CN106078741B - Limited performance flexible mechanical arm control method based on the definite theories of learning - Google Patents

Limited performance flexible mechanical arm control method based on the definite theories of learning Download PDF

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CN106078741B
CN106078741B CN201610457032.3A CN201610457032A CN106078741B CN 106078741 B CN106078741 B CN 106078741B CN 201610457032 A CN201610457032 A CN 201610457032A CN 106078741 B CN106078741 B CN 106078741B
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CN106078741A (en
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王敏
杨安乐
方冲
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South China University of Technology SCUT
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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
    • B25J9/1656—Programme controls characterised by programming, planning systems for manipulators
    • B25J9/1664—Programme controls characterised by programming, planning systems for manipulators characterised by motion, path, trajectory planning
    • 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
    • B25J9/1628—Programme controls characterised by the control loop
    • B25J9/1635—Programme controls characterised by the control loop flexible-arm control

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  • Engineering & Computer Science (AREA)
  • Robotics (AREA)
  • Mechanical Engineering (AREA)
  • Feedback Control In General (AREA)

Abstract

The invention discloses a kind of limited performance flexible mechanical arm control method based on the definite theories of learning, this method is directed to the uncertainty of flexible mechanical arm dynamic model, designs tracking error, is allowed to meet that constraints limits, and form error controller.Step of the present invention includes：Establish flexible mechanical arm dynamic model；Establish State Observer；Design tracking error performance constraints；Nerve network controller is designed based on the definite theories of learning；Utilize Heuristics modifier controller.Control method designed by the present invention can realize Fast Convergent, the dynamic property of low overshoot, meet the constraints limitation of setting, while avoid neural network weight on-line control, shorten control time.In addition, this method can utilize the Heuristics learnt to be directly realized by quick control to identical control task afterwards.

Description

Performance limited flexible mechanical arm control method based on determined learning theory
Technical Field
The invention belongs to the field of track tracking control of flexible mechanical arms, and particularly relates to a method for designing a tracking error aiming at uncertainty of a dynamic model of a flexible mechanical arm system so as to meet constraint condition limitation. A performance limited flexible mechanical arm control method based on a definite learning theory is provided.
Background
Nowadays, as robots are developed in a direction of high precision and rapidity, mechanical parts tend to be more and more lightweight. Compared with the traditional rigid mechanical arm, the flexible mechanical arm has the characteristics of light structure, large working space and high efficiency, so that the application of the flexible element in the fields of manufacturing, aerospace and the like is wider and wider, but the flexible effect caused by the flexible arm rod can cause mechanical vibration, the stability and the accuracy required by the track control are more difficult to achieve, and the track tracking control at the tail end of the flexible mechanical arm is more difficult. In practical industrial application, due to constraint limits of an industrial environment, such as a limit of a maximum output torque of a motor, a limit of a track tracking error at a tail end in a steady state stage, a limit of a maximum overshoot of a tracking error in an initial dynamic stage and a limit of a convergence speed, the control performance of a system is often required to meet a certain constraint condition, so that a designed control scheme not only needs to ensure the stability of the track tracking error system, but also needs to meet the constraint of the tracking error in an actual situation, and great challenges are brought to the design of the control scheme.
The method aims at the intelligent control methods such as track tracking control, inversion control, sliding mode control and dynamic surface control of the flexible mechanical arm system and combines with neural network control, and can solve the problems of stability and accuracy required by track control under the condition that the system model is dynamically uncertain. However, under the condition of specific constraint on the tracking performance, the control methods cannot control the tracking error to meet the requirement of specific performance limitation. A performance error conversion self-adaptive neural network control method is characterized in that a performance limited function is introduced, a tracking error performance limited condition in an actual environment is specifically expressed as a mathematical function expression, an original constrained tracking error control problem is converted into an unconstrained conversion error stability control problem through performance error conversion, self-adaptive neural network control is designed aiming at the unconstrained conversion error, a state observer is designed aiming at an unmeasurable state in a system, and finally the stability of the unconstrained conversion error can enable the tracking of a tail end track to meet the performance limited condition limit. Therefore, by introducing the performance error conversion adaptive neural network control method of the performance constraint function, the corresponding performance constraint function can be designed according to the specific actual performance constraint, and the track tracking control under the condition that the performance of the flexible mechanical arm is limited is realized.
