CN108972560B - Layered sliding mode control method for under-actuated mechanical arm based on fuzzy optimization - Google Patents

Layered sliding mode control method for under-actuated mechanical arm based on fuzzy optimization Download PDF

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CN108972560B
CN108972560B CN201810965656.5A CN201810965656A CN108972560B CN 108972560 B CN108972560 B CN 108972560B CN 201810965656 A CN201810965656 A CN 201810965656A CN 108972560 B CN108972560 B CN 108972560B
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sliding mode
control
input
mechanical arm
switching
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CN108972560A (en
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高欣
翟林
任泽宇
刘惠禾
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Beijing University of Posts and Telecommunications
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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

Abstract

The embodiment of the invention discloses a hierarchical sliding mode control method for an underactuated mechanical arm based on fuzzy optimization. The method comprises the following steps: establishing a planar two-degree-of-freedom active-passive (AP type) under-actuated mechanical arm dynamic model, simplifying the nonlinear coupling dynamic model into an affine nonlinear system form, and taking two joint angles as control targets; then, designing a layered sliding mode controller, forming an angle and an angular speed of each joint into a subsystem, solving equivalent input of the two subsystems, and constructing a sliding mode total switching surface by using a Lyapunov feedback function method to obtain a control rate; and finally, designing a fuzzy rule to dynamically optimize a switching robust item in the sliding mode control rate, and solving the buffeting problem existing in the conventional control system. Compared with other planar two-degree-of-freedom AP type mechanical arm control methods, the technical scheme provided by the embodiment of the invention can reduce the steady-state time of the control process and improve the control precision.

Description

Layered sliding mode control method for under-actuated mechanical arm based on fuzzy optimization
[ technical field ] A method for producing a semiconductor device
The invention belongs to the technical field of automatic control, relates to fuzzy control and layered sliding mode control, and particularly relates to a fuzzy optimization-based layered sliding mode control method for an underactuated mechanical arm.
[ background of the invention ]
Space operations often require sufficient flexibility and redundancy of the robot. In a microgravity environment, the mechanical arm can be made of light materials such as high-strength carbon fibers, but the driving motor cannot be made to be light at present. And the weight can be greatly reduced by using the non-driving joint on the premise of finishing the required action, so that the cost of sending the effective load to the satellite orbit is greatly reduced.
In the controller design of the under-actuated arm system, it is difficult to perform control by a general feedback method because the requirement of Brockett for smooth feedback stabilization is not satisfied. The friction force between joints is utilized to control the non-driving joint to be only suitable for slow motion; layered sliding mode control techniques can enable the system to reach the total sliding mode surface in a limited time when the total control input includes equivalent inputs for all subsystems and appropriate switching terms. However, the layered sliding mode control method still needs to be optimized in terms of the steady-state time and the control precision of the control.
[ summary of the invention ]
In view of the above problems existing in the layered sliding mode control method, the present invention provides a fuzzy optimization-based layered sliding mode control method for an underactuated mechanical arm, including:
establishing a planar two-degree-of-freedom active-passive AP type under-actuated mechanical arm dynamic model, simplifying the nonlinear coupling dynamic model into an affine nonlinear system form, and taking two joint angles as control targets;
designing a layered sliding mode controller, forming an angle and an angular speed of each joint into a subsystem, solving equivalent input of the two subsystems, and constructing a sliding mode total switching surface by using a Lyapunov feedback function method to obtain a control rate;
a switching robust item in the sliding mode control rate is dynamically optimized by designing a fuzzy rule, and the buffeting problem existing in the conventional control system is solved.
