CN107422693A - Center line is reconstructed to space bent pipe inner surface polishing locus generation method - Google Patents
Center line is reconstructed to space bent pipe inner surface polishing locus generation method Download PDFInfo
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- CN107422693A CN107422693A CN201710650198.1A CN201710650198A CN107422693A CN 107422693 A CN107422693 A CN 107422693A CN 201710650198 A CN201710650198 A CN 201710650198A CN 107422693 A CN107422693 A CN 107422693A
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- G—PHYSICS
- G05—CONTROLLING; REGULATING
- G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
- G05B19/00—Programme-control systems
- G05B19/02—Programme-control systems electric
- G05B19/18—Numerical control [NC], i.e. automatically operating machines, in particular machine tools, e.g. in a manufacturing environment, so as to execute positioning, movement or co-ordinated operations by means of programme data in numerical form
- G05B19/19—Numerical control [NC], i.e. automatically operating machines, in particular machine tools, e.g. in a manufacturing environment, so as to execute positioning, movement or co-ordinated operations by means of programme data in numerical form characterised by positioning or contouring control systems, e.g. to control position from one programmed point to another or to control movement along a programmed continuous path
- G05B19/33—Numerical control [NC], i.e. automatically operating machines, in particular machine tools, e.g. in a manufacturing environment, so as to execute positioning, movement or co-ordinated operations by means of programme data in numerical form characterised by positioning or contouring control systems, e.g. to control position from one programmed point to another or to control movement along a programmed continuous path using an analogue measuring device
- G05B19/37—Numerical control [NC], i.e. automatically operating machines, in particular machine tools, e.g. in a manufacturing environment, so as to execute positioning, movement or co-ordinated operations by means of programme data in numerical form characterised by positioning or contouring control systems, e.g. to control position from one programmed point to another or to control movement along a programmed continuous path using an analogue measuring device for continuous-path control
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- G—PHYSICS
- G05—CONTROLLING; REGULATING
- G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
- G05B2219/00—Program-control systems
- G05B2219/30—Nc systems
- G05B2219/45—Nc applications
- G05B2219/45096—Polishing manipulator
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- G—PHYSICS
- G05—CONTROLLING; REGULATING
- G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
- G05B2219/00—Program-control systems
- G05B2219/30—Nc systems
- G05B2219/45—Nc applications
- G05B2219/45133—Lapping
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- Finish Polishing, Edge Sharpening, And Grinding By Specific Grinding Devices (AREA)
Abstract
The invention belongs to magnetically grinding technical field, is reconstructed more particularly, to a kind of center line to space bent pipe inner surface polishing locus generation method, it is characterised in that carry out as follows：Step 1：Obtain the outline point cloud of space bent pipe；Step 2：Three contour lines are obtained by curve matching；Step 3：The method intersected using the outer wheels profile of fitting by section is generated；Step 4：The discretization that the bend pipe center line of generation and the outer wheels profile as caused by measurement are put；Step 5：Establish rectangular coordinate system in space O0X0Y0Z0, it is set to robot tool coordinate system；Step 6：By by workpiece coordinate system OXYZ translation, translating rotation to tool coordinates system；Step 7：By conversion of the homogeneous coordinate transformation matrix to coordinate system, the pose coordinate in the machining path at this processing moment can be obtained；Step 8：Obtain final mechanical polishing track.The present invention can accurately determine machining locus to the space bent pipe of complexity, improve operating efficiency.
Description
Technical field
The invention belongs to magnetically grinding technical field, is reconstructed more particularly, to a kind of center line to space bent pipe inner surface
Polishing locus generation method.
Background technology
It is strict all the more to engine volume requirement in Aero-Space, automobile and other mechanical fields, it is unnecessary to reduce
Space waste, the transporting of oil gas is carried out using the less space bent pipe of space occupancy rate.But due to state-of-art etc. because
Element influences, can make bend pipe inner surface crack with the surface defect such as pit, be particularly acute, cause inside bend pipe in knee
By turbulization when flowing gas or liquid, cause vibration, then influence the even running of engine, reduce service life.
