Method for determining a torque acting on a rotational device or a force acting on a rotational device

Abstract

A method for determining a force acting on a rotational device transverse to an axis of rotation of a rotor of the rotational device. The rotational device has a measuring system comprising a measurement body and detection devices for detecting a relative position of the detection device and the measurement body. The method comprises the following steps: producing a force which acts on the rotational device transverse to the axis of rotation of the rotor in a set rotational position, wherein the force causes a deflection of the rotor; determining the deflection of the rotor and/or a position error of the rotor in the set rotational position of the rotor from the relative position of the detection devices and the measurement body; and determining the force in the set rotational position of the rotor using a predetermined relationship between the force and the deflection and/or position error.

Claims

1. A method for ascertaining a torque and/or a force, which acts on a rotary apparatus of a coordinate measuring machine and which is directed across an axis of rotation of a rotor of the rotary apparatus, wherein the rotary apparatus comprises a measuring system, the latter comprising: a measuring body; and at least three detection devices for detecting a relative position of the detection devices in relation to the measuring body and/or for detecting a change in the relative position of the detection devices in relation to the measuring body; and wherein the method comprises the following steps: producing a torque and/or a force, which acts on the rotary apparatus and which is directed across the axis of rotation of the rotor of the rotary apparatus, in a set rotational position, wherein the torque and/or the force brings about a deflection of the rotor; ascertaining the deflection of the rotor and/or a position error of the rotor in the one set rotational position of the rotor, from the relative position of the at least three detection devices and of the measuring body; and ascertaining a value of the torque and/or the force at the one set rotational position of the rotor using a relationship, which is known or ascertained for the rotary apparatus and which is an assignment or a functional relationship, between i) torque and/or force, and ii) deflection and/or position error.

2. The method as claimed in claim 1, wherein the at least three detection devices comprise four detection devices.

3. The method as claimed in claim 2, wherein the four detection devices are arranged distributed about the axis of rotation, offset from one another by 80-110.

4. The method as claimed in claim 1, wherein the torque is produced by: positioning a mass on the rotor, wherein the center of gravity of the mass is eccentric in relation to the axis of rotation, and/or exerting a force on the rotor by contacting the rotor or a workpiece positioned on the rotor with a measuring system of the coordinate measuring machine.

5. The method as claimed in claim 1, wherein the method is used to ascertain a positioning of a mass on the rotor in which the torque and/or the force is substantially zero, wherein the mass is positioned on the rotor in the method in such a way that the deflection and/or the ascertained position error is substantially zero.

6. The method as claimed in claim 1, further including the step of outputting a warning and/or preventing a rotational movement of the rotor if a predetermined threshold of the torque and/or of the force is reached or exceeded.

7. The method as claimed in claim 1, wherein the relationship is obtained by: a) producing at least one torque and/or at least one force, which is directed across the axis of rotation of the rotor of the rotary apparatus, at at least one rotational position of the rotor; b) ascertaining a deflection and/or a position error of the rotary apparatus, which is caused by the at least one torque and/or the at least one force, at the at least one rotational position; and c)ascertaining a relationship between (i) torque and/or force and(ii) deflection and/or position error.

8. The method as claimed in claim 1, further comprising the steps of: ascertaining a phase angle of the center of gravity of a mass positioned on the rotor in a coordinate system of a stator of the rotary apparatus and/or in a coordinate system of the rotor from the deflection of the rotor or the position error of the rotor.

9. The method as claimed in claim 8, further comprising the steps of: ascertaining a deflection of the rotor in a first spatial direction and a deflection of the rotor in a second spatial direction, at a rotational position of the rotor, and ascertaining the phase angle of the center of gravity of the mass from the deflection in the first spatial direction and the deflection in the second spatial direction.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) The invention is described below on the basis of exemplary embodiments. In the figures:

(2) FIG. 1 shows a position error of a rotary table depending on the rotor rotational angle and the applied tilting moment;

(3) FIG. 2 shows the amplitude of an error of the fundamental wave (half maximum error margin) depending on the applied tilting moment;

(4) FIG. 3 shows the schematic plan view of a rotary apparatus with an applied mass;

(5) FIG. 4 shows an introduction of weight in the 0 direction in the case of a rotary apparatus with a standard;

(6) FIG. 5 shows an introduction of weight in the 90 direction in the case of a rotary apparatus with a standard;

(7) FIG. 6 shows a deflection of the rotor with an applied weight, depending on the rotational angle, measured in increments of the standards;

(8) FIGS. 7a and 7b show an alternative embodiment of a rotary apparatus for carrying out a method according to the invention, a rotary apparatus with measuring body and distance sensors;

(9) FIG. 8 shows an illustration of the atan2 relationship for determining the phase angle of a mass when four reading heads are used;

(10) FIG. 9 shows a procedure of a method for ascertaining a torque and/or a force;

(11) FIG. 10 shows a procedure of a method for ascertaining a phase angle of a center of gravity; and

(12) FIG. 11 shows an arrangement of standard and three detection devices.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

(13) 1. General Considerations.

