CN112276954B - Multi-joint mechanical arm impedance control method based on limited time output state limitation - Google Patents

Multi-joint mechanical arm impedance control method based on limited time output state limitation Download PDF

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CN112276954B
CN112276954B CN202011180514.1A CN202011180514A CN112276954B CN 112276954 B CN112276954 B CN 112276954B CN 202011180514 A CN202011180514 A CN 202011180514A CN 112276954 B CN112276954 B CN 112276954B
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CN112276954A (en
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于金鹏
田新诚
李刚
刘加朋
赵恩亮
徐雨梦
马玉梅
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Qingdao University
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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
    • 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

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Abstract

The invention belongs to the technical field of robot control, and particularly discloses a multi-joint mechanical arm impedance control method based on limited time output state limitation, which is based on a backstepping principle and realizes the force/position control of a mechanical arm through an impedance control technology. The control method provided by the invention can enable the force/position track of the tail end of the mechanical arm to quickly and effectively track the expected track.

Description

Multi-joint mechanical arm impedance control method based on limited time output state limitation
Technical Field
The invention belongs to the technical field of robot control, and particularly relates to a multi-joint mechanical arm impedance control method based on limited time output state limitation.
Background
In recent years, the application of multi-joint robot technology has become more widespread, and the structure thereof has become more sophisticated and precise. Therefore, to ensure the safety and compliance of this technology, a more precise force/position control method is urgently needed. In this field, numerous scholars have made extensive research work and have given a large number of control methods, such as a force/bit hybrid control method, an impedance control method, and the like.
Among them, impedance control is an important method for solving the problem of force/position control, and the method has the advantages of strong disturbance resistance, relatively small calculation amount, easiness in force control of the mechanical arm, and the like, and is widely concerned.
The impedance control method applied to the multi-joint mechanical arm and designed based on the backstepping method is developed rapidly, is applied to the control of a multi-mechanical arm system, obtains a better force/position control effect, still has the defects and is mainly reflected in that:
(1) the multi-joint mechanical arm is a complex nonlinear system; (2) the output force/position of the mechanical arm has the limit of various constraint conditions, the mechanical arm working performance is reduced and the product quality is influenced if the output force/position of the mechanical arm exceeds the range of the constraint condition limit of the mechanical arm, meanwhile, the service life of the mechanical arm is shortened, the hardware of the mechanical arm is easily damaged, even disastrous consequences can be caused, and the personal safety of operators is threatened. The presence of the above-mentioned problems makes the use of the backstepping method highly limited.
Aiming at the technical problem (1), the fuzzy logic system provides a solution for the problem that some functions of the system are required to be linear, wherein the fuzzy logic system approaches nonlinear functions in a complex nonlinear system through an approximate theory, and the anti-interference performance and robustness are improved. However, for the above technical problem (2), that is, the problem that the output state may exceed the allowable range in the current controller design, no good solution exists yet, and therefore, the safe operation of the system cannot be ensured.
Disclosure of Invention
The invention provides a multi-joint mechanical arm impedance control method based on limited time output state limitation, which ensures the safe operation of a system on the premise of ensuring that the output force/position state of a mechanical arm is limited in a bounded interval by restricting an output signal. In order to achieve the purpose, the invention adopts the following technical scheme:
the multi-joint mechanical arm impedance control method based on limited time output state comprises the following steps:
a. establishing a dynamic model of the multi-joint mechanical arm, as shown in formula (1):
Figure GDA0003254130400000011
wherein q ∈ Rn×1The angle of each joint of the multi-joint mechanical arm,
Figure GDA0003254130400000012
respectively representing the angle change speed and the acceleration of each joint of the multi-joint mechanical arm; d (q) is a multi-joint mechanical arm inertia matrix;
Figure GDA0003254130400000021
the method comprises the following steps of (1) forming a multi-joint mechanical arm centrifugal force and Coriolis force matrix; g (q) is a multi-joint mechanical arm gravity term vector; tau epsilon to Rn×1Is a real control law; tau isf∈Rn×1The friction force vector borne by each joint of the multi-joint mechanical arm is obtained; tau isd∈Rn×1The external interference term vector of each joint of the multi-joint mechanical arm is obtained; j (q) ε Rn×nA Jacobian matrix of the multi-joint mechanical arm; fe∈Rn×1Applying a contact force to the end of the multi-joint mechanical arm for the environment; rn×1Is a real number set of n dimensions, Rn×nIs an n multiplied by n dimensional real number set, and n is the joint number of the multi-joint mechanical arm;
