CN114619446B - Track tracking control method and system based on double observers - Google Patents
Track tracking control method and system based on double observers Download PDFInfo
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- B—PERFORMING OPERATIONS; TRANSPORTING
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- B—PERFORMING OPERATIONS; TRANSPORTING
- B25—HAND TOOLS; PORTABLE POWER-DRIVEN TOOLS; MANIPULATORS
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- B—PERFORMING OPERATIONS; TRANSPORTING
- B25—HAND TOOLS; PORTABLE POWER-DRIVEN TOOLS; MANIPULATORS
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Abstract
The invention provides a track tracking control method and a track tracking control system based on double observers, wherein the track tracking control method 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; observing errors of the kinematic model and the dynamic model through a kinematic uncertainty observer and a dynamic uncertainty observer respectively; setting a tracking control law for the unit to be controlled according to the kinematic uncertainty observer and the dynamic uncertainty observer, and realizing tracking control for the unit to be controlled; simulation tests were performed. The method of the invention introduces a method of respectively observing errors in kinematics and dynamics by using a double observer and designing a corresponding controller to finally realize high-precision track tracking control, thereby changing the method of tracking the track of the traditional mechanical arm, which is usually the expected track of the given joint space.
Description
Technical Field
The invention relates to the technical field of robot control, in particular to a mechanical arm track tracking control method and system based on double observers and an electronic device.
Background
Nowadays, the mechanical arm is an indispensable ring in the modern intelligent manufacturing industry chain, and common application scenarios include: parts assembly, workpiece polishing, seal coating, precision welding, etc., which often require the ability of the robot arm end to accurately track a particular spatial trajectory. At present, the most widely used rigid mechanical arms are industrially used, and the mechanical arms have mature dynamic and kinematic models in theory, however, errors and uncertainties are unavoidable in the modeling and parameter identification processes of the mechanical arms, and the factors are accumulated and amplified, so that the control accuracy of the mechanical arms is finally affected in a non-negligible way. Along with the continuous progress of the processing technology, the precision requirement on the mechanical arm is also becoming severe, and how to use the advanced control technology to solve the contradiction between the control precision and the modeling error and uncertainty is a problem which needs to be solved by researchers in the field.
Disclosure of Invention
In order to solve the defects in the prior art, the invention provides a track tracking control method and a track tracking control system based on double observers, which are used for at least solving one technical problem in the background art.
The technical scheme adopted by the invention is as follows:
a track tracking control method based on double observers 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;
observing errors of the kinematic model and the dynamic model through a kinematic uncertainty observer and a dynamic uncertainty observer respectively;
setting a tracking control law for the unit to be controlled according to the kinematic uncertainty observer and the dynamic uncertainty observer, and realizing tracking control for the unit to be controlled;
simulation tests were performed.
The "given the desired reference track of the unit to be controlled, the tracking error of the unit to be controlled" includes:
the tracking error of the unit to be controlled is represented by the following formula:
ε(t)＝X d (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 a kinematic model of the unit to be controlled" includes:
the kinematic model of the unit to be controlled is as follows:
wherein ,a first derivative of an end effector position of the unit to be controlled;A first derivative of each joint angle of the unit to be controlled; j (J) 0 (q) is a nominal jacobian; u is the error of the kinematic model, +.>Wherein Δj (q) is an uncertain jacobian.
The "establishing a dynamics model of the unit to be controlled" includes:
the dynamics model of the unit to be controlled is as follows:
wherein ,a nominal inertial matrix for the unit to be controlled;A nominal coriolis force matrix for the unit to be controlled;A first derivative of each joint angle of the unit to be controlled;Second derivatives of the respective joint angles of the units to be controlled;
wherein , is an uncertain inertial matrix;Is an uncertain coriolis force matrix;An indeterminate gravitational moment to which the unit to be controlled is subjected;An uncertain friction moment applied to each joint of the unit to be controlled; τ d External disturbance is applied to each joint of the unit to be controlled.
