WO2016183618A1 - Means and method for 6 degrees of freedom displacement sensor - Google Patents

Abstract

A displacement sensor comprising at least 3 interferometers and a three dimensional mirror in which the three dimensional mirror is attached to a solid body and interferometer fixed components are independently attached to a solid body and displacement of a solid body to 6 DoF is derived from the displacement of the three dimensional mirror. The three dimensional mirror is mounted on one part of a solid body and the interferometer fixed components on another part of said solid body or a part of a different solid body so that movement of one part relative to the other can be measured to 6DoF. Three interferometers are preferably orthogonally arranged about a three dimensional mirror to derive displacement of the three dimensional mirror to 6 DoF. A method of measuring displacement of a solid body includes the steps of deriving the three 3 dimensional position vectors of the three dimensional mirror from the three 3 dimensional mirror normal angles and the three 3 dimensional mirror tilt axis angles obtained from the 3 interferograms, and the linear displacement and direction of linear displacement of the three dimensional mirror is derived from the direction of movement of the fringe lines across the 3 interferograms and the three dimensional mirror position vectors. A method of overcoming linear displacement errors in a first direction as a consequence of linear displacement in a second direction when using interferometers arranged about a three dimensional mirror to measure displacement to 6 degrees of freedom.

Description

Means and method for 6 degrees of freedom displacement sensor

This invention relates to a sensor for measuring displacement to 6 degrees of freedom using optical interferometry. More particularly, the invention relates to apparatus and means for measuring and recording the displacement of an object in a static or dynamic reference frame due to linear and/or angular displacements applied to the object to six degrees of freedom (6 DoF).

Background to the invention

One use of a displacement sensor is when a torque and/or force is applied to a solid body, the extent to which it is displaced in terms of angle and/or distance is called deflection and in the real world, deflection is 3 dimensional.

Sensors operating with six degrees of freedom usually do not use interferometers.

Multi-axis load cell sensors are readily available that measure deflection to 2 and 3 degrees of freedom (DoF) but commercially available sensors that can accurately measure deflection to 6 DoF as a single device are difficult to obtain and are expensive. In addition, they generally use strain gauge technology to reduce cost and physical size, which compromises sensitivity and accuracy, in comparison to other superior technologies such as optical interferometry.

Optical interferometry is generally more sensitive than the above mentioned solutions. It is based on detecting the interference pattern (interferogram) of two overlapping beams of light, usually from the same monochromatic source, and counting the number of instances of maximum constructive interference, known as fringes, which result from the deflection.

Another use of a displacement sensor is when an object has to be accurately positioned to a resolution of nanometres such as the head or the stage of a semiconductor photolithography apparatus.

Some systems with 6 DoF displacement measurement do use interferometers. USA patent 7948695 discloses an optical element positioning mechanism that uses interferometry with 6 light sources.

USA patent 6876453 uses 6 interferometers to derive 6DoF for measuring 3 D motion. WO 2007011970 discloses a measurement and servo control system that uses one 1 light source to project interference patterns onto multiple surfaces of the object and measuring the fringe pattern.

WO 2007106167 discloses pixel mapping interferometry using one light source but can only measure 1 DoF at a time.

USA 2015/0103356 discloses a compact interferometer able to measure up to 3 degrees of freedom.

A typical Michelson interferometer enables displacement to be accurately measured by detecting the phase difference between two monochromatic beams of light that overlap in space as well as direction. Because the measurement is based on the wavelength of light, displacement using this technology is in the order of

nanometres.

The interferometer has a laser, beam splitter and fixed mirror (collectively called the fixed components) that are mounted in a housing that fixes their relative positions to one another in space. The housing is mounted on a solid body that serves as a reference frame. A moving mirror is mounted to a solid body (can be the same or different to the former) and aligned with the fixed components. As a consequence of an applied action on one or more of the solid bodies, the moving mirror undergoes displacement relative to the fixed components.

Ironically, sensors capable of measuring displacement to 6 DoF using optical interferometry are abundant in academic literature but such devices are not readily available commercially. These sensors are limited to laboratory and highly specialised applications like position sensing and control in silicon wafer lithography. Physically, they are large and too cumbersome for mass industrial applications and their cost of manufacture makes them commercially unattractive. In addition, they are not conducive to long term battery operation due to the large number of optoelectronic components required.

It is an object of this invention to provide a cost effective sensor and method for measuring displacement to 6 DoF.

Brief description of the invention

The present invention provides a sensor comprising 3 interferometers and a 3 dimensional mirror having 3 reflective sides that are adjacent to each other. For the following text, "cube mirror" will be used generically for a 3 dimensional mirror having at least 3 reflective sides that are preferably orthogonal to one another, however, orthogonality is not physically essential as they can be made orthogonal to one another from a mathematical perspective. For the following text the term "orthogonal" means physically and/or mathematically orthogonal. The cube mirror is attached to a solid body and the fixed sensor components are independently attached to the same or a different solid body and displacement of the solid body to 6 DoF is derived from the displacement of the cube mirror. Algorithms for deriving the displacement are also provided.

The displacement sensor simultaneous measures six degrees of freedom (3 translations or linear displacements, 3 rotations) of a solid body using optical interferometry. Preferably 3 Michelson interferometers are orthogonally arranged about a cube mirror to derive displacement of the cube mirror to 6 DoF. The cube mirror is mounted on one solid body and the fixed sensor components on the same or another solid body so that movement of one relative to the other can be measured to 6DoF.