In the method, a planar two-degree-of-freedom active-passive AP type under-actuated mechanical arm dynamic model is established, a nonlinear coupling dynamic model is simplified into an affine nonlinear system form, and two joint angles are used as control targets, wherein the method comprises the following steps:
a planar two-degree-of-freedom AP type under-actuated mechanical arm dynamics model is established as follows:
Figure GDA0002999870140000021
as shown in the formula (1), this is a single-input multi-output nonlinear coupling system, in which θ,
Figure GDA0002999870140000022
Respectively representing a joint angle sequence, a joint angular velocity sequence and a joint angular acceleration sequence, which are 2-dimensional column vectors; m (theta) belongs to R2×2Is an inertial matrix in joint space;
Figure GDA0002999870140000023
is a matrix of coriolis forces and centrifugal forces; τ ═ t (τ)1,0)TIs a joint moment vector of which τ1The input torque of the active joint is 0, and the input torque of the passive joint is 0; two sets of state variables
Figure GDA0002999870140000024
As two subsystems, transforming the above expression can obtain:
Figure GDA0002999870140000025
wherein M is-1(θ) is an inverse matrix of the inertia matrix M (θ), and the equation (2) is developed into the form of an affine nonlinear system to obtain:
Figure GDA0002999870140000026
wherein, a1,a2,a3The parameters related to the mechanical arm are positive constants;
Figure GDA0002999870140000027
Figure GDA0002999870140000028
a3＝m2l1r2
I1、I2the moment of inertia, m, of two joints respectively1、m2Mass of two connecting rods, r1、r2Respectively, the position of the center of gravity of the connecting rod of the mechanical arm1Is the length of the first link, c2Represents cos θ2；x1，x2，x3，x4For the state variable of the system, the corresponding nonlinear function f can be obtained by contrasting the standard formula of the single-input multi-output nonlinear coupling system1(x)，f2(x)，b1(x)，b2(x) (ii) a Selecting two joint angles of the mechanical arm as a control target, namely y ═ x1，x3]。
In the method, designing a layered sliding mode controller, regarding two groups of joint angles as two subsystems to solve equivalent input, and constructing a sliding mode control total switching surface by using a Lyapunov feedback function method to obtain a control rate includes:
two sets of state variables
Figure GDA0002999870140000031
When two subsystems are used, a first-order sub sliding mode surface s is defined for each subsystem respectively1And s2：
s1＝c1x1+x2 (4)
s2＝c2x3+x4 (5)
Wherein, c1,c2Is a positive constant, and according to the Filipov equivalent control theory, the following is obtained:
Figure GDA0002999870140000032
Figure GDA0002999870140000033
by bringing the mechanical arm affine nonlinear system formula (3) into the formula (5) and the formula (6), the equivalent input u of the two subsystems can be solvedeq1And ueq2：
Figure GDA0002999870140000034
Figure GDA0002999870140000035
The input of the system should have a sliding mode switching control part according to the arrival condition of the sliding mode, and the method adopts
Figure GDA0002999870140000036
Figure GDA0002999870140000037
As a switching function, where W, K is a constant and sgn is a sign function; because the number of input control quantities of the under-actuated system is less than the number of output quantities needing to be controlled, for the control of the whole system, even if all subsystems are ensured to move towards the sliding mode surfaces of the subsystems, the whole system cannot be ensured to move towards the total sliding mode surface, and therefore, the system is redesigned by adopting a Lyapunov feedback function method to meet the requirements of all sliding mode surface stable control:
let the switching term of the system be uswDefining the total inputs to the system as:
u＝ueq1+ueq2+usw
define S as the system total slip surface:
S＝z1s1+z2s2
defining a positive lyapunov function V as:
V＝S2/2
wherein z is1,z2And (3) obtaining positive constants by derivation of two ends of the Lyapunov function V and substituting the sub-sliding mode surface formulas (4) and (5) and the total system input u into the Lyapunov derivative formula:
Figure GDA0002999870140000041
defined by the arrival rule of the sliding mode, if the total system is regarded as a first-order system, the system has
Figure GDA0002999870140000042
The switching function part u of the control input can then be derivedswComprises the following steps:
Figure GDA0002999870140000043
thus yielding the total input u to the system.
