CN113799126B - Robot machining path optimization method based on deformation and flexibility - Google Patents

Robot machining path optimization method based on deformation and flexibility Download PDF

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CN113799126B
CN113799126B CN202111054001.0A CN202111054001A CN113799126B CN 113799126 B CN113799126 B CN 113799126B CN 202111054001 A CN202111054001 A CN 202111054001A CN 113799126 B CN113799126 B CN 113799126B
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coordinate system
robot
joint
tool
path
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CN113799126A (en
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李文龙
王刚
王东方
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Huazhong University of Science and 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
    • B25J11/00—Manipulators not otherwise provided for
    • B25J11/005—Manipulators for mechanical processing tasks
    • 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
    • B25J9/1612—Programme controls characterised by the hand, wrist, grip control

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  • Engineering & Computer Science (AREA)
  • Robotics (AREA)
  • Mechanical Engineering (AREA)
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Abstract

The invention belongs to the technical field related to machining, and discloses a robot machining path optimization method based on deformation and dexterity, which comprises the following steps: acquiring the dexterity of the robot at a path point to establish a functional relation between the dexterity and a rotation angle as well as a joint angle of a robot joint; obtaining cutting force obtained by a cutter at each path point, and obtaining the deformation of the robot according to the cutting force; distributing weight values for the dexterity and the deformation to obtain an objective function; solving an objective function in the constraint ranges of joint angles, dexterity and deformation, and further obtaining the optimal rotation angle of the robot at a path point; and converting the posture of the robot under the optimal rotation angle of the path point according to an inverse kinematics calculation method of the robot to obtain the optimal rotation angle of all joints of the robot under the optimal rotation angle. This application can obtain the machining path of robot under deflection and dexterity restraint, and machining path is reasonable accurate more.

Description

Robot machining path optimization method based on deformation and flexibility
Technical Field
The invention belongs to the technical field of machining, and particularly relates to a robot machining path optimization method based on deformation and dexterity.
Background
The industrial robot has the advantages of high flexibility, large working range, low cost and the like, and the robot is used for clamping executive tools such as an electric spindle, a cutter and the like to replace manual or numerical control machine tool machining, so that the teaching workers such as milling, grinding, polishing, riveting and the like of large-scale complex parts with small allowance are realized, and the industrial robot becomes a new trend of intelligent manufacturing. The robot processing planning method has the advantages that deformation is easy to occur in the processing, clamping and assembling processes due to the characteristic of weak rigidity of the thin wall of a large aircraft skin part, a design model does not have reference value, and robot processing planning is intelligently performed based on field measurement point cloud. The existing processing path planning research based on the measurement point cloud mainly aims at the processing of a numerical control machine tool, and rarely relates to the processing path planning of a robot, and compared with the processing of the machine tool, the processing of the robot has the differences of redundant freedom degree, rigidity, flexibility and the like, so that the research on the processing path planning of the robot based on the field measurement point cloud is necessarily conducted around.
The measured point cloud has inherent defects of noise, isolated points and the like, and the discrete points have intervals, so that the problems of local jitter, sudden change and the like of a generated processing track are easily caused, and all joints of the robot exceed the bearing limit of kinematics and dynamics in the processing process. Therefore, a feasible robot processing track needs to satisfy three key conditions: (1) the path points are smooth; (2) the cutter axis vector is smooth; (3) The robot pose changes caused by the robot tip pose rotating about the tool axis are smooth. The existing research mainly focuses on fairing optimization of path points and cutter axis vectors, but only considers the change of the robot attitude caused by the rotation of the robot end attitude around the cutter axis.
Disclosure of Invention
Aiming at the defects or improvement requirements of the prior art, the invention provides a robot machining path optimization method based on deformation and dexterity, so that the machining path of the robot under the constraints of the deformation and the dexterity is obtained, and the machining path is more reasonable and accurate.
