CN111489437B - Adjacent surface tooth preparation curve generation method for robot auxiliary tooth preparation - Google Patents
Adjacent surface tooth preparation curve generation method for robot auxiliary tooth preparation Download PDFInfo
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- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06T—IMAGE DATA PROCESSING OR GENERATION, IN GENERAL
- G06T17/00—Three dimensional [3D] modelling, e.g. data description of 3D objects
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- A—HUMAN NECESSITIES
- A61—MEDICAL OR VETERINARY SCIENCE; HYGIENE
- A61C—DENTISTRY; APPARATUS OR METHODS FOR ORAL OR DENTAL HYGIENE
- A61C7/00—Orthodontics, i.e. obtaining or maintaining the desired position of teeth, e.g. by straightening, evening, regulating, separating, or by correcting malocclusions
- A61C7/002—Orthodontic computer assisted systems
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- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
- G06F17/10—Complex mathematical operations
- G06F17/16—Matrix or vector computation, e.g. matrix-matrix or matrix-vector multiplication, matrix factorization
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- A—HUMAN NECESSITIES
- A61—MEDICAL OR VETERINARY SCIENCE; HYGIENE
- A61C—DENTISTRY; APPARATUS OR METHODS FOR ORAL OR DENTAL HYGIENE
- A61C7/00—Orthodontics, i.e. obtaining or maintaining the desired position of teeth, e.g. by straightening, evening, regulating, separating, or by correcting malocclusions
- A61C7/002—Orthodontic computer assisted systems
- A61C2007/004—Automatic construction of a set of axes for a tooth or a plurality of teeth
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- Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
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Abstract
The invention discloses a generating method of an adjacent surface tooth preparation curve for robot auxiliary tooth preparation, which relates to the technical field of robot auxiliary tooth preparation. The technical key points are as follows: pre-treating a preparation body model; extracting a standardized adjacent surface tooth preparation curve and discrete points; calculating a node vector U and a control point P; and (5) curve interpolation. According to the invention, back calculation and interpolation of the adjacent surface curve are completed based on the NURBS curve according to the extracted curve data of the adjacent surface tooth preparation, so that three-dimensional curve track planning of robot auxiliary tooth preparation is realized, and a foundation is laid for realizing the robot auxiliary tooth preparation.
Description
Technical Field
The invention relates to a generating method of an adjacent surface tooth preparation curve for robot auxiliary tooth preparation, and belongs to the technical field of robot auxiliary tooth preparation.
Background
Caries is an important cause of tooth defect, and seriously affects oral health of people. Dental restoration is an important means for treating defects of teeth, and dental preparation is an essential treatment link in the restoration process, which means an operation process that a doctor quantitatively removes teeth at a caries site and forms a desired three-dimensional shape. In the traditional tooth preparation process, a great deal of repeated fine adjustment of caries teeth is required depending on manual operation of doctors and combining with abundant clinical experience. However, the current situation of unbalanced doctor-patient ratio in China cannot meet the current extremely large dental preparation demand, so that a robot and auxiliary software are required to be introduced to replace or assist doctors to efficiently complete dental preparation work, the problem of unbalanced doctor-patient can be relieved, and the dental preparation quality and the oral cavity treatment effect can be effectively improved.
In the process of finishing tooth preparation by machine assistance, the movement track of the tail end of the robot determines the accuracy of tooth preparation, thereby affecting the treatment effect and success rate. Therefore, reasonable adjacent surface preparation tooth tracks are obtained, and the fitting degree of the preparation body and the rear tooth restoration body is considered seriously, so that the condition of oral secondary diseases caused by improper tooth preparation is reduced, and the method is a research hot spot in the field of auxiliary tooth preparation of the robot at present.
Disclosure of Invention
Aiming at the problems, the invention provides a method for generating the adjacent surface preparation tooth curve for the robot auxiliary tooth preparation, which solves the problem that the prior art for preparing the robot auxiliary tooth lacks a method for obtaining the adjacent surface preparation tooth curve so as to improve the accuracy and success rate of tooth preparation and further realize the robot auxiliary tooth preparation.
