CN112947066A - Manipulator improved finite time inversion control method - Google Patents

Manipulator improved finite time inversion control method Download PDF

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CN112947066A
CN112947066A CN202110105530.2A CN202110105530A CN112947066A CN 112947066 A CN112947066 A CN 112947066A CN 202110105530 A CN202110105530 A CN 202110105530A CN 112947066 A CN112947066 A CN 112947066A
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manipulator
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CN112947066B (en
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祁兴华
张川丰
侯世玺
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Suzhou Liankai Automation Co ltd
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Abstract

The invention discloses a manipulator improved finite time reversal control method, which comprises the following steps: s1, building a manipulator power model; s2, designing a manipulator finite time reversal controller; and S3, inverting the unknown state in the controller by using fuzzy neural network approximation. The control method of the invention utilizes the finite time inversion method to design the manipulator controller, so that the position of each joint of the manipulator reaches the expected target position, and adopts the structure and parameter full-regulation fuzzy neural network structure to design the self-adaptive fuzzy neural network control system, thereby overcoming the defect that the finite time inversion control strategy needs the accurate information of the system and further improving the robustness of the system.

Description

Manipulator improved finite time inversion control method
Technical Field
The invention relates to a manipulator control technology, in particular to an improved finite time reversal control method for a manipulator.
Background
The manipulator is a mechanical device which has the action function similar to that of a human arm, can grab and place objects in space or perform other operations, can replace heavy labor of people to realize mechanization and automation of production, can operate in a harmful environment to protect personal safety, and is widely applied to departments of mechanical manufacturing, metallurgy, electronics, light industry, atomic energy and the like.
Due to the fact that factors such as uncertainty of joint parameters of the manipulator exist, model uncertainty exists in a manipulator dynamic model, and the manipulator control accuracy based on model control is affected. At present, aiming at the problem that uncertainty factors such as model uncertainty and external interference influence the control precision of a manipulator joint, the uncertainty factors are mostly approximated or compensated through an uncertainty estimator, and the adaptive law of parameters in the estimator is determined through stability analysis. A neural network and a fuzzy system with universal approximation capability are widely applied to uncertainty approximation compensation, but the controller is used for manipulator control and needs to estimate system uncertainty upper bound information in advance, so that certain limitations exist.
Disclosure of Invention
The invention aims to overcome the defects of the prior art and provides a manipulator improved finite time reversal control method.
In order to achieve the purpose, the invention adopts the technical scheme that: a manipulator improved finite time reversal control method comprises the following steps:
step 1, building a mechanical arm power model
Consider a rigid manipulator with n joints, whose kinetic equation is:
Figure BDA0002917253980000011
wherein the ratio of q,
Figure BDA0002917253980000012
the position, speed and acceleration vector of each joint of the manipulator, M (q) epsilon Rn×nIn order to include an inertial matrix of the additional mass,
Figure BDA0002917253980000013
g (q) e R as a coupling matrix of centrifugal force and Coriolis forcen×1In the form of a gravity matrix vector,
Figure BDA0002917253980000014
for friction torque, τd∈Rn×1For unknown applied interference, tau epsilon Rn×1For the control input of the movements of the individual joints,n is the joint degree of freedom;
step 2, designing a manipulator finite time reversal controller
2.1 define the tracking error as
e1＝q-qd (2)
Wherein q isdIs the desired position of the joint;
derived from (2)
Figure BDA0002917253980000021
Selecting virtual control quantities
Figure BDA00029172539800000210
Wherein c is1Is a non-zero normal number;
taking the Lyapunov function as
Figure BDA0002917253980000022
Definition of
Figure BDA0002917253980000023
Derived from formula (5)
Figure BDA0002917253980000024
If e2When the value is equal to 0, then
Figure BDA0002917253980000025
The stability requirement is met, and the design is continued;
2.1 e2can be expressed as
Figure BDA0002917253980000026
Then designing a nonsingular terminal sliding mode surface as
Figure BDA0002917253980000027
Wherein λ is1>0 is a constant, p1,p2Is odd number, 1<p2/p1<2，
Further, we can get
Figure BDA0002917253980000028
The finite time inversion controller can be designed as
τBSC＝u1+u2 (10)
Figure BDA0002917253980000029
Figure BDA0002917253980000031
Wherein, tauM≥||τd||；
S3, aiming at the formula (10), approximating the unknown state in the inversion controller by using the fuzzy neural network, and defining the input-output relationship of the fuzzy neural network as follows:
Figure BDA0002917253980000032
wherein the content of the first and second substances,
Figure BDA0002917253980000033
Figure BDA0002917253980000034
lk(k＝1,...,Ny) The k-th output of the rule layer is represented,
Figure BDA0002917253980000035
representing the connection weight matrix between the fuzzification layer and the regular layer, here taken as the unit vector, NyIs the total number of rule layers.
