CN102661742A - Self-adaptive mark point layout method based on curvature characteristic weighting centroid point constraint - Google Patents

Abstract

The invention provides a self-adaptive mark point layout method based on curvature characteristic weighting centroid point constraint. According to a characteristic mark point planning and layout principle, spatial position information of transition points is reasonably planned and arranged as few as possible in a specified measurement space to enable the distribution condition of the transition points to meet the pre-given requirement on transition precision while planning and layout efficiency of characteristic mark points can be remarkably improved.

Description

Self-adaptation monumented point layout method based on the constraint of curvature characteristic weighing center of mass point

Technical field

The invention belongs to the photoelectric measurement field, relate to a kind of self-adaptive features monumented point layout method, be particularly useful for the characteristic indication point programming and distribution design method in the coordinate system conversion based on the constraint of curvature characteristic weighing center of mass point.

Background technology

For many years, the coordinate conversion problem is one of ubiquitous problem in numerous research fields such as geographical mapping, remote sensing, vision measurement, biomedicine, robot navigation always.Especially assemble the field in the manufacturing of main equipments such as aircraft, steamer, automobile, turbine, generator; In order to obtain the complete 3D face shape of large scale mechanical part, just need multiple measuring equipments such as transit, Digital Photogrammetric System, articulated type gage beam, laser tracker and instrument to form the large-scale metrology system.This is a very complicated and loaded down with trivial details process; Wherein related to many research aspect; For example: the splicing of camera calibration, different 3D cloud datas, measurement point cloud and cad model registration, Multi-sensor Fusion etc., in view of only considering rigid motion here, therefore; Above-mentioned all problems is exactly the transfer problem of different coordinates, promptly finds the solution rotation matrix and translation vector between the coordinate system.

Coordinate conversion key of problem part is how to calculate conversion parameter accurately; And the common needs of finding the solution of conversion parameter artificially are provided with some characteristic indication points that under each different coordinates, have the known coordinate value, try to achieve conversion parameter through the simultaneous linear equations group.According to different computing functions, characteristic indication point mainly can be divided into two parts: conversion point set and test point set.Wherein, the conversion point set is mainly used in the coordinate transformation parameter that calculates between the different coordinates; Then, the transfer parameter value substitution test point set with gained according to the error assessment model, utilizes different coordinate conversion error evaluation methods, can obtain corresponding transformed error evaluation index and correlation parameter.

Usually, lay a large amount of monumented points in industry spot and participate in measurement links, this just need utilize least square method to obtain the coordinate conversion optimal value on the basis that all monumented point is known.Its advantage is, obtains globally optimal solution based on mass data, can effectively eliminate stochastic error and disturb; And shortcoming is to need a large amount of monumented points, has not only improved the measurement cost, has increased calculated amount simultaneously.Yet, in the practical set scene,, very easily cause, thereby limited the quantity of survey mark point the blocking of monumented point because assembling instrument, measuring equipment is numerous; Simultaneously, the distribution that the monumented point of equal number is laid in the operation site is different, also can produce different influences for measuring accuracy.

Therefore, the programming and distribution of characteristic indication point research has crucial Research Significance and using value, and as the primary prerequisite of coordinate conversion problem, it will directly determine the coordinate conversion error level, and then influence final measuring accuracy.

At present, many documents all are devoted to study how to distribute monumented point to improve the problem of coordinate conversion precision.Continue and expensive etc. point out that in " based on the co-registration of coordinate systems used theory and the algorithm of rigid body kinematics " monumented point mainly is the measuring error of monumented point and the space layout mode of monumented point to the influence of coordinate conversion precision in Geng Na, Zhu, a state in the Zhou Dynasty.The document is in conversion parameter is found the solution; Derive the relation between monumented point measuring error and the transformed error, and be directed in the in-site measurement monumented point very complicated situation that distributes, provided monumented point and laid principle: monumented point is selected near the transfer point; And be evenly distributed in the wider scope; Constitute irregular shape, suitably increase monumented point quantity, can effectively improve conversion accuracy.The document has been analyzed the relation between monumented point measuring error and the transformed error, but for the monumented point location problem, does not carry out quantitative analysis and derivation, and only the mode with conclusion provides.

To the coordinate system transfer problem under the rotation minute angle situation, it is obviously important in the influence of monumented point quantity to conversion accuracy to the influence of conversion accuracy that Hakan S.Kutoglu etc. has pointed out that monumented point distributes.In addition, correlativity between the conversion parameter and conversion accuracy are irrelevant.Yet this research is only applicable to rotate minute angle, and some assumed conditions in the derivation can't be satisfied with generalized case, therefore, has certain limitation.

Wang Yucheng etc. have proposed the influence that the coordinate conversion precision receives monumented point measuring error and distribution thereof; From the measuring error propagation angle; The influence of the monumented point measuring error of having derived for conversion accuracy, and, advise that then monumented point should be uniformly distributed in the measurement range for the distribution situation of monumented point; But this suggestion is conceptual prompting, does not have tight theoretical derivation as foundation.

Tevfik Ayan etc. points out: in the calculating of coordinate transformation parameter, the estimative transfer parameter value of wherein a part of error absorbs, and the uptake of error depends on the geometric distributions of monumented point.Utilize extrinsic reliabilities adjustment identification and reject the influence of gross error conversion parameter.Tevfik Ayan etc. points out simultaneously, the amount of redundancy that how much mainly depends on the monumented point number of error uptake.Here, the geometric distributions and the gross error of monumented point linked together, mainly solved the influence of how to discern and to reject gross error, but deeply do not inquired into of the influence of the geometric distributions of monumented point transformed error.

Can know that from above-mentioned document most literature has been stressed the influence of the measuring accuracy of monumented point to the coordinate conversion precision more; And, only rely on practical experience to provide some qualitative conclusions analyses for the space distribution situation of monumented point self, lack theoretical derivation process system, complete.Exist the main cause of this phenomenon to be: the measuring error of monumented point is very directly perceived to the influence of conversion accuracy, is easy to carry out the derivation of equation; And there is multiple complicated factor in the distribution of sign point set itself, the multiple often factor weave in of its influence to conversion accuracy, and this has greatly increased the difficulty of deriving.Yet the measuring accuracy of monumented point and the space distribution of monumented point are a kind of subordinate relation, and it can't characterize other characteristic that landmark space distributes fully.

Summary of the invention

The technical matters that the present invention will solve is: guarantee that the least possible transfer point of the layout of making rational planning for just can satisfy given in advance measuring accuracy requirement in the measurement space scope of regulation.

According to an aspect of the present invention; A kind of self-adaptation monumented point layout method based on the constraint of curvature characteristic weighing center of mass point is provided; Said method comprises: (1) with the actual measurement scene as cubic space; Set up the measurement space coordinate system with said cubical geometric center as true origin; And select two the first different observation erect-positions and the second observation erect-position independently to obtain the three-dimensional appearance characteristic of cubic space respectively, second coordinate system and the measurement space coordinate system of first coordinate system of the first observation erect-position and the second observation erect-position are different; (2) four summits choosing two body diagonal direction places in the cube constitute the initial conversion point sets, make the center of mass point coordinate of initial conversion point set be positioned at the true origin place of measurement space coordinate system; (3) select to be different from transfer point and be included in the inner a plurality of monumented points of initial conversion point set to constitute the test point set, and make the center of mass point coordinate of test point set overlap each other with the center of mass point coordinate of initial conversion point set; (4) utilize the coordinate figure of initial conversion point set under first coordinate system and second coordinate system to find the solution and be tied to the rotation matrix of second coordinate system and the initial value of translation vector, coordinate figure substitution rotation matrix and the initial value of translation vector of test point set under first coordinate system come error of calculation evaluation index initial value from first coordinate; (5) in the cube measurement space, lay equally spaced intensive conversion point set; Calculate the curvature fundamental function at each transfer point place; Utilize the coordinate figure of intensive conversion point set under first coordinate system and second coordinate system to find the solution rotation matrix and the translation vector that is tied to second coordinate system from first coordinate, coordinate figure substitution rotation matrix and the translation vector of test point set under first coordinate system come error of calculation evaluation index desired value; (6) keep cube geometric center invariant position, as predetermined step-length, dwindle three coordinate components values of cube measurement space successively, obtain a plurality of by big extremely little sub-cube space with the spacing between concentrated each monumented point of intensive transfer point; In each sub-cube; The curvature fundamental function of each the intensive monumented point that is comprised in the said sub-cube is found the solution the weighted mass center point coordinate value of said sub-cube as weight factor, and weighted mass center point coordinate value and measurement space coordinate origin are compared; If have side-play amount between weighted mass center point coordinate value and the measurement space coordinate origin; Then with the weighted mass center point of trying to achieve as the center; Continuation is progressively compressed said sub-cube space according to said step-length; Be contracted to up to said sub-cube space till the area of space of predetermined size, with thus obtained sub-cube space as the localized cubic body region; If do not have side-play amount between weighted mass center point coordinate value and the measurement space coordinate origin, then search for the localized cubic body region in the next sub-cube space; (7) for a localized cubic body region that in step (6), is obtained; Weighted mass center point with said localized cubic body region is the center; Integral multiple with said step-length is the radius of a ball, construct one group by little to big search ball until said localized cubic body region edge; According to the radius of a ball by little to big order, successively traversal search is carried out on each ball surface and intensive monumented point inner and that do not belong to previous ball and comprised; In the search procedure of each ball; The symmetric flag point that is centrosymmetric with each intensive monumented point and about true origin is as a pair of monumented point; Each a pair of monumented point is introduced the initial conversion point set and constitute current conversion point set; And according to the mode of step (4) to current conversion point set error of calculation evaluation index currency, form the conversion point set of renewal if the error assessment index currency that calculates less than error assessment index initial value, is then introduced the conversion point set with this to monumented point; Utilize the conversion point set that upgrades; A pair of monumented point is according to identical mode error of calculation evaluation index currency under continuing to be directed against; Each monumented point until this spheroid comprised finishes; The monumented point that meets the demands in this time search is incorporated into the concentrated and new initial conversion point set of changing point set as next spheroid search of formation of initial conversion point, and calculating error assessment index at this moment is as the error assessment index initial value of next spheroid search; Repeat above-mentioned search procedure, till the whole search of each spheroid of this localized cubic body region finish; (8) traversal search of all local cube zone repeated execution of steps (7) is operated; It is right that the transition flag point of condition of step (7) is satisfied in searching; Until the error assessment index convergence that calculates or be lower than error assessment target goals value, the number of the monumented point that lay this moment and corresponding coordinate position constitute the optimum translation point set.

