CN109807902B - Double-mechanical-arm force/position fuzzy hybrid control method based on backstepping method - Google Patents

Double-mechanical-arm force/position fuzzy hybrid control method based on backstepping method Download PDF

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CN109807902B
CN109807902B CN201910274053.5A CN201910274053A CN109807902B CN 109807902 B CN109807902 B CN 109807902B CN 201910274053 A CN201910274053 A CN 201910274053A CN 109807902 B CN109807902 B CN 109807902B
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于金鹏
马玉梅
柳建良
刘占杰
胡成江
张国斌
王博
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Qingdao University
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Abstract

The invention discloses a double-mechanical-arm force/position fuzzy hybrid control method based on a backstepping method. The method aims at the force/position control precision requirement of the cooperative robot and the nonlinear problem in the system, the control quantity of the robot system is constrained based on the Lyapunov function, and meanwhile, the nonlinear function in the system is approximated by a fuzzy logic system, so that a fuzzy self-adaptive position tracking controller is constructed. The method can ensure that the tracking error of the system can be converged in a small enough neighborhood around the origin, and can control the contact force of the end effector and the target object in a limited range, and simulation results show that the novel control method ensures that each state quantity of the multiple mechanical arms is in a constrained space of the system, and the input of the controller is stabilized in a bounded region. The method of the invention realizes quick and effective response to the force/position tracking control of the double mechanical arms.

Description

Double-mechanical-arm force/position fuzzy hybrid control method based on backstepping method
Technical Field
The invention belongs to the technical field of force/position tracking control of multiple mechanical arms, and particularly relates to a force/position fuzzy hybrid control method for the two mechanical arms based on a backstepping method.
Background
The automation of the manufacturing industry has made great progress since the introduction of robotics into industrial production in the 60's of the 20 th century. Many industrial processes, however, require complex tasks to be performed, which presents significant challenges to the development of single-robot systems. The cooperative robot has more advantages such as transporting large and heavy loads, assembling and manipulating objects of arbitrary shape and weight, construction and maintenance in a severe environment, etc., than a single robot. However, when the cooperative robot works, the cooperative robot is affected by uncertain factors such as friction and external load disturbance, and the situation of internal force applied to an object grabbed by the mechanical arm needs to be considered. Therefore, it is a challenging issue to realize multi-robot force/bit hybrid control. The direction of recent research in conjunction with robotics has been primarily that the end effectors of multiple robotic arms must track the desired trajectory of an object in synchrony, and that there is a need to control the undesirable internal forces they produce when in contact with the object. In order to solve these problems, the related technologists propose advanced nonlinear control methods, such as sliding mode control, dynamic surface control, hamilton control and other control methods, and achieve better results. These control methods have been able to meet the rapidity and stability of the system response. However, in practical engineering, the main problems of these works are complex adaptation rules, huge computation volumes and unpredictable external disturbances.
Disclosure of Invention
The invention aims to provide a double-mechanical-arm force/position fuzzy hybrid control method based on a backstepping method, so as to solve the problem of force/position hybrid control of multiple robots under the conditions of external disturbance and uncertain parameters.
