CN109514558A - Flexible mechanical arm time-scale separation robust control method based on singular perturbation - Google Patents
Flexible mechanical arm time-scale separation robust control method based on singular perturbation Download PDFInfo
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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
- B25J9/1605—Simulation of manipulator lay-out, design, modelling of manipulator
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Abstract
The invention proposes a kind of flexible mechanical arm time-scale separation robust control method based on singular perturbation.This method comprises: building flexible mechanical arm kinetic model；Hard and soft modal separation processing is carried out to flexible mechanical arm kinetic model；For the slow subsystem of characterization system rigidity mode, robust control method is designed；For the fast subsystem of characterization system resilience mode, design point feedback；Obtain the first control input；Obtain the second control input；First control input and the second control input are combined, combined data is obtained；Combined data is input to flexible mechanical arm kinetic model, obtains output data；Output data is obtained comparing difference compared with the expected data of the joint angle of flexible mechanical arm and elastic vibration data respectively；When comparing difference in relatively threshold range, output is for indicating the effective mark of control.Joint of mechanical arm angle may be implemented to the tracking of expectation instruction and the inhibition of Elastic mode in inventive embodiments, reduces vibration.
Description
Technical field
The present invention relates to the technical fields of aviation robot control, more particularly, to a kind of flexible machine based on singular perturbation
Tool arm time-scale separation robust control method.
Background technique
Flexible mechanical arm kinetic model characterizes the characteristics of hard and soft modal coupling, and there are stronger not true for model parameter
It is qualitative, it is unfavorable for control algorithm design.For model elastic part, have scholar by its at model parameter indeterminate lump
Reason realizes compensation control by the way of disturbance-observer, however this roadmap lacks the in-depth analysis to elastic part；Needle
To parameter uncertainty, ART network algorithm can realize approximation timates by on-line study, however preferable Approximation effect
More online updating parameters are needed, so that algorithm computing speed is partially slow, conventional proportional differential control method is simple, however externally
The resistivity of boundary's interference is poor.
《Composite Learning Control of Flexible-Link Manipulator Using NN and
DOB " (BIN XU, " IEEE Transactions on Systems, Man, and Cybernetics:Systems, 2017,
DOI:10.1109/TSMC.2017.2700433 ") literary grace carries out flexible mechanical arm kinetic model with singular perturbation theory
Time-scale separation processing, makes model dynamics be divided into rigid element and elastic part, for rigid element, using neural network
The method combined with disturbance observation handles indeterminate, and it is preferable that learning information improves control effect, however due to
Algorithm is complicated, and Neural Network Online learning parameter causes computing speed partially slow more, is unfavorable for engineer application.
Summary of the invention
To solve deficiency of the existing control method in flexible mechanical arm control process, the invention proposes one kind based on surprise
The flexible mechanical arm time-scale separation robust control method of different perturbation.
(a) consider n freedom degree flexible mechanical arm kinetic model:
Wherein M is the symmetrical inertial matrix of positive definite,It is related with coriolis force and centripetal force
, D1、D2For damping matrix, K2For stiffness matrix, u is joint input torque；It is by joint of mechanical arm angle and flexibility
The generalized vector of mode composition, wherein θiFor i-th of joint angle variable, δi,jIt is j-th of mode variable of i-th of connecting rod；
(b) it definesModel (1) is further written as:
If k is K2Least member in matrix defines h=1/k, introduces new variables h ε=δ, N=hK2, formula (2) further writes
At following form:
Definition control input:
U=us+uf (5)
Wherein, usIt controls and inputs for slow subsystem；ufIt controls and inputs for fast subsystem；Enable h=0, formula (3), (4) into
One step is written as formula (6), (7)
Wherein, footmark " s " indicates slow component；Convolution (6) formula (7) can obtain slow subsystem such as (8) formula:
Further slow subsystem can be written as follow equivalent form
Wherein, μ1=θs,It is logical
Model parameter nominal information is crossed, the nominal information A of A, B are further calculated0、B0, i.e. A=A0+ Δ A, B=B0+ΔB；
Slow subsystem (9) may further be write:
Enable D=Δ A+ Δ Bus, then formula (10) is further write
Define new state variable: ζ1=ε-εs,Formula (4) becomes:
In conjunction with formula (7), (12), fast subsystem model is obtained:
That is:
Wherein,
(c) error signal e is defined1=μ1-yr, wherein yrFor desired joint angles, error dynamics equation isDesign virtual controlling amount
Wherein, k1∈Rn*nFor the symmetrical nonsingular matrix of positive definite；
Define error signal e2=μ2-u2d, then e2Error dynamics equation are as follows:
Design controller
Wherein, k2∈Rn*nFor the symmetrical nonsingular matrix of positive definite；ω0Artificially to design normal number, e2=[e21,e22,…e2n]T, For the estimated value and satisfaction of D maximum value
It is provided by formula (17)
Wherein, ρ ∈ Rn*n, γ ∈ Rn*nFor the symmetrical nonsingular matrix of positive definite；
(d) dynamic subsystem adoption status feedback controller in:Control gain kδWithFor using pole
The R that point configuration obtainsn*mnRank matrix, controller always input:
U=us+uf (18)
(e) u is inputted according to obtained control, returned in flexible mechanical arm model (1), carrying out control to joint angle makes it
Expectation instruction in tracking, while realizing that mechanical arm Elastic mode inhibits, reduce vibration.
