CN111872933A - SCARA robot trajectory tracking control method based on improved quadratic iterative learning control - Google Patents

SCARA robot trajectory tracking control method based on improved quadratic iterative learning control Download PDF

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CN111872933A
CN111872933A CN201911163254.4A CN201911163254A CN111872933A CN 111872933 A CN111872933 A CN 111872933A CN 201911163254 A CN201911163254 A CN 201911163254A CN 111872933 A CN111872933 A CN 111872933A
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vector
track state
moment
error
iterative learning
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CN111872933B (en
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叶凌箭
朱敏峰
马修水
沈非凡
李英道
钟伟红
李园
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Ningbo Institute of Technology of ZJU
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    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B25—HAND TOOLS; PORTABLE POWER-DRIVEN TOOLS; MANIPULATORS
    • B25J—MANIPULATORS; CHAMBERS PROVIDED WITH MANIPULATION DEVICES
    • B25J9/00—Programme-controlled manipulators
    • B25J9/16—Programme controls
    • B25J9/1602—Programme controls characterised by the control system, structure, architecture
    • B25J9/1605—Simulation of manipulator lay-out, design, modelling of manipulator
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B25—HAND TOOLS; PORTABLE POWER-DRIVEN TOOLS; MANIPULATORS
    • B25J—MANIPULATORS; CHAMBERS PROVIDED WITH MANIPULATION DEVICES
    • B25J9/00—Programme-controlled manipulators
    • B25J9/16—Programme controls
    • B25J9/1602—Programme controls characterised by the control system, structure, architecture
    • B25J9/1607—Calculation of inertia, jacobian matrixes and inverses
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B25—HAND TOOLS; PORTABLE POWER-DRIVEN TOOLS; MANIPULATORS
    • B25J—MANIPULATORS; CHAMBERS PROVIDED WITH MANIPULATION DEVICES
    • B25J9/00—Programme-controlled manipulators
    • B25J9/16—Programme controls
    • B25J9/1628—Programme controls characterised by the control loop
    • B25J9/163—Programme controls characterised by the control loop learning, adaptive, model based, rule based expert control
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B25—HAND TOOLS; PORTABLE POWER-DRIVEN TOOLS; MANIPULATORS
    • B25J—MANIPULATORS; CHAMBERS PROVIDED WITH MANIPULATION DEVICES
    • B25J9/00—Programme-controlled manipulators
    • B25J9/16—Programme controls
    • B25J9/1656—Programme controls characterised by programming, planning systems for manipulators
    • B25J9/1664—Programme controls characterised by programming, planning systems for manipulators characterised by motion, path, trajectory planning

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Abstract

Compared with the original QILC algorithm, the robust control method of the SCARA industrial robot based on the improved Quadratic Iterative Learning Control (QILC) adds the speed error term into the optimization equation, so that a new optimization equation is constructed, and the control quantity of the next batch is obtained by solving the optimization equation. In addition, in practical application, an accurate system model cannot be obtained generally, the invention provides a design method for combining the UKF and an improved QILC algorithm, estimating uncertain parameters and system states of a system simultaneously, and providing a dynamic weight matrix, so that the convergence speed is accelerated, and robust trajectory tracking control is realized. In order to inhibit the influence of real-time disturbance and noise, the invention combines PD feedback control, and adds real-time feedback control quantity on the basis of improving feedforward control quantity calculated by QILC algorithm to be used as final control moment. The method provided by the invention can inhibit the influence of uncertain parameters of a system model, has quicker error convergence performance and better transient performance compared with the original QILC algorithm, can realize accurate track tracking even under the conditions of uncertain system parameters and the existence of measurement noise, and has better robustness and stability.

Description

SCARA robot trajectory tracking control method based on improved quadratic iterative learning control
Technical Field
The invention relates to the technical field of industrial robots, in particular to a SCARA robot trajectory tracking control method based on improved quadratic iterative learning control.
Background
With the increasing urgent need for economic structure transformation in China and the acceleration of global market economy, industrial intelligence has become an important subject of manufacturing industry in China. Industrial robots have several advantages in modern industry: 1. compared with the traditional numerical control machine tool, the investment cost of the industrial robot is lower. 2. The flexibility of the industrial robot to reconfigure for different manufacturing tasks is higher.
