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CN108628172B - Mechanical arm high-precision motion control method based on extended state observer - Google Patents

Mechanical arm high-precision motion control method based on extended state observer Download PDF

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CN108628172B
CN108628172B CN201810658080.8A CN201810658080A CN108628172B CN 108628172 B CN108628172 B CN 108628172B CN 201810658080 A CN201810658080 A CN 201810658080A CN 108628172 B CN108628172 B CN 108628172B
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mechanical arm
model
uncertainty
arm system
extended state
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CN108628172A (en
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胡健
段理想
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Nanjing University of Science and Technology
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Abstract

本发明提出了一种基于扩张状态观测器的机械臂高精度运动控制方法，首选建立具有模型不确定性的机器臂系统状态方程：建立机械臂系统名义模型；考虑外界干扰因数造成的不确定项，建立机械臂系统名义模型；建立具有模型不确定性的机器臂系统状态方程；基于反步法设计机械臂控制器；基于扩张状态观测器设计机械臂控制器：设计一个状态观测器对不确定集合进行观测从而在控制器中加以补偿，设计扩张状态观测器对模型不确定性及干扰进行估计；基于扩张状态观测器设计机械臂系统控制器。本发明的方法具有良好的鲁棒作用，并能够保证对跟关节的角度进行很好的跟踪。

Figure 201810658080

The invention proposes a high-precision motion control method of a manipulator based on an expanded state observer. The first choice is to establish a state equation of the manipulator system with model uncertainty: to establish a nominal model of the manipulator system; to consider the uncertain items caused by external disturbance factors. , establish the nominal model of the manipulator system; establish the state equation of the manipulator system with model uncertainty; design the manipulator controller based on the backstepping method; design the manipulator controller based on the extended state observer: design a state observer for uncertainty Collect observations to compensate in the controller, design an extended state observer to estimate the model uncertainty and disturbance; design the manipulator system controller based on the extended state observer. The method of the invention has good robustness and can ensure good tracking of the angle of the heel joint.

Figure 201810658080

Description

Mechanical arm high-precision motion control method based on extended state observer
Technical Field
The invention belongs to the field of mechanical arm control, and particularly relates to a mechanical arm high-precision motion control method based on an extended state observer.
Background
The mechanical arm is used as mechanical-electrical integrated equipment, can efficiently finish various complex and dangerous operations, improves the production efficiency, and is widely applied to industry and daily life. The rapid development in this field in recent years has led to higher demands on the high precision motion control of the robotic arm. However, as a complex nonlinear system, the mechanical arm system has structural and non-structural uncertainties, such as unmodeled interference, nonlinear friction, parameter uncertainty, external interference and the like. The existence of these uncertainties has a great influence on the motion control accuracy of the mechanical arm, thereby increasing the design difficulty of the controller.
For the motion control of the mechanical arm, common control methods include feedforward compensation control, moment calculation method, adaptive robust control method and the like; feedforward compensation control and moment calculation methods need to be based on accurate mechanical arm models. In actual engineering, due to the uncertainty, an accurate mechanical arm mathematical model is difficult to obtain, so that the control methods are difficult to apply in actual engineering; aiming at parameter uncertainty in a system, the adaptive robust control method designs a proper online estimation strategy to estimate the parameter uncertainty; and for uncertain nonlinearity such as external interference and the like which possibly occur, the nonlinear feedback gain is improved to inhibit the uncertain nonlinearity, so that the system performance is improved. This makes engineering difficult because large nonlinear feedback gains tend to result in design conservatism (i.e., high gain feedback). However, when the non-structural uncertainty such as external interference gradually increases, the designed adaptive robust controller may cause the tracking performance to deteriorate and even cause instability.
Aiming at model uncertainty and external disturbance existing in a mechanical arm, a robust controller based on an Extended State Observer (ESO) is designed for mechanical arm system control. The method has better tracking performance under the condition that the system has structural uncertainty and unstructured uncertainty.
Disclosure of Invention
The invention aims to provide a mechanical arm high-precision motion control method based on an extended state observer, so as to improve the control precision of a mechanical arm.
