CN111604908B - Force and position hybrid control method for mechanical arm of overhead line maintenance robot - Google Patents

Force and position hybrid control method for mechanical arm of overhead line maintenance robot Download PDF

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CN111604908B
CN111604908B CN202010499199.2A CN202010499199A CN111604908B CN 111604908 B CN111604908 B CN 111604908B CN 202010499199 A CN202010499199 A CN 202010499199A CN 111604908 B CN111604908 B CN 111604908B
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mechanical arm
force
sliding mode
decomposition
joint
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CN111604908A (en
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刘荣海
郑欣
郭新良
蔡晓斌
沈锋
杨迎春
焦宗寒
许宏伟
周静波
虞鸿江
陈国坤
代克顺
何运华
孔旭晖
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Electric Power Research Institute of Yunnan Power Grid Co Ltd
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Electric Power Research Institute of Yunnan Power Grid Co Ltd
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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
    • B25J9/1602—Programme controls characterised by the control system, structure, architecture
    • B25J9/1605—Simulation of manipulator lay-out, design, modelling of manipulator
    • 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
    • B25J9/1628—Programme controls characterised by the control loop
    • B25J9/163—Programme controls characterised by the control loop learning, adaptive, model based, rule based expert control
    • 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
    • B25J9/1628—Programme controls characterised by the control loop
    • B25J9/1633—Programme controls characterised by the control loop compliant, force, torque control, e.g. combined with position control
    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02G—INSTALLATION OF ELECTRIC CABLES OR LINES, OR OF COMBINED OPTICAL AND ELECTRIC CABLES OR LINES
    • H02G1/00—Methods or apparatus specially adapted for installing, maintaining, repairing or dismantling electric cables or lines
    • H02G1/02—Methods or apparatus specially adapted for installing, maintaining, repairing or dismantling electric cables or lines for overhead lines or cables
    • Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
    • Y02—TECHNOLOGIES OR APPLICATIONS FOR MITIGATION OR ADAPTATION AGAINST CLIMATE CHANGE
    • Y02P—CLIMATE CHANGE MITIGATION TECHNOLOGIES IN THE PRODUCTION OR PROCESSING OF GOODS
    • Y02P90/00—Enabling technologies with a potential contribution to greenhouse gas [GHG] emissions mitigation
    • Y02P90/02—Total factory control, e.g. smart factories, flexible manufacturing systems [FMS] or integrated manufacturing systems [IMS]

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Abstract

The application discloses a force and position hybrid control method for an overhead line maintenance robot mechanical arm, which is completely independent of dynamic modeling of a system and can determine the uncertainty limit a priori. In actual use, the sliding mode controller decomposes a dynamic model of the mechanical arm executing operation on the curved surface into force, position and redundant joint subspace; in the design of a sliding mode controller, an ideal part and a robust part of a sliding mode control method are approached by a TSK fuzzy system and an adaptive PI controller respectively; and adjusting parameters of the controller on line according to an adaptive rule in the Lyapunov second stability determination. Compared with the traditional force and position hybrid control method, the force and position hybrid control method is independent of a robot model, and only needs a small amount of mechanical and geometric characteristic information about the robot and the environment.

Description

Force and position hybrid control method for mechanical arm of overhead line maintenance robot
Technical Field
The application relates to the technical field of electrical equipment, in particular to a force and position hybrid control method for an overhead line maintenance robot mechanical arm.
Background
The operation condition of the high-voltage overhead transmission line directly influences the distribution condition of the power system, and plays a key role in the safe and stable operation of the power system, so that the operation condition of the high-voltage overhead transmission line needs to be regularly checked. At present, the inspection mode of the high-voltage overhead transmission line is converted from manual inspection to special robot inspection, and the overhead line inspection robot is used for inspecting the high-voltage transmission line so as to improve the operation safety of the transmission line.
The overhead line maintenance robot needs to simultaneously adjust the contact force of the mechanical arm end effector on an operation object and the position of the end effector in the task execution, and the control problem is the force and position mixed control problem of the mechanical arm. At present, the force position hybrid control of the mechanical arm of the overhead line maintenance robot generally adopts a mechanical arm force control method with robustness on uncertainty and parameter change.