This invention includes a method of deriving the 3 position vectors of the cube mirror from the 3 mirror normal angles and 3 moving mirror tilt axis angles obtained from the 3 interferograms.

The direction of linear displacement of the cube mirror is derived from the direction of movement of the fringe lines across the 3 interferograms and the cube mirror position vectors.

A method of correcting linear displacement errors generated when using

interferometers arranged about a cube mirror is also provided. These new methods may be used to derive displacement of the cube mirror to 6 DoF.

The displacement of a solid body to 6 DoF is derived from the 6 DoF displacement of the cube mirror when the fixed components and cube mirror are mounted to the same or different solid bodies.

The output from each interferometer is a fringe pattern (interferogram), which is projected onto an image sensor from which the following four elements of data can be derived: linear displacement of the moving mirror relative to the fixed components - measured in terms of the number of fringe lines that pass across a chosen point on the interferogram as a consequence of the translation

- direction of linear displacement of the moving mirror - obtained in terms of the direction of movement of the fringe lines across the interferogram

- tilt angle of the normal of the moving mirror relative to the primary axis of the interferometer - measured in terms of the distance between successive fringe lines

- the angle of the axis about which the moving mirror normal is tilted - measured from the slope of the fringe lines relative to the horizontal or the vertical axes of the interferogram

This invention is the development of a single sensor for measuring displacement of a solid object to 6 DoF using optical interferometry. It employs 3 interferometers orthogonally arranged about a cube mirror and utilises low cost optoelectronics and optics. The sensor is inexpensive to mass produce including electronics and software.

Applications

Aerospace Helicopter rotor/ swash plate mechanicals, aircraft engine pylon/fuselage/wing interface, landing gear mechanicals load monitoring,

Mining Drilling bit and shaft load monitoring, mine head shaft load monitoring,

Civil Core structure load monitoring

Building Core structure load monitoring

Marine Superstructure and engine mechanicals load monitoring Robotics Articulated linkage arm load monitoring

Power generation Turbine and shaft load monitoring, wind turbines

Sports performance Force plate measurement, climbing wall hand hold, mast of sailing boats

Seismology Seismometer

Automotive Racing, heavy transport

Rail transport Traction vehicle mechanical load monitoring

Military Army & navy heavy artillery Nano positioning Semiconductor manufacturing, medical Detailed description of the invention

Preferred embodiments of the invention will be described with reference to the drawings in which:

Figure 1 is a diagram of an interferometer that is the basic technology upon which the invention is based;

Figure 2 is an illustration of an interferogram that is produced due to the interference generated by the interferometer;

Figure 3 shows an interferogram projected onto an image sensor of an

interferometer, the detail of which is used to derive displacement data;

Figure 4 depicts a simple block diagram of electronics that controls the sensor system;

Figure 5 illustrates 3 interferometers arranged orthogonally about a cube mirror that derives data for measuring displacement to six degrees of freedom;

Figure 6 provides an example of a method of mounting the fixed interferometer components rigidly in space;

Figure 7 shows two methods and means of mounting the fixed and moving components of the interferometers to a solid body reference and a solid body that undergoes displacement with respect to the reference;

Figure 8 is a 3 dimensional illustration of the first and second positions of the cube mirror and the associated X, Y and Z position vectors thereof;

Figure 9 is a vector diagram of the first and second position vectors of the cube mirror;

Figure 10 is a 3 dimensional depiction of the first and second cube mirrors superimposed on one another to derive the wave front angles and moving mirror tilt axis angles;

Figure 11 is a vector diagram of the X interferometer vectors superimposed on the X interferometer interferogram;

Figure 12 is a vector diagram of the Y interferometer vectors superimposed on the Y interferometer interferogram;

Figure 13 is a vector diagram of the Z interferometer vectors superimposed on the Z interferometer interferogram; Figure 14 illustrates a method of determining the direction in which the fringes lines are moving over the interferogram;

Figure 15 is a 2 dimensional view of a tilted cube mirror undergoing linear displacement along an orthogonal axis to demonstrate a method of overcoming a source of linear displacement error.

A typical interferometer is shown in Figure 1 that enables displacement to be accurately measured by detecting the phase difference between two monochromatic beams of light that overlap in space as well as direction. The interferometer has laser 101 , beam splitter 103 and fixed mirror 106 preferably mounted on a mounting that fixes their relative positions to one another in space. The mounting serves as a reference for moving mirror 107. Moving mirror 107 is mounted to a solid body that undergoes displacement relative to the reference.

The interferometer illustrated in Figure 1 provides an output from image sensor 109 that contains information regarding the magnitude of translation of moving mirror 107; the angle at which moving mirror 107 is tilted relative to the reference; the axis about which moving mirror 107 is tilted; and, the direction in which the fringe lines are transitioning across image sensor 109.

When three such interferometers are arranged orthogonally about a 3 dimensional mirror with 3 adjacent reflective surfaces, each of which serves as the moving mirror for respective interferometers (as depicted in Figure 5), the twelve elements of data obtained from the 3 image sensors enable translations along and rotations about the x, y and z axes to be determined, i.e. to 6 DoF.

This invention provides a method of deriving the 3 position vectors of a cube mirror from the 3 wave front angles and 3 moving mirror tilt axis angles generated from 3 orthogonally arranged interferometers about the cube mirror.

In another aspect this invention provides a method of deriving direction of linear displacement of the cube mirror from the direction of transition of fringe lines across the interferogram AND the cube mirror position vectors derived from the 3 moving mirror tilt angles and the 3 moving mirror tilt axis angles generated from the 3 interferometers arranged about the cube mirror. This invention also includes a method of correcting linear displacement errors generated when using at least 3 interferometers arranged about a cube mirror to measure displacement.