To achieve the above object, according to one aspect of the present invention, there is provided a method for optimizing a machining path of a robot based on deformation and dexterity, the robot including a plurality of joints connected in series, a machining tool connected to an end of each joint, the tool machining a workpiece to be machined, the method comprising: s1: method for acquiring robot path points by MVR indexes B p Ti Dexterity of (k) i To establish a dexterity k i Angle of rotation gamma i And joint angle theta of robot joint i，k (k =1, \8230;, n), where k is the kth joint and n is the total number of robot joints; s2: obtaining cutting force obtained by a cutter at each path point, and obtaining the deformation of the robot according to the cutting force; s3: distributing weight values for the dexterity and the deformation to obtain an objective function; s4: solving the objective function in the constraint ranges of the joint angle, the dexterity and the deformation to obtain the minimum value of the objective function, and further obtaining the path point of the robot B p Ti At an optimum rotation angle gamma i，best (ii) a S5: according to the inverse kinematics calculation method of the robot, the robot is positioned at the path point B p Ti Of optimum rotation angle gamma i，best Posture of falling
Figure BDA0003253862980000021
The optimal rotation angle of the robot can be obtained by conversionOptimal articulated angle θ of the joints i，k，best 。
Preferably, the path points in step S5 B p Ti Of optimum rotation angle gamma i，best Posture of falling down
Figure BDA0003253862980000022
The coordinate system of the base coordinate system { B }, the coordinate system of the joint end { E }, the coordinate system of the tool { T }, and the coordinate system of the tool path point { P } i The conversion relationship between the two is obtained,
Figure BDA0003253862980000023
wherein the content of the first and second substances,
Figure BDA0003253862980000024
as a tool path point coordinate system { P } i The transformation relation with respect to the tool coordinate system T,
Figure BDA0003253862980000025
is the transformation relation of the tool coordinate system { T } relative to the joint end coordinate system { E },
Figure BDA0003253862980000026
is the transformation relation of the joint end coordinate system { E } relative to the base coordinate system { B },
Figure BDA0003253862980000031
as a tool path point coordinate system { P } i The transformation relation with respect to the base coordinate system B.
Preferably, the dexterity κ i The calculation formula of (c) is:
Figure BDA0003253862980000032
wherein, the first and the second end of the pipe are connected with each other,
Figure BDA0003253862980000033
u v，i is a speed
Figure BDA0003253862980000034
The unit vector of (a) is,
Figure BDA0003253862980000035
to correspond to the velocity
Figure BDA0003253862980000036
The diagonalized weight matrix of (a) is,
Figure BDA0003253862980000037
to correspond to joint velocity
Figure BDA0003253862980000038
The diagonalized weight matrix of (a) is,
Figure BDA0003253862980000039
wherein, the first and the second end of the pipe are connected with each other,
Figure BDA00032538629800000310
as a waypoint B p T(i+1) The transformation relation of the terminal coordinate system of the robot relative to the base coordinate system,
Figure BDA00032538629800000311
as a path point B p Ti The transformation relation of the terminal coordinate system of the robot relative to the base coordinate system, and delta t is the slave path point B p Ti To B p T(i+1) Time taken, J (θ) i ) As a waypoint B p Ti Jacobian matrix of the robot, theta i For the robot at the path point B p Ti And processing a joint angle matrix of a plurality of joints of the robot.
Preferably, step S2 specifically includes the following steps:
s21: obtaining the cutting force of the cutter under the base coordinate system B F i ：
Figure BDA00032538629800000312
Wherein, the first and the second end of the pipe are connected with each other, B f i and B m i is a 3 x 1 vector, representing force and moment, respectively; B f i ＝f n B n Ti +f f B t Ti ， B t Ti which is the feed direction of the cutting of the tool, B n Ti ＝ B v Ti × B t Ti ， B v Ti is the cutter shaft direction of the cutter; f. of f And f n Are respectively as B t Ti And B n Ti a die length of a directional cutting force component;
s22: cutting force of the tool in the basic coordinate system B F i Conversion into cutting force in a waypoint coordinate system
Figure BDA00032538629800000411
The transformation formula is as follows:
Figure BDA0003253862980000041
wherein the content of the first and second substances,
Figure BDA0003253862980000042
for the robot base coordinate system relative to the path point coordinate system { P } i The rotation matrix of Sr is the operator of the rotation matrix, if R is a 3 x 3 rotation matrix,
Figure BDA0003253862980000043
s23: according to the force accompanying transformation, the cutting force of the end of the cutter under the joint end coordinate system is obtained E F i ：
Figure BDA0003253862980000044
Wherein, the first and the second end of the pipe are connected with each other,
Figure BDA0003253862980000045
Figure BDA0003253862980000046
is the coordinate of the origin of the path point coordinate system in the joint end coordinate system,
Figure BDA0003253862980000047
