CN106959666A - A kind of space free curve approximating method based on NURBS - Google Patents

Abstract

The invention discloses a kind of space free curve approximating method based on NURBS, including step：Obtaining one group needs data point, the corresponding nurbs curve parameter of data point is calculated using Chord Length Parameterization or centripetal parameterization, NURBS knot vectors are calculated, the control point of nurbs curve is calculated using least square method combination system of linear equations, weight factor is determined, finally nurbs curve is can obtain.The present invention is using the relative Solving Linear NURBS methods advised, amount of calculation is relatively small, and solution is difficult to the problem of NURBS space-curve fittings are difficult to real-time calculate.

Description

A kind of space free curve approximating method based on NURBS

Technical field

The present invention relates to space free curve approximating method, more particularly to a kind of space free curve based on NURBS is intended Conjunction method.

Background technology

Space-curve fitting has in terms of robot and numerical control machine tool motion coordinates measurement, points cloud processing widely should With.The main method of space-curve fitting has lagrangian fit method, is segmented three Hermite fitting process, arc-spline curve It is fitted fitting process, NURBS spline curve fitting methods etc..But imperial lattice phenomenon, and error of fitting occurs in Lagrange's interpolation Larger segmentation；The not high only continuous first derivative of three Hermite interpolation slickness；Arc-spline curve curvature is in jump Formula changes, and illustrates that fairness is poor, and poor with circular fitting easement curve adaptability.And existing nurbs curve fitting process Because calculating complicated, time-consuming, needs the occasion compared with hard real-time not apply to for motion planning and robot control etc..

The content of the invention

It is an object of the invention to provide a kind of space free curve approximating method based on NURBS, it is intended to solves space certainly By curve matching flatness and real-time computational problem.

The purpose of the present invention is realized by following technical proposals：

1st, a kind of space free curve approximating method based on NURBS, it is characterised in that comprise the following steps：

S1, one group of discrete point sequence { Q of acquisition_k, k=0,1 ..., n；

S2, Chord Length Parameterization or centripetal parameterization is used to calculate the corresponding nurbs curve parameter of data point；

S3, calculating NURBS knot vectors；

S4, the control point using least square method combination system of linear equations calculating nurbs curve；

S5, weight factor is determined, it is general all to take 1, finally give nurbs curve.

2nd, a kind of space free curve approximating method based on NURBS according to claim 1, it is characterised in that The Chord Length Parameterization of the step S2 is comprised the following steps：

S2.1, make d be two neighboring discrete point chord length (space length) sum, then have

In formula, Q_kFor discrete point vector；K=0,1 ..., n；

The corresponding parameter of S2.2, then discrete pointFor

3rd, a kind of space free curve approximating method based on NURBS according to claim 1, it is characterised in that The centripetal parametrization of the step S2 is comprised the following steps：

S2.1, order

In formula, Q_kFor discrete point vector；K=0,1 ..., n；

S2.2, then discrete point Q_kCorresponding parameterFor

4th, a kind of robotic joint space smooth track planing method according to claim 1, it is characterised in that institute Step S3 is stated to comprise the following steps：

S3.1, order

In formula, m+1 is data point { Q_k, k=0,1 ..., m number；N+1 is control point P_k, k=0,1 ..., n numbers Mesh；P is NURBS SPL number of times；

S3.2, the knot vector U=[u for then having NURBS SPLs₀ u₁ ... u_n+p+1] be

In formula, n counts out for control subtracts 1；P is NURBS SPL number of times；For data point Q_iCorresponding parameter；α= jd-i；I=int (jd), int (m) represent to take the maximum integer less than or equal to m.

