The method of claim 3, wherein the transformation of step S2 is as follows:
xT(t)Q(t)x(t)/2+μT(t)x(t)
Jx(t)＝y(t)
Figure FDA0003201525270000029
Q(t)＝(1-σ)I+σMT(θ)M(θ)
Figure FDA0003201525270000031
Figure FDA0003201525270000032
wherein x isT(t)Q(t)x(t)/2+μT(t) x (t) is expressed as a performance index function, Jx (t) y (t) is expressed as a constraint function borne by the performance index function, and x (t) is expressed as a joint angular acceleration vector to be solved
Figure FDA0003201525270000033
T is matrix transpose, I is belonged to Rn×nIs an identity matrix, Rn×nThe method comprises the steps of representing an n x n dimensional space, representing the dimension of a joint space of the mechanical arm by n, respectively representing J (theta) and theta (t) by J and theta, respectively representing a rotational inertia matrix of the mechanical arm by M (theta), representing that a Hessian matrix is a semi-positive definite matrix by Q (t), and representing a quadratic programming primary term coefficient vector by mu (t).
5. The method according to claim 4, wherein step S3 is a transformation using lagrangian equation, and the transformation is a combination of the constraint function and the performance indicator function to obtain the following lagrangian function L (x (t), λ (t), t):
Figure FDA0003201525270000034
where λ (t) is the Lagrangian multiplier vector, λ (t) is ∈ RmOptimized according to Carlo-Cohen-TakThe chemical conditions are as follows:
Figure FDA0003201525270000035
wherein the content of the first and second substances,
Figure FDA0003201525270000036
is a partial differential sign; the system of partial derivative equations obtained according to the optimization conditions of Carlo-Cohen-Tak is written in a matrix form:
A(t)X(t)＝B(t)
wherein the content of the first and second substances,
Figure FDA0003201525270000037
while assuming A (t) as a nonsingular matrix, R(n+m)×(n+m)Representing a space of (n + m) × (n + m), x (t) representing a combined vector of an angular acceleration vector of the mechanical arm joint to be solved and a lagrange multiplier vector; b (t) represents
Figure FDA0003201525270000038
Of the matrix of (a).
6. The method of claim 5, wherein the solving process of step S4 is as follows:
let the error equation be:
∈(t)＝A(t)X(t)-B(t)
the error vector e (t) represents the difference value of A (t) X (t) and B (t), in order to obtain the X (t) vector to be solved, the periodic rhythm neural network is used to lead the e (t) to trend to a zero vector, and the kinetic equation of the periodic rhythm neural network is as follows:
Figure FDA0003201525270000041
φ(t)＝φ(t-T)+ρ∈(t)
wherein the content of the first and second substances,
Figure FDA0003201525270000042
denotes the derivative of ∈ (t) with respect to time, γ is a positive real number, φ (t) and ω (t) are the compensation matrix and the noise vector, respectively, ω (t) ∈ R(n+m)ρ is a positive real number; the kinetic equation enables the error to be converged at a rate exceeding the exponential, the convergence rate is determined by gamma and rho, and the error can be prompted to be from any initial error e0The convergence is 0 along with the time, namely the track tracking of the robot can eliminate the disturbance and error suffered in the control process;
and (3) simultaneous error equations and kinetic equations obtain a continuous time differential equation:
Figure FDA0003201525270000043
wherein A, B, X is abbreviated as A (t), B (t), X (t), respectively,
Figure FDA0003201525270000044
and
Figure FDA0003201525270000045
a, B and X first derivatives with respect to time, A-1Is the inverse of the matrix of a,
Figure FDA0003201525270000046
the slope or rate of change of X is considered.
7. The method of claim 6, wherein the linear one-step method in step S5 is a Runge Kutta method, and the discretization of the continuous time differential equation in step S4 yields the following discretized periodic rhythm neural network:
Figure FDA0003201525270000047
wherein k is an index value, k is a non-negative integer, τ is a step length, and t is k × τ; xkDenotes the value of x (t) at time t ═ k τ, and the next timeValue of Xk+1From the value X of the current momentkPlus a step τ and an estimated slope (K)1+2K2+2K3+K4) The product of/6; k1Represents the slope of X
Figure FDA0003201525270000051
Taking a value at the time t; k2Representing the time period t, t + tau]The slope of the midpoint is taken, wherein X is the slope K by the Euler method1To determine the value of (t + tau/2), i.e. X ═ Xk+τK1/2；K3Also represents the time period t, t + tau]The slope of the midpoint is taken, but X is taken as the slope K by the Euler method2To determine the value of (t + tau/2), i.e. X ═ Xk+τK2/2；K4Represents the slope of X
Figure FDA0003201525270000052
Taking the value at time t + T, and taking the value X as K3Determining; the value of X at the initial moment is X0Namely X (0), X (0) is a zero vector;
the discretization periodic rhythm neural network is iterated, and the solution X of the moment t ═ k τ is arbitrarily indexedkAnd optimizing the discrete solution of the model for the quadratic form at the index moment.
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