CN116276962A - Limited time anti-interference control method for universal pneumatic flexible mechanical arm - Google Patents
Limited time anti-interference control method for universal pneumatic flexible mechanical arm Download PDFInfo
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Abstract
The limited-time anti-interference control method for the universal pneumatic flexible mechanical arm comprises the following steps: simplifying the universal pneumatic flexible mechanical arm into a connecting rod model by using a D-H method, and performing kinematic analysis to obtain a kinematic analysis model and the positions of the tail end points of the universal pneumatic flexible mechanical arm; based on a kinematic analysis model of the universal pneumatic flexible mechanical arm, establishing a dynamic model of the universal pneumatic flexible mechanical arm by using an Euler-Lagrange method; taking uncertainty of the universal pneumatic flexible mechanical arm as disturbance, and establishing a second-order mathematical model of the pneumatic flexible mechanical arm; designing a finite time expansion state observer to estimate disturbance, and designing a finite time backstepping controller based on a disturbance estimation value to compensate the influence of the disturbance on a system; aiming at the uncertainty of the model, the finite time control method is designed to compensate the influence on the position control precision of the pneumatic flexible mechanical arm, so that the method has stronger robustness and higher control precision.
Description
Technical field:
the invention belongs to the field of pneumatic servo systems and the field of anti-interference control, and particularly relates to a limited-time anti-interference control method for a universal pneumatic flexible mechanical arm.
The background technology is as follows:
with rapid development of robot technology, the mechanical arm is widely applied to the field of intelligent manufacturing. In order to meet the requirements of the intelligent manufacturing site on flexibility and flexibility of the mechanical arm, the universal pneumatic flexible mechanical arm driven by pneumatic artificial muscles is designed, and the functions of grabbing, carrying and the like are realized.
The universal pneumatic flexible mechanical arm is a multi-degree-of-freedom connecting rod mechanism connected by a universal joint, and the kinematic analysis model and the dynamic model of the universal pneumatic flexible mechanical arm are established based on the structural characteristics of the universal pneumatic flexible mechanical arm, so that the universal pneumatic flexible mechanical arm has higher research significance and research value; however, the uncertainty of the model in the model has a large influence on the accurate control of the universal pneumatic flexible mechanical arm, and in most of the existing researches, the detailed modeling is rarely performed on the universal pneumatic flexible mechanical arm, and meanwhile, the influence of the uncertainty of the model on the position control of the pneumatic flexible mechanical arm is considered, so that the steady-state performance and the transient-state performance of the system are reduced, and the expected control effect cannot be achieved.
The invention comprises the following steps:
the invention provides a limited-time anti-interference control method for a universal pneumatic flexible mechanical arm, which establishes a kinematic analysis model and a dynamic model of the universal pneumatic flexible mechanical arm, and aims at the uncertainty of the model, the limited-time control method is designed to compensate the influence of the model on the position control precision of the pneumatic flexible mechanical arm, so that the accurate space track tracking of the pneumatic flexible mechanical arm in the limited time is realized.
In order to achieve the above purpose, the technical scheme adopted by the invention is as follows: 1. the limited-time anti-interference control method for the universal pneumatic flexible mechanical arm is characterized by comprising the following steps of:
simplifying the universal pneumatic flexible mechanical arm into a connecting rod model by using a D-H method, and performing kinematic analysis to obtain a kinematic analysis model and the positions of the tail end points of the universal pneumatic flexible mechanical arm;
based on a kinematic analysis model of the universal pneumatic flexible mechanical arm, establishing a dynamic model of the universal pneumatic flexible mechanical arm by using an Euler-Lagrange method;
taking uncertainty of the universal pneumatic flexible mechanical arm as disturbance, and establishing a second-order mathematical model of the pneumatic flexible mechanical arm;
and designing a finite time expansion state observer to estimate disturbance, and designing a finite time backstepping controller based on a disturbance estimated value to compensate the influence of the disturbance on the system.