for the purpose of the corresponding anti-symmetric matrix,
Figure BDA0003253862980000048
is a coordinate system of path points { P i Rotating matrix relative to the coordinate system of the joint end, ad is force accompanying transformation;
s24: cutting force of the end of the tool in the joint end coordinate system E F i Conversion into cutting force in the base coordinate system B F Ei The transformation formula is:
Figure BDA0003253862980000049
wherein the content of the first and second substances,
Figure BDA00032538629800000410
to be at the path point B p Ti A rotation matrix of the joint end coordinate system relative to the robot base coordinate system, E f E is composed of E F i The force component of (a) is, E m E is composed of E F i A moment portion of (a);
s25: according to cutting force of tool tip B F Ei Obtaining the deformation of the robot in the coordinate system of the joint end E D Ei ：
Figure BDA0003253862980000051
Wherein, the first and the second end of the pipe are connected with each other,
Figure BDA0003253862980000052
to be at the path point B p Ti A rotation matrix of the robot base coordinate system relative to the joint end coordinate system; k D，i To be at the path point B p Ti Cartesian space stiffness matrix, K, of the robot D，i ＝J(θ i ) -T K θ J(θ i ) -1 In which K is θ ＝diag(K 1 ，K 2 ，…，K 6 )，K k Representing the joint stiffness of the k-th joint of the robot;
s26: the deformation of the robot in the coordinate system of the joint end E D Ei Transformed into deformation in a waypoint coordinate system
Figure BDA0003253862980000053
The transformation formula is:
Figure BDA0003253862980000054
wherein the content of the first and second substances,
Figure BDA0003253862980000055
preferably, step S3 further comprises assigning said dexterity κ i And amount of deformation
Figure BDA0003253862980000056
Carrying out non-dimensionalization, wherein the non-dimensionalization formula is as follows:
Figure BDA0003253862980000057
Figure BDA0003253862980000058
wherein, min (1/kappa) i ) And max (1/κ) i ) Respectively 1/k in all path points i Maximum and minimum values of;
Figure BDA0003253862980000061
and
Figure BDA0003253862980000062
respectively the maximum and minimum of all waypoints.
Preferably, the objective function is:
Figure BDA0003253862980000063
wherein w 1 And w 2 Are weight coefficients.
Preferably, in step S4, the constraint range of the joint angle is:
θ i，k，max ≥θ i，k ≥θ i，k，min
wherein, theta i，k，max And theta i，k，min Represents the maximum and minimum values of the k-th joint angle;
the restricted range of the dexterity is as follows:
σ i，1 ≥κ i ≥σ i，6
κ i ≥κ min
wherein σ i，1 And σ i，6 Are respectively J v (θ i ) Maximum and minimum singular values of, k min Presetting a minimum value according to the processing requirement;
the constrained range of the deformation amount is as follows:
Figure BDA0003253862980000064
wherein the content of the first and second substances,
Figure BDA0003253862980000065
is a preset upper processing limit.
Preferably, the following steps are adopted to obtain the path points of the robot B p Ti ：
(1) Respectively fitting the original path points and the cutter axis vectors by adopting an NURBS curve to obtain a path point NURBS curve
Figure BDA0003253862980000066
Sum-axis vector NURBS curve
Figure BDA0003253862980000067
(2) To path point NURBS curve
Figure BDA0003253862980000068
NURBS curve of sum-cutter axis vector point
Figure BDA0003253862980000069
And performing parameter synchronization processing:
Figure BDA0003253862980000071
wherein u is Q (u P ) Is composed of
Figure BDA0003253862980000072
Arbitrary parameter u of P Corresponding curve
Figure BDA0003253862980000073
Parameter u of Q ，
Figure BDA0003253862980000074
Is composed of
Figure BDA0003253862980000075
Parameter set of (3)
Figure BDA0003253862980000076
Is known to be present at the known point in (a),
Figure BDA0003253862980000077
is composed of
Figure BDA0003253862980000078
Parameter set of
Figure BDA0003253862980000079
A known point of (a);
(3) For the NURBS curve
Figure BDA00032538629800000710
And NURBS curve
Figure BDA00032538629800000711
The curves after being respectively offset by the preset distance are as follows:
Figure BDA00032538629800000713
wherein, the first and the second end of the pipe are connected with each other,
Figure BDA00032538629800000714
as NURBS curve
Figure BDA00032538629800000715
The curve after the offset of the preset distance,
Figure BDA00032538629800000716
as NURBS curve
Figure BDA00032538629800000717
Offsetting the curve after the preset distance, wherein d is a second preset distance, d = R-a, R is the radius of the cutter, a is the boundary machining allowance, and c is the vector offset distance of the final cutter end central point relative to the initial cutter end central point along the cutter central axis;
Figure BDA00032538629800000718
is the NURBS curve
Figure BDA00032538629800000719
The tangent vector of (a) is,
Figure BDA00032538629800000720
Figure BDA00032538629800000721
is the NURBS curve
Figure BDA00032538629800000722
The tangent vector of (a) is,
Figure BDA00032538629800000723
v(u P ) Is u P The vector of the central axis of the tool in (a),
Figure BDA00032538629800000724
(4) And uniformly sampling the biased NURBS curve to obtain a final path point and a corresponding cutter axis vector point, and further obtaining a corresponding cutter axis vector.
