CN109465825A - The adaptive dynamic surface control method of the RBF neural of mechanical arm flexible joint - Google Patents
The adaptive dynamic surface control method of the RBF neural of mechanical arm flexible joint 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
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Abstract
The present invention relates to a kind of adaptive dynamic surface control methods of the RBF neural based on mechanical arm flexible joint, belong to artificial intelligence and field of intelligent control technology.The present invention is modeled for flexible joint existing for mechanical arm, in conjunction with RBF neural and the adaptive design of control method controller of dynamic surface technology.Using RBF neural come compensation system parameter uncertainty, it is adjusted using weight of the adaptive law to neural network, RBF neural is improved to the approximation capability of nonlinear function, the needs to mechanical arm precise kinetic model are eliminated, research is suitable for having the Position Tracking Control algorithm of the light-duty mechanical arm of flexible joint.Design controller is verified finally by simulation example, illustrates that the present invention can guarantee that joint can effectively track Setting signal under conditions of robotic arm limited torque output, tracking error is without constraining in a certain range, and all signals all half global boundeds.
Description
Technical field
The invention belongs to artificial intelligence and control fields, and in particular to a kind of RBF neural of mechanical arm flexible joint
Adaptive dynamic surface control method.
Background technique
Since the 1960s, robot is industrially widely applied, such as machining, arc-welding, spot welding,
Spraying, assembly, detection, space flight, space exploration etc..Industrial robot is on automatic industrial manufacturing line for a long time, in occupation of
Important position.But as robot application range expands, it has been found that this fixation of industrial robot repeatedly operates not
More more neatly demands of task are able to satisfy, if task is with position change, the different operation of mission requirements.Cause
This, engineering design goes out to be able to satisfy the light-duty mechanical arm of this kind of particular/special requirement.Light-duty mechanical arm is applied to work unknown in advance mostly
Make under environment, exact position of the surrounding objects relative to itself can not be obtained, need higher positioning accuracy, is based on human-computer exchange
The considerations of safety, therefore the flexibility of equipment is considered in the design process.Light-duty mechanical arm is generally led to by multiple connecting rods
It crosses rotation or linear joint is formed by connecting, the flexibility of robot is mainly derived from flexibility of linking rod and flexibility of joint, the flexibility in joint
Referring to the torsional deformation of transmission mechanism and joint shaft, general flexible joint can use flexible article as attachment,
Such as spring etc..Although can reduce the flexibility in joint by Machine Design, from the point of view of existing result, effect is not very
It is good.
In the industrial production, only consider the movement of rigid bar without in view of the elasticity in movement, being less sufficiently
, or even in some designs, the flexibility of mechanical arm is not accounted for, it is unstable so as to cause closed-loop system.It is asked for this
Topic, in recent years, the motor control problems of the robot with flexible joint have caused the research of many experts and scholars,
Certain development is obtained.Firstly, Spong proposed the simplified model of flexible joint in 1987, that is, it is regarded as linear bullet
Spring, the simplification have pushed the control of flexible joint robot to study significantly.Referring to the control thought of rigid machine people, by PD or
PID and flexible joint compensating controller are applied on flexible robot's joint control.But this method is demonstrate,proved in stability
Bright aspect is more complicated, although and controller it is simpler, control precision it is not high；With going deep into for research, feedback linearization
Change and Robust Adaptive Control is noticed to control flexible joint robot by numerous scholars.Feedback linearization method is to control
Though flexible joint robot to be a kind of feasible method, the tracking performance heavy dependence system model of this method it is accurate
Degree, however, more difficult for the Accurate Model of flexible joint robot.Therefore, feedback linearization and adaptive technique, Fuzzy Control
System, the methods of neural network combine the exact requirements eliminated to model, can preferably to the robot with flexible joint into
Row control, and achieve the effect that ideal, Sira-Ramirez and Spong devise sliding mode controller, and Zeman and Ge etc. are mentioned
Go out using neural network come the design method of compensation system parameter uncertainty.For flexible joint robot, in addition to
Except uncertainty present in system and output feedback ontrol research, due to the limitation of joint motor itself output power, make
Robot articular driver output torque deposits saturation limitation, although many documents are studied, there is presently no compared with
Good solution.
Summary of the invention
Aiming at the problems existing in the prior art, the present invention by flexibility of joint, output torque is limited and external disturbance etc.
