CN111489437A - Adjacent tooth preparation curve generation method for robot-assisted tooth preparation - Google Patents

Adjacent tooth preparation curve generation method for robot-assisted tooth preparation Download PDF

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CN111489437A
CN111489437A CN202010261544.9A CN202010261544A CN111489437A CN 111489437 A CN111489437 A CN 111489437A CN 202010261544 A CN202010261544 A CN 202010261544A CN 111489437 A CN111489437 A CN 111489437A
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CN111489437B (en
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姜金刚
钱伟
黄致远
张永德
唐德栋
曾阳
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Harbin University of Science and Technology
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Abstract

The invention discloses an adjacent tooth preparation curve generation method for robot-assisted tooth preparation, and relates to the technical field of robot-assisted tooth preparation. The technical points are as follows: preprocessing a preparation body model; extracting a standard adjacent tooth preparation curve and discrete points; calculating a node vector U and a control point P; and (6) curve interpolation. According to the invention, the inverse 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 the three-dimensional curve track planning of the robot-assisted tooth preparation is realized, and a foundation is laid for realizing the robot-assisted tooth preparation.

Description

Adjacent tooth preparation curve generation method for robot-assisted tooth preparation
Technical Field
The invention relates to an adjacent surface tooth preparation curve generation method for robot-assisted tooth preparation, and belongs to the technical field of robot-assisted tooth preparation.
Background
Dental caries is an important cause of tooth defect, and seriously influences the oral health of people. Dental restoration is an important means for treating tooth defects, and dental preparation is an essential treatment link in the restoration process, which refers to an operation process of quantitatively removing teeth at the positions suffering from dental caries by a doctor and forming expected three-dimensional shapes. In conventional dental preparation procedures, a great number of repeated fine adjustments of carious teeth are required depending on manual operations of a doctor in combination with abundant clinical experience. However, the current situation of imbalance of the doctor-patient ratio in China cannot meet the current great demand of tooth preparation, so that a robot and auxiliary software need to be introduced to replace or assist doctors to efficiently complete tooth preparation, the problem of imbalance of the doctor and the patient can be relieved, and the quality of prepared teeth and the effect of oral treatment can be effectively improved.
In the process of completing the tooth preparation with the aid of the machine, the motion trail of the tail end of the robot determines the tooth preparation precision, and then the treatment effect and success rate are influenced. Therefore, a reasonable adjacent tooth preparation track is obtained, and the attaching degree of the preparation body and the posterior tooth restoration body is considered emphatically, so that the condition of oral secondary diseases caused by improper preparation of the tooth body is reduced, and the method is a research hotspot in the field of robot-assisted tooth preparation at present.
Disclosure of Invention
Aiming at the problems, the invention provides an adjacent surface tooth preparation curve generation method for robot-assisted tooth preparation, which solves the problem that the existing technical field of robot-assisted tooth preparation lacks a method for obtaining an adjacent surface tooth preparation curve so as to improve the precision and success rate of tooth preparation and further realize the robot-assisted tooth preparation.