The main outcome of this invention is an apparatus and method of deriving the displacement of a solid body from the 3 position vectors of a cube mirror when the fixed components and cube mirror are mounted to the same or different solid bodies.

A typical interferometer is shown in Figure 1 that enables displacement to be accurately measured by detecting the phase difference between two monochromatic beams of light that overlap in space as well as direction. Laser 101 produces a collimated monochromatic light beam 102 giving a wave front that is rectilinear and orthogonal to the beam axis. Light beam 102 is split by means of beamsplitter 103 into a transmitted 104 and reflected 105 beam of equal amplitude. Beamsplitter 103 has a dielectric coating which ideally absorbs very little of the incident light beam 102, causes no phase shift between the transmitted 104 and reflected 105 beams nor affects polarisation of the electromagnetic field. The transmitted beam 104 is projected onto fixed mirror 106 and the reflected beam 105 is projected onto moving mirror 107. Each mirror is orientated to reflect the beam directly back along its incoming path to beam splitter 103 where they recombine with one another as interference beam 108. Interference beam 108 is projected as an interferogram onto image sensor 109 that captures the magnitude of the radiant flux of the

interferogram.

The proposed sensor design lends itself to inexpensive mass production.

Preferably it includes electronics with maximum dimensions 60 mm (L) x 60 mm (dia.). The sensor preferably achieves correlation > 99.5% between the theoretical and physical data. In addition, an accuracy of < 0.05% of Full Scale Output for the sensor is equivalent to currently available strain gauge type sensors. With low power electronics and opto-electronics being used, the sensor power consumption may be operated remotely for many months.

If moving mirror 107 is slightly tilted at angle a_x, as illustrated in Figure 1 , the two wave fronts making up interference beam 108 are no longer parallel to one another but intersect each other at twice the moving mirror 107 tilt angle, i.e. at angle 2a_x, called the wave front angle. Under these conditions, as depicted in Figure 2, straight lines called fringe lines 201 are observed on the interferogram. The brightest part of each fringe line indicates the point of maximum constructive interference of the two oblique wave fronts. By nature, fringe lines are always projected parallel to the axis at which moving mirror 107 is tilted. In Figure 2, the tilted fringe lines 201 describe angle φχ with the horizontal y-axis and is designated moving mirror 107 tilt axis angle. The distance between fringe lines 201 is designated Adfringe.

Deriving moving mirror tilt axis angle φχ

In Figure 1 , interference beam 108 is shown projecting onto image sensor 109 in plan view.

In Figure 3, image sensor 109, which is an array of photosensitive pixels 301 arranged in Y columns and Z rows, is shown in elevation view with the fringe line interferogram superimposed over it. Each pixel 301 captures the magnitude of the radiant flux incident on it, which is digitally quantised in terms of a proportional voltage and transmitted to a digital signal processor 406 (see Figure 4). Pixels 302 with dots indicate pixels that are coincident with maximum radiant flux from fringe lines 201. The digital signal processor 406 analyses each pixel voltage and records the (y, z) address of pixels 302 with maximum radiant flux. The pattern of pixels 302 is digitally processed in real time to analyse the movement of fringe lines 201 across image sensor 109.

The coordinate system is shown in Figure 3 with the positive x-axis pointing towards the reader, the y-axis horizontal and the z-axis vertical following the right hand rule convention. Image sensor 109 has been divided into 4 quadrants as shown in the figure. The width and height of each pixel 301 on image sensor 109 is known from the manufacturer's product specification, therefore trigonometry is applied to derive the moving mirror 107 tilt axis angle φχ as well as the fringe line spacing Adfringe. A method of deriving φχ is to find the apexes of a right angled triangle that includes Adfringe as one of the sides of the triangle. This can be performed by digital signal processor 406, which first selects one pixel 302, e.g. P3 (P3y, P3z), and then respectively searches along that pixel's row and column to find the next pixel 302 with maximum radiant flux, i.e. P1 (P1y, P1z) and P2 (P2y, P2z), where Pn is an apex pixel, Pny is pixel Pn's column address and Pnz is the pixel Pn's row address. The physical position of P1 , P2, and P3 on image sensor 109 is calculated from the size of the pixels. The moving mirror 107 tilt axis angle φχ of the fringe lines 201 is calculated by digital signal processor 406 usin the following equation:

Deriving distance Adfringe between fringe lines

One of the sides of the right angle triangle determined by digital signal processor 406 is Adfringe. The distance Adfringe between fringe lines 201 is calculated by digital signal processor 406 from the following equation (2):

^fringe = (^3y ~ Ply) Sin <p_x (2) Deriving moving mirror tilt angle ax

In Figure 1 , the two wave fronts making up interference beam 108 intersect each other at angle 2ax, i.e. twice the moving mirror 107 tilt angle.

From interferometer theory, the distance between fringe lines 201 is given by the equation

Ad fringe = ^ (3) where Adfringe is the perpendicular distance between the fringe lines 201 in metres, λ is the laser 101 wavelength in metres and 2ax is the wave front angle in radians between the two beams that constitute interference beam 108.