In the existing process of approximating the unknown model dynamics in the flexible mechanical arm system by using the neural network, continuous online adjustment is needed, the weight of the neural network needs to be adjusted again by the controller each time the system is started, the approximation error of the neural network to the unknown model dynamics is larger in the initial stage of weight adjustment, and the adjustment process is very time-consuming, so that the overall control effect is influenced. For the same control task, the unknown model dynamics of the neural network approximation are substantially consistent, so repeated neural network weight adjustments are redundant operations. In order to solve the time-consuming redundant online adjustment process of the neural network, the final convergence of the weights of the neural network is required, which is difficult to achieve. It is determined that the Learning theory (Wang C.and Hill D.J.. Learning From Neural Control [ J ]. IEEE Transactionnon Neural Networks,2006,17(1):130 (145)) has demonstrated that Neural network weights can eventually converge when Neural network approximations are made to a periodic or cycle-like trajectory. The track tracking control method of the performance-limited flexible mechanical arm based on the determined learning theory stores the converged neural network weight, directly utilizes the stored neural network weight in the next same control task, avoids the repeated online adjustment process of the neural network, and realizes the constant neural network control based on the experience knowledge.
Disclosure of Invention
The invention mainly aims to overcome the defects and shortcomings of the prior art, and provides a track tracking control method of a performance-limited flexible mechanical arm based on a definite learning theory, so that the track tracking control task under the condition of specific tracking error performance constraint in an actual control environment is avoided for the same control task for multiple times.
In order to achieve the purpose, the invention adopts the following technical scheme:
the invention discloses a performance-limited flexible mechanical arm control method based on a definite learning theory, which comprises the following steps of:
step (1): establishing a dynamic model of the flexible mechanical arm: establishing a dynamic model in a four-order standard system form for decoupling the single-connecting-rod flexible mechanical arm through state transformation;
step (2): establishing a system state observer: designing a state observer for the system state which cannot be directly measured in the model:
wherein, p and riFor design parameters, i ═ 1, …,4,respectively, in an unmeasurable state [ upsilon ] in the system2υ3υ4]The state observer of (1);
and (3): designing a tracking error performance constraint condition: designing a performance function to constrain the transient and steady performance of the tracking error between the output angle of the connecting rod and the output angle of the periodic reference track, and specifically comprising the following steps:
where ρ (t) is the performance function, ρ0、ρ∞、s、σ、To design the constant, xdReferencing a periodic trajectory for joint link angles, e1Is a track tracking error;
designing a strictly monotone increasing smooth function psi (epsilon)1) Will be limited by the tracking error e1(t) conversion to conversion error ε1(t)：
And (4): designing a neural network controller based on a deterministic learning theory: designing a self-adaptive RBF neural network learning controller by utilizing a determined learning theory:
wherein,for neural network output, k4For the designed gain constant of the controller, epsilon4Is an intermediate error amount calculated by the following design process:
the state observer output established according to step (2)And the unlimited conversion error epsilon after the conversion in the step (3)1Design of virtual controller α1，α2，α3：
Wherein k isi(i ═ 1,2,3) is the designed virtual controller gain constant, xdReferencing a periodic track for a joint connecting rod angle, wherein gamma and B are intermediate control variables involved in virtual control;
intermediate error amount occurring in controller u
And (5): modifying the controller using empirical knowledge: according to the determined learning theory, weighting the neural network in the step (4)Is saved asRealizing constant RBF neural network by using expression experience knowledgeThe correction controller, namely the controller form is:
as a preferred technical solution, in the step (1), the dynamic model in the form of a four-order normative system for flexible manipulator decoupling is as follows:
wherein x is1And x2Respectively, the angle of the joint connecting rod and the rotation angle of the motor, I and J respectively are the inertia of the connecting rod and the motor, M is the mass of the connecting rod, L is the length of the connecting rod, g is the gravity acceleration, K is the elastic coefficient of the spring of the flexible part, and u isAnd a controller, namely the control output of the motor.
