CN111618858B - Manipulator robust tracking control algorithm based on self-adaptive fuzzy sliding mode - Google Patents

Manipulator robust tracking control algorithm based on self-adaptive fuzzy sliding mode Download PDF

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CN111618858B
CN111618858B CN202010492721.4A CN202010492721A CN111618858B CN 111618858 B CN111618858 B CN 111618858B CN 202010492721 A CN202010492721 A CN 202010492721A CN 111618858 B CN111618858 B CN 111618858B
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manipulator
sliding mode
fuzzy
joint
adaptive
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CN111618858A (en
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王三秀
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Taizhou 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
    • B25J9/1602—Programme controls characterised by the control system, structure, architecture
    • B25J9/161—Hardware, e.g. neural networks, fuzzy logic, interfaces, processor
    • 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
    • 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
    • B25J9/1656—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

Abstract

The invention discloses a manipulator robust tracking control algorithm based on a self-adaptive fuzzy sliding mode, which is characterized in that the sliding mode control is adopted to realize the tracking of a manipulator track, the switching gain of the sliding mode control algorithm is adjusted through a self-adaptive fuzzy logic system, and the chattering of the sliding mode control is reduced; and aiming at the influence of unmodeled dynamics and external disturbance, a robust controller is adopted for compensation. Simulation experiments on the two-degree-of-freedom manipulator show that under the action of the manipulator robust tracking control algorithm based on the self-adaptive fuzzy sliding mode, the sliding mode control input signal is smooth, and the manipulator has high track tracking precision.

Description

Manipulator robust tracking control algorithm based on self-adaptive fuzzy sliding mode
Technical Field
The invention belongs to the technical field of manipulator system control, and particularly relates to a manipulator robust tracking control algorithm based on a self-adaptive fuzzy sliding mode.
Background
The multi-joint manipulator system has the characteristics of strong coupling, time variation, nonlinearity and the like, and in recent years, the manipulator high-precision control attracts wide attention in academia and industry. At present, a series of achievements have been obtained in the research on the problem of tracking the space of the manipulator joint, and control algorithms such as sliding mode control, adaptive control and robust control are provided. The sliding mode controller has simple algorithm and stronger robustness to parameter change and disturbance, and is particularly suitable for high-precision tracking control of nonlinear systems such as manipulators. However, the sliding mode control has a high-frequency buffeting problem, the magnitude of vibration is affected by the magnitude of switching gain of the sliding mode controller, and the switching gain which is large enough is usually selected to ensure the stability of a system, so that the buffeting phenomenon of the sliding mode control is aggravated. Such buffeting can cause unmodeled high frequency components to be present in the system and even cause the system to be unstable.
The fuzzy control has the universal approximation characteristic, can approximate any continuous function in a compact set, does not depend on a system model, and is widely applied to the self-adaptive control of the robot.
Disclosure of Invention
The invention aims to provide a manipulator robust tracking control algorithm based on a self-adaptive fuzzy sliding mode, which can effectively eliminate buffeting generated by the sliding mode control algorithm and realize accurate trajectory tracking of a manipulator.
Generally, the manipulator robust tracking control algorithm based on the self-adaptive fuzzy sliding mode adopts a fuzzy logic system to self-adaptively adjust the switching gain of the sliding mode controller on the basis of the sliding mode controller, and effectively eliminates buffeting of the sliding mode control algorithm. Secondly, the performance of the controller is reduced by considering the factors such as uncertainty of parameters of the manipulator, unmodeled dynamics and external disturbance, a robust controller is introduced to compensate the uncertainty of the external disturbance, the unmodeled dynamics and the like, and the accurate track tracking of the manipulator is realized.