(14) Tilting moments acting on the rotor of the rotary shaft cause a tilt of the rotor. Since a lever is usually present between rotational point (bearing site) and standard, the tilt causes an eccentricity of the standard. An eccentricity of the standard causes a so-called eccentric error in the measured angle positions. The position error is greatest if the displacement of the standard is perpendicular to the angle sensor or reading head, and may be calculated as follows:
=s/r
where =position error, s=eccentric amplitude r=radius of the standard

(15) Trials have shown that the relationship between the amplitude of the so-called eccentric error and the causative tilting moment is often good-natured and, for example, describable by a straight line or a polynomial. The trials were carried out using a rotary table mounted on an air bearing. The assumption is made that the results may also be transferred to rotary tables mounted on a rolling bearing. The results in FIGS. 1 and 2 were produced within the scope of a calibration process, by comparing the position values with a reference rotary table, as described below.

(16) 2. Ascertaining a Relationship Between Tilting Moment and Position Error.

(17) Below, ascertaining a relationship between tilting moment and position error is shown in an exemplary manner in points 1 and 2.

(18) 2.1. Standard CAA Data Recording Without Tilting Moment.

(19) For the purposes of measuring the position error without tilting moment, a setup was used as depicted schematically in FIG. 1 of the international patent application PCT/EP 2013/050328. This international patent application also describes a method for ascertaining the rotational position error of a rotary table with the aid of a reference rotary table in detail, in particular in the example part starting on page 81.

(20) 2.2. Computer Aided Accuracy (CAA) Data Recording With a Tilting Moment. Use is made of the same trial setup as in point 1 or in FIG. 1 of PCT/EP 2013/050328. The rotary table had bearing monitoring, by means of which an applied tilting moment may be ascertained. The bearing of the rotary table has a sensor system described in an exemplary manner in the patent application DE 10 2008 058 198 A1. Various angle positions of the rotary table were set in the case of different masses applied to the rotary disk of the rotary table to be measured and tilting moments emerging from the weight thereof. The masses were positioned on the rotor rotary disk in an eccentric fashion in relation to the axis of rotation. The tilting moment and the phase between the zero position of the stator coordinate system of the rotary table and the applied mass were ascertained or recorded from monitoring the bearing of the rotary table to be measured (test object). Here, 12 data records a)-f) with different, increasing tilting moments and, if a tilting moment is present, with the same phase were created. FIG. 1 shows the arising error pattern.

(21) All position error curves shown in FIG. 1 may be described by a cosine function in the radian measure. The relationship between position error and tilting moment is
position error=M*c*cos (t+pMpRH)
where M=current tilting moment in the rotary table-stator coordinate system [Nm] c=scaling factor position error per newton-meter in [Rad/Nm] t=rotary table position in the rotary table-stator coordinate system [Rad] pM=phase angle of the applied mass and the tilting moment produced thereby in the rotary table-stator coordinate system [Rad], relative to the set zero point thereof pRH=reading head position [Rad] in the rotary table-stator coordinate system relative to the set zero point of the rotary table-stator coordinate system.

(22) By way of the variables pM and pRH, it is possible to map various positions of both a mass (which produces a tilting moment) and a detection device, in this case a reading head of the angle encoder, in the rotary table-stator coordinate system, related to a zero point of the coordinate system. As a result of there being a linear relationship between tilting moment and amplitude, as shown in FIG. 2, the tilting moment M with a scaling factor c may be placed in front of the cosine function. The linear relationship is given in this example, but need not be present in other rotary tables. An influence of the tilting moment on an angle measuring system with integral bearing was examined and could be demonstrated. The tilting moment influence or the caused error is smaller than in the case of the system without an integral bearing, as examined above, by one order of magnitude and nonlinear. In this case, the influence of the tilting moment on the amplitude may be described by a polynomial.

