CN107877511B - Multi-double-connecting-rod mechanical arm containing controller based on output position and design method - Google Patents

Multi-double-connecting-rod mechanical arm containing controller based on output position and design method Download PDF

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CN107877511B
CN107877511B CN201710896354.2A CN201710896354A CN107877511B CN 107877511 B CN107877511 B CN 107877511B CN 201710896354 A CN201710896354 A CN 201710896354A CN 107877511 B CN107877511 B CN 107877511B
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follower
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CN107877511A (en
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杨杨
谈杰
岳东
徐闯
缪松涛
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Nanjing University of Posts and Telecommunications
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Abstract

The invention discloses a multi-double-link mechanical arm containing controller based on output positions and a design method thereof, wherein N double-link mechanical arms containing unknown dynamics are taken as followers, a networked system formed by connecting the followers and M leaders through a one-way topological graph is taken as a controlled object, the leaders form a static convex hull, and the containing controller is designed by utilizing the positions output by respective joints 1 and 2 of the followers, so that the output of the followers is converged in the static convex hull. Compared with the prior art, the invention has the beneficial effects that: an extended state observer is designed in each sub-controller by utilizing the input and the output of a follower to reconstruct the system state of the double-link mechanical arm, and unknown dynamics are compensated in real time through the estimation of an extended state quantity, so that the designed inclusion controller for outputting feedback has immunity; the problem of explosion of computational complexity is overcome; the method can effectively solve the problems of unknown dynamics in the system, simplified complex derivation operation and the like.

Description

Multi-double-connecting-rod mechanical arm containing controller based on output position and design method
Technical Field
The invention belongs to the technical field of industrial process control, and particularly relates to a multi-double-connecting-rod mechanical arm containing controller based on an output position and a design method.
Background
In recent years, coordinated control of multi-robot systems is a research focus in the field of robots today. Compared with a single mechanical arm system, the multi-mechanical arm system has the advantages of higher flexibility, higher reliability and capability of completing more complex tasks. The master/slave control coordination mode of the multi-robot arm system proposed by Kosuge, Ahmad and the like is the earliest control scheme, and is also the simplest and most effective coordination strategy currently applied to the field of industrial robot arms. The method only needs to coordinate one mechanical arm in the system to be designated as a leader, and the other mechanical arms tracking the relative pose of the leader are called a follower group. When a plurality of mechanical arms act as leader roles to guide group movement, the controlled group is enabled to enter the range formed by the leader group or the controlled group is surrounded by the leader, and the control problem is called as a containment control problem. In 2008, m.ji et al proposed a stop-and-go control method for multi-robot involved control problems, which enables multi-robots to enter a target area consisting of multiple leaders. And in a coordinated control network for the multi-double-link mechanical arms, the angular displacement of the joints 1 and 2 of the mechanical arms is used as output quantity, the output quantities of a plurality of leaders form a static convex hull, and the output quantities of other double-link mechanical arms are moved to the static convex hull formed by the leaders according to information transmitted by the leaders and followers. However, in many practical applications, the accurate model of the dual link arm system cannot be obtained, and only part of the system status can be detected. Therefore, the invention utilizes Active Disturbance Rejection Control (ADRC) and inversion technology, so that the research on the output feedback containing Control of the multi-double-linkage mechanical arm has the most direct practical significance.
In the n-order affine system control having an unknown dynamic state and a partial system state that is not available, zhao et al proposes an n + 1-order nonlinear Extended State Observer (ESO) that estimates the unknown dynamic state in the system and reconstructs the system state by input and output of a controlled object, taking the unknown dynamic state as the system Extended state. The ESO was originally proposed by the korean kyo professor, whereby it is possible to estimate the total disturbance composed of various unknown uncertainty factors in the controlled object in real time and compensate the system to the form of the nominal model of the integral serializer for control. Along with that, Huangyi et al perfect ESO convergence and stability, and Yangxianxia et al discuss the range of perturbations that can be observed by ESO and that present bounded observation errors. In 2011, guobaozhu et al designed an ESO for a class of nonlinear systems with uncertainty and strictly demonstrated the convergence of the ESO. In 2002, guobaozhu et al also proposed a simple linear Tracking Differentiator (TD), which was also proposed by professor kyoto koreani in 1989 at the earliest time and given an analysis of its convergence. TD makes a function indistinguishable in the classical sense but has its generalized derivative, a property that is widely applied to the derivation of complex nonlinear functions. In 2014, cheng chun hua et al utilized TD to obtain the derivative of the virtual control law in the inversion design, thereby overcoming the problem of computational complexity explosion. The inversion technique is proposed by Kanellakopoulos et al in 1991, and is to construct a corresponding Lyapunov function for each subsystem and design a virtual control law to stabilize the previous subsystem, but the difficulty in deriving the virtual control law cannot be avoided. Yoo et al studied the inclusion control strategy of an uncertain nonlinear strict feedback system under a directed graph topology, and designed a distributed adaptive inclusion controller to drive a follower into the convex hull of a leader by using an inversion technique. Wang Wei in 2015 designed a state feedback controller for an uncertain nonlinear multi-agent system by utilizing an inversion technique, and avoided the difficult problem of complex derivation of a virtual control law by a first-order filter.
In practical application, when the multi-double-link mechanical arm is subjected to inclusion control, because unknown dynamic and part of system states are unavailable in each double-link mechanical arm system and the difficulty that derivation of a virtual control law is complex in inversion design is utilized, the research on inclusion control of output feedback of the multi-double-link mechanical arm has important theoretical significance and practical significance.
Disclosure of Invention
According to the invention, according to the imperfection and the deficiency of the prior background technology, the inclusion control of output feedback of the multi-double-link mechanical arm under a directed communication network is realized based on the active disturbance rejection and inversion technology, an extended state observer is designed in each sub-controller by utilizing the input and the output of a follower to reconstruct the system state of the double-link mechanical arm, and unknown dynamics are compensated in real time through the estimation of the extended state quantity, so that the designed inclusion controller of the output feedback has disturbance rejection. Further, the derivative of the virtual control law in the sub-controllers is estimated by means of a tracking differentiator, so that the problem of computation complexity explosion is overcome. The controller of the multi-double-connecting-rod mechanical arm based on active disturbance rejection and inversion technology only depends on the output position, and the problems of unknown dynamics in a system, simplified complex derivation operation and the like can be effectively solved.