Distance Adfringe is one of the sides of the right angled triangle and its length is derived by digital signal processor 406 from equation (2). As λ is already known, the moving mirror tilt angle ax is calculated from equation

Deriving φχ, Adfringe and ax is not limited to the above methodology as there are several methods using mathematics and digital signal processing that will arrive at the same result. For example, using Fast Fourier Transformation of the rows and columns of pixel outputs in image sensor 109, the periodicity of the fringe pattern in the vertical and horizontal directions can be determined. Using trigonometry, φχ can be determined from the vertical and horizontal periodicity using the inverse tangent function. Adfringe can be derived by dividing the height or width of the image sensor by the respective number of vertical or horizontal periods to give the length of the hypotenuse of a right angled triangle. Then multiplying by the sine or cosine of φχ, Adfringe is obtained, ax is determined using equation (4).

Sensor control system

Referring to Figure 4, a block diagram is illustrated that is one means and method of controlling the interferometer system as a sensor to measure displacement to 6 DoF.

The man-machine interface 401 provides the means for giving inputs and receiving outputs from the system to an operator. The communication interface 402 manages the man machine interaction with the system and provides the means for

communicating instructions to/from other subsystems in the system. The power management subsystem 403 monitors and controls the power supplies for the sensor and provides signals to drive indicators showing the state of the power supplies. The calibration subsystem 404 executes manual and automated

calibration requests to zero or set the sensor in a specific state. The laser control subsystem 405 monitors and controls the laser performance and triggers the lasers at the requisite sampling rate under instruction from digital signal processor 406. The digital signal processor 406 receives data from image sensor 109, runs analysis algorithms and outputs measurement data to communication interface 402, calibration subsystem 404, laser control subsystem 405 and image sensor controller 407. Image sensor controller 407 systematically scans the pixel raster of image sensors 109 with specific regions of interest provided by digital signal processor 406.

Arranging 3 interferometers orthogonally to derive the components of the position vectors of a cube mirror

Three interferometers as depicted in Figure 1 and which operate as described above are arranged orthogonally as shown in Figure 5 about cube mirror 50 .

Preferably, the X, Y and Z interferometers define the x, y and z Cartesian axes in 3 dimensional space with the origin being the point at which the 3 beams incident on X moving mirror 505, Y moving mirror 510 and Z moving mirror 515 would intersect one another were it not for the beams being reflected by the respective moving mirror surfaces. X, Y and Z moving mirrors (505, 510, 515) are mounted on or integrated with cube mirror 501 , which is mounted on cube mirror mounting 517. X interferometer fixed components comprise X laser 502, X beamsplitter 503, X fixed mirror 504 and X image sensor 506, which are firmly fixed in space relative to one another on fixed components mounting 601 depicted in Figure 6. Similarly; Y interferometer fixed components comprise Y laser 507, Y beamsplitter 508, Y fixed mirror 509 and Y image sensor 511 firmly fixed in space relative to one another; and Z interferometer fixed components comprise Z laser 512, Z beamsplitter 513, Z fixed mirror 514 and Z image sensor 516 firmly fixed in space relative to one another; on fixed components mounting 601.

Fixed components mounting 601 is firmly attached to a solid body which provides a reference against which displacement is to be measured. In one embodiment shown in Figure 7a, fixed components mounting 601 is firmly affixed to flange 701 , which is the reference. Whereas cube mirror mounting 517 is mounted to plate 702, which undergoes displacement relative to flange 701 when a force/torque F/T is applied to plate 702. In another embodiment shown in Figure 7b, fixed components mounting 601 is firmly affixed to rod 703 and at an adjacent location on rod 703, cube mirror mounting 517 is mounted. When a force/torque F/T is applied to rod 703 the rod deflects and cube mirror mounting 517 undergoes displacement relative to fixed components mounting 601.

As cube mirror mounting 517 displaces, so does cube mirror 501 and consequently X, Y and Z moving mirrors (505, 510, 515) (not shown in Figure 7) generate independent interferograms projected onto X, Y and Z image sensors (506, 511 , 516) respectively (not shown in figure 7).

X, Y and Z interferometers measure cube mirror 501 's position or change in position in terms of cube mirror 501 's three position vectors by deriving; a. the three interferogram wave front angles and moving mirror tilt axis angles; and b., translation of the cube mirror 501 along the x, y and z axes. Deriving the 3 moving mirror tilt axis angles and wave front angles of the cube mirror

As previously mentioned, the origin for the X, Y and Z interferometers is the point at which the 3 beams incident on X, Y and Z moving mirrors (505, 510, 515) would intersect one another were it not for the beams being reflected by the respective moving mirror surfaces. This origin is relevant to translation of cube mirror 501 , which will be discussed subsequently. However, the origin of the reference frame for derivation of the 3 moving mirror tilt axis angles and wave front angles is the centre of the cube mirror. This is because:

1 the 3 preferably orthogonal X, Y and Z interferometer beams physically and/or mathematically define the direction of the 3 Cartesian axes within the space they intersect

2 moving mirror tilt axis angle and distance Adfringe between fringe lines 201 on each interferogram is unchanged for unchanged cube mirror 501 orientation despite cube mirror 501 varying in translation within the space

3 the centre of cube mirror 501 is therefore a theoretical reference frame that has consistent axis direction and the instantaneous position of the centre of cube mirror 501 within the space is therefore irrelevant This idiosyncrasy simplifies the vector analysis as the position vectors of cube mirror 501 always have magnitude unity and the vector components reduce to

trigonometric functions. When measuring change in moving mirror tilt axis angle and distance Adfringe, orthogonal alignment of cube mirror 501 with X, Y and Z interferometers is not necessary as the initial cube mirror orientation can be taken as an offset to any subsequent change in orientation.

To begin with, Figure 8a) defines the first position of the cube mirror 501 with origin

(0,0,0) and three known points.