In a specific embodiment, the fuzzy neural network consists of an input layer, a fuzzy inference layer and an output layer, wherein the network input is the tracking deviation e1Output as control force tauFNNThe signal propagation and the function of each layer in the fuzzy neural network are represented as follows:
a first layer: input layer
Each node of the layer is directly connected with each component of the input quantity, and the input quantity is transmitted to the next layer;
a second layer: blurring layer
A gaussian-type function is used as the membership function,
Figure BDA0002917253980000036
representing tracking offset vector e1The elements (A) and (B) in (B),
Figure BDA0002917253980000037
and
Figure BDA0002917253980000038
the centre vector and the base width of the membership function of the ith input variable, the jth fuzzy set, respectively, i.e.
Figure BDA0002917253980000039
Easy to calculate, using NpiRepresenting individual numbers of membership functions, and we define the sets of adaptive parameter vectors b and c representing all the basis width and center vectors of Gaussian membership functions, respectively, i.e.
Figure BDA00029172539800000310
Figure BDA00029172539800000311
Wherein
Figure BDA00029172539800000312
Representing the total number of membership functions;
and a third layer: rule layer
The layer adopts a fuzzy inference mechanism, and the output of each node is the product of all input signals of the node, namely
Figure BDA0002917253980000041
In the formula Ik(k＝1,...,Ny) The k-th output of the rule layer is represented,
Figure BDA0002917253980000042
representing the connection weight matrix between the fuzzification layer and the regular layer, here taken as the unit vector, NyIs the total number of rules;
a fourth layer: output layer
The nodes of the layer represent output variables, each node yo (o 1.., N)o) The output of (2) being the sum of all input signals of the node, i.e.
Figure BDA0002917253980000043
Further, the input-output relationship of the fuzzy neural network is defined as follows:
Figure BDA0002917253980000044
wherein the content of the first and second substances,
Figure BDA0002917253980000045
Figure BDA0002917253980000046
as a specific embodiment, the fuzzy neural network includes two self-regulation strategies of data learning and data deletion, specifically:
data learning strategy
The fuzzy rule is determined step by step based on the condition that
Figure BDA0002917253980000047
Where ψ is a spherical potential energy, representing the novelty of the input data, given by the following equation:
Figure BDA0002917253980000048
wherein E issAnd EaIs the novelty and the addition of a threshold value,
Figure BDA0002917253980000049
when a new fuzzy rule needs to be added, its parameters are initialized,
Figure BDA0002917253980000051
where k is the overlap parameter of a pre-given fuzzy rule,
data deletion policy
According to the universal approximation theory, there is an optimal control force
Figure BDA0002917253980000052
Satisfy the requirement of
Figure BDA0002917253980000053
Where ε is the minimum reconstruction error vector, W*,b*And c*Of W, b and c, respectivelyOptimizing parameters;
the output control force of the fuzzy neural network is assumed to be of the following form:
Figure BDA0002917253980000054
wherein the content of the first and second substances,
Figure BDA0002917253980000055
and
Figure BDA0002917253980000056
are respectively W*,b*And c*Is determined by the estimated value of (c),
defining an approximation error:
Figure BDA0002917253980000057
using Taylor series expansion, one can obtain
Figure BDA0002917253980000058
Wherein the content of the first and second substances,
Figure BDA0002917253980000059
b*and c*Is the optimum value for b and c,
Figure BDA00029172539800000510
and
Figure BDA00029172539800000511
is b*And c*Estimated value of, OnvIs a high-order term of the magnetic field,
Figure BDA00029172539800000512
Figure BDA00029172539800000513
then (25) is substituted into (24) to obtain
Figure BDA00029172539800000514
Wherein the content of the first and second substances,
Figure BDA0002917253980000061
further, the formula (9) can be rewritten as
Figure BDA0002917253980000062
Wherein the content of the first and second substances,
Figure BDA0002917253980000063
d＝-E-τdand d is the total uncertainty,
the self-adaptive law of the weight, the base width and the central vector of the designed fuzzy neural network can be designed as follows:
Figure BDA0002917253980000064
Figure BDA0002917253980000065
Figure BDA0002917253980000066
wherein S isiIs an element of S, betaiIs an element in beta, sigmaω,σb,σcIs a normal number, and is,
Figure BDA0002917253980000067
is that
Figure BDA0002917253980000068
Is determined by the estimated value of (c),
Figure BDA0002917253980000069
is omegaiIs determined to be the optimum value of (c),
defining the lyapunov function as
Figure BDA00029172539800000610
Derivative and substitute (27) into
Figure BDA00029172539800000611
Bringing (28) - (30) into (32) to obtain
Figure BDA0002917253980000071
If τ is satisfiedMAnd the designed manipulator improved finite time inversion control method is stable.