In order to achieve the purpose, the invention adopts the following technical scheme:
a force/position fuzzy hybrid control method for double mechanical arms based on a backstepping method comprises the following steps:
a, establishing a dynamic model of the ith mechanical arm, as shown in formula (1):
Figure GDA0002671070120000011
wherein:
qi＝[qi,1,qi,2,qi,3]T，qi,nrepresents the nth joint vector on the ith mechanical arm; n is 1,2, 3;
τi＝[τi,1,τi,2,τi,3]T，τi,nan input vector representing an nth joint force exerted on an ith robot arm;
Figure GDA0002671070120000012
is a symmetrical positive moment of inertia for the ith armArraying;
Figure GDA0002671070120000013
is the coriolis and centrifugal force matrix for the ith arm;
Gi(qi) Is the gravity vector of the ith robot arm;
Figure GDA0002671070120000021
is a Jacobian matrix of the ith robot arm;
Figure GDA0002671070120000022
are dynamic and static friction vectors;
di(t) is a vector of external interference;
Figure GDA0002671070120000023
is the force exerted by the ith end effector on the target object;
b, establishing a dynamic model of the target object, as shown in formula (2):
Figure GDA0002671070120000024
wherein:
Figure GDA0002671070120000025
is the position and orientation vector of the target object;
pris the position vector of the target object and,
Figure GDA0002671070120000026
is the direction vector of the target object;
Mo(p) is a symmetric positive definite inertial matrix of the target object;
Figure GDA0002671070120000027
is the coriolis and centrifugal force matrix of the target object;
Go(p) is the gravity vector of the target object;
Fois the resultant moment vector of the center of mass of the target object, FoExpressed by the following formula (3):
Figure GDA0002671070120000028
wherein,
Figure GDA0002671070120000029
is a jacobian matrix from the ith end effector to the target object;
Figure GDA00026710701200000210
expressed as internal force fiAnd external force EiAnd, thus, obtaining:
Figure GDA00026710701200000211
substituting equation (3) and equation (4) into equation (2) yields:
Figure GDA00026710701200000212
external force EiIs expressed as:
Figure GDA00026710701200000213
wherein,
Figure GDA00026710701200000215
the positive definite diagonal matrix represents the load distribution of the target object on the ith mechanical arm;
substituting equation (6) into equation (4) yields:
Figure GDA00026710701200000214
c, establishing a dynamic model of the cooperation of the two mechanical arms, as shown in a formula (8):
p＝φi(qi) (8)
wherein phi isi(qi) Is p and qiThe kinematic relationship of (a), obtained by using forward kinematics;
the following Jacobian matrix can be obtained by differentiating equation (8) to the first order:
Figure GDA0002671070120000031
wherein, Joq(qi) Representing a joint space variable q from the ith robot armiA jacobian matrix to cartesian space variables;
the following Jacobian matrix can be obtained by first differentiating equation (9):
Figure GDA0002671070120000032
substituting the formulas (3), (7), (9) and (10) into the formula (1) to obtain the dynamic model of the double mechanical arms:
Figure GDA0002671070120000033
wherein,
Figure GDA0002671070120000034
Figure GDA0002671070120000035
Figure GDA0002671070120000036
Figure GDA0002671070120000037
is an oblique symmetric matrix;
therefore, when taking any L-order real column vector x,
Figure GDA0002671070120000038
in order to simplify the dynamic model of the two mechanical arm cooperation, new variables are defined as follows:
Figure GDA0002671070120000039
the dynamic mathematical model of the force/position hybrid control of the two robots can be expressed as:
Figure GDA00026710701200000310
d, designing a double-mechanical-arm force/position fuzzy hybrid control method based on a backstepping method based on a Lyapunov function;
suppose f (Z) is in tight set ΩZIs a continuous function, for arbitrary constants > 0, there is always a fuzzy logic system WTS (Z) satisfies:
Figure GDA00026710701200000311
in the formula, input vector
Figure GDA00026710701200000312
Q is the fuzzy input dimension, RQA real number vector set;
W∈Rlis a fuzzy weight vector, the number of fuzzy nodes is a positive integer, l is greater than 1, RlA real number vector set;
S(Z)＝[s1(Z),...,sl(Z)]T∈Rlfor basis function vectors, the basis is usually chosenFunction sj(Z) is a Gaussian function as follows:
Figure GDA0002671070120000041
wherein, muj＝[μj1,...,μjv]TIs the central position of the distribution curve of the Gaussian function, and ηjThen its width;
defining an unknown constant θ | | | W | | luminance2，
Figure GDA0002671070120000042
Wherein,
Figure GDA0002671070120000043
an estimated value representing theta;
for the
Figure GDA0002671070120000044
R2Representing a set of 2-dimensional real vectors, there is always an inequality:
Figure GDA0002671070120000045
wherein > 0, r > 1, P1> 0, s > 1 and (r-1) (s-1) ═ 1;