In order to achieve the technical purpose, the invention adopts the following technical scheme:
multiple-double-connection based on output positionThe rod mechanical arm comprises a controller, N double-link mechanical arms containing unknown dynamics are regarded as followers, a networked system formed by connecting the N double-link mechanical arms with M leaders through a one-way topological graph is used as a controlled object, the leaders form a static convex hull, the controllers are designed by utilizing the positions output by joints 1 and joints 2 of the followers, so that the output of the followers is converged in the static convex hull, the controller structure of the ith (i is not less than 1 and not more than N) follower comprises a1 st sub-controller and a2 nd sub-controller, and the input ends of the 1 st and the 2 nd sub-controllers and a directed graph
Figure BDA0001422240940000021
The output ends of the 2 sub-controllers are all connected with the input end of the ith follower,
the 1 st sub-controller comprises a first extended state observer unit, an error surface si,1,1The device comprises an arithmetic unit, a first nonlinear arithmetic unit, a first comparator unit, a first tracking differentiator unit and a second nonlinear arithmetic unit; two input ends of the first extended state observer unit are respectively the output y of the ith follower joint 1i,1And the output u of the second non-linear operation uniti,1(ii) a Error surface si,1,1The input ends of the arithmetic units are directed graphs respectively
Figure BDA0001422240940000022
Middle observer state
Figure BDA0001422240940000023
Output y of kth leader double link robot armk,1,dAdjacent communication aijAnd the output of the first extended state observer unit
Figure BDA0001422240940000031
The input ends of the first nonlinear operation units are directed graphs respectively
Figure BDA0001422240940000032
Middle observer state
Figure BDA0001422240940000033
Output y of jth follower (j is more than or equal to 1 and less than or equal to N)j,1Adjacent communication aijOutput y of i-th follower joint 1i,1Output of the first extended state observer unit
Figure BDA0001422240940000034
Sum error surface si,1,1An output of the arithmetic unit; the first comparator unit being an error surface si,1,2The input end of which is the output of the first extended state observer unit
Figure BDA0001422240940000035
And the output x of the first non-linear operation uniti,1,2,d(ii) a The input end of the first tracking differentiator unit is the output x of the first nonlinear operation uniti,1,2,d(ii) a The input end of the second nonlinear operation unit is a directed graph
Figure BDA0001422240940000036
Intermediate adjacent communication aijOutput y of i-th follower joint 1i,1Error surface si,1,1Output of arithmetic unit, output v of first tracking differentiator uniti,1,2An output s of the first comparator uniti,1,2And the output of the first extended state observer unit
Figure BDA0001422240940000037
The 2 nd sub-controller comprises a second extended state observer unit, an error surface si,2,1The device comprises an arithmetic unit, a third nonlinear arithmetic unit, a second comparator unit, a second tracking differentiator unit and a fourth nonlinear arithmetic unit; two input ends of the second extended state observer unit are respectively the output y of the ith follower joint 2i,2And the output u of the fourth non-linear operation uniti,2(ii) a Error surface si,2,1The input ends of the arithmetic units are directed graphs respectively
Figure BDA0001422240940000038
Middle observer state
Figure BDA0001422240940000039
Output y of kth leader double link robot armk,2,dAdjacent communication aijAnd the output of the second extended state observer unit
Figure BDA00014222409400000310
The input ends of the third nonlinear operation units are directed graphs respectively
Figure BDA00014222409400000311
Middle observer state
Figure BDA00014222409400000312
Output y of jth follower joint 2j,2Adjacent communication aijOutput y of the i-th follower joint 2i,2The output of the second extended state observer unit
Figure BDA00014222409400000313
Sum error surface si,2,1An output of the arithmetic unit; the second comparator unit being an error surface si,2,2The input end of which is the output of the second extended state observer unit
Figure BDA00014222409400000314
And the output x of the third non-linear operation uniti,2,2,d(ii) a The input end of the second tracking differentiator unit is the output x of the third nonlinear operation uniti,2,2,d(ii) a The input end of the fourth nonlinear operation unit is a directed graph
Figure BDA00014222409400000315
Intermediate adjacent communication aijOutput y of the i-th follower joint 2i,2Error surface si,2,1Output of arithmetic unit, output v of second tracking differentiator uniti,2,2The output s of the second comparator uniti,2,2And the output of the second extended state observer unit
Figure BDA00014222409400000316
The multi-double-link mechanical arm based on the output position comprises a controller, a multi-agent network formed by connecting follower groups consisting of N double-link mechanical arms containing unknown dynamic states and M leaders through a one-way topological graph is considered, and each follower is communicated with at least one leader; the information communication between them can be made by directed graphs
Figure BDA00014222409400000317
Is shown in which
Figure BDA0001422240940000041
Is a set of nodes, niIndicating a double link arm i, njA double link robot arm j is shown,
Figure BDA0001422240940000042
is a collection of edges that are to be considered,
Figure BDA0001422240940000043
indicating that agent j can directly obtain the information of agent i;
Figure BDA0001422240940000044
is a contiguous matrix, aijThe definition is as follows:
Figure BDA0001422240940000045
the contiguous set of nodes i is defined as
Figure BDA0001422240940000046
Directed graph
Figure BDA0001422240940000047
Laplacian matrix of
Figure BDA0001422240940000048
The definition is as follows:
Figure BDA0001422240940000049
laplace matrix
Figure BDA00014222409400000410
Wherein D ═ diag [ D [ ]1,…,dN]As directed graphs
Figure BDA00014222409400000411
The degree matrix of (c) is,
Figure BDA00014222409400000412
the follower has at least one adjacent node, and the leader has no adjacent node, the Laplace matrix can
Figure BDA00014222409400000413
And (3) decomposition:
Figure BDA00014222409400000414
wherein,
Figure BDA00014222409400000415
the mathematical model of the ith double-link mechanical arm in the follower is:
Figure BDA00014222409400000416
in the formula, qi,1Is the angular position, q, of the ith double link robot arm joint 1i,2Is the angular position, q, of the ith double link robot arm joint 2i＝[qi,1 qi,2]TIs the connection angular position;
Figure BDA00014222409400000417
respectively, connecting angular velocity and acceleration, taui＝[τi,1 τi,2]TIs a control input, the system parameters are selected as follows：
Figure BDA00014222409400000418
In the formula,
Figure BDA00014222409400000419
a3＝m2l1lc2cos(δ)，a4＝m2l1lc2sin(δ)，I1viscous coefficient of friction for turning the joint 1 to2Viscous coefficient of friction to which the joint 2 turns,/1Is the length of the rod 1, m1Is the mass of the rod 1, m2Is the mass of the rod 2, delta is the angle between the initial joint 2 and the longitudinal axis, lc1Is the distance of the joint 1 from the center of mass of the bar 1, lc2Distance of joint 2 to the center of mass of rod 2, further, let [ x [ ]i,1,1 xi,2,1]T＝[qi,1 qi,2]T，
Figure BDA0001422240940000051
[ui,1 ui,2]T＝[τi,1 τi,2]TThe equation of state for the ith follower can be found:
Figure BDA0001422240940000052
in the formula, yi，1、yi，2Output of the i-th follower joint 1 and joint 2, W, respectivelyi，11＝a1+2a2cos(yi，1)+2a4sin(yi，2)，
Figure BDA0001422240940000053
And
Figure BDA0001422240940000054
the inner unknown dynamic part of the ith follower dual link robotic arm.
The design method of the multi-double-connecting-rod mechanical arm containing controller based on the output position comprises the following steps:
A. design of the 1 st sub-controller:
a1, design of first extended state observer unit:
the two inputs of the first extended state observer unit are divided into the output y of the ith follower joint 1i,1And the output u of the second non-linear operation uniti,1The output signal of the first extended state observer unit is obtained by calculation of the following formula
Figure BDA0001422240940000055
And
Figure BDA0001422240940000056
and
Figure BDA0001422240940000057
estimated signals of the first, second and third order state variables of the i-th follower's joint 1, respectively:
Figure BDA0001422240940000058
wherein T is time, Ti,1,1、Ti,1,2And Ti,1,3All are the gains of the extended state observer, riE (1, ∞) is the parameter to be designed, function gi(τ) is fal composed of linear and fractional power functionsi(τ，δi，j1) function, j ═ 1,2,3, expressed as:
Figure BDA0001422240940000061
wherein, deltai,jE (0,1) is an adjustable parameter, sign () is a symbolic function, and the expression is as follows:
Figure BDA0001422240940000062
a2, error surface si,1,1Designing an arithmetic unit:
error surface si,1,1The input ends of the arithmetic units are directed graphs respectively
Figure BDA0001422240940000063
Middle observer state
Figure BDA0001422240940000064
Output y of kth leader double link robot armk,1,dAdjacent communication aijAnd the output of the first extended state observer unit
Figure BDA0001422240940000065
Obtaining an error surface s by calculation of the following formulai,1,1Output s of arithmetic uniti,1,1：
Figure BDA0001422240940000066
A3, design of first nonlinear arithmetic unit:
the input ends of the first nonlinear operation units are directed graphs respectively
Figure BDA0001422240940000067
Middle observer state
Figure BDA0001422240940000068
Output y of jth follower joint 1j,1Adjacent communication aijOutput y of i-th follower joint 1i,1Output of the first extended state observer unit
Figure BDA0001422240940000069
Sum error surface si,1,1The output of the operation unit is calculated by the following formula to obtain the virtual control law xi,1,2,d：
Figure BDA00014222409400000610
In the formula, Ti,1,1And Tj,1,1In order to extend the gain of the state observer,
Figure BDA00014222409400000611
ri∈(1,∞)、rj∈(1,∞)、qi,1,1parameters to be designed are more than 0;
a4, design of first comparator unit:
the input of the first comparator unit is the output of the first extended state observer unit
Figure BDA00014222409400000612
