CN107246866A - A kind of high-precision six-freedom degree measuring system and method - Google Patents

A kind of high-precision six-freedom degree measuring system and method Download PDF

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CN107246866A
CN107246866A CN201710489410.0A CN201710489410A CN107246866A CN 107246866 A CN107246866 A CN 107246866A CN 201710489410 A CN201710489410 A CN 201710489410A CN 107246866 A CN107246866 A CN 107246866A
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mrow
msup
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mtd
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林嘉睿
邾继贵
郭寅
任永杰
杨凌辉
高凯
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Tianjin University
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Tianjin University
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    • GPHYSICS
    • G01MEASURING; TESTING
    • G01CMEASURING DISTANCES, LEVELS OR BEARINGS; SURVEYING; NAVIGATION; GYROSCOPIC INSTRUMENTS; PHOTOGRAMMETRY OR VIDEOGRAMMETRY
    • G01C11/00Photogrammetry or videogrammetry, e.g. stereogrammetry; Photographic surveying
    • G01C11/02Picture taking arrangements specially adapted for photogrammetry or photographic surveying, e.g. controlling overlapping of pictures

Abstract

The invention discloses a kind of high-precision six-freedom degree measuring system and method, system is used for the relative pose for measuring preceding measured target and rear measured target, the characteristic point target being made up of cuboid housing structure is fixed with preceding measured target, what the front and side of characteristic point target were stretched out respectively is provided with support, the end of each support is mounted on magnetic support, it is placed with magnetic support on optical characteristic point, characteristic point target and is additionally provided with the first double-shaft tilt angle sensor and power supply circuit;Vision imaging system is fixed with measured target afterwards, Vision imaging system includes digital camera, the second double-shaft tilt angle sensor, data processing unit and power supply circuit.The image that digital camera during measurement in Vision imaging system gathers optical characteristic point in real time is sent to data processing unit, the first double-shaft tilt angle sensor and the second double-shaft tilt angle sensor will constantly export the angle of itself respectively simultaneously, data processing unit carries out pose resolving to the data of collection, and most pose sends host computer at last.

Description

A kind of high-precision six-freedom degree measuring system and method
Technical field
Asked the present invention relates to space precise geometric measurement field, more particularly to space object high-precision six-freedom degree measurement Topic.
Background technology
With the high speed development of science and technology, the demand of pose measurement is continuously increased.For example during ship assembly, need Will be by monitoring position and the attitude information of each sub-unit in real time, to instruct control unit to complete assembling process.With respect to six certainly The a wide range of circumstance of occlusion occurred in commercial measurement space is effectively have been directed to by degree measuring system, by measuring between multiple target With respect to six degree of freedom, by coordinate system change can all targets of indirect gain six degree of freedom.Six degree of freedom is obtained in real time Vision measurement model is mainly made up of optical characteristic point and Vision imaging system, and optical characteristic point is arranged on measured target, depending on Feel that imaging system is arranged on measured target rear end, in order to ensure that laser via is not blocked, it is desirable to be capable of coverage goal in visual field Range of movement.Vision imaging system is imaged to optical characteristic point in real time, and accurately extracts optical characteristic point center, passes through light Learn the real space topological relation of characteristic point and combine the characteristic point center relation in image, you can calculate characteristic point knot Relative six-degree-of-freedom information between structure and Vision imaging system, completes measured target relative to relative the six of Vision imaging system The estimation of the free degree.
Six degree of freedom vision measurement system can be applied to the measurement in shipbuilding between each part with respect to six degree of freedom, will Feature dot system is arranged on measured target, and the real-time acquisition characteristics dot image of Vision imaging system obtains the relative of measured target Pose is estimated, but still there are many deficiencies:
1) optical characteristic point space topological is distributed for plane non-colinear, and characteristic point is directly installed on plane target drone.Demarcation When, laser tracker must be tuned into non-prism pattern, and precision is relatively low.
2) in the range of 2.5m-5m measurement spaces, the measurement accuracy of monocular vision relative pose measuring system is 5mm, essence Degree is not high, it is difficult to meet the target positioning and attitude measurement requirement of engineering in practice.