The "kinematic uncertainty observer" includes:
for X e Linear system of (c)An observer was designed as follows:
and the estimation law of u is designed as follows:
wherein the observer parametersAre all positive scalar quantities, and y o1 ＝h 2 X e ；sgn(x)＝[sgn(x 1 ) sgn(x 2 ) ... sgn(x n )] T Where sgn (x) is a scalar sign function, which is specifically defined as:Recording deviceObservation error of uncertainty item u>Will be in a limited time +.>The internal index converges to zero;
then there are:
parameter h of a kinematic uncertainty observer 1 ，h 2 ，h 3 ，h 4 The selection method of (2) is as follows:
a) Observer parameter h 1 ，h 2 ，h 3 ，h 4 Should be set to a scalar greater than zero, i.e. h 1 ＞0，h 2 ＞0，h 3 ＞0，h 4 ＞0；
b) Observer parameter h 1 A smaller positive number should be chosen to reduce h theoretically 1 The observation error will be reduced;
c) Set a larger h 2 ，h 3 While selecting smaller h 4 The method has the advantages that the faster convergence speed is obtained, and meanwhile, the observation error is ensured not to oscillate.
The "dynamic uncertainty observer" comprises:
linear system for state variable χAn observer was designed as follows:
and design τ u The estimation law of (2) is as follows:
wherein observer parameter l 1 ，l 2 ，l 3 Are all positive scalar quantities, and y o2 ＝l 2 Chi; uncertainty term τ u Is of (2)Will be in a limited time +.>The internal index converges to zero;
recording deviceThen there are:
parameter of dynamic uncertainty observer 1 ，l 2 ，l 3 ，l 4 The selection method of (2) is as follows:
a) Observer parameter l 1 ，l 2 ，l 3 ，l 4 Should be set to a scalar greater than zero, i.e. l 1 ＞0，l 2 ＞0，l 3 ＞0，l 4 ＞0；
b) Observer parameter l 1 Any positive number may be selected;
c) Observer parameter l 2 ，l 3 The choice of (2) must satisfy inequality 2l 2 l 3 -1＞0；
d) Observer parameter l 2 ，l 3 ，l 4 Will jointly determine the convergence speed of the observer observation errors, and when the constraint is satisfied
2l 2 l 3 On the premise that-1 is more than 0, settingLarger l 2 ，l 3 While selecting a smaller l 4 The method has the advantages that the faster convergence speed is obtained, and meanwhile, the observation error is ensured not to oscillate.
The setting of the tracking control law comprises the following steps:
a new state variable z is first defined: wherein η1 ，Is a positive scalar;
the tracking control law is as follows:
wherein, defineAnd->In the form of a matrix->Is a positive scalar; the control gain must be chosen to meet lambda m K c -0.5I is positive, < - > 0->Is a unit matrix; the observer parameters are chosen to be 2l 2 l 3 -1 > 0, the tracking error epsilon of the end of the unit to be controlled with respect to the reference trajectory can be ensured at a finite time T < T c The internal index converges to zero, and:
||·|| 1 is a norm of the vectorIt is specifically defined as||·|| 2 Is the two norms of the vector, which is specifically defined as +.>
In the design process of the control law, the parameter K c ，K t ，η 1 ，η 2 The selection method of (2) is as follows:
a) To ensure convergence of the state variable z, K in the control law c Must satisfy the constraint lambda m K c -0.5I > 0, wherein > represents a positive sign;
b) K in control law c ，K t Together determine the rate at which the state variable z converges to zero, choosing a larger k c And a smaller K t So that z can converge to zero in a short time and can be ensured not to oscillate;
c) η in control law 1 ，η 2 Together determine the speed at which the tracking error epsilon converges to zero, select a larger eta 1 And a smaller eta 2 。
The application of the track tracking control method based on the double observers in the track tracking control direction of the mechanical arm is disclosed.
A dual observer-based robotic arm trajectory tracking control system, comprising:
the track tracking controller is connected with the outside and is used for acquiring a reference track at the tail end of the mechanical arm;
the actual mechanical arm dynamics system is used for carrying out data interaction with the track tracking controller and acquiring a mechanical arm dynamics model;
the actual mechanical arm kinematic system is used for carrying out data interaction with the actual mechanical arm kinematic system and is used for acquiring a mechanical arm kinematic model and outputting the expected position of the tail end of the mechanical arm;
the uncertain kinematic observer is in data interaction with the track tracking controller and is used for sending errors of the kinematic model to the track 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 track tracking controller and is used for sending an estimation law of the kinematic model;
the uncertain dynamics observer is connected with the input end of the actual mechanical arm kinematics system;
the uncertain kinematic observer is connected with the output end of the actual mechanical arm kinematic system.
An electronic device for dual observer-based robotic arm trajectory tracking control, comprising:
a storage medium storing a computer program;
and the processing unit is used for carrying out data exchange with the storage medium and executing the computer program by the processing unit when the track of the mechanical arm is controlled, so as to carry out the steps of the track tracking control method based on the double observers.