And the output x of the first non-linear operation uniti,1,2,dAn error surface s is obtained by calculation of the following formulai,1,2：
Figure BDA00014222409400000613
A5, design of the first tracking differentiator unit:
the input end of the first tracking differentiator is the output x of the first nonlinear operation uniti,1,2,dThe output v of the tracking differentiator is obtained by calculation of the following formulai,1,2：
Figure BDA0001422240940000071
In the formula, vi,1,2Is the first non-linear operation unit output xi,1,2,dIs estimated, α ∈ (0,1) is the filter factor, λ > 0 is the speed factor;
design of a6 and second nonlinear operation unit:
the input end of the second nonlinear operation unit is a directed graph
Figure BDA0001422240940000072
Intermediate adjacent communication aijI th follower joint 1Output yi,1Error surface si,1,1Output of arithmetic unit, output v of first tracking differentiator uniti,1,2An output s of the first comparator uniti,1,2And the output of the first extended state observer unit
Figure BDA0001422240940000073
The control law u is obtained by calculation of the following formulai,1：
Figure BDA0001422240940000074
In the formula, Ti,1,2In order to extend the gain of the state observer,
Figure BDA0001422240940000075
ri∈(1,∞)、qi,2,1parameters to be designed are more than 0;
B. design of the 2 nd sub-controller:
b1, design of second extended state observer unit:
the two inputs of the second extended state observer unit are divided into the output y of the ith follower joint 1i,2And the output u of the fourth non-linear operation uniti,2The output signal of the second extended state observer unit is obtained by calculation of the following formula
Figure BDA0001422240940000076
And
Figure BDA0001422240940000077
and
Figure BDA0001422240940000078
estimated signals of the first, second and third order state variables of the i-th follower's joint 2, respectively:
Figure BDA0001422240940000079
in the formulaT is time, Ti,2,1、Ti,2,2And Ti,2,3All are the gains of the extended state observer, riE (1, infinity) is a parameter to be designed; b2, error surface si,2,1Designing an arithmetic unit:
error surface si,2,1The input ends of the arithmetic units are directed graphs respectively
Figure BDA00014222409400000710
Middle observer state
Figure BDA00014222409400000711
Output y of kth leader double link robot armk,2,dAdjacent communication aijAnd the output of the second extended state observer unit
Figure BDA00014222409400000712
Obtaining an error surface s by calculation of the following formulai,2,1Output s of arithmetic uniti,2,1：
Figure BDA0001422240940000081
B3, design of third nonlinear operation unit:
the input ends of the third nonlinear operation units are directed graphs respectively
Figure BDA0001422240940000082
Middle observer state
Figure BDA0001422240940000083
Output y of jth follower joint 2j,2Adjacent communication aijOutput y of the i-th follower joint 2i,2The output of the second extended state observer unit
Figure BDA0001422240940000084
Sum error surface si,2,1The output of the operation unit is calculated by the following formula to obtain the virtual control law xi,2,2,d：
Figure BDA0001422240940000085
In the formula, Ti,2,1And Tj,2,1Are the gains of the extended state observer,
Figure BDA0001422240940000086
ri∈(1,∞)、rj∈(1,∞)、qi,2,1parameters to be designed are more than 0;
b4, design of second comparator cell:
the input of the second comparator unit is the output of the second extended state observer unit
Figure BDA0001422240940000087
And the output x of the third non-linear operation uniti,2,2,dAn error surface s is obtained by calculation of the following formulai,2,2：
Figure BDA0001422240940000088
B5, design of second tracking differentiator unit:
the input end of the second tracking differentiator is the output x of the third nonlinear operation uniti,2,2,dThe output v of the tracking differentiator is obtained by calculation of the following formulai,2,2：
Figure BDA0001422240940000089
In the formula, vi,2,2Is the third non-linear operation unit output xi,2,2,dIs estimated, α ∈ (0,1) is the filter factor, λ > 0 is the speed factor;
b6, design of fourth nonlinear operation unit:
the input end of the fourth nonlinear operation unit is a directed graph
Figure BDA00014222409400000810
Adjacent communication aijOutput y of the i-th follower joint 2i,2Error surface si,2,1Output of arithmetic unit, output v of second tracking differentiator uniti,2,2The output s of the second comparator uniti,2,2And the output of the second extended state observer unit
Figure BDA00014222409400000811
The control law u is obtained by calculation of the following formulai,2：
Figure BDA00014222409400000812
In the formula, Ti,2,2In order to extend the gain of the state observer,
Figure BDA0001422240940000091
ri∈(1,∞)、qi,2,2more than 0 is a parameter to be designed;
to this end, the control input u of the ith follower is obtainedi,1And ui,2。
Compared with the prior art, the technical scheme of the invention has the beneficial effects that:
according to the invention, according to the imperfection and the deficiency of the prior background technology, the inclusion control of output feedback of the multi-double-link mechanical arm under a directed communication network is realized based on the active disturbance rejection and inversion technology, an extended state observer is designed in each sub-controller by utilizing the input and the output of a follower to reconstruct the system state of the double-link mechanical arm, and unknown dynamics are compensated in real time through the estimation of the extended state quantity, so that the designed inclusion controller of the output feedback has disturbance rejection. Further, the derivative of the virtual control law in the sub-controllers is estimated by means of a tracking differentiator, so that the problem of computation complexity explosion is overcome. The controller of the multi-double-connecting-rod mechanical arm based on active disturbance rejection and inversion technology only depends on the output position, and the problems of unknown dynamics in a system, simplified complex derivation operation and the like can be effectively solved.
(1) The invention relates to a multi-double-link mechanical arm containing controller based on an output position, aiming at the difficult problem that part of system states of followers in the multi-double-link mechanical arm are unavailable, an extended state observer is designed through the input and the output of the followers to reconstruct the system state of each double-link mechanical arm, and the multi-double-link mechanical arm containing controller for outputting feedback is designed by utilizing the system states observed in real time.
(2) The invention relates to a multi-double-connecting-rod mechanical arm containing controller based on an output position, which solves the problem of complexity in calculation by utilizing the characteristic that a tracking differentiator can effectively estimate a complex nonlinear function derivative aiming at the problem of complex derivation of a virtual control law in the containing controller, and effectively avoids the complexity of design of the containing controller for output feedback of the multi-double-connecting-rod mechanical arm
(3) The invention relates to a multi-double-link mechanical arm containing controller based on an output position, aiming at the influence of unknown dynamics existing in the multi-double-link mechanical arm on the design of the containing controller, by means of the characteristic that an extended state observer does not depend on an accurate double-link mechanical arm model, and the unknown dynamics are compensated in real time through the estimation of an extended state quantity, so that the designed containing controller outputting feedback has the immunity.
Drawings
FIG. 1 is a schematic diagram of an ith follower output feedback structure including a controller according to the present invention;
FIG. 2 is a schematic view of the structure of the ith dual link robotic arm of the follower of the present invention;
FIG. 3 is a one-way topological diagram between a follower dual-link robotic arm and a leader of the present invention;
FIG. 4 is a diagram of a convex hull formed by a controller according to the present invention that converges follower output to a leader;
FIG. 5. Inclusion error e in the output of the joint 1 of four followers in the present invention i11,2,3, 4;
FIG. 6. Inclusion error e in the output of the joint 2 of four followers in the present invention i21,2,3, 4;
FIG. 7 eight control laws u for four followers in the present invention ij1,2,3,4, j is 1, 2;
FIG. 8a is a graph of the observed effect of a first state in a first extended state observer of a second follower of the present invention;
FIG. 8b is a graph of the observed effect of a second state in the first extended state observer of a second follower of the present invention;
FIG. 9a is a graph of the observed effect of a first state in a second extended state observer of a second follower of the present invention;
FIG. 9b is a graph of the observed effect of a second state in a second extended state observer of a second follower of the present invention;
FIG. 10a shows the parameter r in the present inventioniA plot of the absolute value of the estimated error for the first state in the first extended state observer for the second follower at 1,2,3,4 changes;
FIG. 10b shows the parameter r in the present inventioniA plot of the absolute value of the estimated error for the second state in the first extended state observer for the second follower at 1,2,3,4 changes;
FIG. 11a shows the parameter r in the present inventioniA plot of the absolute value of the first state estimate error in the second extended state observer of the second follower at 1,2,3,4 changes;
FIG. 11b shows the parameter r in the present inventioniA second plot of absolute state estimation error in a second extended state observer of a second follower at 1,2,3,4 changes;
FIG. 12a controller parameter q in the present inventioni,2,1、qi,2,2The first follower's first comparison graph contains the absolute value of the error when i changes to 1,2,3, 4;
FIG. 12b illustrates the controller parameter q of the present inventioni,2,1、qi,2,2The second of the first followers contains a comparison graph of absolute error values when i is changed to 1,2,3, 4.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention will be described in further detail with reference to the accompanying drawings and simulations.