But the acquisition of the machining locus of space bent pipe is more difficult, therefore the planning to machining locus is particularly important, one
As in process, carry out the center line determination to bend pipe usually using the method for manual calibration point, because this method is behaviour manually
Make, therefore exist cumbersome, fixed point is easily disturbed, it is possible to create larger error, influences processing effect, reduction processing efficiency.
The content of the invention
It is an object of the invention to overcome deficiency of the prior art, there is provided a kind of center line reconstruct is in space bent pipe
Surface polishing locus generation method, machining locus can be accurately determined to the space bent pipe of complexity, improve operating efficiency.
The purpose of the present invention is realized by following technical proposals：
The center line of the present invention is reconstructed to space bent pipe inner surface polishing locus generation method, it is characterised in that by as follows
Step is carried out：
Step 1：Obtain the outline point cloud of space bent pipe；Space bent pipe outline is carried out by three-coordinates measuring machine
Scanning, obtain the point cloud of three outer contours of space bent pipe；
Step 2：Calculation system is carried out to a cloud, three contour lines are obtained by curve matching；
Step 3：The generation of bend pipe center line is that the method intersected using the outer wheels profile of fitting by section is generated；
Step 4：The discretization that the bend pipe center line of generation and the outer wheels profile as caused by measurement are put, and two
It is equal that person reveals number；
Step 5：Select any point O on center line0Generation two more corresponding with more adjacent and contour line is altogether
The orthogonal vector of point, rectangular coordinate system in space O can be established according to this two vectors by right-hand rule0X0Y0Z0, determined
For robot tool coordinate system；
Step 6：By by workpiece coordinate system OXYZ translation, translating rotation to tool coordinates system；
Step 7：By conversion of the homogeneous coordinate transformation matrix to coordinate system, this processing road for processing the moment can be obtained
Pose coordinate in footpath；
Step 8：Pose coordinate in a series of machining path is together in series and is achieved with final mechanical polishing
Track.
Described step two specifically refers to：
Step 2 carries out calculation system to a cloud to keep parameter accuracy to be asked, then to unlimited section in data
Point place offset be smoothed, smoothly after offset formed by prototype value linear superposition, correlation formula (1) is as follows：
In formula (1), { LvIt is weight factor, L-v=LvIt is even series.
In order that the data { Pn } after must handling are than { Pv } smoothly, while the pattern of former data can be preferably kept, also
Need its two groups of data " deviation " less, carry out computing from Gauss (Gaussian) filtering algorithm, correlation formula (2) is as follows
It is shown：
In formula (2), t is space domain variable.Its Fourier transformation correlation formula (3) is as follows：
In formula (3), λ is wavelength, λcFor cutoff wavelength, α is constant.Discrete type cloud data is directed to, formula should be utilized
(1) discrete to turn to formula (3), it is as follows to finally give formula (4), with the needs of this adaptive filtering,
In formula (4), gkFor the discrete representation of Gauss weight function, Δ x is the distance between consecutive points,
Original point cloud carries out needing to carry out curve fitting after gaussian filtering, is intended by the method for uniform curve
Close, smooth and uniformity parameter node can be obtained after fitting.
Described step three specifically refers to：
The method that outer wheels profile in step 3 is intersected by section generates center line, the cross―section line of perfect condition
It is a line to be handed over shown in such as figure two (a), but is caused because measuring instrument has certain error shown in intersecting lens such as figure two (b)
For three intersections；
When three intersections occur, it is necessary to the actual solid formed for this three intersections of intersection center line, pass through
The solid is subjected to differential, the plane where 3 points of the same position for obtaining three intersections, crosses in its plane at this 3 points
Its center of circle can be asked by doing circle, then be fitted in N number of center of circle by way of integration, obtained the center line of solid, exactly needed
The center line for the space bent pipe wanted.
