CN114619446A - Trajectory tracking control method and system based on double observers - Google Patents

Trajectory tracking control method and system based on double observers Download PDF

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CN114619446A
CN114619446A CN202210252635.5A CN202210252635A CN114619446A CN 114619446 A CN114619446 A CN 114619446A CN 202210252635 A CN202210252635 A CN 202210252635A CN 114619446 A CN114619446 A CN 114619446A
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observer
unit
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mechanical arm
uncertainty
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CN114619446B (en
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陈学松
林士淇
孙剑峰
张丽丽
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Guangdong University of Technology
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    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B25—HAND TOOLS; PORTABLE POWER-DRIVEN TOOLS; MANIPULATORS
    • B25J—MANIPULATORS; CHAMBERS PROVIDED WITH MANIPULATION DEVICES
    • B25J9/00—Programme-controlled manipulators
    • B25J9/16—Programme controls
    • B25J9/1656—Programme controls characterised by programming, planning systems for manipulators
    • B25J9/1664—Programme controls characterised by programming, planning systems for manipulators characterised by motion, path, trajectory planning
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B25—HAND TOOLS; PORTABLE POWER-DRIVEN TOOLS; MANIPULATORS
    • B25J—MANIPULATORS; CHAMBERS PROVIDED WITH MANIPULATION DEVICES
    • B25J13/00—Controls for manipulators
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B25—HAND TOOLS; PORTABLE POWER-DRIVEN TOOLS; MANIPULATORS
    • B25J—MANIPULATORS; CHAMBERS PROVIDED WITH MANIPULATION DEVICES
    • B25J9/00—Programme-controlled manipulators
    • B25J9/16—Programme controls
    • B25J9/1602—Programme controls characterised by the control system, structure, architecture
    • B25J9/161—Hardware, e.g. neural networks, fuzzy logic, interfaces, processor

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Abstract

The invention provides a trajectory tracking control method and system based on a double observer, comprising the following steps: giving an expected reference track of a unit to be controlled, and acquiring a tracking error of the unit to be controlled; establishing a kinematic model and a dynamic model of the unit to be controlled; respectively observing errors of the kinematic model and the dynamic model through a kinematic uncertainty observer and a dynamic uncertainty observer; setting a tracking control law for the unit to be controlled according to the kinematics uncertainty observer and the dynamics uncertainty observer, and realizing tracking control for the unit to be controlled; and (5) carrying out a simulation test. The method introduces double observers to respectively observe errors in kinematics and dynamics, designs a corresponding controller to finally realize high-precision trajectory tracking control, and changes the conventional method that the trajectory tracking of the mechanical arm is usually the expected trajectory of a given joint space.

Description

Trajectory tracking control method and system based on double observers
Technical Field
The invention relates to the technical field of robot control, in particular to a mechanical arm trajectory tracking control method and system based on a double observer and an electronic device.
Background
Nowadays, a mechanical arm is an indispensable part in a modern intelligent manufacturing industry chain, and common application scenarios are as follows: the method comprises the following steps of part assembly, workpiece polishing, sealing and gluing, precision welding and the like, wherein the tail end of a mechanical arm is generally required to have the capability of accurately tracking a certain specific space track in use scenes. At present, the most widely used mechanical arms in industry are rigid mechanical arms, which have mature dynamics and kinematics models in theory, however, errors and uncertainties are inevitably generated in the modeling and parameter identification processes of the mechanical arms, and the factors are accumulated and amplified to finally generate non-negligible negative influence on the control precision of the mechanical arms. With the continuous progress of the processing technology, the precision requirement on the mechanical arm is increasingly strict, and how to solve the contradiction between the control precision and the modeling error and uncertainty by using the advanced control technology is a problem to be solved by researchers in the field.
Disclosure of Invention
In order to solve the defects of the prior art, the invention provides a trajectory tracking control method and system based on a double observer, which are used for solving at least one technical problem in the background art.
The technical scheme adopted by the invention is as follows:
a trajectory tracking control method based on a double observer comprises the following steps:
giving an expected reference track of a unit to be controlled, and acquiring a tracking error of the unit to be controlled;
establishing a kinematic model and a dynamic model of the unit to be controlled;
respectively observing errors of the kinematic model and the dynamic model through a kinematic uncertainty observer and a dynamic uncertainty observer;
setting a tracking control law for the unit to be controlled according to the kinematics uncertainty observer and the dynamics uncertainty observer, and realizing tracking control for the unit to be controlled;
and (5) carrying out a simulation test.
The step of obtaining the tracking error of the unit to be controlled given the expected reference track of the unit to be controlled comprises the following steps:
the tracking error of the unit to be controlled is represented by the following formula:
ε(t)＝Xd(t)-X(t)；
wherein ,Xd(t) is a reference trajectory of an end effector of the unit to be controlled in a workspace; x (t) is the true position of the end effector of the unit to be controlled.
The establishing of the kinematic model of the unit to be controlled comprises the following steps:
the kinematic model of the unit to be controlled is as follows:
Figure BDA0003542598960000021
wherein ,
Figure BDA0003542598960000022
is a first derivative of the end effector position of the unit to be controlled;
Figure BDA0003542598960000023
the first derivative of each joint angle of the unit to be controlled is obtained; j. the design is a square0(q) is a nominal jacobian; u is the error of the kinematic model,
Figure BDA0003542598960000024
where Δ j (q) is uncertain jacobian.