Due to the application of the technical scheme, compared with the prior art, the invention has the following advantages: according to the manipulator-improved finite-time inversion control method, the adaptive fuzzy neural network control system is designed to approach the finite-time inversion controller by adopting the structure and parameter full-regulation fuzzy neural network structure, and the limit that uncertain functions and interference upper bound need to be predicted is relaxed, so that the defect that a finite-time inversion control strategy needs system accurate information is overcome, and the system robustness is further improved.
Drawings
FIG. 1 is a block diagram of a manipulator-improved finite time inversion control method according to the present invention.
Detailed Description
The technical solution of the present invention is further explained with reference to the drawings and the specific embodiments.
A manipulator improved finite time reversal control method comprises the following steps:
firstly, building a mechanical hand power model
Consider a rigid manipulator with n joints, whose kinetic equation is:
Figure BDA0002917253980000072
wherein the ratio of q,
Figure BDA0002917253980000073
the position, speed and acceleration vector of each joint of the manipulator, M (q) epsilon Rn×nIn order to include an inertial matrix of the additional mass,
Figure BDA0002917253980000074
g (q) e R as a coupling matrix of centrifugal force and Coriolis forcen×1In the form of a gravity matrix vector,
Figure BDA0002917253980000075
for friction torque, τd∈Rn×1For unknown applied interference, tau epsilon Rn×1N is the degree of freedom of the joint;
(II) finite time reversal controller of design manipulator
The control object of the robot position tracking control system is to design an appropriate control input τ so that each joint position q of the robot tracks the desired position qdThe design steps are as follows:
2.1 define the tracking error as
e1＝q-qd (2)
Derived from (2)
Figure BDA0002917253980000081
Selecting virtual control quantities
Figure BDA0002917253980000082
Wherein c is1Is a non-zero normal number;
taking the Lyapunov function as
Figure BDA0002917253980000083
Definition of
Figure BDA0002917253980000084
Derived from formula (5)
Figure BDA0002917253980000085
If e2When the value is equal to 0, then
Figure BDA0002917253980000086
The stability requirement is met, so the design is required to be continued;
2.1 e2can be expressed as
Figure BDA0002917253980000087
Then designing a nonsingular terminal sliding mode surface as
Figure BDA0002917253980000088
Wherein λ is1>0 is a constant, p1,p2Is odd number, 1<p2/p1<2，
Further, we can get
Figure BDA0002917253980000089
The finite time inversion controller can be designed as
τBSC＝u1+u2 (10)
Figure BDA00029172539800000810
Figure BDA00029172539800000811
Wherein, tauM≥||τd||；
Constructing a Lyapunov function as
Figure BDA0002917253980000091
To V2Performing a time derivative may yield:
Figure BDA0002917253980000092
by substituting formula (9) for formula (14)
Figure BDA0002917253980000093
From the above formula, the designed manipulator finite time reversal controller is stable. However, the above-mentioned controllers require detailed system information and an upper uncertainty bound τ in practical systemsMIt is difficult to determine, and these problems indicate that the above controller is difficult to implement in practical applications. Therefore, to overcome these problems, an improved finite time inversion controller has been proposed.