The content of the invention
The invention aims to overcome deficiency of the prior art, there is provided a kind of high-precision six-freedom degree measuring system And method, the present invention makes the plane figure of optical characteristic point into space layout, while the optics that will be arranged on characteristic point target Characteristic point makes the canonical measure steel ball model with magnetic support into, and demarcation and measurement accuracy is greatly improved;Solve and use six The problem of free degree vision measuring method time space position error is greatly and precision and stability is low in long range measurements, improves The stability and precision of whole six degree of freedom measuring system.
The purpose of the present invention is achieved through the following technical solutions:
A kind of high-precision six-freedom degree measuring system, the relative pose for measuring preceding measured target and rear measured target, The characteristic point target being made up of cuboid housing structure, the front and side of characteristic point target are fixed with the preceding measured target That stretches out respectively is provided with support, and the end of each support is mounted on being placed with optical characteristic point, feature on magnetic support, the magnetic support The first double-shaft tilt angle sensor and power supply circuit are additionally provided with point target;Visual imaging system is fixed with the rear measured target System, the Vision imaging system includes digital camera, the second double-shaft tilt angle sensor, data processing unit and power supply circuit;Institute Vision imaging system is stated by gathering the optical signature dot image in real time, preceding measured target is calculated relative to rear measured target Relative six-degree-of-freedom information.
The magnetic support is 1.5 inches of canonical measure steel ball, and the optical characteristic point is infrarede emitting diode, infrared hair The centre of sphere of the optical centre of optical diode and canonical measure steel ball steel ball is co-located.
The optical characteristic point distribution is in the non-co-planar type in space.
A kind of measuring method of high-precision six-freedom degree measuring system, comprises the following steps:Number in Vision imaging system The image that word camera gathers optical characteristic point in real time is sent to data processing unit, meanwhile, the first double-shaft tilt angle sensor and The angle value that two double-shaft tilt angle sensors respectively constantly export itself is sent to data processing unit, data processing unit pair The data collected carry out pose resolving, and the pose most calculated at last sends host computer to.
The step of resolving, is as follows:
Characteristic point target co-ordinates system (OTXTYTZT) and digital camera coordinates system (OCXCYCZC) meet following relation:
Wherein characteristic point target co-ordinates are tied to the rotation of digital camera coordinate system, translation matrix and are respectively
Relation between characteristic point target co-ordinates system, the first double-shaft tilt angle sensor coordinate system, world coordinate system, and The relation of spin matrix and the anglec of rotation, and rigid body translation principle try to achieve rotation translation matrix RTC、TTC, it is used as further optimization The initial value of solution;
The imaging model of digital camera is set to:
By spin matrix RTCIt is that unit orthogonal matrix is set to:
By RcIt can set:
Construct object function:
Least square solution is sought above formula with L-M (Levenberg-Marquardt) algorithms, R is finally obtainedTC、TTCIt is optimal Solution.
Compared with prior art, the beneficial effect that technical scheme is brought is:
Optical characteristic point and Vision imaging system of the invention by being distributed on design feature point target in space topological, and It will be combined by single vision sensor with the monocular vision model that characteristic point target is combined into high-precision dual-axis obliquity sensor Use, and set double-shaft tilt angle sensor constrained each other respectively on former and later two testees, substantially increase relative pose Measurement accuracy and the stability of a system.Simultaneity factor is simple in construction, easy for installation, is conducive to job site to use.
Brief description of the drawings
Fig. 1 and Fig. 2 is the structural representation of measuring system of the present invention;
Fig. 3 is characterized a target construction schematic diagram;
Fig. 4 is magnetic support and the structural representation of optical characteristic point;
Fig. 5 is that measuring system builds schematic diagram;
The five kinds of coordinate systems and its correlation schematic diagram of Fig. 6 measuring system measurement models of the present invention.
Reference:Measured target after measured target before 1-, 2-, 3- characteristic point targets, 4- supports, 5- optical characteristic points, The double-shaft tilt angle sensors of 6- first, 7- digital cameras, the double-shaft tilt angle sensors of 8- second, 9- magnetic supports, 10- laser trackers, 11- Stem center
Embodiment
The invention will be further described below in conjunction with the accompanying drawings.