the relation of the multi-joint mechanical arm on a Cartesian coordinate system is shown as the formula (2):
Figure GDA0003254130400000022
wherein x is the tail end position of the multi-joint mechanical arm,
Figure GDA0003254130400000023
converting the joint angle of the multi-joint mechanical arm into a functional relation of the tail end position of the mechanical arm under Cartesian coordinates; the following formula is obtained by collation:
Figure GDA0003254130400000024
the impedance control relation between the tail end position and the tail end force of the multi-joint mechanical arm is shown as the formula (3):
Figure GDA0003254130400000025
wherein E ═ x-xd，FdExpecting force for multi-joint robot arm tip, MdExpecting an inertia matrix, B, for a multi-joint robot armdExpecting a damping matrix, K, for a multi-joint robot armdA rigid matrix of the multi-joint mechanical arm;
xd＝[xd1,xd2,…,xdn]Texpecting a trajectory, x, for a multi-joint robotic arm tipd1,xd2,…,xdnDenotes xdA component of (a);
substituting equation (2) into equation (1) yields:
Figure GDA0003254130400000026
wherein the content of the first and second substances,
Figure GDA0003254130400000027
let x1＝x，
Figure GDA0003254130400000028
Then equation (5) is written from equation (4):
Figure GDA0003254130400000029
wherein x is1Indicating the position of the end of a multi-jointed arm, x1＝[x11,x12,…,x1n]T；x2Representing the velocity of movement, x, of the end of a multi-jointed arm2＝[x21,x22,…,x2n]T；
b. According toThe limited technology of the limited time output state and the self-adaptive backstepping principle design the real control law tau to ensure that the tail end position x of the multi-joint mechanical arm1Tracking a multi-joint robotic arm tip expected trajectory xd；
Suppose f (Z) is in tight set ΩZIs a continuous function, and for an arbitrary constant ε > 0, there is always a fuzzy logic function WTS (Z) satisfies:
Figure GDA0003254130400000031
wherein the vector is input
Figure GDA0003254130400000032
Q is the fuzzy input dimension, RQA real number vector set; w is formed as ROIs a fuzzy weight vector, the number of fuzzy nodes O is a positive integer, and O is more than 1, ROA real number vector set;
S(Z)＝[s1(Z),...,so(Z)]T∈ROis a vector of basis functions, s1(Z),...,so(Z) are the basis functions of S (Z), respectively; selecting a basis function sj(Z) is a Gaussian function as follows:
Figure GDA0003254130400000033
wherein, muj＝[μj1,μj2,…,μjr]Is the central position of the distribution curve of the Gaussian function, and ηjThen the width of the gaussian function; mu.sj1,μj2,…,μjrIs mujThe basis vector of (2);
define a finite time stability:
considering non-linear systems
Figure GDA0003254130400000034
For a smooth positive definite function V (ξ), there are a > 0, b > 0, and 0 < γ < 1, such that
Figure GDA0003254130400000035
If it is true, then the non-linear system
Figure GDA0003254130400000036
Is semi-global finite time stable;
convergence time TsBy the formula:
Ts＝1/(va-γva)[V1-γ(ξ(0))-(b/(a-va))(1-γ)/γ]to estimate, and
Figure GDA0003254130400000037
wherein, v is more than 0 and less than 1, and xi (0) represents the initial value of the nonlinear system;
for any vector k ∈ RnThe following inequality holds for any vector l e R that satisfies | l | < | k |nBoth are true:
Figure GDA0003254130400000038
defining a virtual control law alpha1＝[α11,α12,…,α1n]T；
Defining error variables
Figure GDA0003254130400000039
Wherein z is1i＝x1i-xdiZ represents an error between the position of the end of the multi-joint robot arm in the i direction and the expected trajectory2i＝x2i-α1iThe error between the moving speed of the tail end of the multi-joint mechanical arm in the direction i and the virtual control law is shown, wherein i is 1,2, …, n;
the following two tight sets are defined:
Ωa＝{|z1i|＜kai}，kaiis a normal number; omegac＝{|x1i|＜kci}，kciIs a normal number, i ═ 1,2, …, n;
error variable z1iBound k of a restricted intervalai＝kc1i-Xdi，Xdi＝max{|xdi|}；
Wherein k isc1iFor multi-joint machinesArm end position x1iBoundary of the set restricted interval, set ka＝[ka1,ka2,…,kan]T；
The method comprises the steps that a barrier Lyapunov function is designed and selected to construct a real control law based on a multi-joint mechanical arm impedance control method with a limited time output state, when an error variable tends to a limit of a limited interval, a barrier Lyapunov function value tends to be infinite, and therefore the error variable is guaranteed to be kept in the limited interval all the time; the method specifically comprises the following steps:
b.1. expected trajectory x for multi-joint robot arm tipdSelecting a barrier Lyapunov function
Figure GDA0003254130400000041
To V1The derivation yields:
Figure GDA0003254130400000042
selection of a virtual control law alpha1＝[α11,α12,…,α1n]TWherein:
Figure GDA0003254130400000043
in the formula, k1iIs a normal number, i ═ 1,2,. cndot, n; definition k1＝[k11,k12,…,k1n]T；
Law of virtual control
Figure GDA0003254130400000044
Wherein the content of the first and second substances,
Figure GDA0003254130400000045
substituting equation (8) and equation (9) into equation (7) yields:
Figure GDA0003254130400000046
derived from the young inequality:
Figure GDA0003254130400000047
by substituting formula (11) for formula (10), we obtain:
Figure GDA0003254130400000048
b.2. selecting Lyapunov function
Figure GDA0003254130400000049
To V2The derivation yields:
Figure GDA00032541304000000410
error variable z2The time derivative yields:
Figure GDA0003254130400000051
let f (Z) ═ Δ-1(q)(τf+τd) Defining a non-linear function f (Z) ═ f1(Z),…,fn(Z)]TFor arbitrarily small constants ε according to the universal approximation theoremi> 0, there is a fuzzy logic function Wi TS (Z) is such that fi(Z)＝Wi TS(Z)+δi；