The beneficial effects of the invention are as follows:
according to the method, errors in kinematics and dynamics are observed by the double observers respectively, and the corresponding controllers are designed to finally realize high-precision track tracking control, so that the method for tracking the track of the traditional mechanical arm, which is usually the expected track of a given joint space, is changed, the expected track in the given working space can be directly tracked, and the conversion from the working space track to the joint space track is not needed;
the system disclosed by the invention utilizes the track tracking controller to perform data interaction with the uncertain kinematic observer and the uncertain kinematic observer, is used for controlling an actual mechanical arm dynamics system and an actual mechanical arm kinematics system, realizes track control of the mechanical arm, and has the advantage of simple structure.
Drawings
FIG. 1 is a control block diagram of the present invention applied to a robotic arm;
FIG. 2 is a flow chart of the invention applied to a mechanical arm;
FIG. 3 is a block diagram of a kinematic uncertainty observer of the present invention applied to a robotic arm;
FIG. 4 is a block diagram of a dynamic uncertainty observer of the present invention applied to a robotic arm;
FIG. 5 is a diagram of a trace tracking control law applied to a mechanical arm;
FIG. 6 is a graph showing the tracking error response on a macro scale for an embodiment of the present invention;
FIG. 7 shows the trace error response on a microscopic scale for an embodiment of the present invention;
wherein fig. 6 (a) and 6 (b) together constitute fig. 6; fig. 7 (a) and 7 (b) collectively constitute fig. 7.
Detailed Description
The invention is further described below with reference to the accompanying drawings.
The invention provides a high-precision mechanical arm track tracking control method based on a double observer, and aims to realize high-precision track tracking control on a mechanical arm control system under the condition that a dynamic and kinematic mathematical model of the mechanical arm control system has uncertainty. The method comprises the steps of respectively designing and constructing a dynamic uncertainty observer and a kinematic uncertainty observer, obtaining an accurate observation value of a mechanical arm mathematical model error in a limited convergence time, and designing a control law to realize exponential convergence of a mechanical arm tracking error in the limited time.
In order to achieve the above object, the present invention adopts the following steps:
s1, giving expected reference track X d (t)
The conventional mechanical arm track tracking is usually the expected track of a given joint space, but the method provided by the invention can directly track the expected track in the given working space without converting the working space track into the joint space track. Thus establishing a reference trajectory of the end effector in the workspaceThe reference track X d (t)＝[x d (t) y d (t) z d (t)] T The three elements in (a) represent three Cartesian coordinate values of the end effector of the robotic arm, respectively, that characterize the desired position of the end position of the robotic arm at various times. Let the true position of the robot arm tip be X (t) = [ X (t) y (t) z (t)] T Wherein three elements respectively represent the real positions of the tail end of the mechanical arm at all times, and the definition of the tracking error of the mechanical arm is as follows:
ε(t)＝X d (t)-X(t)
s2, establishing a kinematic and dynamic model of the mechanical arm.
The mechanical arm kinematics is analyzed, and the mathematical model is written as a jacobian form as follows:
wherein Is the first derivative of the position of the end effector of the mechanical arm, i.e., the linear velocity.Is the first derivative of the angle of each joint of the mechanical arm, namely the angular velocity. J (q) is a jacobian matrix of the manipulator at joint configuration q. In the case of errors and uncertainties in the mathematical model of the robotic arm, J (q) may be written as follows:
J(q)＝J 0 (q)+ΔJ(q)
wherein J0 (q) is the nominal jacobian, i.e. the known jacobian from the mechanical arm modeling, and Δj (q) is the uncertain jacobian, i.e. the deviation between the nominal jacobian and the true jacobian. So the kinematic model of the mechanical arm can be written as:
recording deviceThe kinematic model of the mechanical arm can finally be written as follows:
the mechanical arm dynamics is analyzed by utilizing Lagrange mechanics, and a mathematical model is written as follows:
wherein The matrix is an inertial matrix of the mechanical arm, and the matrix is a symmetrical nonsingular matrix.Is the coriolis force matrix of the mechanical arm.Representing the weight moment experienced by each joint of the robotic arm.Representing the friction moment experienced by each joint of the mechanical arm.Representing errors or uncertainties in the dynamics of the robotic arm. τ is the input torque of each joint of the mechanical arm. In practice, errors or uncertainties in the dynamics of the mechanical arm are derived from errors 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 to the mechanical arm, so that the mechanical arm kinematics model is implementedThe rewrites to the following form:
G 0 (q)+ΔG(q)＝G(q)；
wherein ,is the nominal inertial matrix of the mechanical arm, +.>Is an uncertain inertial matrix, i.e. a deviation between a nominal inertial matrix and a real inertial matrix.For the nominal coriolis force matrix of the mechanical arm, < +.>For an uncertain coriolis force matrix, i.e. a deviation between the nominal coriolis force matrix and the real coriolis force matrix.Representing the nominal gravitational moment to which the arm is subjected, +.>Indicating an indeterminate weight moment experienced by the robotic arm.Indicating the nominal friction moment to which each joint of the arm is subjected, < +.>Indicating the uncertain friction torque experienced by each joint of the robotic arm. τ d Indicating the external disturbance experienced by each joint of the robotic arm.