In step (4); The step of utilizing the initial conversion point set to find the solution the initial value of the rotation matrix that is tied to second coordinate system from first coordinate and translation vector at the coordinate figure under first coordinate system and second coordinate system comprises: calculate the coordinate figure of center of mass point under first coordinate system and second coordinate system of initial conversion point set, as the reference point coordinate figure of initial conversion point set under first coordinate system and second coordinate system; Calculate initial conversion point and concentrate each monumented point and the coordinate difference of reference point under first coordinate system, calculate initial conversion point and concentrate each monumented point and the coordinate difference of reference point under second coordinate system; Utilize coordinate difference and the coordinate difference under second coordinate system under said first coordinate system, calculate the initial value of rotation matrix through least square method; According to the initial value and the reference point coordinate of initial conversion point set under first coordinate system and second coordinate system of the rotation matrix that calculates, calculate the initial value of translation vector.

In step (4), come the step of error of calculation evaluation index initial value to comprise at the initial value of coordinate figure substitution rotation matrix under first coordinate system and translation vector the test point set: will test point set and calculate the concentrated coordinates computed value of each test point under second coordinate system of test point at the initial value of coordinate figure substitution rotation matrix under first coordinate system and translation vector; Concentrated each test point of calculating test point is poor coordinates computed value under second coordinate system and the actual coordinate value of said test point under second coordinate system; Come error of calculation evaluation index initial value according to a kind of in the following manner: (a1) calculate the standard deviation and the maximum absolute error of the difference of said coordinate figure, as error assessment index initial value; (a2) root-mean-square error of the difference of the said coordinate figure of calculating is as error assessment index initial value; (a3) calculate the standard deviation and the maximum absolute error of the difference of said coordinate figure, and calculate the root-mean-square error of the difference of said coordinate figure, with the standard deviation of the difference of said coordinate figure and maximum absolute error and root-mean-square error as error assessment index initial value.

In step (5); Utilize intensive conversion point set to find the solution at the coordinate figure under first coordinate system and second coordinate system that the rotation matrix that is tied to second coordinate system from first coordinate and translation vector are found the solution the rotation matrix that is tied to second coordinate system from first coordinate and the step of translation vector comprises: the coordinate figure of center of mass point under first coordinate system and second coordinate system of computation-intensive conversion point set, as the reference point coordinate figure of intensive conversion point set under first coordinate system and second coordinate system; The computation-intensive transfer point is concentrated each monumented point and the coordinate difference of reference point under first coordinate system, and the computation-intensive transfer point is concentrated each monumented point and the coordinate difference of reference point under second coordinate system; Utilize coordinate difference and the coordinate difference under second coordinate system under said first coordinate system, calculate rotation matrix through least square method; According to the initial value and the reference point coordinate of intensive conversion point set under first coordinate system and second coordinate system of the rotation matrix that calculates, calculate translation vector.

In step (5), come the step of error of calculation evaluation index desired value to comprise at coordinate figure substitution rotation matrix under first coordinate system and translation vector the test point set: will test point set and calculate the concentrated coordinates computed value of each test point under second coordinate system of test point at coordinate figure substitution rotation matrix under first coordinate system and translation vector; Concentrated each test point of calculating test point is poor coordinates computed value under second coordinate system and the actual coordinate value of said test point under second coordinate system; Come error of calculation evaluation index desired value according to a kind of in the following manner: (b1) calculate the standard deviation and the maximum absolute error of the difference of said coordinate figure, as error assessment target goals value; (b2) root-mean-square error of the difference of the said coordinate figure of calculating is as error assessment target goals value; (b3) calculate the standard deviation and the maximum absolute error of the difference of said coordinate figure, and calculate the root-mean-square error of the difference of said coordinate figure, with the standard deviation of the difference of said coordinate figure and maximum absolute error and root-mean-square error as error assessment target goals value.

In step (6), the length of side of the area of space of said predetermined size be the cube measurement space the length of side 1/5.

In step (6), if two or more sub-cubes space comprises the transition flag point that is equal to or greater than predetermined percentage simultaneously, then with said two or more sub-cubes space as same sub-cube space.

Said predetermined percentage is 85%.

Said self-adaptation monumented point layout method also comprises: (9) utilize the coordinate figure of optimum translation point set under first coordinate system and second coordinate system to find the solution the optimal value that is tied to the rotation matrix and the translation vector of second coordinate system from first coordinate.

In step (9); The step of utilizing the optimum translation point set to find the solution the optimal value of the rotation matrix that is tied to second coordinate system from first coordinate and translation vector at the coordinate figure under first coordinate system and second coordinate system comprises: the coordinate figure of center of mass point under first coordinate system and second coordinate system of compute optimal conversion point set, as the reference point coordinate figure of optimum translation point set under first coordinate system and second coordinate system; The compute optimal transfer point is concentrated each monumented point and the coordinate difference of reference point under first coordinate system, and the compute optimal transfer point is concentrated each monumented point and the coordinate difference of reference point under second coordinate system; Utilize coordinate difference and the coordinate difference under second coordinate system under said first coordinate system, calculate the optimal value of rotation matrix through least square method; According to the optimal value and the reference point coordinate of optimum translation point set under first coordinate system and second coordinate system of the rotation matrix that calculates, calculate the optimal value of translation vector.

Description of drawings

In conjunction with the drawings, from the description of following embodiment, the present invention these and/or others and advantage will become clear, and are easier to understand, wherein:

Fig. 1 is according to coordinate conversion synoptic diagram of the present invention;

Fig. 2 is the process flow diagram according to the self-adaptive features monumented point layout method based on curvature characteristic weighing center of mass point constraint of the present invention.

Embodiment

Below, specify embodiments of the invention with reference to accompanying drawing.

Fig. 1 is according to coordinate conversion synoptic diagram of the present invention.Stain representation feature monumented point among Fig. 1.

As shown in Figure 1, suppose to exist two rectangular coordinate system in space to be respectively the first coordinate system C _AWith the second coordinate system C _B, from C _ATo C _BCoordinate transformation parameter can be summed up as rotation matrix R and translation vector T; In view of characteristic indication point mainly comprises conversion point set and test point set, suppose that the conversion point set comprises N transition flag point, they have different separately coordinate figures and can be expressed as respectively in above-mentioned two coordinate systems: X _CAiAnd X _CBi(i=1,2 ..., N); The test point set comprises M check mark point, and their coordinate figures in these two coordinate systems all can be expressed as: X _TAjAnd X _TBj(i=1,2 ..., M).Therefore, under above-mentioned two coordinate systems, whole monumented points that conversion point set and test point are concentrated all satisfy the rigid body translation coordinate conversion model in the equality (1), and its concrete form can be expressed as:

X _CBi＝T+R·X _CAi，X _TBj＝T+R·X _TAj (1)

1. layout parameter

The layout parameter of conversion point set and test point set, it mainly can be divided into parameter between point set intrinsic parameter and point set: the former comprises a number, reference point coordinate and coordinate difference; The latter then defines the envelope degrees of fusion between initial conversion point set and the test point set.

1.1 point set intrinsic parameter

1.1.1 some number

Hypothesis according to the front can know that it is N that transfer point is concentrated the some number that comprises, and the some number that the test point set comprises is M.

1.1.2 reference point coordinate

Owing to utilize the conversion point set to estimate coordinate transformation parameter, this center of mass point that just needs to change point set is as reference point.This center of mass point is at the first coordinate system C _AWith the second coordinate system C _BUnder coordinate figure be expressed as respectively:

$X_{\mathrm{AO}}=\frac{\underset{i=1}{\overset{N}{\mathrm{\Σ}}}{X}_{\mathrm{CAi}}}{N},$ $X_{\mathrm{BO}}=\frac{\underset{i=1}{\overset{N}{\mathrm{\Σ}}}{X}_{\mathrm{CBi}}}{N}---\left(2\right)$

And above-mentioned coordinate figure can satisfy following relation:

X _BO＝T+R·X _AO (3)

1.1.3 coordinate difference

The coordinate difference is meant the poor of the coordinate figure between each monumented point and reference point under the same coordinate system, and its symbol is promptly represented the relative position relation between monumented point and the reference point.

At the first coordinate system C _ADown, conversion point set and test point concentrate the coordinate difference between each monumented point and the reference point to be expressed as:

X _cai＝X _CAi-X _AO，X _taj＝X _TAj-X _AO (4)

In like manner, at the second coordinate system C _BDown, conversion point set and test point concentrate the coordinate difference between each monumented point and the reference point to be expressed as:

X _cbi＝X _CBi-X _BO，X _tbj＝X _TBj-X _BO (5)

Equality (4) and (5) substitution equality (3) can be derived:

X _cbi＝R·X _cai (6)

Equality (6) has reflected rotation matrix R and has changed point set respectively at the first coordinate system C _AWith the second coordinate system C _BUnder coordinate difference X _CaiAnd X _CbiBetween funtcional relationship.

1.2 layout parameter between point set

Layout parameter then mainly shows as conversion point set and the envelope degrees of fusion of test between the point set between point set, and this parameter is only in close relations relevant with these two point sets distribution situation and relative position thereof separately, and has nothing to do with coordinate system.Existing only with the first coordinate system C _AFor example describes.