However, this force-level hybrid control method relies on complete or partial knowledge of the system dynamics, i.e., a vector machine that decomposes a mathematical model into a regression matrix and some unknown parameters using the structure of the dynamics. When the established mathematical model is complex, the identification of unknown parameters in the vector machine becomes difficult, the controllability of the mechanical arm of the overhead line maintenance robot of the system is reduced, and the smooth completion of the routing inspection task of the mechanical arm of the overhead line maintenance robot on the overhead transmission line is further influenced.
Disclosure of Invention
The application provides a force and position hybrid control method for an overhead line maintenance robot mechanical arm, which aims to solve the problem that the existing force and position hybrid control method depends on complete or partial system dynamics knowledge, so that the controllability of the overhead line maintenance robot mechanical arm is reduced, and the smooth completion of the inspection task of the overhead line maintenance robot mechanical arm on an overhead transmission line is influenced.
The application provides a force and position hybrid control method of an overhead line maintenance robot mechanical arm, which is characterized by comprising the following steps:
establishing a mechanical arm dynamic model, and performing dynamic decomposition to obtain a decomposition dynamic equation of mechanical arm motion;
designing a sliding mode controller by utilizing a decomposition kinetic equation;
respectively approaching an ideal part and a robust part of a sliding mode controller by adopting a TSK fuzzy system and an adaptive PI controller;
and adjusting parameters of the sliding mode controller on line according to a self-adaptive rule in the Lyapunov second stability determination.
Optionally, a mechanical arm dynamics model is established, and dynamics decomposition is performed to obtain a decomposition dynamics equation of mechanical arm motion, including,
establishing a mechanical arm dynamic model, wherein the mechanical arm kinematic equation is as follows:
Figure BDA0002524055190000011
wherein q is a joint displacement vector, M (q) represents an inertia matrix of the robot,
Figure BDA0002524055190000012
representing the coincidence of the coriolis matrix and the term associated with centripetal force, g (q) is a vector containing terms corresponding to the gravitational effect, τ is the vector of the input torque acting on the joint,
Figure BDA0002524055190000013
is the interaction force between the end effector and the environment, and J is the associated joint velocity vector
Figure BDA0002524055190000014
And task space velocity vector
Figure BDA0002524055190000015
Jacobian matrix of (a):
Figure BDA0002524055190000016
in the formula (I), the compound is shown in the specification,
Figure BDA0002524055190000021
is the position and direction vector of the end effector;
the environment is assumed to be a deformable plane without friction,
Figure BDA0002524055190000022
proportional to the environmental deformation:
Figure BDA0002524055190000023
in the formula (I), the compound is shown in the specification,
Figure BDA0002524055190000024
is an environmental stiffness matrix, x e Representing a point on a stationary environment;
rigidity matrix
Figure BDA0002524055190000025
The decomposition is as follows:
Figure BDA0002524055190000026
the manipulation dynamics are decomposed into three orthogonal subspaces, namely force, position, redundant joint subspaces, and the following elements are introduced:
I＝J + J+J - ,I＝K + K+K - ，
wherein the content of the first and second substances,
Figure BDA0002524055190000027
velocity vector of joint
Figure BDA0002524055190000028
And task space velocity vector
Figure BDA0002524055190000029
The decomposition is as follows:
Figure BDA00025240551900000210
Figure BDA00025240551900000211
by using
Figure BDA00025240551900000212
The second derivative of the pose vector of the end effector of (a) is:
Figure BDA00025240551900000213
by using
Figure BDA00025240551900000214
And the second derivative of the manipulator joint displacement vector to obtain:
Figure BDA00025240551900000215
get
Figure BDA00025240551900000216
To obtain a derivative function as:
Figure BDA00025240551900000217
and obtaining a decomposition kinetic equation of the mechanical arm motion by using the decomposition vectors of the joint space and the task space:
Figure BDA00025240551900000218
optionally, a sliding-mode controller is designed using the equations of decomposed dynamics, including,
define the slip plane and tracking error as:
Figure BDA00025240551900000219
where λ is a constant and subscript d represents the expected value of the variable;
defining the reference speed as:
Figure BDA00025240551900000220
by using
Figure BDA00025240551900000221
Converting the sliding surface into:
Figure BDA00025240551900000222
by using
Figure BDA00025240551900000223
The mechanical arm dynamics equation is converted into:
Figure BDA00025240551900000224
wherein:
Figure BDA0002524055190000031
designing a sliding mode controller as follows:
τ＝-(τ eq +τ rb ),τ eq ＝-(τ e +z)，
wherein, tau eq And τ rb Representing the ideal and robust parts of the control input, respectively.