In general, compared with the prior art, the method for optimizing the machining path of the robot considering the rigidity and the dexterity, which is provided by the invention, has the following beneficial effects:
1. by establishing a combined objective function of dexterity and deformation, the optimal rotation angle of a corresponding tool can be reversely calculated, and the rotation angle, namely the pose, of each joint of the robot can be obtained through change, so that the pose is solved more accurately, and the obtained track is smoother;
2. based on strict coordinate transformation and matrix calculation, the calculation precision is high, the error is small, the obtained robot posture accuracy is high, and engineering application is facilitated;
3. through non-dimensionalization processing, the dexterity variables and the deformation of different dimensions can be unified, and further the establishment of an objective function considering the two variables jointly becomes possible.
Drawings
Fig. 1 is a schematic view of the robot processing according to the embodiment.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is described in further detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention. In addition, the technical features involved in the embodiments of the present invention described below may be combined with each other as long as they do not conflict with each other.
The invention provides a robot machining path optimization method based on deformation and dexterity, which comprises the following steps S1-S5.
The robot comprises a plurality of joints which are connected in sequence, the tail ends of the joints are connected with a processing cutter, and the cutter processes a workpiece to be processed.
Before describing the method in the present application in detail, the coordinate system and its transformation relationship in the present application are described below, as shown in fig. 1.
a) At each path point B p Ti Where a coordinate system is established, the unit vector in the z-axis direction can be expressed as:
t zi ＝ B v Ti
wherein the content of the first and second substances, B v Ti as a waypoint B p Ti The knife axis unit vector of (2);
the unit vector in the X-axis direction is:
Figure BDA0003253862980000091
wherein the content of the first and second substances,
Figure BDA0003253862980000092
to be slave path points B p Ti The direction vector to the origin of the base coordinate system { B }, according to the right hand rule, the unit vector in the y-direction can be expressed as:
t yi ＝t zi ×t xi
b) The tool coordinate system (T) rotates around the z-axis of the tool coordinate system without influencing the processing result, so the tool coordinate system (T) has a redundant degree of freedom of rotating around the z-axis, and therefore, the path point B p Ti Odd transformation matrix of position coordinate system relative to tool coordinate system
Figure BDA0003253862980000093
Can be expressed in relation to the angle of rotationMatrix function of degree γ:
Figure BDA0003253862980000094
in the formula, gamma i The value range of (a) is 0-360 degrees, the rotation direction is anticlockwise rotation around the z axis of the cutter coordinate system (T), and x is i 、y i 、z i Respectively representing points B p Ti Relative to the orientation of the coordinate axes of the robot base coordinate system { B }.
c) Attitude matrix of robot
Figure BDA0003253862980000095
Can be expressed as:
Figure BDA0003253862980000096
wherein the content of the first and second substances,
Figure BDA0003253862980000097
representing a transformation matrix, theta, between adjacent joint coordinate systems of the robot k Expressing the joint angle of the kth joint of the robot to obtain the transformation relation of the tail end of the robot relative to a base coordinate system;
d) And finally establishing a transformation relation of the current path point coordinate system relative to the robot base coordinate system by using the coordinate transformation relation:
Figure BDA0003253862980000098
the tool setting device can be obtained by calibrating a tool by using a tool setting gauge.
The robot milling mathematical model is established through the steps to obtain a transformation matrix from the path point to the robot base coordinate system
Figure BDA0003253862980000101
S1: MVR index acquisition machineRobot at path point B p Ti Dexterity of (k) i To establish a dexterity k i Angle of rotation gamma i And joint angle theta of robot joint i，k (k =1, \8230;, n), where k is the kth joint and n is the total number of robot joints;
wherein the path point B p Ti The path point may be the original path point or the processed path point. The treatment method is as follows:
(1) Respectively fitting the original path points and the cutter axis vectors by adopting an NURBS curve to obtain a path point NURBS curve