1.2.1 conversion point set envelope ball

At the first coordinate system C _ADown, the conversion point set comprises N transfer point, and wherein the coordinate figure of each transfer point is X _CAi(i=1,2 ..., N).Existing center of mass point O with the conversion point set _CABeing the center, is radius R with each transfer point to the maximum Euclidean distance of center of mass point _C, the minimum envelop ball that can include all transfer points that it generated is defined as conversion point set envelope ball, it is at the first coordinate system C _AUnder sphere centre coordinate and radius of a ball R _CCan be write as following form respectively:

X _A0＝(x _A0，y _A0，z _A0) ^T；

$R_C=\underset{i}{\max }\left(\sqrt{{(x_{\mathrm{CAi}}-x_{A0})}^2+{(y_{\mathrm{CAi}}-y_{A0})}^2+{(z_{\mathrm{CAi}}-z_{A0})}^2}\right),(i=\mathrm{1,2},...,N)---\left(7\right)$

1.2.2 test point set envelope ball

At the first coordinate system C _ADown, the test point set comprises M transfer point, and wherein the coordinate figure of each test point is X _TAj(i=1,2 ..., M).Here, to test the barycenter O of point set _TABe the center, with each test point-to-point O _TAMaximum Euclidean distance be radius R _T, generating a minimum envelop ball that can comprise all test points and may be defined as test point set envelope ball, it is at the first coordinate system C _AUnder sphere centre coordinate value and radius of a ball RT be expressed as respectively:

$X_{\mathrm{TA}0}={(x_{\mathrm{TA}0},y_{\mathrm{TA}0},z_{\mathrm{TA}0})}^T={(\frac{\underset{j=1}{\overset{M}{\mathrm{\Σ}}}{x}_{\mathrm{TAj}}}{M},\frac{\underset{j=1}{\overset{M}{\mathrm{\Σ}}}{y}_{\mathrm{TAj}}}{M},\frac{\underset{j=1}{\overset{M}{\mathrm{\Σ}}}{z}_{\mathrm{TAj}}}{M})}^T;$

$R_T=\underset{j}{\max }\left(\sqrt{{(x_{\mathrm{TAj}}-x_{\mathrm{TA}0})}^2+{(y_{\mathrm{TAj}}-y_{\mathrm{TA}0})}^2+{(z_{\mathrm{TAj}}-z_{\mathrm{TA}0})}^2}\right),(j=\mathrm{1,2},...,M)---\left(8\right)$

1.2.3 envelope degrees of fusion

Envelope degrees of fusion problem between conversion point set and the test point set can be converted into crossing, overlapping degree between two envelope balls, and define two envelope blending degree between the point set is φ at present, and mathematical notation is following:

$\mathrm{\φ}=\ln \left(\frac{R_C+R_T}{\stackrel{\‾}{O_{\mathrm{CA}}{O}_{\mathrm{TA}}}}\right)=\ln \left(\frac{R_C+R_T}{\sqrt{{(x_{\mathrm{AO}}-x_{\mathrm{TAO}})}^2+{(y_{\mathrm{AO}}-y_{\mathrm{TAO}})}^2+{(z_{\mathrm{AO}}-z_{\mathrm{TAO}})}^2}}\right)---\left(9\right)$

In the equality (9), the Euclidean distance between 2 envelope centre of sphere points of

expression.

Envelope degrees of fusion parameter Φ can change point set and test the degree that point set blends each other by pragmatize.Can know by equality (9), the span of envelope degrees of fusion parameter Φ be (∞ ,+∞).Envelope degrees of fusion parameter Φ has following character:

(1) works as R _C=R _T=0 o'clock, φ →-∞;

(2) work as R _C+ R _T＞0, $\stackrel{\‾}{O_{\mathrm{CA}}{O}_{\mathrm{TA}}}=0$ The time, $\mathrm{\φ}=\mathrm{Ln}\left(\frac{R_C+R_T}{\stackrel{\‾}{O_{\mathrm{CA}}{O}_{\mathrm{TA}}}}\right)=\mathrm{Lg}\left(\frac{R_C+R_T}{0}\right)\→+\∞;$

(3) when $R_C+R_T=\stackrel{\‾}{O_{\mathrm{CA}}{O}_{\mathrm{TA}}}$ The time, φ=ln (1)=0;

(4) when $R_C+R_T>\stackrel{\‾}{O_{\mathrm{CA}}{O}_{\mathrm{TA}}}$ The time, $\frac{R_C+R_T}{\stackrel{\‾}{O_{\mathrm{CA}}{O}_{\mathrm{TA}}}}>1,$ $\mathrm{\φ}=\mathrm{Ln}\left(\frac{R_C+R_T}{\stackrel{\‾}{O_{\mathrm{CA}}{O}_{\mathrm{TA}}}}\right)>0;$

(5) when $R_C+R_T<\stackrel{\&OverBar;}{O_{\mathrm{CA}}{O}_{\mathrm{TA}}}$ The time, $0<\frac{R_C+R_T}{\stackrel{\&OverBar;}{O_{\mathrm{CA}}{O}_{\mathrm{TA}}}}<1,$ $\mathrm{\&phi;}=\mathrm{Ln}\left(\frac{R_C+R_T}{\stackrel{\&OverBar;}{O_{\mathrm{CA}}{O}_{\mathrm{TA}}}}\right)<0.$

Definition and relevant nature according to the envelope degrees of fusion can be known; When if conversion point set envelope ball mutually disjoints with test point set envelope ball; Then the envelope degrees of fusion is a negative, and along with the Euclidean distance between the centre of sphere of two envelope balls constantly increases and progressively reduces, until φ →-∞; If conversion point set envelope ball presents tangent state with test point set envelope ball, then the envelope degrees of fusion is zero; If when conversion point set envelope ball occurred intersecting with test point set envelope ball, then the envelope degrees of fusion was a positive number, and increases gradually along with constantly reducing of Euclidean distance between the centre of sphere of two envelope balls.And if only if when two envelope centre ofs sphere overlap fully, promptly change point set and can merge fully each other this moment with the test point set, then envelope degrees of fusion φ →+∞.

2. coordinate transformation parameter is found the solution

Finding the solution the numerical value that relates generally to rotation matrix R and translation vector T based on the coordinate transformation parameter of rigid body translation estimates.Usually, will change point set at the first coordinate system C _AWith the second coordinate system C _BUnder coordinate figure as known conditions, resolve coordinate transformation parameter R and T.In computation process, need resolve rotation matrix R earlier, and then ask for translation vector T.

2.1 rotation matrix

Mainly utilize least square method to find the solution rotation matrix R, this method is directly utilized principle of least square method, finds the solution the globally optimal solution of overdetermination system of linear equations, and its concrete solution procedure is described below at present:

Equality (6) is carried out vectorial expansion can be got:

$\left(\begin{array}{c}{x}_{\mathrm{cbi}}\\ y_{\mathrm{cbi}}\\ z_{\mathrm{cbi}}\end{array}\right)=\left(\begin{array}{ccc}{r}_1& r_2& r_3\\ r_4& r_5& r_6\\ r_7& r_8& r_9\end{array}\right)\&CenterDot;\left(\begin{array}{c}{x}_{\mathrm{cai}}\\ y_{\mathrm{cai}}\\ z_{\mathrm{cai}}\end{array}\right)---\left(10\right)$

Being rewritten into system of linear equations can be expressed as:

$\left\{\begin{array}{c}{x}_{\mathrm{cbi}}=r_1\&CenterDot;x_{\mathrm{cai}}+r_2\&CenterDot;y_{\mathrm{cai}}+r_3\&CenterDot;z_{\mathrm{cai}}\\ y_{\mathrm{cbi}}=r_4\&CenterDot;x_{\mathrm{cai}}+r_5\&CenterDot;y_{\mathrm{cai}}+r_6\&CenterDot;z_{\mathrm{cai}}\\ z_{\mathrm{cbi}}=r_7\&CenterDot;x_{\mathrm{cai}}+r_8\&CenterDot;y_{\mathrm{cai}}+r_9\&CenterDot;z_{\mathrm{cai}}\end{array}\right.---\left(11\right)$

It is P that existing hypothesis waits to ask unknown column vector, can make P=(r ₁, r ₂, r ₃, r ₄, r ₅, r ₆, r ₇, r ₈, r ₉) ^T, then equality (11) can be put in order immediately and be following form:

$\left(\begin{array}{ccccccccc}{x}_{\mathrm{ca}1}& y_{\mathrm{ca}1}& z_{\mathrm{ca}1}& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& x_{\mathrm{ca}1}& y_{\mathrm{ca}1}& z_{\mathrm{ca}1}& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& x_{\mathrm{ca}1}& y_{\mathrm{ca}1}& z_{\mathrm{ca}1}\\ ...& ...& ...& ...& ...& ...& ...& ...& ...\\ x_{\mathrm{caN}}& y_{\mathrm{caN}}& z_{\mathrm{caN}}& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& x_{\mathrm{caN}}& y_{\mathrm{caN}}& z_{\mathrm{caN}}& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& x_{\mathrm{caN}}& y_{\mathrm{caN}}& z_{\mathrm{caN}}\end{array}\right)\&CenterDot;\left(\begin{array}{c}{r}_1\\ r_2\\ r_3\\ r_4\\ r_5\\ r_6\\ r_7\\ r_8\\ r_9\end{array}\right)=\left(\begin{array}{c}{x}_{\mathrm{cb}1}\\ y_{\mathrm{cb}1}\\ z_{\mathrm{cb}1}\\ ...\\ x_{\mathrm{cbN}}\\ y_{\mathrm{cbN}}\\ z_{\mathrm{cbN}}\end{array}\right)---\left(12\right)$

The matrix form of equality (12) can be expressed as:

U·P＝F (13)

Wherein, U represents the constant coefficient matrix, and its matrix dimension is (3N * 9), this matrix only with each transfer point at the first coordinate system C _AUnder the coordinate difference relevant; And the dimension of column vector F is (3N * 1), it only with transfer point at the second coordinate system C _BUnder the coordinate difference relevant; And column vector P is unknown vector to be asked.Have only when 3N＞9, that is to say N＞3 o'clock, according to least square method, the overdetermination system of linear equations optimum solution that can obtain expression in the equality (11) is:

P＝(U ^T·U) ^-1·U ^T·F (14)

Subsequently, should optimum solution P be rearranged and can get according to the representation of rotation matrix:

$R=\left(\begin{array}{ccc}P\left(1\right)& P\left(2\right)& P\left(3\right)\\ P\left(4\right)& P\left(5\right)& P\left(6\right)\\ P\left(7\right)& P\left(8\right)& P\left(9\right)\end{array}\right)---\left(15\right)$

Wherein, i element of P (i) expression column vector.