Described step six specifically refers to：
In step 6, the conversion of workpiece coordinate system to tool coordinates system is as follows：
The origin of workpiece coordinate system is passed through into the translation transformation in homogeneous transformation first so that with tool coordinates system system
Origin overlaps；
Further, after Two coordinate system origin overlaps, the new coordinate system X-axis after translation is entered around the Z axis of local Coordinate System
Co-planar where row rotation alpha angle, with the X-axis of original tool coordinates system；
So that Y-axis of the postrotational new coordinate system X-axis around local Coordinate System is carried out into rotation β angle so that two overlapping of axles；
And then the Z axis rotation alpha angle around local Coordinate System similarly is carried out to postrotational new coordinate system Y-axis, make two seats
Mark system Y-axis is coplanar, finally by the change for carrying out rotating γ angles to Y-axis in postrotational new coordinate system around the X-axis of local Coordinate System
Change, so that two overlapping of axles,
Finally, both workpiece coordinate system and tool coordinates system is obtained to overlap.
Described step seven specifically refers to：
The correlation formula (5) of homogeneous coordinate transformation matrix is as follows in step 7：
In formula (5), nxCosine of an angle is pressed from both sides for X-axis in moving coordinate system and position fixing system X-axis；
nyCosine of an angle is pressed from both sides for X-axis in moving coordinate system and position fixing system Y-axis；
nzCosine of an angle is pressed from both sides for X-axis in moving coordinate system and position fixing system Z axis；
oxCosine of an angle is pressed from both sides for Y-axis in moving coordinate system and position fixing system X-axis；
oyCosine of an angle is pressed from both sides for Y-axis in moving coordinate system and position fixing system Y-axis；
ozCosine of an angle is pressed from both sides for Y-axis in moving coordinate system and position fixing system Z axis；
axCosine of an angle is pressed from both sides for Z axis in moving coordinate system and position fixing system X-axis；
ayCosine of an angle is pressed from both sides for Z axis in moving coordinate system and position fixing system Y-axis；
azCosine of an angle is pressed from both sides for Z axis in moving coordinate system and position fixing system Z axis；
Wherein the anglec of rotation can try to achieve according to below equation：
The pose coordinate that can be obtained by by formula (5)-(8) in machining locus is： (x0, y0, z0, α, beta, gamma), most
The pose track walked during processing space bend pipe is drawn eventually.
Advantages of the present invention：
Center line proposed by the present invention reconstruct is in space bent pipe inner surface polishing locus generation method, its data processing
Method employs the method that homogeneous coordinate transformation matrix replaces original manual calibration point by mechanical measurement, can be to complexity
Space bent pipe accurately determine machining locus, improve operating efficiency, the method for the center line reconstruct of proposition, pass through measure it is empty
Between outer surface of bent pipe contour line, generate bend pipe outline spatial point cloud, and then irregular geometric center lines of reconstruction attractor bend pipe,
Calculation optimization obtains bend pipe polishing locus, effectively increases operating efficiency, and calculation optimization obtains bend pipe polishing locus and reduced
The preparation of early stage.
Brief description of the drawings
Fig. 1, which is that polishing locus coordinate system transformation of the present invention is front and rear, to scheme.
A is cross―section line figure ideally in Fig. 2, and b is the cross―section line figure under virtual condition.
Fig. 3 is workpiece coordinate system of the present invention by being translated towards tool coordinates system conversion figure.
Fig. 4 is that workpiece coordinate system X-axis of the present invention is schemed before and after itself Z axis rotation alpha angular transformation.
Fig. 5 is that workpiece coordinate system X-axis of the present invention is schemed before and after itself Y-axis rotation β angular transformation.
Fig. 6 is that workpiece coordinate system Y-axis of the present invention is schemed before and after the rotation γ angular transformations of itself X-axis.
Embodiment
The embodiment of the present invention is further illustrated below in conjunction with the accompanying drawings.