The scheme adopted by the invention to solve the problems is as follows:
an adjacent tooth preparation curve generation method for robot-assisted tooth preparation is realized by the following specific steps:
step one, preparing a body model preprocessing
Scanning according to a clinical tooth preparation model of a doctor to obtain an obj-format standardized preparation body model, selecting Geomagic Wrap reverse engineering software to process the obj-format standardized preparation three-dimensional model, repairing triangular sheet problems such as non-fluid edges, self-intersection, high refraction edges or nails in the model by using a 'gridding doctor' command, enabling the surface of the model to be smoother and convenient for further curving by increasing the number of triangles by using a 'loose polygon' command, setting 'geometric figure types' to 'mechanical' by using 'automatic curving', setting 'curved sheet counting' to 'automatic evaluation' and 'moderate curved surface details', the adaptive tolerance of the surface fitting is 0.0097, the generated surface sheet passes through a 'construction grid' command, constructing a subdivision mesh in each curved surface slice, and finishing outputting a step-format three-dimensional model with curved surface characteristics and boundary characteristics from an obj-format three-dimensional model consisting of triangular slices;
step two, extracting a standard adjacent tooth preparation curve and discrete points
The obtained step-format preparation body model is imported into Creo three-dimensional software, 2/3 parts of the diameter D of the needle are arranged in the shoulder to be prepared, the rest D/3 parts are arranged outside the tooth body, and the curve C of the axis of the needle from the edge of the tooth body is arrangedLIs D/6, reproduces the dental edge curve C of the standardized preparationLAs a reference curve, the "offset" command is again used to input the offset D/6 vs CLThe offset is completed to obtain an adjacent tooth preparation curve CGPreparing tooth curve C on adjacent surfaceGNumber of upper discrete points NQAnd discrete dot spacing LQTwo parameters obtain a plurality of discrete points Q;
step three, back calculation of tooth preparing curve of adjacent surface
Obtaining an adjacent surface tooth preparation curve by carrying out inverse calculation on the discrete points Q in the preprocessing stage based on NURBS, introducing a NURBS curve mathematical model into the inverse calculation targets of a calculation node vector U and a control vertex P,
Figure BDA0002439489010000011
ωi(i ═ 0, 1.., n) is a weighting factor; pi(i ═ 0,1,. and n) are control vertices, the number of which is n + 1; p is NURBS curve number; n is a radical ofi,p(u) is a basis function; n in formula (1)i,p(u) is:
Figure BDA0002439489010000021
in the formula: u. ofiIs an element of the non-uniform node vector U, as shown in equation (3);
Figure BDA0002439489010000022
m +1 is the length of the node vector U; the relation among m, p and n is that m is n + p + 1; a and b are typically 0 and 1;
because the adjacent tooth preparing curve is a closed three-dimensional curve, the first data point and the last data point are coincided with each other0＝QmThe 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: p0＝Pn-2，P1＝Pn-1，P2＝PnThe rational fraction of the NURBS curve is expressed in matrix form:
Figure BDA0002439489010000023
Figure BDA0002439489010000024
in the formula, t ∈ [0,1]Let us order
Figure BDA0002439489010000025
Specific Deltai＝ui+1-ui,
Figure BDA0002439489010000026
m12＝1-m11-m13，
Figure BDA0002439489010000027
Obtaining a calculation formula of a control point P under the condition of a closed curve according to the determined conditions of the head end and the tail end:
Figure BDA0002439489010000029
the parameters in the formula are shown in formula (7):
Figure BDA00024394890100000210
in the formula: 1,2, n-2;
respectively calculating node vectors U of adjacent surface curves of the discrete points Q obtained in the preprocessing stage by using a uniform parameterization method, an accumulation chord length parameterization method and a centripetal parameterization method, wherein the curves obtained by the accumulation chord length parameterization method reflect the distribution condition of the discrete points Q according to chord lengths, and the tooth preparing curves are closer to a standard adjacent surface tooth preparing curve C extracted by a preparation modelGThe preparation precision of the tooth preparation curve is improved.
Step four, interpolation of tooth preparing curve of adjacent surface
The interpolation principle of the tooth-preparing curve of the adjacent surface is to use a time sequence t1,t2,...,tk,...,tn-1,tnDividing parameter sequence u1,u2,...,uk,uk+1,...,un-1,unGet the sequence of interpolation points { C (u) }1),C(u2),...,C(uk),...,C(un-1),C(un) The core calculation of interpolation is to use the interpolation period T to obtain ukAnd uk+1Further by C (u)k) To obtain C (u)k+1)；
Firstly, the derivative of the NURBS curve is deduced to obtain a first derivative C of the adjacent tooth curve pair u based on the NURBS(1)(u)；
Figure BDA0002439489010000031
In the formula: c(1)(u) is the first derivative of the curve with respect to u;
continuously calculating the second derivative of the curve to obtain C(2)(u)：
Figure BDA0002439489010000032
In the formula: c(2)(u) is the second derivative of the curve with respect to u, the basis function Ni,p (m)The expression of (u) is:
Figure BDA0002439489010000033
the feed rate on the NURBS curve is expressed as:
Figure BDA0002439489010000034
and (3) obtaining a formula (12) by arranging and converting the formula (11):
Figure BDA0002439489010000035
in the formula: cx (1)(uk) Is a first derivative of the curve in the x direction; cy (1)(uk) Is a first derivative of the curve in the y direction; cz (1)(uk) Is the first derivative of the curve in the z-direction.