As shown in Figures 1 to 4, a kind of high-precision six-freedom degree measuring system, for measuring preceding measured target 1 and being tested afterwards The characteristic point target 3 being made up of cuboid housing structure, characteristic point target are fixed with the relative pose of target 2, preceding measured target 1 What the front and side of mark 3 were stretched out respectively is provided with support 4, and the end of each support 4 is mounted on magnetic support in magnetic support, the present embodiment For 1.5 inches of canonical measure steel ball, the distribution that optical characteristic point 5 in optical characteristic point 5, the present embodiment is placed with magnetic support is in The non-co-planar type in space;Optical characteristic point 5 is infrarede emitting diode, carries out frock adjustment by high-precision image instrument, makes red UV light-emitting diode optical centre and the canonical measure steel ball centre of sphere are co-located.First pair is additionally provided with characteristic point target 3 Axial rake sensor 6 and power supply circuit;Characteristic point target 3 is used for the characteristic parameter of measured target 1 before expressing, to preceding tested mesh When topology demarcation is carried out between the stem center 11 of mark 1 and optical characteristic point 5, the three-dimensional at optical characteristic point 5 and stem center 11 The three-dimensional coordinate data that coordinate can directly read canonical measure steel ball center by laser tracker is obtained.And then obtain optics The relative position relation between relative position relation and characteristic point target 3 and stem center 11 between characteristic point 5.
Vision imaging system is fixed with measured target 2 afterwards, concrete model is shown in that Fig. 4 Vision imaging systems include digital camera 7th, the second double-shaft tilt angle sensor 8, data processing unit and power supply circuit;Vision imaging system is used to enter characteristic point target 3 Row imaging, is then analyzed and is handled to acquired image, so as to obtain the posture information of target.
Relative six degree of freedom measuring system provided in an embodiment of the present invention, with novel feature point model, optics in model Three-dimensional distribution is presented in characteristic point, and each characteristic point is designed as canonical measure steel ball structure, will carry infraluminescence two The acceptor circuit plate of pole pipe is arranged in steel ball housing, and the precision of demarcation and measurement is greatly improved in this.
The whole measurement process of measuring system is described in detail with reference to Fig. 5 and Fig. 6, it is described below:
The Vision imaging system that the present invention is completed with demarcation obtains Pixel Information of the optical characteristic point on camera imaging face, Pixel Information is handled, the spot center of five characteristic points is extracted respectively, and match one by one with each characteristic point.
Secondly using the coordinate system (O of laser tracker 11W,XW,YW,ZW) as benchmark, characteristic point target is obtained by demarcation Mark coordinate system (OT,XT,YT,ZT) and the first double-shaft tilt angle sensor coordinate system and digital camera coordinate system (OC,XC,YC,ZC) with Relation between second double-shaft tilt angle sensor coordinate system.The model Leica companies for the laser tracker 11 that the present invention is used AT901.
By the spot center position accurately extracted and obliquity sensor data, with reference to calibration result, initial value is linearly tried to achieve, The rotation translation matrix that characteristic point target co-ordinates are tied to digital camera coordinate system is obtained, after initial value is tried to achieve, is recycled non-linear The method of optimization obtains characteristic point to the exact value of the rotation translation matrix of digital camera coordinate system.
Slave computer is obtained by the spot center position calculated in IMAQ and processing thread and in serial port communication thread The forward and backward measured target of obliquity sensor data calculation between relative pose, measurement result is returned into host computer.
Target is with respect to the process of solution of six-degree-of-freedom information before and after obtaining:First obtain characteristic point target co-ordinates system and numeral The initial value of the rotation translation relation of camera coordinates system, because the imaging of camera has radial distortion and tangential distortion, so coordinate Conversion between system is nonlinear, therefore the exact value of rotation translation relation need to be tried to achieve according to nonlinear optimization method.
Spin matrix R around the anglec of rotation of X, Y, Z axis be respectively ω,κ.Obliquity sensor X, the output of Y-axis be respectively β, γ。
Characteristic point target co-ordinates system (OTXTYTZT) and digital camera coordinates system (OCXCYCZC) meet following relation:
Wherein characteristic point target co-ordinates are tied to the rotation of digital camera coordinate system, translation matrix and are respectively
By characteristic point coordinate system, the relation between camera coordinate system and world coordinate system is known that:
RTC=(RCW)-1RTW
Because spin matrix R=RκRφRω, R is made herea=Rκ,Rb=RφRω, then formula (4-5) can be converted into:
OrderTherefore above formula can be changed into:
Wherein,
And it is known that by rigid body translation principle:
Wherein, α22=1.