Positive x-axis vector A, (Ax, y,)=(1 ,0,0)

Positive y-axis vector B,

λ ,ΰ)

Positive z-axis vector C, {Cx, Cy,)=(0,0, 1),

Thus, the non-zero components are: Ax, By, Cz = (1 ,1 , 1 ); i.e. the resultant of the three positive axes Figure 8b) defines the second position of the cube mirror 501 with origin (0,0,0) and three unknown points.

Vector X, (Xx, Xy)

Vector Y, (Yx, Yy)

Vector Z, (Zx, Zy Zz)

The first and second position vectors are depicted on a single vector diagram in Figure 9 where t, /^"and k are unit vectors in the respective x, y and z directions such that t=A, f=B and k=C

With the first and second positions of cube mirror 501 superimposed on one another as shown in Figure 10;

- the moving mirror tilt axis angles φχ, φν and φζ are defined as the angles the respective X, Y and Z moving mirrors (505, 510, 515) in the second position make with the moving mirrors in the first position and are determined from measuring the slope of fringe lines 201 on the interferograms superimposed on X, Y and Z image sensors (506, 511 , 516) respectively;

- the moving mirror tilt angles ax, py and γζ are defined as the angles the X, Y and Z moving mirrors (505, 510, 515) make with the respective x, y and z axes, and are determined by measuring the distance Adfringe between fringe lines 201 on the interferograms superimposed on X, Y and Z image sensors (506, 51 1 , 516) respectively.

Two methods of deriving the angular displacement position vectors of cube mirror 501 from the 3 wave front angles and 3 moving mirror tilt axis angles of X, Y and Z interferometers are now described. Method 1

Vectors A, B and C are orthogonal to one another, therefore vector C can be derived from the cross product of vectors A and B:

C = A x B = i x j = k (5) Similarly, vectors X, Y and Z in cube mirror 501 second position (Figure 8b) are orthogonal to one another therefore vector Z can be derived from the cross product of X and Y:

Z = (XyYz - YyXz)i + (XzYx - YzXx)j + (XxYy - YxXy)k (7) The moving mirror tilt angles αχ, βν and vz can be determined from the dot product of second position vectors and the respective Cartesian axes unit vectors. Therefore: i · X = |X| cos oc (8)

·"· Xx ^~ Gx (9) j - Y = \Y\ cos fi_y (10)

.·. Yy - Gy (11) k - Z = \Z\ cos y_z (12) XxYy - YxXy = Gz (13)

Note

|X| = |Y| = |Z| = 1 (14) Moving mirror tilt axis angles φχ, cpy and φζ can be determined from the

product of the first and second ositions of cube mirror 501. Therefore:

T = -Xzj + X k (16)

T_y = Yzi - Yxk (18) k Z = 0 0 1 (19)

(XyYz - YyXz) {XzYx - YzXx) (XxYy - - YYxxXXyy))\

T_z = -(XzYx - YzXx)i + (XyYz - YyXz)j (20) Where vectors Tx, Ty and Tz are the moving mirror tilt axis vectors in the respective y-z, x-z and x-y planes.

Moving mirror tilt axis angles φχ, cpy and φζ are defined by their tangent values, therefore: tan tp^ = -^- = Fx (21 )

-Xz tan (p_y = -^- = Fy (22)

(XyYz - YyXz) XyYz - YyXz

tan φ_ν =— ; = = Fz (2.0)

^Ψζ -(XzYx - YzXx) YzXx - XzYx ^K '

Solving equations (9), (1 1 ), (13), (21 ), (22) and (23) for the unknowns Xx, Xy, Xz, Yx, Yy and Yz gives:

Xx = Gx = cos o _x (24) -(-/ + #)

^Xy = 2Gy— <²⁵>

J + H

^Xz = 2GyTx- W J - H

Yx = _ _ _ (27)

IGxFxFyFz ^/

Yy = Gy = cos _y (28)

₌ -(-J - H)

IGxFxFz ' where

H = GzFz - GxGyFz + GzFxFy - GxGyFxFy (30) and

/ = ±(Gx²Gy²Fx²Fy² - 2Gx²Gy²FxFyFz + Gx²Gy²Fz²

- 2GxGyGzFx²Fy² - IGxGyGzFz² + GzFx²Fy² (31 )

+ 2Gz²FxFyFz + Gz²Fz²)⁰⁵

Consequently, the components of position vector Z can be determined from equation (7).

Recall from equations (1 ) and (3) and Figure 3 that the X, Y and Z moving mirror tilt axis angles φχ, (py and φζ and moving mirror tilt angles ax, y and γζ are

determined from measurement from image sensors (506, 51 1 and 516).

From equation (31 ), 2 solutions result for each of the coefficients Xy, Xz, Yx and Yz due to the positive and negative sign of the variable J, which gives rise to 16 permutations of x, Xy, Xz, Yx, Yy and Yz. However, only one permutation of vectors X and Y are perpendicular to one another AND they are of unit length. Therefore, that permutation consists of the correct position vectors X and Y of the cube mirror. To find which of the 16 is the correct permutation:

Firstly, each permutation of vectors X and Y have to in turn be tested for

orthogonality using the dot product and,

Secondly, each vectors X and Y has to be tested for unit length by taking the square root of the sum of the squares of the coefficients.

Only one permutation will have its set of X and Y vectors orthogonal to one another AND each vector being of unit length.

Having found the correct permutation of position vectors X and Y, the coefficients of position vector Z are derived using equation (7) by substituting coefficients Xx, Xy, Xz, Yx, Yy and Yz.