In the method, the designing of the fuzzy rule dynamically optimizes the switching robust item in the sliding mode control rate, so as to solve the buffeting problem existing in the conventional control system, and the method comprises the following steps:
analysis on the sliding mode variable structure control shows that when the switching item is increased, the buffeting of the system is increased, meanwhile, the system state is rapidly converged to the sliding mode surface, and on the contrary, the buffeting is reduced, and the steady-state time is increased; in the layered sliding mode control rate, only a proper control rule u is selected, and the system state can reach the total sliding mode surface; thus, let the optimized control inputs u' be:
u′＝ueq1+ueq2+λ×usw (13)
wherein, λ is a coefficient for dynamically adjusting a switching term, a total sliding mode surface S is used as an input of a fuzzy controller, a switching robust term coefficient of the system is used as an output variable, and a fuzzy inference form is as follows:
Ri:If S is Fi Thenλis Ui
wherein R isiFor the ith fuzzy control rule in the fuzzy controller, i is 1,2iAs fuzzy sets of input variables S, UiIs a fuzzy set of output variables λ; the input adopts single-point fuzzification, the output adopts a single-value membership function, and the output of the fuzzy controller can be obtained by adopting a gravity center method defuzzification method as follows:
Figure GDA0002999870140000051
wherein
Figure GDA0002999870140000052
The excitation strength of the ith rule;
the method utilizes fuzzy reasoning to adjust the sliding mode function of the under-actuated system in real time, and when the total sliding mode function S of the system is far away from the sliding mode surface S and is equal to 0, the switching item is increased so as to accelerate convergence and reduce the steady-state time; when the total sliding mode function S of the system moves near the sliding mode surface S which is 0, the switching term is reduced, so that buffeting is reduced, the control precision is increased, and finally the position control of the planar two-degree-of-freedom active-passive AP type under-actuated mechanical arm is realized.
[ description of the drawings ]
In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the drawings needed to be used in the embodiments will be briefly described below, and it is obvious that the drawings in the following description are only some embodiments of the present invention, and it is obvious for those skilled in the art to obtain other drawings based on these drawings without inventive labor.
FIG. 1 shows a design flow diagram of an embodiment of the present invention;
FIG. 2 is a block diagram of a layered sliding mode control system according to an embodiment of the present invention;
FIG. 3 shows a membership function of an input S;
FIG. 4 shows a membership function for the output λ;
FIG. 5 shows a joint angle control curve of the drive joint;
FIG. 6 shows a joint angular velocity control curve of the drive joint;
FIG. 7 shows a joint angle control curve for the drive joint;
fig. 8 shows a joint angular velocity control curve of the drive joint.
[ detailed description ] embodiments
For better understanding of the technical solutions of the present invention, the following detailed description of the embodiments of the present invention is provided with reference to the accompanying drawings.
It should be understood that the described embodiments are only some embodiments of the invention, and not all embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
The layered sliding-mode control method of the underactuated mechanical arm based on fuzzy optimization specifically comprises the following steps:
1. establishing a planar two-degree-of-freedom active-passive AP type under-actuated mechanical arm dynamic model, simplifying the nonlinear coupling dynamic model into an affine nonlinear system form, and taking two joint angles as control targets; 2. designing a layered sliding mode controller, forming an angle and an angular speed of each joint into a subsystem, solving equivalent input of the two subsystems, and constructing a sliding mode total switching surface by using a Lyapunov feedback function method to obtain a control rate; 3. a switching robust item in the sliding mode control rate is dynamically optimized by designing a fuzzy rule, and the buffeting problem existing in the conventional control system is solved.
The embodiment of the present invention provides a fuzzy optimization-based layered sliding-mode control method for an underactuated mechanical arm, and please refer to fig. 1, which is a schematic flow diagram of the fuzzy optimization-based layered sliding-mode control method for an underactuated mechanical arm according to the embodiment of the present invention, as shown in fig. 1, the method includes the following steps:
101, establishing a planar two-degree-of-freedom active-passive AP type under-actuated mechanical arm dynamic model, simplifying a nonlinear coupling dynamic model into an affine nonlinear system form, and taking two joint angles as control targets;
specifically, a planar two-degree-of-freedom active-passive AP type under-actuated mechanical arm dynamic model is established as follows:
Figure GDA0002999870140000061
as shown in the formula (1), this is a single-input multi-output nonlinear coupling system, in which θ,
Figure GDA0002999870140000062
Respectively representing a joint angle sequence, a joint angular velocity sequence and a joint angular acceleration sequence, which are 2-dimensional column vectors; m (theta) belongs to R2×2Is an inertial matrix in joint space;
Figure GDA0002999870140000063
is a matrix of coriolis forces and centrifugal forces; τ ═ τ [ τ ]1，0]TIs a joint moment vector of which τ1The input torque of the active joint is 0, and the input torque of the passive joint is 0; (ii) a Two in shapeSet of state variables
Figure GDA0002999870140000064
Considering two subsystems, transforming the above expression can obtain:
Figure GDA0002999870140000065
wherein M is-1(θ) is an inverse matrix of the inertia matrix M (θ), and the equation (2) is developed into the form of an affine nonlinear system to obtain:
Figure GDA0002999870140000071
wherein, a1,a2,a3The parameters related to the mechanical arm are positive constants;
Figure GDA0002999870140000072
Figure GDA0002999870140000073
a3＝m2l1r2
I1、I2is the moment of inertia of two joints, m1、m2Is the mass of two connecting rods, r1、r2Is the position of the center of gravity of the connecting rod of the mechanical arm1Is the length of the first link, c2Represents cos θ2；x1,x2,x3，x4Comparing the state variable of the system with the standard form of the nonlinear coupling system with single input and multiple outputs
Figure GDA0002999870140000074
Figure GDA0002999870140000075
Figure GDA0002999870140000076
Figure GDA0002999870140000077
Corresponding non-linear function f can be obtained1(x)，f2(x)，b1(x)，b2(x) (ii) a Selecting two joint angles of the mechanical arm as a control target, namely y ═ x1,x3]。
102, designing a layered sliding mode controller, regarding two groups of joint angles as two subsystems to solve equivalent input, constructing a sliding mode control total switching surface by using a Lyapunov feedback function method, and obtaining a control rate, wherein a flow chart is shown in FIG. 2.