As shown in Fig. 1,2,3,4,5 and 6, center line of the invention reconstruct generates to space bent pipe inner surface polishing locus
Method, it is characterised in that carry out as follows：
Step 1：Obtain the outline point cloud of space bent pipe；Space bent pipe outline is carried out by three-coordinates measuring machine
Scanning, obtain the point cloud of three outer contours of space bent pipe；
Step 2：Calculation system is carried out to a cloud, three contour lines are obtained by curve matching；
Step 3：The generation of bend pipe center line is that the method intersected using the outer wheels profile of fitting by section is generated；
Step 4：The discretization that the bend pipe center line of generation and the outer wheels profile as caused by measurement are put, and two
It is equal that person reveals number；
Step 5：Select any point O on center line0Generation two more corresponding with more adjacent and contour line is altogether
The orthogonal vector of point, rectangular coordinate system in space O can be established according to this two vectors by right-hand rule0X0Y0Z0, determined
For robot tool coordinate system；
Step 6：By by workpiece coordinate system OXYZ translation, translating rotation to tool coordinates system；
Step 7：By conversion of the homogeneous coordinate transformation matrix to coordinate system, this processing road for processing the moment can be obtained
Pose coordinate in footpath；
Step 8：Pose coordinate in a series of machining path is together in series and is achieved with final mechanical polishing
Track.
Described step two specifically refers to：
Step 2 carries out calculation system to a cloud to keep parameter accuracy to be asked, then to unlimited section in data
Point place offset be smoothed, smoothly after offset formed by prototype value linear superposition, correlation formula (1) is as follows：
In formula (1), { LvIt is weight factor, L-v=LvIt is even series.
In order that the data { Pn } after must handling are than { Pv } smoothly, while the pattern of former data can be preferably kept, also
Need its two groups of data " deviation " less, carry out computing from Gauss (Gaussian) filtering algorithm, correlation formula (2) is as follows
It is shown：
In formula (2), t is space domain variable.Its Fourier transformation correlation formula (3) is as follows：
In formula (3), λ is wavelength, λcFor cutoff wavelength, α is constant.Discrete type cloud data is directed to, formula should be utilized
(1) discrete to turn to formula (3), it is as follows to finally give formula (4), with the needs of this adaptive filtering,
In formula (4), gkFor the discrete representation of Gauss weight function, Δ x is the distance between consecutive points,
Original point cloud carries out needing to carry out curve fitting after gaussian filtering, is intended by the method for uniform curve
Close, smooth and uniformity parameter node can be obtained after fitting.
Described step three specifically refers to：
The method that outer wheels profile in step 3 is intersected by section generates center line, the cross―section line of perfect condition
It is a line to be handed over shown in such as figure two (a), but is caused because measuring instrument has certain error shown in intersecting lens such as figure two (b)
For three intersections；
When three intersections occur, it is necessary to the actual solid formed for this three intersections of intersection center line, pass through
The solid is subjected to differential, the plane where 3 points of the same position for obtaining three intersections, crosses in its plane at this 3 points
Its center of circle can be asked by doing circle, then be fitted in N number of center of circle by way of integration, obtained the center line of solid, exactly needed
The center line for the space bent pipe wanted.
Described step six specifically refers to：
In step 6, the conversion of workpiece coordinate system to tool coordinates system is as follows：
The origin of workpiece coordinate system is passed through into the translation transformation in homogeneous transformation first so that with tool coordinates system system
Origin overlaps；
Further, after Two coordinate system origin overlaps, the new coordinate system X-axis after translation is entered around the Z axis of local Coordinate System
Co-planar where row rotation alpha angle, with the X-axis of original tool coordinates system；
So that Y-axis of the postrotational new coordinate system X-axis around local Coordinate System is carried out into rotation β angle so that two overlapping of axles；
And then the Z axis rotation alpha angle around local Coordinate System similarly is carried out to postrotational new coordinate system Y-axis, make two seats
Mark system Y-axis is coplanar, finally by the change for carrying out rotating γ angles to Y-axis in postrotational new coordinate system around the X-axis of local Coordinate System
Change, so that two overlapping of axles,
Finally, both workpiece coordinate system and tool coordinates system is obtained to overlap.