Method 2

Vectors X, Y and Z in cube mirror 501 second position (Figure 8b) can be expressed in terms of direction angles as defined in Figure 1 1 , Figure 12 and Figure 13.

X = |X| (cos <x._x ϊ + cos β_χ } + cos γ_χ k) = cos o ^. ϊ + cos β_χ j + cos γ_χ k (32)

Y = |K| (cos oc_y i + cos _yj + cos y_y k) = cos <x_y i + cos ?_y / + cos y_y k (33)

Z = \Z\ (cos o _z i + cos β_ζ) + cos y_z k = cos o _z i + cos β_ζ] + cos y_z k (34) Referring to Figure 11 , the rejection vector of vector X onto the y-z plane is vector Rx with angle θχ relative to the y-axis.

Therefore,

R_x = sin o _x cos B_x j + sin o ^. sin B_x k (35) Substituting the components of equation (35) into equation (32) gives

X = cos o ^. i + sin o ^. cos B_x j + sin o ^. sin B_x k (36) X moving mirror 505 tilt axis vector Tx is the cross product of the unit vector C and second position cube vector X and consequently also lies on the y-z plane at angle φχ relative to the y-axis. Figure 1 1 shows the slope of the fringe lines, which are aligned with the tilt axis vector Tx therefore;

Substituting for θχ in equation (37) into equation (36) gives

X = cos c_x i + sin o ^. cos (<p_x -— ) j + sin <x_x sin (<p_x - k (38) = cos oc_x i + sin oc_x sin <p_x j - sin o ^ cos <p_x k

Referring to Figure 12, the rejection vector of vector Y onto the x-z plane is vector Ry with angle Qy relative to the z-axis.

Therefore,

R_y = sin βγ sin 9_y i + sin β_ν cos G_y k (39) Substituting the components of equation (39) into equation (33) gives

Y = sin p_y sin 0_y i + cos β_γ j + sin fi_y cos B_y k (40) Y moving mirror 510 tilt axis vector Ty is the cross product of the unit vector y~and second position cube vector Y and consequently also lies on the x-z plane at angle <py relative to the z-axis. The same conditions apply to Qy as described for Qx in equation (37) above, therefore:

π

(41 ) 2

Substituting for Qy in equation (41 ) into equation (40) gives

Y = sin fiy sin (φ_ν - ϊ + cos fi_yj + sinfi_y cos (φ_ν - ^ k ^

=— sin βγ cos (py i + cos _yJ + sin β_γ sin <p_y k

Referring to Figure 13, the rejection vector of vector Z onto the x-y plane is vector Rz with angle θζ relative to the z-axis.

Therefore,

R_z = sin y_z cos θ_ζ i + sin y_z sin θ_ζ j (43) Substituting the components of equation (43) into equation (34) gives

Z = sin γ_ζ cos θ_ζ i + sin y_z sin θ_ζ j + cos γ_ζ k (44) Z moving mirror 515 tilt axis vector Tz is the cross product of the unit vector k and second position cube vector Z and consequently also lies on the x-y plane at angle <pz relative to the x-axis. The same conditions apply to θζ as described for θχ in equation (37) above, therefore:

Q_x = <p_x - (45) Substituting for θζ in equation (45) into equation (44) gives

Z = sin y_z cos (φ_ζ - i + sin y_z sin (φ_ζ - J + cos y_z k

= sin y_z sin φ_ζ ϊ - sin y_z cos <p_z j + cos y_z k The components of second position cube vectors X, Y and Z are given in equations (38), (42) and (46) and are calculated from the X, Y and Z interferometer moving mirror tilt angles ax, βγ and vz and moving mirror tilt axis angles cpx, cpy and cpz measured from image sensors (506, 51 1 , 516). Moving mirror tilt angles ax, y and γζ can range from 0 to π radians, however, by limiting these angles to small angles their sine and cosine values are always positive.

The moving mirror tilt axis angles φχ, cpy and φζ are all measured from the inclination of the fringe lines with respect to their respective primary axes. The tilt vectors T_x, T_y and T_z lie along a line parallel to their fringe lines but their direction is unknown, therefore until it is known which direction is correct, φχ, cpy and φζ must initially be treated as having two values, i.e.:

<p_x = a or <p_y = a + π

(47) φ_ν = b or φ_ν = b + π (48) φ_ζ = c or φ_ζ = c + π (49)

This means there are eight permutations of the components of second position cube position vectors X, Y and Z. A method is therefore required to determine which one of the eight permutations is correct.

Firstly, all eight permutations of vectors X, Y and Z are calculated.

Secondly, each permutation needs to be tested for orthogonality. One method of doing this for each permutation is to take the dot product of each pair of vectors, i.e. ΧΎ, X-Z and Y-Z and determine which permutation has all three dot products equal to zero.

Thirdly, each permutation of X, Y and Z has to be tested for unit length by taking the square root of the sum of the squares of each vector's the coefficients.

Only one permutation will have its set of X, Y and Z vectors orthogonal to one another AND each vector being of unit length.

Deriving the angular displacement position vectors of cube mirror 501 from fringe slopes φχ, cpy & φζ and respective fringe spacing Adfringe of the X, Y & Z

interferometers can also be realised using other known mathematical procedures. Deriving moving mirror translation Admirror

Methods 1 and 2 above are examples of methods to derive the orientation of cube mirror 501 in terms of the components of first and second position vectors A, B and C and X, Y and Z respectively, which is purely due tilt of cube mirror 501.

