CN114310894B - Measurement output feedback control method of fourth-order uncertain nonlinear mechanical arm system - Google Patents
Measurement output feedback control method of fourth-order uncertain nonlinear mechanical arm system Download PDFInfo
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Abstract
The invention discloses a measurement output feedback control method of a fourth-order uncertain nonlinear mechanical arm system. Then, combining these matrix inequalities with the improved dynamic scaling technique solves the problem of the robotic arm system performing global adaptive state asymptotic adjustment through output feedback.
Description
Technical Field
A new hybrid gain scaling method involving static and dynamic gains is presented for a robotic arm system with both unknown continuous measurement sensitivity and uncertain nonlinearity. First, a solution to a pair of uncertainty matrix inequalities is explicitly constructed using only the lower bound information of the time-varying parameters. Then, combining these matrix inequalities with the improved dynamic scaling technique solves the problem of the robotic arm system performing global adaptive state asymptotic adjustment through output feedback.
Background
Based on continuous innovation of science and technology and development towards intelligent direction, the application field of robots is widened continuously, the application of robots to industry is becoming trend and mainstream, and the robots play an increasingly important role in future production and social development. Robots are typical representatives of advanced manufacturing techniques and automation equipment, being "final" representatives of man-made machines. The robot relates to a plurality of subjects and fields of machinery, electronics, automatic control, computers, artificial intelligence, sensors, communication, networks and the like, and is the comprehensive integration of various high and new technological development achievements, so that the development of the robot is closely related to the development of a plurality of subjects. In recent years, as science and technology are continuously developed into China, the development and research of robot technology in China are more and more emphasized and supported by government, and a great deal of scientific research results are obtained through the continuous and intensive research of scientific researchers in China. The mechanical arm is a typical representative of an industrial robot, can simulate certain action functions of hands and arms, can replace heavy labor of people to realize mechanization and automation of production, can operate in harmful environments to protect personal safety, and is widely applied to departments of electronics, light industry, atomic energy and the like.
Disclosure of Invention
Aiming at the defects of the prior art, the invention provides a measurement output feedback control method of a fourth-order uncertain nonlinear mechanical arm system.
Step one: and analyzing the mechanical arm system and establishing a corresponding system model.
Step two: the analysis system establishes a corresponding observer;
step three: the dynamic scaling technique and Lyapunov function and linear matrix inequality method are utilized to deduce that the system state is bounded and that the output is ultimately maintained within a preset range.
Step four: and (5) simulating and verifying a result.
Compared with the prior art, the invention has the following effects: the mechanical arm system is widely applied to industrial production, and the problem of asymptotic adjustment of the output feedback global self-adaptive state of a nonlinear mechanical arm system with large measurement uncertainty is solved. At the same time, the proposed control method is generalized to a more general nonlinear mechanical arm system with polynomial growth conditions of unknown parameters. Notably, a non-fallback design approach is presented herein, all of which are readily determined by a set of explicit constraints.
Drawings
FIG. 1 shows the actual x 1 ，x 2 And observed valueState response curves of (2);
FIG. 2 shows the actual x 3 ，x 4 And observed valueState response curves of (2);
fig. 3 is a state response curve of L, u.
Detailed Description
The invention discloses a measurement output feedback control method of a fourth-order uncertain nonlinear mechanical arm system, which specifically comprises the following steps:
step one:
the kinetic model of the robotic arm system is as follows:
as shown above, the unknown number q 1 Representing the displacement of the connecting rod in the system, q 2 Equal to the displacement of the rotor, q1 represents the displacement of the connecting rod, J 1 Representing the inertia of the connecting rod, J m Equal to the inertia, k, of the rotor of the electric machine 0 Represents the elastic constant, g is the gravitational constant, m is equal to the mass, l 0 Represents the mass center, F 1 Representing the viscous friction coefficient of the connecting rod, F m Representing the viscous friction coefficient of the motor rotor, u being the torque transmitted by the motor, of which variables only q 1 Is measurable.
The following are specific parameters of the mechanical arm:
table 1 parameters of the robotic arm system
The state space model of a four-stage mechanical arm system is as follows:
y＝θ(t)x 1
the following is a model with specific parameters:
y＝|1+2sin10t|x 1
there are some errors in the actual measurement, so the concept of sensitivity θ (t) is introduced here, and it is assumed that sensitivity θ (t) is continuous to 0+.θ 1 ≤θ(t)≤θ 2 Wherein θ is 1 ，θ 2 Is a known positive constant.
Step two:
one of the two quotations presented below is a new quotation that was not used in the design.
First, one identity matrix is I epsilon R 4×4 To represent, then, the matrices a, B, D are defined.
θ (t) is the unknown measurement sensitivity, and the lower bound is the constant θ 1 ，σ>0 is also a constant, h i >0，k i >0 is provided withAnd (5) calculating the degree of freedom.
Lemma 1: for an arbitrary constant alpha>0 all have a constant h i >0，V>0, and a digital matrix such that p=p T >0. So that A is T P+PA≤-αI,DP+PD≥VI；
And (4) lemma 2: for an arbitrary constant alpha>0 all have a constant k i >0，β>0, and a digital matrix such that q=q T >0. So that B is T Q+QB≤-βI,DQ+QD＞0.