In addition, most industrial manipulators complete operation in a task space through an end effector, and the expected track of the manipulator is also described by the task space, so that joint space control needs to be implemented by converting the task space coordinates of the manipulator into the joint space and then realizing the requirement of high-precision tracking control of the manipulator through the action of a high-performance manipulator joint space tracking control algorithm
In order to solve the technical problems, the invention adopts the following technical scheme:
a manipulator robust tracking control algorithm based on a self-adaptive fuzzy sliding mode comprises the following steps:
1) establishing manipulator task space trajectory model
Taking a two-joint manipulator as an example, taking a manipulator starting end node as the origin of a rectangular coordinate system, taking a task space track model as the original point of the rectangular coordinate system, taking a manipulator located in the rectangular coordinate system as a task space track model, taking the motion coordinates of a manipulator tail end node as (x, y), and taking the mass of a first connecting rod of the manipulator as m1The mass of the second connecting rod of the manipulator is m2The length of the first connecting rod of the manipulator is l1The length of the second connecting rod of the manipulator is l2The first joint angle is q1The second joint angle is q2。
2) Converting manipulator task space trajectory into joint space trajectory
Converting the task space track of the manipulator into a joint space track, namely converting the motion coordinates (x, y) of the tail end node of the manipulator in the task space into two joint angle positions (q)1,q2) Obtaining:
Figure BDA0002521338120000021
Figure BDA0002521338120000022
wherein the content of the first and second substances,
Figure BDA0002521338120000023
3) establishing mechanical arm dynamic model
The mechanical arm dynamic equation is as follows:
Figure BDA0002521338120000024
wherein the content of the first and second substances,q,
Figure BDA0002521338120000025
and
Figure BDA0002521338120000026
respectively representing the angular displacement, velocity and acceleration vectors of each joint. M (q) is an n x n order symmetric positive definite inertia matrix,
Figure BDA0002521338120000027
is a matrix of centrifugal and coriolis forces of order n × 1, and G (q) is a matrix of gravitational forces of order n × 1. d is equal to RnDenotes an external disturbance, τ ∈ RnThe torque vector, i.e. the control input, is controlled for each joint.
4) Sliding mode controller based on self-adaptive fuzzy control
Defining a sliding mode function as:
Figure BDA0002521338120000028
wherein e ═ qd-q，
Figure BDA0002521338120000029
qdA desired trajectory for the joint. And Λ is a positive definite diagonal constant matrix.
Defining the auxiliary signal:
Figure BDA00025213381200000210
Figure BDA0002521338120000031
the sliding mode controller is designed as
Figure BDA0002521338120000032
Wherein the content of the first and second substances,
Figure BDA0002521338120000033
are respectively M (q),
Figure BDA00025213381200000313
the estimated value of G (q), K, A is a positive definite matrix. Combining the equations (10-11), and substituting the equation (12) into the mechanical arm dynamics equation (6) to obtain:
Figure BDA0002521338120000034
order to
Figure BDA0002521338120000035
Figure BDA0002521338120000036
Figure BDA00025213381200000314
Finishing formula (13) to obtain
Figure BDA0002521338120000037
Wherein
Figure BDA0002521338120000038
Unmodeled dynamics and perturbation terms.
Defining Lyapunov functions
Figure BDA0002521338120000039
Wherein M represents an inertia matrix M (q) in formula (6),
derivation, bringing formula (8) in to obtain
Figure BDA00025213381200000310
Bringing formula (17) into the above formula to obtain
Figure BDA00025213381200000311
Wherein C represents the centrifugal force and the Goldson force matrix
Figure BDA00025213381200000315
Suppose that
Figure BDA00025213381200000316
Has a bounded property, and satisfies that | | | delta f | | | is less than or equal to K
Then
Figure BDA00025213381200000312
5) Design fuzzy system
The invention adopts a fuzzy system designed based on a product reasoning method and a central average defuzzifier to adaptively adjust the switching gain K of sliding mode control. Let K be [ K ]1,…,ki,…kn]T，kiIs the output of the ith fuzzy system.
The output of the fuzzy system is
Figure BDA0002521338120000041
Wherein θ ═ y1,…,ym]TIs a parameter vector, xi (x) ═ xi1(x),…,ξm(x)]TM is the number of fuzzy rules
Definition of
Figure BDA0002521338120000042
As can be seen from the formula (21), to ensure
Figure BDA0002521338120000043
Should make sTK≥0，s＝[s1,…,si,…sn]TAnd should satisfy sTΔf-sTKsgn(s) is less than or equal to 0, then siAnd k isiShould take the same number, and | siI and I kiThe | should change consistently.