As a preferred technical solution, in the step (3), the limited tracking error e1(t) by designing a strictly monotonically increasing smoothing function Ψ (. epsilon.)1) Conversion to conversion error e1(t) by controlling ε1(t) stabilization of e1(t) satisfies the constraint conditionThereby realizing the track tracking control under the constraint condition.
As a preferred technical solution, in the step (4), the intermediate signal variable γ, B related to the virtual controller is specifically:neural network outputUnknown dynamic information for approximating a robotic arm system, input to a neural network isAnd the weight vector of the neural network is updated on line.
As a preferred technical solution, in the step (4), the weight of the neural network is converged, based on a deterministic learning theory, the neural network input by the periodic trajectory is tracked, the regression vector s (x) of the neural network satisfies the continuous excitation condition, and finally the weight of the neural network is obtainedConverge to an optimum value
Preferably, in step (5), the modified controller includes determining empirical knowledge during learningTherefore, the controller can realize the dynamic performance of quick convergence and low overshoot for the same control task.
Compared with the prior art, the invention has the following advantages and beneficial effects:
1. compared with the existing track tracking control method of the flexible mechanical arm, the control method provided by the invention can realize that the track tracking error meets a specific performance constraint condition, not only realizes that the track tracking error finally approaches to zero in one neighborhood, but also can realize the limitation on the overshoot and the convergence speed of the error.
2. The method of the invention can embody the performance constraint condition in a mathematical function mode by designing the performance function and adjusting the parameters of the performance function, thereby further designing the controller.
3. The method of the invention designs a strictly monotone increasing smooth function psi (epsilon)1) Limited tracking error e1(t) conversion to conversion error ε1And (t) the essence of the method is to convert the problem of limited tracking error control into the problem of stability of unlimited error, thereby facilitating the design of the controller.
4. The method can utilize a definite learning theory to learn the uncertain model of the system dynamically, and stores the learned empirical knowledge in the form of a constant neural network weight, so that the stored knowledge can be directly used for control when the same control task is carried out later, the redundant online adjustment process is avoided, the offline control of the neural network is realized, the time is saved, and the dynamic tracking performance at the initial stage is improved.
Drawings
FIG. 1 is a schematic view of a flexible arm system of the present invention.
Fig. 2 is a block diagram of the overall control of the flexible robot arm of the present invention.
FIG. 3 is a simulation diagram of the convergence of the tracking error in the learning phase of the neural network of the present invention.
FIG. 4 is a simulation of the error between the state observer and the observed state in accordance with the present invention.
FIG. 5 is a diagram of neural network weight convergence simulation in accordance with the present invention.
FIG. 6 is a graph of a simulation of the controller output during the neural network learning phase of the present invention.
FIG. 7 is an approximate simulation of the unknown dynamics of the system using a constant neural network in accordance with the present invention.
FIG. 8 is a comparison simulation graph of the convergence of tracking errors during the neural network learning phase and the knowledge reuse control phase of the present invention.
Detailed Description
The present invention will be described in further detail with reference to examples and drawings, but the present invention is not limited thereto.
Examples
In this embodiment, track tracking control of the flexible mechanical arm under the condition of limited tracking performance is mainly studied, and fig. 1 is a schematic diagram of a flexible mechanical arm system.
The overall control block diagram of the performance-limited flexible mechanical arm control method based on the determined learning theory is shown in fig. 2, and the detailed implementation process comprises the following steps:
step (1): and establishing a dynamic model of the flexible mechanical arm.
