CN113446960A - Tooth surface point cloud theoretical distribution modeling method and measuring method - Google Patents
Tooth surface point cloud theoretical distribution modeling method and measuring method Download PDFInfo
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Abstract
The invention discloses a tooth surface point cloud theoretical distribution modeling method and a measuring method, wherein an established model is a method for measuring gear precision based on line structure light. The modeling method comprises the steps of firstly establishing a gear coordinate system and a sensor coordinate system respectively according to the position relation of a gear and a linear structured light sensor during actual measurement, deducing a light path equation of linear structured light by using a coordinate transformation method, and then solving the light path equation in series with a standard tooth surface equation to obtain an accurate tooth surface point cloud theoretical distribution position. The modeling method can calculate the theoretical distribution point cloud of the tooth surface before actual measurement, is convenient for optimizing the measurement position and angle of the line structured light sensor in advance, and can also be used for comparing the measured actual point cloud with the theoretical point cloud point by point to realize rapid denoising and rapid calculation of gear approximation error.
Description
Technical Field
The invention belongs to the technical field of gear detection, and particularly relates to a tooth surface point cloud theoretical distribution modeling method and a measuring method.
Background
The current trend in gear measurement is fast, fully digital. In order to quickly acquire three-dimensional shape information of a gear tooth surface, more and more students begin to acquire a gear tooth surface point cloud by using various optical measurement methods to calculate gear errors, and linear structured light measurement is one of the methods. However, the following problems often exist in the process of actually measuring and calculating gear errors:
1. the measurement method is empirical. The measurement effect of the tooth surface point cloud cannot be accurately predicted before measurement, and the method (the position where a measuring head is placed and the measurement angle) used in the measurement of the linear structured light mainly depends on experience and test;
2. the method of calculating the error using the actual measured point cloud is complex. The current error calculation is mainly performed by calculating the difference between each point and the theoretical position of each point. However, because the analytic formulas of all parts of the gear tooth profile are different, the part to which the point belongs needs to be judged before point-by-point cloud calculation, so that the operations of judging an involute starting point, fitting the involute tooth profile and the like need to be carried out first, and the requirement on operators is high;
3. the method for preprocessing the inclinometer gear point cloud is complex. When the straight-tooth cylindrical gear is measured by linear structured light parallel to the axis of the gear, the obtained data has strong relevance, and because the intersection points of the light path and the tooth surface are all positioned on a straight line vertical to the axis, the measured point cloud is easy to fit the intersected straight line, and noise points are removed. However, when the helical gear is measured, the intersection point is located on the curve, the intersection curves of different corners are different, and the data correlation is not strong, so that the algorithm required when the noise point of the helical gear is removed is complex.
Disclosure of Invention
The invention aims to overcome the defects of the prior art and provide a tooth surface point cloud theoretical distribution modeling method and a measuring method
The technical scheme adopted by the invention is as follows: a tooth surface point cloud theoretical distribution modeling method for measuring gear precision based on line structured light comprises the following steps:
w1, modeling an optical path equation of linear structured light:
establishing a gear coordinate system Og-XgYgOptical path coordinate system OL-XLYL(ii) a Wherein, OLThe point is the initial position of the measuring head, and the light ray is from OLStarting to XLThe negative axis direction; the distance between the measuring head and the tooth center is set as D; xgThe included angle between the shaft and the connecting line of the measuring head and the gear center is sigma1And XLAngle of axis being sigma2(ii) a Then O isLPoint-on-gear coordinate system Og-xgygThe coordinates in (d) can be expressed as (Dcos σ @1,Dsinσ1) The initial slope of the optical path is k ═ tan σ2；
The equation for the initial position of the optical path can be expressed as:
y-D sinσ1＝-tanσ2(x-Dcosσ1) (1)；
O′L-X′LY′Lis O after a rotation time tL-XLYLThe position is defined as the position with the rotation speed of omega and the rotation angle of W-omega; then O'LPoint-on-gear coordinate system Og-xgygThe coordinate in (D cos (W + σ) is1),D sin(W+σ1) The slope of the optical path is k' ═ -tan (σ)2-W)；