Notice that according to making demands before the least square method, N should consider that the represented transfer point number of N should be integer greater than 3, this just explains needs 4 transfer points at least, could guarantee (the U in the equality (14) ^TU) there is inverse matrix, thereby solves optimum solution.

2.2 translation vector

Can know that according to equality (3) translation vector T mainly receives the restriction of reference point coordinate and these two aspects of rotation matrix R.With reference to equality (3), translation vector T can calculate according to following equality:

T＝X _BO-R·X _AO (16)

3. the foundation of coordinate conversion error appraisement system

For the precision level of the coordinate transformation parameter that can effectively weigh the front and calculated, need set up a cover coordinate conversion error appraisement system, this coordinate conversion error appraisement system mainly comprises: coordinate conversion error evaluation model and error assessment method.

3.1 coordinate conversion error evaluation model

3.1.1 conversion point set-conversion point set evaluation model

Conversion point set-conversion point set evaluation model is a kind of coordinate conversion precision evaluation mode of widespread usage.The conversion point set is as wherein unique central factor, and relate generally to two aspects: promptly rotation matrix, translation vector finds the solution and the coordinate conversion precision evaluation.The former adopts least square method to obtain the conversion parameter optimum solution; The latter then is on the former basis, utilizes three kinds of different error assessment methods describing subsequently, carries out the coordinate conversion precision evaluation.

In fact, conversion point set-conversion point set evaluation model is through being optimized following objective function, and to solve a nonlinear least square problem, then this objective function can be expressed as:

$\min H(R,T)=\min \underset{i=1}{\overset{N}{\mathrm{\&Sigma;}}}{\left|\right|R\left(X_{\mathrm{CAi}}\right)+T-X_{\mathrm{CBi}}\left|\right|}^2---\left(17\right)$

Wherein, H is the objective function that needs optimization, || || represent the European norm of this matrix.

This model is simple and clear; Easy to operate; This model can be unified in finding the solution with the coordinate conversion precision evaluation of coordinate transformation parameter among the objective function that shows in the equality (17) effectively; But the evaluation result of coordinate conversion precision receives the influence of conversion point set oneself factor (for example, factors such as the locus distribution of transfer point number, transfer point) easily, has certain limitation.

3.1.2 conversion point set-test point set evaluation model

Conversion point set-test point set evaluation model has reflected a kind of coordinate conversion evaluation method commonly used equally; It is on the basis of preceding a kind of evaluation model, and the fixing and test point set that be different from the conversion point set of one group of extra introducing is devoted to the appraisal of coordinate conversion precision emphatically.This model relates generally to two parts: on the one hand, utilize methods such as least square method, singular value decomposition method and unit quaternion method to resolve rotation matrix and translation vector based on the conversion point set; On the other hand, the conversion parameter substitution test point set with calculating utilizes three kinds of error assessment methods describing subsequently, calculates corresponding error assessment index, promptly estimates error, standard deviation and maximum absolute error etc.

The mathematic(al) representation of conversion point set-test point set evaluation model is following:

$\min H(R,T)=\mathrm{\&alpha;}\&CenterDot;\min \underset{i=1}{\overset{N}{\mathrm{\&Sigma;}}}{\left|\right|R\&CenterDot;\left(X_{\mathrm{CAi}}\right)+T-X_{\mathrm{CBi}}\left|\right|}^2+(1-\mathrm{\&alpha;})\&CenterDot;\min \underset{j=1}{\overset{M}{\mathrm{\&Sigma;}}}[R\left(X_{\mathrm{TAj}}\right)+T-X_{\mathrm{TBj}}]---\left(18\right)$

Equality (18) has embodied a concentrated reflection of above-mentioned two parts content, and wherein, α representes above-mentioned two weight coefficients shared in evaluation model, and 0＜α＜1 is got α=0.5 usually; H is the objective function that needs optimization; || || represent the European norm of this matrix; First of equal sign the right mainly shows as the principle of least square of utilizing the conversion point set to find the solution the rotation translation matrix, and second has then been embodied the test point set and estimated the coordinate conversion precision.This model can obtain the conversion parameter optimum solution based on the least square principle on all transition flag point bases, and this group optimum solution coordinate re-projection error that can make test point concentrate demonstrates and minimizes.

The advantage of this model is mainly reflected in: fixedly choose one group of test point set that is different from the conversion point set as metewand; The corresponding conversion parameter that can be tried to achieve the different switching point set; Realize unified precision evaluation and comparison, thus the influence of having avoided conversion point set self that transformed error is estimated.

3.2 coordinate conversion error evaluation method

The height of coordinate conversion precision mainly depends on to be weighed and estimates the size of coordinate conversion error.Here, three kinds of evaluation methods are mainly adopted in the evaluation of coordinate conversion error: the coordinate figure theory of error, the root-mean-square error method and the relative Euclidean distance theory of error.

Be convenient and derive, make following hypothesis at present: through computational solution get from the first coordinate system C _ATo the second coordinate system C _BConversion parameter be respectively: rotation matrix R _CalWith translation vector T _CalWith each test point at the first coordinate system C _AUnder coordinate figure X _TAjThe conversion parameter that substitution calculates can draw it at the second coordinate system C _BUnder the coordinates computed value, promptly

$X_{\mathrm{TBj}}^{\mathrm{cal}}=T_{\mathrm{cal}}+R_{\mathrm{cal}}\&CenterDot;X_{\mathrm{TAj}},(j=\mathrm{1,2},...,M)---\left(19\right)$

3.2.1 the coordinate figure theory of error

When solving from the first coordinate system C _ATo the second coordinate system C _BConversion parameter R _CalAnd T _CalAfter, can be with the test point set at the first coordinate system C _AUnder the above-mentioned transfer parameter value of coordinate figure substitution, thereby try to achieve each test point at the second coordinate system C _BUnder the coordinates computed value Get the coordinates computed value and the actual coordinate value X of test point again _TBjDifference be the test point set the coordinate figure error matrix, can be expressed as:

${\mathrm{error}}_{\mathrm{co}}={({\mathrm{error}}_{\mathrm{co}1},{\mathrm{error}}_{\mathrm{co}2},...,{\mathrm{error}}_{\mathrm{coj}},...,{\mathrm{error}}_{\mathrm{coM}})}^T$

$={(X_{\mathrm{TB}1}^{\mathrm{cal}}-X_{\mathrm{TB}1},X_{\mathrm{TB}2}^{\mathrm{cal}}-X_{\mathrm{TB}2},...,X_{\mathrm{TBj}}^{\mathrm{cal}}-X_{\mathrm{TBj}},...,X_{\mathrm{TBM}}^{\mathrm{cal}}-X_{\mathrm{TBM}})}^T---\left(20\right)$

In the equality (20), coordinate figure error matrix error _CoIt is the matrix of M * 3; Wherein each row has been represented the coordinate figure error of test point about X, Y, Z axle respectively, and is existing to each row use average error, standard deviation, maximum error, least error and the maximum absolute error parameter index as evaluation coordinate conversion precision.Existing is example with the X axle, lists the computing formula of above-mentioned parameter index, and all the other each coordinate axis all similarly.

${\mathrm{xmean}}_{\mathrm{error}}^{\mathrm{co}}=\frac{\underset{j=1}{\overset{M}{\mathrm{\&Sigma;}}}{\mathrm{error}}_{\mathrm{co}}(j,1)}{M};$

${\mathrm{xstd}}_{\mathrm{error}}^{\mathrm{co}}=\sqrt{\frac{\underset{j=1}{\overset{M}{\mathrm{\&Sigma;}}}{[{\mathrm{error}}_{\mathrm{co}}(j,1)-{\mathrm{xmean}}_{\mathrm{error}}^{\mathrm{co}}]}^2}{M}};$

${x\max }_{\mathrm{error}}^{\mathrm{co}}=\underset{j}{\max }\left[{\mathrm{error}}_{\mathrm{co}}(j,1)\right];$

${x\min }_{\mathrm{error}}^{\mathrm{co}}=\underset{j}{\min }\left[{\mathrm{error}}_{\mathrm{co}}(j,1)\right];$

${\mathrm{xabsm}}_{\mathrm{error}}^{\mathrm{co}}=\underset{j}{\max }\left|{\mathrm{error}}_{\mathrm{co}}(j,1)\right|;---\left(21\right)$

3.2.2 root-mean-square error method

Root-mean-square error (RMS Error) is as a kind of important parameter of weighing precision, and it has embodied a concentrated reflection of the dispersion degree of error.Its calculation equation is represented as follows:

${\mathrm{error}}_{\mathrm{rms}}=\sqrt{\frac{{(X_{\mathrm{TB}1}^{\mathrm{cal}}-X_{\mathrm{TB}1})}^2+{(X_{\mathrm{TB}2}^{\mathrm{cal}}-X_{\mathrm{TB}2})}^2+...+{(X_{\mathrm{TBj}}^{\mathrm{cal}}-X_{\mathrm{TBj}})}^2+...+{(X_{\mathrm{TBM}}^{\mathrm{cal}}-X_{\mathrm{TBM}})}^2}{M}}---\left(22\right)$