defining the system error variables as:
Figure GDA0002671070120000046
wherein x is1dIs a desired joint vector signal; alpha is a virtual control signal which is,
Figure GDA0002671070120000047
wherein A is0,A1,bc1Is a normal number;
in each step of the control method design, a Lyapunov function is selected to construct a virtual control function or a real control law, and the control method design specifically comprises the following steps:
d1 for the desired position signal x1dSetting an error variable z1＝x1-x1d；
Selecting a Lyapunov function as follows:
Figure GDA0002671070120000048
to V1Derived by derivation
Figure GDA0002671070120000049
Selection of virtual control laws
Figure GDA00026710701200000410
Constant k1If > 0, then:
Figure GDA00026710701200000411
d2, selecting a Lyapunov function as follows:
Figure GDA00026710701200000412
due to z2＝x2α, then derived from equation (15):
Figure GDA00026710701200000413
because of the fact that
Figure GDA00026710701200000414
The following can be obtained:
Figure GDA0002671070120000051
wherein:
Figure GDA0002671070120000052
for arbitrarily small positive numbers by the universal approximation theorem2Existence of a fuzzy logic system WTS2(Z) is such that f (Z) is WTS2(Z)+2Wherein2representing approximation error and satisfying inequality2|≤2；
From young's inequality and equation (17), we can obtain:
Figure GDA0002671070120000053
wherein, | W | | is norm of vector W, l2Is a normal number;
substituting equation (18) into equation (16) yields:
Figure GDA0002671070120000054
selecting actual control inputs as:
Figure GDA0002671070120000055
wherein constant k2＞0；
From equation (19) we can obtain:
Figure GDA0002671070120000056
e from the above analysis, choose the Lyapunov function as follows:
Figure GDA0002671070120000057
in the formula eta > 0, is known
Figure GDA0002671070120000058
The derivation of equation (21) can be:
Figure GDA0002671070120000059
let θ | | W | | non-woven phosphor2Obtaining:
Figure GDA00026710701200000510
construct the law of adaptation
Figure GDA00026710701200000511
Wherein m is1Is a normal number, will adapt to law
Figure GDA00026710701200000512
Substituting into formula (23) to obtain:
Figure GDA00026710701200000513
wherein,
Figure GDA00026710701200000514
substituting equation (24) yields:
Figure GDA0002671070120000061
wherein:
a0＝min{2k1,2k2,m1}，
Figure GDA0002671070120000062
assume an initial time of t0For any time t satisfies
Figure GDA0002671070120000063
Equation (25) can be expressed as:
Figure GDA0002671070120000064
wherein, V (t)0) A value representing an initial time Lyapunov function;
is obviously provided with
Figure GDA0002671070120000065
Therefore, the error variable z1Converge into the desired neighborhood around the origin.
The invention has the following advantages:
(1) according to the method, the state quantity and the control quantity of the double-mechanical-arm system are restrained based on the Lyapunov function, so that the error is reduced, and the control precision is improved;
(2) the method utilizes a fuzzy logic system to approximate a nonlinear function in a double-mechanical-arm force/position mixing system, constructs a fuzzy self-adaptive tracking controller, and effectively solves the nonlinear item in the system;
(3) the method of the invention utilizes a back-stepping method to make the tracking error converge to a sufficiently small neighborhood of the origin;
(4) the method has good robustness and stronger load disturbance resistance, and realizes an ideal control effect.
Drawings
FIG. 1 is a schematic diagram of k robots controlling a common target object and establishing corresponding rectangular coordinates to derive a dynamic model of the entire system.
Fig. 2 is a schematic view of two three-link robot arms gripping the same target object together.
FIG. 3 is a graph of a tracking error simulation of a target object after the control method of the present invention is applied.
FIG. 4 is a simulation of a first internal force experienced by a target object using the control method of the present invention.
FIG. 5 is a simulation of a second internal force experienced by the target object using the control method of the present invention.
FIG. 6 is a simulation of a third internal force experienced by the target object using the control method of the present invention.
FIG. 7 is a simulation of a fourth internal force experienced by the target object using the control method of the present invention.
Fig. 8 is a simulation diagram of a fifth internal force applied to the target object by the control method of the present invention.
FIG. 9 is a simulation of a sixth internal force experienced by the target object using the control method of the present invention.
FIG. 10 shows the joint force τ of the controller after the control method of the present invention is adoptediA simulation diagram of (1).