The equation of the optical path after the rotation time t can be expressed as:
y-D sin(W+σ1)＝-tan(σ2-W)[x-Dcos(W+σ1)] (2)；
the equation for the optical path plane can be expressed as:
wherein B is the tooth width, tbIs the tooth width factor;
w2. three-dimensional mathematical model of standard involute gear:
a tooth surface equation can be deduced according to a generating mode of a spiral surface and an end surface tooth profile equation, theta represents the angle of the rotation of a bus around a rotating shaft, the right-hand rotation is positive, the left-hand rotation is negative, delta represents the initial included angle between the Yw axis of a fixed coordinate system Ow and the tooth thickness symmetrical line of the measured gear,
the equation for the involute helicoid is:
the flank equation for the root transition curve is:
the tooth surface equation of the addendum circle is:
the tooth surface equation of the root circle is:
ωAcan be taken from the value of alpha' in formula (5)
Calculating to obtain;
in the above four equations, B is the tooth width, tbV is more than or equal to 0 and less than or equal to 1.0; theta denotes an angle through which the bus bar rotates around the rotation axis,
theta is positive when the gear rotates rightwards, and theta is negative when the gear rotates leftwards; delta represents an angle for rotating the Yw axis of the fixed coordinate system Ow to the tooth thickness symmetrical line of the measured gear, and clockwise is positive and anticlockwise is negative;
when any tooth surface is worked out, the delta values in the above four groups of equations can be used
Substituting to obtain; the meanings and the value ranges of other variables in the formula are the same as those in the corresponding end face tooth profile equation, and only the end face parameters are required to replace the normal face parameters;in the formula, the method for taking +/-and +/-m' is also the same as the equation of the tooth profile of the end face;
w3. calculating the tooth surface distribution point cloud:
the intersection line of the light path and the tooth surface is a complex curve, so that the z-axis coordinate of each sampling point is firstly determined when the point cloud of the intersection point is calculated, and then the formula (3) and the formulas (4), (5), (6) and (7) are sequentially connected and solved according to the intersection sequence of the light path and each part of the tooth surface to obtain the point cloud of the tooth surface;
the number of equally spaced sampling points on the intersecting curve of the light path and the tooth surface is ntbThen the sampling interval is
The z-axis coordinate of the sampling point in each sampling is
Let the sampling frequency be f, and the tooth surface point cloud number measured after the rotation time t be Nt＝ωtfntb；
Because each tooth shape of the standard gear is the same, only the rotation angle needs to be calculated in the process of calculating the point cloud of the tooth surface
Namely, it is
And (3) carrying out point counting, rotating to obtain a tooth surface point cloud of another z-1 tooth, thus obtaining a complete tooth surface point cloud, wherein N is 2 pi fntbAnd (5) rotating to obtain the tooth surface point cloud of the other z-1 tooth, thus obtaining the complete tooth surface point cloud.
A gear precision detection method adopting the tooth surface point cloud theoretical distribution modeling method for measuring the gear precision based on the line structured light comprises the following steps:
the method comprises the following steps: establishing a tooth surface point cloud theoretical distribution model by adopting the tooth surface point cloud theoretical distribution modeling method based on the line structured light measurement gear precision;
step two: the tooth surface is measured through line structured light, and the specific process is as follows: the plane where the light is located is parallel to the axis of the gear, and the light path covers all the tooth sides; in the measuring process, the measuring head is fixed, the gear rotates for a circle at a constant speed, and the measuring head collects points on an intersection line segment of a light path and a tooth surface according to the frequency f to obtain a tooth surface point cloud;
step two: removing obvious noise points, which comprises the following specific steps: calculating the Euclidean distance between each actually measured point cloud and a theoretical point cloud in a tooth surface point cloud theoretical distribution model, wherein the difference value exceeding a threshold value is a noise point;
step three: the parameters which are completely the same when the method is used for actual measurement are input into a tooth surface point cloud theoretical distribution model for calculation, and the point cloud which is completely the same as the distribution of the actually measured point cloud can be obtained, so that the difference value between the actually measured point cloud and the theoretically distributed point cloud can be completely obtained point by point after the point cloud is registered, and the difference value can be approximately used as tooth profile deviation because each point and the corresponding point are completely positioned at the same z-axis height.