5th, a kind of robotic joint space smooth track planing method according to claim 1, it is characterised in that institute Step S4 is stated to comprise the following steps：

S4.1, then according to the requirement of least square fitting method, that is, want NURBS SPL C (u) to meet following condition：

①Q₀=C (0), Q_m=C (1)；

2. remainder data point { Q_k, k=1,2 ..., m-1 and the corresponding points on curveMistake Poor quadratic sum is minimum

S4.2, according to formula (6), order

In formula,

S4.3, the object function f of formula (7) are on n-1 variable P_k, k=1,2 ..., n-1 scalar function will make Object function f is minimum, i.e., to make f is on n-1 variable P_k, k=1,2 ..., n-1 partial derivative is all zero, that is, is had

In formula, l=1,2 ..., n-1；

It can be obtained by formula (8) with abbreviation

In formula, l=1,2 ..., n-1；

S4.4, the system of linear equations containing n-1 unknown number and n-1 equation can be obtained

(N^TN) P=R (10)

In formula,

P=[P₁ … P_n-1]^T；

Due to the knot vector definition mode using step S3, it is ensured that each node interval comprises at least a data point Corresponding parameterSo that the coefficient matrix of system of linear equations is the sparse matrix of positive definite, it can be solved using Gaussian reduction, Solve the control point P of NURBS SPLs_k, k=0,1 ..., n.

The present invention has the following advantages and effect relative to prior art：

NURBS SPLs of the present invention, with lagrangian fit method, are segmented three Hermite fitting process, The curve of arc-spline curve fitting fitting process etc. method generation is compared, with higher derivative it is continuous the characteristics of, flatness is more preferable. And fitting algorithm calculates simple, it is adaptable to the high occasion of motion planning and robot control energy requirement of real-time.

Brief description of the drawings

Fig. 1 is the space free curve approximating method schematic flow sheet based on NURBS of the embodiment of the present invention.

Embodiment

With reference to embodiment and accompanying drawing, the present invention is described in further detail, but embodiments of the present invention are not limited In this.

Now to be fitted certain six degree of freedom drag articulation serial manipulator motion path as object, base is followed the steps below In NURBS space free curve fitting：

1st, a kind of space free curve approximating method based on NURBS, it is characterised in that comprise the following steps：

S1, one group of discrete point sequence { Q of acquisition_k, k=0,1 ..., n；

S2, Chord Length Parameterization or centripetal parameterization is used to calculate the corresponding nurbs curve parameter of data point；

S3, calculating NURBS knot vectors；

S4, the control point using least square method combination system of linear equations calculating nurbs curve；

S5, weight factor is determined, it is general all to take 1, finally give nurbs curve.

2nd, a kind of space free curve approximating method based on NURBS according to claim 1, it is characterised in that The Chord Length Parameterization of the step S2 is comprised the following steps：

S2.1, make d be two neighboring discrete point chord length (space length) sum, then have

In formula, Q_kFor discrete point vector；K=0,1 ..., n；

The corresponding parameter of S2.2, then discrete pointFor

3rd, a kind of space free curve approximating method based on NURBS according to claim 1, it is characterised in that The centripetal parametrization of the step S2 is comprised the following steps：

S2.1, order

In formula, Q_kFor discrete point vector；K=0,1 ..., n；

S2.2, then discrete point Q_kCorresponding parameterFor

4th, a kind of robotic joint space smooth track planing method according to claim 1, it is characterised in that institute Step S3 is stated to comprise the following steps：

S3.1, order

In formula, m+1 is data point { Q_k, k=0,1 ..., m number；N+1 is control point P_k, k=0,1 ..., n numbers Mesh；P is NURBS SPL number of times；

S3.2, the knot vector U=[u for then having NURBS SPLs₀ u₁ ... u_n+p+1] be

In formula, n counts out for control subtracts 1；P is NURBS SPL number of times；For data point Q_iCorresponding parameter；α= jd-i；I=int (jd), int (m) represent to take the maximum integer less than or equal to m.