The step of establishing the dynamic model of the unit to be controlled comprises the following steps:
the dynamic model of the unit to be controlled is as follows:
Figure BDA0003542598960000025
wherein ,
Figure BDA0003542598960000026
a nominal inertia matrix of the unit to be controlled;
Figure BDA0003542598960000027
to be controlled byA nominal coriolis force matrix of the cell;
Figure BDA0003542598960000028
the first derivative of each joint angle of the unit to be controlled;
Figure BDA0003542598960000029
the second derivative of each joint angle of the unit to be controlled;
wherein ,
Figure BDA00035425989600000210
Figure BDA00035425989600000211
an uncertain inertia matrix;
Figure BDA00035425989600000212
an uncertain Coriolis force matrix;
Figure BDA00035425989600000213
is the uncertain gravitational moment to which the unit to be controlled is subjected;
Figure BDA0003542598960000031
uncertain friction torque suffered by each joint of the unit to be controlled; tau isdThe external disturbance to each joint of the unit to be controlled.
The kinematic uncertainty observer comprises:
to about XeLinear system of
Figure BDA0003542598960000032
An observer was devised as follows:
Figure BDA0003542598960000033
and the estimation law for design u is as follows:
Figure BDA0003542598960000034
wherein observer parameter h1，h2，h3，
Figure BDA0003542598960000035
Are all positive scalar quantities, and yo1＝h2Xe；sgn(x)＝[sgn(x1) sgn(x2)…sgn(xn)]TWherein sgn (x) is a scalar sign function, which is specifically defined as:
Figure BDA0003542598960000036
note the book
Figure BDA0003542598960000037
Observation error of uncertainty u
Figure BDA0003542598960000038
Will be in a limited time
Figure BDA0003542598960000039
The inner index converges to zero;
then there are:
Figure BDA00035425989600000310
parameter h of the kinematic uncertainty observer1，h2，h3，h4The selection method comprises the following steps:
a) observer parameter h1，h2，h3，h4Should be set to a scalar greater than zero, i.e. h1＞0，h2＞0，h3＞0，h4＞0；
b) Observer parameter h1A smaller positive number should be chosen to reduce h theoretically1The observation error will be reduced;
C) set a larger h2，h3While choosing a smaller h4At the time of obtaining a faster convergence rateAnd the observation error is ensured not to vibrate while the degree is increased.
The 'dynamics uncertainty observer' comprises:
linear system with respect to state variable χ
Figure BDA0003542598960000041
An observer was designed as follows:
Figure BDA0003542598960000042
and design τuThe estimation law of (c) is as follows:
Figure BDA0003542598960000043
wherein observer parameter l1，l2，l3，
Figure BDA0003542598960000044
Are all positive scalar quantities, and yo2＝l2X; uncertainty term τuObservation error of
Figure BDA0003542598960000045
Will be in a limited time
Figure BDA0003542598960000046
The inner index converges to zero;
note the book
Figure BDA0003542598960000047
Then there are:
Figure BDA0003542598960000048
parameter l of a dynamic uncertainty observer1，l2，l3，l4The selection method comprises the following steps:
a) observation ofParameter l of the device1，l2，l3，l4Should be set to a scalar greater than zero, i.e./1＞0，l2＞0，l3＞0，l4＞0；
b) Observer parameter l1Any positive number may be selected;
c) observer parameter l2，l3Must satisfy inequality 2l2l3-1＞0；
d) Observer parameter l2，l3，l4The convergence speed of the observation error of the observer is determined together and the constraint 2l is satisfied2l3On the premise that-1 is greater than 0, set larger l2，l3While the smaller l is selected4And the observation error is ensured not to vibrate while a faster convergence rate is obtained.
The setting of the tracking control law comprises the following steps:
first a new state variable z is defined:
Figure BDA0003542598960000049
wherein η1，
Figure BDA00035425989600000410
Is a positive scalar quantity;
the tracking control law is as follows:
Figure BDA00035425989600000411
wherein, define
Figure BDA00035425989600000412
And is provided with
Figure BDA00035425989600000413
In the form of a matrix, the matrix is,
Figure BDA00035425989600000414
is a positive scalar quantity; the control gain must be chosen to satisfy lambdamKc-0.5I isIn the case of a positive-going condition,
Figure BDA0003542598960000051
is an identity matrix; the observer parameters are chosen to satisfy 2l2l3-1 > 0, the tracking error epsilon of the end of the unit to be controlled to the reference trajectory can be guaranteed for a limited time T < TcThe inner exponent converges to zero, and:
Figure BDA0003542598960000052
||·||1is a norm of a vector, which is specifically defined as
Figure BDA0003542598960000053
||·||2Is a two-norm of a vector, which is specifically defined as
Figure BDA0003542598960000054
During the design of the control law, the parameter Kc，Kt，η1，η2The selection method comprises the following steps:
a) k in the control law in order to guarantee convergence of the state variable zcThe constraint λ must be satisfiedmKc-0.5I > 0, wherein > represents a positive definite sign;
b) k in control lawc，KtThe speed of the state variable z converging to zero is determined together, and the larger k is selectedcAnd a smaller KtSo that z can be converged to zero in a short time and simultaneously is ensured not to vibrate;
c) eta in control law1，η2The speed of the convergence of the tracking error epsilon to zero is determined together, and a larger eta is selected1And a smaller η2。
The application of the double-observer-based trajectory tracking control method in the trajectory tracking control direction of the mechanical arm is disclosed.