Figure BDA0003253862980000102
Sum arbor vector NURBS curve
Figure BDA0003253862980000103
(2) NURBS curve for path points
Figure BDA0003253862980000104
NURBS curve of summation axis vector point
Figure BDA0003253862980000105
And performing parameter synchronization processing:
Figure BDA0003253862980000106
wherein u is Q (u P ) Is composed of
Figure BDA0003253862980000107
Arbitrary parameter u of P Corresponding curve
Figure BDA0003253862980000108
Parameter u of Q ，
Figure BDA0003253862980000109
Is composed of
Figure BDA00032538629800001010
Parameter set of
Figure BDA00032538629800001011
Is known to be present at the known point in (a),
Figure BDA00032538629800001012
is composed of
Figure BDA00032538629800001013
Parameter set of
Figure BDA00032538629800001014
A known point of (a);
(3) For the NURBS curve
Figure BDA00032538629800001015
And NURBS curve
Figure BDA00032538629800001016
The curves after being respectively offset by the preset distance are as follows:
Figure BDA00032538629800001018
wherein the content of the first and second substances,
Figure BDA00032538629800001019
as NURBS curve
Figure BDA00032538629800001020
The curve after the offset of the preset distance,
Figure BDA00032538629800001021
as NURBS curve
Figure BDA00032538629800001022
Offsetting the curve after the preset distance, wherein d is a second preset distance, d = R-a, R is the radius of the cutter, a is the boundary machining allowance, and C is the final cutter end central point phaseThe distance of vector offset of the central point of the tail end of the initial cutter along the central axis of the cutter;
Figure BDA00032538629800001023
as the NURBS curve
Figure BDA0003253862980000111
The tangent vector of (a) is,
Figure BDA0003253862980000112
Figure BDA0003253862980000113
is the NURBS curve
Figure BDA0003253862980000114
The tangent vector of (a) is,
Figure BDA0003253862980000115
v(u P ) Is u P The vector of the central axis of the tool in (a),
Figure BDA0003253862980000116
(4) Uniformly sampling the biased NURBS curve to obtain a final path point P T ＝{ B p T1 ， B p T2 ，…， B p To And the corresponding arbor vector point Q T ＝{ B q T1 ， B q T2 ，…， B q To Get the corresponding arbor vector V T ＝{ B v T1 ， B v T2 ，…， B v To }。
Said dexterity κ i The calculation formula of (2) is as follows:
Figure BDA0003253862980000117
wherein the content of the first and second substances,
Figure BDA0003253862980000118
u v，i is speed
Figure BDA0003253862980000119
The unit vector of (a) is calculated,
Figure BDA00032538629800001110
to correspond to the speed
Figure BDA00032538629800001111
The diagonalized weight matrix of (a) is,
Figure BDA00032538629800001112
to correspond to joint velocity
Figure BDA00032538629800001113
The diagonalized weight matrix of (a) is,
Figure BDA00032538629800001114
wherein the content of the first and second substances,
Figure BDA00032538629800001115
as a waypoint B p T(i+1) The transformation relation of the terminal coordinate system of the robot relative to the base coordinate system,
Figure BDA00032538629800001116
as a waypoint B p Ti The transformation relation of the terminal coordinate system of the robot relative to the base coordinate system, and delta t is the slave path point B p Ti To B p T(i+1) Time spent, J (θ) i ) As a waypoint B p Ti Jacobian matrix of the robot, theta i For the robot at the path point B p Ti And processing a joint angle matrix of a plurality of joints of the robot.
S2: and acquiring the cutting force obtained by the tool at each path point, and obtaining the deformation of the robot according to the cutting force. The disclosed device is provided with:
s21: obtaining the cutting force of the cutter under the base coordinate system B F i ：
Figure BDA0003253862980000121
Wherein, the first and the second end of the pipe are connected with each other, B f i and B m i is a 3 x 1 vector, representing force and moment, respectively; B f i ＝f n B n Ti +f f B t Ti ， B t Ti which is the feed direction of the cutting of the tool, B n Ti ＝ B v Ti × B t Ti ， B v Ti is the cutter shaft direction of the cutter; f. of f And f n Are respectively as B t Ti And B n Ti a die length of a directional cutting force component;
s22: cutting force of the tool in the basic coordinate system B F i Converted into cutting force in a path point coordinate system
Figure BDA0003253862980000122
The transformation formula is as follows:
Figure BDA0003253862980000123
wherein the content of the first and second substances,
Figure BDA0003253862980000124
for the robot base coordinate system relative to the path point coordinate system { P } i The rotation matrix of Sr, the operator of Sr to the rotation matrix, if R is a 3 x 3 rotation matrix,
Figure BDA0003253862980000125