SCARA industrial robots, also known as planar articulated robots, are one of many types of industrial robots, and are robot arms used in assembly work. The SCARA robot has 3 revolute joints, whose axes are parallel to each other, and is positioned and oriented in a plane. The other joint is a prismatic joint for performing motion of the tip in a direction perpendicular to the plane. The robot has the advantages of light structure and quick response, and the movement speed of the SCARA robot can reach 10m/s, which is several times faster than that of a common articulated robot. It is most suitable for plane positioning and assembling in vertical direction. SCARA robots are also widely used in the fields of plastics industry, automobile industry, electronic product industry, pharmaceutical industry, and food industry. Its main function is to remove parts and assemble them.
In the actual use process, an industrial robot usually has a large number of uncertain factors such as model errors, external disturbances, measurement noises and the like, and the trajectory tracking precision of each joint of the industrial robot is seriously influenced. In this regard, the trajectory tracking control effect can be improved to some extent using external position sensors and direction sensors of high accuracy and high resolution. However, expensive sensors result in excessive investment costs, reducing economic efficiency.
Although conventional control methods such as PID have achieved good control effects, there are still disadvantages, such as difficulty in achieving the desired control effects in the case of external disturbances, measurement noise, and model parameter uncertainty. The iterative learning control can continuously and iteratively learn in the repetitive motion control of the robot, and the position and the tracking error in the previous operation process are utilized to update the control input of the next operation according to the iterative learning control algorithm, so that the tracking error is continuously reduced, and the expected track is finally accurately tracked.
However, in practical application, the current iterative learning model, such as the QILC algorithm, often has the disadvantages of inaccurate system modeling, poor robustness, low convergence rate, and the like.
Disclosure of Invention
The invention aims to solve the technical problem of providing the SCARA robot trajectory tracking control method based on the improved quadratic iterative learning control, which does not need precise modeling, has high convergence speed and good robustness.
The invention adopts the technical scheme that an SCARA robot track tracking control method based on improved quadratic iterative learning control comprises the following steps:
step 1: according to the parameters of the industrial robot model and a dynamics analysis method, performing dynamics modeling on the SCARA robot to obtain a dynamics equation and a system model of the robot;
step 2: determining the output track state of the robot at each moment through a kinetic equation, namely the parameter dimension of an output track state unit, so as to obtain the vector representation of the moment, wherein the parameter dimension comprises a joint angle position and a joint angular velocity, and meanwhile, the vector representation of the expected track state unit is given on the basis of the vector dimension of the output track state unit;
and step 3: taking the output track state of one period as a group of batches, when the previous batch is finished, calculating a first output track state obtained by measuring the motion track of the previous batch by adopting a UKF algorithm, estimating and obtaining an uncertainty parameter and a second output track state according to a system model, and comparing the second output track state with an expected track state to obtain a first error of the previous batch, wherein the expected track state is the expected track state of one period;
and 4, step 4: calculating to obtain a first control input of iterative learning of the current batch by utilizing the stored first error of the previous batch and combining a weight matrix with least square optimization;
and 5: when the operation of the current batch starts, a third output track state unit at the current moment is obtained through measurement, then a fourth output track state unit at the current moment is obtained according to the third output track state unit at the current moment through a UKF algorithm, the fourth output track state unit at the moment is compared with an expected track state unit at the moment to obtain a second error, the second error is input to a PD type feedback controller, the output of the feedback controller at the moment is calculated, and the iterative learning second control input at the moment is obtained;
step 6: adding the first control input at the current moment and the second control input at the current moment to obtain the actual control input at the current moment;
and 7: and continuously iterating the steps 1 to 6 until the output track state accurately tracks the expected track state.
Compared with the existing iterative learning control, the method has the following advantages:
(1) the method adopts iterative learning control based on the improved quadratic criterion, compared with the original quadratic criterion iterative learning control, the algorithm provided by the invention models the mechanical arm through a kinetic equation, compared with the original quadratic criterion iterative learning control, the speed error is added as an optimization index, so that a new optimization equation is constructed and obtained, and a control law of the improved quadratic criterion iterative learning control is obtained through deduction.
(2) The invention does not need to change the control mode of the existing robot, does not need an expensive position measuring system and a position capturing system, does not need particularly accurate kinematic modeling, and accurately tracks the expected output track while reducing the cost.