5th, a kind of robotic joint space smooth track planing method according to claim 1, it is characterised in that institute Step S4 is stated to comprise the following steps：

S4.1, then according to the requirement of least square fitting method, that is, want NURBS SPL C (u) to meet following condition：

①Q₀=C (0), Q_m=C (1)；

2. remainder data point { Q_k, k=1,2 ..., m-1 and the corresponding points on curveMistake Poor quadratic sum is minimum

S4.2, according to formula (6), order

In formula,

S4.3, the object function f of formula (7) are on n-1 variable P_k, k=1,2 ..., n-1 scalar function will make Object function f is minimum, i.e., to make f is on n-1 variable P_k, k=1,2 ..., n-1 partial derivative is all zero, that is, is had

In formula, l=1,2 ..., n-1；

It can be obtained by formula (8) with abbreviation

In formula, l=1,2 ..., n-1；

S4.4, the system of linear equations containing n-1 unknown number and n-1 equation can be obtained

(N^TN) P=R (10)

In formula,

P=[P₁ … P_n-1]^T；

Due to the knot vector definition mode using step S3, it is ensured that each node interval comprises at least a data point Corresponding parameterSo that the coefficient matrix of system of linear equations is the sparse matrix of positive definite, it can be solved using Gaussian reduction, Solve the control point P of NURBS SPLs_k, k=0,1 ..., n.

Above-described embodiment is present aspect preferably embodiment, but the bright embodiment of we is not by above-described embodiment Limitation, other any Spirit Essences away from the present invention and the change made under principle, modification, replacement, combine, it is simplified, Equivalent substitute mode is should be, is included within protection scope of the present invention.

Claims (5)

1. a kind of space free curve approximating method based on NURBS, it is characterised in that comprise the following steps：

S1, one group of discrete point sequence { Q of acquisition_k, k=0,1 ..., n；

S2, Chord Length Parameterization or centripetal parameterization is used to calculate the corresponding nurbs curve parameter of data point；

S3, calculating NURBS knot vectors；

S4, the control point using least square method combination system of linear equations calculating nurbs curve；

S5, weight factor is determined, it is general all to take 1, finally give nurbs curve.

2. a kind of space free curve approximating method based on NURBS according to claim 1, it is characterised in that described Step S2 Chord Length Parameterization is comprised the following steps：

S2.1, make d be two neighboring discrete point chord length (space length) sum, then have

$d=\underset{k=1}{\overset{n}{\Σ}}\left|Q_k-Q_{k-1}\right|,---\left(1\right)$

In formula, Q_kFor discrete point vector；K=0,1 ..., n；

The corresponding parameter of S2.2, then discrete pointFor

$\{\begin{array}{c}{\stackrel{\‾}{u}}_0=0\\ {\stackrel{\‾}{u}}_n=1\\ {\stackrel{\‾}{u}}_k={\stackrel{\‾}{u}}_{k-1}+\frac{\left|Q_k-Q_{k-1}\right|}{d},k=1,2,...,n-1\end{array}.---\left(2\right)$

3. a kind of space free curve approximating method based on NURBS according to claim 1, it is characterised in that described Step S2 centripetal parametrization is comprised the following steps：

S2.1, order

$d=\underset{k=1}{\overset{n}{\Σ}}\sqrt{\left|Q_k-Q_{k-1}\right|},---\left(3\right)$

In formula, Q_kFor discrete point vector；K=0,1 ..., n；

S2.2, then discrete point Q_kCorresponding parameterFor

$\{\begin{array}{c}{\stackrel{\‾}{u}}_0=0\\ {\stackrel{\‾}{u}}_n=1\\ {\stackrel{\‾}{u}}_k={\stackrel{\‾}{u}}_{k-1}+\frac{\sqrt{\left|Q_k-Q_{k-1}\right|}}{d},k=1,2,...,n-1\end{array}.---\left(4\right)$

4. a kind of space free curve approximating method based on NURBS according to claim 1, it is characterised in that described Step S3 is comprised the following steps：

S3.1, order

$d=\frac{m+1}{n+1-p},---\left(5\right)$

In formula, m+1 is data point { Q_k, k=0,1 ..., m number；N+1 is control point P_k, k=0,1 ..., n numbers；P is NURBS SPL number of times；

S3.2, the knot vector U=[u for then having NURBS SPLs₀ u₁ ... u_n+p+1] be

$\left\{\begin{array}{c}{u}_0=u_1=\mathrm{...}=u_p=0\\ u_n=u_{n+1}=\mathrm{...}=u_{n+p+1}=1\\ u_{j+p}=(1-\mathrm{\α}){\stackrel{\‾}{u}}_{i-1}+\mathrm{\α}{\stackrel{\‾}{u}}_i,j=1,2,...,n-p\end{array}\right.,---(2-51)$

In formula, n counts out for control subtracts 1；P is NURBS SPL number of times；For data point Q_iCorresponding parameter；α=jd-i； I=int (jd), int (m) represent to take the maximum integer less than or equal to m.