The multi-double-link mechanical arm based on the output position comprises a controller, N double-link mechanical arms containing unknown dynamics are regarded as followers, a networked system formed by connecting the N double-link mechanical arms with M leaders through a unidirectional topological graph is used as a controlled object, the leaders form a static convex hull, the controller is designed by utilizing the positions of respective joint 1 and joint 2 outputs of the followers, the output of the followers is converged in the static convex hull, the controller structure of the ith (i is more than or equal to 1 and less than or equal to N) follower comprises a1 st sub-controller and a2 nd sub-controller, and the input ends of the 1 st and the 2 nd sub-controllers and the directed graph are connected
Figure BDA0001422240940000101
The output ends of the 2 sub-controllers are all connected with the input end of the ith follower,
the 1 st sub-controller comprises a first extended state observer unit, an error surface si,1,1The device comprises an arithmetic unit, a first nonlinear arithmetic unit, a first comparator unit, a first tracking differentiator unit and a second nonlinear arithmetic unit; two input ends of the first extended state observer unit are respectively the output y of the ith follower joint 1i,1And the output u of the second non-linear operation uniti,1(ii) a Error surface si,1,1The input ends of the arithmetic units are directed graphs respectively
Figure BDA0001422240940000111
Middle observer state
Figure BDA0001422240940000112
Output y of kth leader double link robot armk,1,dAdjacent communication aijAnd the output of the first extended state observer unit
Figure BDA0001422240940000113
The input ends of the first nonlinear operation units are directed graphs respectively
Figure BDA0001422240940000114
Middle observer state
Figure BDA0001422240940000115
Output y of jth follower (j is more than or equal to 1 and less than or equal to N)j,1Adjacent communication aijOutput y of i-th follower joint 1i,1Output of the first extended state observer unit
Figure BDA0001422240940000116
Sum error surface si,1,1An output of the arithmetic unit; the first comparator unit being an error surface si,1,2The input end of which is the output of the first extended state observer unit
Figure BDA0001422240940000117
And the output x of the first non-linear operation uniti,1,2,d(ii) a The input end of the first tracking differentiator unit is the output x of the first nonlinear operation uniti,1,2,d(ii) a The input end of the second nonlinear operation unit is a directed graph
Figure BDA0001422240940000118
Intermediate adjacent communication aijOutput y of i-th follower joint 1i,1Error surface si,1,1Output of arithmetic unit, output v of first tracking differentiator uniti,1,2An output s of the first comparator uniti,1,2And the output of the first extended state observer unit
Figure BDA0001422240940000119
The 2 nd sub-controller comprises a second extended state observer unit, an error surface si,2,1The device comprises an arithmetic unit, a third nonlinear arithmetic unit, a second comparator unit, a second tracking differentiator unit and a fourth nonlinear arithmetic unit; two input ends of the second extended state observer unit are respectively the output y of the ith follower joint 2i,2And the output u of the fourth non-linear operation uniti,2(ii) a Error surface si,2,1The input ends of the arithmetic units are directed graphs respectively
Figure BDA00014222409400001110
Middle observer state
Figure BDA00014222409400001111
Output y of kth leader double link robot armk,2,dAdjacent communication aijAnd the output of the second extended state observer unit
Figure BDA00014222409400001112
The input ends of the third nonlinear operation units are directed graphs respectively
Figure BDA00014222409400001113
Middle observer state
Figure BDA00014222409400001114
Output y of jth follower joint 2j,2Adjacent communication aijOutput y of the i-th follower joint 2i,2The output of the second extended state observer unit
Figure BDA00014222409400001115
Sum error surface si,2,1An output of the arithmetic unit; the second comparator unit being an error surface si,2,2The input end of which is the output of the second extended state observer unit
Figure BDA00014222409400001116
And the output x of the third non-linear operation uniti,2,2,d(ii) a The input end of the second tracking differentiator unit is the output x of the third nonlinear operation uniti,2,2,d(ii) a The input end of the fourth nonlinear operation unit is a directed graph
Figure BDA00014222409400001117
Intermediate adjacent communication aijOutput y of the i-th follower joint 2i,2Error surface si,2,1Output of arithmetic unit, output v of second tracking differentiator uniti,2,2The output s of the second comparator uniti,2,2And a second extended state observer unitOutput of
Figure BDA00014222409400001118
The multi-double-link mechanical arm based on the output position comprises a controller, a multi-agent network formed by connecting follower groups consisting of N double-link mechanical arms containing unknown dynamic states and M leaders through a one-way topological graph is considered, and each follower is communicated with at least one leader; the information communication between them can be made by directed graphs
Figure BDA0001422240940000121
Is shown in which
Figure BDA0001422240940000122
Is a set of nodes, niIndicating a double link arm i, njA double link robot arm j is shown,
Figure BDA0001422240940000123
is a collection of edges that are to be considered,
Figure BDA0001422240940000124
indicating that agent j can directly obtain the information of agent i;
Figure BDA0001422240940000125
is a contiguous matrix, aijThe definition is as follows:
Figure BDA0001422240940000126
the contiguous set of nodes i is defined as
Figure BDA0001422240940000127
Directed graph
Figure BDA0001422240940000128
Laplacian matrix of
Figure BDA0001422240940000129
The definition is as follows:
Figure BDA00014222409400001210
laplace matrix
Figure BDA00014222409400001211
Wherein D ═ diag [ D [ ]1,…,dN]As directed graphs
Figure BDA00014222409400001212
The degree matrix of (c) is,
Figure BDA00014222409400001213
the follower has at least one adjacent node, and the leader has no adjacent node, the Laplace matrix can
Figure BDA00014222409400001214
And (3) decomposition:
Figure BDA00014222409400001215
wherein,
Figure BDA00014222409400001216
many double-link arms based on output position contain the controller, and the mathematical model of the ith double-link arm in the follower is:
Figure BDA00014222409400001217
in the formula, qi,1Is the angular position, q, of the ith double link robot arm joint 1i,2Is the angular position, q, of the ith double link robot arm joint 2i＝[qi,1 qi,2]TIs the connection angular position;
Figure BDA00014222409400001218
respectively, connecting angular velocity and acceleration, taui＝[τi,1 τi,2]TControl input is performed, and system parameters are selected as follows:
Figure BDA00014222409400001219
in the formula,
Figure BDA00014222409400001220
a3＝m2l1lc2cos(δ)，a4＝m2l1lc2sin(δ)，I1viscous coefficient of friction for turning the joint 1 to2Viscous coefficient of friction to which the joint 2 turns,/1Is the length of the rod 1, m1Is the mass of the rod 1, m2Is the mass of the rod 2, delta is the angle between the initial joint 2 and the longitudinal axis, lc1Is the distance of the joint 1 from the center of mass of the bar 1, lc2Distance of joint 2 to the center of mass of rod 2, further, let [ x [ ]i,1,1 xi,2,1]T＝[qi,1 qi,2]T，
Figure BDA0001422240940000131
[ui,1 ui,2]T＝[τi,1 τi,2]TThe equation of state for the ith follower can be found:
Figure BDA0001422240940000132
in the formula, yi,1、yi,2Output of the i-th follower joint 1 and joint 2, W, respectivelyi，11＝a1+2a2cos(yi，1)+2a4sin(yi，2)，
Figure BDA0001422240940000133
And
Figure BDA0001422240940000134
the inner unknown dynamic part of the ith follower dual link robotic arm.