(23) The cosine function illustrated above is obtained when the standard of the angle encoder, for example a graduated disk guided along a detection device (here a reading head), circulates. In FIG. 1, all position error curves are set to the position error of 0 at the rotary table position 0, which is an arbitrary setting and serves the purposes of a comparison in this case.

(24) FIG. 2 shows the amplitude of the fundamental wave of the deflection of the rotor or of the displacement of the angle measuring system relative to the reading head depending on the applied tilting moment. The amplitude of the fundamental wave of FIG. 2 is also ascertainable from FIG. 1 if the maximum of the curves shown therein is halved. In addition to the amplitudes of the measurements from FIG. 1, the plot of FIG. 2 also includes amplitudes of further measurements. The evaluation yields a very good linear relationship between tilting moment and amplitude of the fundamental wave of the position error.

(25) The error measurements in FIG. 1, with tilting moment load, have already been corrected by the error of the rotary table without a load (see point 2.1 above, CAA data recording without tilting moment), i.e. this error has already been removed by calculation. Hence, FIG. 1 only shows the errors additionally caused by the tilting moment.

(26) 3. Setup of a Rotary Apparatus Comprising a Standard and a Detection Device.

(27) FIG. 3 shows the rotary apparatus 1 with the rotor 2, in this case as a rotary table with the rotor in the form of a rotary disk 2. The mass m has been placed onto the rotary disk 2 at the rotational position of zero in the stator coordinate system. In the equation above, pM would be zero in this case. The mass m is positioned at the distance r (distance of the center of gravity) from the axis of rotation D. The axis of rotation D is perpendicular to the plane of the drawing.

(28) The observer views the rotary apparatus 1 from above. A weight F=m*g acts on the mass m, said weight acting downward, into the plane of the drawing, in the direction of view of the observer. Here, the tilting moment M acting perpendicular to the axis of rotation D results as a product M=rF.

(29) Further, FIG. 3 depicts the standard 3 in the form of a graduated disk and the detection devices 4a, 4b, 4c, and 4d (which are also referred to as detector or reading head in the example).

(30) The detection devices 4a, 4b, 4c, and 4d are each offset by 90 from one another. Two pairs of detection devices 4a/4c and 4b/4d are formed, with the detection devices in a pair lying opposite one another, i.e. being offset by 180 from one another.

(31) In the equation above pRH is 90 or /2, relative to the zero position in the stator coordinate system, at the reading head 4a in this setup. The standard 3 is mounted together with the rotary disk/rotor 2 and tilted together with the latter. A deflection as a result of the weight of the mass m on the rotor has the effect of a translation of the standard 3 relative to the detector 4a and 4c at the location of the detector 4a and 4c, from which a position error results. No translation of the standard 3 is detected relative to the detector 4b and 4d in the detection devices 4b and 4d in the shown position of the mass, but this changes depending on the position of the mass (see the explanations in respect of FIGS. 4 and 5). If the phase angle of the mass is zero in the stator coordinate system, as shown here, or if the phase angle is 180 () (mass m at 12 o'clock), the standard 3 is deflected to the greatest extent relative to the detectors 4a and 4c, and the position error is at a maximum here, wherein the error at 180 assumes the respective opposite sign of the two detectors 4a and 4c since the standard is deflected precisely in the opposite direction relative to the detector (upward instead of downward, as in the case of the phase angle of zero, in the shown view). As a result of the maximums at zero=>=>2 emerging from the set zero point and detector position in this configuration, the position error may advantageously be described by the cosine function shown above.

(32) In a method for ascertaining an assignment between a torque M which is across, in particular perpendicular to, an axis of rotation D of the rotary apparatus 1, the mass may be a test mass. The mass m may be the mass of a workpiece W in a method for operating a CMM.

(33) 4. Ascertaining a Relationship Between Tilting Moment and Deflection; Ascertaining a Tilting Moment from the Relationship.

(34) Initially, FIG. 9 highlights a procedure of a method for ascertaining a torque and/or a force, which acts on a rotary apparatus and which is directed across an axis of rotation of a rotor of the rotary apparatus. Step S1 is producing a torque and/or a force, which acts on the rotary apparatus and which is directed across the axis of rotation of the rotor of the rotary apparatus. Step S2 is ascertaining a deflection of the rotor and/or a position error of the rotor at a rotational position of the rotor. Step S3 is ascertaining the torque and/or the force at the at least one rotational position of the rotor, using a relationship, known or ascertained for the rotary apparatus, between torque and/or force, and deflection and/or position error.