The technical solution for realizing the purpose of the invention is as follows:
a mechanical arm high-precision motion control method based on an extended state observer comprises the following steps:
step 1, establishing a state equation of a robot arm system with model uncertainty;
firstly, establishing a nominal model of the mechanical arm system; considering an uncertain item caused by an external interference factor, establishing a nominal model of the mechanical arm system; establishing a state equation of the robot arm system with model uncertainty;
step 2, designing a mechanical arm controller based on a back stepping method;
step 3, designing a mechanical arm controller based on the extended state observer: designing a state observer to observe the uncertain set so as to compensate in the controller, and designing an extended state observer to estimate model uncertainty and interference; and designing a mechanical arm system controller based on the extended state observer.
Compared with the prior art, the invention has the following remarkable advantages:
the mechanical arm high-precision motion control method based on the extended state observer has good robust effect on structural uncertainty such as simultaneous existence of parameters and the like and non-structural uncertainty such as external interference and the like, and can ensure that the angle of the joint can be well tracked.
The present invention is described in further detail below with reference to the attached drawing figures.
Drawings
FIG. 1 is a flow chart of the method of the present invention.
Fig. 2 is a structural view of the double-joint mechanical arm in the embodiment.
FIG. 3 is a comparison curve of angle tracking of each joint of the mechanical arm system along with time under the respective actions of the linear feedback controller based on the extended state observer, the linear feedback controller and the conventional PID controller designed by the present invention; (a) and (b) are partial enlarged views; (c) the (e) and (g) are angle tracking curve graphs of the joint 1; (d) the angle tracking curves of the joint 2 are (f), (h).
FIG. 4 is a comparison curve of tracking errors of the angles of the joints of the robot arm system over time under the respective actions of the controller, the linear feedback controller (indicated by BFDL in the figure) and the conventional PID controller; (a) the (c) and (e) are angle tracking error curve graphs of the joint 1; (b) the angle tracking error graphs of the joint 2 are shown in (d) and (f).
FIG. 5 is a graph of the controller's estimation and estimation error for the uncertainty of the manipulator model and external disturbances in accordance with the present invention; (a) and (c) are respectively an estimation and estimation error curve diagram of the external disturbance of the joint 1; (b) and (d) are the estimated and estimated error plots of the external disturbance of the joint 2, respectively.
Fig. 6 is a graph showing control input curves of the controller according to the present invention to the joints of the robot arm.
Detailed Description
For the purpose of illustrating the technical solutions and technical objects of the present invention, the present invention will be further described with reference to the accompanying drawings and specific embodiments.
The invention discloses a mechanical arm high-precision motion control method based on an extended state observer, which comprises the following steps of:
step 1, establishing a state equation of a robot arm system with model uncertainty:
step 1.1, establishing a dynamic model of the robot arm system with uncertainty:
in order to realize high-precision control of the robot arm, various uncertain factors including model uncertainty and external interference must be comprehensively considered, and a robot arm dynamic model with uncertainty is established:
Figure BDA0001706032280000031
wherein q ∈ RnD (q) is a positive definite inertia matrix of order n x n,
Figure BDA0001706032280000032
is an n x n order inertia matrix which represents the centrifugal force and the Coriolis force of the mechanical arm, G (q) epsilon RnTau epsilon R is the gravity term of the mechanical armnTo control the moment, τd∈RnFor externally applied disturbance; and n is the number of the mechanical arm joints.
Step 1.2, establishing a nominal model of the mechanical arm system:
in actual work, due to the influence of measurement errors, load changes and external interference factors, the dynamic parameter values of the robot arm can be changed, so that the accurate values of the dynamic parameters of the robot arm are difficult or impossible to obtain, and only an ideal nominal model can be built.
Representing each parameter of the mechanical arm in the nominal model of the mechanical arm as D0(q)，
Figure BDA0001706032280000033
G0(q), therefore, the actual kinematic model terms of the robot arm are expressed in the form:
Figure BDA0001706032280000034
wherein the molar ratio of [ Delta ] D (q),
Figure BDA00017060322800000311
Δ g (q) is an uncertainty term caused by external interference factors, and therefore, the kinetic model of the robot arm can be expressed as:
Figure BDA0001706032280000035
wherein
Figure RE-GDA0001776304250000036
Is a set function of uncertain terms of a mechanical arm system model,
Figure RE-GDA0001776304250000037
is bounded.