The second derivative continues to be solved for equation (12) to obtain:
Figure BDA0002439489010000036
in the formula: cx (2)(uk) Is the second derivative of the curve in the x direction; cy (2)(uk) Is the second derivative of the curve in the y direction; cz (2)(uk) Is the second derivative of the curve in the z direction;
the relationship between the geometric characteristics and the motion characteristics of the NURBS-based adjacent tooth preparation curve is obtained through arrangement:
Figure BDA0002439489010000041
the invention has the beneficial effects that:
1. the invention provides an adjacent tooth preparing curve generating method for robot-assisted tooth preparation, which is characterized in that discretized surface modeling is carried out on a model in a preprocessing stage of a standardized preparation model, original triangular pieces are connected and reconstructed, and surface fitting is further carried out, so that the processed preparation model has surface characteristics and boundary characteristics, and the precision and the fitting degree of an extracted tooth preparing curve are improved.
2. In the invention, during the back calculation process of the adjacent tooth preparation curve, NURBS is selected to express the adjacent tooth preparation curve, and the control point obtains a more flexible control range for the adjacent tooth preparation curve by utilizing different node distances, and the adjacent tooth preparation curve is a 4-freedom-degree 3-time NURBS curve, so that the first and last 3 control points are sequentially equal, the calculation rate of the adjacent tooth preparation curve is improved, and the number of inflection points of the interpolated curve is increased.
3. In the back calculation process of the adjacent tooth preparing curve, the invalid contact of the adjacent teeth of the target tooth in the preparation process is considered, the characteristic of the curve obtained by the cumulative chord length parameterization method is utilized, the distribution condition of discrete points Q according to the chord length is reflected, the preparation precision of the tooth preparing curve is effectively improved, and the adjacent tooth preparing curve is kept consistent, smooth and continuous.
4. The method takes a standardized posterior tooth preparation model as a reference, plans the adjacent tooth preparation stage into a three-dimensional curve track of the robot, extracts curve data of the adjacent tooth preparation through preprocessing the standard posterior tooth full-crown preparation model, and completes inverse calculation and interpolation of the adjacent tooth curve based on a NURBS curve according to the extracted curve data of the adjacent tooth preparation, thereby realizing the three-dimensional curve track planning of the robot-assisted tooth preparation and laying a foundation for realizing the robot-assisted tooth preparation.
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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 prepared tooth curve for robot-assisted tooth preparation;
FIG. 2 is a flow chart of a pre-process for normalizing a preparation model;
FIG. 3 is a calculation method of discrete points and three node vectors U of an adjacent tooth preparing curve;
FIG. 4 is a robot trajectory plan of an adjacent tooth preparation curve;
FIG. 5 is an interpolation result of the adjacent tooth preparing curve;
FIG. 6 is a robotic-assisted dental preparation experiment system;
FIG. 7 is an experimental procedure of the robot completing the preparation phase of the adjacent surfaces;
fig. 8 is a selection of feature points in the preparation stage of the adjacent surface.
Detailed Description
For the purposes of promoting a clear understanding of the objects, aspects and advantages of the invention, reference will now be made to the following description of the preferred embodiments illustrated in the accompanying drawings, with the understanding that the description is illustrative only and is not intended to limit the scope of the invention, and that the following description will omit descriptions of well-known structures and techniques in order to avoid unnecessarily obscuring the concepts of the invention.