3.2.3 the relative Euclidean distance theory of error

At the second coordinate system C _BDown, concentrate the some test points of selection arbitrarily, utilize the actual coordinate value X of each test point respectively in test point _TBjAnd coordinates computed value separately

Ask for the Euclidean distance vector of all the other test points, can be expressed as with respect to selected test point:

${\mathrm{dis}}_t=\sqrt{{(x_{\mathrm{TBj}}-x_{\mathrm{TB}1})}^2+{(y_{\mathrm{TBj}}-y_{\mathrm{TB}1})}^2+{(z_{\mathrm{TBj}}-z_{\mathrm{TB}1})}^2},$ j＝2，3，…，M，t＝j-1；

dis＝(dis ₁，dis ₂，…，dis _t，…，dis _M-1) ^T；

${\mathrm{dis}}_t^{\mathrm{cal}}=\sqrt{{(x_{\mathrm{TBj}}^{\mathrm{cal}}-x_{\mathrm{TB}1}^{\mathrm{cal}})}^2+{(y_{\mathrm{TBj}}^{\mathrm{cal}}-y_{\mathrm{TB}1}^{\mathrm{cal}})}^2+{(z_{\mathrm{TBj}}^{\mathrm{cal}}-z_{\mathrm{TB}1}^{\mathrm{ca}1})}^2},$ j＝2，3，…，M，t＝j-1；

${\mathrm{dis}}^{\mathrm{cal}}={({\mathrm{dis}}_1^{\mathrm{cal}},{\mathrm{dis}}_2^{\mathrm{cal}},...,d_t^{\mathrm{cal}},...,{\mathrm{dis}}_{M-1}^{\mathrm{cal}})}^T;---\left(23\right)$

Then the Euclidean distance error vector can be expressed as relatively:

${\mathrm{error}}_{\mathrm{dis}}=({\mathrm{dis}}_1^{\mathrm{cal}}-{\mathrm{dis}}_1,{\mathrm{dis}}_2^{\mathrm{cal}}-{\mathrm{dis}}_2,...,{\mathrm{dis}}_t^{\mathrm{cal}}-{\mathrm{dis}}_t,...,{\mathrm{dis}}_{M-1}^{\mathrm{cal}}-{\mathrm{dis}}_{M-1})---\left(24\right)$

The dimension of this error vector is (M-1) * 1, and this vector is made even equal error, standard deviation, maximum error, least error and maximum absolute error as the parameter index of estimating the coordinate conversion precision, and expression is as follows respectively:

${\mathrm{mean}}_{\mathrm{error}}^{\mathrm{dis}}=\frac{\underset{t=1}{\overset{M-1}{\mathrm{\&Sigma;}}}{\mathrm{error}}_{\mathrm{dis}}(t,1)}{M-1};$

${\mathrm{std}}_{\mathrm{error}}^{\mathrm{dis}}=\sqrt{\frac{\underset{t=1}{\overset{M-1}{\mathrm{\&Sigma;}}}{[{\mathrm{error}}_{\mathrm{dis}}(t,1)-{\mathrm{mean}}_{\mathrm{error}}^{\mathrm{dis}}]}^2}{M-1}};$

${\max }_{\mathrm{error}}^{\mathrm{dis}}=\underset{t}{\max }\left[{\mathrm{error}}_{\mathrm{dis}}(t,1)\right];$

${\min }_{\mathrm{error}}^{\mathrm{dis}}=\underset{t}{\min }\left[{\mathrm{error}}_{\mathrm{dis}}(t,1)\right];$

${\mathrm{dbsm}}_{\mathrm{error}}^{\mathrm{dis}}=\underset{t}{\max }\left|{\mathrm{error}}_{\mathrm{dis}}(t,1)\right|;---\left(25\right)$

Yet; In practical operation, find; Relatively the Euclidean distance theory of error generally is applicable to conversion point set-conversion point set error assessment model, but it has certain scope of application for conversion point set-test point set error assessment model: when the relative tertiary location of testing point set fixedly the time, no matter change what kind of variation of point set generation; The Euclidean distance theory of error is invalid relatively, and error assessment parameter index wherein is steady state value.Below, will make following proof to above-mentioned conclusion:

On the one hand, at the first coordinate system C _ADown, select wherein some test points point as a reference, its coordinate figure is X _TA0, all the other M-1 test point respectively and the relative Euclidean distance between this RP

Can be expressed as:

${\left(d_{\mathrm{jj}}^A\right)}^2={(X_{\mathrm{TAjj}}-X_{\mathrm{TA}0})}^T\&CenterDot;(X_{\mathrm{TAjj}}-X_{\mathrm{TA}0}),(\mathrm{jj}=\mathrm{1,2},...,M-1)---\left(26\right)$

On the other hand, at the second coordinate system C _BDown, the estimated coordinates value of selected RP does

In like manner, all the other M-1 test point respectively and the relative Euclidean distance between this RP

Then can be expressed as:

${\left(d_{\mathrm{jj}}^{\mathrm{Bcal}}\right)}^2={(X_{\mathrm{TBjj}}^{\mathrm{cal}}-X_{\mathrm{TB}0}^{\mathrm{cal}})}^T\&CenterDot;(X_{\mathrm{TBjj}}^{\mathrm{cal}}-X_{\mathrm{TB}0}^{\mathrm{cal}}),(\mathrm{jj}=\mathrm{1,2},...,M-1)---\left(27\right)$

Now on the Euclidean distance between two opposing and make the following derivation of the relationship between:

${\left(d_{\mathrm{jj}}^{\mathrm{Bcal}}\right)}^2={(X_{\mathrm{TBjj}}^{\mathrm{cal}}-X_{\mathrm{TB}0}^{\mathrm{cal}})}^T\&CenterDot;(X_{\mathrm{TBjj}}^{\mathrm{cal}}-X_{\mathrm{TB}0}^{\mathrm{cal}})$

$={[(T_{\mathrm{cal}}+R_{\mathrm{cal}}\&CenterDot;X_{\mathrm{TAjj}})-(T_{\mathrm{cal}}+R_{\mathrm{cal}}\&CenterDot;X_{\mathrm{TA}0})]}^T\&CenterDot;[(T_{\mathrm{cal}}+R_{\mathrm{cal}}\&CenterDot;X_{\mathrm{TAjj}})-(T_{\mathrm{cal}}+R_{\mathrm{cal}}\&CenterDot;X_{\mathrm{TA}0})]$

$={[R_{\mathrm{cal}}\&CenterDot;(X_{\mathrm{TAjj}}-X_{\mathrm{TA}0})]}^T\&CenterDot;[R_{\mathrm{cal}}\&CenterDot;(X_{\mathrm{TAjj}}-X_{\mathrm{TA}0})]$

$={(X_{\mathrm{TAjj}}-X_{\mathrm{TA}0})}^T\&CenterDot;{\left(R_{\mathrm{cal}}\right)}^T\&CenterDot;R_{\mathrm{cal}}\&CenterDot;(X_{\mathrm{TAjj}}-X_{\mathrm{TA}0})$

$={(X_{\mathrm{TAjj}}-X_{\mathrm{TA}0})}^T\&CenterDot;{(R_{\mathrm{cal}}\&CenterDot;{\left(R_{\mathrm{cal}}\right)}^T)}^T\&CenterDot;(X_{\mathrm{TAjj}}-X_{\mathrm{TA}0})$

$={(X_{\mathrm{TAjj}}-X_{\mathrm{TA}0})}^T\&CenterDot;I\&CenterDot;(X_{\mathrm{TAjj}}-X_{\mathrm{TA}0})={\left(d_{\mathrm{jj}}^A\right)}^2---\left(28\right)$

In above-mentioned derivation, relate to one of them character of rotation matrix, i.e. R _Cal(R _Cal) ^T=I.I is a unit matrix.

In addition, at the second coordinate system C _BDown, selected actual coordinate value with reference to test point is X _TB0, then all the other M-1 test point respectively and the relative Euclidean distance between this RP

Can be expressed as:

${\left(d_{\mathrm{jj}}^B\right)}^2={(X_{\mathrm{TBjj}}-X_{\mathrm{TB}0})}^T\&CenterDot;(X_{\mathrm{TBjj}}-X_{\mathrm{TB}0}),(\mathrm{jj}=\mathrm{1,2},...,M-1)---\left(29\right)$

According to the Error Calculation formula of relative Euclidean distance, can obtain relative Euclidean distance error parameter and be:

${\mathrm{error}}_{\mathrm{jj}}=d_{\mathrm{jj}}^B-d_{\mathrm{jj}}^{\mathrm{Bcal}}=d_{\mathrm{jj}}^B-d_{\mathrm{jj}}^A---\left(30\right)$

Can find out that from equality (30) only the coordinate figure under two coordinate systems is closely related respectively with test point for the relative Euclidean distance error of M-1 the test point that test point is concentrated, and irrelevant with coordinate transformation parameter.

Can find out that based on above-mentioned derivation which kind of rotation matrix, translation vector that no matter the transfer point collected explanations or commentaries is calculated have separates, then as long as the test point set is definite; Then its relative Euclidean distance just can obtain; That is to say no matter change point set and whether change, can not influence the relative Euclidean distance error assessment parameter value that finally calculates; That is, Euclidean distance error assessment method is invalid relatively.

4. the governing principle of characteristic indication point programming and distribution designs

The governing principle of characteristic indication point programming and distribution mainly comprises initial conversion point set selection principle, tests the point set selection principle, newly changes point set programming and distribution principle and four aspects of monumented point programming and distribution constraint condition.

4.1 the selection principle of initial conversion point set

(1) the concentrated some number that comprises of initial conversion point is 4

(2) will change the reference point of the center of mass point of point set, thereby the reference point coordinate of conversion point set is positioned at the true origin place as the conversion point set.That is to say that each initial conversion point is about the true origin distribution that is centrosymmetric.