Question synthesis considers, proposes the adaptive dynamic surface control method based on RBF neural of mechanical arm flexible joint.
In order to realize above-mentioned task, the invention adopts the following technical scheme:
A kind of adaptive dynamic surface control method of RBF neural of mechanical arm flexible joint, comprising the following steps:
Step 1, mechanical arm flexible joint is modeled；
Step 2, according to the physical characteristic of tool arm flexible joint, the equation model that modeling obtains is converted into state equation；
Step 3, it defines first tracking error and designs first Virtual Controller, by the letter of first Virtual Controller
Number input first low-pass first order filter, obtain new state variable x2dNext step meter is carried out instead of first Virtual Controller
It calculates, reduces calculation amount；
Step 4, second tracking error is defined, using the first RBF neural to containing the non-linear of uncertain parameter
Function is approached, and the adaptive law of the first RBF neural weight is designed；Design second Virtual Controller, second void
The signal of quasi- controller obtains new variable x by second low-pass first order filter3d；
Step 5, third tracking error is defined, third Virtual Controller is designed, by the letter of third Virtual Controller
Number it is input to third low-pass first order filter, obtains new state variable x4d；
Step 6, define the 4th tracking error, according to the state equation, using the second RBF neural to its into
Row Compensate approximate designs the adaptive law of the second RBF neural weight；
Step 7, practical controller is designed.
Further, the equation model of the mechanical arm flexible joint are as follows:
In above formula, q, θ respectively indicate the angular position of connecting rod, rotor,It is expressed as the velocity of rotation of connecting rod corner, I, J divide
Not Biao Shi connecting rod, rotor rotary inertia, CHIndicate the gravitational vectors of mechanical arm, BmIndicate that viscous rub of motor shaft examines coefficient, K
Represent joint of mechanical arm stiffness coefficient, M, g, l are respectively connecting rod quality, acceleration of gravity and joint to connecting rod centroid distance, τm
It is motor torque input；The respectively single order of q, second dervative,For the second dervative of θ.
Further, state equation described in step 2 are as follows:
In above formula, x=[x1,x2,x3,x4]TFor state variable,Respectively x1,x2,x3,x4First derivative, Δ (t) is extraneous BOUNDED DISTURBANCES and meets Δ (t) < ρ, and ρ is a constant；g(x1,
x2), f (x1,x3) be concrete form unknown but bounded nonlinear function；Assuming that mechanical arm flexible joint needs to track input letter
Number be x1d, first derivative, second dervativeExist, and meetsξ is a constant.
Further, the step 3 specifically includes:
Step 3.1, first tracking error: S is defined1=x1-x1d, derivation obtainsTake Virtual Controller are as follows:Wherein c1For design parameter, and meet c1> 0；
Step 3.2, willIt is input to first low-pass first order filter, obtains new state variable x2d；
Step 3.3, the filtering error e of first Virtual Controller is defined2, expression formula are as follows:It can then obtain
Its derivative are as follows:
Further, the step 4 specifically includes:
Step 4.1, second tracking error: S is defined2=x2-x2d, derivation obtains:
Step 4.2, the first RBF neural is established；
Step 4.3, unknown function is approached using the first RBF neural described in step 4.2 WhereinIndicate optimal value when the first RBF neural weight approaches unknown function, h1
(x1,x2) be neural network neuron activation functions, ε1 *Indicate the error approached；
Step 4.4, second Virtual Controller is designedWhereinIt is
Estimated value, k2, λ2、c2It is design parameter, and meets k2> 0, λ2> 0, c2> 0；
Step 4.5, design adaptive lawWherein Γ1It is symmetrical for a positive definite
Matrix, σ1For design parameter, and meet σ1> 0, K2For to tracking error S2The constraint of size；
Step 4.6, willIt is input to second low-pass first order filter, obtains new variable x3d；
Step 4.7, second Virtual Controller filtering error e is defined3, expression formula are as follows:It can then be obtained
Derivative are as follows:
Further, the step 5 specifically includes:
Step 5.1, third tracking error: S is defined3=x3-x3d, derivation obtainsDesign Virtual Controllerλ3、c3It is design parameter, and meets λ3> 0, c3> 0；
Step 5.2, willThird low-pass first order filter is inputted, new state variable x is obtained4d；
Step 5.3, third Virtual Controller filtering error e is defined4, expression formula are as follows:It can then be obtained
Derivative are as follows:
Further, the step 6 specifically includes:
Step 6.1, the 4th tracking error: S is defined4=x4-x4d, derivation obtains
Step 6.2, the second RBF neural of construction is to nonlinear functionIt is approached:WhereinIt indicates using the second RBF neural weight Nonlinear Function Approximation most
The figure of merit, ε2 *Indicate the error approached, h2(x1,x3) be the second RBF neural neuron activation primitive；
Step 6.3, definition vectorWherein ρ is dry outside Δ (t)
The upper bound disturbed, ω are design constant, and meet ω > 0, c4For design constant, and meet c4> 0, design adaptive law are as follows:Wherein Γ2For a positive definite symmetric matrices, σ2For design parameter, meet σ2> 0, K4It is right
Error S4The constraint of size；
Further, the step 7 specifically includes:
Design practical controller are as follows:WhereinIt is θ2Estimated value, λ4It is design parameter, and
Meet λ4> 0.