s23: according to the force accompanying transformation, the cutting force of the end of the cutter under the joint end coordinate system is obtained E F i ：
Figure BDA0003253862980000126
Wherein the content of the first and second substances,
Figure BDA0003253862980000127
Figure BDA0003253862980000128
is the coordinate of the origin of the coordinate system of the path point in the coordinate system of the joint end,
Figure BDA0003253862980000131
is a corresponding anti-symmetric matrix, and,
Figure BDA0003253862980000132
is a coordinate system of path points { P i Rotation matrix relative to the joint end coordinate system, ad is force adjoint transformation;
s24: cutting force of the end of the tool in the joint end coordinate system E F i Conversion into cutting force in the base coordinate system B F Ei The transformation formula is:
Figure BDA0003253862980000133
wherein the content of the first and second substances,
Figure BDA0003253862980000134
to be at the path point B p Ti A rotation matrix of the joint end coordinate system relative to the robot base coordinate system, E f E is composed of E F i The force component of (a) is, E m E is composed of E F i A moment portion of (a);
s25: according to cutting force of tool tip B F Ei Obtaining the deformation of the robot in the joint end coordinate system E D Ei ：
Figure BDA0003253862980000135
Wherein the content of the first and second substances,
Figure BDA0003253862980000136
to be at the path point B p Ti A rotation matrix of the robot base coordinate system relative to the joint base coordinate system; k D，i To be at the path point B p Ti Cartesian space stiffness matrix, K, of the robot D，i ＝J(θ i ) -T K θ J(θ i ) -1 In which K is θ ＝diag(K 1 ，K 2 ，…，K 6 )，K k Representing the joint stiffness of the k-th joint of the robot;
s26: the deformation of the robot in the coordinate system of the joint end E D Ei Transformed into deformation in a waypoint coordinate system
Figure BDA0003253862980000137
The transformation formula is:
Figure BDA0003253862980000138
wherein the content of the first and second substances,
Figure BDA0003253862980000139
s3: distributing weight values to the dexterity and the deformation to obtain an objective function;
step S3 also comprises the step of determining said dexterity k i And amount of deformation
Figure BDA0003253862980000141
Carrying out non-dimensionalization, wherein the non-dimensionalization formula is as follows:
Figure BDA0003253862980000142
Figure BDA0003253862980000143
wherein, min (1/kappa) i ) And max (1/κ) i ) Respectively 1/k in all path points i Maximum and minimum values of;
Figure BDA0003253862980000144
and
Figure BDA0003253862980000145
respectively the maximum and minimum of all waypoints.
The objective function is:
Figure BDA0003253862980000146
wherein w 1 And w 2 Are weight coefficients.
S4: solving the objective function in the constraint ranges of the joint angle, the dexterity and the deformation to ensure that the objective function obtains the minimum value, and further obtains the path point of the robot B p Ti At an optimum angle of rotation gamma i，best 。
The constraint range of the joint angle is as follows:
θ i，k，max ≥θ i，k ≥θ i，k，min
wherein, theta i，k，max And theta i，k，min Represents the maximum value and the minimum value of the k joint angle;
the restricted range of dexterity is:
σ i，1 ≥κ i ≥σ i，6
κ i ≥κ min
wherein σ i，1 And σ i，6 Are respectively J v (θ i ) Maximum and minimum singular values of, k min Presetting a minimum value according to the processing requirement;
the constraint range of the deformation amount is as follows:
Figure BDA0003253862980000151
wherein the content of the first and second substances,
Figure BDA0003253862980000152
is a preset upper limit of processing.
When the rotation angles at the respective path points are equal, the posture of the robot relative to the path points is kept constant, and the trajectory is smooth, the objective function is converted into an objective function about the optimal rotation angle as follows:
Figure BDA0003253862980000153
s.t.θ k，max ≥θ k ≥θ k，min
κ i ≥κ min
Figure BDA0003253862980000154
s5: according to the inverse kinematics calculation method of the robot, the robot is positioned at the path point B p Ti Of optimum rotation angle gamma i，best Posture of falling
Figure BDA0003253862980000155
The optimal joint angle theta of all joints of the robot under the optimal rotation angle can be obtained through conversion i，k，best 。
In conclusion, the processing path of the robot under the constraint of the deformation and the dexterity can be obtained, and the processing path is more reasonable and accurate.
It will be understood by those skilled in the art that the foregoing is only a preferred embodiment of the present invention, and is not intended to limit the invention, and that any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the scope of the present invention.