Specifically, two sets of state variables are combined
Figure GDA0002999870140000078
Viewed as two subsystems, a first-order sliding-mode surface s is defined for each subsystem1And s2：
s1＝c1x1+x2 (4)
s2＝c2x3+x4 (5)
Wherein, c1,c2Is a positive constant, and according to the Filipov equivalent control theory, the following is obtained:
Figure GDA0002999870140000079
Figure GDA00029998701400000710
the input u can be obtained by bringing the mechanical arm affine nonlinear system formula (3) into the formula (6) and the formula (7)eq1And ueq2：
Figure GDA0002999870140000081
Figure GDA0002999870140000082
The input of the system should have a sliding mode switching control part according to the arrival condition of the sliding mode, and the method adopts
Figure GDA0002999870140000083
Figure GDA0002999870140000084
As a switching function; because the number of input control quantities of the under-actuated system is less than the number of output quantities needing to be controlled, for the control of the whole system, even if all subsystems are ensured to move towards the sliding mode surfaces of the subsystems, the whole system cannot be ensured to move towards the total sliding mode surface, and therefore, the system is redesigned by adopting a Lyapunov feedback function method to meet the requirements of all sliding mode surface stable control:
let the switching term of the system be uswDefining the total input u of the system as:
u＝ueq1+ueq2+usw
define S as the system total slip surface:
S＝z1s1+z2s2
defining a positive lyapunov function V as:
V＝S2/2
wherein z is1,z2And (3) obtaining positive constants by derivation of two ends of the Lyapunov function V and substituting the sub-sliding mode surface formulas (4) and (5) and the total system input u into the Lyapunov derivative formula:
Figure GDA0002999870140000085
defined by the arrival rule of the sliding mode, and the total system is regarded as a first-order system, then
Figure GDA0002999870140000086
The switching function part u of the control input can then be derivedswComprises the following steps:
Figure GDA0002999870140000087
where W, K is a constant and sgn is a sign function, thus yielding the total input u to the system.
And 103, designing a fuzzy rule to dynamically optimize a switching robust item in the sliding mode control rate, and solving the buffeting problem existing in the conventional control system.
Specifically, analysis of sliding mode variable structure control shows that when the switching term is increased, the buffeting of the system is increased, and meanwhile, the system state is rapidly converged towards the sliding mode surface, and on the contrary, the buffeting is reduced, and the steady-state time is increased; in the layered sliding mode control rate, only a proper control rule u is selected, and the system state can reach the total sliding mode surface; thus, let the optimized control inputs u' be:
u′＝ueq1+ueq2+λ×usw (13)
wherein, λ is a coefficient for dynamically adjusting a switching term, a total sliding mode surface S is used as an input of the fuzzy controller, a switching robust term coefficient of the system is used as an output variable, and a specific fuzzy inference form is as follows:
R1:If S is PB Thenλis PB
R2:If S is P Thenλis P
R3:If S is Z Thenλis Z
R4:If S is N Thenλis P
R5:If S is NB Thenλis PB
wherein R isiFor the ith fuzzy control rule in the fuzzy controller, i is 1,2, 5, the input adopts single-point fuzzification, the output adopts a single-value membership function, membership functions of the input S and the output lambda are shown in fig. 3 and 4, and the output of the fuzzy controller can be obtained by adopting a gravity center method defuzzification method.