If recordThe mechanical arm dynamics model is further written as:
s3, designing an observer for the mechanical arm
From the above, there are errors or uncertainty terms u and τ on the kinematic model and the kinetic model, respectively, in the mechanical arm mathematical model u The observer is designed to converge the observation error index of these errors or uncertainty items to zero in a limited time, i.e. to obtain a precise observation in a limited time.
S31, designing a kinematic uncertainty observer
Considering a mechanical arm kinematic model (1) with errors or uncertainty terms, an auxiliary system is introduced:
wherein X is defined as e ＝X-X a And (2) andis a positive scalar. Then the difference between (1) and (3) is made:
thereby obtaining a reference X e For this linear system (4) an observer is designed as follows:
and the estimation law of u is designed as follows:
wherein observer parameter h 1 ，h 2 ，h 3 ，Are all positive scalar quantities, and y o1 ＝h 2 X e . Then it can be ensured that errors of the kinematic model or observation errors of the uncertainty item u +.>Will be in a limited time +.>The internal index converges to zero. The appearance of sgn (x) in the observer is a sign function of a vector, which is specifically defined as sgn (x) = [ sgn (x) 1 ) sgn(x 2 ) ... sgn(x n )] T Where sgn (x) is a scalar sign function, which is specifically defined as:
Recording deviceThen there are:
parameter h of a kinematic uncertainty observer 1 ，h 2 ，h 3 ，h 4 The selection method of (2) is as follows:
c) Observer parameter h 1 ，h 2 ，h 3 ，h 4 Should be set to a scalar greater than zero, i.e. h 1 ＞0，h 2 ＞0，h 3 ＞0，h 4 ＞0。
d) Observer parameter h 1 A smaller positive number should be chosen to reduce h theoretically 1 The observation error will be reduced.
e) Observer parameter h 2 ，h 3 ，h 4 Will jointly determine the convergence rate of observer observation errors, select the proper h 2 ，h 3 Let the product h of the two 2 ，h 3 An increase will result in a faster convergence speed. Although h is increased 4 The value of (2) will also increase the convergence speed, but too large a value will also cause severe oscillations of the observed error before convergence. So a larger h should be set 2 ，h 3 While selecting smaller h 4 The method has the advantages that the faster convergence speed is obtained, and meanwhile, the observation error is ensured not to oscillate.
S32, designing a dynamic uncertainty observer
Considering the mechanical arm dynamics model (2) with error or uncertainty term, a new state variable is introduced as follows:
wherein Is a positive scalar. The mechanical arm dynamic model and the properties thereof are as follows:
it is therefore possible to obtain from (8) (9) and the kinetic model (2):
(7) The time on the left side and the right side is derived, and (10) is substituted to obtain:
a linear system is thus obtained with respect to the state variable χ. An observer is designed for this linear system (11) as follows:
and design τ u The estimation law of (2) is as follows:
wherein observer parameter l 1 ,l 2 ,l 3 Are all positive scalar quantities, and y o2 ＝l 2 And χ. Then it can be ensured that the error or uncertainty term τ of the kinetic model u Is +.>Will be in a limited time +.>The internal index converges to zero.