By siFor input of fuzzy system, gain k is switchediAs an output, the input and output variables are fuzzified. The fuzzy quantity of the system input and output is described by negative middle, negative small, zero, positive small and positive middle, these 5 variables are described. The fuzzy inference rule is shown in table 1.
TABLE 1 fuzzy inference rules
Figure BDA0002521338120000044
Membership functions for representing fuzzy sets are designed as
Figure BDA0002521338120000045
The output of the ith fuzzy system is then:
Figure BDA0002521338120000046
get
Figure BDA0002521338120000047
Is Δ fiThe approximation of (1) exists according to the universal approximation theorem of (omega)iGreater than 0, has
Figure BDA0002521338120000048
Selecting an adaptation law as
Figure BDA0002521338120000051
Then the sliding mode control law based on the adaptive fuzzy switching gain control is as follows:
Figure BDA0002521338120000052
6) designing an adaptive robust controller
The robust controller is designed as
Figure BDA0002521338120000053
In the formula, epsilon is a very small normal number, beta is a disturbance and uncertain upper bound, and the following conditions are satisfied:
||Δf||≤β＝ρμ (30)
where ρ ═ max (1, | | e | | | | non-woven phosphor)2) Is a coefficient vector; mu isThe uncertainty of the system, its value adopts the following self-adapting algorithm to adjust automatically:
Figure BDA0002521338120000054
in the formula, γ is a positive constant matrix. And satisfy
Figure BDA0002521338120000055
Thus, the adaptive robust controller urCan be re-described as
Figure BDA0002521338120000056
The overall control law is
u＝u0+ur (33)
The invention discloses a manipulator robust tracking control algorithm based on a self-adaptive fuzzy sliding mode, and provides the self-adaptive fuzzy sliding mode robust tracking control algorithm aiming at a manipulator control system with uncertain parameters and external disturbance. The algorithm of the invention combines the self-adaptive fuzzy control, the sliding mode control and the robust control algorithm, realizes the compensation of modeling errors and interferences, weakens the buffeting of the sliding mode control and ensures the accurate track tracking of an uncertain manipulator system.
Drawings
FIG. 1 is a schematic diagram of a two-joint manipulator task space trajectory model in an embodiment of the invention;
FIG. 2 is a block diagram of a sliding mode robust control system based on adaptive fuzzy control according to the present invention;
FIG. 3 is a graph of a fixed gain based sliding mode control input signal according to an embodiment of the present invention;
FIG. 4 is a graph of a sliding mode control input signal based on a fuzzy adaptive gain in an embodiment of the present invention;
FIG. 5 is a graph of adaptive gain variation for two joints according to an embodiment of the present invention;
FIG. 6 is a graph of position tracking of a first joint and a second joint in an embodiment of the present invention;
FIG. 7 is a graph of tracking error for a first joint and a second joint in an embodiment of the present invention;
FIG. 8 is a graph of a trajectory trace of the end of the robot arm in an embodiment of the present invention;
FIG. 9 is a diagram of the pose movement locus of the robot arm in the embodiment of the invention.
Detailed Description
The following describes a control algorithm for servo system profile error according to the present invention in detail with reference to the accompanying drawings. In the description of the present invention, it is to be understood that the terms "left side", "right side", "upper", "lower", "bottom", etc., indicate orientations or positional relationships based on those shown in the drawings, and are only for convenience of describing the present invention and simplifying the description, but do not indicate or imply that the device or element referred to must have a specific orientation, be constructed and operated in a specific orientation, "first", "second", etc., do not represent an important degree of the component parts, and thus are not to be construed as limiting the present invention. The specific dimensions used in the present example are only for illustrating the technical solution and do not limit the scope of protection of the present invention.
The invention discloses a manipulator robust tracking control algorithm based on a self-adaptive fuzzy sliding mode, which comprises the following steps of:
1) establishing manipulator task space trajectory model
As shown in FIG. 1, m1And m2For the mass of two connecting rods of a manipulator, |1And l2Is the length of the two connecting rods, s1And s2Respectively, the distance from the center of mass to the revolute joint, q1And q is2Two joint angles.