The beneficial effect of foregoing invention embodiment can be with are as follows:
By introducing singular perturbation theory to flexible mechanical arm kinetic model, model is made to be divided into characterization system rigidity mode
Slow subsystem and characterize system resilience mode fast subsystem, asked for the parameter uncertainty in slow subsystem
Topic, is handled by way of estimating its upper bound, and Estimation of Upper-Bound value is introduced controller and realizes compensation control, algorithm letter
Single, pace of learning is fast, is conducive to Project Realization.
Detailed description of the invention
In order to illustrate more clearly of the technical solution implemented of the present invention, below by it is of the invention be said in need using
Attached drawing make simple resolve.It is clear that drawings described below is only some embodiments of the present invention, for ability
For the technical staff in domain, without creative efforts, it is also possible to obtain other drawings based on these drawings.
Fig. 1 is the stream of the flexible mechanical arm time-scale separation robust control method based on singular perturbation of one embodiment of the invention
Journey schematic diagram.
Specific embodiment
In order to make the object, technical scheme and advantages of the embodiment of the invention clearer, below in conjunction with the embodiment of the present invention
In attached drawing, technical scheme in the embodiment of the invention is clearly and completely described.Obviously, described embodiment is
A part of the embodiments of the present invention, rather than whole embodiments.Based on the embodiments of the present invention, the ordinary skill people of this field
Without making creative work, every other embodiment obtained shall fall within the protection scope of the present invention member.
The feature of the various aspects of the embodiment of the present invention is described more fully below.In the following detailed description, it proposes
Many concrete details, so as to complete understanding of the present invention.But for those of ordinary skill in the art, very bright
Aobvious, the present invention can also be implemented in the case where not needing these details.Below to the description of embodiment
Just for the sake of being better understood by showing example of the invention to the present invention.The present invention is not limited to presented below any
Specific setting and method, but cover all product structures covered without departing from the spirit of the invention, method
Any improvement, replacement etc..In each attached drawing and following description, well known structure and technology is not shown, to avoid right
The present invention causes unnecessary fuzzy.
It should be noted that in the absence of conflict, the feature in the embodiment of the present invention and embodiment can be tied mutually
It closes, each embodiment mutually can be referred to and be quoted.The present invention will be described in detail below with reference to the accompanying drawings and embodiments.
Fig. 1 is the stream of the flexible mechanical arm time-scale separation robust control method based on singular perturbation of one embodiment of the invention
Journey schematic diagram.
As shown in Figure 1, method includes the following steps: S110, builds flexible mechanical arm kinetic model；S120 is used
Singular perturbation theory carries out hard and soft modal separation processing to model；S130, for the fast subsystem of characterization system resilience mode
Design point feedback control algorithm；S140 is calculated for the fast subsystem design point feedback control of characterization system resilience mode
Method；S150, two kinds of control inputs pass to kinetic model after combining, realize joint angle tracking expectation instruction target and elasticity
Vibration suppression target.
The embodiment of the present invention can solve the problem of hard and soft modal coupling kinetic model modal separation control.Firstly, this
Inventive embodiments can introduce the time-scale separation that singular perturbation theory realizes controlled device, be divided into Rigid-flexible Coupling Dynamics model
Rigid expression formula and elastic expression formula；Secondly, designing robust control side for the slow subsystem of characterization system rigidity mode
Method, the Estimation of Upper-Bound of implementation model indeterminate, and complete compensation control；For the fast subsystem of characterization system resilience mode
Design point feedback；Most latter two control method is combined into one collective effect in flexible mechanical arm kinetic model.
Invention aims at tracking of the joint of mechanical arm angle to expectation instruction, while realizing the inhibition of Elastic mode, reduces vibration.