Alternatively, the ideal part and the robust part of the sliding-mode controller are approximated separately using a TSK fuzzy system and an adaptive PI controller, including,
blurring system and s with TSK j Is input, tau j fuzz For output, defining the j' th joint of the mechanical arm to meet the following conditions:
when in use
Figure BDA0002524055190000032
When the temperature of the water is higher than the set temperature,
Figure BDA0002524055190000033
by using a univariate fuzzy generator, process reasoning and central mean deblurring, the output of the jth joint of the mechanical arm maintains the following relationship:
Figure BDA0002524055190000034
consider the following minimum input as s 1 ,...,s m And n is output as
Figure BDA0002524055190000035
The fuzzy sliding mode controller:
Figure BDA0002524055190000036
to pair
Figure BDA0002524055190000037
Uncertainty, error bound, and θ in j Making an estimate of e j Replacement of
Figure BDA0002524055190000038
Estimate error of
Figure BDA0002524055190000039
Equivalent part tau to be input by fuzzy sliding mode controller eq Estimated as
Figure BDA00025240551900000310
The governing equation is
Figure BDA00025240551900000311
The kinematic equation of the mechanical arm is used as the dynamic equation of the manipulator arm contacting with the environment, and the control moment is assumed to be
Figure BDA00025240551900000312
Wherein, the first and the second end of the pipe are connected with each other,
Figure BDA00025240551900000313
expressing the law of adaptation as
Figure BDA00025240551900000314
Wherein γ is a common principle.
Optionally, parameters of the sliding mode controller are adjusted online according to an adaptive rule in the Lyapunov second stability determination, including,
ideal controllers, τ, under precise and well-known conditions of the object model, according to Lyapunov's second stability theorem id And a stabilizing system existsThe following approximate relationship: tau is id ＝-τ eq -K D s＝-(τ eq +K D s)，
Using tau id ＝-τ eq -K D s＝-(τ eq +K D And s) adjusting parameters of the sliding mode controller on line.
The application provides a force and position hybrid control method for an overhead line maintenance robot mechanical arm, which is completely independent of dynamic modeling of a system and can determine the uncertainty limit a priori. During actual use, the sliding mode controller decomposes a dynamic model of the mechanical arm executing operation on the curved surface into force, position and redundant joint subspaces; in the design of a sliding mode controller, an ideal part and a robust part of a sliding mode control method are approached by a TSK fuzzy system and an adaptive PI controller respectively; and then, adjusting the parameters of the controller on line according to an adaptive rule in the Lyapunov second stability determination. Compared with the traditional force and position hybrid control method, the force and position hybrid control method for the mechanical arm of the overhead line maintenance robot does not depend on a robot model, and only needs a small amount of mechanical and geometric characteristic information about the robot and the environment. In addition, the method eliminates the need to determine uncertainty bounds, which are evaluated using the proposed strategy.
Drawings
In order to more clearly explain the technical solution of the present application, the drawings needed to be used in the embodiments are briefly described below, and it is obvious for those skilled in the art to obtain other drawings without creative efforts.
FIG. 1 is a flow chart of a force-position hybrid control method for an overhead line inspection robot mechanical arm according to the present application;
FIG. 2 is a schematic diagram of a sliding mode controller design;
FIG. 3 is an idealized part of the sliding mode controller and the robust part replacement principle;
fig. 4 is a parameter schematic diagram of an online adjustment sliding mode controller.
Detailed Description
The application provides a force and position hybrid control method for an overhead line maintenance robot mechanical arm, which is used for adjusting the contact force of an end effector of the mechanical arm on an operation object and the position of the end effector.
Fig. 1 is a flowchart of a force-position hybrid control method for an overhead line inspection robot mechanical arm according to the present application, and as shown in fig. 1, the force-position hybrid control method for an overhead line inspection robot mechanical arm includes:
and S100, establishing a mechanical arm dynamic model, and performing dynamic decomposition to obtain a decomposition dynamic equation of mechanical arm motion.