Wherein, deltaiRepresents an approximation error and satisfies δi≤εi，i＝1,…,n；
Due to z2＝[z21,z22,…,z2n]TDerived from the young inequality:
Figure GDA0003254130400000052
wherein l is a constant greater than zero, WiRepresents the fuzzy weight vector, | WiI represents WiNorm of (d);
definition θ ═ max { | | | W1||2,||W2||2,…,||Wn||2Equation (15) is written as:
Figure GDA0003254130400000053
definition of
Figure GDA0003254130400000054
Then:
Figure GDA0003254130400000055
the real control law τ is selected as follows:
Figure GDA0003254130400000056
wherein k is2A constant greater than zero is represented by a constant,
Figure GDA0003254130400000057
an estimated value representing theta;
substituting equations (16), (17) and (18) into equation (13) defines the estimation error as
Figure GDA0003254130400000058
Obtaining:
Figure GDA0003254130400000059
b.3. selecting a Lyapunov function:
Figure GDA00032541304000000510
wherein r isA constant greater than zero, derived from V:
Figure GDA00032541304000000511
selecting a self-adaptive law:
Figure GDA00032541304000000512
wherein m is a constant greater than zero;
c. performing stability analysis on the constructed multi-joint mechanical arm impedance control method based on limited time output state limitation;
substituting equation (21) into equation (20) yields equation (22):
Figure GDA0003254130400000061
derived from the young inequality:
Figure GDA0003254130400000062
substituting formula (23) for formula (22) to obtain:
Figure GDA0003254130400000063
when in use
Figure GDA0003254130400000064
Obtaining:
Figure GDA0003254130400000065
when in use
Figure GDA0003254130400000066
Obtaining:
Figure GDA0003254130400000067
formula (27) is derived from formulae (25) and (26):
Figure GDA0003254130400000068
when the condition | z is satisfied1i|≤kaiAnd when gamma is more than 0 and less than 1, the following inequality is satisfied:
Figure GDA0003254130400000069
by substituting equations (27) and (28) into equation (24), we obtain:
Figure GDA00032541304000000610
wherein, a is min { min {2 ═ min { (2) }γk1i},2γk2,mγ}，
Figure GDA00032541304000000611
As can be seen from equation (29), when the selection parameter satisfies k1i＞0，k2When m is more than 0, a is more than 0 and b is more than 0;
determining T according to the definition of finite time stabilitysAs shown in equation (30):
Figure GDA00032541304000000612
wherein V is more than 0 and less than 1, V1-γ(0) Representing that the power of 1-gamma of the initial value of the Lyapunov function V is selected;
for the
Figure GDA0003254130400000071
Satisfy the requirement of
Figure GDA0003254130400000072
The above inequality indicates that the error signal in the closed loop system is semi-global time-limited stable; it is further known that the initial condition satisfies z1(0)∈Ω0＝{z1∈Rn:|z1|＜|kaIn | z }, get | z1i|＜|kai|，i＝1,2,…n，
Figure GDA0003254130400000073
From the formula x1＝z1+xdAnd formula | xdi(t)|≤XdiI 1,2, …, n, yields | x1i|＜kai+Xdi＝kc1i，i＝1,2,…,n，
Figure GDA0003254130400000074
Thus, the robot output force/position state is constrained within a set bounded interval.
The invention has the following advantages:
as described above, the invention relates to an impedance control method of a multi-joint mechanical arm based on limited time output state, which combines the limited time technology and the impedance control technology, realizes better force/position tracking of the tail end of the multi-joint mechanical arm in limited time, and reduces the force/position tracking error of the mechanical arm; the fuzzy logic system is utilized to approximate the unknown nonlinear function in the multi-joint mechanical arm system, so that the unknown nonlinear item in the mechanical arm system is effectively processed, and the mechanical arm can realize better force/position tracking control under the condition that the friction force function is uncertain; in addition, the system output signal is restrained by introducing the application of the barrier Lyapunov function, so that the output force/position state of the mechanical arm can be restrained in a bounded interval, and the running safety of the system is guaranteed. The control method has stronger robustness and is more in line with the practical engineering application.
Drawings
FIG. 1 is a schematic view of a model employing a two-degree-of-freedom robot arm according to an embodiment of the present invention;
FIG. 2 is a simulation diagram of a tracking curve in the X-axis direction at the tail end of a mechanical arm after the control method of the invention is adopted;
FIG. 3 is a simulation diagram of tracking error in the X-axis direction at the tail end of the mechanical arm after the control method of the invention is adopted;
FIG. 4 is a simulation diagram of a Y-axis direction tracking curve of the tail end of the mechanical arm after the control method of the invention is adopted;
FIG. 5 is a simulation diagram of tracking error in the Y-axis direction at the tail end of the mechanical arm after the control method of the present invention is adopted;
FIG. 6 is a simulation diagram of the X-axis force tracking at the end of the mechanical arm after the control method of the present invention is adopted;
FIG. 7 is a simulation diagram of the X-axis force tracking error at the end of the mechanical arm after the control method of the present invention is adopted;
FIG. 8 is a simulation diagram of the Y-axis force tracking at the end of the mechanical arm after the control method of the present invention is adopted;
FIG. 9 is a simulation diagram of the Y-axis force tracking error at the end of the mechanical arm after the control method of the present invention is adopted;
FIG. 10 is a diagram showing the moment simulation of each joint of the two-joint robot arm after the control method of the present invention is adopted.