Step 1.3, establishing a state equation of the robot arm system with model uncertainty:
defining a tracking error: angle error e, angular velocity error
Figure BDA0001706032280000038
The following were used:
Figure BDA0001706032280000039
wherein q isdThe desired angle for each joint and the second order derivative, q is the actual angle for each joint.
Defining a state variable x of a mechanical arm system1＝e，
Figure BDA00017060322800000310
The robot arm system (3) with model uncertainty can be expressed as:
Figure BDA0001706032280000041
wherein w ═ D0 -1(q)(ρ+τd) An uncertainty set containing model uncertainty and external interference. By dynamics of the robot armIllustratively, w is bounded.
Wherein, for the convenience of the design and analysis of the controller, the following definitions are made: let the disturbance moment be bounded, i.e. | | τdD is less than or equal to | l, wherein D is more than 0, known from the dynamic characteristics of the mechanical arm, D0(q) is positively bounded, and thus w is bounded, provided
Figure BDA0001706032280000042
Wherein
Figure BDA0001706032280000043
Step 2, designing a mechanical arm controller based on a back stepping method:
step 2.1, design virtual control input
Figure BDA0001706032280000044
To obtain stabilization of the system, a state variable x is introduced2Virtual control input of
Figure RE-GDA0001776304250000046
Order to
Figure RE-GDA0001776304250000047
And is
Figure RE-GDA0001776304250000048
Defining an error variable z:
Figure BDA0001706032280000048
equation (5) can be expressed as:
Figure BDA0001706032280000049
to ensure system stability, virtual control inputs are designed
Figure BDA00017060322800000410
Comprises the following steps:
Figure BDA00017060322800000411
wherein x1＝[x11,...,x1n]T∈Rn，x1nIndicates the angular error of the joint n, k1A coefficient greater than 0.
Step 2.2, designing a controller tau:
defining the Lyapunov function V as:
Figure BDA00017060322800000412
then
Figure BDA00017060322800000413
The controller τ is designed based on equation (10) as:
Figure BDA0001706032280000051
wherein k is2For a coefficient greater than 0, formula (11) is substituted for formula (10) to obtain:
Figure BDA0001706032280000052
then the robotic arm system becomes progressively more stable as shown in equation (12).
Step 3, designing a mechanical arm controller based on an Extended State Observer (ESO):
step 3.1, designing the extended state observer:
in the above controller design, the uncertain set w is regarded as a known quantity, but in practice, the uncertain set w is usually not known accurately, so a state observer is designed to observe the uncertain set w so as to compensate in the controller. Considering the advantage that the extended state observer ESO does not need too much model information, the extended state observer ESO is designed to estimate model uncertainty and disturbance.
Let the state variable x3＝w，
Figure BDA0001706032280000053
And | h (t) | is less than or equal to δ; equation (5) can be expressed as:
Figure BDA0001706032280000054
from equation (13), the ESO structure is designed as follows:
Figure BDA0001706032280000055
wherein
Figure BDA0001706032280000056
Is x1Is estimated by the estimation of (a) a,
Figure BDA0001706032280000057
is x2Is estimated by the estimation of (a) a,
Figure BDA0001706032280000058
is x3Estimate of (a), ω0> 0 represents the bandwidth of the ESO.