Preferably, the step of simplifying the universal pneumatic flexible mechanical arm into a connecting rod model by using a D-H method and performing kinematic analysis to obtain the kinematic analysis model and the position of the end point of the universal pneumatic flexible mechanical arm includes:
the mechanical structure based on the universal pneumatic flexible mechanical arm simplifies the universal pneumatic flexible mechanical arm into a six-link model by using a D-H method, and a transfer matrix A is introduced for acquiring the position of the tail end point of the mechanical arm i ；
Wherein alpha is i-1 Is the torsion angle of the ith-1 connecting rod; d, d i Is the offset distance of the ith link; θ i Is the ith joint angle; according to the transfer matrix, the transfer matrix of each connecting rod of the universal pneumatic flexible mechanical arm is obtained as follows:
multiplying the seven transfer matrices to obtain the position of the tail end point of the mechanical arm as follows:
wherein,,
wherein: p (P) x (t)、P y (t)、P z (t) is the coordinate of X, Y, Z axis of the terminal point in the space coordinate system, c i ＝cosθ i (t)，s i ＝sinθ i (t)，θ i (t) is the deflection angle of the ith joint, where i=1, 2,3,4,5,6, l is the arm link length.
Preferably, the step of establishing the dynamic model of the universal pneumatic flexible mechanical arm by using the euler-lagrangian method based on the kinematic analysis model of the universal pneumatic flexible mechanical arm comprises the following steps:
the Euler-Lagrangian equation is expressed as follows:
based on Euler-Lagrange equation and kinematic analysis, the dynamic model of the universal pneumatic flexible mechanical arm is expressed as:
wherein:
is an inertial matrix, ++>
Is a coriolis force matrix,/->
Is a gravity matrix; m is the mass of the connecting rod; g T ＝[-g，0，0，0]Is a gravitational acceleration matrix;
is a centroid distance matrix;
wherein: upj = a1a2. Qaj. Ap,
upjk=a1a2..qaj..qak..ap are three intermediate variables;
is two parameter matrices;
based on the driving mode of the universal pneumatic flexible mechanical arm, the dynamic model is transformed, and each element is written into a vector form:
wherein:
wherein θ (t) is the vector representation of the deflection angle, D-1 (t) is the vector representation of Dij, C (t) is the vector representation of Cijk, g (t) is the vector representation of Gi, k0 is the scaling factor of the electrical proportional valve, u0 is the preloaded voltage, L0 is the pneumatic artificial muscle length, R is the force arm, b is the total length of the mesh, N is the number of turns around the outer mesh, ui (t) is the control voltage of the ith joint, where i=1, 2,3,4,5,6; i6 is the identity matrix.
Preferably, the step of establishing the second-order mathematical model of the pneumatic flexible mechanical arm taking the uncertainty of the universal pneumatic flexible mechanical arm as a disturbance includes:
the second-order mathematical model of the universal pneumatic flexible mechanical arm is formed by combining the kinematic analysis model and the dynamic model:
in the method, in the process of the invention,
wherein,,
is D -1 Elements of (t)/(x)>
Is->
Elements b of (b) kk (t)/>
Element f of (2) k Is P x First order partial derivative of (t), g k Is P y First order partial derivative of (t), f kj Is P x First derivative of (t), g kj Is P x (t) a first derivative; omega (t) is the uncertainty of the universal pneumatic flexible mechanical arm.
