DE10254778B3 - Surface topography determination method has surface topography reconstructed from combined values obtained from measured distances to surface provided by relatively spaced sensors - Google Patents

Abstract

The method uses a measuring head (14) for scanning the surface (12), provided with at least 3 sensors (1,2,3) in a fixed relationship to one another, for measuring respective distances (d1,d2,d3) to the surface, combined for each scanning step, with reconstruction of the surface topography from the combined values. An independent claim for a device for determining the surface topography of an object is also included.

Description

The The invention relates to a method for highly accurate determination of Topography of a surface, at the surface by method of a plurality of sensors for determining distances to surface scanning head is scanned.

The The invention further relates to a device for highly accurate determination the topography of a surface by means of a plurality of sensors for determining distances to surface having a measuring head, which can be moved to scan the surface is arranged.

It It is known to determine the topography of a surface Measuring head with at least one sensor substantially parallel to surface to proceed to the surface scanning with the measuring head (scanning). Go as geometric mistakes thereby distance error of the measuring head from the surface and Angle error during movement of the measuring head into the measurement.

It is known, measurements with two, in fixed spatial relationship to each other to perform arranged sensors and make a difference of the measured values, so positioning error to eliminate the measuring head. While first has been tried, the distance of the sensors small compared to the changes to choose the measured values are later mathematical methods have become known, which is a reconstruction also allow the topography when the distances of the Sensors (Shear) significantly larger than the resolution the measured value changes are. This procedure offers the advantage of being differentiated resulting measured values are large across from the inevitable noise of the sensors.

Other Known methods for determining the topography use angle measuring devices for Determination of the angle of a surface reflected by the surface to be determined Taststrahls, for example by means of autocollimation telescopes (AKF).

Out EP 0 174 168 A1 It is known to scan a three-dimensional elongated body prefetched on a belt with light sensors located above and to the side of the body to determine the respective distance of the sensor from the scanned surface of the body. From the data obtained, a computer calculates a cross-sectional area in order to determine the weight of the pre-conveyed body.

DE 198 33 207 A1 describes the generation of three-dimensional distance images of spatial objects, wherein a short-time exposure is performed with laser diodes. The backscattered by the object laser pulses are detected and evaluated in two integration windows with different integration times and by averaging over several laser pulses with a pixel-resolving opto-electronic sensor.

A Highly accurate determination of the topography of a surface in the sense The present invention is characterized by the two cited documents not possible.

Of the Invention is based on the object, the highly accurate determination the topography of a surface using sensors to determine the distance to the surface and at least the influence of position and angle errors to eliminate first and second order.

to solution This object is a method according to the invention the aforementioned Art characterized in that the measuring head has a number of n Sensors (n ≥ 3) which has a fixed spatial Are arranged relative to each other, and that each for a Scanning step recorded measured values of the sensors by difference formations to be combined with one another and from the combined ones Values the topography is reconstructed.

The inventive method thus represents a surprising Continuing known difference method, in which by Difference measurements instead of location coordinates slopes are determined. In the method according to the invention By evaluating and combining at least three, of the Sensors determined distance values by at least two differences a combined value is formed and from the combined values the Topography reconstructed.

to Facilitating the evaluation, it is advantageous if in the scanning direction successively arranged sensors are used.

It turns out that when using three sensors one clear evaluation of the measured values to determine the topography clearly then possible is, if at least one additional information is used to To exclude ambiguity.

The Clear evaluation is in a particularly preferred practical embodiment thereby possible an arrangement of at least five, aligned in the scan direction Sensors is used in which the sensors are pairwise symmetrical are arranged to a central sensor.

In a development of the invention can be a large number of sensors, for example more than 100 sensors, in the arrangement of an array or a Matrix are used, expediently equal distances between the sensors are manufactured.

to solution the above object is also a device of the type mentioned according to the invention thereby characterized in that the measuring head has a number of n sensors (n ≥ 3), the in solid spatial Relation to each other are arranged and for measuring value recording during each a scanning step are set up and that an evaluation device for combining the measured values determined by differences to one value per scan step and reconstruction of the topography from the combined values.

The Invention will be described below with reference to in the drawing schematically illustrated embodiments be explained in more detail. Show it:

1 a schematic representation of a measuring head with three sensors, which is moved by scanning over a surface to be measured

2 a schematic representation according to 1 a measuring head with five, in a defined manner to each other arranged sensors.