(III) design mechanical arm improved finite time inversion controller
The designed manipulator-improved finite-time inversion controller adopts a fuzzy neural network to approximate the inversion controller (10), thereby overcoming the defect that the inversion controller needs detailed system information. Fuzzy neural network adoptedThe network tracking error detection method is composed of an input layer, a fuzzy inference layer and an output layer, wherein the network input is tracking error e1Output as control force tauFNN. The signal propagation and the function of the layers in the fuzzy neural network are represented as follows:
a first layer: input layer
Each node of the layer is directly connected with each component of the input quantity, and the input quantity is transmitted to the next layer;
a second layer: blurring layer
A gaussian-type function is used as the membership function,
Figure BDA0002917253980000094
representing tracking offset vector e1The elements (A) and (B) in (B),
Figure BDA0002917253980000095
and
Figure BDA0002917253980000096
the centre vector and the base width of the membership function of the ith input variable, the jth fuzzy set, respectively, i.e.
Figure BDA0002917253980000097
Easy to calculate, using NpiRepresenting individual numbers of membership functions, and we define the sets of adaptive parameter vectors b and c representing all the basis width and center vectors of Gaussian membership functions, respectively, i.e.
Figure BDA0002917253980000101
Figure BDA0002917253980000102
Wherein
Figure BDA0002917253980000103
Representing the total number of membership functions;
and a third layer: rule layer
The layer adopts a fuzzy inference mechanism, and the output of each node is the product of all input signals of the node, namely
Figure BDA0002917253980000104
In the formula Ik(k＝1,...,Ny) The k-th output of the rule layer is represented,
Figure BDA0002917253980000105
representing the connection weight matrix between the fuzzification layer and the regular layer, here taken as the unit vector, NyIs the total number of rules;
a fourth layer: output layer
The nodes of the layer represent output variables, each node yo(o＝1,...,No) The output of (2) being the sum of all input signals of the node, i.e.
Figure BDA0002917253980000106
Further, the input-output relationship of the fuzzy neural network is defined as follows:
y＝[y1 y2…yNo]＝τFNN＝Wl
wherein the content of the first and second substances,
Figure BDA0002917253980000107
Figure BDA0002917253980000108
particularly, the meta-cognition fuzzy neural model provided by the invention considers two self-regulation strategies of data learning and data deletion, and is beneficial to efficiently executing a real-time control task.
First is a data learning strategy. The data learning process of the meta-cognitive fuzzy neural network involves online evolution and parameter updating of rules closest to current input data.
The fuzzy rule is based on the following conditionsDetermined stepwise, i.e.
Figure BDA0002917253980000109
And psi<EsWhere ψ is a spherical potential energy, representing the novelty of the input data, given by the following equation:
Figure BDA0002917253980000111
wherein E issAnd EaIs the novelty and the addition of a threshold value,
Figure BDA0002917253980000112
when a new fuzzy rule needs to be added, its parameters are initialized,
Figure BDA0002917253980000113
where κ is an overlap parameter of a predefined fuzzy rule
Figure BDA0002917253980000114
When the time is needed, the rule parameters are adjusted,
in the learning process, the contribution degree of a certain rule to the output may be reduced. In this case, insignificant rules should be removed from the rule base to avoid overcomputing. The degree of contribution of the qth rule is given by the following equation:
Figure BDA0002917253980000115
wherein the content of the first and second substances,
Figure BDA0002917253980000116
n represents the dimension of the input.
If the contribution degree of the q rule to the input is lower than the threshold value EpThen the rule is deleted.
Data deletion policy
Specifically, when the current tracking error is close to the error in the last iterative calculation process of the neural network, network parameters do not need to be updated, excessive learning is avoided, and the calculation burden is reduced.