Claims (8)

1. A robot machining path optimization method based on deformation and dexterity is characterized in that the robot comprises a plurality of joints which are sequentially connected, a machining tool is connected to the tail ends of the joints, and the tool machines a workpiece to be machined, the method comprises the following steps:
s1: method for acquiring robot path points by MVR indexes B p Ti Dexterity of i To establish a dexterity k i Angle of rotation gamma i And joint angle theta of robot joint i,k (k =1, \8230;, n), where k is the kth joint; n is the total number of robot joints; gamma ray i The value range of (a) is 0-360 degrees, and the rotation direction of the tool is anticlockwise rotation around the z axis of a tool coordinate system { T };
s2: obtaining cutting force obtained by a cutter at each path point, and obtaining the deformation of the robot according to the cutting force;
s3: distributing weight values to the dexterity and the deformation to obtain an objective function;
s4: solving the objective function in the constraint ranges of the joint angle, the dexterity and the deformation to obtain the minimum value of the objective function, and further obtaining the path point of the robot B p Ti At an optimum rotation angle gamma i,best ；
S5: according to the inverse kinematics calculation method of the robot, the robot is positioned at the path point B p Ti Is optimum rotation angle gamma i,best Posture of falling down
Figure FDA0003900810250000011
The optimal joint angle theta of all joints of the robot under the optimal rotation angle is obtained through conversion i,k,best 。
2. Method according to claim 1, characterized in that the path points in step S5 B p Ti Is optimum rotation angle gamma i,best Posture of falling down
Figure FDA0003900810250000012
By the base coordinate system { B }) a joint tip coordinate system { E }),Tool coordinate system { T } and tool path point coordinate system { P } i A conversion relation between (a) and (b) is obtained,
Figure FDA0003900810250000013
wherein the content of the first and second substances,
Figure FDA0003900810250000014
as a tool path point coordinate system { P } i The transformation relation with respect to the tool coordinate system T,
Figure FDA0003900810250000021
is the transformation relation of the tool coordinate system { T } relative to the joint end coordinate system { E },
Figure FDA0003900810250000022
is the transformation relation of the joint end coordinate system { E } relative to the base coordinate system { B },
Figure FDA0003900810250000023
as a tool path point coordinate system { P } i The transformation relation with respect to the base coordinate system B.
3. The method of claim 1, wherein said dexterity k is i The calculation formula of (c) is:
Figure FDA0003900810250000024
wherein, the first and the second end of the pipe are connected with each other,
Figure FDA0003900810250000025
u v,i is speed
Figure FDA0003900810250000026
The unit vector of (a) is,
Figure FDA0003900810250000027
to correspond to the speed
Figure FDA0003900810250000028
The diagonalized weight matrix of (a) is,
Figure FDA0003900810250000029
to correspond to joint velocity
Figure FDA00039008102500000210
The diagonalized weight matrix of (a) is,
Figure FDA00039008102500000211
wherein the content of the first and second substances,
Figure FDA00039008102500000212
as a waypoint B p T(i+1) The transformation relation of the terminal coordinate system of the robot relative to the base coordinate system,
Figure FDA00039008102500000213
as a waypoint B p Ti The transformation relation of the terminal coordinate system of the robot relative to the base coordinate system, and delta t is a slave path point B p Ti To B p T(i+1) Time taken, J (θ) i ) As a waypoint B p Ti Jacobian matrix of the robot, theta i For robots at waypoints B p Ti Joint angle matrix, theta, of multiple joints of the robot at the time of treatment i ＝[θ i,1 ,θ i,2 ,…,θ i,n ] T 。
4. The method according to claim 3, wherein step S2 comprises in particular the steps of:
s21: obtaining the cutting force of the cutter under the base coordinate system B F i ：
Figure FDA0003900810250000031
Wherein, the first and the second end of the pipe are connected with each other, B f i and B m i is a 3 x 1 vector, representing force and moment, respectively; B f i ＝f n B n Ti +f f B t Ti ， B t Ti which is the feed direction of the cutting of the tool, B n Ti ＝ B v Ti × B t Ti ， B v Ti is the cutter shaft direction of the cutter; f. of f And f n Are respectively as B t Ti And B n Ti a die length of a directional cutting force component;
s22: cutting force of the tool in the base coordinate system B F i Converted into cutting force in a path point coordinate system
Figure FDA00039008102500000310
The transformation formula is as follows:
Figure FDA0003900810250000032
wherein the content of the first and second substances,
Figure FDA0003900810250000033
for the robot base coordinate system relative to the path point coordinate system { P } i The rotation matrix of, sr is the operator of the rotation matrix, if R is a 3 x 3 rotation matrix,
Figure FDA0003900810250000034
s23: according to the force accompanying transformation, the cutting force of the end of the cutter in the joint end coordinate system is obtained E F i ：
Figure FDA0003900810250000035
Wherein, the first and the second end of the pipe are connected with each other,
Figure FDA0003900810250000036
Figure FDA0003900810250000037
is the coordinate of the origin of the path point coordinate system in the joint end coordinate system,
Figure FDA0003900810250000038