What is now considered is deriving the translation of cube mirror 501 along each of the Cartesian x, y and z axes. Referring back to Figure 1 and a single

interferometer, the distance from beam splitter 103 to fixed and moving mirrors (106, 107) is designated d1 and d2 respectively. The return paths of transmitted and reflected beams (104, 105) are therefore 2d1 and 2d2 respectively. The difference in distance between the return paths is called the optical path difference, OPD, and is represented by equation (50),

OPD = 2d_x - 2d₂ = 2(d_x - d₂) = 2Ad_mirror (50) where d1 - d2=Admirror, which is the difference in distance in metres between the fixed 106 and moving 107 mirrors. Because fixed mirror 106 remains static, the magnitude of d1 is constant, therefore any variation in Admirror is as a

consequence of translation of moving mirror 107.

From interferometer theory, the number of wavelengths that equate to Admirror is given by equation (51 ),

ηλ . Δd_mfrror = y (51) where, λ is the wavelength of the laser 101 in metres and n is an integer that represents the number of full wavelengths equal to the OPD.

Fringe lines 201 on the interferogram (Figure 3) represent instances of maximum constructive interference one wavelength λ in phase apart. As moving mirror 107 translates, fringe lines 201 move across the interferogram in concert with the translation, therefore, by counting the number of fringe lines n passing over a pixel 301 , Admirror can be calculated from equation (51).

A method of counting the number of fringe lines n passing over a pixel is by selecting a pixel 301 close to the centre of image sensor 109, e.g. P3 (P3y, P3z), and every time image sensor 109 registers maximum constructive interference at P3, a counter is stepped. This process is undertaken in digital signal processor 406. Although one pixel can be used to count the number of fringe lines passing over it, one pixel cannot determine the direction in which the fringe lines are moving. Deriving direction of moving mirror linear displacement

When the tilted moving mirror 107 translates, fringe lines 201 move over the interferogram orthogonally to the moving mirror 107 tilt axis angle φχ. The direction in which moving mirror 107 translates is determined by:

1 knowing towards which quadrant (see Figure 3) on image sensor 109 the fringe lines are moving, and

2 knowing whether wave front angle ax is tilted is towards the same quadrant or towards the opposite quadrant.

Determining towards which quadrant the fringe lines are moving

Figure 14 depicts image sensor 109 on which the interferogram is projected. A method of determining towards which quadrant the fringe lines are moving is by selecting one or more pixels, e.g. pixel P4, in close proximity to the pixel closest the centre of image sensor 109, i.e. pixel P3 (P3y, P3z). If the fringe lines are moving toward quadrant 2, then digital signal processor 406 detects maximum constructive interference occurring at pixel P3 before pixel P4. Similarly, maximum constructive interference occurring at pixel P4 before pixel P3 indicates the fringe lines moving towards quadrant 4.

Determining direction of linear displacement Linear wave theory is applicable to the interferometers utilised in this invention and defines the behaviour of the interferogram generated by interference beam 108 (Figure 1). The behaviour of the interferogram as a function of translation of moving mirror 107 and wave front angle 2ax is specified by the following set of rules: 1 If fringe lines 201 are moving across the interferogram in the direction of the quadrant towards which moving mirror 107 normal is tilted, then moving mirror 107 is moving towards beamsplitter103 and the value n is increased, i.e. the fringe line counter is incremented 2 If fringe lines 201 are moving across the interferogram in the direction of the quadrant away from which moving mirror 107 normal is tilted, then moving mirror 107 is moving away from beamsplitter! 03 and the value n is decreased, i.e. the fringe line counter is decremented

The method for: deriving moving mirror translation Admirror; deriving the direction of translation and; determining towards which quadrant the fringe lines are moving, as described above for a single interferometer, applies to each of the 3 orthogonally arranged interferometers depicted in Figure 5 and Figure 6.

To be able to determine the quadrant towards which the wave front is tilted requires knowledge of the position vector of moving mirror 107. For the X, Y and Z

interferometers, the position vectors X, Y & Z of cube mirror 501 were determined by methods 1 or 2. Therefore, the quadrant towards which the wave front angle is tilted can be determined from the sign of the position vector components and the rules defined in Table 1 :

Table 1 Determining towards which quadrant X, Y and Z moving mirror normals are tilted

Having determined the quadrant towards which the fringe lines are moving and the direction towards which the moving mirror normal is tilted (from Table 1 ), digital signal processor 406 increments or decrements the respective fringe counter in accordance with the above defined rules. Consequently, translation of X, Y and Z moving mirrors (505, 510, 515) towards or away from respective beamsplitters (503, 508, 513) is now known.

Linear displacement error of a single interferometer with tilted moving mirror

Returning to the topic of translation, equation (51) gave the equation for translation of moving mirror 107 as an ideal case for a single interferometer. When moving mirror 107 is tilted, a wave front with angle 2ax results as illustrated in Figure 1 and this introduces an error in translation measurement. Taking this error into consideration, equation (51 ) then becomes:

n λ

&d_mirror = - (52) cos 2a_x 2

which implies that the greater the moving mirror tilt angle ax, the greater the magnitude of Admirror for each fringe line n counted.

However, when using 3 orthogonal interferometers with a cube mirror, additional translation errors to equation (52) arise that require mathematical treatment to correct. Deriving the corrected values for linear displacement of the cube mirror

The origin from which linear displacement of cube mirror 501 is measured is the point at which the 3 beams incident on X, Y and Z moving mirrors (505, 510, 515) would intersect one another were it not for the beams being reflected by the respective moving mirror surfaces.