A mechanical arm trajectory tracking control system based on a double observer comprises:
the track tracking controller is connected with the outside and used for acquiring a reference track of the tail end of the mechanical arm;
the actual mechanical arm dynamics system is in data interaction with the track tracking controller and used for acquiring a mechanical arm dynamics model;
the actual mechanical arm kinematics system is in data interaction with the actual mechanical arm dynamics system and is used for acquiring a mechanical arm kinematics model and outputting an expected position of the tail end of the mechanical arm;
the uncertain kinematics observer is in data interaction with the trajectory tracking controller and is used for sending the error of the kinematics model to the trajectory tracking controller;
the uncertain dynamics observer is used for carrying out data interaction with the actual mechanical arm dynamics system;
the uncertain dynamics observer is connected with the output end of the trajectory tracking controller and used for sending an estimation law of a kinematics model;
the uncertain dynamics observer is connected with the input end of the actual mechanical arm kinematics system;
and the uncertain kinematics observer is connected with the output end of the actual mechanical arm kinematics system.
An electronic device for mechanical arm trajectory tracking control based on a double observer comprises:
a storage medium for storing a computer program;
and the processing unit is used for exchanging data with the storage medium and executing the computer program through the processing unit when the mechanical arm trajectory tracking control is performed so as to perform the steps of the trajectory tracking control method based on the double observer.
The invention has the beneficial effects that:
the method introduces the double observers to respectively observe errors in kinematics and dynamics, designs the corresponding controller to finally realize high-precision track tracking control, changes the method that the traditional mechanical arm track tracking is usually the expected track of a given joint space, can directly track the expected track in the given working space, and does not need to convert the working space track into the joint space track;
the system provided by the invention utilizes the trajectory tracking controller to perform data interaction with the uncertain kinematics observer and the uncertain dynamics observer, is used for controlling the actual mechanical arm kinematics system and the actual mechanical arm kinematics system, realizes trajectory control on the mechanical arm, and has the advantage of simple structure.
Drawings
FIG. 1 is a control structure diagram of the present invention applied to a robot arm;
FIG. 2 is a flow chart of the present invention applied to a robotic arm;
FIG. 3 is a view of a kinematic uncertainty observer of the present invention applied to a robotic arm;
FIG. 4 is a structural diagram of a dynamic uncertainty observer applied to a mechanical arm according to the present invention;
FIG. 5 is a diagram of a trajectory tracking control law of the present invention applied to a robotic arm;
FIG. 6 shows the result of the tracking error response on a macro scale when the present invention is applied to an embodiment;
FIG. 7 is a result of a tracking error response at a microscopic scale when the present invention is applied to an embodiment;
wherein, fig. 6(a) and fig. 6(b) together constitute fig. 6; fig. 7(a) and 7(b) jointly constitute fig. 7.
Detailed Description
The invention will be further described with reference to the accompanying drawings.
The invention provides a high-precision mechanical arm track tracking control method based on double observers, and aims to realize high-precision track tracking control on a mechanical arm control system under the condition that dynamics and kinematics mathematical models of the mechanical arm control system have uncertainty. The method comprises the steps of designing and constructing a dynamics uncertainty observer and a kinematics uncertainty observer respectively, obtaining an accurate observation value of a mechanical arm mathematical model error in limited convergence time, and designing a control law to achieve exponential convergence of a mechanical arm tracking error in limited time.
In order to achieve the above purpose, the invention adopts the following steps:
s1, giving a desired reference track Xd(t)
The traditional mechanical arm trajectory tracking is usually the expected trajectory of a given joint space, but the method provided by the invention can directly track the expected trajectory in the given working space without converting the working space trajectory into the joint space trajectory. Thus establishing a reference trajectory of the end effector in the workspace
Figure BDA0003542598960000071
The reference track Xd(t)＝[xd(t) yd(t) zd(t)]TThe three elements in (a) represent three cartesian coordinate values of the end effector of the robot arm that characterize the desired position of the end position of the robot arm at various times. Let the real position of the end of the arm be X (t) ([ x (t) y (t) z (t))]TWhere three elements represent the actual position of the end of the robot arm at each moment, the tracking error of the robot arm is defined as follows:
ε(t)＝Xd(t)-X(t)
and S2, establishing a kinematics and dynamics model of the mechanical arm.
The kinematics of the mechanical arm is analyzed, and a mathematical model of the kinematics is written as the form of Jacobian:
Figure BDA0003542598960000081
wherein
Figure BDA0003542598960000082
The first derivative of the position of the end effector of the robot arm, i.e., the linear velocity.
Figure BDA0003542598960000083
The first derivative of the angle of each joint of the mechanical arm, namely the angular velocity. J (q) for a robotic arm in joint configuration qA jacobian matrix. In the case of errors and uncertainties in the mathematical model of the arm, j (q) can be written as follows:
J(q)＝J0(q)+ΔJ(q)
wherein J0(q) is the nominal jacobian, i.e., the known jacobian from modeling the robotic arm, and Δ j (q) is the uncertain jacobian, i.e., the deviation between the nominal jacobian and the true jacobian. The kinematic model of the mechanical arm can be written as:
Figure BDA0003542598960000084
note the book
Figure BDA0003542598960000085
The kinematic model of the mechanical arm can finally be written in the form:
Figure BDA0003542598960000086
lagrange mechanics is used for analyzing mechanical arm dynamics, and a mathematical model of the mechanical arm dynamics is written as follows:
Figure BDA0003542598960000087
wherein
Figure BDA0003542598960000088
The matrix is an inertia matrix of the mechanical arm, and the matrix is a symmetrical nonsingular matrix.
Figure BDA0003542598960000089
Is the coriolis force matrix of the robotic arm.
Figure BDA00035425989600000810
Representing the gravitational moment experienced by each joint of the robotic arm.
Figure BDA00035425989600000811
Representing the friction torque experienced by each joint of the mechanical arm.
