2.2 Synchronization of the double parallellogram

Abstract

For the measurement machine NANOMEFOS (Nanometer Accuracy NOn-contact MEa-surement of Freeform Optical Surfaces), currently developed by Rens Henselmans, an elas-tic guidance for the optical probe is needed. This guidance has to full following demands:no/minimal friction and hysteresis, an optimal inherent straightness, high bandwidth andminimal heat dissipation. Two different guidance types with matching actuator have beendesigned that can meet these requirements. The guidance based on six folded plate springs,three on each side under 120o, turned out to be a better solution than the one based onthe double parallellogram principle. Although they have, by coincidence, the same heatdissipation during a normal measurement, the guidance using a double parallellogram willbe more dicult to manufacture. In this setup a proper synchronization is designed butthe correct mechanical connection with the parallellogram will be hard to realize. The rstdisturbing eigenfrequency of this guidance is 1030 Hz, which is much lower than the 1920Hz of the guidance using the folded plate springs.To synchronize the double parallellogram different solutions are possible, but they all have to meet the requirement of a straightness (after calibration) of 25 nm over a stroke of 5 mm, or they must be adjustable to do so, or the movement has to be calibrated and the error must stay beneath it.

Chapter 1

Introduction

In this traineeship the goal is to design an elastic guidance with minimal friction andhysteresis, optimal straightness, high bandwidth and minimal heat dissipation for themeasurement machine Nanomefos (Nanometer Accuracy NOn-contact MEasurement ofFreeform Optical Surfaces). This machine is designed at the moment by Ir. R. Henselmansas basis of his doctorate. The objective is to measure optical surfaces that have non-axial-symmetrical shapes, called freeform (optics), in a measurement time of approximately 15minutes. To do so, the freeform is placed on a turntable below a measurement machinewith an axis in radial (r) and an axis in height direction (z), see gure 1.1.

Below the z-axis the  -axis is mounted on the machine to rotate the measurement headabove the freeform. This  -axis can rotate between an angle of  45oto 135oto measureconvex and concave freeform optics. To measure the position of the  -axis, two laserbeams are used as shown in gure 1.1 and a rotational encoder on one end. The elasticguidance that has to be designed is mounted on that  -axis and preferably has to twithin the dimensions of 55 mm in diameter and 50 mm in height. To be able to measurea large variety of freeform optics the stroke of the guidance has to be 5 mm. Othermain requirements of the guidance are listed in table 1.1. The uncertainty goal of the measurement of a freeform is 10 nm at an perpendicular measurement, and 35 nm with a5otilt of the measurement head w.r.t the optical surface.

1.1 Optical measurement

 

The measurement itself is done by a beam with a diameter of 2 mm. This beam comesfrom a corner cube in the middle of the -axis. The design of the non-contact probe (see[Cac06]) is shown in gure 1.2. A cilinder of 5 mm round that beam is kept free to be ableto make some adjustments and alignments in the optical layout. The beam is focused onthe freeform using an a-spherical lens. On both sides of the beam two beams of 3 mm areplaced to measure the position of the guidance. At the bottom a mirror is placed with inthe middle the aspherical lens. The two outer beams re ect on the mirror and enable aoptical interference measurement of the position of the lens, the beam in middle is focusedon the surface of the freeform. Using the focus error signal during a whole rotation of thefreeform, the height of the lens and mirror are known and thus the shape of the freeform.The measured data of the outer two beams are used to actuate the guidance and thus tomove the mirror with lens.

The design space is shown in gure 1.3, where the outer distance of the beams is 13.5 mmand the overall diameter of the guidance is 55 mm. It is also possible to place a squareshaped guidance in the design space of 55 ¢ 55 mm. To minimize measuring errors dueto diused light, it is preferable to leave the space between the beams open. So a tubewith some inner spacing around the three beams is allowed but something around onlythe middle beam is rejected.The guidance has to translates along a (nearly) straight line i.e. it should be a parallelguidance for the mirror and focal lens. Two potential best solutions that meet this propertyare one based on the double-parallellogram principle and one based on six folded plate-springs. Both types of the guidance will be explained further on in this report dealing with the pros and cons of the specic solution and what type of actuator will be used to drivethat kind of guidance.