Preferably, the step of designing the finite time expansion state observer estimates the disturbance and designing the finite time backstepping controller based on the disturbance estimation value, and compensating the influence of the disturbance on the system includes:
the finite time extended state observer is:
in the method, in the process of the invention,
the finite time backstepping controller is:
wherein B is 0 Is a system parameter; k (k) p Is a positive adjustable parameter; sliding die face sigma 1 (t) is designed to:
wherein sigma 2 (t) is sigma 1 The first derivative of (t),
is a virtual variable of the back-stepping method, wherein +.>
Is a slip form surface sigma 1 An error between (t) and 0;
wherein e (t) =x 1 (t)-v 1 (t) is the position error, k of the universal pneumatic flexible mechanical arm 1 、k 2 、k 3 、k 4 、ζ、ξ 1 Adjustable parameters, v, with gamma, p and q being positive 0 (t) is the desired position, v 0 (t) obtaining v via a tracking differentiator 1 (t)。
Preferably, the step of designing the finite time expansion state observer to estimate disturbance and designing the finite time backstepping controller based on the disturbance estimation value to compensate the influence of the disturbance on the system further comprises: and carrying out convergence analysis on the limited-time extended state observer by adopting a Liepunov function and carrying out convergence analysis on the limited-time backstepping controller by adopting the Liepunov function.
Preferably, the convergence analysis of the finite time extended state observer by adopting the Liapunov function specifically includes:
the design of the Lyapunov equation for the finite time extended state observer proves the finite time convergence, and the error system of the finite time extended state observer is given as follows:
design demonstration of required related variables
The derivative is obtained by:
wherein,,
the lyapunov equation was designed as follows:
v(t)＝ε T (t)Pε(t)
deriving the lisapunov function equation v (t):
will be
The following is carried into the above formula:
the convergence time satisfies:
wherein:
η 1 ∈(0，r 1 )，η 2 ∈(0，r 2 ) Is two bounded constants; v (e) 0 ) Is the initial value of the lyapunov function;
preferably, the convergence analysis of the finite time back-step controller by using the Liapunov function specifically includes:
two lyapunov functions are designed using the back-stepping method, the first lyapunov function being designed as:
for V 1 (t) derivative and combination
Obtaining:
wherein,,
is a slip form surfaceFirst derivative sigma of (2) 2 (t) and virtual variable alpha 1 An error between (t);
when (when)
When there is +.>
I.e. < ->
The second lyapunov function is designed based on this as:
for V 2 (t) finite time back-step controllers that derive and bring into design are available:
the convergence time satisfies:
compared with the prior art, the invention has the following beneficial effects:
the invention establishes the kinematic analysis model and the dynamic model of the universal pneumatic flexible mechanical arm, and designs the finite time control method to compensate the influence of the model uncertainty on the position control precision of the pneumatic flexible mechanical arm, thereby having stronger robustness, being easy for engineering realization, having higher control precision and ensuring the stable, accurate and fast control performance.
Description of the drawings:
FIG. 1 is a schematic diagram of a limited time anti-interference control method for a universal pneumatic flexible mechanical arm.
FIG. 2 is a flow chart of a method for controlling the limited time anti-interference of a flexible pneumatic mechanical arm according to a preferred embodiment of the present invention.
Fig. 3 is a kinematic analysis diagram of the universal pneumatic flexible mechanical arm of the invention.
Fig. 4 is a graph of position tracking for a gimbaled pneumatic flexible mechanical arm of the present invention.
Fig. 5 is a graph of the state 1 estimated signal of the finite time extended state observer according to the present invention.
The specific embodiment is as follows:
for the purpose of clarifying the technical objects of the present invention, the technical scheme of the present invention is explained in detail with reference to the following drawings and specific embodiments.
Fig. 1 is a schematic diagram of the method of the invention, and illustrates a limited-time anti-interference control method for a universal pneumatic flexible mechanical arm.
The following describes in detail, but not by way of limitation, modeling of a universal pneumatic flexible mechanical arm and a control algorithm designed for model uncertainty of the mechanical arm according to the present invention with reference to fig. 1-4.