Example 1 was carried out:
as shown in fig. 1, fig. 2, fig. 3, and fig. 4, the following technical solutions are adopted in the present embodiment: an adjacent tooth preparation curve generation method for robot-assisted tooth preparation is realized by the following specific steps:
step one, preparing a body model preprocessing
Scanning according to a clinical tooth preparation model of a doctor to obtain an obj-format standardized preparation model, processing the obj-format standardized preparation three-dimensional model by utilizing reverse engineering software, constructing subdivision meshes in each curved plate, and outputting the obj-format preparation model consisting of triangular plates into a step-format preparation model with curved surface characteristics and boundary characteristics;
step two, extracting a standard adjacent tooth preparation curve and discrete points
The obtained step-format preparation body model is imported into Creo three-dimensional software, 2/3 parts of the diameter D of the needle are arranged in the shoulder to be prepared, the rest D/3 parts are arranged outside the tooth body, and the curve C of the axis of the needle from the edge of the tooth body is arrangedLIs D/6, reproduces the dental edge curve C of the standardized preparationLAs a reference curve, the "offset" command is again used to input the offset D/6 vs CLThe offset is completed to obtain an adjacent tooth preparation curve CGPreparing tooth curve C on adjacent surfaceGNumber of upper discrete points NQAnd discrete dot spacing LQTwo parameters obtain a plurality of discrete points Q;
step three, back calculation of tooth preparing curve of adjacent surface
Obtaining an adjacent surface tooth preparation curve by carrying out inverse calculation on the discrete points Q in the preprocessing stage based on NURBS, introducing a NURBS curve mathematical model into the inverse calculation targets of a calculation node vector U and a control vertex P,
Figure BDA0002439489010000051
ωi(i ═ 0, 1.., n) is a weighting factor; pi(i ═ 0,1,. and n) are control vertices, the number of which is n + 1; p is NURBS curve number; n is a radical ofi,p(u) is a basis function; n in formula (1)i,p(u) is:
Figure BDA0002439489010000052
in the formula: u. ofiIs an element of the non-uniform node vector U, as shown in equation (3);
Figure BDA0002439489010000053
m +1 is the length of the node vector U; the relation among m, p and n is that m is n + p + 1; a and b are typically 0 and 1;
due to the fact thatThe adjacent tooth preparing curve is a closed three-dimensional curve, so that the first data point and the last data point are coincided with each other0＝QmThe 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: p0＝Pn-2，P1＝Pn-1，P2＝PnThe rational fraction of the NURBS curve is expressed in matrix form:
Figure BDA0002439489010000054
Figure BDA0002439489010000055
in the formula, t ∈ [0,1]Let us order
Figure BDA0002439489010000056
Is specially characterized in that
Figure BDA0002439489010000057
m12＝1-m11-m13，
Figure BDA0002439489010000058
Obtaining a calculation formula of a control point P under the condition of a closed curve according to the determined conditions of the head end and the tail end:
Figure BDA00024394890100000510
the parameters in the formula are shown in formula (7):
Figure BDA0002439489010000061
in the formula: 1,2, n-2;
respectively calculating node vectors U of adjacent surface curves of the discrete points Q obtained in the preprocessing stage by using a uniform parameterization method, an accumulated chord length parameterization method and a centripetal parameterization method, and reacting curves obtained by the accumulated chord length parameterization methodThe distribution of the discrete points Q according to chord length is realized, and the tooth preparing curve is closer to the standard adjacent tooth preparing curve C extracted by the preparation modelGThe preparation precision of the tooth preparation curve is improved.