The invention has the following beneficial effects:
1. the tooth surface theoretical distribution point cloud of the specified measurement parameters can be directly calculated before actual measurement, and the measurement scheme (position and angle) can be conveniently optimized. For example, aiming at the shielding phenomenon of the peripheral teeth to the measuring teeth, parameters can be adjusted in the model, measuring parameters capable of measuring more tooth surfaces are selected, and then actual measurement is carried out.
2. The method is simple and convenient to remove obvious noise points in the process of calculating the precision of the linear structure light detection gear.
3. The error calculation after the actual point cloud measurement is simplified.
Drawings
In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below, and it is obvious that the drawings in the following description are only some embodiments of the present invention, and it is within the scope of the present invention for those skilled in the art to obtain other drawings based on the drawings without inventive exercise.
FIG. 1 is a schematic diagram of a three-dimensional point cloud measurement system;
FIG. 2 is a schematic view of a structured light detection gear;
FIG. 3 is a cross-section of the measuring device at the lower end face of the gear;
FIG. 4 is a three-dimensional mathematical model of the tooth surface of the gear under test;
FIG. 5 is a cross point of a light path and a tooth surface when t is more than or equal to 0.6s and less than or equal to 0.9 s;
FIG. 6 is a complete tooth surface point cloud;
fig. 7 is a flow chart of a method.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention will be described in further detail with reference to the accompanying drawings.
The invention provides a tooth surface point cloud theoretical distribution modeling method for measuring gear precision based on line structured light, which comprises the following steps:
w1, modeling an optical path equation of linear structured light:
the method for linear structured light measurement of the tooth surface comprises the following steps: the plane of the light is parallel to the gear axis, and the light path covers all the tooth sides. In the actual measurement process, the measuring head is fixed, the gear rotates for a circle at a constant speed, and the measuring head collects points on the intersection line segment of the light path and the tooth surface according to a certain frequency f to obtain the tooth surface point cloud. The process is subjected to mathematical modeling, in order to facilitate calculation, the measuring method is equivalent to the fact that the gear is still in the calculating process, and the measuring head rotates around the axis of the gear at a constant speed.
As shown in fig. 1, a conventional optical gear measuring table is taken as an example. The measuring table comprises a displacement platform, a rotary platform, a displacement control PC, a line structure light sensor and a measuring head parameter control PC. The linear structure light sensor is connected with the displacement platform, and the PC can be used for controlling X, Y, Z to move in three axial directions. The sensor is horizontally arranged, so that the plane of the line-structured light is parallel to the axis of the gear. The gear is horizontally placed on the rotary platform, and the C shaft can be controlled to rotate by the PC. The linear structure measures the positional relationship of the tooth surfaces as shown in fig. 2, the plane of the light is parallel to the gear axis, and the light path covers all the tooth sides.
FIG. 3 is a cross section of the measuring device at the lower end face of the gear, establishing a gear coordinate system Og-XgYgOptical path coordinate system OL-XLYL(ii) a Wherein, OLThe point is the initial position of the measuring head, and the light ray is from OLStarting to XLThe negative axis direction; the distance between the measuring head and the tooth center is set as D; xgThe included angle between the shaft and the connecting line of the measuring head and the gear center is sigma1And XLAngle of axis being sigma2(ii) a Then O isLPoint-on-gear coordinate system Og-xgygThe coordinates in (d) can be expressed as (Dcos σ @1,Dsinσ1) The initial slope of the optical path is k ═ tan σ2；
The equation for the initial position of the optical path can be expressed as:
y-Dsinσ1＝-tanσ2(x-Dcosσ1) (1)；
O′L-X′LY′Lis O after a rotation time tL-XLYLThe position is defined as the position with the rotation speed of omega and the rotation angle of W-omega; then O'LPoint-on-gear coordinate system Og-xgygThe coordinate in (D cos (W + σ) is1),D sin(W+σ1) The slope of the optical path is k' ═ -tan (σ)2-W)；
The equation of the optical path after the rotation time t can be expressed as:
y-D sin(W+σ1)＝-tan(σ2-W)[x-D cos(W+σ1)] (2)；
the equation for the optical path plane can be expressed as:
wherein B is the tooth width, tbIs the tooth width factor;
w2. three-dimensional mathematical model of standard involute gear:
the tooth surface equation of the involute gear can be derived from the tooth surface profile equation. The end tooth profile consists of the following parts: the AB section is a transition curve formed by cutting a tool fillet into a base circle, and the BC section is an involute generated by conjugate of generating motion of the tool and a workpiece; the other parts are tooth top circular arcs and tooth root circular arcs.