In the application, a mechanical arm dynamic model is established, and dynamic decomposition is carried out to obtain a decomposition dynamic equation of mechanical arm motion, which comprises,
and (3) establishing a mechanical arm dynamic model, wherein a mechanical arm kinematic equation is as follows:
Figure BDA0002524055190000041
wherein q is a joint displacement vector, M (q) represents an inertia matrix of the robot,
Figure BDA0002524055190000042
representing the coincidence of the coriolis matrix and the term associated with centripetal force, g (q) is a vector containing terms corresponding to the gravitational effect, τ is the vector of the input torque acting on the joint,
Figure BDA0002524055190000043
is the interaction force between the end effector and the environment, and J is the coupled joint velocity vector
Figure BDA0002524055190000044
And task space velocity vector
Figure BDA0002524055190000045
Jacobian matrix of (a):
Figure BDA0002524055190000046
in the formula (I), the compound is shown in the specification,
Figure BDA0002524055190000047
is the position and orientation vector of the end effector.
The environment is assumed to be a deformable plane without friction,
Figure BDA0002524055190000048
proportional to the environmental deformation:
Figure BDA0002524055190000049
in the formula (I), the compound is shown in the specification,
Figure BDA00025240551900000410
is an environmental stiffness matrix, x e Representing points on a stationary environment.
In this application, a stiffness matrix is formed
Figure BDA00025240551900000411
The decomposition is as follows:
Figure BDA0002524055190000051
the manipulation dynamics are decomposed into three orthogonal subspaces, namely force, position, redundant joint subspaces, and the following elements are introduced:
I＝J + J+J - ,I＝K + K+K - (5)
wherein the content of the first and second substances,
Figure BDA0002524055190000052
velocity vector of joint
Figure BDA0002524055190000053
And task space velocity vector
Figure BDA0002524055190000054
The decomposition is as follows:
Figure BDA0002524055190000055
Figure BDA0002524055190000056
by using
Figure BDA0002524055190000057
The second derivative of the pose vector of the end effector of (a) is:
Figure BDA0002524055190000058
by using
Figure BDA0002524055190000059
And the second derivative of the manipulator joint displacement vector to obtain:
Figure BDA00025240551900000510
get
Figure BDA00025240551900000511
A derivative function is obtained as:
Figure BDA00025240551900000512
and obtaining a decomposition kinetic equation of the motion of the mechanical arm by using the decomposition vectors of the joint space and the task space:
Figure BDA00025240551900000513
where the components of F, x and q are decomposed into three orthogonal subspaces to facilitate the design of the sliding-mode controller.
And S200, designing a sliding mode controller by using a decomposition kinetic equation.
Fig. 2 is a schematic diagram of a design of a sliding mode controller, as shown in fig. 2, which is designed by using a decomposition kinetic equation in the present application, including,
define the slip plane and tracking error as:
Figure BDA00025240551900000514
where λ is a constant and the subscript d represents the expected value of the variable.
Defining the reference speed as:
Figure BDA00025240551900000515
by using
Figure BDA00025240551900000516
Converting the sliding surface into:
Figure BDA00025240551900000517
by using
Figure BDA00025240551900000518
The mechanical arm dynamics equation is converted into:
Figure BDA00025240551900000519
wherein:
Figure BDA0002524055190000061
Figure BDA0002524055190000062
designing a sliding mode controller as follows:
τ＝-(τ eq +τ rb ),τ eq ＝-(τ e +z) (17)
wherein, tau eq And τ rb Representing the ideal and robust parts of the control input, respectively.
And step S300, respectively approaching an ideal part and a robust part of the sliding mode controller by adopting a TSK (Takagi-Sugeno-Kang) fuzzy system and an adaptive PI controller.