According to the universal approximation theory, there is an optimal control force
Figure BDA0002917253980000117
Satisfy the requirement of
Figure BDA0002917253980000118
Where ε is the minimum reconstruction error vector, W*,b*And c*Optimal parameters of W, b and c respectively;
the output control force of the fuzzy neural network is assumed to be of the following form:
Figure BDA0002917253980000119
wherein the content of the first and second substances,
Figure BDA00029172539800001110
and
Figure BDA00029172539800001111
are respectively W*,b*And c*Is determined by the estimated value of (c),
defining an approximation error:
Figure BDA00029172539800001112
using Taylor series expansion, one can obtain
Figure BDA0002917253980000121
Wherein the content of the first and second substances,
Figure BDA0002917253980000122
b*and c*Is the optimum value for b and c,
Figure BDA0002917253980000123
and
Figure BDA0002917253980000124
is b*And c*Estimated value of, OnvIs a high-order term of the magnetic field,
Figure BDA0002917253980000125
Figure BDA0002917253980000126
then (25) is substituted into (24) to obtain
Figure BDA0002917253980000127
Wherein the content of the first and second substances,
Figure BDA0002917253980000128
further, the formula (9) can be rewritten as
Figure BDA0002917253980000129
Wherein the content of the first and second substances,
Figure BDA00029172539800001210
d＝-E-τdand d is the total uncertainty,
the self-adaptive law of the weight, the base width and the central vector of the designed fuzzy neural network can be designed as follows:
Figure BDA00029172539800001211
Figure BDA00029172539800001212
Figure BDA00029172539800001213
wherein S isiIs an element of S, betaiIs an element in beta, sigmaω,σb,σcIs a normal number, and is,
Figure BDA00029172539800001214
is that
Figure BDA00029172539800001215
Is determined by the estimated value of (c),
Figure BDA00029172539800001216
is omegaiIs determined to be the optimum value of (c),
defining the lyapunov function as
Figure BDA0002917253980000131
Derivative and substitute (27) into
Figure BDA0002917253980000132
Bringing (28) - (30) into (32) to obtain
Figure BDA0002917253980000133
If τ is satisfiedMAnd the designed manipulator improved finite time inversion control method is stable.
The above embodiments are merely illustrative of the technical ideas and features of the present invention, and the purpose thereof is to enable those skilled in the art to understand the contents of the present invention and implement the present invention, and not to limit the protection scope of the present invention. All equivalent changes and modifications made according to the spirit of the present invention should be covered within the protection scope of the present invention.

Claims (3)

1. A manipulator improved finite time reversal control method is characterized by comprising the following steps:
step 1, building a mechanical arm power model
Consider a rigid manipulator with n joints, whose kinetic equation is:
Figure FDA0002917253970000011
wherein the ratio of q,
Figure FDA0002917253970000012
the position, speed and acceleration vector of each joint of the manipulator, M (q) epsilon Rn×nIn order to include an inertial matrix of the additional mass,
Figure FDA0002917253970000013
g (q) e R as a coupling matrix of centrifugal force and Coriolis forcen×1In the form of a gravity matrix vector,
Figure FDA0002917253970000014
for friction torque, τd∈Rn×1For unknown applied interference, tau epsilon Rn×1N is the degree of freedom of the joint;
step 2, designing a manipulator finite time reversal controller
2.1 define the tracking error as
e1＝q-qd (2)
Wherein q isdIs the desired position of the joint;
derived from (2)
Figure FDA0002917253970000015
Selecting virtual control quantities
Figure FDA0002917253970000016
Wherein c is1Is a non-zero normal number;
taking the Lyapunov function as
Figure FDA0002917253970000017
Definition of
Figure FDA0002917253970000018
Derived from formula (5)
Figure FDA0002917253970000019
If e2When the value is equal to 0, then
Figure FDA00029172539700000110
The stability requirement is met, and the design is continued;
2.1 e2can be expressed as
Figure FDA00029172539700000111
Then designing a nonsingular terminal sliding mode surface as
Figure FDA0002917253970000021
Wherein λ is1>0 is a constant, p1,p2Is odd number, 1<p2/p1<2，
Further, we can get
Figure FDA0002917253970000022
The finite time inversion controller can be designed as
τBSC＝u1+u2 (10)
Figure FDA0002917253970000023
Figure FDA0002917253970000024
Wherein, tauM≥||τd||；
S3, aiming at the formula (10), approximating the unknown state in the inversion controller by using the fuzzy neural network, and defining the input-output relationship of the fuzzy neural network as follows:
Figure FDA0002917253970000025
wherein the content of the first and second substances,
Figure FDA0002917253970000026
Figure FDA0002917253970000027
lk(k＝1,...,Ny) The k-th output of the rule layer is represented,
Figure FDA0002917253970000028
representing a matrix of connection weights between the obfuscated layer and the regular layer, taken here as unit directionAmount, NyIs the total number of rule layers.
2. The manipulator-improved finite time reversal control method according to claim 1, wherein the fuzzy neural network is composed of an input layer, a fuzzy inference layer and an output layer, wherein the network input is a tracking deviation e1Output as control force tauFNNThe signal propagation and the function of each layer in the fuzzy neural network are represented as follows:
a first layer: input layer
Each node of the layer is directly connected with each component of the input quantity, and the input quantity is transmitted to the next layer;
a second layer: blurring layer
A gaussian-type function is used as the membership function,
Figure FDA0002917253970000031
representing tracking offset vector e1The elements (A) and (B) in (B),
Figure FDA0002917253970000032
and
Figure FDA0002917253970000033
the centre vector and the base width of the membership function of the ith input variable, the jth fuzzy set, respectively, i.e.