Step four, interpolation of tooth preparing curve of adjacent surface
The interpolation principle of the tooth-preparing curve of the adjacent surface is to use a time sequence t1,t2,...,tk,...,tn-1,tnDividing parameter sequence u1,u2,...,uk,uk+1,...,un-1,unGet the sequence of interpolation points { C (u) }1),C(u2),...,C(uk),...,C(un-1),C(un) The core calculation of interpolation is to use the interpolation period T to obtain ukAnd uk+1Further by C (u)k) To obtain C (u)k+1)；
Firstly, the derivative of the NURBS curve is deduced to obtain a first derivative C of the adjacent tooth curve pair u based on the NURBS(1)(u)；
Figure BDA0002439489010000062
In the formula: c(1)(u) is the first derivative of the curve with respect to u;
continuously calculating the second derivative of the curve to obtain C(2)(u)：
Figure BDA0002439489010000063
In the formula: c(2)(u) is the second derivative of the curve with respect to u, the basis function Ni,p (m)The expression of (u) is:
Figure BDA0002439489010000064
the feed rate on the NURBS curve is expressed as:
Figure BDA0002439489010000065
and (3) obtaining a formula (12) by arranging and converting the formula (11):
Figure BDA0002439489010000071
in the formula: cx (1)(uk) Is a first derivative of the curve in the x direction; cy (1)(uk) Is a first derivative of the curve in the y direction; cz (1)(uk) Is the first derivative of the curve in the z-direction.
The second derivative continues to be solved for equation (12) to obtain:
Figure BDA0002439489010000072
in the formula: cx (2)(uk) Is the second derivative of the curve in the x direction; cy (2)(uk) Is the second derivative of the curve in the y direction; cz (2)(uk) Is the second derivative of the curve in the z direction;
the relationship between the geometric characteristics and the motion characteristics of the NURBS-based adjacent tooth preparation curve is obtained through arrangement:
Figure BDA0002439489010000073
example 2 was carried out:
as shown in fig. 5, 6, 7 and 8, the method of the present invention is exemplified by taking specific experimental data as an example.
Scanning according to a clinical tooth preparation model of a doctor to obtain an obj-format standardized preparation body model, selecting Geomagic Wrap reverse engineering software to process the obj-format standardized preparation three-dimensional model, repairing triangular sheet problems such as non-fluid edges, self-intersection, high refraction edges or nails in the model by using a 'gridding doctor' command, enabling the surface of the model to be smoother and convenient for further curving by increasing the number of triangles by using a 'loose polygon' command, setting 'geometric figure types' to 'mechanical' by using 'automatic curving', setting 'curved sheet counting' to 'automatic evaluation' and 'moderate curved surface details', the adaptive tolerance of the surface fitting is 0.0097, the generated surface sheet passes through a 'construction grid' command, and constructing a subdivision mesh in each curved surface slice, and finishing outputting a step-format three-dimensional model with curved surface characteristics and boundary characteristics from an obj-format three-dimensional model consisting of triangular slices.
The method is realized by MAT L AB software programming, the selected needle diameter D is 1.6mm, the curve offset is 0.265, and the number N of discrete pointsQTake 100, discrete point increment LQThe interpolation result shown in fig. 5 was obtained by taking 0.01 and the interpolation period T equal to 0.002 s.
And (4) converting the interpolation points of the planned robot adjacent preparation track through space coordinates to obtain a tail end coordinate point of the robot under a joint coordinate system, and inputting the tail end coordinate point into software of an upper computer for the robot to finish tooth preparation. In the experiment process, the diameter of a tail end machine needle is 1.6mm, the experiment voltage is 12V, the current is 0.25A, the rotating speed of the tail end is 46000r/min, and the moving speed of the robot is 11.1mm/s, which is 15% of the set point playback speed.
Analyzing the known interpolation points and obtaining the characteristic points in the preparation stage of the adjacent surface, wherein the inflection points between the curve of the interpolation points and the inflection points between the X, Y and Z coordinate values of the interpolation points reflect the change of the whole curve, so that the inflection point of the curve of the tooth is selected as the characteristic point, and the theoretical numerical values of the characteristic points in the preparation stage of the adjacent surface are shown in table 1. Each point was measured 5 times and recorded, but the close proximity of the two sets of characteristic points (II and V, IV and VI) resulted in similar results for the multiple measurements, thus reducing the two characteristic points (V and VI) resulted in a total of 30 experimental data points as shown in table 2.