(35) The amplitude of the fundamental wave produced by the tilting moment may be ascertained as follows: Assumption: the measuring system (scale) is ideally without errors. By way of example, this may be achieved by a calibration of the rotational position measuring system with a computational correction. Alternatively, the amplitude of the fundamental wave may be detected by one or more known tilting moments for each angle position and corrected by computation. This corresponds to the procedure of FIG. 1, wherein the tilting moments must be introduced from different directions for the data record. An optionally rotational-angle-dependent scaling factor c, by means of which a conversion may be carried out from the amplitude of the eccentric error to the current tilting moment, emerges. The rotational-angle-dependence of the scaling factor is qualified as follows: The resilience in relation to a tilting moment possibly depends on the rotational position of the rotor. Thus, c is not necessarily constant over the entire considered range of movement. This often applies to rotary tables having an internal three-point bearing. These are usually resilient in a non-uniform manner.
The explanations in relation to FIGS. 4 and 5 relate to the production of a torque by a mass and the ascertainment of a deflection of the rotor in accordance with step S1 and S2 of the procedure from FIG. 9. In comparison with FIG. 3, only two of the total of four detection devices are considered in FIGS. 4 and 5 and the setup is depicted schematically therein. FIGS. 4 and 5 consider the pair of detection devices 4b, 4d, wherein the setup is twisted by 180 (4d is at the bottom in FIG. 3 and at the top in FIGS. 4 and 5, at 0 on the angle scale of the stator). The explanations apply in an analogous fashion to a further pair of detection devices 4a, 4c, which is offset in relation to the pair 4b, 4d by 90, naturally with an appropriately adapted selection of the rotor position and phase position of the mass.

(36) In the following example, the reading heads 4b, 4d of angle measuring systems are considered to be counters. FIGS. 4 and 5 show selected angle values of the angle scale of the stator of the rotary table. The assumption is made that the rotor 2 of the rotary table stands at 100 on this angle scale. The zero mark of the rotor 2 is shown using a thick line at 100. In the unloaded case, both reading heads 4b, 4b would indicate e.g. 1000 increments (inc).

(37) A tilting moment about the X-axis arises if, as shown in FIG. 4, a weight is introduced by the mass m as a result of an eccentric workpiece mass oriented at 0. The rotor 2 tilts and this tilt emerges as a deflection or translation ty in the Y-direction in the plane of the angle measuring system. In the case shown in FIG. 4, the displacement of the scale occurs in a non-sensitive direction of the reading heads 4b, 4d. Thus, the displacement is not registered by the reading heads 4b, 4d; both reading heads continue to show 1000 increments (inc) as a position value.

(38) In FIG. 5, the rotor 2 of the rotary table is still situated at the 100 position. The weights as a result of the mass m are now introduced from the 90 direction. As a result of this, a tilting moment arises about the Y-axis. The rotor tilts and this tilt emerges as a translation tx in the X-direction in the plane of the angle measuring system. In FIG. 5, the displacement of the scale occurs in the sensitive direction of the reading heads 4b, 4d. The displacement which is identical in terms of absolute value in FIG. 4 and FIG. 5, tx corresponds to ty in terms of absolute value, is therefore registered in the entirety thereof by the reading heads 4b, 4d. Here, on account of the opposite arrangement, the reading heads 4b, 4d count in the opposite direction such that, for example, the one reading head 4d indicates 900 increments (inc) and the other reading head 4b indicates 1100 increments (inc).

(39) The setups in FIGS. 4 and 5 have a standard which is analogous to the setup in FIG. 3, with the standard being scanned by the detection devices 4b, 4d in FIGS. 4 and 5.

(40) At the top, FIG. 6 shows the absolute values of the deflection of the rotor 2, measured in increments of the standards for the detection devices 4b, 4d, depending on the direction (angle) of the introduced force.

(41) The lower curve in FIG. 6 shows the resultant deflection after combining the values by calculation. Using the example of the value at 90 from FIG. 5, a value of 900 increments is obtained at detection device 4d and a value of 1100 increments is obtained at detection device 4b, and so the resultant deflection at each detection device is ascertained to be 100.