(3) The method combines improved quadratic criterion iterative learning control with UKF, can inhibit the influence of uncertainty of system model parameters and measurement noise, and improves the robustness of a control system.
(4) The invention combines PD type feedback control and iterative learning control, and improves the robustness of the control system to the external real-time disturbance.
Preferably, the kinetic equation in step 1 is:
Figure BDA0002286723830000031
where theta is the joint angle position vector,
Figure BDA0002286723830000032
is a joint angular velocity vector, D (theta) is an inertia matrix of the robot, D (theta) represents a centrifugal force and a Coriolis force, tau is a moment vector acting on a double joint, B (theta) is a Coriolis matrix,
Figure BDA0002286723830000033
is the quadratic derivative of the joint angle position vector.
Preferably, the system model in step 1 is:
θk＝f(τk,θk-1)+ωk
zk＝h(θk)+υk
where the index k represents the kth iteration, θkFor the dual joint angle position vector at the kth iteration, ωkIs a process disturbance vector upsilon at the kth iterationkFor the measurement noise vector at the kth iteration, τkIs the moment vector acting on the double joint at the k-th iteration, zkIs a vector of observations.
Preferably, the output trajectory state unit in step 2 is represented by a vector as:
S＝[θ1θ2v1v2]T
wherein, theta1And theta2Representing a first and a second joint angle position, v, respectively, of the robot1And v2Representing a first joint angular velocity and a second joint angular velocity of the robot.
Preferably, the calculation method for obtaining the first control input in step 4 includes:
step 41: obtaining an objective function J based on the iterative learning control of the improved QILC, wherein the objective function J comprises a position error vector e of the current batch iterationθ,k+1Velocity error vector ev,k+1And control input variation Deltauk+1，
Step 42: respectively distributing certain weight matrixes to the position error vector, the speed error vector and the control input change vector of the current batch, and finally constructing a least square optimization problem:
Figure BDA0002286723830000034
wherein Q, R and W are both positive definite matrices.
Preferably, the weight matrix is formed by the position error vector e of the previous batchθ,kVelocity error vector ev,kAnd dynamically calculating to obtain a designed weight matrix as follows:
Figure BDA0002286723830000041
Figure BDA0002286723830000042
Figure BDA0002286723830000043
wherein I represents an identity matrix, norm (e)θ,k) And norm: (ev,k) The weight matrix self-adaptive adjustment performance can dynamically adjust the weight matrix according to the tracking error, accelerate the convergence rate and reduce the tracking error
Drawings
FIG. 1 is a schematic view of the flow structure of the present invention;
FIG. 2 is a diagram illustrating the convergence rate of the first joint angle error according to the present invention;
FIG. 3 is a graph illustrating the convergence rate of a second joint angle error according to the present invention;
FIG. 4 is a graph of the error convergence effect of the first joint angle modified algorithm after adding parameter uncertainty and noise in accordance with the present invention;
FIG. 5 is a graph of the effect of error convergence of the second joint angle post-parametric uncertainty and noise improvement algorithm of the present invention.
Detailed Description
The present invention is further described with reference to the accompanying drawings in combination with the embodiments so that those skilled in the art can implement the invention by referring to the description, and the scope of the present invention is not limited to the embodiments.
As shown in fig. 1, the invention relates to a SCARA robot trajectory tracking control method based on improved quadratic iterative learning control, which comprises the following steps:
step 1: the SCARA robot comprises a first joint angle and a second joint angle, and dynamic modeling is carried out on the SCARA robot according to industrial robot model parameters and kinematic analysis, wherein a system model is as follows:
θk＝f(τk,θk-1)+ωk
zk＝h(θk)+υk
the kinetic equation is as follows:
Figure BDA0002286723830000044
wherein θ is offThe pitch position vector is a vector of the pitch position,
Figure BDA0002286723830000045
is a joint angular velocity vector, D (theta) is an inertia matrix of the robot, D (theta) represents a centrifugal force and a Coriolis force, tau is a moment vector acting on a double joint, B (theta) is a Coriolis matrix,
Figure BDA0002286723830000046
is the quadratic derivative of the joint angle position vector. The parameters in the formula can be expressed as:
Figure BDA0002286723830000051
Figure BDA0002286723830000052
wherein i denotes a reference numeral of a joint angle, i denotes a first joint angle when i is 1, i denotes a second joint angle when i is 2, and m denotes a reference numeral of a joint angleiRepresents the mass of each connecting rod; liRepresenting the length of each connecting rod; r is the radius of the rotating guide rail; c. CiAnd siRespectively represent cos (. theta.)i) And sin (theta)i)。
Figure BDA0002286723830000053
Figure BDA0002286723830000054
Figure BDA0002286723830000055
Step 2: and determining the output track state of the robot at each moment through a kinetic equation, namely the parameter dimension of the output track state unit, so as to obtain the vector representation of the moment, and simultaneously giving the vector representation of the expected track state unit on the basis of the vector dimension of the output track state unit.