Figure BDA00035425989600000812
Representing errors or uncertainties in the robot arm dynamics. Tau is the input torque of each joint of the mechanical arm. In practice, errors or uncertainties existing in the dynamics of the mechanical arm come from errors existing in the modeling process of an inertia matrix, a coriolis force matrix, a gravity moment and a friction moment of the mechanical arm and external disturbance applied to the mechanical arm, so that a kinematics model of the mechanical arm is rewritten into the following form:
Figure BDA00035425989600000813
wherein
Figure BDA0003542598960000091
Is a nominal inertia matrix of the robot arm,
Figure BDA0003542598960000092
is the uncertainty inertia matrix, i.e., the deviation between the nominal inertia matrix and the true inertia matrix.
Figure BDA0003542598960000093
Is the nominal coriolis force matrix of the robot arm,
Figure BDA0003542598960000094
is an uncertain coriolis force matrix, i.e., the deviation between the nominal coriolis force matrix and the true coriolis force matrix.
Figure BDA0003542598960000095
Representing the nominal gravitational moment to which the robot arm is subjected,
Figure BDA0003542598960000096
representing an uncertain gravitational moment to which the mechanical arm is subjected.
Figure BDA0003542598960000097
Represents the nominal friction torque of each joint of the mechanical arm,
Figure BDA0003542598960000098
representing the uncertain friction torque experienced by the various joints of the mechanical arm. Tau isdIndicating the external disturbances to which the various joints of the robotic arm are subjected.
If remember
Figure BDA0003542598960000099
The mechanical arm dynamics model is further written as:
Figure BDA00035425989600000910
s3, designing an observer for the mechanical arm
From the above, the mechanical arm mathematical model has errors or uncertainty u and tau on the kinematic model and the dynamic model respectivelyuThe observer is designed to converge the observed error index of these errors or uncertainties to zero in a finite time, i.e., to obtain accurate observed values in a finite time.
S31. design kinematics uncertainty observer
Considering a kinematic model (1) of a mechanical arm with errors or uncertainties, an auxiliary system is introduced:
Figure BDA00035425989600000911
wherein is defined ase＝X-XaAnd is and
Figure BDA00035425989600000912
is a positive scalar quantity. Then the difference between (1) and (3) can be obtained:
Figure BDA00035425989600000913
thereby obtaining a value related to XeFor this linear system (4), an observer is designed as follows:
Figure BDA0003542598960000101
and the estimation law for design u is as follows:
Figure BDA0003542598960000102
wherein observer parameter h1，h2，h3，
Figure BDA0003542598960000103
Are all positive scalar quantities, and yo1＝h2Xe. It is then possible to ensure errors of the kinematic model or observation errors of the uncertainty u
Figure BDA0003542598960000104
Will be in a limited time
Figure BDA0003542598960000105
The inner exponent converges to zero. Sgn (x) appears in the observer as a symbolic function of a vector, which is specifically defined as sgn (x) ═ sgn (x)1) sgn(x2)…sgn(xn)]TWherein sgn (x) is a scalar sign function, which is specifically defined as:
Figure BDA0003542598960000106
note the book
Figure BDA0003542598960000107
Then there are:
Figure BDA0003542598960000108
parameter h of the kinematic uncertainty observer1，h2，h3，h4The selection method comprises the following steps:
c) observer parameter h1，h2，h3，h4Should be set to a scalar greater than zero, i.e. h1＞0，h2＞0，h3＞0，h4＞0。
d) Observer parameter h1A smaller positive number should be chosen to reduce h theoretically1The observation error will be reduced.
e) Observer parameter h2，h3，h4The convergence rate of the observation error of the observer is determined together, and a proper h is selected2，h3Make the product h of the two2h3Increasing, a faster convergence speed will be obtained. Although increasing h4Will also increase the convergence speed, but too large a value will also cause a strong oscillation of the observation error before convergence. So that a large h should be set2，h3While choosing a smaller h4And the observation error is ensured not to vibrate while a faster convergence rate is obtained.
S32. design dynamics uncertainty observer
Considering the mechanical arm dynamics model (2) with error or uncertainty terms, a new state variable is introduced as follows:
Figure BDA0003542598960000111
wherein
Figure BDA0003542598960000112
Is a positive scalar quantity. From the mechanical arm dynamics model and its properties:
Figure BDA0003542598960000113
Figure BDA0003542598960000114
therefore, the following results can be obtained from (8), (9) and the kinetic model (2):
Figure BDA0003542598960000115
(7) the left side and the right side of the formula are derived according to time, and (10) is substituted to obtain:
Figure BDA0003542598960000116
a linear system is thus obtained with respect to the state variable χ. An observer is designed for this linear system (11) as follows:
Figure BDA0003542598960000117
and design τuThe estimation law of (c) is as follows:
Figure BDA0003542598960000118
wherein observer parameter l1，l2，l3，
Figure BDA0003542598960000119
Are all positive scalar quantities, and yo2＝l2χ. Then it can be ensured that the error or uncertainty term tau of the dynamical modeluOf (2) observation error
Figure BDA00035425989600001110
Will be in a limited time
Figure BDA00035425989600001111
The inner exponent converges to zero.
Note the book
Figure BDA0003542598960000121
Then there are:
Figure BDA0003542598960000122
parameter l of dynamics uncertainty observer1，l2，l3，l4The selection method comprises the following steps:
e) observer parameter l1，l2，l3，l4Should be set to a scalar greater than zero, i.e./1＞0，l2＞0，l3＞0，l4＞0。
f) Observer parameter l1Any positive number may be selected.
g) Observer parameter l2，l3Must satisfy inequality 2l2l3-1 > 0 to ensure the convergence of the trajectory tracking control law.
h) Observer parameter l2，l3，l4The convergence rate of the observation error of the observer is determined together and 2l of constraint is satisfied2l3On the premise that-1 is more than 0, selecting proper l2，l3Make the product of the two l2l3Increasing, a faster convergence speed will be obtained. Although increasing l4Will also increase the convergence speed, but too large a value will also cause a strong oscillation of the observation error before convergence. So a larger l should be set2，l3While the smaller l is selected4And the observation error is ensured not to vibrate while the faster convergence speed is obtained.