4.2 the selection principle of test point set

(1) test point coordinate and transfer point coordinate must be different;

(2) test point must be included within the initial point set coordinate figure scope;

(3) test point set center-of-mass coordinate overlaps with conversion point set center-of-mass coordinate;

(4) the test point set is tending towards infinitely great with the envelope degrees of fusion of conversion point set, promptly φ →+∞.

4.3 newly change the programming and distribution principle of point set

(1) the new transfer point number Δ N that introduces should be the positive integer more than or equal to 1;

(2) the coordinate average of a new Δ N transfer point of introducing should overlap with the reference point coordinate of initial conversion point set as much as possible, makes newly to change point set still about the true origin distribution that is centrosymmetric; Wherein, when Δ N is odd number, have at least a new transfer point of introducing to be positioned at the true origin place; When Δ N is even number, can find the transfer point of the new introducing of its correspondence that is centrosymmetric about true origin for the transfer point of each new introducing.

4.4 the constraint condition of monumented point programming and distribution

4.4.1 curvature characteristic weighing center of mass point constraint

In view of the curvature characteristic be 3D measurement space self intrinsic attribute, irrelevant with extraneous factor, therefore, the curvature characteristic distribution will directly determine the density degree of monumented point programming and distribution.For the influence that embodies the curvature characteristic to the monumented point programming and distribution directly perceived, introduce curvature characteristic weighing center of mass point notion at present, can be used as one of them important constraint condition of monumented point programming and distribution.

The definition procedure of curvature characteristic weighing center of mass point is following: suppose to exist a group mark point set can be expressed as X={X _i| i=1,2 ..., n}, X _i=(x _i, y _i, z _i) ^T, wherein the principal curvatures at each monumented point place is made as k ₁, k ₂, and the Gaussian curvature at each monumented point place and mean curvature can be set at k respectively _Gas, k _Avg, the curvature characteristic weighing center of mass point coordinate X of this group mark point set then _OCan calculate to as follows:

$X_O=\frac{\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}Q\left(X_i\right)\&CenterDot;X_i}{\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}Q\left(X_i\right)}---\left(31\right)$

Wherein, Q (X _i) represent the curvature fundamental function at each monumented point place, its mathematical definition formula is:

$Q\left(X_i\right)=k_1^2+k_2^2={(2\&CenterDot;k_{\mathrm{avg}})}^2-{2k}_{\mathrm{gas}}$

$k_{\mathrm{avg}}=\frac{k_1+k_2}{2}$

k _gas＝k ₁·k ₂ (32)

Can find out from equality (31) and formula (32), with the curvature characteristic at each monumented point place as the weight coefficient of finding the solution the center of mass point coordinate, weighted mass center point position will be obviously near the tangible zone of curved transition.Based on the rigid body kinematics principle; Need to indicate that point set is regarded as rigid body; And the monumented point position with violent curved transition is more remarkable for the influence of coordinate conversion precision, and this just need be at the more relatively monumented point of the tangible area arrangements of curved transition, to guarantee the coordinate conversion precision.Therefore, the area of space that curvature characteristic weighing center of mass point constraint can be used for confirming to have remarkable curvature characteristic, thus establish favourable basis for the monumented point layout.

4.4.2 the quadratic sum least commitment of coordinate conversion error

Can know that according to the least square method ultimate principle in finding the solution the process of coordinate transformation parameter, the optimum solution that require to obtain is based on all and participates in transfer points of calculating, make that the quadratic sum of coordinate conversion error of whole transfer points is minimum.

4.4.3 the absolute error index least commitment of test point

Here mainly adopt three kinds of error assessment methods such as the coordinate figure theory of error, root-mean-square error method and the relative Euclidean distance theory of error that the coordinate conversion precision is evaluated.The coordinate figure theory of error relates generally to two parameter indexs (that is, standard deviation and maximum absolute error) with the relative Euclidean distance theory of error, and the root-mean-square error method relates generally to root-mean-square error.Here the final purpose of studying the programming and distribution of characteristic indication point just is to make standard deviation and maximum absolute error and/or root-mean-square error to reduce to minimum value.Yet characteristic indication point programming and distribution designs is based on that same group of test point set carry out, therefore; The Euclidean distance theory of error is in disarmed state relatively; Can not participate in the coordinate conversion error evaluation, therefore, the present invention only adopts the coordinate figure theory of error and root-mean-square error method.

To describe in detail below according to the self-adaptive features monumented point layout method based on the constraint of curvature characteristic weighing center of mass point of the present invention.Purpose based on the self-adaptive features monumented point layout method of curvature characteristic weighing center of mass point constraint is; The principle of the characteristic indication point programming and distribution of summarizing according to the front; In the measurement space of regulation, the make rational planning for spatial positional information of the least possible transfer point of layout; Make the distribution situation of these transfer points can satisfy given in advance conversion accuracy requirement (that is, being positioned within the transformed error scope of permission), can improve characteristic indication point programming and distribution efficient significantly again.On this basis, utilize least square method to confirm that the conversion parameter between the coordinate system promptly can be considered optimum solution.

Fig. 2 is the process flow diagram according to the self-adaptive features monumented point layout method based on curvature characteristic weighing center of mass point constraint of the present invention.

With reference to Fig. 2, in operation 201, the measure field of reality as cubic space, is set up the measurement space coordinate system with this cubical geometric center as true origin, and confirm the span of each coordinate components of this cubic space.Then, select two the first different observation erect-positions and the second observation erect-position independently to obtain the 3D shape characteristic of cubic space respectively, the coordinate system of the first observation erect-position is the first coordinate system C _A, the coordinate system of the second observation erect-position is the second coordinate system C _B, and the first coordinate system C _AWith the second coordinate system C _BDifferent with the measurement space coordinate system, from the first coordinate system C _ATo the second coordinate system C _BCoordinate transformation parameter be the institute ask, coordinate transformation parameter mainly comprises rotation matrix R and translation vector T.

In operation 202, choose four summits formation initial conversion point sets at two body diagonal direction places in the cube, thereby make the center of mass point coordinate of initial conversion point set be positioned at the true origin place of measurement space coordinate system.Therefore, the transfer point concentrated of initial conversion point is at the first coordinate system C _AWith the second coordinate system C _BUnder coordinate figure be expressed as X respectively _CAiAnd X _CBi(i=1,2,3,4), and satisfy the mathematical relation in the equality (1).

In operation 203, select to be different from transfer point and be contained in M inner monumented point of initial conversion point set to constitute the test point set, and make the center of mass point coordinate of test point set overlap each other with the center of mass point coordinate of initial conversion point set.

Each test point is at the first coordinate system C _AWith the second coordinate system C _BUnder coordinate figure be expressed as X respectively _TAjAnd X _TBj(i=1,2 ..., M), these two groups of coordinate figures have one-to-one relationship, and they satisfy the relational expression in the equality (1) equally.

In operation 204, utilize the initial conversion point set at the first coordinate system C _AWith the second coordinate system C _BUnder coordinate figure solve the first coordinate system C _AWith the second coordinate system C _BBetween rotation matrix R and the initial value of translation vector T, will test point set at the first coordinate system C _AUnder coordinate figure substitution rotation matrix R and the initial value of translation vector T come error of calculation evaluation index initial value.

Can calculate the layout parameter of initial conversion point set and test point set respectively, layout parameter mainly can be divided into parameter between point set intrinsic parameter and point set: the former comprises monumented point number, reference point coordinate and coordinate difference; The latter defines the envelope degrees of fusion between initial conversion point set and the test point set.Wherein, the some number that the initial conversion point set comprises is 4, and the some number that the test point set comprises is M; The reference point coordinate is at the first coordinate system C _AWith the second coordinate system C _BUnder coordinate figure respectively shown in equality (2), and satisfy the relation in the equality (3) between the two equally; Initial conversion point set, test point set are separately at the first coordinate system C _AWith the second coordinate system C _BUnder the coordinate difference should calculate according to equality (4) and equality (5) respectively; Envelope degrees of fusion between initial conversion point set and the test point set then should resolve according to equality (9).Then, according to the layout parameter of the initial conversion point set that calculates, utilize least square method to calculate rotation matrix R (referring to equality (14)), and calculate translation vector T (referring to equality (16)) according to rotation matrix R.

Utilize the initial conversion point set at the first coordinate system C _AWith the second coordinate system C _BUnder coordinate figure solve from the first coordinate system C _ATo the second coordinate system C _BThe concrete operations of initial value of rotation matrix R and translation vector T following: the center of mass point of calculating the initial conversion point set is at the first coordinate system C _AWith the second coordinate system C _BUnder coordinate figure, as the initial conversion point set at the first coordinate system C _AWith the second coordinate system C _BUnder the reference point coordinate figure; Calculating initial conversion point concentrates each monumented point and reference point at the first coordinate system C _AUnder the coordinate difference, calculate concentrated each monumented point of initial conversion point and reference point at the second coordinate system C _BUnder the coordinate difference; Utilize the said first coordinate system C _AUnder the coordinate difference and the second coordinate system C _BUnder the coordinate difference, calculate the initial value (referring to equality (14)) of rotation matrix R through least square method; According to the initial value of the rotation matrix R that calculates and initial conversion point set at the first coordinate system C _AWith the second coordinate system C _BUnder the reference point coordinate, calculate the initial value (referring to equality (16)) of translation vector T.

In the process of error of calculation evaluation index initial value; Can be according to conversion point set test point collection error assessment model (referring to equality (18)); Adopt the coordinate figure theory of error and/or root-mean-square error method; Calculate corresponding standard difference and maximum absolute error and/or root-mean-square error, as error assessment index initial value.