The present invention has following technical characterstic compared with prior art:
1. the present invention introduces firstorder filter by " stepping type " design method in conjunction with Reverse Step Control, in each step to count
The derivative for calculating virtual controlling item, obtains new state variable, so as to avoid backstepping control method to virtual controlling derivation repeatedly
Caused item number expansion issues；The controller and adaptive law of design, may further ensure that the tracking error of joint rotation angle about
Beam is in a certain range.
2. the mathematical models due to soft articulated mechanical arm can not be getable, in the flexible joint that design construction is stable
While robot controller, need to consider that external interference and compensating parameter uncertainty bring influence, research is suitable for having
The positional control algorithm of the light-duty mechanical arm in flexible joint.The present invention makes full use of RBF neural to nonlinear function
Approximation capability designs controller using approximate error and adaptive law is adjusted the weight of neural network, eliminates to machine
The needs of tool arm precise kinetic model.Design controller is verified finally by simulation example, it was demonstrated that side of the present invention
The validity of method guarantees that joint can effectively track Setting signal, and all signals under conditions of robotic arm limited torque output
All half global boundeds.
Detailed description of the invention
Fig. 1 is the structural schematic diagram of mechanical arm flexible joint；
Fig. 2 is neural networks principles schematic diagram；
Fig. 3 is neural network identification nonlinear organization schematic diagram；
Fig. 4 is RBF neural self adaptive control schematic diagram；
Fig. 5 is tracking effect schematic diagram；
Fig. 6 is tracking error schematic diagram；
Fig. 7 is neural network Nonlinear Function ApproximationSchematic diagram；
Fig. 8 is neural network Nonlinear Function ApproximationSchematic diagram；
Fig. 9 is that output control moment shows τmIt is intended to.
Specific embodiment
The present invention is directed to the mechanical arm with flexible joint, proposes a kind of based on the adaptive of Liapunov stability analysis
RBF neural dynamic surface control method is answered, this method makes full use of the approximation capability compensating Modeling of RBF neural inaccurate
The problem of bringing eliminates the needs to accurate kinetic model, solves traditional self adaptive control using dynamic surface technology and uses
Backstepping bring calculate expansion issues, and consider that the torque output of mechanical arm is limited and extraneous uncertain dry
It disturbs, proposes suitable controller；One complicated mechanical arm system is virtually disassembled into multiple subsystems by Virtual Controller, will
Modeling, controller design and the stability analysis of complication system decompose subsystems, will be each finally by " fictitious power stream "
Between a subsystem dynamic connect.It is of the invention that the specific method is as follows:
A kind of adaptive dynamic surface control method of RBF neural of tool arm flexible joint, comprising the following steps:
Step 1, mechanical arm flexible joint is modeled
This programme uses the generally accepted flexible mechanical arm joint structure in current this field: " spring-connecting rod " model, such as schemes
Shown in 1, the equation model of mechanical arm flexible joint is established are as follows:
In above formula, q, θ respectively indicate the angular position of connecting rod, rotor,It is expressed as the velocity of rotation of connecting rod corner, I, J divide
Not Biao Shi connecting rod, rotor rotary inertia, CHIndicate the gravitational vectors of mechanical arm, BmIndicate that viscous rub of motor shaft examines coefficient, K
Represent joint of mechanical arm stiffness coefficient, M, g, l are respectively connecting rod quality, acceleration of gravity and joint to connecting rod centroid distance, τm
It is motor torque input；The respectively single order of q, second dervative,For the second dervative of θ；In following formula, in parameter
Mark dot indicates the derivative of the parameter, and dot number is derivative order number, repeats no more.