1 schematically shows an object 11 whose surface 12 should be determined. This should be done from a reference plane 13 the function of the distance from the reference plane 13 are determined as the topography t (x) as a function of the location x.

For this purpose, a measuring head 14 provided in a scanned in a double arrow scan direction S parallel to the reference plane 13 is moved. For this purpose, the measuring head 14 at a distance d from the reference plane 13 arranged, wherein an adjustment error d ₀ (x) is to be considered as a constant offset.

Another, exaggerated in the drawing error results from an angular deviation α _{0 of} the measuring head 14 from the parallels to the reference plane 13 ,

For a location x be through the three sensors 1 . 2 . 3 the distances d ₁ , d ₂ and d _{3 of} the sensors 1 . 2 . 3 from the surface 12 certainly. With the distance d ₁ , the value t (x ₁ ) adds to the distance of the sensor 1 from the reference plane 13 ,

1 lets recognize that applies d 1 (x) = d + d 0 (x) - t (x 1 ) + (x 1 - x) tanα 0 (x) (01) d 2 (x) = d + d 0 (x) - t (x 2 ) + (x 2 - x) tanα 0 (x) (02) d 3 (x) = d + d 0 (x) - t (x 3 ) + (x 3 - x) tanα 0 (x) (03) with the definition of difference values s 0 = x 1 - x (04) s 1 = x 2 - x 1 (05) s 2 = x 3 - x 1 (06) the coordinates in the scanning direction x ₁ , x ₂ , x ₃ result x 1 = s 0 + x (07) x 2 = s 0 + s 1 + x (08) x 3 = s 0 + s 2 + x (09).

A difference of the of the sensors 1 . 2 . 3 measured distance values, taking into account the equations (07) to (09): d 12 (x) = d 1 (x) - d 2 (x) (10) d 13 (x) = d 1 (x) - d 3 (x) (11) d 12 (x) = -t (s 0 + x) + t (s 0 + s 1 + x) - s 1 tanα 0 (x) (12) d 13 (x) = -t (s 0 + x) + t (s 0 + s 2 + x) - s 2 tanα 0 (x) (13) simple transformation (: s ₁ ) results d 12 (X) / s 1 = -T (s 0 + x) / s 1 + t (s 0 + s 1 + x) / s 1 - tanα 0 (x) (14) d 13 (X) / s 1 = -T (s 0 + x) / s 2 + t (s 0 + s 2 + x) / s 2 - tanα 0 (X).

A new difference leads to the coupled difference d 123 (x) = d 12 (x) - d 13 (x) (15). d 123 (x) = -t (s 0 + x) / s 1 + t (s 0 + s 1 + x) / s 1 + t (s 0 + x) / s 2 - t (s 0 + s 2 + x) / s 2

It can be seen from equation (15) that the influences of the offset d ₀ (x) and the angular error α ₀ (x) are omitted, so that these errors no longer enter the formed coupled difference d ₁₂₃ (x).

Here in is the essential starting point of the present invention.

The above consideration has been made only for the inclination angle α ₀ (x). On the one hand, the other angle influences (roll angle, yaw angle) have only a significantly minor significance; on the other hand, the same applies to these angle influences as to the angle of inclination, namely, the angle influences fall out due to the formation of the coupled difference d ₁₂₃ (x).

The solution to the mathematical problem of determining the topographical function t (x) from the coupled differences d ₁₂₃ (x) is achieved in a manner known per se by the transfer functions of the system. However, the transfer functions can not be used in a conventional manner for evaluation, because due to the present incomplete information on the differences of differences no Fourier transformation can be performed. It is also not possible to use the method of natural extension here, as is usual with other shearing methods. However, the evaluation succeeds with other evaluation strategies, for example evaluations using power series polynomials or Zernike polynomials, spline polynomials, Fourier polynomials etc., that is with so-called modal methods. After performing the evaluation, the transfer functions then provide complete information about the underlying transmission channel from the coupled distance differences to the final topographies.

durch Einsetzen von (15)

In a solution example, a transfer function of the system is indicated

by inserting (15)

By integration of the constant values s ₀ , s ₁ and s ₂ results

so dass sich ergibtThe remaining integral is defined as

so that results

Taking equation (18) into consideration, this results

In the general case of many sensors, the transfer function can be written in the following way:

Here, N is the number of shears s _j and R is the spatial frequency.