The robot trajectory tracking means that the robot end effector tracks a desired trajectory in a task space, however, most of the robot execution structures are mounted on the robot body and each joint, so that the robot trajectory tracking in the joint space is to make each joint angle of the robot track each desired joint angle one by one. Therefore, in order to realize control in the joint space, it is necessary to convert the task space trajectory into the joint space trajectory.
2) Converting manipulator task space trajectory into joint space trajectory
Converting robot end node motion coordinates (x, y) in task space to two joint angle positions (q)1,q2)。
From FIG. 1, it can be seen that:
x＝l1cosq1+l2cos(q1+q2) (1)
y＝l1sinq1+l2sin(q1+q2) (2)
the sum of the square of formula (1) and the square of formula (2) is obtained
x2+y2＝l1 2+l2 2+2l1l2cosq2 (3)
Thereby can obtain
Figure BDA0002521338120000071
Get
Figure BDA0002521338120000072
Then
Figure BDA0002521338120000073
3) Establishing mechanical arm dynamic model
The mechanical arm is a complex multi-input multi-output nonlinear time-varying system, and the kinetic equation of the mechanical arm can be described as
Figure BDA0002521338120000074
Wherein the content of the first and second substances,q,
Figure BDA0002521338120000075
and
Figure BDA0002521338120000076
m (q) is an n multiplied by n order symmetric positive definite inertia matrix,
Figure BDA0002521338120000077
is a matrix of centrifugal and coriolis forces of order n × 1, and G (q) is a matrix of gravitational forces of order n × 1. d is equal to RnDenotes an external disturbance, τ ∈ RnThe torque vector, i.e. the control input, is controlled for each joint.
The robot dynamics system has the following dynamics characteristics:
properties 1: the inertial matrix M (q) is a symmetric positive definite matrix and is uniformly bounded for all q, i.e., there is a positive number λm，λMTo satisfy
0＜λmI≤M(q)＜λMI (7)
Properties 2:
Figure BDA00025213381200000710
as a diagonally symmetric matrix, i.e. for arbitrary vectors
Figure BDA0002521338120000078
Is provided with
Figure BDA0002521338120000079
Characteristic 3: the gravity term G (q) satisfies the condition that all q epsilon RnIs always bounded.
Controller design is performed as follows:
aiming at the tracking of the manipulator track, a sliding mode robust control algorithm based on self-adaptive fuzzy is designed. The control system block diagram is shown in fig. 2.
4) Sliding mode controller based on self-adaptive fuzzy control
Defining a sliding mode function as
Figure BDA0002521338120000081
Wherein e ═ qd-q，
Figure BDA0002521338120000082
qdThe trajectory is formed by the robot arm end coordinate movement for the desired trajectory of the joint. And Λ is a positive definite diagonal constant matrix.
Defining auxiliary signals
Figure BDA0002521338120000083
Figure BDA0002521338120000084
The sliding mode controller is designed as
Figure BDA0002521338120000085
Wherein the content of the first and second substances,
Figure BDA0002521338120000086
are respectively M (q),
Figure BDA00025213381200000815
the estimated value of G (q), K, A is a positive definite matrix.
Combining the equations (10-11), and substituting the equation (12) into the mechanical arm dynamics equation (6) to obtain:
Figure BDA0002521338120000087
order to
Figure BDA0002521338120000088
Figure BDA0002521338120000089
Figure BDA00025213381200000810
Finishing formula (13) to obtain
Figure BDA00025213381200000811
Wherein
Figure BDA00025213381200000812
Unmodeled dynamics and perturbation terms.
Defining Lyapunov functions
Figure BDA00025213381200000813
Wherein M represents an inertia matrix M (q) in formula (6),
derivation, bringing formula (8) in to obtain
Figure BDA00025213381200000814
Bringing formula (17) into the above formula to obtain
Figure BDA0002521338120000091
Wherein C represents the centrifugal force and the Goldson force matrix
Figure BDA0002521338120000092
Suppose that
Figure BDA0002521338120000093
Has a bounded property, and satisfies that | | | delta f | | | is less than or equal to K
Then
Figure BDA0002521338120000094
The system is stable.