Detailed Description
The basic idea of the invention is as follows:
based on an impedance control technology, constructing a middle virtual control signal by using a backstepping method, and gradually recurrently obtaining a control law so as to control an end effector of the multi-joint robot arm; unknown friction and external disturbance in a multi-joint mechanical arm system are approximated by using a fuzzy self-adaptive technology, the anti-interference performance and robustness of the system are improved, a barrier Lyapunov equation is introduced to constrain an output control signal of a mechanical arm, so that the output force/position state of the mechanical arm is in a limited interval range to ensure the operation safety of the system, the mechanical arm force/position tracking signal is converged in limited time by using limited time control, and the mechanical arm force/position tracking error is ensured to be converged in a small enough field of an origin point in the limited time. The invention concept ensures that the method can lead the tail end force/position of the mechanical arm to quickly and accurately track the expected track and lead the control error to be in a reasonable range.
The invention is described in further detail below with reference to the following figures and detailed description:
the multi-joint mechanical arm impedance control method based on limited time output state comprises the following steps:
a. establishing a dynamic model of the multi-joint mechanical arm, as shown in formula (1):
Figure GDA0003254130400000081
wherein q ∈ Rn×1The angle of each joint of the multi-joint mechanical arm,
Figure GDA0003254130400000082
respectively representing the angle change speed and the acceleration of each joint of the multi-joint mechanical arm; d (q) is a multi-joint mechanical arm inertia matrix;
Figure GDA0003254130400000083
the method comprises the following steps of (1) forming a multi-joint mechanical arm centrifugal force and Coriolis force matrix; g (q) is a multi-joint mechanical arm gravity term vector; tau epsilon to Rn×1Is a real control law; tau isf∈Rn×1The friction force vector borne by each joint of the multi-joint mechanical arm is obtained; tau isd∈Rn×1The external interference term vector of each joint of the multi-joint mechanical arm is obtained; j (q) ε Rn×nA Jacobian matrix of the multi-joint mechanical arm; fe∈Rn×1Applying a contact force to the end of the multi-joint mechanical arm for the environment; rn×1Is a real number set of n dimensions, Rn×nIs an n multiplied by n dimensional real number set, and n is the joint number of the multi-joint mechanical arm.
The relation of the multi-joint mechanical arm on a Cartesian coordinate system is shown as the formula (2):
Figure GDA0003254130400000084
wherein x is the tail end position of the multi-joint mechanical arm,
Figure GDA0003254130400000085
the method is a functional relation formula for converting the joint angle of the multi-joint mechanical arm into the tail end position of the mechanical arm under Cartesian coordinates. The following formula is obtained by collation:
Figure GDA0003254130400000086
the impedance control relation between the tail end position and the tail end force of the multi-joint mechanical arm is shown as the formula (3):
Figure GDA0003254130400000087
wherein E ═ x-xd，FdExpecting force for multi-joint robot arm tip, MdExpecting an inertia matrix, B, for a multi-joint robot armdExpecting a damping matrix, K, for a multi-joint robot armdIs a rigid matrix of the multi-joint mechanical arm.
xd＝[xd1,xd2,…,xdn]TExpecting a trajectory, x, for a multi-joint robotic arm tipd1,xd2,…,xdnDenotes xdA component of (a);
substituting equation (2) into equation (1) yields:
Figure GDA0003254130400000091
wherein the content of the first and second substances,
Figure GDA0003254130400000092
let x1＝x，
Figure GDA0003254130400000093
Then equation (5) is written from equation (4):
Figure GDA0003254130400000094
wherein x is1Indicating the position of the end of a multi-jointed arm, x1＝[x11,x12,…,x1n]T；x2Representing the velocity of movement, x, of the end of a multi-jointed arm2＝[x21,x22,…,x2n]T。
b. According to the limited technology of the limited time output state and the principle of the self-adaptive backstepping method, a real control law tau is designed to ensure that the tail end position x of the multi-joint mechanical arm1Tracking a multi-joint robotic arm tip expected trajectory xd。
Suppose f (Z) is in tight set ΩZIs a continuous function, and for an arbitrary constant ε > 0, there is always a fuzzy logic function WTS (Z) satisfies:
Figure GDA0003254130400000095
wherein the vector is input
Figure GDA0003254130400000096
Q is the fuzzy input dimension, RQA real number vector set; w is formed as ROIs a fuzzy weight vector, the number of fuzzy nodes O is a positive integer, and O is more than 1, ROIs a set of real vectors.
S(Z)＝[s1(Z),...,so(Z)]T∈ROIs a vector of basis functions, s1(Z),...,so(Z) are the basis functions of S (Z), respectively; selecting a basis function sj(Z) is a Gaussian function as follows:
Figure GDA0003254130400000097
wherein, muj＝[μj1,μj2,…,μjr]Is the central position of the distribution curve of the Gaussian function, and ηjThen the width of the gaussian function; mu.sj1,μj2,…,μjrIs mujThe basis vector of (2).
Define a finite time stability:
considering non-linear systems
Figure GDA0003254130400000098
For a smooth positive definite function V (ξ), there are a > 0, b > 0, and 0 < γ < 1, such that
Figure GDA0003254130400000099
If it is true, then the non-linear system
Figure GDA00032541304000000910
Is semi-global finite time stable.
Convergence time TsBy the formula:
Ts＝1/(va-γva)[V1-γ(ξ(0))-(b/(a-va))(1-γ)/γ]to estimate, and
Figure GDA00032541304000000911
wherein, 0 < v < 1, and xi (0) represents the initial value of the nonlinear system.