Described step seven specifically refers to：
The correlation formula (5) of homogeneous coordinate transformation matrix is as follows in step 7：
In formula (5), nxCosine of an angle is pressed from both sides for X-axis in moving coordinate system and position fixing system X-axis；
nyCosine of an angle is pressed from both sides for X-axis in moving coordinate system and position fixing system Y-axis；
nzCosine of an angle is pressed from both sides for X-axis in moving coordinate system and position fixing system Z axis；
oxCosine of an angle is pressed from both sides for Y-axis in moving coordinate system and position fixing system X-axis；
oyCosine of an angle is pressed from both sides for Y-axis in moving coordinate system and position fixing system Y-axis；
ozCosine of an angle is pressed from both sides for Y-axis in moving coordinate system and position fixing system Z axis；
axCosine of an angle is pressed from both sides for Z axis in moving coordinate system and position fixing system X-axis；
ayCosine of an angle is pressed from both sides for Z axis in moving coordinate system and position fixing system Y-axis；
azCosine of an angle is pressed from both sides for Z axis in moving coordinate system and position fixing system Z axis；
Wherein the anglec of rotation can try to achieve according to below equation：
The pose coordinate that can be obtained by by formula (5)-(8) in machining locus is： (x0, y0, z0, α, beta, gamma), most
The pose track walked during processing space bend pipe is drawn eventually.
The conversion of final coordinate system is as shown in Figure 1.
Therefore the present invention provides a kind of center line reconstruct to space bent pipe inner surface polishing locus generation method, instead of original
The method for the manual calibration point having, machining locus can be accurately determined to the space bent pipe of complexity, improve operating efficiency, meter
Optimization is calculated to obtain bend pipe polishing locus and reduce the preparation of early stage.
Claims (5)
1. a kind of center line reconstruct is to space bent pipe inner surface polishing locus generation method, it is characterised in that enters as follows
OK：
Step 1：Obtain the outline point cloud of space bent pipe；Space bent pipe outline is scanned by three-coordinates measuring machine,
Obtain the point cloud of three outer contours of space bent pipe；
Step 2：Calculation system is carried out to a cloud, three contour lines are obtained by curve matching；
Step 3：The generation of bend pipe center line is that the method intersected using the outer wheels profile of fitting by section is generated；
Step 4：The discretization that the bend pipe center line of generation and the outer wheels profile as caused by measurement are put, and both points
It is equal to change number；
Step 5：Select any point O on center line0Two concurrents of generation more corresponding with more adjacent and contour line are mutual
Vertical vector, rectangular coordinate system in space O can be established by right-hand rule according to this two vectors0X0Y0Z0, it is set to robot
Tool coordinates system；
Step 6：By by workpiece coordinate system OXYZ translation, translating rotation to tool coordinates system；
Step 7：By conversion of the homogeneous coordinate transformation matrix to coordinate system, in the machining path that this processing moment can be obtained
Pose coordinate；
Step 8：Pose coordinate in a series of machining path is together in series and is achieved with final mechanical polishing track.
2. center line reconstruct according to claim 1 exists to space bent pipe inner surface polishing locus generation method, its feature
Specifically referred in described step two：
Step 2 carries out calculation system to cloud to keep parameter accuracy to be asked, then to type at unlimited node in data
Value is smoothed, it is smooth after offset formed by prototype value linear superposition, correlation formula (1) is as follows：
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In order that the data { Pn } after must handling are than { Pv } smoothly, while it can preferably keep the pattern of former data, it is also necessary to
Its two groups of data " deviation " less, from Gauss (Gaussian) filtering algorithm carry out computing, and correlation formula (2) is as follows：
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In formula (2), t is space domain variable.Its Fourier transformation correlation formula (3) is as follows：
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In formula (4), gkFor the discrete representation of Gauss weight function, Δ x is the distance between consecutive points,
Original point cloud carries out needing to carry out curve fitting after gaussian filtering, is fitted by the method for uniform curve, intends
Smooth and uniformity parameter node can be obtained after conjunction.