S4, designing a tracking control law for the mechanical arm
Before designing the tracking control law of the mechanical arm, a new state variable z needs to be defined:
Figure BDA0003542598960000123
wherein η1，
Figure BDA0003542598960000124
Is a positive scalar quantity. For a mechanical arm to which the observers (5) (12) and the estimation laws (6) (13) are applied, the following control laws are designed:
Figure BDA0003542598960000125
wherein is defined in
Figure BDA0003542598960000126
And K iscE is the matrix of the matrix and,
Figure BDA0003542598960000127
is a positive scalar quantity. The control gain must be chosen to satisfy lambdamKc-0.5I is positive, wherein
Figure BDA0003542598960000128
Is an identity matrix. The observer parameters are chosen to satisfy 2l2l3-1 > 0. The tracking error epsilon of the tail end of the mechanical arm to the reference track can ensure that the limited time T is less than TcThe inner exponent converges to zero, and:
Figure BDA0003542598960000131
||·||1is a norm of a vector, which is specifically defined as
Figure BDA0003542598960000132
||·||2Is a two-norm of a vector, which is specifically defined as
Figure BDA0003542598960000133
K in the design of the control lawc，Kt，η1，η2Are all parameters that need to be set reasonably, and the selection method of the parameters is as follows:
d) k in the control law in order to guarantee convergence of the state variable zcThe constraint λ must be satisfiedmKc-0.5I > 0, wherein > represents a positive definite sign.
e) K in the control lawc，KtTogether determine the speed at which the state variable z converges to zero. To simplify KcIs assumed to be set up
Figure BDA0003542598960000134
When larger k is selectedcWhen, i.e. satisfies the constraint lambdamKcFaster convergence rates can also be achieved with-0.5I > 0. Although K is increasedtWill also increase the convergence speed, but too large a value will cause the state variable z to oscillate strongly before convergence. For good trajectory tracking control, z should be converged to zero as fast as possible. So a larger k should be selectedcAnd a smaller KtSo that z can converge to zero in a short time while ensuring that it does not oscillate.
f) Eta in control law1，η2Together determining the speed at which the tracking error epsilon converges to zero. When selecting larger eta1The convergence rate of epsilon will be increased, and eta that is too large2Will cause epsilon to oscillate strongly before convergence. Therefore, in order to obtain stable and high-precision tracking control, a larger η should be selected1And a smaller η2。
And S5, applying the observer and the controller to the mechanical arm to perform a track tracking simulation experiment.
The present invention provides an embodiment:
referring to fig. 1-7, taking a two-degree-of-freedom mechanical arm as an example, the kinematics observer, the dynamics observer and the trajectory tracking controller described above are applied to a two-link mechanical arm, a simulation experiment is performed by using a simulation tool SImulink in math software MATLAB, an arc trajectory is tracked, and experimental data is collected and drawn.
The real kinematic and kinetic parameters of the arm are assumed to be: the length of the connecting rod is respectively a1＝0.5，a2＝0.5In meters; the mass of the connecting rod is m1＝0.98，m20.98 in kilograms; the joint friction coefficients are respectively b1＝b20.00148. The inertia matrix of the connecting rod is respectively:
Figure BDA0003542598960000141
the center of mass is located at the midpoint of the connecting rod.
S1, giving a desired reference track Xd(t)
In this implementation example, the desired trajectory is given as follows:
Figure BDA0003542598960000142
the reference track Xd(t)＝[xd(t) yd(t) zd(t)]TThe three elements in (a) represent three cartesian coordinate values of the end effector of the robot arm that characterize the desired position of the end position of the robot arm at various times. The reference trajectory is a circular arc trajectory which takes (0.2, 0, 0) as a center and has a radius of 0.5 m and is positioned on the xoy plane in geometric view. Let the real position of the end of the arm be X (t) ([ x (t) y (t) z (t))]TWhere three elements represent the actual position of the end of the robot arm at each moment, the tracking error of the robot arm is defined as follows:
ε(t)＝Xd(t)-X(t)
and S2, establishing a kinematics and dynamics model of the mechanical arm.
The kinematics of the two-link mechanical arm are analyzed, and the real D-H parameters are shown in the following table:
TABLE 1 true D-H parameters of the robot arm
Figure BDA0003542598960000143
Assuming that the measured lengths of the connecting rods are respectively a due to the measurement error of the lengths of the connecting rods during the modeling of the mechanical arm1＝0.51，a2＝0.52 in meters. The D-H parameters of the modeled arm are shown in the following table:
TABLE 2D-H parameters for mechanical arm modeling
Figure BDA0003542598960000151
Therefore, a kinematic model of the mechanical arm can be obtained:
Figure BDA0003542598960000152
further written is the kinematic jacobian form:
Figure BDA0003542598960000153
if the D-H parameter obtained by modeling is taken as a nominal kinematic parameter, a Jacobian kinematic model with an error term can be written:
Figure BDA0003542598960000154
or directly substituting the kinematic parameters and calling MATLAB Robotics Toolbox to obtain the mechanical arm kinematics.