The tooth profile equation of the straight gear is deduced, and the tooth profile equation is directly written without being described in detail.
The coordinate of any point k on the tooth profile involute is as follows:
pressure angle alpha with independent variable of k pointkThe value range is (alpha)B，αC) The values are:
the method for taking the plus or minus sign in the formula is to take an upper end sign when the right side tooth profile of the y axis is calculated and calculate a lower end sign of the left side tooth profile.
The transition curve tooth profile equation is as follows:
the value range of alpha' is
When alpha' is taken, the transition curve is connected with the involute and is tangent to the point B; get
The transition curve is tangent to the dedendum circle at the point A. The method for taking the plus or minus sign in the formula is to take an upper end sign when the right side tooth profile of the y axis is calculated and calculate a lower end sign of the left side tooth profile.
The tooth form of the end surface of the helical gear is similar to that of a straight gear, but each parameter is no longer a standard value. The section perpendicular to the tangent line of the spiral line on the indexing cylinder of the bevel gear is called as a normal surface, and because the gear is cut along the spiral line direction, the tooth profile parameters of the normal surface are consistent with those of the cutter and are standard values. However, since the end face parameters are generally used for geometric calculation of the helical gear model, the end face parameters and the normal face parameters are converted as shown in table 1.
TABLE 1 calculation formula of bevel gear end face parameters
| Bevel gear end face parameters | (symbol) | Formula (II) |
| Modulus of elasticity | mt | mt＝mn/cosβ |
| Reference circle pressure angle | αt | αt＝arctan(tanαn/cosβ) |
| Radius of reference circle | rt | rt＝mtz＝(mn/cosβ)z |
| Radius of base circle | rbt | rbt＝rtcosαt |
| Coefficient of tooth crest height | hat | hat＝hancosβ |
| Coefficient of tooth tip clearance | ct | ct＝cncosβ |
| Radius of addendum circle | rat | rat＝rt+hatmt |
| Root circle radius | rft | rat＝rt-(hat+ct)mt |
In the table, subscripts "n", "t" denote a normal plane and an end plane, respectively, and β is a pitch angle on the pitch circle.
Calculating the cutter parameters of the end face of the bevel gear as follows:
a tooth surface equation can be deduced by a generating mode of a spiral surface and an end surface tooth profile equation, as shown in figure 4, theta represents the angle of a bus rotating around a rotating shaft, right-hand rotation is positive, left-hand rotation is negative, delta represents an initial included angle between an Yw axis of a fixed coordinate system Ow and a tooth thickness symmetrical line of a measured gear,
the equation for the involute helicoid is:
the flank equation for the root transition curve is:
the tooth surface equation of the addendum circle is:
the tooth surface equation of the root circle is:
ωAcan be taken from the value of alpha' in formula (5)
Calculating to obtain;
in the above four equations, B is the tooth width, tbThe increment of the tooth width (v is more than or equal to 0 and less than or equal to 1.0); theta denotes an angle through which the bus bar rotates around the rotation axis,
theta is positive when the gear rotates rightwards, and theta is negative when the gear rotates leftwards; delta represents an angle for rotating the Yw axis of the fixed coordinate system Ow to the tooth thickness symmetrical line of the measured gear, and clockwise is positive and anticlockwise is negative;
when any tooth surface is worked out, the delta values in the above four groups of equations can be used
Substituting to obtain; the meanings and the value ranges of other variables in the formula are the same as those in the corresponding end face tooth profile equation, and only the end face parameters are required to replace the normal face parameters; in the formula, the method for taking +/-and +/-m' is also the same as the equation of the tooth profile of the end face;
w3. calculating the tooth surface distribution point cloud:
the intersection line of the light path and the tooth surface is a complex curve, so that the z-axis coordinate of each sampling point is firstly determined when the point cloud of the intersection point is calculated, and then the formula (3) and the formulas (4), (5), (6) and (7) are sequentially connected and solved according to the intersection sequence of the light path and each part of the tooth surface to obtain the point cloud of the tooth surface;
the number of equally spaced sampling points on the intersecting curve of the light path and the tooth surface is ntbThen the sampling interval is
The z-axis coordinate of the sampling point in each sampling is
Let the sampling frequency be f, and the tooth surface point cloud number measured after the rotation time t be Nt＝ωtfntb；
Because each tooth shape of the standard gear is the same, only the rotation angle needs to be calculated in the process of calculating the point cloud of the tooth surface
Namely, it is
And (3) carrying out point counting, rotating to obtain a tooth surface point cloud of another z-1 tooth, thus obtaining a complete tooth surface point cloud, wherein N is 2 pi fntbAnd (5) rotating to obtain the tooth surface point cloud of the other z-1 tooth, thus obtaining the complete tooth surface point cloud.