The method utilizes fuzzy reasoning to adjust the sliding mode function of the under-actuated system in real time, and when the total sliding mode function S of the system is far away from the sliding mode surface S and is equal to 0, the switching item is increased so as to accelerate convergence and reduce the steady-state time; when the total sliding mode function S of the system moves near the sliding mode surface S which is 0, the switching term is reduced, so that buffeting is reduced, the control precision is increased, and finally the position control of the planar two-degree-of-freedom active-passive AP type under-actuated mechanical arm is realized.
In the specific implementation, the kinetic parameters and control parameters are shown in the following table:
a1 a2 a3 c1 c2 W K z1 z2
3.375 1.396 0.25 3.5 0.27 3 13 8.6 6.3
assume an initial angle of θ10＝95°,θ20At-5 DEG, the initial angular velocity is
Figure GDA0002999870140000091
The desired point to be reached by the robot arm is theta1d＝90°,θ2dAt 0 deg., the angular velocity is still 0. The simulation time was set to 120 s. Fig. 5 and 6 show a joint angle control curve of the drive joint and a joint angular velocity control curve of the drive joint, respectively. It can be seen that the drive arm responds very quickly, and a change in state occurs around 1 second, indicating that the system is moving in the reverse direction very quickly after responding to the initial control input; the overshoot is small, the steady state error is only about 0.2 degrees, and the steady state is entered about 80 seconds and basically enters the steady state by 120 seconds. Fig. 7 and 8 show the joint angle control curve of the passive joint and the joint angular velocity control curve of the passive joint, respectively, and the motion of the non-driving arm is larger than the amplitude of the driving arm because the non-driving arm is subjected to the inertia effect when the driving arm runs. The change in state also occurs around 1 second and enters substantially steady state around 120 seconds. As can be seen from the simulation results, the method provided by the invention can reduce the steady-state time of the control process and improve the control precision.
The above-mentioned contents are only for illustrating the technical idea of the present invention, and the protection scope of the present invention is not limited thereby, and any modification made on the basis of the technical idea of the present invention falls within the protection scope of the claims of the present invention.

Claims (1)

1. A hierarchical sliding mode control method of an underactuated mechanical arm based on fuzzy optimization is characterized by comprising the following steps:
(1) the method comprises the following steps of establishing a planar two-degree-of-freedom active-passive AP type under-actuated mechanical arm dynamic model, simplifying a nonlinear coupling dynamic model into an affine nonlinear system form, and taking two joint angles as control targets, wherein the method specifically comprises the following steps:
a two-degree-of-freedom planar active-passive AP type under-actuated mechanical arm dynamic model is established as follows:
Figure FDA0003171849980000011
as shown in the formula (1), this is a single-input multi-output nonlinear coupling system, in which θ,
Figure FDA0003171849980000012
Respectively representing a joint angle sequence, a joint angular velocity sequence and a joint angular acceleration sequence, which are 2-dimensional column vectors; m (theta) belongs to R2×2Is an inertial matrix in joint space;
Figure FDA0003171849980000013
is a matrix of coriolis forces and centrifugal forces; τ ═ t (τ)1,0)TIs a joint moment vector of which τ1The input torque of the active joint is 0, and the input torque of the passive joint is 0; two sets of state variables
Figure FDA0003171849980000014
As two subsystems, transforming the above expression can obtain:
Figure FDA0003171849980000015
wherein M is-1(θ) is an inverse matrix of the inertia matrix M (θ), and the equation (2) is developed into the form of an affine nonlinear system to obtain:
Figure FDA0003171849980000016
wherein, a1,a2,a3The parameters related to the mechanical arm are positive constants;
Figure FDA0003171849980000021
Figure FDA0003171849980000022
a3＝m2l1r2
I1、I2the moment of inertia, m, of two joints respectively1、m2Mass of two connecting rods, r1、r2Respectively, the position of the center of gravity of the connecting rod of the mechanical arm1Is the length of the first link, c2Represents cos θ2；x1,x2,x3,x4For the state variable of the system, the corresponding nonlinear function f can be obtained by contrasting the standard formula of the single-input multi-output nonlinear coupling system1(x)，f2(x)，b1(x)，b2(x) (ii) a Selecting two joint angles of the mechanical arm as a control target, namely y ═ x1,x3]；