5. a kind of space free curve approximating method based on NURBS according to claim 1, it is characterised in that described Step S4 is comprised the following steps：

S4.1, then according to the requirement of least square fitting method, that is, want NURBS SPL C (u) to meet following condition：

①Q₀=C (0), Q_m=C (1)；

2. remainder data point { Q_k, k=1,2 ..., m-1 and the corresponding points on curveError put down Side and minimum

$\underset{k=1}{\overset{m-1}{\Σ}}{\left|{\mathrm{\δ}}_k\right|}^2=\min \underset{k=1}{\overset{m-1}{\Σ}}{|Q_k-C\left({\stackrel{\‾}{u}}_k\right)|}^2;---\left(6\right)$

S4.2, according to formula (6), order

$\begin{array}{c}f=\underset{k=1}{\overset{m-1}{\Σ}}{|Q_k-C\left({\stackrel{\‾}{u}}_k\right)|}^2\\ f=\underset{k=1}{\overset{m-1}{\Σ}}{|R_k-\underset{i=1}{\overset{n-1}{\Σ}}{N}_{i,p}\left({\stackrel{\‾}{u}}_k\right)P_i|}^2\\ f=\underset{k=1}{\overset{m-1}{\Σ}}\[R_k^2-2\underset{i=1}{\overset{n-1}{\Σ}}{N}_{i,p}\left({\stackrel{\‾}{u}}_k\right)P_i{R}_k+{\left(\underset{i=1}{\overset{n-1}{\Σ}}{N}_{i,p}\right({\stackrel{\‾}{u}}_k\left)P_i\right)}^2\]\end{array},---\left(7\right)$

In formula,

S4.3, the object function f of formula (7) are on n-1 variable P_k, k=1,2 ..., n-1 scalar function will make target letter Number f is minimum, i.e., to make f is on n-1 variable P_k, k=1,2 ..., n-1 partial derivative is all zero, that is, is had

$\frac{\∂f}{\∂P_l}=\underset{k=1}{\overset{m-1}{\Σ}}\[-2N_{l,p}\left({\stackrel{\‾}{u}}_k\right)R_k+2N_{l,p}\left({\stackrel{\‾}{u}}_k\right)\underset{i=1}{\overset{n-1}{\Σ}}{N}_{i,p}\left({\stackrel{\‾}{u}}_k\right)P_i\]=0,---\left(8\right)$

In formula, l=1,2 ..., n-1；

It can be obtained by formula (8) with abbreviation

$\underset{k=1}{\overset{m-1}{\Σ}}\left(\underset{i=1}{\overset{n-1}{\Σ}}{N}_{l,p}\right({\stackrel{\‾}{u}}_k\left)N_{i,p}\right({\stackrel{\‾}{u}}_k\left)\right)P_i=\underset{k=1}{\overset{m-1}{\Σ}}{N}_{l,p}\left({\stackrel{\‾}{u}}_k\right)R_k,---\left(9\right)$

In formula, l=1,2 ..., n-1；

S4.4, the system of linear equations containing n-1 unknown number and n-1 equation can be obtained

(N^TN) P=R (10)

In formula,

P=[P₁ … P_n-1]^T；

Due to the knot vector definition mode using step S3, it is ensured that each node interval comprises at least a data point correspondence ParameterSo that the coefficient matrix of system of linear equations is the sparse matrix of positive definite, it can be solved, solved using Gaussian reduction The control point P of NURBS SPLs_k, k=0,1 ..., n.