The invention adopts the scheme for solving the problems that:
a method for generating an adjacent surface tooth preparation curve for robot assisted tooth preparation comprises the following specific implementation processes:
step one, preparation of a body model pretreatment
Scanning according to a tooth preparation model in a doctor clinic to obtain a standardized preparation model in an obj format, processing the standardized tooth preparation three-dimensional model in the obj format by using Geomagic Wrap reverse engineering software, repairing triangle sheet problems of non-fluid edges, self-intersecting edges, highly refractive edges or nails in the model by using a grid doctor command, enabling the surface of the model to be smoother and more convenient for further curved surface formation by increasing the number of triangles by using a relaxed polygon command, setting a geometric figure type as a machine by using automatic curved surface formation, setting a curved sheet count as automatic evaluation, setting a curved surface detail as moderate, setting an adaptive tolerance of curved surface fitting as 0.0097, constructing subdivision grids in each curved sheet by using a grid constructing command, and finishing outputting a step-format three-dimensional model with curved surface characteristics and boundary characteristics from the three-dimensional model in the obj format consisting of triangle sheets;
step two, extracting a standardized adjacent surface tooth preparation curve and discrete points
The obtained step-format preparation model is imported into Creo three-dimensional software, 2/3 of the diameter D of the needle is arranged in the shoulder to be prepared, the rest D/3 is positioned outside the tooth body, and the axis of the needle is away from the tooth edge curve C L Is D/6, replicates the dental margin curve C of the standardized preparation L As a reference curve, the offset command is again used to input the offset D/6 versus C L Obtaining the adjacent surface tooth preparation curve C by completing offset G Tooth preparation curve C on the adjacent surface G Up by the number N of discrete points Q And a discrete dot pitch L Q Obtaining a plurality of discrete points Q by two parameters;
step three, back calculation of adjacent surface tooth preparation curve
The discrete point Q in the pretreatment stage is calculated by inverse calculation based on NURBS to obtain the adjacent surface tooth preparation curve, the inverse calculation target is to calculate the node vector U and control vertex P, a NURBS curve mathematical model is introduced,
ω i is a weight factor, where i=0, 1,..n; p (P) i To control vertices, the number is n+1, where i=0, 1,..n; p is NURBS curve number; n (N) i,p (u) is a basis function; n in formula (1) i,p (u) is:
wherein: u (u) i Is an element of a non-uniform node vector U, as shown in equation (3);
m+1 is the length of the node vector U; the relation among m, p and n is m=n+p+1; a and b are 0 and 1;
because the adjacent surface tooth preparation curve is a closed three-dimensional curve, the first and the last data points are coincided with each other to form Q 0 ＝Q m The adjacent tooth preparation curve is a 4-degree-of-freedom 3-time NURBS curve, and the first and last 3 control points are sequentially equal to obtain an equation: p (P) 0 ＝P n-2 ，P 1 ＝P n-1 ，P 2 ＝P n The rational division of NURBS curves is represented in the form of a matrix:
wherein: t E [0,1 ]]Order-makingΔ i ＝u i+1 -u i ,/>m 12 ＝1-m 11 -m 13 ，/>
According to the determined conditions of the first end and the last end, a calculation formula of a control point P under the condition of a closed curve is obtained:
wherein each parameter is shown in formula (7):
wherein: i=1, 2,/i., n-2;
the node vector U of the adjacent surface curve is calculated by the discrete point Q obtained in the pretreatment stage through a uniform parameterization method, an accumulated chord length parameterization method and a centripetal parameterization method, the curve obtained by the accumulated chord length parameterization method reflects the distribution condition of the discrete point Q according to the chord length, and the tooth preparation curve is more similar to the standardized adjacent surface tooth preparation curve C extracted by the preparation model G The preparation precision of the tooth preparation curve is improved.