The design method of the multi-double-connecting-rod mechanical arm containing controller based on the output position comprises the following steps:
A. design of the 1 st sub-controller:
a1, design of first extended state observer unit:
the two inputs of the first extended state observer unit are divided into the output y of the ith follower joint 1i,1And the output u of the second non-linear operation uniti,1The output signal of the first extended state observer unit is obtained by calculation of the following formula
Figure BDA0001422240940000135
And
Figure BDA0001422240940000136
and
Figure BDA0001422240940000137
estimated signals of the first, second and third order state variables of the i-th follower's joint 1, respectively:
Figure BDA0001422240940000138
wherein T is time, Ti,1,1、Ti,1,2And Ti,1,3All are the gains of the extended state observer, riE (1, ∞) is the parameter to be designed, function gi(τ) is fal composed of linear and fractional power functionsi(τ，δi，j1) function, j ═ 1,2,3, expressed as:
Figure BDA0001422240940000141
wherein, deltai,jE (0,1) is a tunable parameter, sign (. cndot.) is a sign functionThe expression is:
Figure BDA0001422240940000142
a2, error surface si,1,1Designing an arithmetic unit:
error surface si,1,1The input ends of the arithmetic units are directed graphs respectively
Figure BDA0001422240940000143
Middle observer state
Figure BDA0001422240940000144
Output y of kth leader double link robot armk,1,dAdjacent communication aijAnd the output of the first extended state observer unit
Figure BDA0001422240940000145
Obtaining an error surface s by calculation of the following formulai,1,1Output s of arithmetic uniti,1,1：
Figure BDA0001422240940000146
A3, design of first nonlinear arithmetic unit:
the input ends of the first nonlinear operation units are directed graphs respectively
Figure BDA0001422240940000147
Middle observer state
Figure BDA0001422240940000148
Output y of jth follower joint 1j,1Adjacent communication aijOutput y of i-th follower joint 1i,1Output of the first extended state observer unit
Figure BDA0001422240940000149
Sum error surface si,1,1The output of the arithmetic unit is calculated byThe calculation of the formula yields the virtual control law xi,1,2,d：
Figure BDA00014222409400001410
In the formula, Ti,1,1And Tj,1,1In order to extend the gain of the state observer,
Figure BDA00014222409400001411
ri∈(1,∞)、rj∈(1,∞)、qi,1,1parameters to be designed are more than 0;
a4, design of first comparator unit:
the input of the first comparator unit is the output of the first extended state observer unit
Figure BDA00014222409400001412
And the output x of the first non-linear operation uniti,1,2,dAn error surface s is obtained by calculation of the following formulai,1,2：
Figure BDA00014222409400001413
A5, design of the first tracking differentiator unit:
the input end of the first tracking differentiator is the output x of the first nonlinear operation uniti,1,2,dThe output v of the tracking differentiator is obtained by calculation of the following formulai,1,2：
Figure BDA0001422240940000151
In the formula, vi,1,2Is the first non-linear operation unit output xi,1,2,dIs estimated, α ∈ (0,1) is the filter factor, λ > 0 is the speed factor;
design of a6 and second nonlinear operation unit:
the input end of the second nonlinear operation unit is a directed graph
Figure BDA0001422240940000152
Intermediate adjacent communication aijOutput y of i-th follower joint 1i,1Error surface si,1,1Output of arithmetic unit, output v of first tracking differentiator uniti,1,2An output s of the first comparator uniti,1,2And the output of the first extended state observer unit
Figure BDA0001422240940000153
The control law u is obtained by calculation of the following formulai,1：
Figure BDA0001422240940000154
In the formula, Ti,1,2In order to extend the gain of the state observer,
Figure BDA0001422240940000155
ri∈(1,∞)、qi,2,1parameters to be designed are more than 0;
B. design of the 2 nd sub-controller:
b1, design of second extended state observer unit:
the two inputs of the second extended state observer unit are divided into the output y of the ith follower joint 1i,2And the output u of the fourth non-linear operation uniti,2The output signal of the second extended state observer unit is obtained by calculation of the following formula
Figure BDA0001422240940000156
And
Figure BDA0001422240940000157
and
Figure BDA0001422240940000158
estimated signals of the first, second and third order state variables of the i-th follower's joint 2, respectively:
Figure BDA0001422240940000159
wherein T is time, Ti,2,1、Ti,2,2And Ti,2,3All are the gains of the extended state observer, riE (1, infinity) is a parameter to be designed;
b2, error surface si,2,1Designing an arithmetic unit:
error surface si,2,1The input ends of the arithmetic units are directed graphs respectively
Figure BDA00014222409400001510
Middle observer state
Figure BDA00014222409400001511
Output y of kth leader double link robot armk,2,dAdjacent communication aijAnd the output of the second extended state observer unit
Figure BDA00014222409400001512
Obtaining an error surface s by calculation of the following formulai,2,1Output s of arithmetic uniti,2,1：
Figure BDA0001422240940000161
B3, design of third nonlinear operation unit:
the input ends of the third nonlinear operation units are directed graphs respectively
Figure BDA0001422240940000162
Middle observer state
Figure BDA0001422240940000163
Output y of jth follower joint 2j,2Adjacent communication aijOutput y of the i-th follower joint 2i,2The output of the second extended state observer unit
Figure BDA0001422240940000164
Sum error surface si,2,1The output of the operation unit is calculated by the following formula to obtain the virtual control law xi,2,2,d：
Figure BDA0001422240940000165
In the formula, Ti,2,1And Tj,2,1Are the gains of the extended state observer,
Figure BDA0001422240940000166
ri∈(1,∞)、rj∈(1,∞)、qi,2,1parameters to be designed are more than 0;
b4, design of second comparator cell:
the input of the second comparator unit is the output of the second extended state observer unit
Figure BDA0001422240940000167
And the output x of the third non-linear operation uniti,2,2,dAn error surface s is obtained by calculation of the following formulai,2,2：
Figure BDA0001422240940000168
B5, design of second tracking differentiator unit:
the input end of the second tracking differentiator is the output x of the third nonlinear operation uniti,2,2,dThe output v of the tracking differentiator is obtained by calculation of the following formulai,2,2：
Figure BDA0001422240940000169
In the formula, vi,2,2Is the third non-linear operation unit output xi,2,2,dIs estimated, α ∈ (0,1) is the filter factor, λ > 0 is the speed factor;
b6, design of fourth nonlinear operation unit:
the input end of the fourth nonlinear operation unit is directed graph G adjacent communication aijOutput y of the i-th follower joint 2i,2Error surface si,2,1Output of arithmetic unit, output v of second tracking differentiator uniti,2,2The output s of the second comparator uniti,2,2And the output of the second extended state observer unit
Figure BDA00014222409400001610
The control law u is obtained by calculation of the following formulai,2：
Figure BDA0001422240940000171
In the formula, Ti,2,2In order to extend the gain of the state observer,
Figure BDA0001422240940000172
ri∈(1,∞)、qi,2,2more than 0 is a parameter to be designed;
to this end, the control input u of the ith follower is obtainedi,1And ui,2。
Examples
The invention selects the state equation of the ith double-link mechanical arm system in the follower:
Figure BDA0001422240940000173
in the formula, yi,1、yi,2Connecting angular position, u, output for i-th follower joint 1 and joint 2i,1、ui,2Control law, W, input for i- th follower joints 1 and 2i,11＝a1+2a2cos(yi,1)+2a4sin(yi,2)，
Figure BDA0001422240940000174
I1 is the viscous molar movement to which the joint 1 is transferredCoefficient of friction of 0.12, I2The viscous friction coefficient of the joint 2 is 0.25 m1The mass of the rod 1 is 1kg, m2The mass of the rod 2 is 2kg, Ic2The distance from the joint 2 to the mass center of the rod 2 is 0.6m,
Figure BDA0001422240940000175
Figure BDA0001422240940000176
a4＝m2l1lc2sin(δ)，l1is a rod 1 having a length of 1m, Ic1The distance from the joint 1 to the barycenter of the rod 1 is 0.5m, and delta is the included angle between the initial joint 2 and the longitudinal axis pi/6 rad. The initial values of the four follower system states are respectively:
Figure BDA0001422240940000177
the outputs of the three leaders are: y is5,d＝[0.8,2]T、y6,d＝[1,4]T、y7,d＝[1.6,3]T。
Considering a multi-agent network consisting of 4 followers and 3 leaders, with each follower having communication with at least one leader, the communication topology of which is shown in fig. 3, wherein 1,2,3,4 are the numbers of four followers, and 5, 6, 7 are the numbers of three leaders, a laplacian matrix is further obtained
Figure BDA0001422240940000181
From the laplace matrix, we know that:
Figure BDA0001422240940000182
in the embodiment, the purpose of system control is to effectively solve the problems of unknown dynamic and partial system state failure existing in a multi-double-link mechanical arm system when the output positions of four followers are controlled under a one-way topological diagram by the aid of the controller designed hereinCan obtain and virtually control the difficult problem of complex derivation in the inversion design, so that the output y of the followeri＝[yi,1,yi,2]TCan converge within the static convex hull formed by the leader outputs.
For this system, the controller comprising the ith dual link robot arm of the follower can be designed according to fig. 1 as follows:
Figure BDA0001422240940000191
the following parameters were selected depending on the repeated simulation experiments to determine the controller parameters: r isi＝3、Ti,1,1＝10、Ti,1,2＝20、Ti,1,3＝50、Ti,2,1＝10、Ti,2,2＝20、Ti,2,3＝50、qi,1,1＝1.1、qi,1,2＝15、qi,2,1＝1.1、qi,2,2In fig. 3 to 7, the initial value of each state in the extended state observer is the same as the initial value of the observed system, and the initial values of the remaining extended state observer and tracking differentiator are all 0.
As shown in FIGS. 3-7, the output positions of the follower groups consisting of four two-link arms are designed to include the control law ui,j(j-1, 2) into the convex hull formed by the leader, four followers' inclusion ei,j(j 1,2) the error converges to a very small neighborhood of zero. As can be seen in FIGS. 8a-9 b: in the case where the extended state observers are both 0, the two extended state observers of the second follower can effectively estimate the state in the system. As can be seen in FIGS. 10a-11 b: in the extended state, the observer is initialized to 0 and divides the parameter riExcept for the case where all parameters are unchanged, the observer follows the expansion of the parameter r in the stateiAs the value becomes larger, the absolute value of the estimation error of the two extended state observers in the second follower becomes smaller. As can be seen from fig. 12a and 12 b: in-division controller gain qi,2,1、qi,2,2With all other parameters being constant, with the controller gain qi,2,1、qi,2,2When it becomes largeThe absolute value of the first follower including the error becomes smaller. The simulation shows that the controller of the multi-double-connecting-rod mechanical arm based on the active disturbance rejection and inversion technology only depends on the output position, and the problems of unknown dynamics in a system, simplified complex derivation operation and the like can be effectively solved.