A limited-time anti-interference control method for a universal pneumatic flexible mechanical arm comprises the following steps:
s100, simplifying the universal pneumatic flexible mechanical arm into a connecting rod model by using a D-H method, and performing kinematic analysis to obtain a kinematic analysis model and the positions of the tail end points of the universal pneumatic flexible mechanical arm;
s200, based on a kinematic analysis model of the universal pneumatic flexible mechanical arm, establishing a dynamic model of the universal pneumatic flexible mechanical arm by using an Euler-Lagrange method;
s300, taking uncertainty of the universal pneumatic flexible mechanical arm as disturbance, and establishing a second-order mathematical model of the pneumatic flexible mechanical arm;
s400, designing a finite time expansion state observer to estimate disturbance, and designing a finite time backstepping controller based on a disturbance estimated value to compensate the influence of the disturbance on a system. The invention establishes the kinematic analysis model and the dynamic model of the universal pneumatic flexible mechanical arm, and designs the finite time control method to compensate the influence of the model uncertainty on the position control precision of the pneumatic flexible mechanical arm, thereby having stronger robustness, being easy for engineering realization, having higher control precision and ensuring the stable, accurate and fast control performance.
In this embodiment, the step of simplifying the universal pneumatic flexible mechanical arm into the connecting rod model by using the D-H method and performing the kinematic analysis to obtain the kinematic analysis model and the position of the end point of the universal pneumatic flexible mechanical arm includes:
the mechanical structure based on the universal pneumatic flexible mechanical arm is simplified into a six-link model by using a D-H method, and the D-H table is listed as follows:
| connecting rod | a i | a i | d i | θ i |
| 1 | 0 | 0 | 0 | |
| 2 | 0 | -90° | 0 | θ 2 |
| 3 | l | 90° | 0 | |
| 4 | 0 | -90° | 0 | θ 4 |
| 5 | l | 90° | 0 | θ 5 |
| 6 | 0 | -90° | 0 | θ 6 |
In the above formula, a i Is the i-th link length; alpha i Is the torsion angle of the ith connecting rod; d, d i Is the offset distance of the ith link; θ i Is the i-th joint angle.
In order to obtain the position of the tail end point of the mechanical arm, a transfer matrix A is introduced i ；
Wherein alpha is i-1 Is the torsion angle of the ith-1 connecting rod; d, d i Is the offset distance of the ith link; θ i Is the ith joint angle; root of Chinese characterAccording to the D-H table and the transfer matrix, the transfer matrix of each connecting rod of the universal pneumatic flexible mechanical arm can be obtained as follows:
multiplying the seven transfer matrices to obtain the position of the tail end point of the mechanical arm as follows:
wherein,,
wherein: p (t) is the position of the end point of the universal pneumatic flexible mechanical arm, px (t), py (t), pz (t) are the coordinates of the end point on the X, Y, Z axis in the spatial coordinate system, ci=cos θi (t), si=sin θi (t), θi (t) is the deflection angle of the ith joint, where i=1, 2,3,4,5,6, l is the mechanical arm link length.
In this embodiment, S200, based on a kinematic analysis model of a universal pneumatic flexible mechanical arm, the step of establishing a dynamic model of the universal pneumatic flexible mechanical arm by using the euler-lagrangian method includes:
the Euler-Lagrangian equation is expressed as follows:
based on Euler-Lagrange equation and kinematic analysis, the dynamic model of the universal pneumatic flexible mechanical arm is expressed as:
wherein:
is an inertial matrix, ++>
Is a coriolis force matrix,/->
Is a gravity matrix; m is the mass of the connecting rod; g T ＝[-g，0，0，0]Is a gravitational acceleration matrix;
is a centroid distance matrix;
wherein:
U pjk ＝A 1 A 2 L QA j L QA k LA p is three intermediate variables;
is two parameter matrices;
based on the driving mode of the universal pneumatic flexible mechanical arm, the dynamic model is transformed, and each element is written into a vector form:
wherein:
U 0 (t)＝[u 1 (t)，u 2 (t)，u 3 (t)+u 4 (t)，u 4 (t)-u 3 (t)，u 5 (t)，u 6 (t)] T ，
wherein θ (t) is a vector representation of the deflection angle, D -1 (t) is D ij Is c (t) is c ijk G (t) is G i Is represented by a vector of k 0 Is the proportionality coefficient of the electric proportioning valve, u 0 To preload the voltage, L 0 The pneumatic artificial muscle is the original length, R is a force arm, b is the total length of the woven mesh, and N is the number of circles of the outer woven mesh; u (u) i (t) is the control voltage of the ith joint, where i = 1,2,3,4,5,6; i 6 Is an identity matrix.