Flexible robotic arm system model according to:
the dynamic model form converted into the decoupled fourth-order canonical system form is as follows:
wherein x is1And x2Respectively a joint connecting rod angle and a motor rotation angle, I and J respectively represent inertia of the connecting rod and the motor, M represents mass of the connecting rod, L represents length of the connecting rod, g represents gravity acceleration, K represents elastic coefficient of a flexible part spring, and u represents control output of the motor;
in this example, the system parameters of the flexible robot arm are selected as follows:
M＝0.2kg，L＝1m，I＝2.3kg·m2，K＝15N·m/rad，J＝0.5kg·m2，g＝9.8m/s2
step (2): and establishing a system state observer.
Since the measurable state in the system is x1And upsilon2，υ3，υ4When the system is not measurable, the state observer is designed as follows:
wherein p is 1250, r1＝r3＝-2p,r2＝-3p,r4＝-p,Respectively, in an unmeasurable state [ upsilon ] in the system2υ3υ4]The state observer of (1).
And (3): and designing a tracking error performance constraint condition.
The following periodic reference trajectories are selected:
wherein x isdIs a reference angle trajectory of the articulation link tip angle and there is a continuous derivativee1＝x1-xdIs the trajectory tracking error of the end of the connecting rod.
The tracking error is required in this example to meet the following constrained performance constraints: e.g. of the type1The upper limit and the lower limit of the maximum overshoot are 1.44 and-1.2 respectively, e1Must not be lower than e-t，e1Is constrained between-0.05 and 0.05, and the performance limiting function is designed according to the above constraint conditions as follows:
designing a strictly monotone increasing smooth function psi (epsilon) according to a performance limited function1) Will be limited by the tracking error e1Conversion to an unlimited conversion error epsilon1：
And (4): designing a neural network controller based on the deterministic learning theory.
Ensuring the conversion error epsilon by designing the controller1Thereby achieving a limited error e1The method meets the requirements of constraint conditions, and specifically comprises the following steps:
first of all designVirtual controller α1And with a state observerForm an error variable epsilon2：
Wherein k is1＝1，
Further design ofVirtual controller α2And with a state observerForm an error variable epsilon3：
Wherein k is2＝4，γ＝1/(ρ+e1)+1/(1.2ρ-e1)。
Further design ofVirtual controller α3And with a state observerForm an error variable epsilon4：
Wherein k is3＝10，
In this example, the system model dynamics of the flexible robotic arm is completely unknown, using neural networksApproximating unknown dynamics:wherein the input of the neural networkThe adaptive neural network learning controller is designed into the following form:
wherein the controller gain k is selected4＝20。
Selecting weights of neural networkThe online update adjustment rate is:
the system initial state and neural network parameters are selected as follows:
initial conditions of the system:
neural network parameter selection: the number N of the neural network nodes is 9 multiplied by 11, and the initial value of the weight isThe center points are uniformly distributed in [ -0.9,0.9 [)]×[-0.9,0.9]×[-1.5,1.5]×[-1.5,1.5]The neural network update rate parameter Γ is 12, and o is 0.0001.
Controller based on neural networkControlling the switching error epsilon1Stabilization of epsilon1Is stable so that a limited tracking error e is obtained1Satisfies the constraint-rho (t) < e1(t) < 1.2 rho (t), thereby realizing the trajectory tracking control under the performance constraint.
FIG. 3 shows the tracking error e during the learning phase1Converged simulation plot, final tracking error e1Under the constraint-rho (t) < e1Fluctuation in (t) < 1.2 rho (t) and convergence to a small neighborhood of zero in limited time, thereby meeting the requirements of set constraint conditions and realizing the trajectory tracking control under the performance constraint conditions. FIG. 4 is a simulation of the observed error variation between the designed state observer and the observed state in the system. FIG. 5 is a simulation diagram of the learning weight convergence of the neural network on the unknown system dynamics F (X). FIG. 6 is a waveform simulation diagram of the output u of the neural network learning phase controller. According to fig. 5, in the neural network learning phase, the weights of the neural networks finally converge to constant values, and the constant values of the weights of the neural networks are stored as the expression of unknown dynamic information of the system.
And (5): the controller is modified using empirical knowledge.