(2) Designing a layered sliding mode controller, forming an angle and an angular speed of each joint into a subsystem, solving equivalent input of the two subsystems, and constructing a sliding mode total switching surface by using a Lyapunov feedback function method to obtain a control rate, wherein the method specifically comprises the following steps:
two sets of state variables
Figure FDA0003171849980000023
When two subsystems are used, a first-order sub sliding mode surface s is defined for each subsystem respectively1And s2：
s1＝c1x1+x2 (4)
s2＝c2x3+x4 (5)
Wherein, c1,c2Is a positive constant, and according to the Filipov equivalent control theory, the following is obtained:
Figure FDA0003171849980000024
Figure FDA0003171849980000025
by bringing the mechanical arm affine nonlinear system formula (3) into the formula (6) and the formula (7), the equivalent input u of the two subsystems can be solvedeq1And ueq2：
Figure FDA0003171849980000026
Figure FDA0003171849980000027
The input of the system has a sliding mode switching control part by adopting the sliding mode control of the arrival conditions
Figure FDA0003171849980000028
Figure FDA0003171849980000031
As a switching function, where W, K is a constant and sgn isA sign function; because the number of input control quantities of the under-actuated system is less than the number of output quantities required to be controlled, for the control of the whole system, even if all subsystems are ensured to move towards the sliding mode surface of the system, the whole system cannot be ensured to move towards the total sliding mode surface, so that the system is redesigned by adopting a Lyapunov feedback function method, and the requirements of all sliding mode surface stable control are met:
let the switching term of the system be uswDefining the total input u of the system as:
u＝ueq1+ueq2+usw
define S as the system total slip surface:
S＝z1s1+z2s2
defining a positive lyapunov function V as:
V＝S2/2
wherein z is1,z2And (3) obtaining positive constants by derivation of two ends of the Lyapunov function V and substituting the sub-sliding mode surface formulas (4) and (5) and the total system input u into the Lyapunov derivative formula:
Figure FDA0003171849980000032
defined by the arrival rule of the sliding mode, regarding the total system as a first-order system, the following are provided:
-WS-Ksgn(S)＝z2b2(x)ueq1+z1b1(x)ueq2+(z1b1(x)+z2b2(x))usw (11)
the switching function part u of the control input can then be derivedswComprises the following steps:
Figure FDA0003171849980000033
thus obtaining the total input u of the system;
(3) a switching robust item in a sliding mode control rate is dynamically optimized by designing a fuzzy rule, the buffeting problem existing in the conventional control system is solved, and the method specifically comprises the following steps:
analysis on the sliding mode variable structure control shows that when the switching item is increased, the buffeting of the system is increased, meanwhile, the system state is rapidly converged to the sliding mode surface, and on the contrary, the buffeting is reduced, and the steady-state time is increased; in the layered sliding mode control rate, as long as the switching item contains compensation for the equivalent input item, the state of the system can reach the total sliding mode surface; thus, let the optimized control inputs u' be:
u′＝ueq1+ueq2+λ×usw (13)
wherein, λ is a coefficient for dynamically adjusting a switching term, a total sliding mode surface S is used as an input of a fuzzy controller, a switching robust term coefficient λ of the system is used as an output variable, and a fuzzy inference form is as follows:
Ri：If S is FiThenλis Ui
wherein R isiFor the ith fuzzy control rule in fuzzy controller, i is 1,2, … n …, FiAs fuzzy sets of input variables S, UiIs a fuzzy set of output variables λ; the input adopts single-point fuzzification, the output adopts a single-value membership function, and the output of the fuzzy controller can be obtained by adopting a gravity center method defuzzification method as follows:
Figure FDA0003171849980000041
wherein
Figure FDA0003171849980000042
The excitation strength of the ith rule;
the method utilizes fuzzy reasoning to adjust the sliding mode function of the under-actuated system in real time, and when the total sliding mode function S of the system is far away from the sliding mode surface S and is equal to 0, the switching item is increased so as to accelerate convergence and reduce the steady-state time; when the total sliding mode function S of the system moves near the sliding mode surface S which is 0, the switching term is reduced, so that buffeting is reduced, the control precision is increased, and finally the position control of the planar two-degree-of-freedom active-passive AP type under-actuated mechanical arm is realized.
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