In this embodiment, S300, taking uncertainty of the universal pneumatic flexible mechanical arm as disturbance, the step of establishing a second-order mathematical model of the pneumatic flexible mechanical arm includes:
the second-order mathematical model of the universal pneumatic flexible mechanical arm is formed by combining the kinematic analysis model and the dynamic model:
in the method, in the process of the invention,
wherein,,
is D -1 Elements of (t)/(x)>
Is->
Elements b of (b) kk (t)/>
Element f of (2) k Is P x First order partial derivative of (t), g k Is P y First order partial derivative of (t), f kj Is P x First derivative of (t), g kj Is P x (t) a first derivative; omega (t) is the uncertainty of the universal pneumatic flexible mechanical arm. The invention establishes the kinematic analysis model and the dynamic model of the universal pneumatic flexible mechanical arm, and designs the finite time control method to compensate the influence of the model uncertainty on the position control precision of the pneumatic flexible mechanical arm, thereby having stronger robustness, being easy for engineering realization, having higher control precision and ensuring the stable, accurate and fast control performance.
In this embodiment, the step of designing the finite-time expansion state observer to estimate the disturbance and designing the finite-time backstepping controller based on the disturbance estimation value to compensate the influence of the disturbance on the system includes:
the finite time extended state observer is:
in the method, in the process of the invention,
the finite time backstepping controller is:
wherein B is 0 Is a system parameter; k (k) p Is a positive adjustable parameter; sliding die face sigma 1 (t) is designed to:
wherein sigma 2 (t) is sigma 1 The first derivative of (t),
is a virtual variable of the back-stepping method, wherein +.>
Is a slip form surface sigma 1 An error between (t) and 0;
wherein e (t) =x 1 (t)-v 1 (t) is the position error, k of the universal pneumatic flexible mechanical arm 1 、k 2 、k 3 、k 4 、ζ、ξ 1 Adjustable parameters, v, with gamma, p and q being positive 0 (t) is the desired position, v 0 (t) obtaining v via a tracking differentiator 1 (t)。
Preferably, the step of designing the finite time expansion state observer to estimate disturbance and designing the finite time backstepping controller based on the disturbance estimation value to compensate the influence of the disturbance on the system further comprises: s500, carrying out convergence analysis on the finite time expansion state observer by adopting a Liepunov function and carrying out convergence analysis on the finite time backstepping controller by adopting the Liepunov function.
In this embodiment, S501 performs convergence analysis on the finite time extended state observer by using a li-apunov function, and specifically includes:
the design of the Lyapunov equation for the finite time extended state observer proves the finite time convergence, and the error system of the finite time extended state observer is given as follows:
design demonstration of required related variables
The derivative is obtained by:
wherein,,
the lyapunov equation was designed as follows:
v(t)＝ε T (t)Pε(t)
deriving the lisapunov function equation v (t):
will be
The following is carried into the above formula:
the convergence time satisfies:
wherein:
η 1 ∈(0，r 1 )，η 2 ∈(0，r 2 ) Is two bounded constants; v (e) 0 ) Is the initial value of the lyapunov function;
thus demonstrating a limited time convergence of ε (t).
In this embodiment, S502 performs convergence analysis on the finite time backstepping controller by using a li-apunov function, which specifically includes:
two design lyapunov functions are designed using the back-stepping method, the first lyapunov function being designed as:
for V 1 (t) derivative and combination
Obtaining:
wherein,,
for the first derivative sigma of the slide surface 2 (t) and virtual variable alpha 1 An error between (t);
when (when)
When there is +.>
I.e. < ->
The second lyapunov function is designed based on this as:
for V 2 (t) finite time back-step controllers that derive and bring into design are available:
the convergence time satisfies:
thus, a limited time convergence of the limited time backstepping controller can be obtained. Therefore, the limited-time anti-interference control method for the universal pneumatic flexible mechanical arm is stable and effective.