Referring to Figure 15, the 2 dimensional illustration shows tilted cube 1501 prior to translation (dotted lines) and tilted cube 1501' (solid lines) after being translated a distance dy along the y-axis. The linear displacement of tilted cube 1501 in the y direction has also generated translation 6xy in the x direction, which is a source of measurement error if not appropriately treated.

The components of vector X are known, i.e. Xx, Xy, Xz, therefore the angle Pxy is given by:

X

p_Xy = tan-¹ ^ (53)

Therefore

S_Xy = bd_y tan (tan^'1 = Ad_y (^) (54) The same principle applied to translation in the z direction yields: δ_Χζ = Ad_z tan (tan^'1 = Ad₂ (½) (55) Adding equations (52), (54) and (55) results in the overall equation for linear displacement in the x direction: n,

Ad_r + S_r +S_r = (56)

^{X Xz} cos 2a_r 2

_{Mx + M} ( ) _{+ Mz}(h) ₌ ^h_ (57) x ^y\X_rJ ^z X_rJ cos2a_r2

Following the same principle for deriving the overall equations for linear

displacement for the Y and Z interferometers ields:

Putting equations (57), (58) and (59) into matrix form yields;

Simplifying and using matrix algebra to solve for Adx, Ady and Adz renders:

1 A B Ad_x L

c 1 D Ady = M (61)

E F 1. Ad_z .N.

(62)

1-FD -(C-ED) CF-E

[Rco factor] = -(A- FB) 1-EB -(F - EA) (63)

AD-B - - CB) 1-CA

1-FD -(A- FB) AD-B

[Rco factor]⁷ = -(C- ED) 1-EB —{P— CB) (64)

CF-E — ( - EA) 1-CA (65) det[R] 1 + ADE + BCF - EB - FD

Ad_x ^"1 - FD FB -A AD — B L^"

Ad_y ED -C 1 - EB CB -D M (66)

1 + ADE + BCF -EB - FD - CA

Ad, .CF -E EA -F 1 - CA. .N. ^"Ad,^" 1 ^"1 - FD FB - AO - B^' L

Δά_ν y ED - C 1 - Efl CB - D M (67) det[R]

Ad_z -C - E &4 - F 1 - CA . .N.

Therefore, the corrected linear displacement of the cube mirror in the x-, y- and z- directions are given be equations (68) - (70).

1

- FD)L + (FB - A)M + (AD - B)N]

^Ad* ⁼ det[R] (68)

[(ED - C)L + (1 - EB)M + (CB - D)N] (69) y det[R]

[(CF - E)L + (EA - F)M + (1 - CA N] (70) z det[R]

Having obtained the angular displacement position vector components from

Methods 1 and 2 above and the magnitude of the linear displacement of the respective position vectors from equations (68), (69) & (70), the resultant position vectors are simply obtained by summing the respective components for the X, Y and Z interferometers.

From the above it can be seen that this invention provides a unique and inexpensive means of accurately measuring displacement of solid bodies in 6DoF.

Those skilled in the art will realise that this invention may be implemented in embodiments other than those disclosed without departing from the core teachings of the invention.

Claims

1. A displacement sensor including at least one source of light, at least 3

interferometers and a three dimensional mirror in which the three dimensional mirror and the interferometer fixed components are independently attached to one or more solid bodies and displacement of one or more solid bodies to 6 degrees of freedom is derived from the displacement of the three dimensional mirror.

2. A displacement sensor as claimed in claim 1 in which 3 interferometers are arranged about a cube mirror to derive displacement of the three dimensional mirror to 6 degrees of freedom.

3. A displacement sensor as claimed in claim 2 which uses 3 light sources to sense 6 degrees of freedom.

4. A displacement sensor as claimed in claim 1 or 2 in which the three dimensional mirror is mounted on one part of a solid body and the interferometer fixed

components on another part of said solid body so that movement of one part relative to the other can be measured to 6 degrees of freedom.

5. A displacement sensor as claimed in claim 1 or 2 in which the three dimensional mirror is mounted on a solid body and the interferometer fixed components on another solid body so that movement of one solid body relative to the other is measured to 6 degrees of freedom.

6. A method of measuring displacement of a solid body using the sensor of claim 1 which includes the steps of deriving the three 3 dimensional position vectors of the three dimensional mirror from the three 3 dimensional mirror normal angles and the three 3 dimensional mirror tilt axis angles obtained from the at least 3 fringe patterns (interferograms), and the translation and direction of translation of the three dimensional mirror is derived from the direction of movement of the fringe lines across the at least 3 interferograms and from the three 3 dimensional mirror position vectors.

7. A method as claimed in claim 6 wherein output from an interferometer is an interferogram, which is projected onto an image sensor from which the following four elements of data are derived:

- linear displacement of the three dimensional mirror relative to the fixed

components of the said interferometer, measured in terms of the number of fringe lines that pass across a chosen point on the interferogram as a

consequence of the linear displacement

- direction of linear displacement of the three dimensional mirror obtained in terms of the direction of movement of the fringe lines across the interferogram

- tilt angle of the normal of the three dimensional mirror relative to the axis of the said interferometer, measured in terms of the distance between successive fringe lines on the interferogram

- the angle of the axis about which the three dimensional mirror normal is tilted, measured from the slope of the fringe lines of the interferogram relative to the horizontal or vertical axes of the interferogram

8. A method as claimed in claim 6 to overcome linear displacement errors of the three dimensional mirror in a first direction as a consequence of linear displacement of the three dimensional mirror in a second direction when using interferometers arranged about a three dimensional mirror to measure displacement to 6 degrees of freedom.