(42) Absolute value and direction of the torque:

(43) In the case of an isolated treatment of two reading heads lying opposite one another in a pairwise manner, the vector component of the tilting moment in one spatial direction and the sign thereof may be calculated as follows: M=c*(x2x1)/2.

(44) Here, c is a scaling factor between the amplitude of the eccentric error and x1 and x2 are the readings of the two reading heads 4b, 4d.

(45) TABLE-US-00001 Force from Torque Direction of Direction and sign direction: about the torque of the displacement 0 X ty 90 Y +tx 180 X + +ty 270 Y + tx

(46) *Using 2 reading heads it is only possible to determine the vector component of the displacement in one spatial direction, i.e. of tx or ty. The reading heads need not be oriented along a main direction and need not lie exactly opposite one another either. In any case, it is necessary to know the angle between the reading heads.

(47) Proceeding from the treatment of two reading heads, as described above, use is made of at least three reading heads in the present invention instead of only one or two.

(48) A variant with four reading heads 4a-d is shown in FIG. 3 and described below. A variant with three reading heads is shown in FIG. 11 and described at the end of this example part.

(49) If four regularly arranged, i.e. 90 offset, reading heads are used, the absolute value of the resultant total torque may be calculated as
Mv=sqrt(Mx{circumflex over ()}2+My{circumflex over ()}2).

(50) No rotational movement is required. The tilting moment may be calculated immediately in every position. Thus, the treatment of one rotational position is sufficient and there is no need to set a further rotational position by rotating the rotor. This calculation of the torque corresponds to step S3 of the procedure from FIG. 9.

(51) FIG. 7 shows a rotary apparatus, in which use is made of a measuring body 95 and distance sensors 64a, 94a, 94b, which detect a distance from the measuring body 95, instead of angle sensors or reading heads oriented towards a standard.

(52) FIG. 7a shows a combination of two different measuring systems or partial measuring systems in a rotary apparatus. The rotary apparatus 50 comprises a stator 53, a pivot bearing means 44 and a rotor 51.

(53) Measuring body 95, which is a measuring body within the meaning of the present invention, is attached to the rotor by way of a downwardly projecting rod-shaped carrier 73. This measuring body 95 comprises a cylindrical disk, a first sensor 64a for ascertaining the radial relative position between the cylinder disk 95 and the stator 53 being aligned on the external edge of said cylindrical disk extending in the circumferential direction. Further, two sensors 94a, 94b which are aligned in the axial direction, i.e. parallel to the direction of the axis of rotation D, onto a plane surface of the cylinder disk 95 are connected to the stator 53.

(54) A tilting moment onto the rotor 51 is produced if a mass m, for example of a workpiece, the center of gravity of which lies eccentric to the axis of rotation D, is applied. If the center of gravity of the mass m lies e.g. in the plane of the drawing, there is a rotational deflection of the rotor 51 and of the measuring body 95 about a rotational axis which is across the axis of rotation D of the rotor, in this example about the axis Q which is perpendicular to the plane of the drawing. This deflection corresponds to a tilt of the measuring body 95 about the axis Q.

(55) The distance between the sensor 94b and measuring body 95 is reduced as a result of the tilt. The distance between the sensor 94a and measuring body 95 and the distance between the sensor 64a and measuring body 95 are increased.

(56) More distance sensors than the three sensors 64a, 94a, 94b shown in an exemplary manner may be present. FIG. 7b shows a view from below onto the measuring body 95 along the axis of rotation D. Further radially aligned sensors 64b, 64c, 64d and further axial sensors 94c and 94d are shown along the external circumference of the measuring body 95.

(57) Further, a rotational position measuring system comprising the standard 75 in the form of a graduated disk and the angle sensors 74a and 74b is shown in the rotary apparatus 50 from FIG. 7a. Here, this rotational position measuring system is not used for the method according to the invention for ascertaining the deflection of the rotor 51, as occurs, for example, using a setup according to FIG. 3 or FIG. 11. Only two angle sensors 74a and 74b are used in this case. However, it would also be possible to provide a third angle sensor or even a fourth angle sensor in order also to use the rotational position measuring system for ascertaining a deflection of the rotor, with redundant systems being present in this case as a measuring body 95 and distance sensors 64 and 94 are also provided.

(58) 5. Ascertaining the Phase Angle of the Center of Gravity of a Mass on the Rotor.