The doing transmission from an absolute coordinate system to an object-related one Coordinate system gives R = n / L a reference to the number n of the measured data and on the scan length L.

The Transfer function for the determined measurement data is thus too

With Help the transfer function according to Eq. 25 or 25a and later 52 and 52a is generally not a determination of the final topography possible, as was done before Equation 16. The reason is that, as is generally the case with differences with Shears, not over the entire range of the form or topography to be determined the differences measurable or known. Therefore, Fourier transformations can not easily be used which, indeed, the knowledge about the whole to be Fourier transforming Assume area. At earlier Applications could use the so-called natural continuation at this point which then the differences over the entire range of made available to the topography and thus also a Fourier transform applicable. As before mentioned, in the present case is a determination of a natural one Continuation basically not more is possible, thus the procedure with the help of the natural continuation not applied can be.

Much more Here methods are used that fundamentally research mathematically and are known and for it in the practical execution with the help of computer programs also already finished solutions of the mathematical problems in the form of purchasable libraries for various High-level languages (C, C ++, IDL, FORTRAN, etc.) exist (e.g., the IMSL library from Visual Numerics, United States of America). A successful Evaluation of the double differences can be carried out by that a mathematical model of the double differences from a parametric erected topographical function, its results be compared with the experimentally determined double differences. Thus, there is a quality measure for the match of the thus determined parametric model of the final topographical function at a point of topography. By doing so for all points above the Area of topography performed and optimization based on the sum of the least-sgares of the Deviations between model and experiment are carried out can determines the parameters of the parametrically defined topographical function become. This problem is a linear problem in matrix notation can be formulated, and this so formulated mathematical Problem can be with commercially available software libraries (see above) solved become. Corresponding mathematical solutions are available.

However, it turns out that when using three sensors 1 . 2 . 3 the topographical function t (x) to be reconstructed is not necessarily uniquely determinable. However, the unambiguous determination can be made using additional information, so that even with three sensors 1 . 2 . 3 a clear determination of the topography t (x) is possible. The additional information may also be provided by at least one further sensor 4 . 5 be won.

It shows that with a suitable arrangement of the sensors in the present with the shears weighted differences both a definite reconstruction the topography possible as well as one that provides noise propagation with a noise enhancement factor of about one, so that is the noise of the single sensor not reinforced reflected in the topography.

2 shows an embodiment of a measuring head 14 ' with five sensors 1 . 2 . 3 . 4 . 5 , For this results: d 1 (x) = d + d 0 (x) - t (x 1 ) + (x 1 - x) tanα 0 (x) (27) d 2 (x) = d + d 0 (x) - t (x 2 ) + (x 2 - x) tanα 0 (x) (28) d 3 (x) = d + d 0 (x) - t (x 3 ) + (x 3 - x) tanα 0 (x) (29) d 4 (x) = d + d 0 (x) - t (x 4 ) + (x 4 - x) tanα 0 (x) (30) d 5 (x) = d + d 0 (x) - t (x 5 ) + (x 5 - x) tanα 0 (x) (31).

The five sensors are in pairs symmetrical to the middle sensor 3 arranged so that the sensors 2 and 4 the same distance to the sensor 3 exhibit. The same applies to the sensors 1 and 5 , Thus: x 1 = s 0 - s 2 + x (32) x 2 = s 0 - s 1 + x (33) x 3 = s 0 + x (34) x 4 = s 0 + s 1 + x (35) x 5 = s 0 + s 2 + x (36) d 23 (x) = d 2 (x) - d 3 (x) (37).

After some conversions results: d 23 (x) = -t (s 0 - s 1 + x) + t (s 0 + x) - s 1 tanα 0 (x) (38).

In an analogous manner results for d 34 (x) = -t (s 0 + x) + t (s 0 + s 1 + x) - s 1 tanα 0 (x) (39)

The coupled difference for the sensors 2 . 3 . 4 results d 234 (x) = d 23 (x) - d 34 (x) (40) d 234 (x) = t (s 0 + x) - t (s 0 - s 1 + x) + t (s 0 + x) - t (s 0 + s 1 + x) (41) and in a similar way for the sensors 1 . 3 . 5 d 135 (x) = d 13 (x) - d 35 (x) (42) d 135 (x) = t (s 0 + x) -t (s 0 - s 2 + x) + t (s 0 + x) -t (s 0 + s 2 + x) (43).