Let the estimation error
Figure BDA0001706032280000059
i is 1,2, 3; then from (13), (14) the estimated error of the ESO observer can be derived as:
Figure BDA00017060322800000510
definition of
Figure BDA00017060322800000511
i is 1,2, 3; equation (15) can be expressed as:
Figure BDA00017060322800000512
wherein B is [0,0,1 ]]TA is a Helverz matrix having ATAnd P + PA is-I, the matrix P is a symmetrical positive definite matrix, and the matrix I is an identity matrix. From the formula (15), it can be deduced
Figure BDA0001706032280000061
Description of the drawings: assuming h (t) is bounded, the estimated state is always bounded, and there is a constant γi> 0 and a finite time T1> 0, such that:
Figure BDA0001706032280000062
from the above, it can be seen that the proposed extended state observer ESO has good observation performance. After a limited time, the bandwidth ω can be increased by0The estimation error is reduced to a prescribed range. This indicates that the estimated states can be used in controller design
Figure BDA0001706032280000063
To compensate for the total uncertainty x3。
3.2, designing a mechanical arm system controller based on the ESO of the extended state observer:
based on the above description, the controller of the mechanical arm system based on the ESO is designed as follows:
Figure BDA0001706032280000064
and (3) carrying out system stability analysis on the mechanical arm system controller:
defining the Lyapunov function as:
Figure BDA0001706032280000065
then
Figure BDA0001706032280000066
Substituting formula (18) for formula (20) to obtain:
Figure BDA0001706032280000067
the terms of the above formula are simplified:
Figure BDA0001706032280000068
wherein
Figure BDA0001706032280000071
Figure BDA0001706032280000072
Then
Figure BDA0001706032280000073
Obtained by the formula (16):
Figure BDA0001706032280000074
substituting the equations (22), (24) and (25) into (21) yields:
Figure BDA0001706032280000075
where eta ═ x1,z,ε1,ε2,ε3)T，
Figure BDA0001706032280000076
λmin(. is) the minimum value of the characteristic polynomial of the matrix, λmax(. cndot.) is the maximum of the matrix eigenpolynomial.
Order to
Figure BDA0001706032280000077
Then
Figure BDA0001706032280000078
Then
Figure BDA0001706032280000079
From equation (29), the closed-loop system of the mechanical arm is bounded and stable
Figure BDA0001706032280000081
z is defined as x2Is also bounded. Therefore, the closed-loop system of the mechanical arm is guaranteed to be bounded and stable.
Examples
With reference to fig. 2, the present embodiment describes a design flow of a robot arm high-precision motion control method based on an extended state observer according to the present invention with a two-degree-of-freedom robot arm connected in series. The method comprises the following specific steps:
step 1, establishing a state equation of a robot arm system with model uncertainty:
step 1.1, establishing a dynamic model of the robot arm system with uncertainty:
in order to realize high-precision control of the robot arm, various uncertain factors including model uncertainty and external interference must be comprehensively considered, and a robot arm dynamic model with uncertainty is established:
Figure BDA0001706032280000082
wherein q is [ q ]1,q2]D (q) is a positive definite inertia matrix of 2 x 2 order,
Figure BDA0001706032280000083
is an inertia matrix of 2 x 2 orders, representing the centrifugal force and the Coriolis force of the mechanical arm, G (q) epsilon R2Tau epsilon R is the gravity term of the mechanical arm2To control the moment, τd∈R2Is the applied disturbance.
Step 1.2, establishing a nominal model of the mechanical arm system:
in actual work, the values of the dynamic parameters of the robot arm may change due to the influence of measurement errors, load changes and external interference factors, so that the accurate values of the dynamic parameters of the robot arm are difficult or impossible to obtain. Only an ideal nominal model can be built.
Representing each parameter of the mechanical arm in the nominal model of the mechanical arm as D0(q)，
Figure BDA0001706032280000084
G0(q), therefore, the actual kinematic model terms of the robot arm are expressed in the form:
Figure BDA0001706032280000085
wherein the molar ratio of [ Delta ] D (q),
Figure BDA0001706032280000086
Δ g (q) is an uncertainty term caused by external interference factors, and therefore, the kinematic model of the robot arm can be expressed as:
Figure BDA0001706032280000087
wherein
Figure BDA0001706032280000088
Is a collective function of mechanical arm system model uncertainties,
Figure BDA0001706032280000089
there is a limit.
Step 1.3, establishing a state equation of the robot arm system with model uncertainty:
the tracking error e is defined and,
Figure BDA0001706032280000091
the following were used:
Figure BDA0001706032280000092
wherein q isdThe desired angle for each joint and the second order derivative, q is the actual angle for each joint.