Step four, adjacent surface tooth preparation curve interpolation
The principle of interpolation of the adjacent surface tooth preparation curve is to use the time sequence { t } 1 ,t 2 ,...,t k ,...,t n-1 ,t n Sequence of { u } segmentation parameters 1 ,u 2 ,...,u k ,u k+1 ,...,u n-1 ,u n And then obtains the interpolation point sequence { C (u) 1 ),C(u 2 ),...,C(u k ),...,C(u n-1 ),C(u n ) The interpolation core calculation is to calculate u by using the interpolation period T k And u k+1 The relation between them is further defined by C (u k ) Obtain C (u) k+1 )；
Firstly, deriving the derivative of NURBS curve to obtain the first derivative C of the adjacent tooth preparation curve pair u based on NURBS (1) (u)；
Wherein: c (C) (1) (u) is the first derivative of the curve with respect to u;
continuing to calculate the second derivative of the curve to obtain C (2) (u)：
Wherein: c (C) (2) (u) is the second derivative of the curve to u, the basis function N i,p (m) The expression of (u) is:
the feed rate on the NURBS curve is expressed as:
sorting the transformation formula (11) to obtain formula (12):
wherein: c (C) x (1) (u k ) Is the first derivative of the curve x direction; c (C) y (1) (u k ) Is the first derivative of the curve y direction; c (C) z (1) (u k ) Is the first derivative of the z direction of the curve.
Continuing to calculate the second derivative of the formula (12), and obtaining:
wherein: c (C) x (2) (u k ) Is the second derivative of the curve in the x direction; c (C) y (2) (u k ) Is the second derivative of the curve y direction; c (C) z (2) (u k ) Is the second derivative of the curve in the z direction;
the arrangement results in a relationship between the geometrical and movement characteristics of the NURBS-based interproximal dental curves:
the beneficial effects of the invention are as follows:
1. the invention provides a generating method of an adjacent surface tooth preparation curve for robot auxiliary tooth preparation, which comprises the steps of performing discretization curved surface modeling on a model in a pretreatment stage of a standardized preparation model, connecting and reconstructing original triangular plates, and further performing curved surface fitting, so that the processed preparation model has curved surface characteristics and boundary characteristics, and the accuracy and fitting degree of an extracted tooth preparation curve are improved.
2. In the back calculation process of the adjacent tooth preparation curves, NURBS is selected to express the adjacent tooth preparation curves, and the control points obtain more flexible control range for the adjacent tooth preparation curves by utilizing different node distances, wherein the adjacent tooth preparation curves are 4 degrees of freedom and 3 NURBS curves, so that the first and last 3 control points are sequentially equal, the operation rate of the adjacent tooth preparation curves is improved, and the quantity of inflection points of the interpolated curves is increased.
3. In the back calculation process of the adjacent tooth preparation curve, the characteristic of the curve obtained by the cumulative chord length parameterization method is utilized to reflect the distribution condition of the discrete points Q according to the chord length in consideration of the ineffective contact of the adjacent teeth of the target tooth in the preparation process, so that the preparation precision of the tooth preparation curve is effectively improved, and the adjacent tooth preparation curve is consistent, smooth and continuous.
4. According to the invention, the standardized back tooth preparation model is used as a reference, the adjacent tooth preparation stage is planned as the three-dimensional curve track planning of the robot, the curve data of the adjacent tooth preparation is extracted through the pretreatment of the standard back tooth full crown preparation model, the back calculation and the interpolation of the adjacent tooth curve are completed based on the NURBS curve according to the extracted curve data of the adjacent tooth preparation, the three-dimensional curve track planning of the auxiliary tooth preparation of the robot is further realized, and the foundation is laid for realizing the auxiliary tooth preparation of the robot.
Drawings
For ease of illustration, the invention is described in detail by the following detailed description and the accompanying drawings.
FIG. 1 is a flow chart of a method for generating an adjacent dental backup curve for robotic-assisted dental preparation;
FIG. 2 is a preprocessing flow of a normalized preparation model;
FIG. 3 is a graph of the calculation of discrete points of the adjacent tooth preparation curve and three node vectors U;
FIG. 4 is a robot trajectory plan for an adjacent tooth preparation curve;
FIG. 5 shows the interpolation result of adjacent tooth preparation curves;
FIG. 6 is a robotic-assisted dental preparation experiment system;
FIG. 7 is an experimental process of a robot completing a neighbor preparation phase;
fig. 8 shows the selection of feature points in the neighbor preparation stage.