In a further preferred embodiment of the invention, in order to verify that the limited-time anti-interference control method for the universal pneumatic flexible mechanical arm provided by the invention has better control performance, the verification of the invention is provided, so that the universal pneumatic flexible mechanical arm has higher control precision and better anti-interference capability under the control method provided by the invention, and the method specifically comprises the following steps:
the initial position of the end of the universal pneumatic flexible mechanical arm is set to be (0, 67.5). The universal pneumatic flexible mechanical arm is driven by 12 McKibben type pneumatic artificial muscles, an electric proportional valve voltage signal is given to an industrial personal computer board card configured through an experimental platform, the electric proportional valve controls the internal air pressure of the pneumatic artificial muscles, so that the universal pneumatic flexible mechanical arm generates corresponding deflection, the deflection angle is collected by an angle sensor, and the position of an end effector is obtained through space position analysis and is used as the output of the system.
The control target is set as follows:
the two step signals are respectively used as expected signals of two shafts of the end effector of the universal pneumatic flexible mechanical arm, and the amplitudes are respectively: v 01 ＝18cm、v 02 ＝11.25cm；
The position curve of the output of the universal pneumatic flexible mechanical arm is shown in fig. 4 given the different desired position signals of the two axes. Wherein, the solid line represents the expected position signals input by two shafts, and the amplitude is 18cm and 11.25cm respectively; the dashed lines represent the actual position outputs of the two axes at the desired positions of 18cm and 11.25cm, respectively. As can be seen from fig. 4, the actual positions of the two axes of the universal pneumatic flexible mechanical arm can accurately track the desired position without overshoot.
The desired position signals on the two axes are v 01 ＝18cm、v 02 When 11.25cm, the finite time extended state observer outputs the position x 1 Estimate z of (t) 1 (t) is shown in FIG. 4. Wherein z is 11 (t) represents the output position x for the first axis at an 18cm desired position signal input 11 An estimated curve of (t); z 12 (t) represents the output position x for the second axis at 11.25cm desired position signal input 12 An estimated curve of (t). As can be seen from fig. 5, the finite time extended state observer can quickly and stably estimate the actual output position x when different desired position signals are inputted to both axes 1 (t)。
Variations and modifications of the above embodiments will occur to those skilled in the art to which the invention pertains from the foregoing disclosure and teachings. Therefore, the present invention is not limited to the above-described embodiments, but is intended to be capable of modification, substitution or variation in light thereof, which will be apparent to those skilled in the art in light of the present teachings.
Claims (8)
1. The limited-time anti-interference control method for the universal pneumatic flexible mechanical arm is characterized by comprising the following steps of:
simplifying the universal pneumatic flexible mechanical arm into a connecting rod model by using a D-H method, and performing kinematic analysis to obtain a kinematic analysis model and the positions of the tail end points of the universal pneumatic flexible mechanical arm;
based on a kinematic analysis model of the universal pneumatic flexible mechanical arm, establishing a dynamic model of the universal pneumatic flexible mechanical arm by using an Euler-Lagrange method;
taking uncertainty of the universal pneumatic flexible mechanical arm as disturbance, and establishing a second-order mathematical model of the pneumatic flexible mechanical arm;
and designing a finite time expansion state observer to estimate disturbance, and designing a finite time backstepping controller based on a disturbance estimated value to compensate the influence of the disturbance on the system.