Recording deviceThen there are:
parameter of dynamic uncertainty observer 1 ，l 2 ,l 3 ,l 4 The selection method of (2) is as follows:
e) Observer parameter l 1 ，l 2 ,l 3 ,l 4 Should be set to a scalar greater than zero, i.e. l 1 ＞0，l 2 ＞0,l 3 ＞0,l 4 ＞0。
f) Observer parameter l 1 Any positive number may be selected.
g) Observer parameter l 2 ，l 3 The choice of (2) must satisfy inequality 2l 2 l 3 -1 > 0 to ensure convergence of the trajectory tracking control law.
h) Observer parameter l 2 ，l 3 ，l 4 Will jointly determine the convergence rate of observer observation errors, when the constraint of 2l is satisfied 2 l 3 On the premise that-1 > 0, select the appropriate l 2 ，l 3 Let the product of the two l 2 l 3 An increase will result in a faster convergence speed. Although increase l 4 The value of (2) will also increase the convergence speed, but too large a value will also cause severe oscillations of the observed error before convergence. So a larger l should be provided 2 ，l 3 While selecting a smaller l 4 The method has the advantages that the faster convergence speed is obtained, and meanwhile, the observation error is ensured not to oscillate.
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 first:
wherein η1 ，Is a positive scalar. Then for a manipulator to which the observer (5) (12) and the estimation law (6) (13) are applied, the following control law is designed:
wherein is defined asAnd K is c E is a matrix, ">Is a positive scalar. The control gain must be chosen to meet lambda m K c -0.5I is positive, wherein +.>Is an identity matrix. The observer parameters are chosen to be 2l 2 l 3 -1 > 0. The tracking error epsilon of the tail end of the mechanical arm to the reference track can ensure that T is less than T in a finite time c The internal index converges to zero, and:
||·|| 1 is a norm of the vector, which is specifically defined as||·|| 2 Is the two norms of the vector, which is specifically defined as +.>
K in the design process of control law c ，K t ，η 1 ，η 2 Are parameters which need to be reasonably set, and the selection method of the parameters is as follows:
d) To ensure convergence of the state variable z, K in the control law c Must satisfy the constraint lambda m K c -0.5I > 0, wherein > represents a positive sign.
e) K in control law c ，K t Together determine the speed at which the state variable z converges to zero.To simplify K c Is assumed to beWhen a larger k is selected c When meeting the constraint lambda m K c -0.5I > 0, a faster convergence speed can also be obtained. Although K is increased t The value of (2) will also increase the convergence speed, but an excessive value will cause the state variable z to oscillate drastically before convergence. In order to obtain a good trajectory tracking control effect, z should be converged to zero as fast as possible. So a larger k should be chosen c And a smaller K t So that z can converge to zero in a short time while ensuring that it does not oscillate.
f) η in control law 1 ，η 2 Together determine the speed at which the tracking error epsilon converges to zero. When a larger eta is selected 1 Will increase the convergence rate of epsilon and excessively large eta 2 Will cause epsilon to oscillate vigorously before convergence. Therefore, in order to obtain stable and high-precision trajectory tracking control, a large η should be selected 1 And a smaller eta 2 。
S5, applying the observer and the controller to the mechanical arm, and carrying out a track tracking simulation experiment.
The present invention provides an embodiment:
as shown in fig. 1-7, taking a two-degree-of-freedom mechanical arm as an example, the kinematic observer, the kinetic 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 MATLAB, an arc trajectory is tracked, and experimental data is collected and drawn.
The real kinematic and kinetic parameters of the mechanical arm are assumed to be: the lengths of the connecting rods are respectively a 1 ＝0.5，a 2 =0.5, in meters; the mass of the connecting rods is m respectively 1 ＝0.98，m 2 =0.98, unit kg; the joint friction coefficients are b respectively 1 ＝b 2 = 0.00148. The inertia matrix of the connecting rod is respectively as follows:the centroid is located at the midpoint of the connecting rod.
S1, giving expected reference track X d (t)
In this embodiment, the desired trajectory is given as follows:
the reference track X d (t)＝[x d (t) y d (t) z d (t)] T The three elements in (a) represent three Cartesian coordinate values of the end effector of the robotic arm, respectively, that characterize the desired position of the end position of the robotic arm at various times. The reference track is a circular arc track with a radius of 0.5 m and positioned on an xoy plane by taking (0.2,0,0) as a center in a geometric view. Let the true position of the robot arm tip be X (t) = [ X (t) y (t) z (t)] T Wherein three elements respectively represent the real positions of the tail end of the mechanical arm at all times, and the definition of the tracking error of the mechanical arm is as follows:
ε(t)＝X d (t)-X(t)
s2, establishing a kinematic and dynamic model of the mechanical arm.