Particularly, will test point set at the first coordinate system C _AUnder the initial value of coordinate figure substitution rotation matrix R and translation vector T calculate test point and concentrate each test point at the second coordinate system C _BUnder the coordinates computed value; Calculating test point concentrates each test point at the second coordinate system C _BUnder coordinates computed value and said test point at the second coordinate system C _BUnder actual coordinate value poor.Then, come error of calculation evaluation index initial value according to a kind of in the following manner: (a1) adopt the coordinate figure theory of error to calculate the standard deviation and the maximum absolute error (referring to equality (21)) of the difference of said coordinate figure, as error assessment index initial value; (a2) adopt the root-mean-square error method to calculate the root-mean-square error (referring to equality (22)) of the difference of said coordinate figure, as error assessment index initial value; (a3) adopt the coordinate figure theory of error to calculate the standard deviation and the maximum absolute error of the difference of said coordinate figure; Adopt the root-mean-square error method to calculate the root-mean-square error of the difference of said coordinate figure, the standard deviation of the difference of said coordinate figure and maximum absolute error and root-mean-square error are as error assessment index initial value.

If the accuracy requirement to error is higher, can adopt aforesaid way (a1); If the dispersion degree requirement to error is less, can adopt aforesaid way (a2); If the accuracy requirement to error is higher, and less to the dispersion degree requirement of error, can adopt aforesaid way (a3).

In operation 205, in the cube measurement space, lay equally spaced intensive conversion point set, calculate the curvature fundamental function (referring to equality (32)) at each transfer point place, utilize intensive conversion point set at the first coordinate system C _AWith the second coordinate system C _BUnder coordinate figure find the solution from the first coordinate system C _ATo the second coordinate system C _BRotation matrix R and translation vector T, will test point set at the first coordinate system C _AUnder coordinate figure substitution rotation matrix R and translation vector T come error of calculation evaluation index desired value.

Likewise, can distinguish the layout parameter of computation-intensive conversion point set and test point set.

Likewise, utilize intensive conversion point set at the first coordinate system C _AWith the second coordinate system C _BUnder coordinate figure find the solution from the first coordinate system C _ATo the second coordinate system C _BThe concrete operations of rotation matrix R and translation vector T following: the center of mass point of computation-intensive conversion point set is at the first coordinate system C _AWith the second coordinate system C _BUnder coordinate figure, as intensive conversion point set at the first coordinate system C _AWith the second coordinate system C _BUnder the reference point coordinate figure; The computation-intensive transfer point concentrates each monumented point and reference point at the first coordinate system C _AUnder the coordinate difference, the computation-intensive transfer point concentrates each monumented point and reference point at the second coordinate system C _BUnder the coordinate difference; Utilize the said first coordinate system C _AUnder the coordinate difference and the second coordinate system C _BUnder the coordinate difference, calculate rotation matrix R (referring to equality (14)) through least square method; According to the rotation matrix R that calculates and intensive conversion point set at the first coordinate system C _AWith the second coordinate system C _BUnder the reference point coordinate, calculate translation vector T (referring to equality (16)).

Likewise; In the process of error of calculation evaluation index desired value; Can be according to conversion point set-test point set error assessment model (referring to equality (18)); Adopt the coordinate figure theory of error and/or root-mean-square error method, calculate corresponding standard difference and maximum absolute error and/or root-mean-square error, as error assessment target goals value.

Particularly, will test point set at the first coordinate system C _AUnder coordinate figure substitution rotation matrix R and translation vector T calculate test point and concentrate each test point at the second coordinate system C _BUnder the coordinates computed value; Calculating test point concentrates each test point at the second coordinate system C _BUnder coordinates computed value and said test point at the second coordinate system C _BUnder actual coordinate value poor.Then, come error of calculation evaluation index desired value according to a kind of in the following manner: (b1) adopt the coordinate figure theory of error to calculate the standard deviation and the maximum absolute error (referring to equality (21)) of the difference of said coordinate figure, as error assessment target goals value; (b2) adopt the root-mean-square error method to calculate the root-mean-square error (referring to equality (22)) of the difference of said coordinate figure, as error assessment target goals value; (b3) adopt the coordinate figure theory of error to calculate the standard deviation and the maximum absolute error of the difference of said coordinate figure; Adopt the root-mean-square error method to calculate the root-mean-square error of the difference of said coordinate figure, the standard deviation of the difference of said coordinate figure and maximum absolute error and root-mean-square error are as error assessment target goals value.

In operation 206, keep cube geometric center invariant position, as predetermined step-length, dwindle three coordinate components values of cube measurement space with the spacing between concentrated each monumented point of intensive transfer point successively, obtain a plurality of by big extremely little sub-cube space; In each sub-cube; The curvature fundamental function of each the intensive monumented point that is comprised in the said sub-cube is found the solution the weighted mass center point coordinate value (referring to equality (31)) of said sub-cube as weight factor; And weighted mass center point coordinate value and measurement space coordinate origin compared; If have side-play amount between weighted mass center point coordinate value and the measurement space coordinate origin; Then represent to exist in this sub-cube space significant curvature characteristic area, as the center, continue progressively to compress said sub-cube space according to said step-length with the weighted mass center point that is about to try to achieve; Be contracted to up to said sub-cube space till the area of space of predetermined size, the area of space that obtain this moment can be referred to as the localized cubic body region; If do not have side-play amount between weighted mass center point coordinate value and the measurement space coordinate origin, then search for the localized cubic body region that possibly exist in the next sub-cube space.So far,, can confirm to have the different localized cubic body region of remarkable curvature characteristic, and calculate the weighted mass center point coordinate of each localized cubic body region through the aforesaid operations self-adaptation according to the curvature changing features in the given cubic space.

The area of space of said predetermined size also is a sub-cubic space, the length of side of the area of space of said predetermined size can be the cube measurement space the length of side 1/5.Therefore, the threshold range of three coordinate components of the area of space of said predetermined size is a/5≤ξ≤a, and wherein, a is the length of side of cube measurement space, and ξ is the length of side of the area of space of said predetermined size.

Preferably,, promptly can be considered same sub-cube space, therefore can accelerate to compress the speed in sub-cube space if two or more sub-cubes space comprises the transition flag point that is equal to or greater than predetermined percentage simultaneously.Preferably, said predetermined percentage can be 85%.

In step 207; For a localized cubic body region that in step 206, is obtained; Weighted mass center point with said localized cubic body region is the center, is the radius of a ball with the integral multiple of said step-length, construct one group by little to big search ball until said localized cubic body region edge; According to the radius of a ball by little to big order, successively traversal search is carried out on each ball surface and intensive monumented point inner and that do not belong to previous ball and comprised; In the search procedure of each ball; The symmetric flag point that is centrosymmetric with each intensive monumented point and about true origin is as a pair of monumented point; Each a pair of monumented point is introduced the initial conversion point set and constitute current conversion point set; And according to the mode of step 204 to current conversion point set error of calculation evaluation index currency, form the conversion point set of renewal if the error assessment index currency that calculates less than error assessment index initial value, is then introduced the conversion point set with this to monumented point; If the error assessment index currency that calculates is not less than error assessment index initial value, then this is not introduced the conversion point set to monumented point; Utilize the conversion point set that upgrades; A pair of monumented point is according to identical mode error of calculation evaluation index currency under continuing to be directed against; Each monumented point until this spheroid comprised finishes; The monumented point that meets the demands in this time search is incorporated into the concentrated and new initial conversion point set of changing point set as next spheroid search of formation of initial conversion point, and calculating error assessment index at this moment is as the error assessment index initial value of next spheroid search; Repeat above-mentioned search procedure, till the whole search of each spheroid of this localized cubic body region finish.Therefore,, then change point set and be updated, utilize the conversion point set that upgrades then, to descending a pair of monumented point according to identical mode error of calculation evaluation index currency if introduced a pair of monumented point that satisfies condition.

In step 208; Traversal search operation to all local cube zone repeated execution of steps 207; It is right that the transition flag point of condition of step 207 is satisfied in searching; Until the error assessment index convergence that calculates or be lower than error assessment target goals value, and marked change takes place no longer, the number of the monumented point that lay this moment and corresponding coordinate position constitute the optimum translation point set.On this basis, the coordinate figure of optimum translation point set capable of using under first coordinate system and second coordinate system found the solution from the first coordinate system C through equality (14) and (16) _ATo the second coordinate system C _BRotation matrix and the optimal value of translation vector.

Likewise, utilize the optimum translation point set at the first coordinate system C _AWith the second coordinate system C _BUnder coordinate figure find the solution from the first coordinate system C _ATo the second coordinate system C _BThe concrete operations of optimal value of rotation matrix R and translation vector T following: the center of mass point of compute optimal conversion point set is at the first coordinate system C _AWith the second coordinate system C _BUnder coordinate figure, as the optimum translation point set at the first coordinate system C _AWith the second coordinate system C _BUnder the reference point coordinate figure; The compute optimal transfer point concentrates each monumented point and reference point at the first coordinate system C _AUnder the coordinate difference, the compute optimal transfer point concentrates each monumented point and reference point at the second coordinate system C _BUnder the coordinate difference; Utilize the said first coordinate system C _AUnder the coordinate difference and the second coordinate system C _BUnder the coordinate difference, calculate the optimal value (referring to equality (14)) of rotation matrix R through least square method; According to the optimal value of the rotation matrix R that calculates and optimum translation point set at the first coordinate system C _AWith the second coordinate system C _BUnder the reference point coordinate, calculate the optimal value (referring to equality (16)) of translation vector T.

Compared with prior art; Self-adaptation monumented point layout method based on the constraint of curvature characteristic weighing center of mass point according to the present invention is simple to operate, quick; Need not artificially to lay the number of characteristics monumented point, can obtain higher coordinate conversion precision, can be used for the characteristic indication point programming and distribution design of coordinate conversion; And then can be widely used in multiple actual engineering field, have significant Research Significance and using value.

Though the present invention is specifically described with reference to its exemplary embodiment and is shown; But will be understood by those skilled in the art that; Under the situation that does not break away from the spirit and scope of the present invention that are defined by the claims, can carry out the various changes of form and details to it.