Defining a state variable x of a mechanical arm system1＝e，
Figure BDA0001706032280000093
The robot arm system (3) with model uncertainty can be expressed as:
Figure BDA0001706032280000094
wherein w ═ D0 -1(q)(ρ+τd) An uncertainty set containing model uncertainty and external interference. W is bounded by the dynamics of the robot arm.
Wherein, for the convenience of the design and analysis of the controller, the following definitions are made: let the disturbance moment be bounded, i.e. | | τdD is less than or equal to | l, wherein D is more than 0, known from the dynamic characteristics of the mechanical arm, D0(q) is positively bounded, and thus w is bounded, provided
Figure BDA0001706032280000095
Wherein
Figure BDA0001706032280000096
Step 2, designing a mechanical arm controller based on a back stepping method:
step 2.1, design virtual control input
Figure BDA0001706032280000097
To obtain stabilization of the system, a state variable x is introduced2Virtual control input of
Figure BDA0001706032280000098
Order to
Figure BDA0001706032280000099
And is
Figure BDA00017060322800000910
Defining an error variable z:
Figure BDA00017060322800000911
equation (5) can be expressed as:
Figure BDA00017060322800000912
to ensure system stability, virtual control inputs are designed
Figure BDA00017060322800000913
Comprises the following steps:
Figure BDA00017060322800000914
wherein x1＝[x11,x12]T∈R2，x12Indicates the angular error, k, of the joint 21A coefficient greater than 0.
Step 2.2, design controller τ
Defining the Lyapunov function V as:
Figure BDA00017060322800000915
then
Figure BDA0001706032280000101
The controller τ is designed based on equation (10) as:
Figure BDA0001706032280000102
wherein k is2For a coefficient greater than 0, formula (11) is substituted for formula (10) to obtain:
Figure BDA0001706032280000103
then the robotic arm system becomes progressively more stable as shown in equation (12).
Step 3, designing a mechanical arm controller based on an Extended State Observer (ESO)
Step 3.1, design of extended state observer
In the above controller design, the uncertain set w is regarded as a known quantity, but in practice, the uncertain set w is usually not known accurately, so a state observer is designed to observe the uncertain set w so as to compensate in the controller. The Extended State Observer (ESO) is designed to estimate model uncertainty and disturbances, taking into account the advantage that the ESO does not need too much model information. Let the state variable x3＝w，
Figure BDA0001706032280000104
And | h (t) | is less than or equal to δ; equation (5) can be expressed as:
Figure BDA0001706032280000105
from equation (13), the ESO structure is designed as follows:
Figure BDA0001706032280000106
wherein
Figure BDA0001706032280000107
Is x1Is estimated by the estimation of (a) a,
Figure BDA0001706032280000108
is x2Is estimated by the estimation of (a) a,
Figure BDA0001706032280000109
is x3Estimate of (a), ω0> 0 represents the bandwidth of the ESO.
Let the estimation error
Figure BDA00017060322800001010
i＝1,2,3; then from (13), (14) the estimated error of the ESO observer can be derived as:
Figure BDA00017060322800001011
definition of
Figure BDA0001706032280000111
i is 1,2, 3; equation (15) can be expressed as:
Figure BDA0001706032280000112
wherein B is [0,0,1 ]]TA is a Helverz matrix having ATAnd P + PA is-I, the matrix P is a symmetrical positive definite matrix, and the matrix I is an identity matrix. From the formula (15), it can be deduced
Figure BDA0001706032280000113
Introduction 1: assuming h (t) is bounded, the estimated state is always bounded, and there is a constant γi> 0 and finite time T1> 0, such that:
Figure BDA0001706032280000114
description 1: it can be seen from lemma 1 that the proposed extended state observer ESO has good observation performance. After a limited time, the bandwidth ω can be increased by0The estimation error is reduced to a prescribed range. This indicates that the estimated states can be used in the controller design
Figure BDA0001706032280000115
To compensate for the total uncertainty x3。
Step 3.2, designing the mechanical arm system controller based on the Extended State Observer (ESO)
Based on the above description, the controller of the mechanical arm system based on the ESO is designed as follows:
Figure BDA0001706032280000116
and (3) carrying out system stability analysis on the mechanical arm system controller:
defining the Lyapunov function as:
Figure BDA0001706032280000117
then
Figure BDA0001706032280000118
Substituting formula (18) for formula (20) to obtain:
Figure BDA0001706032280000119
the terms of the above formula are simplified:
Figure BDA0001706032280000121
wherein
Figure BDA0001706032280000122
Figure BDA0001706032280000123
Then
Figure BDA0001706032280000124
Obtained by the formula (16):
Figure BDA0001706032280000125
substituting the equations (22), (24) and (25) into (21) yields:
Figure BDA0001706032280000126
where eta ═ x1,z,ε1,ε2,ε3)T，
Figure BDA0001706032280000127
λmin(. is) the minimum value of the characteristic polynomial of the matrix, λmax(. cndot.) is the maximum of the matrix eigenpolynomial.