3. center line reconstruct according to claim 1 exists to space bent pipe inner surface polishing locus generation method, its feature
Specifically referred in described step three：
The method that outer wheels profile in step 3 is intersected by section generates center line, the cross―section line such as figure two of perfect condition
(a) it is a line to be handed over shown in, but causes intersecting lens such as figure two (b) to show three friendships because measuring instrument has certain error
Line；
When three intersections occur, it is necessary to the actual solid formed for this three intersections of intersection center line, by should
Solid carries out differential, the plane where 3 points of the same position for obtaining three intersections, crosses at this 3 points in its plane and does circle
Its center of circle can be asked, is then fitted in N number of center of circle by way of integration, the center line of solid is obtained, exactly needs
The center line of space bent pipe.
4. center line reconstruct according to claim 1 exists to space bent pipe inner surface polishing locus generation method, its feature
Specifically referred in described step six：
In step 6, the conversion of workpiece coordinate system to tool coordinates system is as follows：
The origin of workpiece coordinate system is passed through into the translation transformation in homogeneous transformation first so that the origin weight with tool coordinates system system
Close；
Further, after Two coordinate system origin overlaps, the new coordinate system X-axis after translation is revolved around the Z axis of local Coordinate System
Turn α angles, with co-planar where the X-axis of original tool coordinates system；
So that Y-axis of the postrotational new coordinate system X-axis around local Coordinate System is carried out into rotation β angle so that two overlapping of axles；
And then the Z axis rotation alpha angle around local Coordinate System similarly is carried out to postrotational new coordinate system Y-axis, make Two coordinate system Y
Axle is coplanar, finally by the conversion for carrying out rotating γ angles to Y-axis in postrotational new coordinate system around the X-axis of local Coordinate System, from
And cause two overlapping of axles,
Finally, both workpiece coordinate system and tool coordinates system is obtained to overlap.
5. center line reconstruct according to claim 1 exists to space bent pipe inner surface polishing locus generation method, its feature
Specifically referred in described step seven：
The correlation formula (5) of homogeneous coordinate transformation matrix is as follows in step 7：
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<mtd>
<msub>
<mi>a</mi>
<mi>x</mi>
</msub>
</mtd>
<mtd>
<msub>
<mi>p</mi>
<mi>x</mi>
</msub>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>n</mi>
<mi>y</mi>
</msub>
</mtd>
<mtd>
<msub>
<mi>o</mi>
<mi>y</mi>
</msub>
</mtd>
<mtd>
<msub>
<mi>a</mi>
<mi>y</mi>
</msub>
</mtd>
<mtd>
<msub>
<mi>p</mi>
<mi>y</mi>
</msub>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>n</mi>
<mi>z</mi>
</msub>
</mtd>
<mtd>
<msub>
<mi>o</mi>
<mi>z</mi>
</msub>
</mtd>
<mtd>
<msub>
<mi>a</mi>
<mi>z</mi>
</msub>
</mtd>
<mtd>
<msub>
<mi>p</mi>
<mi>z</mi>
</msub>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>5</mn>
<mo>)</mo>
</mrow>
</mrow>
In formula (5), nxCosine of an angle is pressed from both sides for X-axis in moving coordinate system and position fixing system X-axis；
nyCosine of an angle is pressed from both sides for X-axis in moving coordinate system and position fixing system Y-axis；