Detailed Description
The basic concept of the invention is as follows: a fuzzy logic system is utilized to approximate unknown nonlinear functions in a multi-robot system, meanwhile, a middle virtual control signal is constructed by a back-stepping method based on a Lyapunov function, and a control law is obtained by gradual recursion, so that the end effector of the multi-robot arm is restrained, the end effector of the mechanical arm can be ensured to accurately track an ideal track of a target object, and the internal force borne by the target object can be ignored.
The adaptive back-stepping method is widely regarded and applied because the method can effectively overcome the influence of parameter time variation and load disturbance on the system performance. The backstepping method is a method of controlling a system having uncertainty and nonlinearity, particularly those systems that do not satisfy a given condition. The big advantage of the backstepping method is that the original high-order system can be simplified by using virtual control variables, so that the final output result can be automatically obtained by using a proper Lyapunov equation.
The self-adaptive backstepping control method decomposes a complex nonlinear system into a plurality of simple low-order subsystems, designs the controller step by introducing virtual control variables, and finally determines a control law and a parameter self-adaptive law, thereby realizing the effective control of the system. In addition, the ability of fuzzy logic systems to handle unknown nonlinear functions has attracted extensive attention in the domestic and foreign control community and is used in the design of complex control systems with high degrees of nonlinearity and uncertainty.
The invention is described in further detail below with reference to the following figures and detailed description:
as shown in fig. 1, a double-robot force/position fuzzy hybrid control method based on a backstepping method includes the following steps:
a, establishing a dynamic model of the ith mechanical arm, as shown in formula (1):
Figure GDA0002671070120000071
wherein:
qi＝[qi,1,qi,2,qi,3]T，qi,nrepresents the nth joint vector on the ith mechanical arm; n is 1,2, 3;
τi＝[τi,1,τi,2,τi,3]T，τi,nan input vector representing an nth joint force exerted on an ith robot arm;
Figure GDA0002671070120000072
is a symmetric positive definite inertia matrix of the ith mechanical arm;
Figure GDA0002671070120000073
is the coriolis and centrifugal force matrix for the ith arm;
Gi(qi) Is the gravity vector of the ith robot arm;
Figure GDA0002671070120000074
is a Jacobian matrix of the ith robot arm;
Figure GDA0002671070120000075
are dynamic and static friction vectors;
di(t) is a vector of external interference;
Figure GDA0002671070120000076
is the ith end effector applied toThe force on the target object.
b, establishing a dynamic model of the target object, as shown in formula (2):
Figure GDA0002671070120000077
wherein:
Figure GDA0002671070120000078
is the position and orientation vector of the target object;
pris the position vector of the target object and,
Figure GDA0002671070120000081
is the direction vector of the target object.
Mo(p) is a symmetric positive definite inertial matrix of the target object;
Figure GDA0002671070120000082
is the coriolis and centrifugal force matrix of the target object;
Go(p) is the gravity vector of the target object.
FoIs the resultant moment vector of the center of mass of the target object, FoExpressed by the following formula (3):
Figure GDA0002671070120000083
wherein,
Figure GDA0002671070120000084
is the jacobian matrix from the ith end effector to the target object.
Figure GDA0002671070120000085
Expressed as internal force fiAnd external force EiAnd, thus, obtaining:
Figure GDA0002671070120000086
substituting equation (3) and equation (4) into equation (2) yields:
Figure GDA0002671070120000087
external force EiIs expressed as:
Figure GDA00026710701200000815
wherein,
Figure GDA0002671070120000089
the diagonal matrix is defined positively, and represents the load distribution of the target object on the ith mechanical arm.
Substituting equation (6) into equation (4) yields:
Figure GDA00026710701200000810
c, establishing a dynamic model of the cooperation of the two mechanical arms, as shown in a formula (8):
p＝φi(qi) (8)
wherein phi isi(qi) Is p and qiThe kinematic relationship of (a) is obtained by using forward kinematics.