If the focal length of camera is f, radial distortion (k1,k2,k3), tangential distortion (p1,p2), the characteristic point after distortion correction exists Coordinate on practising physiognomy is (xc,yc), it can be obtained by camera imaging model:
Wherein,
Here above formula can be further write as on x=[α β t1t2t3]TSystem of linear equations:
X, R can be solved by solving above formulaTC、TTCJust it can therewith obtain, be used as the initial value of further Optimization Solution.
After initial value is tried to achieve, in addition it is also necessary to try to achieve R using the method for nonlinear optimizationTC、TTCExact value.
The imaging model of digital camera is set to:
By spin matrix RTCIt is that unit orthogonal matrix is set to:
By RcIt can set:
Construct object function:
Least square solution is asked to above formula with L-M (Levenberg-Marquardt) algorithm, suitable penalty factor M is taken, Finally obtain RTC、TTCOptimal solution.
The embodiment of the present invention is to the model of each device in addition to specified otherwise is done, and the model of other devices is not limited, As long as the device of above-mentioned functions can be completed.
The present invention is not limited to embodiments described above.The description to embodiment is intended to describe and said above Bright technical scheme, above-mentioned embodiment is only schematical, is not restricted.This is not being departed from In the case of invention objective and scope of the claimed protection, one of ordinary skill in the art may be used also under the enlightenment of the present invention The specific conversion of many forms is made, these are belonged within protection scope of the present invention.

Claims (5)

1. a kind of high-precision six-freedom degree measuring system, the relative pose for measuring preceding measured target and rear measured target, its It is characterised by, the characteristic point target being made up of cuboid housing structure is fixed with the preceding measured target, characteristic point target What front and side were stretched out respectively is provided with support, and the end of each support is mounted on being placed with optics on magnetic support, the magnetic support The first double-shaft tilt angle sensor and power supply circuit are additionally provided with characteristic point, characteristic point target;It is fixed with the rear measured target Vision imaging system, the Vision imaging system includes digital camera, the second double-shaft tilt angle sensor, data processing unit and confession Circuit;The Vision imaging system by gathering the optical signature dot image in real time, before calculating measured target relative to The relative six-degree-of-freedom information of measured target afterwards.
2. a kind of high-precision six-freedom degree measuring system according to claim 1, it is characterised in that the magnetic support is 1.5 English Very little canonical measure steel ball, the optical characteristic point is infrarede emitting diode, the optical centre and mark of infrarede emitting diode The centre of sphere of locating tab assembly steel ball steel ball is co-located.
3. a kind of high-precision six-freedom degree measuring system according to claim 1 or claim 2, it is characterised in that the optical signature Point distribution is in the non-co-planar type in space.
4. a kind of measuring method of high-precision six-freedom degree measuring system according to claim 1, it is characterised in that including with Lower step:The image that digital camera in Vision imaging system gathers optical characteristic point in real time is sent to data processing unit, together When, the angle value that the first double-shaft tilt angle sensor and the second double-shaft tilt angle sensor respectively constantly export itself is sent to number According to processing unit, data processing unit carries out pose resolving to the data collected, and the pose most calculated at last is sent to Position machine.