The sliding mode controller is simple and effective, but in the sliding mode control law, buffeting is the main problem to be solved, wherein the switching gain K is the main reason for buffeting, and the larger the K is, the more obvious the buffeting is. K is used to compensate for the effects of external disturbances, and a large enough switching gain value is needed to fully compensate, further aggravating the system chattering.
In order to solve the problem of buffeting of the fixed gain of the sliding mode control, the self-adaptive fuzzy control is added into the sliding mode controller, and the switching gain value is self-adaptively adjusted by the fuzzy controller, so that the switching gain of the sliding mode controller can be adjusted along with time, and the buffeting phenomenon is improved.
5) Design fuzzy system
In this embodiment, a fuzzy system is designed based on a product reasoning method and a central average deblurring device, and the switching gain K of sliding mode control is adaptively adjusted. Let K be [ K ]1,…,ki,…kn]T，kiIs the output of the ith fuzzy system.
The output of the fuzzy system is
Figure BDA0002521338120000095
Wherein θ ═ y1,…,ym]TIs a parameter vector, xi (x) ═ xi1(x),…,ξm(x)]TM is the number of fuzzy rules
Definition of
Figure BDA0002521338120000096
As can be seen from the formula (21), to ensure
Figure BDA0002521338120000097
Should make sTK≥0，s＝[s1,…,si,…sn]TAnd should satisfy sTΔf-sTKsgn(s) is less than or equal to 0, then siAnd k isiShould take the same number, and | siI and I kiThe | should change consistently.
By siFor input of fuzzy system, gain k is switchediAs an output, the input and output variables are fuzzified. The fuzzy quantity of the system input and output is described by negative middle, negative small, zero, positive small and positive middle, these 5 variables are described. The fuzzy inference rule is shown in table 1.
TABLE 1 fuzzy inference rules
Figure BDA0002521338120000101
Membership functions for representing fuzzy sets are designed as
Figure BDA0002521338120000102
The output of the ith fuzzy system is then:
Figure BDA0002521338120000103
get
Figure BDA0002521338120000104
Is Δ fiThe approximation of (1) exists according to the universal approximation theorem of (omega)iGreater than 0, has
Figure BDA0002521338120000105
Selecting an adaptation law as
Figure BDA0002521338120000106
The sliding mode control law based on adaptive fuzzy switching gain control is
Figure BDA0002521338120000107
6) Designing an adaptive robust controller
This section designs robust control terms to eliminate their effects for unmodeled dynamics and external disturbances d.
The robust controller is designed as
Figure BDA0002521338120000108
In the formula, epsilon is a very small normal number, beta is a disturbance and uncertain upper bound, and the following conditions are satisfied:
||Δf||≤β＝ρμ (30)
where ρ ═ max (1, | | e | | | | non-woven phosphor)2) Is a coefficient vector; mu is an uncertainty item of the system, and the value of the uncertainty item is automatically adjusted by adopting the following adaptive algorithm:
Figure BDA0002521338120000111
in the formula, γ is a positive constant matrix. And satisfy
Figure BDA0002521338120000112
Thus, the adaptive robust controller urCan be re-described as
Figure BDA0002521338120000113
The overall control law is
u＝u0+ur (33)
Simulation analysis:
in order to verify the effectiveness of the controller designed in this embodiment, simulation studies were performed with the two-link robot as the target. The specific expression of the kinetic equation is as follows:
Figure BDA0002521338120000114
wherein the content of the first and second substances,
Figure BDA0002521338120000115
Figure BDA0002521338120000116
C12＝C21＝m2l1s2sinq2
Figure BDA0002521338120000117
G2＝2m2l1s2gcos(q1+q2)
wherein g is the acceleration of gravity.
The simulation parameters of the robot object are shown in Table 2
Table 2 two-degree-of-freedom manipulator simulation parameters
Figure BDA0002521338120000118
Selecting manipulator task space tracking motion trail
Figure BDA0002521338120000119
The initial position is [ x ]d0 yd0]＝[-0.25 0]T. Function d is respectively selected by two joint disturbancesx＝sin(πt)，dy＝sin(2πt)。
In order to implement control in the joint space, the desired joint trajectory and the initial joint angle are obtained by inverse solution according to the above.