The following Jacobian matrix can be obtained by differentiating equation (8) to the first order:
Figure GDA00026710701200000811
wherein,
Figure GDA00026710701200000812
representing a joint space variable q from the ith robot armiJacobian matrices to cartesian space variables. The following Jacobian matrix can be obtained by first differentiating equation (9):
Figure GDA00026710701200000813
substituting the formulas (3), (7), (9) and (10) into the formula (1) to obtain the dynamic model of the double mechanical arms:
Figure GDA00026710701200000814
wherein,
Figure GDA0002671070120000091
Figure GDA0002671070120000092
Figure GDA0002671070120000093
Figure GDA0002671070120000094
is a diagonally symmetric matrix.
Therefore, when taking any L-order real column vector x,
Figure GDA0002671070120000095
to simplify the expression of the overall system dynamics, the following conditions need to be set:
1. the arms of all robots are non-redundant and have the same degree of freedom;
2. there is no relative motion between each end effector and the target object, i.e.:
the contact between the object and the end effector is rigid;
3. the kinematics of a collaborative robotic system are fully known. However, the precise kinetic model of the system is not accessible;
4. the kinematic equation of each mechanical arm is non-singular;
5. all joints and target objects are rigid.
In order to simplify the dynamic model of the two mechanical arm cooperation, new variables are defined as follows:
Figure GDA0002671070120000096
the dynamic mathematical model of the force/position hybrid control of the two robots can be expressed as:
Figure GDA0002671070120000097
d, designing a double-mechanical-arm force/position fuzzy hybrid control method based on a backstepping method based on a Lyapunov function.
Suppose f (Z) is in tight set ΩZIs a continuous function, for arbitrary constants > 0, there is always a fuzzy logic system WTS (Z) satisfies:
Figure GDA0002671070120000098
in the formula, input vector
Figure GDA0002671070120000099
Q is the fuzzy input dimension, RQA real number vector set;
W∈Rlis a fuzzy weight vector, the number of fuzzy nodes is a positive integer, l is greater than 1, RlA real number vector set;
S(Z)＝[s1(Z),...,sl(Z)]T∈Rlfor the basis function vector, a basis function s is usually chosenj(Z) is a Gaussian function as follows:
Figure GDA00026710701200000910
wherein, muj＝[μj1,...,μjv]TIs the central position of the distribution curve of the Gaussian function, and ηjIt is its width.
Defining an unknown constant θ | | | W | | luminance2，
Figure GDA0002671070120000101
Wherein,
Figure GDA0002671070120000102
representing an estimate of theta.
For the
Figure GDA0002671070120000103
R2Representing a set of 2-dimensional real vectors, there is always an inequality:
Figure GDA0002671070120000104
wherein > 0, r > 1, P1> 0, s > 1 and (r-1) (s-1) ═ 1.
Defining the system error variables as:
Figure GDA0002671070120000105
wherein x is1dIs a desired joint vector signal; alpha is a virtual control signal which is,
Figure GDA0002671070120000106
wherein A is0,A1,bc1Is a normal number.
In each step of the control method design, a Lyapunov function is selected to construct a virtual control function or a real control law, and the control method design specifically comprises the following steps:
d1 for the desired position signal x1dSetting an error variable z1＝x1-x1d；
Selecting Lyapunov letterThe number is as follows:
Figure GDA0002671070120000107
to V1Derived by derivation
Figure GDA0002671070120000108
Selection of virtual control laws
Figure GDA0002671070120000109
Constant k1If > 0, then:
Figure GDA00026710701200001010
d2, selecting a Lyapunov function as follows:
Figure GDA00026710701200001011
due to z2＝x2α, then derived from equation (15):
Figure GDA00026710701200001012
because of the fact that
Figure GDA00026710701200001013
The following can be obtained:
Figure GDA00026710701200001014
wherein:
Figure GDA00026710701200001015
for arbitrarily small positive numbers by the universal approximation theorem2Existence of a fuzzy logic system WTS2(Z) is such that f (Z) is WTS2(Z)+2Wherein2indicates an approximation error, and is fullFoot inequality as2|≤2。
From young's inequality and equation (17), we can obtain:
Figure GDA0002671070120000111
wherein, | W | | is norm of vector W, l2Is a normal number.