In turn by applying the Fourier transform

ergibt sich:

By defining the transfer function

surrendered:

By inserting R = n / L, the transfer function for the sampled measurement data results

In the general case of many sensors, the transfer function can be written in the following way, since such a system can be realized with many double shears, in individual cases with up to 500 double shears:

Here, N is the number of double sears s _j and R is the spatial frequency.

at the evaluation with Fourier coefficients has to be considered, that this first are non-orthogonal, because information does not need to be Fourier Transformed throughout Interval exists. The solution but is known and possible with leastsquares optimization. There there is a linear problem, too, are fast matrix operations applicable.

There The sensors in the measuring head have to be mounted individually, can be constant Offsets regarding the relative arrangement of the sensors play a role to each other. these can determined and taken into account in the evaluation. The determination The offsets can be determined by direct measurement, but preferably by Calibration of the system with a known topography done.

When Sensors can in principle also mechanical buttons are used, however, regularly because of of influence on the surface not wanted are and above Be slow to use.

Prefers Therefore, optical buttons are used, which in different versions gives. It is possible not the distances but to measure the differences of distances, if interferometric Sensor systems are used. It is even conceivable, not only the individual differences but even the coupled differences over so-called Differential interferometer to measure, taking the different parts the beam paths by polarization optical aids from the information point of view can be separated.

In the described embodiments is the measuring head with a one-dimensional arrangement of the sensors used. This allows one-dimensional scans, the then be guided in different directions over the examinee, from which the topographical function for the surface is formed.

It is according to the invention also possible, to use the sensors as direct two-dimensional measuring sensors and to arrange in the form of an array or a matrix.

Claims (9)

Method for highly accurate determination of the topography of a surface ( 12 ), in which the surface ( 12 ) by moving a plurality of sensors ( 1 . 2 . 3 . 4 . 5 ) for determining distances (d ₁ , d ₂ , d ₃ , d ₄ , d ₅ ) to the surface ( 12 ) measuring head ( 14 . 14 ' ), characterized in that the measuring head ( 14 . 14 ' ) has a number of n sensors (n ≥ 3) which are arranged in a fixed spatial relation to one another and that the measured values of the sensors recorded in each case for one scanning step ( 1 . 2 . 3 . 4 . 5 ) are combined to form a value using difference formations and the topography (t (x)) is reconstructed from the combined values. A method according to claim 1, characterized in that in the scanning direction (S) successively arranged sensors ( 1 . 2 . 3 . 4 . 5 ) be used. Method according to claim 1 or 2, characterized that for the clear evaluation of the measured values and determination of the Topography from at least one additional information is used. A method according to claim 1, 2 or 3, characterized in that an arrangement of at least five, in the scanning direction (S) aligned sensors ( 1 . 2 . 3 . 4 . 5 ) is used, in which the sensors ( 1 . 2 . 3 . 4 . 5 ) in pairs symmetrically to a middle sensor ( 3 ) are arranged. Method according to one of claims 1 to 4, characterized that a big one Number n of sensors (n ≥ 100) is used in the arrangement of an array or a matrix. Device for highly accurate determination of the topography of a surface ( 12 ) by means of a plurality of sensors ( 1 . 2 . 3 . 4 . 5 ) for determining distances (d ₁ , d ₂ , d ₃ , d ₄ , d ₅ ) to the surface ( 12 ) measuring head ( 14 . 14 ' ), which is used to scan the surface ( 12 ) is arranged movable, characterized in that the measuring head ( 14 . 14 ' ) has a number of n sensors (n ≥ 3) which are arranged in fixed spatial relation to one another and are set up for measuring value acquisition during each scanning step, and that an evaluation device for combining the determined measured values by difference formations to a value per scanning step and reconstruction of the topography (t (x)) is provided from the combined values. Device according to claim 6, characterized in that the sensors ( 1 . 2 . 3 . 4 . 5 ) are arranged one behind the other in the scanning direction (S). Apparatus according to claim 7, characterized in that at least five aligned in the scanning direction (S) sensors ( 1 . 2 . 3 . 4 . 5 ) which are pairwise symmetrical to a middle sensor ( 3 ) are arranged. Apparatus according to claim 6, characterized by a big Number n of sensors (n ≥ 100) in the arrangement of an array or a matrix.