2. The method for controlling the limited-time anti-interference of the universal pneumatic flexible mechanical arm according to claim 1, wherein the step of simplifying the universal pneumatic flexible mechanical arm into a connecting rod model by using a D-H method and performing kinematic analysis to obtain a kinematic analysis model and the position of the end point of the universal pneumatic flexible mechanical arm comprises the following steps:
the mechanical structure based on the universal pneumatic flexible mechanical arm simplifies the universal pneumatic flexible mechanical arm into a six-link model by using a D-H method, and a transfer matrix A is introduced for acquiring the position of the tail end point of the mechanical arm i ；
Wherein a is i-1 Is the torsion angle of the ith-1 connecting rod; d, d i Is the offset distance of the ith link; θ i Is the ith joint angle; according to the transfer matrix, the transfer matrix of each connecting rod of the universal pneumatic flexible mechanical arm is obtained as follows:
multiplying the seven transfer matrices to obtain the position of the tail end point of the mechanical arm as follows:
wherein,,
wherein: p (P) x (t)、P y (t)、P z (t) is the coordinate of X, Y, Z axis in the spatial coordinate system of the terminal point, C i ＝cosθ i (t)，S i ＝sinθ i (t)，θ i (t) is the angle of deflection of the ith joint, where i=1, 2,3,4,5,6,
is the length of the connecting rod of the mechanical arm.
3. The method for controlling the limited-time anti-interference of the universal pneumatic flexible mechanical arm according to claim 1, wherein the step of establishing the dynamic model of the universal pneumatic flexible mechanical arm by using the euler-lagrangian method based on the kinematic analysis model of the universal pneumatic flexible mechanical arm comprises the following steps:
the Euler-Lagrangian equation is expressed as follows:
based on Euler-Lagrange equation and kinematic analysis, the dynamic model of the universal pneumatic flexible mechanical arm is expressed as:
wherein:
is an inertial matrix, ++>
Is a matrix of the coriolis force,
is a gravity matrix; m is the mass of the connecting rod; g T ＝[-g,0,0,0]Is a gravitational acceleration matrix;
is a centroid distance matrix;
wherein: u (U) pj ＝A 1 A 2 ...QA j ...A p ,
U pjk ＝A 1 A 2 ...QA j ...QA k ...A p Is three intermediate variables;
is two parameter matrices;
based on the driving mode of the universal pneumatic flexible mechanical arm, the dynamic model is transformed, and each element is written into a vector form:
wherein:
U 0 (t)＝[u 1 (t),u 2 (t),u 3 (t)+u 4 (t),u 4 (t)-u 3 (t),u 5 (t),u 6 (t)] T ,
wherein θ (t) is a vector representation of the deflection angle, D -1 (t) is D ij Is C (t) is C ijk G (t) is G i Is represented by a vector of k 0 Is the proportionality coefficient of the electric proportioning valve, u 0 To preload the voltage, L 0 Is the original length of pneumatic artificial muscle, R is the force arm, b is the total length of the woven mesh, N is the number of circles around the outer woven mesh, u i (t) is the control voltage of the ith joint, where i = 1,2,3,4,5,6; i 6 Is an identity matrix.
4. The method for controlling finite time anti-interference of a universal pneumatic flexible mechanical arm according to claim 1, wherein the step of taking uncertainty of the universal pneumatic flexible mechanical arm into consideration as disturbance and establishing a second-order mathematical model of the pneumatic flexible mechanical arm comprises the steps of:
the second-order mathematical model of the universal pneumatic flexible mechanical arm is formed by combining the kinematic analysis model and the dynamic model:
in the method, in the process of the invention,
wherein,,
is D -1 Elements of (t)/(x)>
Is->
Elements b of (b) kk (t) is->
Element f of (2) k Is P x First order partial derivative of (t), g k Is P y First order partial derivative of (t), f kj Is P x First derivative of (t), g kj Is P x (t) a first derivative; omega (t) is the uncertainty of the universal pneumatic flexible mechanical arm.