Substituting equation (18) into equation (16) yields:
Figure GDA0002671070120000112
selecting actual control inputs as:
Figure GDA0002671070120000113
wherein constant k2＞0。
From equation (19) we can obtain:
Figure GDA0002671070120000114
e from the above analysis, choose the Lyapunov function as follows:
Figure GDA0002671070120000115
in the formula eta > 0, is known
Figure GDA0002671070120000116
The derivation of equation (21) can be:
Figure GDA0002671070120000117
let θ | | W | | non-woven phosphor2Obtaining:
Figure GDA0002671070120000118
construct the law of adaptation
Figure GDA0002671070120000119
Wherein m is1Is a normal number, will adapt to law
Figure GDA00026710701200001110
Substituting into formula (23) to obtain:
Figure GDA00026710701200001111
wherein,
Figure GDA00026710701200001112
substituting equation (24) yields:
Figure GDA00026710701200001113
wherein:
Figure GDA00026710701200001114
assume an initial time of t0For any time t satisfies
Figure GDA00026710701200001115
Equation (25) can be expressed as:
Figure GDA0002671070120000121
wherein, V (t)0) Represents the value of the Lyapunov function at the initial instant.
Is obviously provided with
Figure GDA0002671070120000122
Error variable z1Converge into the desired neighborhood around the origin.
From the above analysis, at the control law τi＝[τi,1,τi,2,τi,3]TUnder the action of (2), the tracking error of the system converges to a sufficient neighborhood of the origin, ensuring that other signals are bounded and not violating state constraints.
According to the steps, the method is based on the Lyapunov function, and the problem of multi-robot force/position hybrid control under the conditions of external disturbance and uncertain parameters is effectively solved by combining a backstepping method and a fuzzy self-adaptive technology.
In fig. 1, { B } denotes a reference coordinate system, { O } denotes a coordinate system established centering on the centroid of the target object, { E }iThe method is characterized in that the method is a rectangular coordinate system established by taking the ith end effector as the center, wherein i is more than or equal to 1 and less than or equal to k.
Simulating the established double-mechanical-arm force/position hybrid controller based on the backstepping method in a virtual environment, and verifying the feasibility of the proposed cooperative robot force/position hybrid control based on the backstepping method:
as shown in fig. 2, the robot parameters and the parameters of the target object are coordinated:
the lengths of the two mechanical arms are respectively: l1,1＝l2,1＝2m,l1,2＝l2,2＝1.5m,l1,3＝l2,3＝0.5m；
Mass m1,1＝m2,1＝1.5kg,m1,2＝m2,2＝1kg,m1,3＝m2,3＝0.5kg；
Moment I1,1＝I2,1＝1.5N·m,I1,2＝I2,2＝0.5N·m,I1,3＝I2,3＝0.3N·m；
The radius, mass and moment of the common target object are respectively l0＝1.5m,m0＝0.3kg,I0＝0.1N·m。
The bases of the mechanical arm are respectively: (x)1,y1)＝(-1.4,0)，(x2,y2)＝(1.4,0)。
Selecting the control law parameters as follows:
k1＝10，k2＝8,l2＝20；
y is obtained from the formula (8)d＝φ(x1d)，
The tracking reference signal is:
Figure GDA0002671070120000123
the fuzzy membership function is:
Figure GDA0002671070120000131
Figure GDA0002671070120000132
Figure GDA0002671070120000133
Figure GDA0002671070120000134
Figure GDA0002671070120000135
Figure GDA0002671070120000136
in FIG. 2, qi＝[qi,1,qi,2,qi,3]T∈R3Is the angle of each joint, dx,1，dx,2Respectively representing the base coordinate position, m, of each robot arm0,r0Respectively representing the mass and radius of the target object.
The simulation is performed on the premise that system parameters and nonlinear functions are unknown. The simulation results for the adaptive fuzzy control method based on the state constraint are shown in the attached drawings. After the method of the invention is applied for control:
the errors of the tracking signal and the desired signal are shown in fig. 3, where Δ x, Δ y, and Δ θ represent the errors of the target object in the x-axis, y-axis, and azimuth θ, respectively. As can be seen from fig. 3, the output of the system tracks the desired signal well.
Fig. 4 to 9 are simulation diagrams of six internal forces to which a target object is subjected after the control method of the present invention is applied. As can be seen from FIGS. 4 to 9, after the control is performed by the method of the invention, the controller inputs tauiStabilized within a bounded area.