Wherein, the output track state unit is represented by a vector as:
S＝[θ1θ2v1v2]T
the expected trajectory state unit is represented by a vector as:
Sq＝[θq1θq2vq1vq2]T
wherein, theta1And theta2Respectively representing two joint angles, v, of the robot1And v2Representing the joint angular velocity of the robot.
And step 3: in an actual situation, the influence of the inability to accurately model, external disturbance and observation error on the mechanical arm model exists, and a measurement error exists between the joint angle position measured by the sensor and the actual position, so that the output data measured at the end of one period needs to be processed.
In the invention, at the end of a period, a first output track state can be obtained through measurement, wherein the period comprises Z sampling nodes, so that the first output track state comprises Z first output track state units, modeling is carried out according to a kinetic equation, and the output track state units comprise a first joint angle position theta of the robot1Second joint angle position theta2First joint angular velocity v1Second joint angular velocity v2We can represent the first output trajectory state in the form of a matrix:
Figure BDA0002286723830000061
wherein the content of the first and second substances,
Figure BDA0002286723830000062
represents the vector formed by the Z first joint angle positions in a cycle measured at the end of the cycle,
Figure BDA0002286723830000063
indicating at the end of a cycle, byThe vector formed by the Z second joint angle positions in the period is measured,
Figure BDA0002286723830000064
represents the vector formed by the Z first joint angular velocities in the period obtained by measurement at the end of the period,
Figure BDA0002286723830000065
representing the vector formed by the angular velocities of Z second joints in a period obtained by measurement at the end of the period
Obtaining a first output trajectory state by measurement
Figure BDA0002286723830000066
After that, we will
Figure BDA0002286723830000067
The uncertainty parameter and the initial value of the uncertainty parameter are used as input quantity of a filter, and the uncertainty parameter and the second output track state are obtained by estimation through UKF
Figure BDA0002286723830000068
The second output trajectory state
Figure BDA0002286723830000069
Can be expressed as:
Figure BDA00022867238300000610
wherein the content of the first and second substances,
Figure BDA00022867238300000611
represents an estimated vector composed of the Z first joint angle positions in the previous cycle by the UKF,
Figure BDA00022867238300000612
represents an estimated vector composed of Z second joint angle positions in the previous cycle by the UKF,
Figure BDA00022867238300000613
represents an estimated vector composed of the Z first joint angular velocities in the previous cycle obtained by the UKF,
Figure BDA00022867238300000614
represents an estimated vector composed of the Z second joint angular velocities in the previous cycle obtained by the UKF.
The desired trajectory state may be represented as:
Figure BDA0002286723830000071
wherein the content of the first and second substances,
Figure BDA0002286723830000072
representing the desired vector made up of the Z first joint angle desired positions,
Figure BDA0002286723830000073
representing the desired vector made up of the Z second joint angle desired positions,
Figure BDA0002286723830000074
representing a desired vector consisting of Z first joint angular desired velocities,
Figure BDA0002286723830000075
and the expected vector composed of the Z second joint angular expected speeds is represented.