Claims (10)

1. self-adaptation monumented point layout method based on curvature characteristic weighing center of mass point constraint, said method comprises:

(1) with the actual measurement scene as cubic space; Set up the measurement space coordinate system with said cubical geometric center as true origin; And select two the first different observation erect-positions and the second observation erect-position independently to obtain the three-dimensional appearance characteristic of cubic space respectively, second coordinate system and the measurement space coordinate system of first coordinate system of the first observation erect-position and the second observation erect-position are different;

(2) four summits choosing two body diagonal direction places in the cube constitute the initial conversion point sets, make the center of mass point coordinate of initial conversion point set be positioned at the true origin place of measurement space coordinate system;

(3) select to be different from transfer point and be included in the inner a plurality of monumented points of initial conversion point set to constitute the test point set, and make the center of mass point coordinate of test point set overlap each other with the center of mass point coordinate of initial conversion point set;

(4) utilize the coordinate figure of initial conversion point set under first coordinate system and second coordinate system to find the solution and be tied to the rotation matrix of second coordinate system and the initial value of translation vector, coordinate figure substitution rotation matrix and the initial value of translation vector of test point set under first coordinate system come error of calculation evaluation index initial value from first coordinate;

(5) in the cube measurement space, lay equally spaced intensive conversion point set; Calculate the curvature fundamental function at each transfer point place; Utilize the coordinate figure of intensive conversion point set under first coordinate system and second coordinate system to find the solution rotation matrix and the translation vector that is tied to second coordinate system from first coordinate, coordinate figure substitution rotation matrix and the translation vector of test point set under first coordinate system come error of calculation evaluation index desired value;

(6) keep cube geometric center invariant position, as predetermined step-length, dwindle three coordinate components values of cube measurement space successively, obtain a plurality of by big extremely little sub-cube space with the spacing between concentrated each monumented point of intensive transfer point; In each sub-cube; The curvature fundamental function of each the intensive monumented point that is comprised in the said sub-cube is found the solution the weighted mass center point coordinate value of said sub-cube as weight factor, and weighted mass center point coordinate value and measurement space coordinate origin are compared; If have side-play amount between weighted mass center point coordinate value and the measurement space coordinate origin; Then with the weighted mass center point of trying to achieve as the center; Continuation is progressively compressed said sub-cube space according to said step-length; Be contracted to up to said sub-cube space till the area of space of predetermined size, with thus obtained sub-cube space as the localized cubic body region; If do not have side-play amount between weighted mass center point coordinate value and the measurement space coordinate origin, then search for the localized cubic body region in the next sub-cube space;

(7) for a localized cubic body region that in step (6), is obtained; Weighted mass center point with said localized cubic body region is the center; Integral multiple with said step-length is the radius of a ball, construct one group by little to big search ball until said localized cubic body region edge; According to the radius of a ball by little to big order, successively traversal search is carried out on each ball surface and intensive monumented point inner and that do not belong to previous ball and comprised; In the search procedure of each ball; The symmetric flag point that is centrosymmetric with each intensive monumented point and about true origin is as a pair of monumented point; Each a pair of monumented point is introduced the initial conversion point set and constitute current conversion point set; And according to the mode of step (4) to current conversion point set error of calculation evaluation index currency, form the conversion point set of renewal if the error assessment index currency that calculates less than error assessment index initial value, is then introduced the conversion point set with this to monumented point; Utilize the conversion point set that upgrades; A pair of monumented point is according to identical mode error of calculation evaluation index currency under continuing to be directed against; Each monumented point until this spheroid comprised finishes; The monumented point that meets the demands in this time search is incorporated into the concentrated and new initial conversion point set of changing point set as next spheroid search of formation of initial conversion point, and calculating error assessment index at this moment is as the error assessment index initial value of next spheroid search; Repeat above-mentioned search procedure, till the whole search of each spheroid of this localized cubic body region finish;

(8) traversal search of all local cube zone repeated execution of steps (7) is operated; It is right that the transition flag point of condition of step (7) is satisfied in searching; Until the error assessment index convergence that calculates or be lower than error assessment target goals value, the number of the monumented point that lay this moment and corresponding coordinate position constitute the optimum translation point set.

2. self-adaptation monumented point layout method according to claim 1; Wherein, In step (4), utilize the coordinate figure of initial conversion point set under first coordinate system and second coordinate system to find the solution the step of initial value that is tied to rotation matrix and the translation vector of second coordinate system from first coordinate and comprise:

Calculate the coordinate figure of center of mass point under first coordinate system and second coordinate system of initial conversion point set, as the reference point coordinate figure of initial conversion point set under first coordinate system and second coordinate system;

Calculate initial conversion point and concentrate each monumented point and the coordinate difference of reference point under first coordinate system, calculate initial conversion point and concentrate each monumented point and the coordinate difference of reference point under second coordinate system;

Utilize coordinate difference and the coordinate difference under second coordinate system under said first coordinate system, calculate the initial value of rotation matrix through least square method;

According to the initial value and the reference point coordinate of initial conversion point set under first coordinate system and second coordinate system of the rotation matrix that calculates, calculate the initial value of translation vector.

3. self-adaptation monumented point layout method according to claim 2, wherein, in step (4), come the step of error of calculation evaluation index initial value to comprise coordinate figure substitution rotation matrix and the initial value of translation vector of test point set under first coordinate system:

The test point set is calculated the concentrated coordinates computed value of each test point under second coordinate system of test point at the initial value of coordinate figure substitution rotation matrix under first coordinate system and translation vector;

Concentrated each test point of calculating test point is poor coordinates computed value under second coordinate system and the actual coordinate value of said test point under second coordinate system;

Come error of calculation evaluation index initial value according to a kind of in the following manner: (a1) calculate the standard deviation and the maximum absolute error of the difference of said coordinate figure, as error assessment index initial value; (a2) root-mean-square error of the difference of the said coordinate figure of calculating is as error assessment index initial value; (a3) calculate the standard deviation and the maximum absolute error of the difference of said coordinate figure, and calculate the root-mean-square error of the difference of said coordinate figure, with the standard deviation of the difference of said coordinate figure and maximum absolute error and root-mean-square error as error assessment index initial value.

4. self-adaptation monumented point layout method according to claim 1; Wherein, In step (5), utilize the coordinate figure of intensive conversion point set under first coordinate system and second coordinate system to find the solution the rotation matrix and the translation vector that are tied to second coordinate system from first coordinate and find the solution from first coordinate and be tied to the rotation matrix of second coordinate system and the step of translation vector comprises:

The coordinate figure of center of mass point under first coordinate system and second coordinate system of computation-intensive conversion point set is as the reference point coordinate figure of intensive conversion point set under first coordinate system and second coordinate system;

The computation-intensive transfer point is concentrated each monumented point and the coordinate difference of reference point under first coordinate system, and the computation-intensive transfer point is concentrated each monumented point and the coordinate difference of reference point under second coordinate system;

Utilize coordinate difference and the coordinate difference under second coordinate system under said first coordinate system, calculate rotation matrix through least square method;

According to the initial value and the reference point coordinate of intensive conversion point set under first coordinate system and second coordinate system of the rotation matrix that calculates, calculate translation vector.

5. self-adaptation monumented point layout method according to claim 4, wherein, in step (5), come the step of error of calculation evaluation index desired value to comprise coordinate figure substitution rotation matrix and the translation vector of test point set under first coordinate system:

The test point set is calculated the concentrated coordinates computed value of each test point under second coordinate system of test point at coordinate figure substitution rotation matrix under first coordinate system and translation vector;

Concentrated each test point of calculating test point is poor coordinates computed value under second coordinate system and the actual coordinate value of said test point under second coordinate system;

Come error of calculation evaluation index desired value according to a kind of in the following manner: (b1) calculate the standard deviation and the maximum absolute error of the difference of said coordinate figure, as error assessment target goals value; (b2) root-mean-square error of the difference of the said coordinate figure of calculating is as error assessment target goals value; (b3) calculate the standard deviation and the maximum absolute error of the difference of said coordinate figure, and calculate the root-mean-square error of the difference of said coordinate figure, with the standard deviation of the difference of said coordinate figure and maximum absolute error and root-mean-square error as error assessment target goals value.

6. self-adaptation monumented point layout method according to claim 1, wherein, in step (6), the length of side of the area of space of said predetermined size be the cube measurement space the length of side 1/5.

7. self-adaptation monumented point layout method according to claim 6; Wherein, In step (6), if two or more sub-cubes space comprises the transition flag point that is equal to or greater than predetermined percentage simultaneously, then with said two or more sub-cubes space as same sub-cube space.

8. self-adaptation monumented point layout method according to claim 7, wherein, said predetermined percentage is 85%.

9. self-adaptation monumented point layout method according to claim 1 also comprises:

(9) utilize the coordinate figure of optimum translation point set under first coordinate system and second coordinate system to find the solution the optimal value that is tied to the rotation matrix and the translation vector of second coordinate system from first coordinate.

10. self-adaptation monumented point layout method according to claim 9; Wherein, In step (9), utilize the coordinate figure of optimum translation point set under first coordinate system and second coordinate system to find the solution the step of optimal value that is tied to rotation matrix and the translation vector of second coordinate system from first coordinate and comprise:

The coordinate figure of center of mass point under first coordinate system and second coordinate system of compute optimal conversion point set is as the reference point coordinate figure of optimum translation point set under first coordinate system and second coordinate system;

The compute optimal transfer point is concentrated each monumented point and the coordinate difference of reference point under first coordinate system, and the compute optimal transfer point is concentrated each monumented point and the coordinate difference of reference point under second coordinate system;

Utilize coordinate difference and the coordinate difference under second coordinate system under said first coordinate system, calculate the optimal value of rotation matrix through least square method;

According to the optimal value and the reference point coordinate of optimum translation point set under first coordinate system and second coordinate system of the rotation matrix that calculates, calculate the optimal value of translation vector.

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