5. The method for controlling finite time anti-interference of a universal pneumatic flexible mechanical arm according to claim 1, wherein the step of designing the finite time extended state observer to estimate the disturbance and designing the finite time backstepping controller based on the disturbance estimated value to compensate the influence of the disturbance on the system comprises:
the finite time extended state observer is:
in the method, in the process of the invention,
and is also provided with
β 1 ，β 2 And beta 3 Positive adjustable parameter e 1 (t)＝z 1 (t)-x 1 (t)、e 2 (t)＝z 2 (t)-x 2 (t)、e 3 (t)＝z 3 (t)-x 3 (t) is an error variable in the finite time extended state observer;
the finite time backstepping controller is:
wherein B is 0 Is a system parameter; k (k) p Is a positive adjustable parameter; sliding die face sigma 1 (t) is designed to:
wherein sigma 2 (t) is sigma 1 The first derivative of (t),
is a virtual variable of a back-stepping method, wherein
Is a slip form surface sigma 1 An error between (t) and 0;
wherein e (t) =x 1 (t)-v 1 (t) is the position error, k of the universal pneumatic flexible mechanical arm 1 、k 2 、k 3 、k 4 、ζ、ξ 1 Adjustable parameters, v, with gamma, p and q being positive 0 (t) is the desired position, v 0 (t) obtaining v via a tracking differentiator 1 (t)。
6. The method for controlling finite time anti-interference of a universal pneumatic flexible mechanical arm according to claim 1, wherein the step of designing the finite time extended state observer to estimate disturbance and designing the finite time backstepping controller based on the disturbance estimated value to compensate the influence of disturbance on the system further comprises: and carrying out convergence analysis on the limited-time extended state observer by adopting a Liepunov function and carrying out convergence analysis on the limited-time backstepping controller by adopting the Liepunov function.
7. The method for controlling finite time anti-interference of a universal pneumatic flexible mechanical arm according to claim 6, wherein the method for performing convergence analysis on the finite time extended state observer by using a liaeprunov function specifically comprises:
the design of the Lyapunov equation for the finite time extended state observer proves the finite time convergence, and the error system of the finite time extended state observer is given as follows:
design demonstration of required related variables
The derivative is obtained by:
wherein,,
Δ(t)＝[0 0 -g(t)] T
the lyapunov equation was designed as follows:
v(t)＝ε T (t)Pε(t)
deriving the lisapunov function equation v (t):
will be
λ min (P)||ε(t)|| 2 ≤v≤λ max (P)||ε(t)|| 2 The following is carried into the above formula:
the convergence time satisfies:
wherein:
η 1 ∈(0，r 1 )，η 2 ∈(0，r 2 ) Is two bounded constants; v (e) 0 ) Is the initial value of the lyapunov function;
8. the method for finite time anti-interference control of a universal pneumatic flexible mechanical arm according to claim 6, wherein the finite time backstepping controller is subjected to convergence analysis by adopting a lisapunov function, and specifically comprises the following steps:
two lyapunov functions are designed using the back-stepping method, the first lyapunov function being designed as:
for V 1 (t) derivative and combination
Obtaining:
wherein,,
for the first derivative sigma of the slide surface 2 (t) and virtual variable alpha 1 An error between (t);
when (when)
When there is +.>
I.e. < ->
The second lyapunov function is designed based on this as:
for V 2 (t) finite time back-step controllers that derive and bring into design are available:
the convergence time satisfies:
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| Publication number | Priority date | Publication date | Assignee | Title |
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| CN117590754A (en) * | 2024-01-18 | 2024-02-23 | 北京理工大学 | Intelligent learning output regulation and control method of robot system |
| CN117885103A (en) * | 2024-03-14 | 2024-04-16 | 山东大学 | Flexible mechanical arm control method and system based on reduced-order expanded state observer |
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| Publication number | Priority date | Publication date | Assignee | Title |
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| CN117590754A (en) * | 2024-01-18 | 2024-02-23 | 北京理工大学 | Intelligent learning output regulation and control method of robot system |
| CN117590754B (en) * | 2024-01-18 | 2024-05-03 | 北京理工大学 | Intelligent learning output regulation and control method of robot system |
| CN117885103A (en) * | 2024-03-14 | 2024-04-16 | 山东大学 | Flexible mechanical arm control method and system based on reduced-order expanded state observer |
| CN117885103B (en) * | 2024-03-14 | 2024-05-17 | 山东大学 | Flexible mechanical arm control method and system based on reduced-order expanded state observer |
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