1.2 Material choice

To be able to make a relatively long stroke, with an elastic guidance that ts in the designspace, a proper selection has to be made to nd a suitable material. For both of the abovementioned elastic guidances a material is needed that has a low elastic modulus (E) tominimize the actuation forces and a high allowable stress (), combining both gives aparameter that denes the allowable strain: =E. Another material parameter (using alsothe density ) that helps to choose the material is =pE, representing the maximal strokethat can be obtained when making a plate spring out of this material. In [Bon95, page64] a group of materials is compared resulting in the titanium alloy TiAl6V4, also calledtialv, as the most suitable material to use in plate springs (E = 114 GPa,  = 880 MPa, = 4430 kg/m3). Using this material in the design of the probe guidance a lifetime of 10years has to be taken into account, resulting in 108cycles. As shown in the fatigue gure1.41the maximal stress at 107cycles is approximately 500 MPa. A more specic valuecan be found on matweb (see appendix A for the data), where the unnotched (meaning nogrooves, holes etc. in the utilized region) value is 510 MPa. To meet the requirements themaximal stresses are restricted to 450 MPa.

To increase the buckling resistance thickened plate springs are used. Normally the lengthof the thin parts is 1=6 of the length of a normal plate spring, and the thick part (6 to 10 times thicker) is 5=6 l long. Doing so the buckling force is 9 times higher, the stressesare the same and the stiness in direction perpendicular to the plate spring is only 20 %increased. An overview of different plate springs and corresponding formulas is given inappendix B.

Chapter 2

Double parallellogram guidance

2.1 Overview

The basic concept of a so called double parallellogram is shown in gure 2.1, where bodyB is called the intermediate body and A is the main body. When B translates over adistance z=2 the plate springs c and d will de ect and shorten a little (for formulas seeappendix B), resulting in a parabolic motion of B. When at the same time A translatesover a distance z the plate springs e and f will shorten the same as c; d and therefore Amoves along a straight line. This operation is only correct when the of motion of bodies Aand B satisfy the proper ratio of 2:1. So only if body B makes exactly half of the strokeof body A a straight movement of A is obtained. This requires a synchronization betweenA and B and will be treated in section 2.2, but the use of a 1 over 2 'lever' is obvious.

Within the design space a few dierent orientations of the double parallellogram are possi-ble, they are shown in gure 2.2. In 2.2a the guidance is placed with a radial distance withrespect to the three beams. An advantage of this setup is that there is enough room forthe actuator placed radial opposite to the guidance. Disadvantage of this setup is the fact that there is little room for the synchronization within the double parallellogram. Thiscan be solved by placing the 1 over 2 lever behind the guidance. However this will resultin a longer stroke of the actuator (1.5 times the translation of the main body) when theactuator is connected to the lever to preform a correct actuation of the guidance. This isnecessary because actuating the plate springs in the wrong way results in moments in thebodies and springs, and thus in a tilt of the guided body A. The correct location of theactuation force is in the middle between the bodies A and B, see [vE85, chapter 5] and[Ein03, chapter 6]. Elongating the lever leads to the location of the pivot of the lever to layoutside the design region which is not preferable. Placing the pivot at the height of bodyB and choosing the length of the lever the same as the plate spring length (approximately17 mm), then the lever will be to small to manufacture.

In gure 2.2b the guidance is placed in the middle of the design space with a hole to allowthe three optical measurement beams to pass. In this setup it is possible to use longerplate springs, a practical length could be 28 mm. Now it is also possible to place thesynchronization (partially) inside the guidance. Two actuators can be placed, as shown,at each side of the double parallellogram. A disadvantage is now that the lever passesthrough the beams and the guidance. This problem will be solved in section 2.2. Figure2.2c could be an improvement of 2.2b, because the stiness could be nearly the same in thetranslation direction but higher for the direction perpendicular to that. A disadvantageof this setup is the fact that it is more dicult to manufacture due to the slopes of theplate springs. Further comparison between the two setups can be based on nite elementanalysis, which is done in section 2.4.