In this example, the constant neural network weight is calculated by averaging the weights converged over a time period [400s,500s ]:
designing a neural network controller based on empirical knowledge:
wherein, the selection of the control parameters in the controller is consistent with the learning stage of the neural network.
FIG. 7 is a constant neural networkAnd an approximation effect simulation graph between the neural network and the unknown system dynamics F (X), wherein the stored constant neural network can approximate the unknown system, so that the learning and recycling process of the neural network on the unknown system dynamics is realized, and a redundant online readjustment process is avoided. Fig. 8 is a simulation diagram of the tracking error convergence comparison effect in the neural network learning stage and the experience reuse control stage, for the same control task, the neural network controller based on the experience knowledge has a smaller overshoot, the dynamic characteristic in the initial stage is superior to that in the neural network learning stage, and the adjustment time is shortened.
The above embodiments are preferred embodiments of the present invention, but the present invention is not limited to the above embodiments, and any other changes, modifications, substitutions, combinations, and simplifications which do not depart from the spirit and principle of the present invention should be construed as equivalents thereof, and all such changes, modifications, substitutions, combinations, and simplifications are intended to be included in the scope of the present invention.

Claims (6)

1. The performance limited flexible mechanical arm control method based on the determined learning theory is characterized by comprising the following steps of:
step (1): establishing a dynamic model of the flexible mechanical arm: establishing a dynamic model in a four-order standard system form for decoupling the single-connecting-rod flexible mechanical arm through state transformation;
step (2): establishing a system state observer: designing a state observer for the system state which cannot be directly measured in the model:
wherein, p and riFor design parameters, i ═ 1, …,4,respectively, in an unmeasurable state [ upsilon ] in the system2υ3υ4]The state observer of (1);
and (3): designing a tracking error performance constraint condition: designing a performance function to constrain the transient and steady performance of the tracking error between the output angle of the connecting rod and the output angle of the periodic reference track, and specifically comprising the following steps:
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where ρ (t) is the performance function, ρ0、ρ∞、s、σ、To design the constant, xdReferencing a periodic trajectory for joint link angles, e1Is a track tracking error;
designing a strictly monotone increasing smooth function psi (epsilon)1) Will be limited by the tracking error e1(t) conversion to conversion error ε1(t)：
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And (4): designing a neural network controller based on a deterministic learning theory: designing a self-adaptive RBF neural network learning controller by utilizing a determined learning theory:
<mrow> <mi>u</mi> <mo>=</mo> <mo>-</mo> <msub> <mi>k</mi> <mn>4</mn> </msub> <msub> <mi>&amp;epsiv;</mi> <mn>4</mn> </msub> <mo>-</mo> <msup> <mover> <mi>W</mi> <mo>^</mo> </mover> <mi>T</mi> </msup> <mi>S</mi> <mrow> <mo>(</mo> <mi>X</mi> <mo>)</mo> </mrow> </mrow>
wherein,for neural network output, k4For the designed gain constant of the controller, epsilon4Is an intermediate error amount calculated by the following design process:
the state observer output established according to step (2)And the unlimited conversion error epsilon after the conversion in the step (3)1Design of virtual controller α1，α2，α3：
Wherein k isi(i ═ 1,2,3) is the designed virtual controller gain constant, xdReferencing a periodic track for a joint connecting rod angle, wherein gamma and B are intermediate control variables involved in virtual control;
intermediate error amount epsilon appearing in controller u4：
And (5): modifying the controller using empirical knowledge: according to the determined learning theory, weighting the neural network in the step (4)Is saved asRealizing constant RBF neural network by using expression experience knowledgeThe correction controller, namely the controller form is:
<mrow> <mi>u</mi> <mo>=</mo> <mo>-</mo> <msub> <mi>k</mi> <mn>4</mn> </msub> <msub> <mi>&amp;epsiv;</mi> <mn>4</mn> </msub> <mo>-</mo> <msup> <mover> <mi>W</mi> <mo>&amp;OverBar;</mo> </mover> <mi>T</mi> </msup> <mi>S</mi> <mrow> <mo>(</mo> <mi>X</mi> <mo>)</mo> </mrow> <mo>.</mo> </mrow>