Detailed Description
For the purpose of making apparent the objects, technical solutions and advantages of the present invention patent, the present invention patent is described below by way of specific embodiments shown in the drawings, but it should be understood that these descriptions are merely exemplary and are not intended to limit the scope of the present invention patent, and furthermore, in the following description, descriptions of well-known structures and techniques are omitted so as not to unnecessarily obscure the concepts of the present invention patent.
Implementation example 1:
as shown in fig. 1,2, 3 and 4, the following technical solutions are adopted in this embodiment: a method for generating an adjacent surface tooth preparation curve for robot assisted tooth preparation comprises the following specific implementation processes:
step one, preparation of a body model pretreatment
Scanning according to a tooth preparation model in a clinic of a doctor to obtain a standardized preparation model in an obj format, processing the standardized tooth preparation three-dimensional model in the obj format by using reverse engineering software, constructing a subdivision grid in each curved surface sheet, and outputting the preparation model in the obj format formed by triangular sheets into a preparation model in a step format with curved surface characteristics and boundary characteristics;
step two, extracting a standardized adjacent surface tooth preparation curve and discrete points
The obtained step-format preparation model is imported into Creo three-dimensional software, 2/3 of the diameter D of the needle is arranged in the shoulder to be prepared, the rest D/3 is positioned outside the tooth body, and the axis of the needle is away from the tooth edge curve C L Is D/6, replicates the dental margin curve C of the standardized preparation L As a reference curve, the offset command is again used to input the offset D/6 versus C L Obtaining the adjacent surface tooth preparation curve C by completing offset G Tooth preparation curve C on the adjacent surface G Up by the number N of discrete points Q And a discrete dot pitch L Q Obtaining a plurality of discrete points Q by two parameters;
step three, back calculation of adjacent surface tooth preparation curve
The discrete point Q in the pretreatment stage is calculated by inverse calculation based on NURBS to obtain the adjacent surface tooth preparation curve, the inverse calculation target is to calculate the node vector U and control vertex P, a NURBS curve mathematical model is introduced,
ω i is a weight factor, where i=0, 1,..n; p (P) i To control vertices, the number is n+1, where i=0, 1,..n; p is NURBS curve number;
N i,p (u) is a basis function; n in formula (1) i,p (u) is:
wherein: u (u) i Is an element of a non-uniform node vector U, as shown in equation (3);
m+1 is the length of the node vector U; the relation among m, p and n is m=n+p+1; a and b are 0 and 1;
because the adjacent surface tooth preparation curve is a closed three-dimensional curve, the first and the last data points are coincided with each other to form Q 0 ＝Q m The adjacent tooth preparation curve is a 4-degree-of-freedom 3-time NURBS curve, and the first and last 3 control points are sequentially equal to obtain an equation: p (P) 0 ＝P n-2 ，P 1 ＝P n-1 ，P 2 ＝P n The rational division of NURBS curves is represented in the form of a matrix:
wherein: t E [0,1 ]]Order-makingΔ i ＝u i+1 -u i ,/>m 12 ＝1-m 11 -m 13 ，/>
According to the determined conditions of the first end and the last end, a calculation formula of a control point P under the condition of a closed curve is obtained:
wherein each parameter is shown in formula (7):
wherein: i=1, 2,/i., n-2;
the node vector U of the adjacent surface curve is calculated by the discrete point Q obtained in the pretreatment stage through a uniform parameterization method, an accumulated chord length parameterization method and a centripetal parameterization method, the curve obtained by the accumulated chord length parameterization method reflects the distribution condition of the discrete point Q according to the chord length, and the tooth preparation curve is more similar to the standardized adjacent surface tooth preparation curve C extracted by the preparation model G The preparation precision of the tooth preparation curve is improved.