Order to
Figure BDA0001706032280000128
Then
Figure BDA0001706032280000131
Then
Figure BDA0001706032280000132
From equation (29), the closed-loop system of the mechanical arm is bounded and stable
Figure BDA0001706032280000133
z is defined as x2Is also bounded. Therefore, the closed-loop system of the mechanical arm is guaranteed to be bounded and stable.
Performing MATLAB simulation on the controller with the design:
taking the expected angles of the three controllers as q1d＝1+0.2sin(0.5πt)，q2d1-0.2cos (0.5 tt); taking external disturbances
Figure BDA0001706032280000134
Wherein d is1＝2，d2＝2，d3＝2，
Figure BDA0001706032280000135
The initial value of each joint angle of the mechanical arm is taken as
Figure BDA0001706032280000136
Comparing simulation results: the parameter selection of the mechanical arm high-precision motion controller based on the extended state observer is the control gain
Figure BDA0001706032280000137
ESO bandwidth is taken as w080; parameter selection of feedback linearization controller based on backstepping method is control gain
Figure BDA0001706032280000138
The parameter of the PID controller is selected as a proportionality coefficient Kp500, integral coefficient K i0, differential coefficient Kd＝380。
The tracking performance of the three controllers is shown in fig. 3(a-g), fig. 4 (a-f). Fig. 3 is a comparison curve of tracking angle of each joint of the mechanical arm system with time under the action of the linear feedback controller (labeled as ESOFDL in the figure), the linear feedback controller (labeled as BFDL in the figure) and the traditional PID controller based on the extended state observer designed by the invention. FIG. 4 is a comparison graph of tracking error of each joint angle of the mechanical arm system with time under the action of the controller designed by the present invention (identified by ESOFDL in the figure), the linear feedback controller (identified by BFDL in the figure) and the traditional PID controller respectively. From fig. 4, it can be seen that the linear feedback controller ESOFDL controller based on the extended state observer has a smaller tracking error on the joint angle (the angle error of the joint 1 is 7.68 × 10-4 °, and the angle error of the joint 2 is 2.76 × 10-4 °) with time, and the transient and final tracking performance of the controller is better than that of the linear feedback controller BFDL and the PID controller. Further, FIG. 5 shows the estimation of system uncertainty and estimation error by the Extended State Observer (ESO). As can be seen from fig. 5, the Extended State Observer (ESO) has a good estimate and compensation for system model uncertainty and external disturbances. Fig. 6(a-b) shows the control moments for both joints.