Wherein, K is joint stiffness, and K is bigger, illustrates that joint stiffness is bigger, and flexibility of joint is with regard to small, and q and θ more connect at this time
Closely；K is smaller, illustrates that joint stiffness is smaller, and flexibility of joint is with regard to big, and q and θ difference is bigger at this time.
The control target of this programme is that the corner based on connecting rod can track given signal, and tracking error is small, is protected
All signals of card system all half global boundeds.
Step 2, according to the physical characteristic of mechanical arm flexible joint, the equation model that step 1 is established is converted to state side
Journey is specifically expressed as follows:
In above formula, x=[x1,x2,x3,x4]TFor state variable,Respectively x1,x2,x3,x4First derivative, Δ (t) is extraneous BOUNDED DISTURBANCES and meets Δ (t) < ρ, and ρ is a constant；g(x1,
x2), f (x1,x3) be concrete form unknown but bounded nonlinear function, it is due to g (x that wherein concrete form is unknown1,x2), f
(x1,x3) C in expression formulaH、Bm, K, J, I concrete form be it is unknown, in the method nonlinear function g (x1,x2), f
(x1,x3) be utilized respectively RBF neural and approached；Assuming that it is x that mechanical arm flexible joint, which needs to track input signal,1d,
First derivative, second dervativeExist, and meetsξ is a constant.
It is directed to the state equation of step foundation, in conjunction with the stepping type of Backstepping, is controlled using dynamic surface Technology design
Device.
Step 3, it defines first tracking error and designs first Virtual Controller, by the letter of first Virtual Controller
Number input first low-pass first order filter, obtain new state variable x2dNext step meter is carried out instead of first Virtual Controller
It calculates, reduces calculation amount；Specific step is as follows:
Step 3.1, first tracking error: S is defined1=x1-x1d, derivation obtainsTake Virtual Controller are as follows:Wherein c1For design parameter, and meet c1> 0.
Step 3.2, willIt is input to first low-pass first order filter, obtains new state variable x2d；Described in wherein
Firstorder filter indicates are as follows:
In above formula, x2dFor the new state variable obtained by firstorder filter, τ2For the time constant filter of filter,
WhereinIt indicates through Virtual ControllerInitial value and filtered signal x2dInitial value it is equal；Use x2d
Instead ofDuring deriving practical controller, next step calculating is carried out.
Step 3.3, the filtering error e of first Virtual Controller is defined2, expression formula are as follows:It can then obtain
Its derivative are as follows:
Step 4, second tracking error is defined, due to containing nonlinear function in system state equation, utilizes the
One RBF neural approaches the nonlinear function containing uncertain parameter, constructs Liapunov function, passes through Li Ya
Pu Nuofu stability analysis designs the adaptive law of neural network weight, keeps approximate error as small as possible；Design second void
Quasi- controller, the signal of second Virtual Controller obtain new variable x by second low-pass first order filter3d；Specifically such as
Under:
Step 4.1, second tracking error: S is defined2=x2-x2d, derivation obtains:
Step 4.2, the first RBF neural is established
Due toIn contain all nonlinear terms g (x1,x2), therefore it is approached using the first RBF neural, institute
The first RBF neural stated is the neural network of multiple input single output, and structure is as shown in Fig. 2, its approximation theory such as Fig. 3 and figure
4.In RBF network structure, X=[x1,x2,…,xi]TFor the input layer vector of network, if RBF network hidden layer radial direction basal orientation
Measure H=[h1,h2,…,hn]T, wherein hjIt is the neuron activation functions of neural network, expression formula for Gaussian bases are as follows:J=1,2 ..., n；Wherein the center vector of j-th of node of network is cj=[cj1,cj2,…,
cjn]T, the sound stage width vector of network are as follows: bj=[bj1,bj2,…,bjn]T, bjFor node j's and width parameter, and have bj> 0.Network
Weight vector are as follows: θj=[θj1,θj2,…,θjn]T.The output of RBF network are as follows: yn(t)=θ1h1+θ2h2+…+θnhn。
Step 4.3, due toIt is unknown, therefore approached not using the first RBF neural described in step 4.2
Know function WhereinIndicate that neural network weight approaches unknown function
When optimal value, h1(x1,x2) be the first RBF neural neuron activation functions, ε1 *Indicate the error approached, neural network
Nonlinear Function Approximation effect picture is as shown in Figure 8.