Step four, adjacent surface tooth preparation curve interpolation
The principle of interpolation of the adjacent surface tooth preparation curve is to use the time sequence { t } 1 ,t 2 ,...,t k ,...,t n-1 ,t n Sequence of { u } segmentation parameters 1 ,u 2 ,...,u k ,u k+1 ,...,u n-1 ,u n And then obtains the interpolation point sequence { C (u) 1 ),C(u 2 ),...,C(u k ),...,C(u n-1 ),C(u n ) The interpolation core calculation is to calculate u by using the interpolation period T k And u k+1 The relation between them is further defined by C (u k ) Obtain C (u) k+1 )；
Firstly, deriving the derivative of NURBS curve to obtain the first derivative C of the adjacent tooth preparation curve pair u based on NURBS (1) (u)；
Wherein: c (C) (1) (u) is the first derivative of the curve with respect to u;
continuing to calculate the second derivative of the curve to obtain C (2) (u)：
Wherein: c (C) (2) (u) is the second derivative of the curve to u, the basis function N i,p (m) The expression of (u) is:
the feed rate on the NURBS curve is expressed as:
sorting the transformation formula (11) to obtain formula (12):
wherein: c (C) x (1) (u k ) Is the first derivative of the curve x direction; c (C) y (1) (u k ) Is the first derivative of the curve y direction; c (C) z (1) (u k ) Is the first derivative of the z direction of the curve.
Continuing to calculate the second derivative of the formula (12), and obtaining:
wherein: c (C) x (2) (u k ) Is the second derivative of the curve in the x direction; c (C) y (2) (u k ) Is the second derivative of the curve y direction; c (C) z (2) (u k ) Is the second derivative of the curve in the z direction;
the arrangement results in a relationship between the geometrical and movement characteristics of the NURBS-based interproximal dental curves:
implementation example 2:
as shown in fig. 5, 6, 7 and 8, the method of the present invention is exemplified by specific experimental data.
Scanning according to a tooth preparation model clinically by a doctor to obtain a standardized preparation model in an obj format, processing the standardized tooth preparation three-dimensional model in the obj format by using Geomagic Wrap reverse engineering software, repairing triangle sheet problems such as non-fluid edges, self-intersecting edges, highly refractive edges or nails in the model by using a grid doctor command, enabling the surface of the model to be smoother and more convenient for further curved surface formation by using a relaxed polygon command through increasing the number of triangles, setting a geometric figure type as a machine by using automatic curved surface formation, setting a curved sheet count as automatic evaluation, setting a curved surface detail as moderate, setting an adaptive tolerance of curved surface fitting as 0.0097, constructing subdivision grids in each curved sheet by using a grid construction command, and finishing outputting a step-format three-dimensional model with curved surface characteristics and boundary characteristics from the three-dimensional model in the obj format consisting of triangle sheets.
Realized by MATLAB software programming, the diameter D of the selected needle is 1.6mm, the curve offset is 0.265, and the number of discrete points N is equal to that of the needle Q Taking 100, discrete point increment L Q Taking 0.01, the interpolation period t=0.002 s, the interpolation result shown in fig. 5 is obtained.
And (3) obtaining the terminal coordinate point of the robot under the joint coordinate system through space coordinate conversion of the interpolation points of the planned adjacent surface preparation track of the robot, and inputting the terminal coordinate point into software of an upper computer for the robot to finish tooth preparation. In the experimental process, the diameter of the tail end needle is 1.6mm, the experimental voltage is 12V, the current is 0.25A, the tail end rotating speed is 46000r/min, and the moving speed of the robot is 11.1mm/s and is 15% of the playback speed of the set storage point.
Analyzing the known interpolation points and obtaining the characteristic points in the adjacent surface preparation stage, wherein the change of the integral curve is reflected by the inflection points between the X, Y and Z coordinate values of the interpolation points and the interpolation point curve, so that the inflection point of the tooth preparation curve is selected as the characteristic point, and the theoretical values of the characteristic points in the adjacent surface preparation stage are shown in table 1. Each point was measured 5 times and recorded, but the values of the two sets of feature points (II and V, IV and VI) were very close resulting in similar results for the multiple measurements, thus reducing the two feature points (V and VI) to a total of 30 experimental data points as shown in table 2.