Let i=1, 2,3 … n satisfy the linear growth condition: i f i (t,x,v)|≤c(|x 1 |+.......+|x n I), wherein c>0 is one
An unknown constant, called unknown growth rate; based on the parameters obtained in quotients 1 and 2 and designing a dynamic input
The output feedback controller designs a standard form of a fourth-order system observer according to the mechanical arm system, wherein the standard form is shown as follows:
L(0)＝1
l is determined primarily by the above equation, wherein σ,is a constant of design and +.>0＜σ＜0.5,τ>1. Wherein the parameters are designed; h is a 1 ＝0.3；h 2 ＝1.8；h 3 ＝0.3；h 4 ＝0.8；k 1 ＝0.6；k 2 ＝1.5；k 3 ＝1.7；σ＝0.45 />
Step three:
when i=1, 2,3,4, letThen introducing a scaling transformation:
according to the scaling and transformingThe system is described as:
wherein ε= [ ε ] 1 ,ε 2 ,ε 3 ,ε 4 ] T ,H＝[h 1 ,h 2 ,h 3 ,h 4 ] T
f 1 (t,v,x)＝0
f 3 (t,v,x)＝0
For the hypothetical system, in the case that the dynamic gain is bounded and the other closed loop state is globally converged to zero, the global adaptive state adjustment is realized through the control scheme constructed as above.
Firstly, selecting a Lyapunov function:
from the following componentsDeducing:
when 0< sigma <0.5, find the design parameter τ to satisfy:
0.5α-m 1 τ -2σ ≥γ 1
βτ-kτ 2σ -m 2 ≥γ 2
γ 1 ,γ 2 is a suitable constant.
Thus from the availability:
γ 1 L-c 1 ,γ 2 L-c 2 possibly negative. Therefore, it is not a standard Lyapunov function, i.e., the asymptotic stability of the closed loop system cannot be directly demonstrated according to the Lyapunov stability theorem.
Assuming that the solution has a maximum interval [0, t f )，t f The product of the process is denoted by E (0, + -infinity) or t f ＝+∞。
Let L (t) be [0, t f ) The inner bound will demonstrate the progressive convergence of the progressive system.
Proving that L is bounded:
let L be at [0, t f ) The above is unbounded. So that the number of the parts to be processed,
wherein the time of existence t 1 ，t 1 ≤t≤t f So that
Υ 2 L-c 2 ≥1,Υ 2 L-c 2 ≥1
Can be obtained by combining the above
By passing throughDefinition of (1) to obtain
As a result, it was obtainedSmaller than a certain constant contradicts the set L-bound, so L-bound is obtained.
Order the
Will be discussed later
I=1, … …, n, introducing a new scaling transform:
similar to the scaling mentioned above
Among these, it must be stated that:
b＝[0,......,1],/>
A *T P * +P * A≤-2I
and symmetrical matrix P * Above 0, introducing a Lieplov function for the above system
Substituting to obtain:
simplifying and obtaining:
right integral of this formula
According to the Barbalat lemma
Step four: and (5) simulating and verifying a result.
As shown in FIG. 1, is the actual x 1 ，x 2 And observed valueState response curves of (2).
As shown in FIG. 2, is the actual x 3 ，x 4 And observed valueState response curve of (2)
As shown in fig. 3, the state response curve is L, u.
Claims (2)
1. The measuring output feedback control method of the fourth-order uncertain nonlinear mechanical arm system is characterized by comprising the following steps of:
step one: analyzing a mechanical arm system and establishing a corresponding system model;
the kinetic model of the robotic arm system is as follows:
the variable q as shown above 1 Representing the displacement of the connecting rod in the system, q 2 Representing the displacement of the rotor of the motor, J 1 Representing the inertia of the connecting rod, J m Representing the inertia, k, of the rotor of the motor 0 Represents the elastic constant, g is the gravitational constant, m represents the mass, l 0 Represents the mass center of the connecting rod, F 1 Representing the viscous friction coefficient of the connecting rod, F m Representing the viscous friction coefficient of the motor rotor, u being the torque transmitted by the motor, of which variables only q is present 1 Is measurable;
the state space model of a four-stage mechanical arm system is as follows:
y＝θ(t)x 1
there are some errors in the actual measurement, so the concept of sensitivity θ (t) is introduced here, and it is assumed that sensitivity θ (t) is continuous to 0+.θ 1 ≤θ(t)≤θ 2 Wherein θ is 1 ，θ 2 Is a known positive constant;
step two: the analysis system establishes a corresponding observer;
first, one identity matrix is I epsilon R 4×4 To represent, then define matrices a, B, D,
θ (t) is the unknown measurement sensitivity, and the lower bound is the constant θ 1 ，σ>0 is also a constant, h i >0，k i >0 is a design parameter;
lemma 1: for an arbitrary constant alpha>0 all have a constant h i >0，V>0, and a digital matrix such that p=p T >0 is A T P+PA≤-αI,DP+PD≥VI；
And (4) lemma 2: for an arbitrary constant alpha>0 all have a constant k i >0，β>0, and a digital matrix such that q=q T >0 is B T Q+QB≤-βI,DQ+QD>0；
Let i=1, 2,3 … n satisfy the linear growth condition: i f i (t,v,x)|≤c(|x 1 |+.......+|x n I), wherein c>0, an unknown constant, called the unknown growth rate; based on the parameters obtained in quotients 1 and 2 and designing a dynamic outputThe feedback controller designs a standard form of a fourth-order system observer according to the mechanical arm system, wherein the standard form is as follows:
L(0)＝1
l is determined primarily by the above equation, wherein σ,is a constant of design and +.>0<σ<0.5,τ>1；
Step three: deducing that the system state is bounded and the output is finally kept in a preset range by using a dynamic scaling technology, a Lyapunov function and a linear matrix inequality method;
step four: and (5) simulating and verifying a result.