Claims (4)

1. The multi-double-connecting-rod mechanical arm based on the output position comprises a controller which is used for controlling the multi-double-connecting-rod mechanical arm to move
Figure DEST_PATH_IMAGE002
A double link robot arm with unknown dynamics is considered as a follower, which is connected with
Figure DEST_PATH_IMAGE004
A networked system formed by connecting leaders through a one-way topological graph is used as a controlled object, the leaders form a static convex hull, and an inclusion controller is designed by utilizing the positions of the outputs of joints 1 and 2 of followers respectively so that the output of the followers is converged in the static convex hull
Figure DEST_PATH_IMAGE006
The controller structure of each follower comprises a1 st sub-controller and a2 nd sub-controller,
Figure DEST_PATH_IMAGE008
the input ends of the 1 st sub-controller and the 2 nd sub-controller are connected with the directed graph
Figure DEST_PATH_IMAGE010
Is connected with the output end of the second sub-controller, and the output ends of the 2 sub-controllers are all connected with the second sub-controller
Figure 898281DEST_PATH_IMAGE006
The input of individual follower is connected, its characterized in that:
the 1 st sub-controller comprises a first extended state observer unit, an error surface
Figure DEST_PATH_IMAGE012
The device comprises an arithmetic unit, a first nonlinear arithmetic unit, a first comparator unit, a first tracking differentiator unit and a second nonlinear arithmetic unit; two input ends of the first extended state observer unit are respectively the first
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Output of the follower joint 1
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And the output of the second non-linear operation unit
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(ii) a Error surface
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The input ends of the arithmetic units are directed graphs respectively
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Middle observer state
Figure DEST_PATH_IMAGE018
The first step
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Output of a leader double link robot arm
Figure DEST_PATH_IMAGE022
Adjacent communication
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And the output of the first extended state observer unit
Figure DEST_PATH_IMAGE026
(ii) a The input ends of the first nonlinear operation units are directed graphs respectively
Figure 371452DEST_PATH_IMAGE010
Middle observer state
Figure 334860DEST_PATH_IMAGE018
、
Figure DEST_PATH_IMAGE028
The first step
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Output of a follower
Figure DEST_PATH_IMAGE032
Adjacent communication
Figure 523264DEST_PATH_IMAGE024
The first step
Figure 876492DEST_PATH_IMAGE006
Output of the follower joint 1
Figure 76529DEST_PATH_IMAGE014
Output of the first extended state observer unit
Figure 210838DEST_PATH_IMAGE026
And error surface
Figure 496326DEST_PATH_IMAGE012
The output of the arithmetic unit is used for calculating the output of the arithmetic unit,
Figure DEST_PATH_IMAGE034
(ii) a The first comparator unit being an error surface
Figure DEST_PATH_IMAGE036
The input end of which is the output of the first extended state observer unit
Figure DEST_PATH_IMAGE038
And a firstOutput of non-linear operation unit
Figure DEST_PATH_IMAGE040
(ii) a The input end of the first tracking differentiator unit is the output of the first nonlinear operation unit
Figure 311967DEST_PATH_IMAGE040
(ii) a The input end of the second nonlinear operation unit is a directed graph
Figure 976297DEST_PATH_IMAGE010
Intermediate adjacent communication
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The first step
Figure 834718DEST_PATH_IMAGE006
Output of the follower joint 1
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Error surface
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Output of arithmetic unit, output of first tracking differentiator unit
Figure DEST_PATH_IMAGE042
The output of the first comparator unit
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And the output of the first extended state observer unit
Figure 395830DEST_PATH_IMAGE026
、
Figure DEST_PATH_IMAGE044
；
The 2 nd sub-controller comprises a second extended state observer unit,Error surface
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The device comprises an arithmetic unit, a third nonlinear arithmetic unit, a second comparator unit, a second tracking differentiator unit and a fourth nonlinear arithmetic unit; two input ends of the second extended state observer unit are respectively the first
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Output of the follower joint 2
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And the output of the fourth non-linear operation unit
Figure DEST_PATH_IMAGE050
(ii) a Error surface
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The input ends of the arithmetic units are directed graphs respectively
Figure 760581DEST_PATH_IMAGE010
Middle observer state
Figure DEST_PATH_IMAGE052
The first step
Figure DEST_PATH_IMAGE054
Output of a leader double link robot arm
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Adjacent communication
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And the output of the second extended state observer unit
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(ii) a Input terminal of third non-linear operation unitAre respectively directed graphs
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Middle observer state
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、
Figure DEST_PATH_IMAGE060
The first step
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Output of the follower joint 2
Figure DEST_PATH_IMAGE062
Adjacent communication
Figure 933549DEST_PATH_IMAGE024
The first step
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Output of the follower joint 2
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The output of the second extended state observer unit
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And error surface
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An output of the arithmetic unit; the second comparator unit being an error surface
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The input end of which is the output of the second extended state observer unit
Figure DEST_PATH_IMAGE066
And a third non-linear operation sheetOutput of elements
Figure DEST_PATH_IMAGE068
(ii) a The input end of the second tracking differentiator unit is the output of the third nonlinear operation unit
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(ii) a The input end of the fourth nonlinear operation unit is a directed graph
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Intermediate adjacent communication
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The first step
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Output of the follower joint 2
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Error surface
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Output of arithmetic unit and output of second tracking differentiator unit
Figure DEST_PATH_IMAGE070
The output of the second comparator unit
Figure 282905DEST_PATH_IMAGE064
And the output of the second extended state observer unit
Figure 339985DEST_PATH_IMAGE058
、
Figure DEST_PATH_IMAGE072
。
2. According to claim 1The multi-double-connecting-rod mechanical arm based on the output position comprises a controller, and is characterized in that: consider that
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Follower group consisting of double-link mechanical arms with unknown dynamics and follower group
Figure 332398DEST_PATH_IMAGE004
The leaders are connected into a multi-agent network through a one-way topological graph, and each follower is communicated with at least one leader; the information communication between them can be made by directed graphs
Figure DEST_PATH_IMAGE074
Is shown in which
Figure DEST_PATH_IMAGE076
Is a set of nodes that are to be considered,
Figure DEST_PATH_IMAGE078