Example 1:
TABLE 2 helical gear parameters
TABLE 3 gauge head parameters
| Parameter(s) | Numerical value |
| Included angle sigma between initial position of measuring head and tooth center1/(°) | 10 |
| Initial angle of light path and xgAngle of axis sigma2/(°) | 20 |
| Distance d/mm between measuring head and |
50 |
| Gear speed omega (anticlockwise)/(rad/s) | π/2 |
| Sampling frequency f/Hz | 2000 |
| Number of sampling points n at a |
10 |
According to the input parameters, the complete tooth surface measurement needs 4 seconds in the actual measurement, and the minimum measurement time is
And second. Fig. 5 is a calculation result when t is set to be 0.6s or more and 0.9s or less, a red line is a light ray of a sampling point, 10 points are taken at equal intervals for each frequency on an intersecting curved surface of a light path and a tooth surface, that is, intersection points of 10 light rays and the tooth surface are calculated, blue points are intersection points of the light rays and the tooth surface, the total number of the intersection points is 6000, 4000 (that is, the number of intersection points of one tooth) of any continuous intersection points are taken, a tooth surface point cloud of the other z-1 tooth is calculated according to an included angle between two teeth, and 80000 complete tooth surface point clouds are finally obtained, as shown in fig. 6.
It will be understood by those skilled in the art that all or part of the steps in the method for implementing the above embodiments may be implemented by relevant hardware instructed by a program, and the program may be stored in a computer-readable storage medium, such as ROM/RAM, magnetic disk, optical disk, etc.
The above disclosure is only for the purpose of illustrating the preferred embodiments of the present invention, and it is therefore to be understood that the invention is not limited by the scope of the appended claims.
Claims (2)
1. A tooth surface point cloud theoretical distribution modeling method for measuring gear precision based on line structured light is characterized by comprising the following steps:
w1, modeling an optical path equation of linear structured light:
establishing a gear coordinate system Og-XgYgOptical path coordinate system OL-XLYL(ii) a Wherein, OLThe point is the initial position of the measuring head, and the light ray is from OLStarting to XLThe negative axis direction; the distance between the measuring head and the tooth center is set as D; xgThe included angle between the shaft and the connecting line of the measuring head and the gear center is sigma1And XLAngle of axis being sigma2(ii) a Then O isLPoint-on-gear coordinate system Og-xgygThe coordinates in (d) can be expressed as (Dcos σ @1,Dsinσ1) The initial slope of the optical path is k ═ tan σ2；
The equation for the initial position of the optical path can be expressed as:
y-Dsinσ1＝-tanσ2(x-Dcosσ1) (1)；
OL′-XL′YL' is O after a rotation time tL-XLYLThe position is defined as the position with the rotation speed of omega and the rotation angle of W-omega; then O isLPoint-on-gear coordinate system Og-xgygThe coordinate in (D) is (Dcos (W + σ)1),Dsin(W+σ1) The slope of the optical path is k' ═ -tan (σ)2-W)；
The equation of the optical path after the rotation time t can be expressed as:
y-Dsin(W+σ1)＝-tan(σ2-W)[x-Dcos(W+σ1)] (2)；
the equation for the optical path plane can be expressed as:
wherein B is the tooth width, tbIs the tooth width factor;
w2. three-dimensional mathematical model of standard involute gear:
a tooth surface equation can be deduced according to a generating mode of a spiral surface and an end surface tooth profile equation, theta represents the angle of the rotation of a bus around a rotating shaft, the right-hand rotation is positive, the left-hand rotation is negative, delta represents the initial included angle between the Yw axis of a fixed coordinate system Ow and the tooth thickness symmetrical line of the measured gear,
the equation for the involute helicoid is:
the flank equation for the root transition curve is:
the tooth surface equation of the addendum circle is:
the tooth surface equation of the root circle is:
ωAcan be taken from the value of alpha' in formula (5)
Calculating to obtain;
in the above four equations, B is the tooth width, tbV is more than or equal to 0 and less than or equal to 1.0; theta denotes an angle through which the bus bar rotates around the rotation axis,