The kinematics of the two-link mechanical arm are analyzed, and the actual D-H parameters are shown in the following table:
TABLE 1 true D-H parameters for robotic arms
Assume that the length of the connecting rod is measured to be a respectively due to the measurement error of the length of the connecting rod when the mechanical arm is modeled 1 ＝0.51，a 2 =0.52, in meters. The D-H parameters of the modeled robot arm are shown in the following table:
TABLE 2D-H parameters for mechanical arm modeling
Therefore, the kinematic model of the mechanical arm can be obtained:
the jacobian form of its kinematics is further written as follows:
if the D-H parameter obtained by modeling is taken as a nominal kinematic parameter, a jacobian kinematic model with error terms can be written:
the kinematic parameters can also be directly substituted, and MATLAB Robotics Toolbox is called to obtain the kinematics of the mechanical arm.
The dynamics of the two-link mechanical arm is analyzed, parameters of an actual mechanical arm system are simplified in consideration of dynamics modeling, for example, the mechanical arm links are regarded as rigid bodies with evenly distributed mass, joint friction is ignored, and the like. In addition, the measurement of the parameters of the mechanical arm in the modeling process is considered, so that the kinetic parameters after modeling are assumed to be: the mass of the connecting rods is m respectively 1 ＝1，m 2 =1, unit kg; the joint friction coefficients are b respectively 1 ＝b 2 =0. The inertia matrix of the connecting rod is respectively as follows:the centroid is located at the midpoint of the connecting rod. The kinetic parameters were substituted separately and the robot dynamics were determined using MATLAB Robotics Toolbox.
S3, designing an observer for the mechanical arm
From the above, there are errors or uncertainty terms u and τ on the kinematic model and the kinetic model, respectively, in the mechanical arm mathematical model u The observer is designed to converge the observation error index of these errors or uncertainty items to zero in a limited time, i.e. to obtain a precise observation in a limited time.
S31, designing a kinematic uncertainty observer
Considering a mechanical arm kinematic model (1) with errors or uncertainty terms, an auxiliary system is introduced:
wherein X is defined as e ＝X-X a And (2) andis a positive scalar. An observer is designed as follows:
and the estimation law of u is designed as follows:
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
Substituting the parameters into the final form of the observer:
s32, designing a dynamic uncertainty observer
Considering the mechanical arm dynamics model (2) with error or uncertainty term, a new state variable is introduced as follows:
wherein Is a positive scalar. An observer is designed as follows:
and design τ u The estimation law of (2) is as follows:
the parameters of the kinetic observer should be configured with reference to the methods given above. The parameter configuration in this embodiment is shown in the following table:
table 4 kinetic observer parameter configuration
Substituting the parameters into the final form of the observer:
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 first:
wherein η1 ，Is a positive scalar. Then for a manipulator to which the observer (5) (12) and the estimation law (6) (13) are applied, the following control law is designed:
wherein is defined asAnd->In the form of a matrix->Is a positive scalar.
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
Substituting parameters yields the final form of the control law:
s5, applying the observer and the controller to the mechanical arm to carry out a track tracking simulation experiment, wherein the track tracking simulation experiment specifically comprises the following steps:
the control block diagram of the observer and the controller shown in fig. 1 is integrated and finally applied to the two-link mechanical arm, simulation experiments are carried out by using a simulation tool Simulink in MATLAB, an arc track is tracked, and experimental data are collected and drawn. The simulation time was set to 10 seconds, fig. 6 shows the tracking error response result on the macro scale when the present invention is applied to the embodiment example, and fig. 7 shows the tracking error response result on the micro scale when the present invention is applied to the embodiment example.
Also from FIG. 6, it can be seen macroscopically that the tracking trajectory error ε for the coordinates x, y 1 ，ε 2 The index converges to zero in a very short time; from fig. 7, it can be seen from the microscopic level that when the mechanical arm tracks an arc track, the track tracking error will oscillate within plus or minus two micrometers, and the calculated tracking mean square error of the x coordinate is 2.35e-4mm, and the tracking mean square error of the y coordinate is 2.59e-4mm. Therefore, the invention has high track tracking precision. Compared with other existing methods, the method can only solve the uncertainty in the dynamics of the mechanical arm, and the kinematic uncertainty observer can accurately estimate the uncertainty and the error in the kinematics. In addition, compared with the other prior art which can only track a given joint track, the invention realizes the direct tracking of the given space track,no additional inverse kinematics solution is required.