For any vector k ∈ RnThe following inequality holds for any vector l e R that satisfies | l | < | k |nBoth are true:
Figure GDA0003254130400000101
defining a virtual control law alpha1＝[α11,α12,…,α1n]T；
Defining error variables
Figure GDA0003254130400000102
Wherein z is1i＝x1i-xdiZ represents an error between the position of the end of the multi-joint robot arm in the i direction and the expected trajectory2i＝x2i-α1iAnd the error between the motion speed of the tail end of the multi-joint mechanical arm in the direction i and the virtual control law is represented, wherein i is 1,2, |, n.
The following two tight sets are defined:
Ωa＝{|z1i|＜kai}，kaiis a normal number; omegac＝{|x1i|＜kci}，kciIs a normal number, i ═ 1,2, …, n;
error variable z1iBound k of a restricted intervalai＝kc1i-Xdi，Xdi＝max{|xdi|}。
Wherein k isc1iFor the end position x of a multi-joint mechanical arm1iBoundary of the set restricted interval, set ka＝[ka1,ka2,…,kan]T。
A barrier Lyapunov function is designed and selected to construct a real control law based on a multi-joint mechanical arm impedance control method with a limited time output state, and when an error variable tends to a limit of a limited interval, a barrier Lyapunov function value tends to infinity, so that the error variable is ensured to be always kept in the limited interval. The method specifically comprises the following steps:
b.1. expected trajectory x for multi-joint robot arm tipdSelecting a barrier Lyapunov function
Figure GDA0003254130400000103
To V1The derivation yields:
Figure GDA0003254130400000104
selection of a virtual control law alpha1＝[α11,α12,…,α1n]TWherein:
Figure GDA0003254130400000105
in the formula, k1iIs a normal number, i ═ 1,2,. cndot, n; definition k1＝[k11,k12,…,k1n]T。
Law of virtual control
Figure GDA0003254130400000106
Wherein the content of the first and second substances,
Figure GDA0003254130400000107
substituting equation (8) and equation (9) into equation (7) yields:
Figure GDA0003254130400000111
derived from the young inequality:
Figure GDA0003254130400000112
by substituting formula (11) for formula (10), we obtain:
Figure GDA0003254130400000113
b.2. selecting Lyapunov function
Figure GDA0003254130400000114
To V2The derivation yields:
Figure GDA0003254130400000115
error variable z2The time derivative yields:
Figure GDA0003254130400000116
let f (Z) ═ Δ-1(q)(τf+τd) Defining a non-linear function f (Z) ═ f1(Z),…,fn(Z)]TFor arbitrarily small constants ε according to the universal approximation theoremi> 0, there is a fuzzy logic function Wi TS (Z) is such that fi(Z)＝Wi TS(Z)+δi。
Wherein the content of the first and second substances,δirepresents an approximation error and satisfies δi≤εi，i＝1,…,n。
Due to z2＝[z21,z22,…,z2n]TDerived from the young inequality:
Figure GDA0003254130400000117
wherein l is a constant greater than zero, WiRepresents the fuzzy weight vector, | WiI represents WiNorm of (d).
Definition θ ═ max { | | | W1||2,||W2||2,…,||Wn||2Equation (15) is written as:
Figure GDA0003254130400000118
definition of
Figure GDA0003254130400000119
Then:
Figure GDA00032541304000001110
the real control law τ is selected as follows:
Figure GDA00032541304000001111
wherein k is2A constant greater than zero is represented by a constant,
Figure GDA00032541304000001112
representing an estimate of theta.
Substituting equations (16), (17) and (18) into equation (13) defines the estimation error as
Figure GDA00032541304000001113
Obtaining:
Figure GDA0003254130400000121
b.3. selecting a Lyapunov function:
Figure GDA0003254130400000122
where r is a constant greater than zero, deriving V:
Figure GDA0003254130400000123
selecting a self-adaptive law:
Figure GDA0003254130400000124
wherein m is a constant greater than zero.
c. And carrying out stability analysis on the constructed multi-joint mechanical arm impedance control method based on limited time output state limitation.
Substituting equation (21) into equation (20) yields equation (22):
Figure GDA0003254130400000125
derived from the young inequality:
Figure GDA0003254130400000126
substituting formula (23) for formula (22) to obtain:
Figure GDA0003254130400000127
when in use
Figure GDA0003254130400000128
Obtaining:
Figure GDA0003254130400000129
when in use
Figure GDA00032541304000001210
Obtaining:
Figure GDA00032541304000001211
formula (27) is derived from formulae (25) and (26):
Figure GDA00032541304000001212
when the condition | z is satisfied1i|≤kaiAnd when gamma is more than 0 and less than 1, the following inequality is satisfied:
Figure GDA00032541304000001213
by substituting equations (27) and (28) into equation (24), we obtain:
Figure GDA0003254130400000131
wherein, a is min { min {2 ═ min { (2) }γk1i},2γk2,mγ}，
Figure GDA0003254130400000132
As can be seen from equation (29), when the selection parameter satisfies k1i＞0，k2When m is greater than 0, a is greater than 0 and b is greater than 0.
Determining T according to the definition of finite time stabilitysAs shown in equation (30):
Figure GDA0003254130400000133
wherein V is more than 0 and less than 1, V1-γ(0) The initial value of the Lyapunov function V is selected to be the power of 1-gamma.