Claims (2)

1. A mechanical arm high-precision motion control method based on an extended state observer is characterized by comprising the following steps:
step 1, establishing a state equation of a robot arm system with model uncertainty;
firstly, establishing a nominal model of the mechanical arm system; considering an uncertain item caused by an external interference factor, establishing a nominal model of the mechanical arm system; establishing a state equation of the robot arm system with model uncertainty;
step 2, designing a mechanical arm control torque based on a backstepping method, and specifically comprising the following steps:
step 2.1, design virtualPseudo control input
Figure FDA0002958216830000011
Introducing a state variable x2Virtual control input of
Figure FDA0002958216830000012
Order to
Figure FDA0002958216830000013
And is
Figure FDA0002958216830000014
Defining an error variable z:
Figure FDA0002958216830000015
then:
Figure FDA0002958216830000016
designing virtual control inputs
Figure FDA0002958216830000017
Comprises the following steps:
Figure FDA0002958216830000018
wherein x1＝[x11,...,x1n]T∈Rn，x1nIndicates the angular error of the joint n, k1A coefficient greater than 0;
step 2.2, designing a control torque:
defining the Lyapunov function V as:
Figure FDA0002958216830000019
then
Figure FDA00029582168300000110
The control torque is designed based on equation (10):
Figure FDA00029582168300000111
step 3, designing a mechanical arm control torque based on the extended state observer: the method specifically comprises the following steps:
step 3.1, designing the extended state observer:
let the state variable x3＝w，
Figure FDA00029582168300000112
And | h (t) | is less than or equal to δ; then
Figure FDA0002958216830000021
From equation (13), the ESO structure is designed as follows:
Figure FDA0002958216830000022
wherein
Figure FDA0002958216830000023
Is x1Is estimated by the estimation of (a) a,
Figure FDA0002958216830000024
is x2Is estimated by the estimation of (a) a,
Figure FDA0002958216830000025
is x3Estimate of (a), ω0> 0 indicates an expanded state(ii) a bandwidth of the observer;
let the estimation error
Figure FDA0002958216830000026
Then from (13), (14) the estimated error of the ESO observer can be derived as:
Figure FDA0002958216830000027
definition of
Figure FDA0002958216830000028
Equation (15) can be expressed as:
Figure FDA0002958216830000029
wherein B is [0,0,1 ]]TA is a Helverz matrix having ATP + PA is-I, the matrix P is a symmetrical positive definite matrix, and the matrix I is a unit matrix;
step 3.2, designing a control moment tau of the mechanical arm system based on the extended state observer ESO:
Figure FDA00029582168300000210
wherein
Figure FDA00029582168300000211
The parameters of the mechanical arm in the nominal model of the mechanical arm are expressed as D0(q)，
Figure FDA00029582168300000212
G0(q)；qdDesired angles for each joint; x is the number of1、x2、x3Is a state variable of the mechanical arm system; k is a radical of2Is a coefficient greater than 0, z is a defining error variable,
Figure FDA00029582168300000213
for introducing a state variable x2A virtual control input of (a); q is the joint angle.
2. The extended state observer-based mechanical arm high-precision motion control method according to claim 1, wherein the step 1 of establishing a mechanical arm system state equation with model uncertainty specifically comprises the following steps:
step 1.1, establishing a dynamic model of the robot arm system with uncertainty:
Figure FDA00029582168300000214
where D (q) is a positive definite inertia matrix of order n x n,
Figure FDA00029582168300000215
is an n x n order inertia matrix representing the centrifugal and Coriolis forces of the robot arm, G (q) epsilon RnIs the gravity term of the arm, taud∈RnFor externally applied disturbance;
step 1.2, establishing a nominal model of the mechanical arm system:
representing each parameter of the mechanical arm in the nominal model of the mechanical arm as D0(q)，
Figure FDA0002958216830000031
G0(q), the actual kinematic model terms of the robot arm are expressed in the form:
Figure FDA0002958216830000032
wherein the molar ratio of [ Delta ] D (q),
Figure FDA0002958216830000033
Δ G (q) is an uncertainty term caused by external interference factors, and therefore, the machineThe kinetic model of the arm can be expressed as:
Figure FDA0002958216830000034
wherein
Figure FDA0002958216830000035
Is a set function of uncertainty terms of the mechanical arm system model;
step 1.3, establishing a state equation of the robot arm system with model uncertainty:
defining a tracking error: angle error e, angular velocity error
Figure FDA0002958216830000036
The following were used:
Figure FDA0002958216830000037
defining a state variable x of a mechanical arm system1＝e，
Figure FDA0002958216830000038
The robot arm system with model uncertainty is expressed as:
Figure FDA0002958216830000039
wherein w ═ D0 -1(q)(ρ+τd) An uncertainty set containing model uncertainty and external interference.
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