Then outputting the second output track state
Figure BDA0002286723830000076
And expected trajectory state
Figure BDA0002286723830000077
Comparing, and calculating to obtain a first error at the end of a period, wherein the first error comprises a position error
Figure BDA0002286723830000078
And speed error
Figure BDA0002286723830000079
Figure BDA00022867238300000710
For ease of calculation, we denote the position error at the end of the kth cycle as eθ,kThe velocity error is denoted as ev,k。
Step 4: according to an improved iterative learning control law, a first control input of the (k + 1) th iterative learning is calculated by utilizing the stored first error of the kth period and combining a weight matrix with least square optimization
Figure BDA00022867238300000711
The calculation method for obtaining the first control input comprises the following steps:
step 41: obtaining an objective function J based on the iterative learning control of the improved QILC, wherein the objective function J comprises a position error vector e of each iterationθ,k+1Velocity error vector ev,k+1And control input variation Deltauk+1，
Step 42: respectively distributing certain weight matrixes to the position error vector, the speed error vector and the control input change vector of the current batch, and finally constructing a least square optimization problem:
Figure BDA00022867238300000712
wherein Q, R and W are positive definite matrixes, and the control input changes
Figure BDA00022867238300000713
Wherein the error transfer equation is:
eθ,k+1＝eθ,k-GkΔuk+1
ev,k+1＝ev,k-HkΔuk+1
solving a least squares optimization problem:
Figure BDA00022867238300000714
let the first order partial derivative be 0:
Figure BDA00022867238300000715
namely:
Figure BDA0002286723830000081
and calculating to obtain the control input increment of the next batch, and obtaining the control input of the next batch:
Figure BDA0002286723830000082
Figure BDA0002286723830000083
Figure BDA0002286723830000084
Figure BDA0002286723830000085
wherein HQRepresents the matrix of the iterative learning gain,
Figure BDA0002286723830000086
and
Figure BDA0002286723830000087
representing two sub-matrices, matrix G, obtained by a linearized calculation along the expected trajectory curve of the angular position, representing the mapping of the inputs and outputs of the angular position of the system, matrix H, obtained by a linearized calculation along the expected trajectory curve of the angular velocity, representing the mapping of the inputs and outputs of the angular velocity of the system,
Figure BDA0002286723830000088
the first error obtained by filtering after the kth iteration is referred to.
The weight matrix is composed of a position error vector e at the end of the k periodθ,kVelocity error vector ev,kAnd dynamically calculating to obtain a designed weight matrix as follows:
Figure BDA0002286723830000089
Figure BDA00022867238300000810
Figure BDA00022867238300000811
wherein I represents an identity matrix, norm (e)θ,k) And norm (e)v,k) Which represent the two-norm position error vector and velocity error vector, respectively, and Th represents a set trajectory tracking error threshold.
And 5: when a (k + 1) th cycle starts, the output track of the current system is measured in real time, a sampling node is counted by t, a third output track state unit of a t-th node is obtained, then a fourth output track state unit of the t-th node is obtained by adopting a UKF algorithm, when the control torque of the t +1 th node is not input, the fourth output track state is compared with an expected output state unit of the t +1 th node, a joint angular position error and a speed error when the control torque of the t +1 th node is not input are obtained as second errors, the second errors are used as the input of a PD type controller, and the output of the PD type controller under the t +1 th node is obtained
Figure BDA00022867238300000812
As a second control input to the t +1 node.
Step 6: the first control input of the t-th node in the k +1 period can be obtained through the calculation
Figure BDA00022867238300000813
Second control input
Figure BDA0002286723830000091
The actual control input at the current time is:
Figure BDA0002286723830000092
and 7: and continuously iterating the steps 1 to 6 until the output track state accurately tracks the expected track state.
The iterative learning control algorithm based on the improved QILC has better robustness and stability on model errors, can inhibit the influence of uncertain parameters of a system model, has a weight self-adaptive adjustment characteristic, and has quicker error convergence performance and better transient performance compared with the original QILC algorithm.
As shown in the experimental data diagrams of fig. 2 to 4, the vertical axis is an error, and the horizontal axis is a sampling node, and experiments prove that the method provided by the invention can suppress the influence of uncertainty of the model parameters of the controlled object, has better robustness and stability, and can realize rapid convergence of the error and accurate tracking of output.
It should be emphasized that the embodiments described herein are illustrative rather than restrictive, and thus the present invention is not limited to the embodiments described in the detailed description, but also includes other embodiments that can be derived from the technical solutions of the present invention by those skilled in the art.