Parameters of the sliding mode controller are Λ ═ diag (10,10), a ═ diag (150 ), parameters of the robust controller are γ ═ 2, and ∈ ═ 0.01. Selecting fuzzy membership function as muNM(xi)＝exp[-((xi+π/6)/(π/24))2],μNS(xi)＝exp[-((xi+π/12)/(π/24))2],μZ(xi)＝exp[-(xi/(π/24))2],μPS(xi)＝exp[-((xi-π/12)/(π/24))2],μPM(xi)＝exp[-((xi-π/6)/(π/24))2]，
Firstly, respectively simulating by adopting a traditional sliding mode control law based on fixed gain (K is diag (15,15)) and a sliding mode control law based on fuzzy adaptive gain adjustment to embody the effect of the self-adaptive fuzzy control adjustment of sliding mode switching gain on improving the phenomenon of sliding mode control buffeting. The two-case sliding mode control input signals are shown in fig. 3 and 4.
As can be seen from fig. 3 and 4, the sliding mode control algorithm with fixed gain is adopted, the sliding mode control input signal has obvious buffeting, and the robot sliding mode control based on the fuzzy adaptive gain adjustment is adopted, so that the buffeting phenomenon is effectively improved, and the input signal is controlled to be smooth. The two-joint gain adaptation changes are shown in fig. 5.
The adaptive fuzzy sliding mode robust controller provided by the embodiment is used for tracking the trajectory of the two-joint manipulator, and simulation results are shown in fig. 6-9. Wherein, fig. 6 is two-joint position tracking, wherein the upper curve in the figure represents the expected joint track, and the lower curve in the figure represents the actual tracking track. Fig. 7 is a two joint tracking error. Fig. 8 is robot task space end trajectory tracking. Fig. 9 is a robot pose motion trajectory.
As can be seen from the simulation result, the manipulator control method has higher track tracking precision, and basically realizes high-precision tracking with zero tracking error except for the fact that the initial stage has a tiny tracking error.
Based upon the foregoing description of the preferred embodiment of the invention, it should be apparent that the invention defined by the appended claims is not limited solely to the specific details set forth in the foregoing description, as many apparent variations thereof are possible without departing from the spirit or scope thereof.

Claims (1)

1. A manipulator robust tracking control algorithm based on an adaptive fuzzy sliding mode is characterized by comprising the following steps:
1) establishing manipulator task space trajectory model
The starting end node of the manipulator is used as the origin of a rectangular coordinate system, the task space track model comprises the manipulator located in the rectangular coordinate system, the motion coordinates of the tail end node of the manipulator are (x, y), and the mass of the first connecting rod of the manipulator is m1The mass of the second connecting rod of the manipulator is m2The length of the first connecting rod of the manipulator is l1The length of the second connecting rod of the manipulator is l2The first joint angle is q1The second joint angle is q2；
2) Converting manipulator task space trajectory into joint space trajectory
Converting the task space track of the manipulator into a joint space track, namely converting the motion coordinates (x, y) of the tail end node of the manipulator in the task space into two joint angle positions (q)1,q2) Obtaining:
Figure FDA0002521338110000011
Figure FDA0002521338110000012
wherein the content of the first and second substances,
Figure FDA0002521338110000013
3) establishing mechanical arm dynamic model
The mechanical arm dynamic equation is as follows:
Figure FDA0002521338110000014
wherein the content of the first and second substances,q,
Figure FDA0002521338110000015
and
Figure FDA0002521338110000016
respectively representing angular displacement, speed and acceleration vectors of each joint; m (q) is an n x n order symmetric positive definite inertia matrix,
Figure FDA0002521338110000017
is a matrix of n × 1 order centrifugal force and coriolis force, and G (q) is a matrix of n × 1 order gravity; d is equal to RnDenotes an external disturbance, τ ∈ RnControlling the torque vector, i.e. the control input, for each joint;