nzCosine of an angle is pressed from both sides for X-axis in moving coordinate system and position fixing system Z axis；
oxCosine of an angle is pressed from both sides for Y-axis in moving coordinate system and position fixing system X-axis；
oyCosine of an angle is pressed from both sides for Y-axis in moving coordinate system and position fixing system Y-axis；
ozCosine of an angle is pressed from both sides for Y-axis in moving coordinate system and position fixing system Z axis；
axCosine of an angle is pressed from both sides for Z axis in moving coordinate system and position fixing system X-axis；
ayCosine of an angle is pressed from both sides for Z axis in moving coordinate system and position fixing system Y-axis；
azCosine of an angle is pressed from both sides for Z axis in moving coordinate system and position fixing system Z axis；
Wherein the anglec of rotation can try to achieve according to below equation：
<mrow>
<mi>c</mi>
<mi>o</mi>
<mi>s</mi>
<mi>&alpha;</mi>
<mo>=</mo>
<mfrac>
<msub>
<mi>n</mi>
<mi>x</mi>
</msub>
<msqrt>
<mrow>
<msup>
<msub>
<mi>n</mi>
<mi>x</mi>
</msub>
<mn>2</mn>
</msup>
<mo>+</mo>
<msup>
<msub>
<mi>n</mi>
<mi>y</mi>
</msub>
<mn>2</mn>
</msup>
<mo>+</mo>
<msup>
<msub>
<mi>n</mi>
<mi>z</mi>
</msub>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</mfrac>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>6</mn>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mi>c</mi>
<mi>o</mi>
<mi>s</mi>
<mi>&beta;</mi>
<mo>=</mo>
<mfrac>
<msqrt>
<mrow>
<msup>
<msub>
<mi>n</mi>
<mi>x</mi>
</msub>
<mn>2</mn>
</msup>
<mo>+</mo>
<msup>
<msub>
<mi>n</mi>
<mi>y</mi>
</msub>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
<msqrt>
<mrow>
<msup>
<msub>
<mi>n</mi>
<mi>x</mi>
</msub>
<mn>2</mn>
</msup>
<mo>+</mo>
<msup>
<msub>
<mi>n</mi>
<mi>y</mi>
</msub>
<mn>2</mn>
</msup>
<mo>+</mo>
<msup>
<msub>
<mi>n</mi>
<mi>z</mi>
</msub>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</mfrac>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>7</mn>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mi>c</mi>
<mi>o</mi>
<mi>s</mi>
<mi>&gamma;</mi>
<mo>=</mo>
<mfrac>
<msqrt>
<mrow>
<msup>
<msub>
<mi>o</mi>
<mi>x</mi>
</msub>
<mn>2</mn>
</msup>
<mo>+</mo>
<msup>
<msub>
<mi>o</mi>
<mi>y</mi>
</msub>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
<msqrt>
<mrow>
<msup>
<msub>
<mi>o</mi>
<mi>x</mi>
</msub>
<mn>2</mn>
</msup>
<mo>+</mo>
<msup>
<msub>
<mi>o</mi>
<mi>y</mi>
</msub>
<mn>2</mn>
</msup>
<mo>+</mo>
<msup>
<msub>
<mi>o</mi>
<mi>z</mi>
</msub>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</mfrac>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>8</mn>
<mo>)</mo>
</mrow>
</mrow>
The pose coordinate that can be obtained by by formula (5)-(8) in machining locus is：(x0, y0, z0, α, beta, gamma), finally draw
The pose track walked during processing space bend pipe.
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| CN108890648A (en) * | 2018-08-08 | 2018-11-27 | 苏州天准科技股份有限公司 | A kind of method for building up of joint series formula robot kinematics' model |
| CN108972160A (en) * | 2018-08-03 | 2018-12-11 | 辽宁科技大学 | A kind of complexity axis bend pipe surfaces externally and internally magnetic abrasive finishing device and method |
| CN111230886A (en) * | 2020-03-05 | 2020-06-05 | 浙江理工大学 | Industrial robot kinematics parameter calibration system and method with tolerance capability |
| CN115415886A (en) * | 2022-08-30 | 2022-12-02 | 天津大学 | Inner wall optical surface polishing path calculation method |
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