In the invention, even if model errors and uncertainties exist in mathematical modeling of the mechanical arm, the track tracking control errors of the mechanical arm can still be ensured to be converged to zero in a limited convergence time theory upper bound index. The high-precision track tracking control is realized under the condition that the dynamic and kinematic mathematical model of the mechanical arm control system has uncertainty.
The invention also discloses an embodiment:
referring to fig. 1, a dual observer-based mechanical arm track tracking control system includes: a trajectory tracking controller 100, an actual robotic arm kinematic system 200, an actual robotic arm kinematic system 300, an uncertain kinematic observer 400, an uncertain kinematic observer 500; the track tracking controller 100 is connected with the outside and is used for acquiring a reference track of the tail end of the mechanical arm; the actual mechanical arm dynamics system 200 performs data interaction with the track tracking controller 100 to obtain a mechanical arm dynamics model; the actual mechanical arm kinematic system 300 performs data interaction with the actual mechanical arm kinematic system 200, acquires a mechanical arm kinematic model and outputs a desired position of the tail end of the mechanical arm; 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 performs data interaction with the actual mechanical arm dynamics system 200; the uncertain dynamics observer 500 is connected with the output end of the track tracking controller 100, and is used for sending an estimation law of a kinematic model; the uncertain dynamics observer 500 is connected with the input end of the actual mechanical arm kinematics system 300; the uncertain kinematic observer 400 is connected with the output end of the actual mechanical arm kinematic system 300; the control structure on the mechanical arm constructed by the structure can accurately estimate the uncertainty and the error of the kinematics and has the advantage of simple structure.
An electronic device for dual observer-based robotic arm trajectory tracking control, comprising: a storage medium and a processing unit; a storage medium 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
When the computer program is executed by the processing unit, the steps of the track tracking control method based on the double observers are performed.
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 shown above. The computer program may be downloaded and installed from a network. The above-described functions defined in the system of the present invention are performed when the computer program is executed by a CPU.
The invention also provides an embodiment:
a computer-readable storage medium having a computer program stored therein; the computer program, when run, performs the steps of the main data acquisition method as described above.
In the context of this document, 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, the computer-readable signal medium may include a data signal propagated in baseband or as part of a carrier wave, with the computer-readable program code embodied therein. Such a propagated data signal may take any of a variety of forms, including, but not limited to, electro-magnetic, optical, or any suitable combination of the foregoing. 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 foregoing disclosure is merely illustrative of some embodiments of the invention, and the invention is not limited thereto, as modifications may be made by those skilled in the art without departing from the scope of the invention.
Claims (7)
1. The track tracking control method based on the double observers is characterized by comprising the following steps of:
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;
observing errors of the kinematic model and the dynamic model through a kinematic uncertainty observer and a dynamic uncertainty observer respectively;
setting a tracking control law for the unit to be controlled according to the kinematic uncertainty observer and the dynamic uncertainty observer, and realizing tracking control for the unit to be controlled;
performing a simulation test;
the kinematic uncertainty observer comprises:
for X e Linear system of (c)An observer was designed as follows:
and the estimation law of u is designed as follows:
wherein the observer parametersAre all positive scalar quantities, and y o1 ＝h 2 X e ；sgn(x)＝[sgn(x 1 ) sgn(x 2 )...sgn(x n )] T Where sgn (x) is a scalar sign function, which is specifically defined as:Recording deviceObservation error of u->Will be in a limited time +.>The internal index converges to zero;Where u is the error of the kinematic model;The first derivative of each joint angle of the unit to be controlled; Δj (q) is an uncertain jacobian;
then there are:
parameter h of a kinematic uncertainty observer 1 ，h 2 ，h 3 ，h 4 The selection method of (2) is as follows:
a) Observer parameter h 1 ，h 2 ，h 3 ，h 4 Should be set to a scalar greater than zero, i.e. h 1 ＞0，h 2 ＞0，h 3 ＞0，h 4 ＞0；
b) Observer parameter h 1 A smaller positive number should be chosen to reduce h theoretically 1 The observation error will be reduced;
c) Set a larger h 2 ，h 3 While selecting smaller h 4 The method has the advantages that the faster convergence speed is obtained, and meanwhile, the observation error is ensured not to oscillate;