FIG. 10 is a graph of controller joint force (contact force of target object and end effector) τ after the method of the present invention has been appliediF1, F2, F3 are the forces exerted by the controller on the first arm, and F4, F5, F6 are the forces exerted by the controller on the second arm. As can be seen in fig. 10, the contact force of the target object with the end effector is maintained within a bounded region.
The experiments show that the control method can efficiently track the reference signal and has good practical implementation significance.
It should be understood, however, that the description herein of specific embodiments is not intended to limit the invention to the particular forms disclosed, but on the contrary, the intention is to cover all modifications, equivalents, and alternatives falling within the spirit and scope of the invention as defined by the appended claims.

Claims (1)

1. A force/position fuzzy mixed control method of two mechanical arms based on a backstepping method is characterized in that,
the method comprises the following steps:
a, establishing a dynamic model of the ith mechanical arm, as shown in formula (1):
Figure FDA0002671070110000011
wherein:
qi＝[qi,1,qi,2,qi,3]T，qi,nrepresents the nth joint vector on the ith mechanical arm; n is 1,2, 3;
τi＝[τi,1,τi,2,τi,3]T，τi,nan input vector representing an nth joint force exerted on an ith robot arm;
Figure FDA0002671070110000012
is a symmetric positive definite inertia matrix of the ith mechanical arm;
Figure FDA0002671070110000013
is the coriolis and centrifugal force matrix for the ith arm;
Gi(qi) Is the gravity vector of the ith robot arm;
Figure FDA0002671070110000014
is a Jacobian matrix of the ith robot arm;
Figure FDA0002671070110000015
are dynamic and static friction vectors;
di(t) is a vector of external interference;
Figure FDA0002671070110000016
is the force exerted by the ith end effector on the target object;
b, establishing a dynamic model of the target object, as shown in formula (2):
Figure FDA0002671070110000017
wherein:
Figure FDA0002671070110000018
is the position and orientation vector of the target object;
pris the position vector of the target object and,
Figure FDA0002671070110000019
is the direction vector of the target object;
Mo(p) is a symmetric positive definite inertial matrix of the target object;
Figure FDA00026710701100000110
is the coriolis and centrifugal force matrix of the target object;
Go(p) is the gravity vector of the target object;
Fois the resultant moment vector of the center of mass of the target object, FoExpressed by the following formula (3):
Figure FDA00026710701100000111
wherein,
Figure FDA00026710701100000112
is a jacobian matrix from the ith end effector to the target object;
Figure FDA0002671070110000021
expressed as internal force fiAnd external force EiAnd, thus, obtaining:
Figure FDA0002671070110000022
substituting equation (3) and equation (4) into equation (2) yields:
Figure FDA0002671070110000023
external force EiIs expressed as:
Figure FDA0002671070110000024
wherein,
Figure FDA0002671070110000025
the positive definite diagonal matrix represents the load distribution of the target object on the ith mechanical arm;
substituting equation (6) into equation (4) yields:
Figure FDA0002671070110000026
c, establishing a dynamic model of the cooperation of the two mechanical arms, as shown in a formula (8):
p＝φi(qi) (8)
wherein phi isi(qi) Is p and qiThe kinematic relationship of (a), obtained by using forward kinematics;
the following Jacobian matrix can be obtained by differentiating equation (8) to the first order:
Figure FDA0002671070110000027
wherein, Joq(qi) Representing a joint space variable q from the ith robot armiA jacobian matrix to cartesian space variables;
the following Jacobian matrix can be obtained by first differentiating equation (9):
Figure FDA0002671070110000028
substituting the formulas (3), (7), (9) and (10) into the formula (1) to obtain the dynamic model of the double mechanical arms:
Figure FDA0002671070110000029
wherein,
Figure FDA00026710701100000210
Figure FDA00026710701100000211
Figure FDA00026710701100000212
Figure FDA00026710701100000213
is an oblique symmetric matrix;
therefore, when taking any L-order real column vector x,
Figure FDA00026710701100000214