TABLE 1 theoretical values of characteristic points in preparation stage of adjacent surfaces
Figure BDA0002439489010000074
TABLE 2 Experimental measurement values for the respective characteristic points
Figure BDA0002439489010000075
And calculating to obtain a measurement mean value mu, a relative fixed point error, a relative standard deviation RSD and a confidence interval, and summarizing all parameter statistical tables of system errors to a table 3.
TABLE 3 statistical table of system error parameters
Figure BDA0002439489010000081
The error range of the relative fixed point of each feature 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 the feature points can be controlled within 0.5mm, so that the robot can accurately reach each key node in the adjacent surface preparation stage to complete preparation; in the aspect 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 errors of each characteristic point in the three directions are basically kept stable and are less than 11%; the width range of the confidence interval of each characteristic point in the X direction is 0.18-0.43 mm, the width range of the confidence interval in the Y direction is 0.03-0.48 mm, and the width range of the confidence interval 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 shows that the system error of each characteristic point in the adjacent surface preparation stage can be stabilized in a small 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 the fundamental principles and essential features of the invention and its advantages, it will be understood by those skilled in the art that the invention is not limited by the embodiments described above, which are given by way of illustration of the principles of the invention and which are within the spirit and scope of the invention as defined by the appended claims. The scope of the invention is defined by the appended claims and equivalents thereof.

Claims (1)

1. An adjacent surface tooth preparation curve generation method for robot-assisted tooth preparation is characterized by comprising the following steps of: the method comprises the following concrete implementation processes:
step one, preparing a body model preprocessing
Scanning according to a clinical tooth preparation model of a doctor to obtain an obj-format standardized preparation body model, selecting GeomagicWrap reverse engineering software to process the obj-format standardized preparation three-dimensional model, repairing triangular sheet problems such as non-fluid edges, self-intersection, high refraction edges or nails in the model by using a 'gridding doctor' command, enabling the surface of the model to be smoother and convenient for further curving by increasing the number of triangles by using a 'loose polygon' command, setting 'geometric figure type' to be 'mechanical' by using 'automatic curving', setting 'curved sheet count' to be 'automatic evaluation' and 'curved surface detail' to be moderate, the adaptive tolerance of the surface fitting is 0.0097, the generated surface sheet passes through a 'construction grid' command, constructing a subdivision mesh in each curved surface slice, and finishing outputting a step-format three-dimensional model with curved surface characteristics and boundary characteristics from an obj-format three-dimensional model consisting of triangular slices;
step two, extracting a standard adjacent tooth preparation curve and discrete points
The obtained step-format preparation body model is imported into Creo three-dimensional software, 2/3 parts of the diameter D of the needle are arranged in the shoulder to be prepared, the rest D/3 parts are arranged outside the tooth body, and the curve C of the axis of the needle from the edge of the tooth body is arrangedLIs D/6, reproduces the dental edge curve C of the standardized preparationLAs a reference curve, the "offset" command is again used to input the offset D/6 vs CLThe offset is completed to obtain an adjacent tooth preparation curve CGPreparing tooth curve C on adjacent surfaceGNumber of upper discrete points NQAnd discrete dot spacing LQTwo parameters obtain a plurality of discrete points Q;
step three, calculating a node vector U and a control point P
Obtaining an adjacent surface tooth preparation curve by carrying out inverse calculation on the discrete points Q in the preprocessing stage based on NURBS, introducing a NURBS curve mathematical model into the inverse calculation targets of a calculation node vector U and a control vertex P,