Claims (6)

1. A SCARA robot track tracking control method based on improved quadratic iterative learning control is characterized by comprising the following steps:
step 1: according to the parameters of the industrial robot model and a dynamics analysis method, performing dynamics modeling on the SCARA robot to obtain a dynamics equation and a system model of the robot;
step 2: determining the output track state of the robot at each moment through a kinetic equation, namely the parameter dimension of an output track state unit, so as to obtain the vector representation of the moment, wherein the parameter dimension comprises a joint angle position and a joint angular velocity, and meanwhile, the vector representation of the expected track state unit is given on the basis of the vector dimension of the output track state unit;
and step 3: taking the output track state of one period as a group of batches, when the previous batch is finished, calculating a first output track state obtained by measuring the motion track of the previous batch by adopting a UKF algorithm, estimating and obtaining an uncertainty parameter and a second output track state according to a system model, and comparing the second output track state with an expected track state to obtain a first error of the previous batch, wherein the expected track state is the expected track state of one period;
and 4, step 4: calculating to obtain a first control input of iterative learning of the current batch by utilizing the stored first error of the previous batch and combining a weight matrix with least square optimization;
and 5: when the operation of the current batch starts, a third output track state unit at the current moment is obtained through measurement, then a fourth output track state unit at the current moment is obtained according to the third output track state unit at the current moment through a UKF algorithm, the fourth output track state unit at the moment is compared with an expected track state unit at the moment to obtain a second error, the second error is input to a PD type feedback controller, the output of the feedback controller at the moment is calculated, and the iterative learning second control input at the moment is obtained;
step 6: adding the first control input at the current moment and the second control input at the current moment to obtain the actual control input at the current moment;
and 7: and continuously iterating the steps 1 to 6 until the output track state accurately tracks the expected track state.
2. The SCARA robot trajectory tracking control method based on the improved quadratic iterative learning control as claimed in claim 1, wherein the dynamical equation in step 1 is:
Figure FDA0002286723820000011
where theta is the joint angle position vector,
Figure FDA0002286723820000012
is a joint angular velocity vector, D (theta) is an inertia matrix of the robot, D (theta) represents a centrifugal force and a Coriolis force, tau is a moment vector acting on a double joint, B (theta) is a Coriolis matrix,
Figure FDA0002286723820000021
is the quadratic derivative of the joint angle position vector.
3. The SCARA robot trajectory tracking control method based on the improved quadratic iterative learning control as claimed in claim 1, wherein the system model in step 1 is,
θk＝f(τk,θk-1)+ωk
zk＝h(θk)+υk
where the index k represents the kth iteration, θkFor the dual joint angle position vector at the kth iteration, ωkIs a process disturbance vector upsilon at the kth iterationkFor the measurement noise vector at the kth iteration, τkIs the moment vector acting on the double joint at the k-th iteration, zkIs a vector of observations.
4. The SCARA robot trajectory tracking control method based on the improved quadratic iterative learning control as claimed in claim 1, wherein the output trajectory state unit in step 2 is represented by a vector as:
S＝[θ1θ2v1v2]T
wherein, theta1And theta2Representing a first and a second joint angle position, v, respectively, of the robot1And v2Representing a first joint angular velocity and a second joint angular velocity of the robot.
5. The SCARA robot trajectory tracking control method based on the improved quadratic iterative learning control as claimed in claim 1, wherein the calculation method for obtaining the first control input in step 4 comprises:
step 41: obtaining an objective function J based on the iterative learning control of the improved QILC, wherein the objective function J comprises a position error vector e of the current batch iterationθ,k+1Velocity error vector ev,k+1And control input variation Deltauk+1，
Step 42: respectively distributing certain weight matrixes to the position error vector, the speed error vector and the control input change vector of the current batch, and finally constructing a least square optimization problem:
Figure FDA0002286723820000022
wherein Q, R and W are both positive definite matrices.
6. The SCARA robot trajectory tracking control method based on improved quadratic iterative learning control as claimed in claim 5, wherein the weight matrix is formed by the position error vector e of the previous batchθ,kVelocity error vector ev,kAnd dynamically calculating to obtain a designed weight matrix as follows:
Figure FDA0002286723820000031
Figure FDA0002286723820000032
Figure FDA0002286723820000033
wherein I represents an identity matrix, norm (e)θ,k) And norm (e)v,k) Which represent the two-norm position error vector and velocity error vector, respectively, and Th represents a set trajectory tracking error threshold.
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CN114167722A (en) * 2021-11-26 2022-03-11 杭州电子科技大学 Parallel robot tracking control method based on super-exponential convergence neural network

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