the dynamic uncertainty observer includes:
linear system for state variable χAn observer was designed as follows:
and design τ u The estimation law of (2) is as follows:
wherein the observer parametersAre all positive scalar quantities, and y o2 ＝l 2 Chi; uncertainty term τ u Is +.>Will be in a limited time +.>The internal index converges to zero;Errors or uncertainties in the dynamics of the mechanical arm are derived from errors 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 to the mechanical arm, so G 0 (q)+ΔG(q)＝G(q)， Is an uncertain inertial matrix;Is an uncertain coriolis force matrix;An indeterminate gravitational moment to which the unit to be controlled is subjected;An uncertain friction moment applied to each joint of the unit to be controlled;Representing the nominal gravitational moment experienced by the robotic arm;The nominal friction moment received by each joint of the mechanical arm is represented; τ d External disturbance to each joint of the unit to be controlled;The second derivative of each joint angle of the unit to be controlled;
recording deviceThen there are:
uncertainty in dynamicsParameters of qualitative observer l 1 ，l 2 ，l 3 ，l 4 The selection method of (2) is as follows:
a) Observer parameter l 1 ，l 2 ，l 3 ，l 4 Should be set to a scalar greater than zero, i.e. l 1 ＞0，l 2 ＞0，l 3 ＞0，l 4 ＞0；
b) Observer parameter l 1 Any positive number may be selected;
c) Observer parameter l 2 ，l 3 The choice of (2) must satisfy inequality 2l 2 l 3 -1＞0；
d) Observer parameter l 2 ，l 3 ，l 4 Will jointly determine the convergence rate of observer observation errors, when the constraint of 2l is satisfied 2 l 3 On the premise that-1 is more than 0, a larger l is set 2 ，l 3 While selecting a smaller l 4 The method has the advantages that the faster convergence speed is obtained, and meanwhile, the observation error is ensured not to oscillate.
2. The method according to claim 1, wherein the step of obtaining the tracking error of the unit to be controlled given the 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)＝X d (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.
3. The method for controlling trajectory tracking based on double observers according to claim 1, wherein said establishing a kinematic model of the unit to be controlled comprises:
the kinematic model of the unit to be controlled is as follows:
wherein ,a first derivative of an end effector position of the unit to be controlled; j (J) 0 (q) is a nominal jacobian.
4. The method for controlling trajectory tracking based on double observers according to claim 1, wherein said establishing a kinetic model of the unit to be controlled comprises:
the dynamics model of the unit to be controlled is as follows:
wherein ,a nominal inertial matrix for the unit to be controlled;And (3) a nominal Coriolis force matrix of the unit to be controlled.
5. The method of claim 1, wherein setting a tracking control law comprises:
a new state variable z is first defined: whereinIs a positive scalar;
the tracking control law is as follows:
wherein, defineAnd->In the form of a matrix->Is a positive scalar; the control gain must be chosen to meet lambda m K c -0.5I is positive, < - > 0->Is a unit matrix; the observer parameters are chosen to be 2l 2 l 3 -1 > 0, the tracking error epsilon of the end of the unit to be controlled with respect to the reference trajectory can be ensured at a finite time T < T c The internal index converges to zero, and:
||·|| 1 is a norm of the vector, which is specifically defined as||·|| 2 Is the two norms of the vector, which is specifically defined as +.> A nominal inertial matrix for the unit to be controlled;A nominal coriolis force matrix for the unit to be controlled; j (J) 0 (q) is a nominal jacobian;A first derivative of a reference trajectory of an end effector of a unit to be controlled in a working space;
in the design process of the control law, the parameter K c ，K t ，η 1 ，η 2 The selection method of (2) is as follows:
a) To ensure convergence of the state variable z, K in the control law c Must satisfy the constraint lambda m K c -0.5I > 0, wherein > represents a positive sign;
b) K in control law c ，K t Together determine the rate at which the state variable z converges to zero, choosing a larger k c And a smaller K t So that z can converge to zero in a short time and can be ensured not to oscillate;
c) η in control law 1 ，η 2 Together determine the speed at which the tracking error epsilon converges to zero, select a larger eta 1 And a smaller eta 2 。
6. Use of a dual observer-based trajectory tracking control method according to any one of claims 1 to 5 in a robot trajectory tracking control direction.
7. An electronic device for tracking and controlling a mechanical arm track based on a double observer is characterized by comprising:
a storage medium storing a computer program;
a processing unit, which is used for exchanging data with the storage medium and executing the computer program by the processing unit when the track of the mechanical arm is controlled, so as to perform the steps of the track tracking control method based on the double observers according to any one of claims 1 to 5.
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