(59) Initially, FIG. 10 shows the procedure of a method for ascertaining the phase angle of the center of gravity of a mass on a rotor of a rotary apparatus in a coordinate system of a stator of the rotary apparatus and/or in a coordinate system of the rotor. Step S1.0 is positioning a mass on a rotor of the rotary apparatus. Step S1.1 is ascertaining a deflection of the rotor at at least one rotational position of the rotor, using a rotational position measuring system of the rotary apparatus comprising a standard and a detection device. Here, the deflection of the rotor is ascertained from the relative position of detection device and standard and/or from the change in the relative position of detection device and standard. Step S1.1 was already explained above on the basis of step S2 from FIG. 9 (see point 4 above in this respect). Step S1.2 is ascertaining the phase angle of the center of gravity of the mass in a coordinate system of a stator of the rotary apparatus and/or in a coordinate system of the rotor from the deflection of the rotor.

(60) Special embodiment variants are described below.

(61) If use is made of four reading heads, the phase of the torque, in step S1.2 of the shown method procedure, may be calculated from the individual torques using
P=a tan 2(Mx/My)*180/pi.

(62) Here, the sign of the torque acting in the Y-direction(=the torque about the X-axis) must be rotated in order to arrive at the position of the center of gravity from the torque (see FIG. 8).

(63) If the position of the center of gravity is intended to be specified in the rotor coordinate system, the phase p must still be combined with the rotational angle by calculation.

(64) The following example shows the calculation of the translation of an error-free rotational angle measuring system on the basis of three reading heads.

(65) FIG. 11 shows the setup for this example. A plurality of reading heads LK1, LK2, LKn, but at least the three reading heads shown here, are arranged around an error-free or already calibrated standard 110 at the angles 1 . . . m in relation to the global coordinate system. The standard 110 is coupled to the rotor of a rotary apparatus (not shown here) and is stationary, i.e. relatively immovable in relation to the rotor. The provided axis of rotation D of the rotor, and hence the standard 110, is perpendicular to the plane of the drawing and passes through the plane of the drawing where indicated by the arrow proceeding from the reference sign D. The standard 110 is likewise deflected as a result of a deflection of the rotor, to be precise into the position 110. During the deflection, the standard and the rotor are rotated about an axis which extends parallel to the plane of the drawing (not in the plane of the drawing) and across D.

(66) The deflection which, in principle, is a rotational movement is detected at the reading heads LK.sub.1, LK.sub.2, LK.sub.n as a translation. The observer of FIG. 11 also sees the movement of the standard 110 into the position 110 in the two-dimensional plane of the drawing as a translation. This detected translation of the standard 110 into the position 110 is described by the absolute value s and the angle .sub.M in relation to the global coordinate system, which is plotted top right in FIG. 11.

(67) In the complex plane, depicted in FIG. 11 by the real axis (Re) and the imaginary axis (Im), the movement of the standard may be described by
M()=s().Math.e.sup.(j.Math.M())(1)
depending on the rotational angle of the standard. The angle positions of the reading heads in the complex plane may be expressed by the relationship
E.sub.n=e.sup.(j.Math.n)(2)

(68) The movement s of the standard results in a different tangential translation component for each reading head. It may be calculated by

(69) $\begin{array}{cc}\begin{array}{c}{s}_{t,n}\left(\right)=\left(\frac{M}{E_n}\right)=\left(s\left(\right).Math.\frac{e^{\left(j.Math.{}_M\left(\right)\right)}}{e^{\left(j.Math.{}_n\right)}}\right)=\\ \left(s.Math.e^{\left(j.Math.\left({}_M\left(\right)-{}_n\right)\right)}\right)\\ =\mathrm{.Math.}\\ =s.Math.\sin \left({}_M\left(\right)-{}_n\right)\end{array}& \left(3\right)\end{array}$
for each reading head. As a result of this, the incorrect count of the reading heads resulting from this translation then emerges as

(70) $\begin{array}{cc}{}_n\left(\right)=\arctan \left(\frac{s_{t,n}\left(\right)}{R}\right),& \left(4\right)\end{array}$
where R denotes the distance of the measuring point of the reading head from the ideal axis of rotation of the standard. Since arctan(x)x applies for very small angles, the preceding equation may be rewritten as