indicating that the double link arm is numbered from 1 to
Figure DEST_PATH_IMAGE080
，
Figure DEST_PATH_IMAGE082
Indicating double-link mechanical arm
Figure 50562DEST_PATH_IMAGE006
，
Figure DEST_PATH_IMAGE084
Indicating double-link mechanical arm
Figure 624632DEST_PATH_IMAGE030
，
Figure DEST_PATH_IMAGE086
Is a collection of edges that are to be considered,
Figure DEST_PATH_IMAGE088
representing an agent
Figure 823794DEST_PATH_IMAGE030
Can directly obtain intelligent agent
Figure 245548DEST_PATH_IMAGE006
The information of (a);
Figure DEST_PATH_IMAGE090
is a contiguous matrix of the neighbors,
Figure 901657DEST_PATH_IMAGE024
the definition is as follows:
Figure DEST_PATH_IMAGE092
；
node point
Figure 448176DEST_PATH_IMAGE006
Is defined as
Figure DEST_PATH_IMAGE094
，
Figure DEST_PATH_IMAGE096
Indicating and double-link mechanical arm
Figure 566875DEST_PATH_IMAGE006
The set of all other adjacent double-link mechanical arms with information interaction and directed graph
Figure 780818DEST_PATH_IMAGE010
Laplacian matrix of
Figure DEST_PATH_IMAGE098
Definition ofThe following were used:
Figure DEST_PATH_IMAGE100
；
laplace matrix
Figure DEST_PATH_IMAGE102
Wherein
Figure DEST_PATH_IMAGE104
As directed graphs
Figure 765086DEST_PATH_IMAGE010
The degree matrix of (c) is,
Figure DEST_PATH_IMAGE106
(ii) a The follower has at least one adjacent node, and the leader has no adjacent node, then the Laplace matrix
Figure DEST_PATH_IMAGE108
Can be decomposed:
Figure DEST_PATH_IMAGE110
；
wherein
Figure DEST_PATH_IMAGE112
，
Figure DEST_PATH_IMAGE114
。
3. The output position based multi-dual link robotic arm of claim 1 comprising a controller, wherein:
the first of the followers
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The mathematical model of the double-connecting-rod mechanical arm is as follows:
Figure DEST_PATH_IMAGE116
；
Figure DEST_PATH_IMAGE118
is as follows
Figure 258308DEST_PATH_IMAGE006
The angular position of the double link mechanical arm joint 1,
Figure DEST_PATH_IMAGE120
is as follows
Figure 107184DEST_PATH_IMAGE006
The angular position of the double link robot arm joint 2,
Figure DEST_PATH_IMAGE122
is the connection angular position;
Figure DEST_PATH_IMAGE124
、
Figure DEST_PATH_IMAGE126
respectively the connection angular velocity and the acceleration,
Figure DEST_PATH_IMAGE128
is as follows
Figure 13085DEST_PATH_IMAGE006
Control input of a double-link mechanical arm joint 1,
Figure DEST_PATH_IMAGE130
is as follows
Figure 580202DEST_PATH_IMAGE006
Control of a double link robot joint 2The input of the input data is carried out,
Figure DEST_PATH_IMAGE132
control input is performed, and system parameters are selected as follows:
Figure DEST_PATH_IMAGE134
；
in the formula,
Figure DEST_PATH_IMAGE136
，
Figure DEST_PATH_IMAGE138
，
Figure DEST_PATH_IMAGE140
，
Figure DEST_PATH_IMAGE142
，
Figure DEST_PATH_IMAGE144
is the viscous coefficient of friction of the rotation of the joint 1,
Figure DEST_PATH_IMAGE146
is the viscous coefficient of friction of the rotation of the joint 2,
Figure DEST_PATH_IMAGE148
in the length of the bar 1, it is,
Figure DEST_PATH_IMAGE150
it is the mass of the rod 1 that,
Figure DEST_PATH_IMAGE152
it is the mass of the rod 2 that,
Figure DEST_PATH_IMAGE154
is the angle between the initial joint 2 and the longitudinal axis,
Figure DEST_PATH_IMAGE156
the distance of the joint 1 from the center of mass of the bar 1,
Figure DEST_PATH_IMAGE158
is the distance of the joint 2 from the center of mass of the rod 2, and further, the
Figure DEST_PATH_IMAGE160
，
Figure DEST_PATH_IMAGE162
，
Figure DEST_PATH_IMAGE164
Can obtain the first
Figure 957481DEST_PATH_IMAGE006
The state equation of each follower:
Figure DEST_PATH_IMAGE166
；
in the formula,
Figure DEST_PATH_IMAGE168
as a matter of time, the time is,
Figure DEST_PATH_IMAGE170
、
Figure DEST_PATH_IMAGE172
、
Figure DEST_PATH_IMAGE174
、
Figure DEST_PATH_IMAGE176
is as follows
Figure 693487DEST_PATH_IMAGE006
The information of the state of each joint is obtained,
Figure 144191DEST_PATH_IMAGE016
、
Figure 746074DEST_PATH_IMAGE050
respectively the output of the second non-linear operation unit and the output of the fourth non-linear operation unit,
Figure 455272DEST_PATH_IMAGE014
、
Figure 826211DEST_PATH_IMAGE048
are respectively the first
Figure 510133DEST_PATH_IMAGE006
The output of the individual follower joints 1 and 2,
Figure DEST_PATH_IMAGE178
and
Figure DEST_PATH_IMAGE180
is as follows
Figure 646584DEST_PATH_IMAGE006
An internal unknown dynamic portion of the follower dual link robotic arm;
Figure DEST_PATH_IMAGE182
，
Figure DEST_PATH_IMAGE184
；
Figure DEST_PATH_IMAGE186
、
Figure DEST_PATH_IMAGE188
、
Figure DEST_PATH_IMAGE190
、
Figure DEST_PATH_IMAGE192
and
Figure DEST_PATH_IMAGE194
no physical significance, table intermediate.
4. The design method of a multi-dual link robot arm including a controller based on output positions as claimed in claim 1, wherein: first, the
Figure 785572DEST_PATH_IMAGE006
The design of the inclusion controller for each follower includes the following steps:
A. design of the 1 st sub-controller:
a1, design of first extended state observer unit:
the two inputs of the first extended state observer unit are divided into
Figure 135651DEST_PATH_IMAGE006
Output of the follower joint 1
Figure 990474DEST_PATH_IMAGE014
And the output of the second non-linear operation unit
Figure 566949DEST_PATH_IMAGE016
The output signal of the first extended state observer unit is obtained by calculation of the following formula
Figure 116486DEST_PATH_IMAGE026
、
Figure 930858DEST_PATH_IMAGE038
And
Figure 425424DEST_PATH_IMAGE044
，
Figure DEST_PATH_IMAGE196
，
Figure 879408DEST_PATH_IMAGE026
、
Figure 281571DEST_PATH_IMAGE038
and
Figure 216029DEST_PATH_IMAGE044
are respectively the first
Figure 632229DEST_PATH_IMAGE006
Estimated signals of the first, second and third state variables of the joint 1 of the individual follower:
Figure DEST_PATH_IMAGE198
；
in the formula,
Figure 58662DEST_PATH_IMAGE168
as a matter of time, the time is,
Figure DEST_PATH_IMAGE200
、
Figure DEST_PATH_IMAGE202
and
Figure DEST_PATH_IMAGE204
are the gains of the extended state observer,
Figure DEST_PATH_IMAGE206
for the parameter, function, to be designed
Figure DEST_PATH_IMAGE208
Is composed of linear and fractional power functions
Figure DEST_PATH_IMAGE210
The function of the function is that of the function,
Figure DEST_PATH_IMAGE212
the expression is as follows:
Figure DEST_PATH_IMAGE214
；
wherein,
Figure DEST_PATH_IMAGE216
is an adjustable parameter that is,
Figure DEST_PATH_IMAGE218
is a symbolic function, and the expression is:
Figure DEST_PATH_IMAGE220
；
a2, error surface
Figure 627965DEST_PATH_IMAGE012
Designing an arithmetic unit:
error surface
Figure 682508DEST_PATH_IMAGE012
The input ends of the arithmetic units are directed graphs respectively
Figure 518877DEST_PATH_IMAGE010
Middle observer state
Figure 760503DEST_PATH_IMAGE018
The first step
Figure 753736DEST_PATH_IMAGE020
Output of a leader double link robot arm
Figure 803731DEST_PATH_IMAGE022
Adjacent communication
Figure 670056DEST_PATH_IMAGE024
And the output of the first extended state observer unit
Figure 287726DEST_PATH_IMAGE026
The error surface is obtained by the calculation of the following formula
Figure 897699DEST_PATH_IMAGE012
Output of arithmetic unit
Figure 802201DEST_PATH_IMAGE012
：
Figure DEST_PATH_IMAGE222
In the formula,
Figure DEST_PATH_IMAGE224
is shown as
Figure DEST_PATH_IMAGE226
Trajectory of the individual leader joints 1;
a3, design of first nonlinear arithmetic unit:
the input ends of the first nonlinear operation units are directed graphs respectively
Figure 145152DEST_PATH_IMAGE010
Middle observer state
Figure 299053DEST_PATH_IMAGE018
、
Figure 634088DEST_PATH_IMAGE028
The first step
Figure 393097DEST_PATH_IMAGE030
Output of the follower joint 1
Figure 866803DEST_PATH_IMAGE032
Adjacent communication
Figure 193486DEST_PATH_IMAGE024
The first step
Figure 145262DEST_PATH_IMAGE006
Output of the follower joint 1
Figure 24356DEST_PATH_IMAGE014
Output of the first extended state observer unit
Figure 324756DEST_PATH_IMAGE026
And error surface
Figure 781145DEST_PATH_IMAGE012