The dynamics of the two-link mechanical arm is analyzed, and the system parameters of the actual mechanical arm are simplified in consideration of the dynamics modeling, for example, the mechanical arm link is regarded as a rigid body with uniformly distributed quality, the joint friction is ignored, and the like. And considering the measurement of the mechanical arm parameters in the modeling process, the kinetic parameters after modeling are assumed to be: the mass of the connecting rod is m1＝1，m 21, unit kg; the joint friction coefficients are respectively b1＝b 20. The inertia matrix of the connecting rod is respectively:
Figure BDA0003542598960000155
the center of mass is located at the midpoint of the connecting rod. Will powerThe mechanical arm dynamics were obtained by using MATLAB Robotics Toolbox by substituting the mathematical parameters.
S3, designing an observer for the mechanical arm
From the above, the mechanical arm mathematical model has errors or uncertainty u and tau on the kinematic model and the dynamic model respectivelyuThe observer is designed to converge the observed error index of these errors or uncertainties to zero in a finite time, i.e., to obtain accurate observed values in a finite time.
S31. design kinematics uncertainty observer
Considering a kinematic model (1) of a mechanical arm with errors or uncertainties, an auxiliary system is introduced:
Figure BDA0003542598960000161
wherein X is definede＝X-XaAnd is made of
Figure BDA0003542598960000162
Is a positive scalar quantity. An observer is designed as follows:
Figure BDA0003542598960000163
and the estimation law of design u is as follows:
Figure BDA0003542598960000164
the parameters of the kinematic observer should be configured with reference to the methods given above. The parameter configuration in this embodiment is shown in the following table:
TABLE 3 kinematic observer parameter configuration
Figure BDA0003542598960000165
Substituting the parameters to obtain the final form of the observer:
Figure BDA0003542598960000166
Figure BDA0003542598960000167
Figure BDA0003542598960000168
s32. design dynamics uncertainty observer
Considering the mechanical arm dynamics model (2) with error or uncertainty terms, a new state variable is introduced as follows:
Figure BDA0003542598960000171
wherein
Figure BDA0003542598960000172
Is a positive scalar quantity. An observer is designed as follows:
Figure BDA0003542598960000173
and designing tauuThe estimation law of (c) is as follows:
Figure BDA0003542598960000174
the parameters of the dynamic observer should be configured with reference to the method given above. The parameter configuration in this embodiment is shown in the following table:
TABLE 4 kinetic observer parameter configuration
Figure BDA0003542598960000175
Substituting the parameters to obtain the final form of the observer:
Figure BDA0003542598960000176
Figure BDA0003542598960000177
Figure BDA0003542598960000178
s4, designing a tracking control law for the mechanical arm
Before designing the tracking control law of the mechanical arm, a new state variable z needs to be defined:
Figure BDA0003542598960000179
wherein
Figure BDA00035425989600001710
Is a positive scalar quantity. For a mechanical arm to which the observers (5) (12) and the estimation laws (6) (13) are applied, the following control laws are designed:
Figure BDA0003542598960000181
wherein is defined
Figure BDA0003542598960000182
And is
Figure BDA0003542598960000183
In the form of a matrix, the matrix is,
Figure BDA0003542598960000184
is a positive scalar quantity.
The parameters of the trajectory tracking control law should be configured with reference to the methods given above. The parameter configuration in this embodiment is shown in the following table:
TABLE 5 control law parameter configuration
Figure BDA0003542598960000185
Substituting the parameters to obtain the final form of the control law:
Figure BDA0003542598960000186
Figure BDA0003542598960000187
s5, applying the observer and the controller to the mechanical arm to perform a track tracking simulation experiment, which specifically comprises the following steps:
the method is characterized in that a control block diagram of an observer and a controller shown in figure 1 is integrated, the method is finally applied to a two-link mechanical arm, a simulation tool Simulink in mathematic software MATLAB is used for carrying out simulation experiments, an arc track is tracked, and experimental data are collected and drawn. The simulation time was set to 10 seconds, fig. 6 is a result of a tracking error response on a macro scale when the present invention is applied to an implementation example, and fig. 7 is a result of a tracking error response on a micro scale when the present invention is applied to an implementation example.
Furthermore, from fig. 6, it can be seen macroscopically that the tracking error epsilon of the coordinates x, y1，ε2The index converges to zero in a very short time; according to fig. 7, it can be seen from the microscopic level that when the mechanical arm tracks an arc track, the track tracking error oscillates within plus and minus two micrometers, and the tracking mean square error of the x coordinate is calculated to be 2.35e-4mm, and the tracking mean square error of the y coordinate is calculated to be 2.59e-4 mm. Therefore, the invention has high track tracking precision. Compared with other existing methods, the method can only solve the problemThe uncertainty on the dynamics of the mechanical arm, and the kinematics uncertainty observer can accurately estimate the uncertainty and the error on the kinematics. In addition, compared with other prior art which can only track the given joint track, the method and the device realize the direct tracking of the given space track without additional inverse kinematics solution.
In the invention, even if model errors and uncertainties exist in the mathematical modeling of the mechanical arm, the track tracking control errors of the mechanical arm can be ensured to be exponentially converged to zero within the limited convergence time theoretical upper bound. The high-precision trajectory tracking control of the mechanical arm control system is realized under the condition that the dynamics and kinematics mathematical model of the mechanical arm control system have uncertainty.