2. the method for controlling the performance-limited flexible mechanical arm based on the deterministic learning theory as claimed in claim 1, wherein in the step (1), the dynamic model in the form of the four-order normative system for decoupling the flexible mechanical arm is as follows:
<mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>&amp;upsi;</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>&amp;upsi;</mi> <mn>2</mn> </msub> <mo>=</mo> <msub> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>&amp;upsi;</mi> <mn>3</mn> </msub> <mo>=</mo> <msub> <mover> <mi>x</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>&amp;upsi;</mi> <mn>4</mn> </msub> <mo>=</mo> <msub> <mover> <mi>x</mi> <mo>&amp;CenterDot;&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mn>1</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mtable> <mtr> <mtd> <mrow> <msub> <mover> <mi>&amp;upsi;</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>i</mi> </msub> <mo>=</mo> <msub> <mi>&amp;upsi;</mi> <mrow> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> </mrow> </mtd> </mtr> </mtable> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mover> <mi>&amp;upsi;</mi> <mo>&amp;CenterDot;</mo> </mover> <mn>4</mn> </msub> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mn>2</mn> </msub> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mfrac> <mi>K</mi> <mrow> <mi>I</mi> <mi>J</mi> </mrow> </mfrac> <mi>u</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mn>2</mn> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mi>M</mi> <mi>g</mi> <mi>L</mi> <msubsup> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mn>1</mn> <mn>2</mn> </msubsup> </mrow> <mi>I</mi> </mfrac> <mi>sin</mi> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mfrac> <mrow> <mi>M</mi> <mi>g</mi> <mi>L</mi> <mi> </mi> <mi>cos</mi> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mi>K</mi> </mrow> <msup> <mi>I</mi> <mn>2</mn> </msup> </mfrac> <mrow> <mo>(</mo> <mrow> <mi>M</mi> <mi>g</mi> <mi>L</mi> <mi> </mi> <mi>sin</mi> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mi>K</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>-</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mfrac> <mrow> <msup> <mi>K</mi> <mn>2</mn> </msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>-</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mi>I</mi> <mi>J</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mfenced>
wherein x is1And x2The angle of the joint connecting rod and the rotation angle of the motor are respectively, I and J are respectively the inertia of the connecting rod and the motor, M is the mass of the connecting rod, L is the length of the connecting rod, g is the gravity acceleration, K is the elastic coefficient of the spring of the flexible part, and u is the controller, namely the control output of the motor.
3. The performance-limited flexible mechanical arm control method based on the deterministic learning theory as claimed in claim 1, wherein in the step (3), the limited tracking error e1(t) by designing a strictly monotonically increasing smoothing function Ψ (. epsilon.)1) Conversion to conversion error e1(t) by controlling ε1(t) stabilization of e1(t) satisfies the constraint conditionThereby realizing the track tracking control under the constraint condition.
4. The method for controlling the performance-limited flexible mechanical arm based on the deterministic learning theory as claimed in claim 1, wherein in the step (4), the intermediate signal variable γ, B involved in the virtual controller is specifically: neural network outputUnknown dynamic information for approximating a robotic arm system, input to a neural network is And the weight vector of the neural network is updated on line.
5. The method for controlling the performance-limited flexible mechanical arm based on the deterministic learning theory as claimed in claim 1, wherein in the step (4), the weight of the neural network is converged, based on the deterministic learning theory, the neural network with the periodic trajectory input is tracked, the regression vector S (X) of the neural network satisfies the continuous excitation condition, and finally the weight of the neural networkConverge to an optimum value
6. The method for controlling a performance limited flexible mechanical arm based on deterministic learning theory as claimed in claim 1, wherein in the step (5), the modified controller comprises empirical knowledge in deterministic learning processTherefore, the controller can realize the dynamic performance of quick convergence and low overshoot for the same control task.
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