The output of the operation unit is calculated by the following formula to obtain the virtual control law
Figure DEST_PATH_IMAGE228
：
Figure DEST_PATH_IMAGE230
；
In the formula,
Figure 38076DEST_PATH_IMAGE200
and
Figure DEST_PATH_IMAGE232
in order to extend the gain of the state observer,
Figure DEST_PATH_IMAGE234
，
Figure 817682DEST_PATH_IMAGE206
、
Figure DEST_PATH_IMAGE236
、
Figure DEST_PATH_IMAGE238
are all parameters to be designed;
a4, design of first comparator unit:
the input of the first comparator unit is the output of the first extended state observer unit
Figure 996641DEST_PATH_IMAGE038
And the output of the first non-linear operation unit
Figure 815692DEST_PATH_IMAGE040
The error surface is obtained by the calculation of the following formula
Figure 109270DEST_PATH_IMAGE036
：
Figure DEST_PATH_IMAGE240
；
A5, design of the first tracking differentiator unit:
the input end of the first tracking differentiator is the output of the first nonlinear operation unit
Figure 212224DEST_PATH_IMAGE040
The output of the tracking differentiator is obtained by calculation of the following formula
Figure 74001DEST_PATH_IMAGE042
：
Figure DEST_PATH_IMAGE242
；
In the formula,
Figure 458977DEST_PATH_IMAGE042
is the output of the first non-linear operation unit
Figure 431612DEST_PATH_IMAGE040
Is estimated from the derivative of (a) a,
Figure DEST_PATH_IMAGE244
in order to be a factor of the filtering,
Figure DEST_PATH_IMAGE246
is a velocity factor;
design of a6 and second nonlinear operation unit:
the input end of the second nonlinear operation unit is a directed graph
Figure DEST_PATH_IMAGE248
Intermediate adjacent communication
Figure 326756DEST_PATH_IMAGE024
The first step
Figure 903975DEST_PATH_IMAGE006
Output of the follower joint 1
Figure 822252DEST_PATH_IMAGE014
Error surface
Figure 332999DEST_PATH_IMAGE012
Output of arithmetic unit, output of first tracking differentiator unit
Figure 754753DEST_PATH_IMAGE042
The output of the first comparator unit
Figure 207600DEST_PATH_IMAGE036
And the output of the first extended state observer unit
Figure 347594DEST_PATH_IMAGE026
、
Figure 724349DEST_PATH_IMAGE044
The control law is obtained by calculation of the following formula
Figure 892287DEST_PATH_IMAGE016
：
Figure DEST_PATH_IMAGE250
；
In the formula,
Figure 532347DEST_PATH_IMAGE202
in order to extend the gain of the state observer,
Figure 159638DEST_PATH_IMAGE234
，
Figure DEST_PATH_IMAGE252
、
Figure DEST_PATH_IMAGE254
are all parameters to be designed, and are,
Figure 902202DEST_PATH_IMAGE046
is an error surface
Figure 767390DEST_PATH_IMAGE046
The output of the arithmetic unit is used for calculating the output of the arithmetic unit,
Figure DEST_PATH_IMAGE256
is a system parameter;
B. design of the 2 nd sub-controller:
b1, design of second extended state observer unit:
the two inputs of the second extended state observer unit are divided into
Figure 234143DEST_PATH_IMAGE006
Output of the follower joint 1
Figure 348730DEST_PATH_IMAGE048
And the output of the fourth non-linear operation unit
Figure 755702DEST_PATH_IMAGE050
The output signal of the second extended state observer unit is obtained by calculation of the following formula
Figure 272134DEST_PATH_IMAGE058
、
Figure 785155DEST_PATH_IMAGE066
And
Figure 511672DEST_PATH_IMAGE072
，
Figure 96237DEST_PATH_IMAGE196
，
Figure 342542DEST_PATH_IMAGE058
、
Figure 88781DEST_PATH_IMAGE066
and
Figure 535549DEST_PATH_IMAGE072
are respectively the first
Figure 923805DEST_PATH_IMAGE006
First, second and third order of the joint 2 of the individual followerEstimation signal of state variable:
Figure DEST_PATH_IMAGE258
；
in the formula,
Figure 555775DEST_PATH_IMAGE168
as a matter of time, the time is,
Figure DEST_PATH_IMAGE260
、
Figure DEST_PATH_IMAGE262
and
Figure DEST_PATH_IMAGE264
are the gains of the extended state observer,
Figure DEST_PATH_IMAGE266
as a result of the parameters of the system,
Figure DEST_PATH_IMAGE268
is a parameter to be designed;
b2, error surface
Figure 692489DEST_PATH_IMAGE046
Designing an arithmetic unit:
error surface
Figure 268964DEST_PATH_IMAGE046
The input ends of the arithmetic units are directed graphs respectively
Figure 319966DEST_PATH_IMAGE248
Middle observer state
Figure 399917DEST_PATH_IMAGE052
The first stepkOutput of a leader double link robot arm
Figure 894483DEST_PATH_IMAGE056
Adjacent communication
Figure 692675DEST_PATH_IMAGE024
And the output of the second extended state observer unit
Figure 786183DEST_PATH_IMAGE058
The error surface is obtained by the calculation of the following formula
Figure 986220DEST_PATH_IMAGE046
Output of arithmetic unit
Figure 651688DEST_PATH_IMAGE046
：
Figure DEST_PATH_IMAGE270
In the formula,
Figure DEST_PATH_IMAGE272
is shown askThe trajectory of the individual leader joints 2;
b3, design of third nonlinear operation unit:
the input ends of the third nonlinear operation units are directed graphs respectively
Figure 265072DEST_PATH_IMAGE010
Middle observer state
Figure 798821DEST_PATH_IMAGE052
、
Figure 479464DEST_PATH_IMAGE060
The first step
Figure 909308DEST_PATH_IMAGE030
Is followed byOutput of the human joint 2
Figure 557458DEST_PATH_IMAGE062
Adjacent communication
Figure 629319DEST_PATH_IMAGE024
The first step
Figure 663003DEST_PATH_IMAGE006
Output of the follower joint 2
Figure 467011DEST_PATH_IMAGE048
The output of the second extended state observer unit
Figure 461512DEST_PATH_IMAGE058
And error surface
Figure 694654DEST_PATH_IMAGE046
The output of the operation unit is calculated by the following formula to obtain the virtual control law
Figure 723790DEST_PATH_IMAGE068
：
Figure DEST_PATH_IMAGE274
；
In the formula,
Figure 620070DEST_PATH_IMAGE260
and
Figure DEST_PATH_IMAGE276
are the gains of the extended state observer,
Figure DEST_PATH_IMAGE278
，
Figure 931228DEST_PATH_IMAGE268
、
Figure 344892DEST_PATH_IMAGE236
、
Figure 838321DEST_PATH_IMAGE254
are all parameters to be designed;
b4, design of second comparator cell:
the input of the second comparator unit is the output of the second extended state observer unit
Figure 46449DEST_PATH_IMAGE066
And the output of the third non-linear operation unit
Figure 140176DEST_PATH_IMAGE068
The error surface is obtained by the calculation of the following formula
Figure 357530DEST_PATH_IMAGE064
：
Figure DEST_PATH_IMAGE280
；
B5, design of second tracking differentiator unit:
the input end of the second tracking differentiator is the output of the third nonlinear operation unit
Figure 916251DEST_PATH_IMAGE068
The output of the tracking differentiator is obtained by calculation of the following formula
Figure 560859DEST_PATH_IMAGE070
：
Figure DEST_PATH_IMAGE282
；
In the formula,
Figure DEST_PATH_IMAGE284
is the output of the third non-linear operation unit
Figure 79565DEST_PATH_IMAGE068
Is estimated from the derivative of (a) a,
Figure DEST_PATH_IMAGE286
in order to be a factor of the filtering,
Figure DEST_PATH_IMAGE288
is a velocity factor;
b6, design of fourth nonlinear operation unit:
the input end of the fourth nonlinear operation unit is a directed graph
Figure 507135DEST_PATH_IMAGE248
Adjacent communication
Figure 991469DEST_PATH_IMAGE024
The first step
Figure 806978DEST_PATH_IMAGE006
Output of the follower joint 2
Figure 157188DEST_PATH_IMAGE048
Error surface
Figure 840979DEST_PATH_IMAGE046
Output of arithmetic unit and output of second tracking differentiator unit
Figure 553720DEST_PATH_IMAGE070
The output of the second comparator unit
Figure 415497DEST_PATH_IMAGE064
And the output of the second extended state observer unit
Figure 846478DEST_PATH_IMAGE058
、
Figure 566916DEST_PATH_IMAGE072
The control law is obtained by calculation of the following formula
Figure 134164DEST_PATH_IMAGE050
：
Figure DEST_PATH_IMAGE290
；
In the formula,
Figure 432421DEST_PATH_IMAGE262
in order to extend the gain of the state observer,
Figure DEST_PATH_IMAGE292
，
Figure 740911DEST_PATH_IMAGE206
、
Figure DEST_PATH_IMAGE294
is a parameter to be designed;
to this end, the first
Figure 205653DEST_PATH_IMAGE006
Control input for a follower
Figure 892986DEST_PATH_IMAGE016
And
Figure 893303DEST_PATH_IMAGE050
。
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