For the
Figure GDA0003254130400000134
Satisfy the requirement of
Figure GDA0003254130400000135
The above inequality indicates that the error signal in the closed loop system is semi-global time-limited stable; it is further known that the initial condition satisfies z1(0)∈Ω0＝{z1∈Rn:|z1|＜|kaIn | z }, get | z1i|＜|kai|，i＝1,2,…n，
Figure GDA0003254130400000136
From the formula x1＝z1+xdAnd formula | xdi(t)|≤XdiI 1,2, …, n, yields | x1i|＜kai+Xdi＝kc1i，i＝1,2,…,n，
Figure GDA0003254130400000137
Thus, the robot output force/position state is constrained within a set bounded interval.
The established multi-joint mechanical arm impedance control method based on the limited time output state is simulated in a virtual environment to verify the feasibility of the control method.
The two-degree-of-freedom mechanical arm on the vertical plane is shown in fig. 1, and a simulation experiment will prove the effectiveness of the proposed control method. A rotary joint two-degree-of-freedom mechanical arm system model of a simulation experiment is represented as follows:
Figure GDA0003254130400000138
wherein the content of the first and second substances,
Figure GDA0003254130400000139
in FIG. 1, miAnd liRespectively the mass and length of the ith section of connecting rod of the mechanical armciThe distance from the I-1 th joint of the mechanical arm to the mass center of the I-section connecting rod, IiThe moment of inertia for joint i based on the coordinate axis passing through the center of mass of the joint.
x1＝[x11,x12]T,
Figure GDA00032541304000001310
Wherein x is11And x12Respectively representing the positions x of the tail end of the mechanical arm on X, Y axes on a Cartesian coordinate system of the two-degree-of-freedom mechanical arm1And y1。
xd＝[xd1,xd2]Wherein x isd1And xd2Respectively show the positions x of the expected tracks of the mechanical arms on the X, Y axes1dAnd y1d。
q＝[q1,q2]TShowing the angle of each joint of the mechanical arm.
Fe＝[Fe1,Fe2]TWherein F ise1And Fe2Respectively representing the moment F of the tail end of the mechanical arm in the direction of X, Y axes on a Cartesian coordinate system of the two-degree-of-freedom mechanical armxAnd Fy。
Fd＝[Fd1,Fd2]TWherein F isd1And Fd2Respectively representing the moment F of the expected force of the tail end of the mechanical arm in the direction of X, Y axesxdAnd Fyd。
Inertia matrix D (q), Coriolis force and centrifugal force matrix of two-degree-of-freedom mechanical arm
Figure GDA0003254130400000141
The gravity term matrix g (q) is defined as follows:
Figure GDA0003254130400000142
Figure GDA0003254130400000143
Figure GDA0003254130400000144
the jacobian matrix j (q) of the two-degree-of-freedom robot arm is defined as follows:
Figure GDA0003254130400000145
parameter joint 1,2 mass m of two-degree-of-freedom mechanical arm1、m2All are 1.00 kg; length l of joints 1,21、l2Are all 1.00 m; moment of inertia I of joints 1,21、I2Are all 0.25kgm2。
The initial parameter of the mechanical arm is x11＝x12＝0.5，
Figure GDA0003254130400000146
The expected tracking trajectory of the two-degree-of-freedom robot arm tip is x as shown belowd＝[0.7+0.2cos(t),0.7+0.2sin(t)]TWherein t ∈ [0,20 ]]。
For the limited time output state limited impedance control of the two-degree-of-freedom mechanical arm, the control parameter is selected to be k1＝[3,3]T，k2＝2，ka＝[0.6,0.6]T，γ＝0.8，l＝0.5，m＝0.25，n＝2。
The expected impedance of the two-degree-of-freedom mechanical arm is selected to be Md＝I，Bd＝diag[10,10]，Kd＝diag[70,70]。
The fuzzy membership function is:
Figure GDA0003254130400000151
Figure GDA0003254130400000152
Figure GDA0003254130400000153
wherein i is 1,2, …, 9.
Fig. 2 and 3 are graphs of an X-axis tracking curve and a tracking error of the tail end of the mechanical arm respectively, and fig. 4 and 5 are graphs of a Y-axis tracking curve and a tracking error of the tail end of the mechanical arm respectively.
As can be seen from fig. 2-5, the control method of the present invention enables the end of the robot arm to follow the desired trajectory quickly and accurately.
Fig. 6 and 7 are a force tracking diagram in the X axis direction and a force tracking error diagram of the end of the robot arm, respectively, according to the control method of the present invention, and fig. 8 and 9 are a force tracking diagram in the Y axis direction and a force tracking error diagram, respectively, according to the control method of the present invention.
As can be seen from fig. 6-9, the control method of the present invention enables the contact force of the robot arm tip to follow the desired contact force well.
FIG. 10 is a moment diagram of each joint of a two-joint robot arm according to the control method of the present invention.
In FIG. 10,. tau.1For moment of the 1 st joint of the arm, tau2The 2 nd joint moment of the mechanical arm.
The analog signals clearly show that the impedance control method based on the multi-joint mechanical arm with limited time output state can efficiently track the expected track of the mechanical arm on the premise of ensuring the system safety.
It should be understood, however, that the description herein of specific embodiments is not intended to limit the invention to the particular forms disclosed, but on the contrary, the intention is to cover all modifications, equivalents, and alternatives falling within the spirit and scope of the invention as defined by the appended claims.