Step 4.4, second Virtual Controller is designedWhereinIt is
Estimated value, k2, λ2、c2It is design parameter, and meets λ2> 0, c2> 0.
Step 4.5, design adaptive lawWherein Γ1It is symmetrical for a positive definite
Matrix, σ1For design parameter, and meet σ1> 0, K2For to tracking error S2The constraint of size.
Step 4.6, willIt is input to second low-pass first order filter, obtains new variable x3d；Wherein filter can be with
It indicates are as follows:
Wherein, τ3For the time constant filter of filter, new state variable x is obtained3d；
Step 4.7, second Virtual Controller filtering error e is defined3, expression formula are as follows:It can then be obtained
Derivative are as follows:
Step 5, third tracking error is defined, third Virtual Controller is designed, by third Virtual Controller signal
It is input to third low-pass first order filter, obtains new state variable x4d, the specific steps are as follows:
Step 5.1, third tracking error: S is defined3=x3-x3d, derivation obtainsDesign Virtual Controllerλ3、c3It is design parameter, and meets λ3> 0, c3> 0；
Step 5.2, willThird low-pass first order filter is inputted, new state variable x is obtained4d；Wherein filter can
To indicate are as follows:
Wherein, τ4For the time constant filter of filter, new state variable x is obtained4d。
Step 5.3, third Virtual Controller filtering error e is defined4, expression formula are as follows:It can then be obtained
Derivative are as follows:
Step 6, it defines the 4th tracking error non-linear letter is occurred according to the state equation in state equation
Number, compensates it using the second RBF neural and approaches, Approximation effect such as Fig. 8, it is contemplated that external interference Δ (t), design
Interference compensation item.Liapunov function is constructed, is analyzed by Liapunov stability, designs the adaptive of neural network weight
Ying Lv keeps the approximate error of nonlinear function as small as possible；Specific step is as follows:
Step 6.1, the 4th tracking error: S is defined4=x4-x4d, derivation obtains
Step 6.2, due to m2With f (x1,x3) unknown, therefore the second RBF neural is constructed to nonlinear functionIt is approached:WhereinIt indicates to weigh using the second RBF neural
It is worth the optimal value of Nonlinear Function Approximation, ε2 *Indicate the error approached, h2(x1,x3) be second RBF neural neuron
Activation primitive, neural network Nonlinear Function Approximation effect picture, such as Fig. 8；First RBF neural, second RBF neural
Network structure is identical, and step 6.2 and step 4.3 are the difference is that input, the output of two neural networks are different, first mind
Input through network is x1, x2, export and beThe input of second neural network is x1, x3, output isIn the present solution, the parameter upper right corner adds * to indicate the optimal value approached.
Step 6.3, definition vectorWherein ρ is dry outside Δ (t)
The upper bound disturbed, ω are design constant, and meet ω > 0, c4For design constant, and meet c4> 0, design adaptive law are as follows:Wherein Γ2For a positive definite symmetric matrices, σ2For design parameter, meet σ2> 0, K4It is right
Error S4The constraint of size.
Step 7, practical controller is designed are as follows:WhereinIt is θ2Estimated value, λ4It is design
Parameter, and meet λ4> 0.
In practical application, the torque controller of mechanical arm flexible joint motor is inputted are as follows: τm, such as Fig. 9, to mechanical arm
The motor torque of flexible joint is controlled, and then influences the rotation of the connecting rod and rotor in joint, keeps robot linkage real
Now track Setting signal x1d, tracking effect is as shown in figure 5, tracking error is as shown in Figure 6.