The invention also discloses an embodiment:
referring to fig. 1, a mechanical arm trajectory tracking control system based on a double observer includes: a trajectory tracking controller 100, an actual mechanical arm dynamics system 200, an actual mechanical arm kinematics system 300, an uncertain kinematics observer 400, and an uncertain dynamics observer 500; the trajectory tracking controller 100 is connected with the outside and used for acquiring a reference trajectory of the tail end of the mechanical arm; the actual mechanical arm dynamics system 200 performs data interaction with the trajectory tracking controller 100 to obtain a mechanical arm dynamics model; the actual mechanical arm kinematics system 300 performs data interaction with the actual mechanical arm dynamics system 200, obtains a mechanical arm kinematics model and outputs an expected position of the mechanical arm end; the uncertain kinematics observer 400 performs data interaction with the trajectory tracking controller 100 and sends an error of the kinematics model to the trajectory tracking controller 100; the uncertain dynamics observer 500 and the actual mechanical arm dynamics system 200 perform data interaction; the uncertain dynamics observer 500 is connected with the output end of the trajectory tracking controller 100 and is used for sending an estimation law of a kinematics model; the uncertain dynamics observer 500 is connected with the input end of the actual mechanical arm kinematics system 300; the uncertain kinematics observer 400 is connected with the output end of the actual mechanical arm kinematics system 300; the control structure on the mechanical arm constructed by the structure can accurately estimate the uncertainty and error in kinematics, and has the advantage of simple structure.
An electronic device for mechanical arm trajectory tracking control based on a double observer comprises: a storage medium and a processing unit; a storage medium for storing a computer program; the processing unit exchanges data with the storage medium and is used for tracking and controlling the track of the mechanical arm
The steps of the dual observer based trajectory tracking control method as described above are performed by the processing unit executing the computer program.
The invention also provides an embodiment:
a computer program product comprising a computer program carried on a computer readable medium, the computer program comprising program code for performing the method as set out above. The computer program may be downloaded and installed from a network. The computer program, when executed by the CPU, performs the above-described functions defined in the system of the present invention.
The invention also provides an embodiment:
a computer-readable storage medium having a computer program stored therein; the computer program, when running, performs the steps of the master data acquisition method as described above.
In the present invention, a computer readable storage medium may be any tangible medium that can contain, or store a program for use by or in connection with an instruction execution system, apparatus, or device. In the present invention, however, a computer readable signal medium may include a propagated data signal with computer readable program code embodied therein, for example, in baseband or as part of a carrier wave. Such a propagated data signal may take many forms, including, but not limited to, electro-magnetic, optical, or any suitable combination thereof. A computer readable signal medium may also be any computer readable medium that is not a computer readable storage medium and that can communicate, propagate, or transport a program for use by or in connection with an instruction execution system, apparatus, or device. Program code embodied on a computer readable medium may be transmitted using any appropriate medium, including but not limited to: wireless, wire, fiber optic cable, RF, etc., or any suitable combination of the foregoing.
The above disclosure is only a few specific implementation scenarios of the present invention, however, the present invention is not limited thereto, and any variations that can be made by those skilled in the art are intended to fall within the scope of the present invention.

Claims (10)

1. A trajectory tracking control method based on a double observer is characterized by comprising the following steps:
giving an expected reference track of a unit to be controlled, and acquiring a tracking error of the unit to be controlled;
establishing a kinematic model and a dynamic model of the unit to be controlled;
respectively observing errors of the kinematic model and the dynamic model through a kinematic uncertainty observer and a dynamic uncertainty observer;
setting a tracking control law for the unit to be controlled according to the kinematics uncertainty observer and the dynamics uncertainty observer, and realizing the tracking control for the unit to be controlled;
and (5) carrying out a simulation test.
2. The method according to claim 1, wherein the obtaining a tracking error of the unit to be controlled given a desired reference trajectory of the unit to be controlled, comprises:
the tracking error of the unit to be controlled is represented by the following formula:
ε(t)＝Xd(t)-X(t)；
wherein ,Xd(t) is a reference trajectory of an end effector of the unit to be controlled in a workspace; xd (t) is the true position of the end effector of the unit to be controlled.
3. The double-observer-based trajectory tracking control method according to claim 1, wherein the establishing a kinematic model of the unit to be controlled comprises:
the kinematic model of the unit to be controlled is as follows:
Figure FDA0003542598950000011
wherein ,
Figure FDA0003542598950000012
is a first derivative of the end effector position of the unit to be controlled;
Figure FDA0003542598950000013
the first derivative of each joint angle of the unit to be controlled is obtained; j. the design is a square0(q) is a nominal jacobi; u is the error of the kinematic model,
Figure FDA0003542598950000014
where Δ j (q) is uncertain jacobian.
4. The double-observer-based trajectory tracking control method according to claim 1, wherein the establishing of the dynamic model of the unit to be controlled comprises:
the dynamic model of the unit to be controlled is as follows:
Figure FDA0003542598950000021
wherein ,
Figure FDA0003542598950000022
a nominal inertia matrix of the unit to be controlled;
Figure FDA0003542598950000023
a nominal Coriolis force matrix of the unit to be controlled;
Figure FDA0003542598950000024
the first derivative of each joint angle of the unit to be controlled;
Figure FDA0003542598950000025
second derivatives of the angles of the joints of the unit to be controlled;
wherein ,
Figure FDA0003542598950000026
an uncertain inertia matrix;
Figure FDA0003542598950000027
an uncertain Coriolis force matrix;
Figure FDA0003542598950000028
is the uncertain gravity moment to which the unit to be controlled is subjected;
Figure FDA0003542598950000029
uncertain friction torque suffered by each joint of the unit to be controlled; tau isdThe external disturbance to each joint of the unit to be controlled.