Figure FDA0002439487000000011
ωi(i ═ 0, 1.., n) is a weighting factor; pi(i ═ 0,1,. and n) are control vertices, the number of which is n + 1; p is NURBS curve number; n is a radical ofi,p(u) is a basis function; n in formula (1)i,p(u) is:
Figure FDA0002439487000000012
in the formula: u. ofiIs an element of the non-uniform node vector U, as shown in equation (3);
Figure FDA0002439487000000013
m +1 is the length of the node vector U; the relation among m, p and n is that m is n + p + 1; a and b are typically 0 and 1;
because the adjacent tooth preparing curve is a closed three-dimensional curve, the first data point and the last data point are coincided with each other0＝QmThe 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: p0＝Pn-2，P1＝Pn-1，P2＝PnThe rational fraction of the NURBS curve is expressed in matrix form:
Figure FDA0002439487000000014
Figure FDA0002439487000000021
in the formula, t ∈ [0,1]Let us order
Figure FDA0002439487000000022
Specific Deltai＝ui+1-ui,
Figure FDA0002439487000000023
m12＝1-m11-m13，
Figure FDA0002439487000000024
Figure FDA0002439487000000029
Figure FDA0002439487000000025
Obtaining a calculation formula of a control point P under the condition of a closed curve according to the determined conditions of the head end and the tail end:
Figure FDA0002439487000000026
the parameters in the formula are shown in formula (7):
Figure FDA0002439487000000027
in the formula: 1,2, n-2;
respectively calculating node vectors U of adjacent surface curves of the discrete points Q obtained in the preprocessing stage by using a uniform parameterization method, an accumulation chord length parameterization method and a centripetal parameterization method, wherein the curves obtained by the accumulation chord length parameterization method reflect the distribution condition of the discrete points Q according to chord lengths, and the tooth preparing curves are closer to a standard adjacent surface tooth preparing curve C extracted by a preparation modelGThe preparation precision of the tooth preparation curve is improved;
step four, interpolation of tooth preparing curve of adjacent surface
The interpolation principle of the tooth-preparing curve of the adjacent surface is to use a time sequence t1,t2,...,tk,...,tn-1,tnDividing parameter sequence u1,u2,...,uk,uk+1,...,un-1,unGet the sequence of interpolation points { C (u) }1),C(u2),...,C(uk),...,C(un-1),C(un) The core calculation of interpolation is to use the interpolation period T to obtain ukAnd uk+1Further by C (u)k) To obtain C (u)k+1)；
Firstly, the derivative of the NURBS curve is deduced to obtain a first derivative C of the adjacent tooth curve pair u based on the NURBS(1)(u)；
Figure FDA0002439487000000028
In the formula: c(1)(u) is the first derivative of the curve with respect to u;
continuously calculating the second derivative of the curve to obtain C(2)(u)：
Figure FDA0002439487000000031
In the formula: c(2)(u) is the second derivative of the curve with respect to u, the basis function Ni,p (m)The expression of (u) is:
Figure FDA0002439487000000032
the feed rate on the NURBS curve is expressed as:
Figure FDA0002439487000000033
and (3) obtaining a formula (12) by arranging and converting the formula (11):
Figure FDA0002439487000000034
in the formula: cx (1)(uk) Is a first derivative of the curve in the x direction; cy (1)(uk) Is a first derivative of the curve in the y direction; cz (1)(uk) Is a first derivative of the curve in the z direction;
the second derivative continues to be solved for equation (12) to obtain:
Figure FDA0002439487000000035
in the formula: cx (2)(uk) Is the second derivative of the curve in the x direction; cy (2)(uk) Is the second derivative of the curve in the y direction; cz (2)(uk) Is the second derivative of the curve in the z direction;
the relationship between the geometric characteristics and the motion characteristics of the NURBS-based adjacent tooth preparation curve is obtained through arrangement:
Figure FDA0002439487000000036
the method is realized by MAT L AB software programming, the selected needle diameter D is 1.6mm, the curve offset is 0.265, and the number N of discrete pointsQTake 100, discrete point increment LQThe interpolation period T was 0.01 s and 0.002 s.
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