Emulation experiment:
In the emulation experiment, control target is that bar is made to connect corner q tracking ideal trajectory x1d=sint, it is assumed that external interference
For Δ=0.2cos (2t).According to real system, the system physical parameter in model that this example uses can be selected as: I=J=
1.0kg·m2, Mgl=5Nm, K=40Nm/rad, CH=1N, Bm=0.001, ω=0.01, K2=0.2, K4=0.5, c1
=6, c2=80, c3=34, c4=5, q=0.35.System initial state is selected as x=[0,0,0,0]T, filter time constant
It is selected as τ2=τ3=τ4=0.005, take Γ1=[2,2,2,2,2,2,2], due to considering external interference in angle of rotor,
Using RBF remove uncertainty error when, Γ in adaptive law2=[10,10,10,10,10,10,10,0.5].For
Two nonlinear functions for needing to approach, wherein first RBF Approximation Network is chosen are as follows: 2-7-1.Due to the function that need to be approachedIt is related with two state variables, then two inputs are selected, 7 neurons is selected, it is defeated to finally obtain a network
Out.Initial value is all selected as 0, according to x1, x3Input range selection Gaussian bases value range are as follows: [- 2.4,2.4],
I.e.Wherein first RBF Approximation Network is chosen are as follows: 2-7-1.Due to needing
The function approachedIt is related with two state variables, then two inputs are selected, 7 neurons is selected, finally obtains
One network output.Initial value is all selected as 0, according to x1, x3Input range selection Gaussian bases value range are as follows:
[- 2.4,2.4], i.e.,
Interpretation of result:
According to the state equation in step 2, consideration is compacted as follows:
Wherein q is any positive number,
Respectively indicating isTo θ1、To θ2Evaluated error when being estimated, and meet
Choose Lyapunov functionAccording to Li Ya
General promise stability theorem meets primary condition V (0)≤q, then adjustment parameter ci, τi, ε, σ1, σ2, Γ1, Γ2The system that can make
All half globally uniformly boundeds of signal, the tracking error of system can converge near origin.Note:Respectively indicate x1d, x2d, x2d, x3d, x4dCorresponding single order lead
Several and second dervative.x2d(0), x3d(0), x4d(0) x is indicated2d, x3d, x4dInitial value.
In the method nonlinear function g (x1,x2), f (x1,x3) due to containing all unknown parameters, and actual machinery
Arm flexible joint model is also increasingly complex, so referred to as concrete form is unknown.But the tracing control proposed in simulating, verifying
When algorithm, need to provide value to unknown parameter, but be still it is nonlinear need to be utilized respectively it neural network into
Row approaches.
The method of the present invention uses two different neural networks and approaches two nonlinear functions respectively, and adaptive law is used for
Adjust weight θ when neural network Nonlinear Function Approximation1, improve approximation capability；Control law is divided into Virtual Controller and reality
Controller, this is that backstepping method is distinctive, and Virtual Controller guarantees the stabilization of system for deriving practical control
Property, it can be understood as virtual controlling is used to guarantee the stabilization of each subsystem, i.e.,The practical control finally derived
It can guarantee the stabilization of whole system, it can be according to Lyapunov theorem of stability proof system stability.
When the program is applied to practical, when designing the electric machine controller of mechanical arm, need to design Virtual Controller Adaptive law:And actual controller: τmMotor torque for control system.Finally control
Device can guarantee that system exports, i.e., the connecting rod of mechanical arm can track given reference signal x1d, tracking effect figure such as Fig. 5,
The error of tracking such as Fig. 6.
Claims (8)
1. a kind of adaptive dynamic surface control method of RBF neural of mechanical arm flexible joint, which is characterized in that including following
Step:
Step 1, mechanical arm flexible joint is modeled；
Step 2, according to the physical characteristic of tool arm flexible joint, the equation model that modeling obtains is converted into state equation；
Step 3, it defines first tracking error and designs first Virtual Controller, the signal of first Virtual Controller is defeated
Enter first low-pass first order filter, obtains new state variable x2dNext step calculating is carried out instead of first Virtual Controller,
Reduce calculation amount；
Step 4, second tracking error is defined, using the first RBF neural to the nonlinear function containing uncertain parameter
It is approached, designs the adaptive law of the first RBF neural weight；Design second Virtual Controller, second virtual control
The signal of device processed obtains new variable x by second low-pass first order filter3d；
Step 5, third tracking error is defined, third Virtual Controller is designed, the signal of third Virtual Controller is defeated
Enter and obtains new state variable x to third low-pass first order filter4d；
Step 6, the 4th tracking error is defined to mend it using the second RBF neural according to the state equation
It repays and approaches, design the adaptive law of the second RBF neural weight；
Step 7, practical controller is designed.