(71) $\begin{array}{cc}{}_n\left(\right)=\frac{s_{t,n}\left(\right)}{R}=\frac{s\left(\right).Math.\sin \left({}_M\left(\right)-{}_n\right)}{R}& \left(5\right)\end{array}$
If all m installed reading heads were referenced to the same reference mark of the calibrated or gradation-fault-free scale, the angle position {tilde over ()}.sub.n represented by the n-th reading head is composed of the actual angle rotation of the standard, the assembly angle n of the reading head and the incorrect count .sub.n caused by the translation:
{tilde over ()}.sub.n=+.sub.n+.sub.n(6)
The angle difference {tilde over ()}.sub.k,l(), which may be measured between the k-th and l-th reading head, can be calculated by
{tilde over ()}.sub.k,l()={tilde over ()}.sub.k{tilde over ()}.sub.l=+.sub.k.sub.l+.sub.k().sub.l() for kl.(7)
Since the angle positions .sub.1 . . . .sub.m are known, the differences .sub.k.sub.l thereof are also known. Therefore, the measured angle difference {tilde over ()}.sub.k,l() may be rewritten into an offset-corrected angle difference .sub.k,l() as
.sub.k,l()={tilde over ()}.sub.k,l()(.sub.k.sub.l)=.sub.k().sub.l() for kl.(8)
If .sub.k and .sub.l in equation (8) are now replaced by the values of equation (5), the offset-corrected angle differences .sub.k,l emerge as

(72) $\begin{array}{cc}\begin{array}{c}{}_{k,l}\left(\right)=\frac{s\left(\right).Math.\sin \left({}_M\left(\right)-{}_k\right)}{R}-\frac{s\left(\right).Math.\sin \left({}_M\left(\right)-{}_l\right)}{R}\\ \frac{s\left(\right)}{R}.Math.\left(\sin \left({}_M\left(\right)-{}_k\right)-\sin \left({}_M\left(\right)-{}_l\right)\right)\end{array}& \left(9\right)\end{array}$
Using the addition the orems sin(xy)=sin x cos ysin y cos x of the trigonometric functions, equation (9) emerges as

(73) $\begin{array}{cc}\begin{array}{c}{}_{k,l}\left(\right)=\frac{s\left(\right)}{R}.Math.\stackrel{\stackrel{\sin \left({}_M\left(\right)-{}_k\right)}{}}{\begin{array}{c}(\sin \left({}_M\left(\right)\right).Math.\cos \left({}_k\right)-\\ \sin \left({}_k\right).Math.\cos \left({}_M\left(\right)\right)\end{array}}.Math.\\ \stackrel{\stackrel{-\sin \left({}_M\left(\right)-{}_l\right)}{}}{\begin{array}{c}-\sin \left({}_M\left(\right)\right).Math.\cos \left({}_l\right)+\\ \sin \left({}_l\right).Math.\cos \left({}_M\left(\right)\right))\end{array}}\\ =\frac{s\left(\right)}{R}.Math.(\sin \left({}_M\left(\right)\right).Math.\cos \left({}_k-\cos {}_l\right)+\\ \cos \left({}_M\left(\right)\right).Math.\sin \left({}_l-\sin {}_k\right))\end{array}& \left(10\right)\end{array}$
In end effect, equation (10) only consists of two unknowns: s() and .sub.M(). .sub.k,l() is known from the measurement (difference measurement of the reading head signals) and .sub.1 . . . .sub.m are known by forming an average (see Geckeler's equation (7)). It is also clear from equation (10) that the translation cannot be calculated uniquely using only two reading heads. However, if m3 reading heads are installed on the standard,
.sub.i=1.sup.m1m
possible instances of forming the difference emerge according to equations (7) and (8) and hence also correspondingly many different equations according to formula (10). Hence, the system of equations is uniquely solvable for each rotational angle .
Calculation example.

(74) The translation back calculation for a specific rotational angle should be shown below on the basis of an example.

(75) Given:

(76) Translation: M=45, s=0.00001 m three reading heads: .sub.1=0, .sub.2=74, and 3=132.85 R=0.075m
Measured angle differences .sub.2,1=1.5892.Math.10.sup.4 .sub.3,1=2.2752.Math.10.sup.4 .sub.3,2=6.859.Math.10.sup.5
Calculated differences of the angle values: K2,1:=cos 2cos 1=0.724 K3,1:=cos 3cos 1=1.6801 K3,2:=cos 3cos 2=0.9552 S2,1:=sin 1sin 2=0.9613 S3,1:=sin 1sin 3=0.7331 S3,2:=sin 2sin 3=0.2281
Initially, equation (10) is rewritten for s for the cases k=2 and l=1, and k=3 and l=1, and these cases are equated:

(77) $\begin{array}{cc}\frac{{}_{2,1}.Math.R}{\sin \left({}_{M,\mathrm{rueck}}\right).Math.K_{2,1}+\cos \left({}_{M,\mathrm{rueck}}\right).Math.S_{2,1}}=\frac{{}_{3,1}.Math.R}{\sin \left({}_{M,\mathrm{rueck}}\right).Math.K_{3,1}+\cos \left({}_{M,\mathrm{rueck}}\right).Math.S_{3,1}}{}_{2,1}.Math.\left(\sin \left({}_{M,\mathrm{rueck}}\right).Math.K_{3,1}+\cos \left({}_{M,\mathrm{rueck}}\right).Math.S_{3,1}\right)={}_{3,1}.Math.\left(\sin \left({}_{M,\mathrm{rueck}}\right).Math.K_{2,1}+\cos \left({}_{M,\mathrm{rueck}}\right).Math.S_{2,1}\right){}_{2,1}.Math.\sin \left({}_{M,\mathrm{rueck}}\right).Math.K_{3,1}+{}_{2,1}.Math.\cos \left({}_{M,\mathrm{rueck}}\right).Math.S_{3,1}={}_{3,1}.Math.\sin \left({}_{M,\mathrm{rueck}}\right).Math.K_{2,1}+{}_{3,1}.Math.\cos \left({}_{M,\mathrm{rueck}}\right).Math.S_{2,1}\sin \left({}_{M,\mathrm{rueck}}\right).Math.\left({}_{2,1}.Math.K_{3,1}-{}_{3,1}.Math.K_{2,1}\right)=\cos \left({}_{M,\mathrm{rueck}}\right).Math.\left({}_{3,1}.Math.S_{2,1}-{}_{2,1}.Math.S_{3.1}\right)\frac{\sin \left({}_{M,\mathrm{rueck}}\right)}{\cos \left({}_{M,\mathrm{rueck}}\right)}=\tan \left({}_{M,\mathrm{rueck}}\right)=\frac{{}_{3,1}.Math.S_{2,1}-{}_{2,1}.Math.S_{3,1}}{{}_{2,1}.Math.K_{3,1}-{}_{3,1}.Math.K_{2,1}}.Math.{}_{M,\mathrm{rueck}}=\arctan \left(\frac{{}_{3,1}.Math.S_{2,1}-{}_{2,1}.Math.S_{3,1}}{{}_{2,1}.Math.K_{3,1}-{}_{3,1}.Math.K_{2,1}}\right)& \left(11\right)\end{array}$
The aforementioned values are inserted into equation (11):

(78) $\begin{array}{cc}{}_{M,\mathrm{rueck}}=\arctan \left(\frac{\begin{array}{c}\left(-2.2752.Math.{10}^{-4}.Math.-0.9613\right)-\\ \left(1.5892.Math.{10}^{-4}.Math.-0.7331\right)\end{array}}{\begin{array}{c}\left(-1.5892.Math.{10}^{-4}.Math.-1.6801\right)-\\ \left(2.2752.Math.{10}^{-4}.Math.-0.724\right)\end{array}}\right){}_{M,\mathrm{rueck}}=\arctan \left(1\right)=\frac{}{4}\stackrel{.Math.}{=}45& \left(12\right)\end{array}$
Then, the angle .sub.M,rueck obtained thus may be inserted into equation (10) for either k=2 and l=1 or k=3 and l=1. Hence, the absolute value of the translation s emerges as

(79) $\begin{array}{cc}\begin{array}{c}{s}_{\mathrm{rueck}}=\frac{{}_{3,1}.Math.R}{\sin \left({}_{M,\mathrm{rueck}}\right).Math.K_{3,1}+\cos \left({}_{M,\mathrm{rueck}}\right).Math.S_{3,1}}\\ =\frac{-2.2752.Math.{10}^{-4}.Math.0.075m}{\sin \left(\frac{}{4}\right).Math.-16801+\cos \left(\frac{}{4}\right).Math.-0.7331}\\ =0.00001m\end{array}& \left(13\right)\end{array}$
It was possible to back calculate the translation in terms of absolute value and phase; the result corresponds to the predetermined values.
The more reading heads are able to be used during the back calculation, the more accurate the calculation should be in relation to influences such as noise in that case, since the back calculation may be carried out using a number of formulae. Here, the translation was only back calculated from an ideal signal.