5. a kind of measuring method of high-precision six-freedom degree measuring system according to claim 4, it is characterised in that the solution The step of calculation, is as follows:
Characteristic point target co-ordinates system (OTXTYTZT) and digital camera coordinates system (OCXCYCZC) meet following relation:
<mrow> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>X</mi> <mi>C</mi> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>Y</mi> <mi>C</mi> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>Z</mi> <mi>C</mi> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <msub> <mi>R</mi> <mrow> <mi>T</mi> <mi>C</mi> </mrow> </msub> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>X</mi> <mi>T</mi> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>Y</mi> <mi>T</mi> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>Z</mi> <mi>T</mi> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>+</mo> <msub> <mi>T</mi> <mrow> <mi>T</mi> <mi>C</mi> </mrow> </msub> </mrow>
Wherein characteristic point target co-ordinates are tied to the rotation of digital camera coordinate system, translation matrix and are respectively
Relation between characteristic point target co-ordinates system, the first double-shaft tilt angle sensor coordinate system, world coordinate system, and rotation The relation of matrix and the anglec of rotation, and rigid body translation principle try to achieve rotation translation matrix RTC、TTC, it is used as further Optimization Solution Initial value;
The imaging model of digital camera is set to:
<mrow> <msub> <mi>f</mi> <mrow> <mi>i</mi> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>x</mi> <mi>c</mi> </msub> <mo>-</mo> <mi>f</mi> <mfrac> <mrow> <msub> <mi>r</mi> <mn>1</mn> </msub> <msub> <mi>X</mi> <mi>T</mi> </msub> <mo>+</mo> <msub> <mi>r</mi> <mn>2</mn> </msub> <msub> <mi>Y</mi> <mi>T</mi> </msub> <mo>+</mo> <msub> <mi>r</mi> <mn>3</mn> </msub> <msub> <mi>Z</mi> <mi>T</mi> </msub> <mo>+</mo> <msub> <mi>t</mi> <mn>1</mn> </msub> </mrow> <mrow> <msub> <mi>r</mi> <mn>7</mn> </msub> <msub> <mi>X</mi> <mi>T</mi> </msub> <mo>+</mo> <msub> <mi>r</mi> <mn>8</mn> </msub> <msub> <mi>Y</mi> <mi>T</mi> </msub> <mo>+</mo> <msub> <mi>r</mi> <mn>9</mn> </msub> <msub> <mi>Z</mi> <mi>T</mi> </msub> <mo>+</mo> <msub> <mi>t</mi> <mn>3</mn> </msub> </mrow> </mfrac> </mrow>
<mrow> <msub> <mi>f</mi> <mrow> <mi>i</mi> <mn>2</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>y</mi> <mi>c</mi> </msub> <mo>-</mo> <mi>f</mi> <mfrac> <mrow> <msub> <mi>r</mi> <mn>4</mn> </msub> <msub> <mi>X</mi> <mi>T</mi> </msub> <mo>+</mo> <msub> <mi>r</mi> <mn>5</mn> </msub> <msub> <mi>Y</mi> <mi>T</mi> </msub> <mo>+</mo> <msub> <mi>r</mi> <mn>6</mn> </msub> <msub> <mi>Z</mi> <mi>T</mi> </msub> <mo>+</mo> <msub> <mi>t</mi> <mn>2</mn> </msub> </mrow> <mrow> <msub> <mi>r</mi> <mn>7</mn> </msub> <msub> <mi>X</mi> <mi>T</mi> </msub> <mo>+</mo> <msub> <mi>r</mi> <mn>8</mn> </msub> <msub> <mi>Y</mi> <mi>T</mi> </msub> <mo>+</mo> <msub> <mi>r</mi> <mn>9</mn> </msub> <msub> <mi>Z</mi> <mi>T</mi> </msub> <mo>+</mo> <msub> <mi>t</mi> <mn>3</mn> </msub> </mrow> </mfrac> </mrow>
By spin matrix RTCIt is that unit orthogonal matrix is set to:
<mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>j</mi> <mn>1</mn> </msub> <mo>=</mo> <msup> <msub> <mi>r</mi> <mn>1</mn> </msub> <mn>2</mn> </msup> <mo>+</mo> <msup> <msub> <mi>r</mi> <mn>2</mn> </msub> <mn>2</mn> </msup> <mo>+</mo> <msup> <msub> <mi>r</mi> <mn>3</mn> </msub> <mn>2</mn> </msup> <mo>-</mo> <mn>1</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>j</mi> <mn>2</mn> </msub> <mo>=</mo> <msup> <msub> <mi>r</mi> <mn>4</mn> </msub> <mn>2</mn> </msup> <mo>+</mo> <msup> <msub> <mi>r</mi> <mn>5</mn> </msub> <mn>2</mn> </msup> <mo>+</mo> <msup> <msub> <mi>r</mi> <mn>6</mn> </msub> <mn>2</mn> </msup> <mo>-</mo> <mn>1</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>j</mi> <mn>3</mn> </msub> <mo>=</mo> <msup> <msub> <mi>r</mi> <mn>7</mn> </msub> <mn>2</mn> </msup> <mo>+</mo> <msup> <msub> <mi>r</mi> <mn>8</mn> </msub> <mn>2</mn> </msup> <mo>+</mo> <msup> <msub> <mi>r</mi> <mn>9</mn> </msub> <mn>2</mn> </msup> <mo>-</mo> <mn>1</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>j</mi> <mn>4</mn> </msub> <mo>=</mo> <msub> <mi>r</mi> <mn>1</mn> </msub> <msub> <mi>r</mi> <mn>4</mn> </msub> <mo>+</mo> <msub> <mi>r</mi> <mn>2</mn> </msub> <msub> <mi>r</mi> <mn>5</mn> </msub> <mo>+</mo> <msub> <mi>r</mi> <mn>3</mn> </msub> <msub> <mi>r</mi> <mn>6</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>j</mi> <mn>5</mn> </msub> <mo>=</mo> <msub> <mi>r</mi> <mn>1</mn> </msub> <msub> <mi>r</mi> <mn>7</mn> </msub> <mo>+</mo> <msub> <mi>r</mi> <mn>2</mn> </msub> <msub> <mi>r</mi> <mn>8</mn> </msub> <mo>+</mo> <msub> <mi>r</mi> <mn>3</mn> </msub> <msub> <mi>r</mi> <mn>9</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>j</mi> <mn>6</mn> </msub> <mo>=</mo> <msub> <mi>r</mi> <mn>4</mn> </msub> <msub> <mi>r</mi> <mn>7</mn> </msub> <mo>+</mo> <msub> <mi>r</mi> <mn>5</mn> </msub> <msub> <mi>r</mi> <mn>8</mn> </msub> <mo>+</mo> <msub> <mi>r</mi> <mn>6</mn> </msub> <msub> <mi>r</mi> <mn>9</mn> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced>
By RcIt can set:
<mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>k</mi> <mn>1</mn> </msub> <mo>=</mo> <msup> <mi>R</mi> <mi>c</mi> </msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>,</mo> <mn>3</mn> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <msub> <mi>k</mi> <mn>2</mn> </msub> <mo>=</mo> <msup> <mi>R</mi> <mi>c</mi> </msup> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mo>,</mo> <mn>3</mn> </mrow> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>k</mi> <mn>3</mn> </msub> <mo>=</mo> <msup> <mi>R</mi> <mi>c</mi> </msup> <mrow> <mo>(</mo> <mrow> <mn>3</mn> <mo>,</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <msub> <mi>k</mi> <mn>4</mn> </msub> <mo>=</mo> <msup> <mi>R</mi> <mi>c</mi> </msup> <mrow> <mo>(</mo> <mrow> <mn>3</mn> <mo>,</mo> <mn>2</mn> </mrow> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>k</mi> <mn>5</mn> </msub> <mo>=</mo> <msup> <mi>R</mi> <mi>c</mi> </msup> <mrow> <mo>(</mo> <mrow> <mn>3</mn> <mo>,</mo> <mn>3</mn> </mrow> <mo>)</mo> </mrow> <mo>-</mo> <mn>1</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>k</mi> <mn>6</mn> </msub> <mo>=</mo> <msup> <mi>R</mi> <mi>c</mi> </msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <msup> <mi>R</mi> <mi>c</mi> </msup> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mo>,</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>k</mi> <mn>7</mn> </msub> <mo>=</mo> <msup> <mi>R</mi> <mi>c</mi> </msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>-</mo> <msup> <mi>R</mi> <mi>c</mi> </msup> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mo>,</mo> <mn>2</mn> </mrow> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>k</mi> <mn>8</mn> </msub> <mo>=</mo> <msup> <mi>R</mi> <mi>c</mi> </msup> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mi>R</mi> <mi>c</mi> </msup> <msup> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mo>,</mo> <mn>2</mn> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>-</mo> <mn>1</mn> </mrow> </mtd> </mtr> </mtable> </mfenced>
Construct object function:
<mrow> <mi>F</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <msubsup> <mi>f</mi> <mrow> <mi>i</mi> <mn>1</mn> </mrow> <mn>2</mn> </msubsup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>+</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <msubsup> <mi>f</mi> <mrow> <mi>i</mi> <mn>2</mn> </mrow> <mn>2</mn> </msubsup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>M</mi> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mn>6</mn> </munderover> <msubsup> <mi>j</mi> <mi>i</mi> <mn>2</mn> </msubsup> <mo>(</mo> <mi>x</mi> <mo>)</mo> <mo>+</mo> <mi>M</mi> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mn>8</mn> </munderover> <msubsup> <mi>k</mi> <mi>i</mi> <mn>2</mn> </msubsup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mrow>
Least square solution is sought above formula with L-M (Levenberg-Marquardt) algorithms, R is finally obtainedTC、TTCOptimal solution.
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