2. the adaptive dynamic surface control method of RBF neural of mechanical arm flexible joint as described in claim 1, feature
It is, the equation model of the mechanical arm flexible joint are as follows:
In above formula, q, θ respectively indicate the angular position of connecting rod, rotor,It is expressed as the velocity of rotation of connecting rod corner, I, J distinguish table
Show the rotary inertia of connecting rod, rotor, CHIndicate the gravitational vectors of mechanical arm, BmIndicate that viscous rub of motor shaft examines coefficient, K is represented
Joint of mechanical arm stiffness coefficient, M, g, l are respectively connecting rod quality, acceleration of gravity and joint to connecting rod centroid distance, τmIt is electricity
Machine torque input；The respectively single order of q, second dervative,For the second dervative of θ.
3. the adaptive dynamic surface control method of RBF neural of mechanical arm flexible joint as described in claim 1, feature
It is, state equation described in step 2 are as follows:
In above formula, x=[x1,x2,x3,x4]TFor state variable,Respectively x1,x2,x3,x4First derivative, Δ (t) is extraneous BOUNDED DISTURBANCES and meets Δ (t) < ρ, and ρ is a constant；g(x1,
x2), f (x1,x3) be concrete form unknown but bounded nonlinear function；Assuming that mechanical arm flexible joint needs to track input letter
Number be x1d, first derivative, second dervativeExist, and meetsξ is a constant.
4. the adaptive dynamic surface control method of RBF neural of mechanical arm flexible joint as described in claim 1, feature
It is, the step 3 specifically includes:
Step 3.1, first tracking error: S is defined1=x1-x1d, derivation obtainsTake Virtual Controller are as follows:Wherein c1For design parameter, and meet c1> 0；
Step 3.2, willIt is input to first low-pass first order filter, obtains new state variable x2d；
Step 3.3, the filtering error e of first Virtual Controller is defined2, expression formula are as follows:It can be then obtained to lead
Number are as follows:
5. the adaptive dynamic surface control method of RBF neural of mechanical arm flexible joint as described in claim 1, feature
It is, the step 4 specifically includes:
Step 4.1, second tracking error: S is defined2=x2-x2d, derivation obtains:
Step 4.2, the first RBF neural is established；
Step 4.3, unknown function is approached using the first RBF neural described in step 4.2 WhereinIndicate optimal value when the first RBF neural weight approaches unknown function, h1
(x1,x2) be neural network neuron activation functions, ε1 *Indicate the error approached；
Step 4.4, second Virtual Controller is designedWhereinIt isEstimate
Evaluation, k2, λ2、c2It is design parameter, and meets λ2> 0, c2> 0；
Step 4.5, design adaptive lawWherein Γ1For a positive definite symmetric matrices,
σ1For design parameter, and meet σ1> 0, K2For to tracking error S2The constraint of size；
Step 4.6, willIt is input to second low-pass first order filter, obtains new variable x3d；
Step 4.7, second Virtual Controller filtering error e is defined3, expression formula are as follows:Its derivative can then be obtained
Are as follows:
6. the adaptive dynamic surface control method of RBF neural of mechanical arm flexible joint as described in claim 1, feature
It is, the step 5 specifically includes:
Step 5.1, third tracking error: S is defined3=x3-x3d, derivation obtainsDesign Virtual Controllerλ3、c3It is design parameter, and meets λ3> 0, c3> 0；
Step 5.2, willThird low-pass first order filter is inputted, new state variable x is obtained4d；
Step 5.3, third Virtual Controller filtering error e is defined4, expression formula are as follows:Its derivative can then be obtained
Are as follows:
7. the adaptive dynamic surface control method of RBF neural of mechanical arm flexible joint as described in claim 1, feature
It is, the step 6 specifically includes:
Step 6.1, the 4th tracking error: S is defined4=x4-x4d, derivation obtains
Step 6.2, the second RBF neural of construction is to nonlinear functionIt is approached:WhereinIt indicates using the second RBF neural weight Nonlinear Function Approximation most
The figure of merit, ε2 *Indicate the error approached, h2(x1,x3) be the second RBF neural neuron activation primitive；
Step 6.3, definition vectorWherein ρ is Δ (t) external disturbance
The upper bound, ω are design constant, and meet ω > 0, c4For design constant, and meet c4> 0, design adaptive law are as follows:Wherein Γ2For a positive definite symmetric matrices, σ2For design parameter, meet σ2> 0, K4It is right
Error S4The constraint of size.
8. the adaptive dynamic surface control method of RBF neural of mechanical arm flexible joint as described in claim 1, feature
It is, the step 7 specifically includes:
Design practical controller are as follows:WhereinIt is θ2Estimated value, λ4It is design parameter, and meets
λ4> 0.
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