in order to simplify the dynamic model of the two mechanical arm cooperation, new variables are defined as follows:
Figure FDA00026710701100000215
the dynamic mathematical model of the force/position hybrid control of the two robots can be expressed as:
Figure FDA0002671070110000031
d, designing a double-mechanical-arm force/position fuzzy hybrid control method based on a backstepping method based on a Lyapunov function;
suppose f (Z) is in tight set ΩZIs a continuous function, for arbitrary constants > 0, there is always a fuzzy logic system WTS (Z) satisfies:
Figure FDA0002671070110000032
in the formula, input vector
Figure FDA0002671070110000033
Q is the fuzzy input dimension, RQA real number vector set;
W∈Rlis a fuzzy weight vector, the number of fuzzy nodes is a positive integer, l is greater than 1, RlA real number vector set;
S(Z)＝[s1(Z),...,sl(Z)]T∈Rlfor the basis function vector, a basis function s is usually chosenj(Z) is a Gaussian function as follows:
Figure FDA0002671070110000034
wherein, muj＝[μj1,...,μjv]TIs the central position of the distribution curve of the Gaussian function, and ηjThen its width;
defining an unknown constant θ | | | W | | luminance2，
Figure FDA0002671070110000035
Wherein,
Figure FDA0002671070110000036
an estimated value representing theta;
for the
Figure FDA0002671070110000037
R2Representing a set of 2-dimensional real vectors, there is always an inequality:
Figure FDA0002671070110000038
wherein > 0, r > 1, P1> 0, s > 1 and (r-1) (s-1) ═ 1;
defining the system error variables as:
Figure FDA0002671070110000039
wherein x is1dIs a desired joint vector signal; alpha is a virtual control signal which is,
Figure FDA00026710701100000310
wherein A is0,A1,bc1Is a normal number;
in each step of the control method design, a Lyapunov function is selected to construct a virtual control function or a real control law, and the control method design specifically comprises the following steps:
d1 for the desired position signal x1dSetting an error variable z1＝x1-x1d；
Selecting a Lyapunov function as follows:
Figure FDA00026710701100000311
to V1Derived by derivation
Figure FDA0002671070110000041
Selection of virtual control laws
Figure FDA0002671070110000042
Constant k1If > 0, then:
Figure FDA0002671070110000043
d2, selecting a Lyapunov function as follows:
Figure FDA0002671070110000044
due to z2＝x2α, then derived from equation (15):
Figure FDA0002671070110000045
because of the fact that
Figure FDA0002671070110000046
The following can be obtained:
Figure FDA0002671070110000047
wherein:
Figure FDA0002671070110000048
for arbitrarily small positive numbers by the universal approximation theorem2Existence of a fuzzy logic system WTS2(Z) is such that f (Z) is WTS2(Z)+2Wherein2representing approximation error and satisfying inequality2|≤2；
From young's inequality and equation (17), we can obtain:
Figure FDA0002671070110000049
wherein, | W | | is norm of vector W, l2Is a normal number;
substituting equation (18) into equation (16) yields:
Figure FDA00026710701100000410
selecting actual control inputs as:
Figure FDA00026710701100000411
wherein constant k2＞0；
From equation (19) we can obtain:
Figure FDA00026710701100000412
e from the above analysis, choose the Lyapunov function as follows:
Figure FDA00026710701100000413
in the formula eta > 0, is known
Figure FDA00026710701100000414
The derivation of equation (21) can be:
Figure FDA0002671070110000051
let θ | | W | | non-woven phosphor2Obtaining:
Figure FDA0002671070110000052
construct the law of adaptation
Figure FDA0002671070110000053
Wherein m is1Is a normal number, will adapt to law
Figure FDA0002671070110000054
Substituting into formula (23) to obtain:
Figure FDA0002671070110000055
wherein,
Figure FDA0002671070110000056
substituting equation (24) yields:
Figure FDA0002671070110000057
wherein:
a0＝min{2k1,2k2,m1}，
Figure FDA0002671070110000058
assume an initial time of t0For any time t satisfies
Figure FDA0002671070110000059
Equation (25) can be expressed as:
Figure FDA00026710701100000510
wherein, V (t)0) A value representing an initial time Lyapunov function;
is obviously provided with
Figure FDA00026710701100000511
Therefore, the error variable z1Converge into the desired neighborhood around the origin.
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