In some embodiments, the flexible mechanical arm time-scale separation robust control method based on singular perturbation may include:
Based on Lagrange's equation, flexible mechanical arm kinetic model is built；
According to singular perturbation method, hard and soft modal separation processing is carried out to flexible mechanical arm kinetic model；
For the slow subsystem of characterization system rigidity mode, robust control method is designed；
For the fast subsystem of characterization system resilience mode, design point feedback；
According to robust control method, the first control input is obtained；
According to state feedback control method, the second control input is obtained；
First control input and the second control input are combined, combined data is obtained；
Combined data is input to flexible mechanical arm kinetic model, obtains output data；
Output data is compared compared with the expected data of the joint angle of flexible mechanical arm and elastic vibration data respectively
Compared with difference；
When comparing difference in relatively threshold range, output is for indicating the effective mark of control.
In some embodiments, the implementation of the flexible mechanical arm time-scale separation robust control method based on singular perturbation
May include:
(a) consider 2DOF flexible mechanical arm kinetic model:
Wherein M is the symmetrical inertial matrix of positive definite,It is related with coriolis force and centripetal force
, D1、D2For damping matrix, K2For stiffness matrix, u is joint input torque.The broad sense arrow being made of joint of mechanical arm angle and flexible mode
It measures, wherein θiFor i-th of joint angle variable, δi,jIt is j-th of mode variable of i-th of connecting rod.
Length of connecting rod l1=l2=0.5m, connecting rod quality m1=m2=0.1kg, connecting rod bending stiffness EI1=EI2=10N
m2, end equivalent mass mp=0.1kg.It is derived with this
I takes 1,2, rank number of mode m=2.
(b) it definesModel (1) is further written as:
If k is K2Least member in matrix defines h=1k, introduces new variables h ε=δ, N=hK2, formula (2) is into one
Step is written as follow form:
Definition control input:
U=us+uf (5)
Wherein, usIt controls and inputs for slow subsystem；ufIt controls and inputs for fast subsystem；Enable h=0, formula (3), (4) into
One step is written as formula (6), (7)
Wherein, footmark " s " indicates slow component；Convolution (6) formula (7) can obtain slow subsystem such as (8) formula:
Further slow subsystem can be written as follow equivalent form:
Wherein, μ1=θs,It is logical
Model parameter nominal information is crossed, the nominal information of A, B are further calculated
Slow subsystem (9) may further be write:
Enable D=Δ A+ Δ Bus, then formula (10) is further write
Define new state variable: ζ1=ε-εs,Formula (4) becomes:
In conjunction with formula (7), (12), fast subsystem model is obtained:
That is:
Wherein,
(c) error signal e is defined1=μ1-yr, wherein yrFor desired joint angles, error dynamics equation isDesign virtual controlling amount:
Wherein, k1∈R2*2For the symmetrical nonsingular matrix of positive definite, when Manipulator Dynamic inputs u in controlexUnder effect
It is expected that final k can be obtained when joint angles in tracking1Matrix.
Define error signal e2=μ2-u2d, then e2Error dynamics equation are as follows:
Design controller:
Wherein, k2∈R2*2For the symmetrical nonsingular matrix of positive definite, with k1Matrix value mode is consistent, e2=[e21,e22]T, ω0
For normal number, value and k1Matrix value mode is consistent,
For the estimated value and satisfaction of D maximum valueBy formula
(17) it provides:
Wherein, ρ ∈ R2*2, γ ∈ R2*2For the symmetrical nonsingular matrix of positive definite, with k1Matrix value mode is consistent.
(d) fast subsystem adoption status feedback controller:
Master control input:
U is inputted according to obtained control, returns in flexible mechanical arm model (1), pass is controlled such that joint angle
Expectation instruction in angle tracking is saved, while realizing that Elastic mode inhibits, reduces elastic vibration.
It should be noted that above-mentioned flow operations can carry out different degrees of combined application, for simplicity, repeating no more
Various combined implementations.Those skilled in the art can by actual conditions by above-mentioned method the step of sequence (or
The position of the component of person's product) it is adjusted flexibly, or the operation such as combination.
It should be noted that the implementation of functional unit shown in above-described embodiment can for hardware, software or
Combination.When realizing in hardware, electronic circuit, specific integrated circuit (ASIC), plug-in unit, function card can be made
Deng.When realizing with software mode, the program or code segment of required task can be used to carry out with thing.Program or code
Section can go out be stored in machine perhaps in readable medium or it can be situated between by data-signal entrained in carrier wave in transmission
Matter or communication links are sent.