5. The dual observer-based trajectory tracking control method according to claim 1, wherein said "kinematic uncertainty observer" comprises:
to about XeLinear system of
Figure FDA00035425989500000210
An observer was devised as follows:
Figure FDA00035425989500000211
and the estimation law of design u is as follows:
Figure FDA00035425989500000212
wherein observer parameters
Figure FDA00035425989500000213
Are all positive scalar quantities, and yo1＝h2Xe；sgn(x)＝[sgn(x1) sgn(x2)…sgn(xn)]TWherein sgn (x) is a scalar sign function, which is specifically defined as:
Figure FDA00035425989500000214
note the book
Figure FDA00035425989500000215
Observation error of uncertainty u
Figure FDA00035425989500000216
Will be in a limited time
Figure FDA00035425989500000217
The inner index converges to zero;
then there are:
Figure FDA00035425989500000218
parameter h of a kinematic uncertainty observer1，h2，h3，h4The selection method comprises the following steps:
a) observer parameter h1，h2，h3，h4Should be set to a scalar greater than zero, i.e. h1＞0，h2＞0，h3＞0，h4＞0；
b) Observer parameter h1A smaller positive number should be chosen to reduce h theoretically1The observation error will be reduced;
C) set a larger h2，h3While choosing a smaller h4And the observation error is ensured not to vibrate while a faster convergence rate is obtained.
6. The dual observer-based trajectory tracking control method according to claim 1, wherein said "dynamics uncertainty observer" comprises:
linear system with respect to state variable χ
Figure FDA0003542598950000038
An observer was designed as follows:
Figure FDA0003542598950000031
and design τuThe estimation law of (c) is as follows:
Figure FDA0003542598950000032
wherein observer parameters
Figure FDA0003542598950000033
Are all positive scalar quantities, and yo2＝l2X; uncertainty term τuOf (2) observation error
Figure FDA0003542598950000034
Will be in a limited time
Figure FDA0003542598950000035
The inner index converges to zero;
note the book
Figure FDA0003542598950000036
Then there are:
Figure FDA0003542598950000037
parameter l of a dynamic uncertainty observer1，l2，l3，l4The selection method comprises the following steps:
a) observer parameter l1，l2，l3，l4Should be set to a scalar greater than zero, i.e./1＞0，l2＞0，l3＞0，l4＞0；
b) Observer parameter l1Any positive number may be selected;
c) observer parameter l2，l3Must satisfy inequality 2l2l3-1＞0；
d) Observer parameter l2，l3，l4The convergence rate of the observation error of the observer is determined together and 2l of constraint is satisfied2l3On the premise that-1 is greater than 0, set larger l2，l3While the smaller l is selected4And the observation error is ensured not to vibrate while a faster convergence rate is obtained.
7. The method for tracking and controlling the trajectory based on the double observer according to claim 1, wherein the setting of the tracking and controlling law comprises:
first a new state variable z is defined:
Figure FDA0003542598950000041
wherein
Figure FDA0003542598950000042
Is a positive scalar quantity;
the tracking control law is as follows:
Figure FDA0003542598950000043
wherein, define
Figure FDA0003542598950000044
And is
Figure FDA0003542598950000045
In the form of a matrix, the matrix is,
Figure FDA0003542598950000046
is a positive scalar quantity; the control gain must be chosen to satisfy lambdamKc-0.5I is positive,
Figure FDA0003542598950000047
is a unit matrix; the observer parameters are chosen to satisfy 2l2l3-1 > 0, the tracking error epsilon of the end of the unit to be controlled to the reference trajectory can be guaranteed to be T < T for a finite time TcThe inner exponent converges to zero, and:
Figure FDA0003542598950000048
||·||1is a norm of a vector, which is specifically defined as
Figure FDA0003542598950000049
||·||2Is a two-norm of a vector, which is specifically defined as
Figure FDA00035425989500000410
During the design of the control law, the parameter Kc，Kt，η1，η2The selection method comprises the following steps:
a) k in the control law in order to guarantee convergence of the state variable zcThe constraint λ must be satisfiedmKc-0.5I > 0, wherein > represents a positive definite sign;
b) k in control lawc，KtJointly determine the speed of the state variable z converging to zero, and selectSelecting larger kcAnd a smaller KtSo that z can be converged to zero in a short time and simultaneously is ensured not to vibrate;
c) eta in control law1，η2The speed of the convergence of the tracking error epsilon to zero is determined together, and a larger eta is selected1And a smaller η2。
8. Use of a dual observer-based trajectory tracking control method according to any one of claims 1 to 7 in a robot trajectory tracking control direction.
9. A mechanical arm track tracking control system based on a double observer is characterized by comprising:
the track tracking controller is connected with the outside and used for acquiring a reference track of the tail end of the mechanical arm;
the actual mechanical arm dynamics system is in data interaction with the track tracking controller and used for acquiring a mechanical arm dynamics model;
the actual mechanical arm kinematics system is in data interaction with the actual mechanical arm dynamics system and is used for acquiring a mechanical arm kinematics model and outputting an expected position of the tail end of the mechanical arm;
the uncertain kinematics observer is in data interaction with the trajectory tracking controller and is used for sending the error of the kinematics model to the trajectory tracking controller;
the uncertain dynamics observer is used for carrying out data interaction with the actual mechanical arm dynamics system;
the uncertain dynamics observer is connected with the output end of the trajectory tracking controller and used for sending an estimation law of a kinematics model;
the uncertain dynamics observer is connected with the input end of the actual mechanical arm kinematics system;
and the uncertain kinematics observer is connected with the output end of the actual mechanical arm kinematics system.
10. An electronic device for tracking and controlling a mechanical arm track based on a double observer is characterized by comprising:
a storage medium for storing a computer program;
a processing unit in data communication with the storage medium, for executing the steps of the dual-observer-based trajectory tracking control method according to any one of claims 1 to 7 by executing the computer program by the processing unit when performing the robot trajectory tracking control.
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