TABLE 1 theoretical values of characteristic points at the preliminary stage of the adjacent surface
Table 2 experimental measurement values of various feature points
The measurement mean mu, the relative fixed point error epsilon, the relative standard deviation RSD and the confidence interval are calculated, and the statistical tables of various parameters of the system error are summarized in Table 3.
Table 3 statistics of various parameters of systematic errors
The relative fixed point error epsilon range of each characteristic point in the X direction is 0.03-0.24 mm, the error range in the Y direction is 0.02-0.29 mm, the error range in the Z direction is 0.03-0.37 mm, and the errors of all characteristic points can be controlled within 0.5mm, so that the robot can accurately reach each key node in the adjacent surface preparation stage to finish preparation; in terms of relative standard error (RSD), the error range of each characteristic point in the X direction is 1.12-4.64%, the error range in the Y direction is 2.35-10.82%, and the error range in the Z direction is 0.59-1.51%, and the relative standard error of each characteristic point in three directions can be found to be basically kept stable and less than 11%; the confidence interval width range of each feature point in the X direction is 0.18-0.43 mm, the confidence interval width range in the Y direction is 0.03-0.48 mm, and the confidence interval width range in the Z direction is 0.29-0.49 mm. The confidence interval widths in different directions under the same characteristic point are stabilized at about 0.31mm on average, which proves that the systematic errors of all the characteristic points in the adjacent surface preparation stage can be stabilized in a smaller range, and the preparation precision of the robot in the stage is ensured.
While there has been shown and described what are at present considered to be the basic principles and the essential features of the invention and the advantages of the invention, it will be understood by those skilled in the art that the invention is not limited by the foregoing embodiments, but is described in the foregoing examples and description merely illustrative of the principles of the invention, and various changes and modifications can be made therein without departing from the spirit and scope of the invention as defined in the appended claims. The scope of the invention is defined by the appended claims and equivalents thereof.
Claims (1)
1. A method for generating an adjacent surface tooth preparation curve for robot assisted tooth preparation is characterized by comprising the following steps of: the method comprises the following specific implementation processes:
step one, preparation of a body model pretreatment
Scanning according to a tooth preparation model in a doctor clinic to obtain a standardized preparation model in an obj format, processing the standardized tooth preparation three-dimensional model in the obj format by using Geomagic Wrap reverse engineering software, repairing triangle sheet problems of non-fluid edges, self-intersecting edges, highly refractive edges or nails in the model by using a grid doctor command, enabling the surface of the model to be smoother and more convenient for further curved surface formation by increasing the number of triangles by using a relaxed polygon command, setting a geometric figure type as a machine by using automatic curved surface formation, setting a curved sheet count as automatic evaluation, setting a curved surface detail as moderate, setting an adaptive tolerance of curved surface fitting as 0.0097, constructing subdivision grids in each curved sheet by using a grid constructing command, and finishing outputting a step-format three-dimensional model with curved surface characteristics and boundary characteristics from the three-dimensional model in the obj format consisting of triangle sheets;
step two, extracting a standardized adjacent surface tooth preparation curve and discrete points
The obtained step-format preparation model is imported into Creo three-dimensional software, 2/3 of the diameter D of the needle is arranged in the shoulder to be prepared, the rest D/3 is positioned outside the tooth body, and the axis of the needle is away from the tooth edge curve C L Is D/6, replicates the dental margin curve C of the standardized preparation L As a reference curve, the offset command is again used to input the offset D/6 versus C L Obtaining the adjacent surface tooth preparation curve C by completing offset G Tooth preparation curve C on the adjacent surface G Up by the number N of discrete points Q And a discrete dot pitch L Q Obtaining a plurality of discrete points Q by two parameters;