for the purpose of the corresponding anti-symmetric matrix,
Figure FDA0003900810250000039
is a coordinate system of path points { P i Rotating matrix relative to the coordinate system of the joint end, ad is force accompanying transformation;
s24: cutting force of the end of the tool in the joint end coordinate system E F i Conversion into cutting force in the base coordinate system B F Ei The transformation formula is:
Figure FDA0003900810250000041
wherein, the first and the second end of the pipe are connected with each other,
Figure FDA0003900810250000042
to be at a waypoint B p Ti A rotation matrix of the joint end coordinate system relative to the robot base coordinate system, E f E is composed of E F i The portion of force of (a) is, E m E is composed of E F i A moment portion of (a);
s25: according to cutting force of tool tip B F Ei Obtaining the deformation of the robot in the coordinate system of the joint end E D Ei ：
Figure FDA0003900810250000043
Wherein the content of the first and second substances,
Figure FDA0003900810250000044
to be at a waypoint B p Ti A rotation matrix of the robot base coordinate system relative to the joint end coordinate system; k D,i To be at the path point B p Ti Cartesian space stiffness matrix, K, of the robot D,i ＝J(θ i ) -T K θ J(θ i ) -1 In which K is θ ＝diag(K 1 ,K 2 ,…,K 6 )，K k Representing the joint stiffness of the k joint of the robot;
s26: the deformation of the robot in the coordinate system of the joint end E D Ei Transformation into deformation in a waypoint coordinate system
Figure FDA0003900810250000048
The transformation formula is as follows:
Figure FDA0003900810250000045
wherein, the first and the second end of the pipe are connected with each other,
Figure FDA0003900810250000046
5. method according to claim 1 or 4, characterized in that step S3 further comprises adapting said dexterity k i And amount of deformation
Figure FDA0003900810250000047
Carrying out non-dimensionalization, wherein the non-dimensionalization formula is as follows:
Figure FDA0003900810250000051
Figure FDA0003900810250000052
wherein, min (1/kappa) i ) And max (1/κ) i ) Respectively 1/k in all path points i Maximum and minimum values of;
Figure FDA0003900810250000053
and
Figure FDA0003900810250000054
respectively the maximum and minimum of all waypoints.
6. The method of claim 5, wherein the objective function is:
Figure FDA0003900810250000055
wherein, w 1 And w 2 Are weight coefficients.
7. The method according to claim 4, wherein in step S4, the constraint range of the joint angle is:
θ i,k,max ≥θ i,k ≥θ i,k,min
wherein, theta i,k,max And theta i,k,min Represents the maximum value and the minimum value of the k joint angle;
the restricted range of dexterity is:
σ i,1 ≥κ i ≥σ i,6
κ i ≥κ min
wherein σ i,1 And σ i,6 Are respectively J v (θ i ) Maximum and minimum singular values of, k min Presetting a minimum value according to the processing requirement;
the constraint range of the deformation amount is as follows:
Figure FDA0003900810250000061
wherein the content of the first and second substances,
Figure FDA0003900810250000062
is a preset upper limit of processing.
8. Method according to claim 1, characterized in that the following steps are used to obtain the path points of the robot B p Ti ：
(1) Respectively fitting the original path points and the cutter axis vectors by adopting an NURBS curve to obtain a path point NURBS curve
Figure FDA0003900810250000063
Sum arbor vector NURBS curve
Figure FDA0003900810250000064
(2) NURBS curve for path points
Figure FDA0003900810250000063
NURBS curve of sum-cutter axis vector point
Figure FDA0003900810250000066
And (3) performing parameter synchronization treatment:
Figure FDA0003900810250000067
wherein u is Q (u P ) Is composed of
Figure FDA0003900810250000068
Arbitrary parameter u of P Corresponding curve
Figure FDA0003900810250000069
Parameter u of Q ，
Figure FDA00039008102500000610
Is composed of
Figure FDA00039008102500000611
Parameter set of
Figure FDA00039008102500000612
Is known to be present at the known point in (a),
Figure FDA00039008102500000613
is composed of
Figure FDA00039008102500000614
Parameter set of
Figure FDA00039008102500000615
A known point of (a);
(3) For the NURBS curve
Figure FDA00039008102500000616
And NURBS curve
Figure FDA00039008102500000617
The curves after being respectively offset by the preset distance are as follows:
Figure FDA00039008102500000618
wherein the content of the first and second substances,
Figure FDA00039008102500000619
as NURBS curve
Figure FDA00039008102500000620
The curve is offset by a preset distance,
Figure FDA00039008102500000621
as NURBS curve
Figure FDA00039008102500000622
Offsetting the curve after the preset distance, wherein d is a second preset distance, d = R-a, R is the radius of the cutter, a is the boundary machining allowance, and c is the vector offset distance of the final cutter end central point relative to the initial cutter end central point along the cutter central axis;
Figure FDA00039008102500000623
as the NURBS curve
Figure FDA00039008102500000624
The tangent vector of (a) is,
Figure FDA00039008102500000625
Figure FDA00039008102500000626
as the NURBS curve
Figure FDA0003900810250000071
The tangential vector of (a) is,
Figure FDA0003900810250000072
v(u P ) Is u P The vector of the central axis of the tool in (a),
Figure FDA0003900810250000073
(4) And uniformly sampling the biased NURBS curve to obtain a final path point and a corresponding cutter axis vector point, and further obtaining a corresponding cutter axis vector.
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