4) sliding mode controller based on self-adaptive fuzzy control
Defining a sliding mode function as:
Figure FDA0002521338110000018
wherein e ═ qd-q，
Figure FDA0002521338110000019
qdA desired trajectory for the joint; Λ is a positive fixed diagonal constant matrix;
defining the auxiliary signal:
Figure FDA0002521338110000021
Figure FDA0002521338110000022
the sliding mode controller is designed as
Figure FDA0002521338110000023
Wherein the content of the first and second substances,
Figure FDA0002521338110000024
are respectively M (q),
Figure FDA0002521338110000025
g (q), K, A is a positive definite matrix;
combining the equations (10-11), and substituting the equation (12) into the mechanical arm dynamics equation (6) to obtain:
Figure FDA0002521338110000026
order to
Figure FDA0002521338110000027
Figure FDA0002521338110000028
Figure FDA0002521338110000029
Finishing formula (13) to obtain
Figure FDA00025213381100000210
Wherein
Figure FDA00025213381100000211
As unmodeled dynamics and perturbation terms;
defining Lyapunov functions
Figure FDA00025213381100000212
Wherein M represents an inertia matrix M (q) in formula (6),
derivation, bringing formula (8) in to obtain
Figure FDA00025213381100000213
Bringing formula (17) into the above formula to obtain
Figure FDA00025213381100000214
Wherein C represents the centrifugal force and the Goldson force matrix
Figure FDA00025213381100000215
Suppose that
Figure FDA00025213381100000216
Has a bounded property, and satisfies that | | | delta f | | | is less than or equal to K
Then
Figure FDA0002521338110000031
5) Design fuzzy system
Designing a fuzzy system by adopting a product reasoning method and a central average defuzzifier, and adaptively adjusting the switching gain K of sliding mode control; let K be [ K ]1,…,ki,…kn]T，kiIs the output of the ith fuzzy system;
the output of the fuzzy system is
Figure FDA0002521338110000032
Wherein θ ═ y1,…,ym]TIs a parameter vector, xi (x) ═ xi1(x),…,ξm(x)]TM is the number of fuzzy rules
Definition of
Figure FDA0002521338110000033
As can be seen from the formula (21), to ensure
Figure FDA0002521338110000037
Should make sTK≥0，s＝[s1,…,si,…sn]TAnd should satisfy sTΔf-sTK sgn(s) is less than or equal to 0, then siAnd k isiShould take the same number, and | siI and I kiThe | should have consistent trend;
by siFor input of fuzzy system, gain k is switchediAs output, fuzzifying the input and output variables; fuzzy quantities input and output by the system are respectively described by 5 variables of negative middle, negative small, zero, positive small and positive middle; the fuzzy inference rule is shown in table 1;
TABLE 1 fuzzy inference rules
Figure FDA0002521338110000034
Membership functions for representing fuzzy sets are designed as
Figure FDA0002521338110000035
The output of the ith fuzzy system is then:
Figure FDA0002521338110000036
get
Figure FDA0002521338110000041
Is Δ fiThe approximation of (1) exists according to the universal approximation theorem of (omega)iGreater than 0, has
Figure FDA0002521338110000042
Selecting an adaptation law as
Figure FDA0002521338110000043
Then the sliding mode control law based on the adaptive fuzzy switching gain control is as follows:
Figure FDA0002521338110000044
6) designing an adaptive robust controller
The robust controller is designed as
Figure FDA0002521338110000045
In the formula, epsilon is a very small normal number, beta is a disturbance and uncertain upper bound, and the following conditions are satisfied:
||Δf||≤β＝ρμ (30)
where ρ ═ max (1, | | e | | | | non-woven phosphor)2) Is a coefficient vector; mu is an uncertainty item of the system, and the value of the uncertainty item is automatically adjusted by adopting the following adaptive algorithm:
Figure FDA0002521338110000046
wherein gamma is a positive definite constant matrix; and satisfy
Figure FDA0002521338110000047
Thus, the adaptive robust controller urCan be re-described as
Figure FDA0002521338110000048
The overall control law is
u＝u0+ur (33)。
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