theta is positive when the gear rotates rightwards, and theta is negative when the gear rotates leftwards; delta represents an angle for rotating the Yw axis of the fixed coordinate system Ow to the tooth thickness symmetrical line of the measured gear, and clockwise is positive and anticlockwise is negative;
when any tooth surface is worked out, the delta values in the above four groups of equations can be used
Substituting to obtain; the meanings and the value ranges of other variables in the formula are the same as those in the corresponding end face tooth profile equation, and only the end face parameters are required to replace the normal face parameters; in the formula, "+/-)"
The method is also the same as the equation of the tooth profile of the end face;
w3. calculating the tooth surface distribution point cloud:
the intersection line of the light path and the tooth surface is a complex curve, so that the z-axis coordinate of each sampling point is firstly determined when the point cloud of the intersection point is calculated, and then the formula (3) and the formulas (4), (5), (6) and (7) are sequentially connected and solved according to the intersection sequence of the light path and each part of the tooth surface to obtain the point cloud of the tooth surface;
the number of equally spaced sampling points on the intersecting curve of the light path and the tooth surface is ntbThen the sampling interval is
The z-axis coordinate of the sampling point in each sampling is
Let the sampling frequency be f, and the rotation time be measuredThe number of tooth surface point clouds is Nt＝ωtfntb；
Because each tooth shape of the standard gear is the same, only the rotation angle needs to be calculated in the process of calculating the point cloud of the tooth surface
Namely, it is
And (3) carrying out point counting, rotating to obtain a tooth surface point cloud of another z-1 tooth, thus obtaining a complete tooth surface point cloud, wherein N is 2 pi fntbAnd (5) rotating to obtain the tooth surface point cloud of the other z-1 tooth, thus obtaining the complete tooth surface point cloud.
2. A gear accuracy detection method using the tooth surface point cloud theoretical distribution modeling method for measuring gear accuracy based on line structured light according to claim 1, characterized by comprising the steps of:
the method comprises the following steps: the tooth surface point cloud theoretical distribution modeling method for measuring the gear precision based on the line structured light is adopted to establish a tooth surface point cloud theoretical distribution model according to claim 1;
step two: the tooth surface is measured through line structured light, and the specific process is as follows: the plane where the light is located is parallel to the axis of the gear, and the light path covers all the tooth sides; in the measuring process, the measuring head is fixed, the gear rotates for a circle at a constant speed, and the measuring head collects points on an intersection line segment of a light path and a tooth surface according to the frequency f to obtain a tooth surface point cloud;
step three: removing obvious noise points, which comprises the following specific steps: calculating the Euclidean distance between each actually measured point cloud and a theoretical point cloud in a tooth surface point cloud theoretical distribution model, wherein the difference value exceeding a threshold value is a noise point;
step four: the parameters which are completely the same when the method is used for actual measurement are input into a tooth surface point cloud theoretical distribution model for calculation, and the point cloud which is completely the same as the distribution of the actually measured point cloud can be obtained, so that the difference value between the actually measured point cloud and the theoretically distributed point cloud can be completely obtained point by point after the point cloud is registered, and the difference value can be approximately used as tooth profile deviation because each point and the corresponding point are completely positioned at the same z-axis height.
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Application publication date: 20210928 Assignee: Wenzhou Kehong Robot Co.,Ltd. Assignor: Wenzhou University Contract record no.: X2024330000007 Denomination of invention: A modeling method and measurement method for theoretical distribution of tooth surface point clouds Granted publication date: 20230103 License type: Common License Record date: 20240124 |