Fig. 3 shows the ideal part and the robust part of the sliding mode controller, and as shown in fig. 3, the TSK fuzzy system and the adaptive PI controller are adopted in the present application to respectively approximate the ideal part and the robust part of the sliding mode controller, including,
blurring System and s with TSK j Is input, tau j fuzz For output, defining the j' th joint of the mechanical arm to meet the following conditions:
when in use
Figure BDA0002524055190000063
When the temperature of the water is higher than the set temperature,
Figure BDA0002524055190000064
by using a univariate fuzzy generator, process reasoning and central mean deblurring, the output of the jth joint of the mechanical arm maintains the following relationship:
Figure BDA0002524055190000065
consider the following minimum input as s 1 ,...,s m And n is output as
Figure BDA0002524055190000066
Fuzzy slide ofA mold controller:
Figure BDA0002524055190000067
for is to
Figure BDA0002524055190000068
Uncertainty, error bound, and θ in j Making an estimate of e j Replacement of
Figure BDA0002524055190000069
The estimation error is:
Figure BDA00025240551900000610
equivalent part tau to be input by fuzzy sliding mode controller eq The estimation is as follows:
Figure BDA00025240551900000611
the control equation is:
Figure BDA00025240551900000612
taking a mechanical arm kinematic equation as a dynamic equation of the manipulator arm in contact with the environment, and assuming that the control moment is as follows:
Figure BDA00025240551900000613
wherein the content of the first and second substances,
Figure BDA00025240551900000614
the adaptation law is expressed as:
Figure BDA00025240551900000615
Figure BDA00025240551900000616
Figure BDA00025240551900000617
and S400, adjusting parameters of the sliding mode controller on line according to the self-adaptive rule in the Lyapunov second stability determination.
Fig. 4 is a schematic diagram of parameters of a sliding mode controller for online adjustment, as shown in fig. 4, according to an adaptive rule in the Lyapunov second stability determination, parameters of the sliding mode controller for online adjustment in the present application, including,
ideal controllers, τ, under precise and well-known conditions of the object model, according to Lyapunov's second stability theorem id The following approximate relationship exists with a stable system:
τ id ＝-τ eq -K D s＝-(τ eq +K D s) (27)
using tau id ＝-τ eq -K D s＝-(τ eq +K D And s) adjusting parameters of the sliding mode controller on line.
The application provides a force and position hybrid control method for an overhead line maintenance robot mechanical arm, which is completely independent of dynamic modeling of a system and can determine the uncertainty limit a priori. During actual use, the sliding mode controller decomposes a dynamic model of the mechanical arm executing operation on the curved surface into force, position and redundant joint subspaces; in the design of a sliding mode controller, an ideal part and a robust part of a sliding mode control method are approached by a TSK fuzzy system and an adaptive PI controller respectively; and then, adjusting the parameters of the controller on line according to an adaptive rule in the Lyapunov second stability determination. Compared with the traditional force position hybrid control method, the force position hybrid control method of the overhead line maintenance robot mechanical arm does not depend on a robot model, and only needs a small amount of mechanical and geometric characteristic information about the robot and the environment. In addition, the method eliminates the need to determine uncertainty bounds, which are evaluated using the proposed strategy.
The above-described embodiments of the present application do not limit the scope of the present application.

Claims (2)

1. A force and position hybrid control method for an overhead line maintenance robot mechanical arm is characterized by comprising the following steps:
establishing a mechanical arm dynamic model, and performing dynamic decomposition to obtain a decomposition dynamic equation of mechanical arm motion;
designing a sliding mode controller by utilizing a decomposition kinetic equation, and specifically defining sliding surfaces and tracking errors as follows:
Figure FDA0003567115140000011
where λ is a constant and subscript d represents the expected value of the variable;
defining the reference speed as:
Figure FDA0003567115140000012
by using
Figure FDA0003567115140000013
Converting the sliding surface into:
Figure FDA0003567115140000014
by using
Figure FDA0003567115140000015
The mechanical arm dynamics equation is converted into:
Figure FDA0003567115140000016
wherein:
Figure FDA0003567115140000017
designing a sliding mode controller as follows:
τ＝-(τ eq +τ rb ),τ eq ＝-(τ e +z)，
wherein, tau eq And τ rb An ideal part and a robust part representing the control input, respectively;