Finally it should be noted that: the above embodiments are merely illustrative of the technical solutions of the present invention, but protection of the invention
Range is not limited thereto, and anyone skilled in the art in the technical scope disclosed by the present invention, can be thought easily
It is modified or replaceed to various equivalent, these, which are modified or replaceed, should all be included within the scope of the present invention.
Claims (6)
1. a kind of flexible mechanical arm time-scale separation robust control method based on singular perturbation characterized by comprising
Based on Lagrange's equation, flexible mechanical arm kinetic model is built；
According to singular perturbation method, hard and soft modal separation processing is carried out to flexible mechanical arm kinetic model；
For the slow subsystem of characterization system rigidity mode, robust control method is designed；
For the fast subsystem of characterization system resilience mode, design point feedback；
According to robust control method, the first control input is obtained；
According to state feedback control method, the second control input is obtained；
First control input and the second control input are combined, combined data is obtained；
Combined data is input to flexible mechanical arm kinetic model, obtains output data；
Output data is obtained poor compared with the expected data of the joint angle of flexible mechanical arm and elastic vibration data respectively
Value；
When comparing difference in relatively threshold range, output is for indicating the effective mark of control.
2. the method according to claim 1, wherein flexible mechanical arm kinetic model is expression formula (1):
Wherein, M is the symmetrical inertial matrix of positive definite,It is item related with coriolis force and centripetal force,
D1、D2For damping matrix, K2For stiffness matrix, u is joint input torque；It is by joint of mechanical arm angle and flexible die
The generalized vector of state composition, θiFor i-th of joint angle variable, δi,jIt is j-th of mode variable of i-th of connecting rod.
3. the method according to claim 1, wherein using singular perturbation theory, to flexible mechanical arm dynamics
Model carries out hard and soft modal separation processing, comprising:
DefinitionAnd expression formula (2) are converted by expression formula (1):
If k is K2Least member in matrix defines h=1/k, introduces new variables h ε=δ, N=hK2, convert expression formula (2) to
Expression formula (3) and expression formula (4):
Definition control input:
U=us+uf (5)
Wherein, usIt controls and inputs for slow subsystem；ufIt controls and inputs for fast subsystem；H=0 is enabled, expression formula (3), (4) turn
Turn to expression formula (6) and expression formula (7):
Wherein, footmark " s " indicates slow component；Slow subsystem such as expression formula (8) can be obtained in conjunction with expression formula (6) expression formula (7):
Slow subsystem is converted into following expression
Wherein, μ1=θs,Pass through mould
Shape parameter nominal information calculates the nominal information A of A, B0、B0, i.e. A=A0+ Δ A, B=B0+ΔB；
Slow subsystem expression formula (9) is converted into expression formula (10):
Enable D=Δ A+ Δ Bus, then expression formula (10) is converted into expression formula (11):
Definition status variable: ζ1=ε-εs,H=1/k, t are the time, and expression formula (4) is converted into expression
Formula (12):
In conjunction with expression formula (7), (12), fast subsystem model is obtained:
Expression formula (13) are converted are as follows:
Wherein,
4. according to the method described in claim 3, it is characterized in that, design robust control method, comprising:
Define error signal e1=μ1-yr, wherein yrFor desired joint angles, error dynamics equation isDesign virtual controlling amount
Wherein, k1∈Rn*nFor the symmetrical nonsingular matrix of positive definite；
Define error signal e2=μ2-u2d, then e2Error dynamics equation are as follows:
Design controller
Wherein, k2∈Rn*nFor the symmetrical nonsingular matrix of positive definite；ω0Artificially to design normal number,
For the estimated value and satisfaction of D maximum valueIt is obtained by expression formula (17):
Wherein, ρ ∈ Rn*n, γ ∈ Rn*nFor the symmetrical nonsingular matrix of positive definite.
5. the method according to claim 1, wherein design point feedback, comprising:
Fast subsystem adoption status feedback controller:Control gain kδWithTo be obtained using POLE PLACEMENT USING
Rn*mnRank matrix,
Wherein, ufIt controls and inputs for fast subsystem.
6. the method according to claim 1, wherein obtaining output data, comprising:
Controller always inputs:
U=us+uf (18)
usIt controls and inputs for slow subsystem；ufIt controls and inputs for fast subsystem.
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| CN112506057A (en) * | 2020-12-02 | 2021-03-16 | 郑州轻工业大学 | Online multi-time-scale quick self-adaptive control method for uncertain singular perturbation system |
| CN112506057B (en) * | 2020-12-02 | 2021-09-17 | 郑州轻工业大学 | Online multi-time-scale quick self-adaptive control method for uncertain singular perturbation system |
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