2. The method for feedback control of metrology output of a fourth order uncertainty nonlinear mechanical arm system in accordance with claim 1, wherein: the third step is specifically as follows:
when i=1, 2,3,4, letThen introducing a scaling transformation:
according to the scaling and transformingThe system is described as:
wherein ε= [ ε ] 1 ,ε 2 ,ε 3 ,ε 4 ] T ,H＝[h 1 ,h 2 ,h 3 ,h 4 ] T
f 1 (t,v,x)＝0
f 3 (t,v,x)＝0
For the assumed system, under the condition that the dynamic gain is bounded and the global convergence of other closed loop states is zero, the global self-adaptive state adjustment is realized through the control scheme formed by the above steps;
firstly, selecting a Lyapunov function:
from the following componentsDeducing:
when 0< σ <0.5, find the design parameter τ satisfies:
0.5α-m 1 τ -2σ ≥γ 1
βτ-kτ 2σ -m 2 ≥γ 2
γ 1 ,γ 2 is a constant value, and is a function of the constant,
thus, it is possible to obtain:
γ 1 L-c 1 ,γ 2 L-c 2 possibly negative; therefore, the method is not a standard Lyapunov function, i.e. the asymptotic stability of the closed-loop system cannot be directly proved according to the Lyapunov stability theorem;
assuming that the solution has a maximum interval [0, t f )，t f ∈(0,+∞)；
Let L (t) be [0, t f ) The inner limit will prove the progressive convergence of the progressive system;
proving that L is bounded:
let L be at [0, t f ) The upper part is unbounded; so that the number of the parts to be processed,
wherein the time of existence t 1 ，t 1 ≤t≤t f So that
γ 1 L-c 1 ≥1，γ 2 L-c 2 ≥1
Combining the above to obtain
By passing throughDefinition of (1) to obtain
As a result, it was obtainedLess than a certain constant, contradicts the set L bounded, so that L is bounded;
order the
I=1, … …, n, introducing a new scaling transform:
A *T P * +P * A≤-2I
and symmetrical matrix P * Above 0, introducing a Lieplov function for the above system
Substituting to obtain:
simplifying and obtaining:
to the right integrate
According to the Barbalat lemma
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| JPH10309684A (en) * | 1997-05-07 | 1998-11-24 | Yaskawa Electric Corp | Compliance control method of manipulator |
| US9296474B1 (en) * | 2012-08-06 | 2016-03-29 | The United States of America as represented by the Administrator of the National Aeronautics & Space Administration (NASA) | Control systems with normalized and covariance adaptation by optimal control modification |
| CN106773694A (en) * | 2016-12-26 | 2017-05-31 | 东北电力大学 | Precision Piezoelectric location platform self adaptation output feedback inverse control method |
| CN112241124A (en) * | 2020-10-27 | 2021-01-19 | 南昌大学 | Design method of self-adaptive inversion integral nonsingular fast terminal sliding mode controller |
| CN113534666A (en) * | 2021-07-29 | 2021-10-22 | 河南科技大学 | Trajectory tracking control method of single-joint mechanical arm system under multi-target constraint |
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| JP6895242B2 (en) * | 2016-11-25 | 2021-06-30 | 株式会社東芝 | Robot control device, robot control method and picking device |
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| JPH10309684A (en) * | 1997-05-07 | 1998-11-24 | Yaskawa Electric Corp | Compliance control method of manipulator |
| US9296474B1 (en) * | 2012-08-06 | 2016-03-29 | The United States of America as represented by the Administrator of the National Aeronautics & Space Administration (NASA) | Control systems with normalized and covariance adaptation by optimal control modification |
| CN106773694A (en) * | 2016-12-26 | 2017-05-31 | 东北电力大学 | Precision Piezoelectric location platform self adaptation output feedback inverse control method |
| CN112241124A (en) * | 2020-10-27 | 2021-01-19 | 南昌大学 | Design method of self-adaptive inversion integral nonsingular fast terminal sliding mode controller |
| CN113534666A (en) * | 2021-07-29 | 2021-10-22 | 河南科技大学 | Trajectory tracking control method of single-joint mechanical arm system under multi-target constraint |
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| 量测不确定性非线性时滞***的自适应控制;邵一鸣等;《杭州电子科技大学学报（自然科学版）》;全文 * |
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