step three, calculating a node vector U and a control point P
The discrete point Q in the pretreatment stage is calculated by inverse calculation based on NURBS to obtain the adjacent surface tooth preparation curve, the inverse calculation target is to calculate the node vector U and control vertex P, a NURBS curve mathematical model is introduced,
ω i is a weight factor, where i=0, 1,..n; p (P) i To control vertices, the number is n+1, where i=0, 1,..n; p is NURBS curve number; n (N) i,p (u) is a basis function; n in formula (1) i,p (u) is:
wherein: u (u) i Is an element of a non-uniform node vector U, as shown in equation (3);
m+1 is the length of the node vector U; the relation among m, p and n is m=n+p+1; a and b are 0 and 1;
because the adjacent surface tooth preparation curve is a closed three-dimensional curve, the first and the last data points are coincided with each other to form Q 0 ＝Q m The adjacent tooth preparation curve is a 4-degree-of-freedom 3-time NURBS curve, and the first and last 3 control points are sequentially equal to obtain an equation: p (P) 0 ＝P n-2 ，P 1 ＝P n-1 ，P 2 ＝P n The rational division of NURBS curves is represented in the form of a matrix:
wherein: t E [0,1 ]]Order-makingΔ i ＝u i+1 -u i ,/>m 12 ＝1-m 11 -m 13 ，
According to the determined conditions of the first end and the last end, a calculation formula of a control point P under the condition of a closed curve is obtained:
wherein each parameter is shown in formula (7):
wherein: i=1, 2,/i., n-2;
the node vector U of the adjacent surface curve is calculated by the discrete point Q obtained in the pretreatment stage through a uniform parameterization method, an accumulated chord length parameterization method and a centripetal parameterization method, the curve obtained by the accumulated chord length parameterization method reflects the distribution condition of the discrete point Q according to the chord length, and the tooth preparation curve is more similar to the standardized adjacent surface tooth preparation curve C extracted by the preparation model G The preparation precision of the tooth preparation curve is improved;
step four, adjacent surface tooth preparation curve interpolation
The principle of interpolation of the adjacent surface tooth preparation curve is to use the time sequence { t } 1 ,t 2 ,...,t k ,...,t n-1 ,t n Sequence of { u } segmentation parameters 1 ,u 2 ,...,u k ,u k+1 ,...,u n-1 ,u n And then obtains the interpolation point sequence { C (u) 1 ),C(u 2 ),...,C(u k ),...,C(u n-1 ),C(u n ) The interpolation core calculation is to calculate u by using the interpolation period T k And u k+1 The relation between them is further defined by C (u k ) Obtain C (u) k+1 )；
Firstly, deriving the derivative of NURBS curve to obtain the first derivative C of the adjacent tooth preparation curve pair u based on NURBS (1) (u)；
Wherein: c (C) (1) (u) is the first derivative of the curve with respect to u;
continuing to calculate the second derivative of the curve to obtain C (2) (u)：
Wherein: c (C) (2) (u) is the second derivative of the curve to u, the basis function N i,p (m) The expression of (u) is:
the feed rate on the NURBS curve is expressed as:
sorting the transformation formula (11) to obtain formula (12):
wherein: c (C) x (1) (u k ) Is the first derivative of the curve x direction; c (C) y (1) (u k ) Is the first derivative of the curve y direction; c (C) z (1) (u k ) Is the first derivative of the curve z direction;
continuing to calculate the second derivative of the formula (12), and obtaining:
wherein: c (C) x (2) (u k ) Is the second derivative of the curve in the x direction; c (C) y (2) (u k ) Is the second derivative of the curve y direction; c (C) z (2) (u k ) Is the second derivative of the curve in the z direction;
the arrangement results in a relationship between the geometrical and movement characteristics of the NURBS-based interproximal dental curves:
through MATLAB software programming, the diameter D of the selected needle is 1.6mm, the curve offset is 0.265, and the number of discrete points N is equal to that of the selected needle Q Taking 100, discrete point increment L Q Taking 0.01, the interpolation period t=0.002 s.
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