respectively approaching to an ideal part and a robust part of the sliding mode controller by adopting a TSK fuzzy system and an adaptive PI controller, and concretely comprises the steps of adopting the TSK fuzzy system and s j Is input, tau j fuzz For output, defining the j' th joint of the mechanical arm to meet the following conditions:
when in use
Figure FDA00035671151400000117
When the utility model is used, the water is discharged,
Figure FDA0003567115140000018
by using a univariate fuzzy generator, process reasoning and central mean deblurring, the output of the jth joint of the mechanical arm maintains the following relationship:
Figure FDA0003567115140000019
consider the following minimum input as s 1 ,...,s m And n is output as
Figure FDA00035671151400000110
The fuzzy sliding mode controller:
Figure FDA00035671151400000111
for is to
Figure FDA00035671151400000112
Uncertainty, error bound, and θ in j Making an estimation using e j Replacement of
Figure FDA00035671151400000113
Estimate the error as
Figure FDA00035671151400000114
Equivalent part tau to be input by fuzzy sliding mode controller eq Estimated as
Figure FDA00035671151400000115
The control equation is
Figure FDA00035671151400000116
The kinematic equation of the mechanical arm is used as the dynamic equation of the manipulator arm contacting with the environment, and the control moment is assumed to be
Figure FDA0003567115140000021
Wherein the content of the first and second substances,
Figure FDA0003567115140000022
express the law of accommodation as
Figure FDA0003567115140000023
Wherein gamma is a common principle;
adjusting parameters of the sliding mode controller on line according to the self-adaptive rule in the Lyapunov second stability determinationThe body consists of an ideal controller, τ, under precise and well-known conditions of the object model, according to the Lyapunov second stability theorem id The following approximate relationship exists with a stable system: tau is id ＝-τ eq -K D s＝-(τ eq +K D s)，
Using tau id ＝-τ eq -K D s＝-(τ eq +K D And s) adjusting parameters of the sliding mode controller on line.
2. The force-position hybrid control method of the overhead line inspection robot manipulator of claim 1, wherein a manipulator dynamics model is established and a dynamics decomposition is performed to obtain a decomposition dynamics equation of the manipulator motion, comprising,
establishing a mechanical arm dynamic model, wherein the mechanical arm kinematic equation is as follows:
Figure FDA0003567115140000024
wherein q is a joint displacement vector, M (q) represents an inertia matrix of the robot,
Figure FDA0003567115140000025
representing the coincidence of the coriolis matrix and the term associated with centripetal force, g (q) is a vector containing terms corresponding to the gravitational effect, τ is the vector of the input torque acting on the joint,
Figure FDA0003567115140000026
is the interaction force between the end effector and the environment, and J is the coupled joint velocity vector
Figure FDA0003567115140000027
And task space velocity vector
Figure FDA0003567115140000028
Jacobian matrix of (a):
Figure FDA0003567115140000029
in the formula (I), the compound is shown in the specification,
Figure FDA00035671151400000210
is the position and direction vector of the end effector;
the environment is assumed to be a deformable plane without friction,
Figure FDA00035671151400000211
proportional to the environmental deformation:
Figure FDA00035671151400000212
in the formula (I), the compound is shown in the specification,
Figure FDA00035671151400000213
is an ambient stiffness matrix, x e Representing a point on a stationary environment;
rigidity matrix
Figure FDA00035671151400000214
The decomposition is as follows:
Figure FDA00035671151400000215
the manipulation dynamics are decomposed into three orthogonal subspaces, namely force, position and redundant joint subspaces, and the following elements are introduced:
I＝J + J+J - ,I＝K + K+K - ，
wherein the content of the first and second substances,
Figure FDA00035671151400000216
velocity vector of joint
Figure FDA00035671151400000217
And task space velocity vector
Figure FDA00035671151400000218
The decomposition is as follows:
Figure FDA00035671151400000219
Figure FDA00035671151400000220
by using
Figure FDA0003567115140000031
The second derivative of the pose vector of the end effector of (a) is:
Figure FDA0003567115140000032
by using
Figure FDA0003567115140000033
And the second derivative of the manipulator joint displacement vector to obtain:
Figure FDA0003567115140000034
get the
Figure FDA0003567115140000035
A derivative function is obtained as:
Figure FDA0003567115140000036
and obtaining a decomposition kinetic equation of the mechanical arm motion by using the decomposition vectors of the joint space and the task space:
Figure FDA0003567115140000037
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