Figure FDA0002917253970000034
Easy to calculate, using NpiRepresenting individual numbers of membership functions, and we define the sets of adaptive parameter vectors b and c representing all the basis width and center vectors of Gaussian membership functions, respectively, i.e.
Figure FDA0002917253970000035
Figure FDA0002917253970000036
Wherein
Figure FDA0002917253970000037
Representing the total number of membership functions;
and a third layer: rule layer
The layer adopts a fuzzy inference mechanism, and the output of each node is the product of all input signals of the node, namely
Figure FDA0002917253970000038
In the formula Ik(k＝1,...,Ny) The k-th output of the rule layer is represented,
Figure FDA0002917253970000039
representing the connection weight matrix between the fuzzification layer and the regular layer, here taken as the unit vector, NyIs the total number of rules;
a fourth layer: output layer
The nodes of the layer represent output variables, each node yo(o＝1,...,No) The output of (2) being the sum of all input signals of the node, i.e.
Figure FDA00029172539700000310
Further, the input-output relationship of the fuzzy neural network is defined as follows:
Figure FDA00029172539700000311
wherein the content of the first and second substances,
Figure FDA0002917253970000041
Figure FDA0002917253970000042
3. the manipulator-improved finite time reversal control method according to claim 2, wherein the fuzzy neural network comprises two self-regulation strategies of data learning and data deletion, specifically:
data learning strategy
The fuzzy rule is determined step by step based on the condition that
Figure FDA0002917253970000043
And psi<EsWhere ψ is a spherical potential energy, representing the novelty of the input data, given by the following equation:
Figure FDA0002917253970000044
wherein E issAnd EaIs the novelty and the addition of a threshold value,
Figure FDA0002917253970000045
when a new fuzzy rule needs to be added, its parameters are initialized,
Figure FDA0002917253970000046
where k is the overlap parameter of a pre-given fuzzy rule,
data deletion policy
According to the universal approximation theory, there is an optimal control force
Figure FDA0002917253970000047
Satisfy the requirement of
Figure FDA0002917253970000048
Where ε is the minimum reconstruction error vector, W*,b*And c*Optimal parameters of W, b and c respectively;
the output control force of the fuzzy neural network is assumed to be of the following form:
Figure FDA0002917253970000049
wherein the content of the first and second substances,
Figure FDA00029172539700000410
and
Figure FDA00029172539700000411
are respectively W*,b*And c*Is determined by the estimated value of (c),
defining an approximation error:
Figure FDA00029172539700000412
using Taylor series expansion, one can obtain
Figure FDA0002917253970000051
Wherein the content of the first and second substances,
Figure FDA0002917253970000052
b*and c*Is the optimum value for b and c,
Figure FDA0002917253970000053
and
Figure FDA0002917253970000054
is b*And c*Estimated value of, OnvIs a high-order term of the magnetic field,
Figure FDA0002917253970000055
Figure FDA0002917253970000056
then (25) is substituted into (24) to obtain
Figure FDA0002917253970000057
Wherein the content of the first and second substances,
Figure FDA0002917253970000058
further, the formula (9) can be rewritten as
Figure FDA0002917253970000059
Wherein the content of the first and second substances,
Figure FDA00029172539700000510
d＝-E-τdand d is the total uncertainty,
the self-adaptive law of the weight, the base width and the central vector of the designed fuzzy neural network can be designed as follows:
Figure FDA00029172539700000511
Figure FDA00029172539700000512
Figure FDA00029172539700000513
wherein S isiIs an element of S, betaiIs an element in beta, sigmaω,σb,σcIs a normal number, and is,
Figure FDA00029172539700000514
is that
Figure FDA00029172539700000515
Is determined by the estimated value of (c),
Figure FDA0002917253970000061
is omegaiIs determined to be the optimum value of (c),
defining the lyapunov function as
Figure FDA0002917253970000062
Derivative and substitute (27) into
Figure FDA0002917253970000063
Bringing (28) - (30) into (32) to obtain
Figure FDA0002917253970000064
If τ is satisfiedMAnd the designed manipulator improved finite time inversion control method is stable.
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