Claims (1)

1. A multi-joint mechanical arm impedance control method based on limited time output state is characterized in that,
the method comprises the following steps:
a. establishing a dynamic model of the multi-joint mechanical arm, as shown in formula (1):
Figure FDA0003254130390000011
wherein q ∈ Rn×1The angle of each joint of the multi-joint mechanical arm,
Figure FDA0003254130390000012
respectively representing the angle change speed and the acceleration of each joint of the multi-joint mechanical arm; d (q) is a multi-joint mechanical arm inertia matrix;
Figure FDA0003254130390000013
the method comprises the following steps of (1) forming a multi-joint mechanical arm centrifugal force and Coriolis force matrix; g (q) is a multi-joint mechanical arm gravity term vector; tau epsilon to Rn×1Is a real control law; tau isf∈Rn×1The friction force vector borne by each joint of the multi-joint mechanical arm is obtained; tau isd∈Rn×1The external interference term vector of each joint of the multi-joint mechanical arm is obtained; j (q) ε Rn×nA Jacobian matrix of the multi-joint mechanical arm; fe∈Rn×1Applying a contact force to the end of the multi-joint mechanical arm for the environment; rn×1Is a real number set of n dimensions, Rn×nIs an n multiplied by n dimensional real number set, and n is the joint number of the multi-joint mechanical arm;
the relation of the multi-joint mechanical arm on a Cartesian coordinate system is shown as the formula (2):
Figure FDA0003254130390000014
wherein x is the tail end position of the multi-joint mechanical arm,
Figure FDA0003254130390000015
converting the joint angle of the multi-joint mechanical arm into a functional relation of the tail end position of the mechanical arm under Cartesian coordinates; the following formula is obtained by collation:
Figure FDA0003254130390000016
the impedance control relation between the tail end position and the tail end force of the multi-joint mechanical arm is shown as the formula (3):
Figure FDA0003254130390000017
wherein E ═ x-xd，FdExpecting force for multi-joint robot arm tip, MdExpecting an inertia matrix, B, for a multi-joint robot armdExpecting a damping matrix, K, for a multi-joint robot armdA rigid matrix of the multi-joint mechanical arm;
xd＝[xd1,xd2,…,xdn]Texpecting a trajectory, x, for a multi-joint robotic arm tipd1,xd2,…,xdnDenotes xdA component of (a);
substituting equation (2) into equation (1) yields:
Figure FDA0003254130390000018
wherein the content of the first and second substances,
Figure FDA0003254130390000019
let x1＝x，
Figure FDA0003254130390000021
Then equation (5) is written from equation (4):
Figure FDA0003254130390000022
wherein x is1Indicating the position of the end of a multi-jointed arm, x1＝[x11,x12,…,x1n]T；x2Representing the velocity of movement, x, of the end of a multi-jointed arm2＝[x21,x22,…,x2n]T；
b. According to the limited technology of the limited time output state and the principle of the self-adaptive backstepping method, a real control law tau is designed to ensure that the tail end position x of the multi-joint mechanical arm1Tracking a multi-joint robotic arm tip expected trajectory xd；
Suppose f (Z) is in tight set ΩZIs a continuous function, and for an arbitrary constant ε > 0, there is always a fuzzy logic function WTS (Z) satisfies:
Figure FDA0003254130390000023
wherein the vector is input
Figure FDA0003254130390000024
Q is the fuzzy input dimension, RQA real number vector set; w is formed as ROIs a fuzzy weight vector, the number of fuzzy nodes O is a positive integer, and O is more than 1, ROA real number vector set;
S(Z)＝[s1(Z),...,so(Z)]T∈ROis a vector of basis functions, s1(Z),...,so(Z) are the basis functions of S (Z), respectively; selecting a basis function sj(Z) is a Gaussian function as follows:
Figure FDA0003254130390000025
wherein, muj＝[μj1,μj2,…,μjr]Is the central position of the distribution curve of the Gaussian function, and ηjThen the width of the gaussian function; mu.sj1,μj2,…,μjrIs mujThe basis vector of (2);
define a finite time stability:
considering non-linear systems
Figure FDA0003254130390000026
For a smooth positive definite function V (ξ), there are a > 0, b > 0, and 0 < γ < 1, such that
Figure FDA0003254130390000027
If it is true, then the non-linear system
Figure FDA0003254130390000028
Is semi-global finite time stable;
convergence time TsBy the formula:
Ts＝1/(va-γva)[V1-γ(ξ(0))-(b/(a-va))(1-γ)/γ]to estimate, and
Figure FDA0003254130390000029
wherein, v is more than 0 and less than 1, and xi (0) represents the initial value of the nonlinear system;
for any vector k ∈ RnThe following inequality holds for any vector l e R that satisfies | l | < | k |nBoth are true:
Figure FDA00032541303900000210
defining a virtual control law alpha1＝[α11,α12,…,α1n]T；
Defining error variables
Figure FDA0003254130390000031
Wherein z is1i＝x1i-xdiZ represents an error between the position of the end of the multi-joint robot arm in the i direction and the expected trajectory2i＝x2i-α1iThe error between the moving speed of the tail end of the multi-joint mechanical arm in the direction i and the virtual control law is shown, wherein i is 1,2, …, n;
the following two tight sets are defined:
Ωa＝{|z1i|＜kai}，kaiis a normal number; omegac＝{|x1i|＜kci}，kciIs a normal number, i ═ 1,2, …, n;
error variable z1iBound k of a restricted intervalai＝kc1i-Xdi，Xdi＝max{|xdi|}；
Wherein k isc1iFor the end position x of a multi-joint mechanical arm1iBoundary of the set restricted interval, set ka＝[ka1,ka2,…,kan]T；
The method comprises the steps that a barrier Lyapunov function is designed and selected to construct a real control law based on a multi-joint mechanical arm impedance control method with a limited time output state, when an error variable tends to a limit of a limited interval, a barrier Lyapunov function value tends to be infinite, and therefore the error variable is guaranteed to be kept in the limited interval all the time; the method specifically comprises the following steps:
b.1. expected trajectory x for multi-joint robot arm tipdSelecting a barrier Lyapunov function
Figure FDA0003254130390000032
To V1The derivation yields:
Figure FDA0003254130390000033
selection of a virtual control law alpha1＝[α11,α12,…,α1n]TWherein:
Figure FDA0003254130390000034
in the formula, k1iIs a normal number, i ═ 1,2,. cndot, n; definition k1＝[k11,k12,…,k1n]T；
Law of virtual control
Figure FDA0003254130390000035
Wherein the content of the first and second substances,
Figure FDA0003254130390000036
substituting equation (8) and equation (9) into equation (7) yields:
Figure FDA0003254130390000037
derived from the young inequality:
Figure FDA0003254130390000041
by substituting formula (11) for formula (10), we obtain:
Figure FDA0003254130390000042
b.2. selecting Lyapunov function
Figure FDA0003254130390000043
To V2The derivation yields:
Figure FDA0003254130390000044
error variable z2The time derivative yields:
Figure FDA0003254130390000045
let f (Z) ═ Δ-1(q)(τf+τd)，Defining a non-linear function f (Z) ═ f1(Z),…,fn(Z)]TFor arbitrarily small constants ε according to the universal approximation theoremi> 0, there is a fuzzy logic function Wi TS (Z) is such that fi(Z)＝Wi TS(Z)+δi；
Wherein, deltaiRepresents an approximation error and satisfies δi≤εi，i＝1,…,n；
Due to z2＝[z21,z22,…,z2n]TDerived from the young inequality:
Figure FDA0003254130390000046
wherein l is a constant greater than zero, WiRepresents the fuzzy weight vector, | WiI represents WiNorm of (d);
definition θ ═ max { | | | W1||2,||W2||2,…,||Wn||2Equation (15) is written as:
Figure FDA0003254130390000047
definition of
Figure FDA0003254130390000048
Then:
Figure FDA0003254130390000049
the real control law τ is selected as follows:
Figure FDA00032541303900000410
wherein k is2Represents a constant greater than zeroThe number of the first and second groups is,
Figure FDA00032541303900000411
an estimated value representing theta;
substituting equations (16), (17) and (18) into equation (13) defines the estimation error as
Figure FDA00032541303900000412
Obtaining:
Figure FDA00032541303900000413
b.3. selecting a Lyapunov function:
Figure FDA0003254130390000051
where r is a constant greater than zero, deriving V:
Figure FDA0003254130390000052
selecting a self-adaptive law:
Figure FDA0003254130390000053
wherein m is a constant greater than zero;
c. performing stability analysis on the constructed multi-joint mechanical arm impedance control method based on limited time output state limitation;
substituting equation (21) into equation (20) yields equation (22):
Figure FDA0003254130390000054
derived from the young inequality:
Figure FDA0003254130390000055
substituting formula (23) for formula (22) to obtain:
Figure FDA0003254130390000056
when in use
Figure FDA0003254130390000057
Obtaining:
Figure FDA0003254130390000058
when in use
Figure FDA0003254130390000059
Obtaining:
Figure FDA00032541303900000510
formula (27) is derived from formulae (25) and (26):
Figure FDA00032541303900000511
when the condition | z is satisfied1i|≤kaiAnd when gamma is more than 0 and less than 1, the following inequality is satisfied:
Figure FDA00032541303900000512
by substituting equations (27) and (28) into equation (24), we obtain:
Figure FDA00032541303900000513
wherein, a is min { min {2 ═ min { (2) }γk1i},2γk2,mγ}，
Figure FDA0003254130390000061
As can be seen from equation (29), when the selection parameter satisfies k1i＞0，k2When m is more than 0, a is more than 0 and b is more than 0;
determining T according to the definition of finite time stabilitysAs shown in equation (30):
Figure FDA0003254130390000062
wherein V is more than 0 and less than 1, V1-γ(0) Representing that the power of 1-gamma of the initial value of the Lyapunov function V is selected;
for the
Figure FDA0003254130390000063
Satisfy the requirement of
Figure FDA0003254130390000064
The above inequality indicates that the error signal in the closed loop system is semi-global time-limited stable; it is further known that the initial condition satisfies z1(0)∈Ω0＝{z1∈Rn:|z1|＜|kaIn | z }, get | z1i|＜|kai|，i＝1,2,…n，
Figure FDA0003254130390000065
From the formula x1＝z1+xdAnd formula | xdi(t)|≤XdiI 1,2, …, n, yields | x1i|＜kai+Xdi＝kc1i，i＝1,2,…,n，
Figure FDA0003254130390000066
Thus, the robot output force/position state is constrained within a set bounded interval.
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