2.2 Synchronization of the double parallellogram

To synchronize the double parallellogram dierent solutions are possible, but they all haveto meet the requirement of a straightness (after calibration) of 25 nm over a stroke of 5mm, or they must be adjustable to do so, or the movement has to be calibrated and theerror must stay beneath it.

 

2.2.1 'Simple' lever

Figure 2.3: Different setups for synchronization of the double parallellogram, using in (a) a lever and in (b) a rocker to prescribe the motion of A and B.

As shown in section 2.1, particularly gure 2.1, a proper synchronization between intermediate and main body is required. Taking a look at the simple lever rst, see gure 2.3a, the connection rods also shorten during the movement. So the main body moves along a nearly straight line (over a distance 2z), the intermediate body along a parabolic curve (over a distance z) and the lever makes a circular motion round the pivot. This behavior given in formulas will result for the shortening of the lever in:

where Llever represents the lever length, and the rotation angle of the lever is given by = 2z=Llever. The vertical displacement of the intermediate body is given by (see also appendix B):

where L is the length of the plate springs. The shortening of the rod connecting the main body to the lever is the same as the motion of the lever, a = lever. The shortening of the rod between the intermediate body and the lever is given by:

build up by the rst term, representing the parabolic motion of the intermediate body,and a term describing half the shortening of the lever itself. The actual displacement of the intermediate body and the main body can now be determined (see also gure 2.4):

Figure 2.4: Shortening at the left half of the double parallellogram, the dotted circles are the desired displacement and the dotted lines the actual one.
The synchronization will be perfect if these two displacements where the same, thus using formula 2.2 for the shortening of the connection rods and choosing the length of the rod connecting the main body as la, the length of the other one (lb) is determined by:

Thus depending on the length of the plate springs in the double parallellogram and the length of the lever the different lengths of the rods can be chosen in such a way that there is a minimal deviation of the main body to the perfect straight translation. In gure 2.5 the position of the main body is shown as function of the lever angle. In this case the full scale represents a motion of 2.5 mm in one direction. For a correct length of la and lb there is a small (below one nanometer) displacement in radial direction due to the mismatch between the circular motion of the lever and the parabolic motion of the intermediate
body even if there is a small deviation of 10 m in length. On the other hand, choosing the same length for la and lb, say 7 mm, then the straightness is 12 nm which is already half of the required value for the guidance, still neglecting the manufacturing tolerances. Actually this deviation is repeatable so there is the possibility to calibrate it and removing the error in software but this is not advisable because there is a way to do it mechanical. In the derivation above, the assumption is made that the pivot of the lever rotates around one pole. But with a normal elastic cross spring pivot this is not the case. During the rotation the pole moves along a straight line, see [Bon95, page 14] and B.1e, so the correct 1 over 2 prescribed motion is disrupted with the linear motion of the pivot. To avoid this a pivot is used where the crossing of the springs is at 1=8 of the length. It is also possible to made this pivot using thickened plate springs, see appendix B for more details. In this application wire electric discharge machining (wire EDM) and  ormal EDM is used to create the lever. A benet of the material used, Tialv, is that it works easily using EDM. The process enables the manufacturing of free standing lms of 100 m thick, when the length of the wire is kept as short as possible (in this case 30 mm) and when the wire is surrounded with material during the erosion. But with a wire length of < 10 mm and very sensitive work free standing lms of 25 m are already made.

2.2.2 Rocker
In gure 2.3b synchronization using a rocker is shown where the concept is based on the motion of a rolling wheel. At the axle of the wheel half the displacement is made with respect to the top of the wheel. The connection to the main body A can be realized using strings under tension as shown in gure 2.6a. Connecting the rocker in this way will introduce no moments around the vertical axis. To link body B with the rocker two folded plate springs can be used at each side of the rocker, as shown in gure 2.6b. The torsion in the springs is not a normal use but the dimensions (length and width) can be optimized to ensure a constraint even under torsion. Using a cilinder at the bottom of the rocker and loading it in vertical direction, the rocker is fully constrained. Another option is by using two rolling cilinders in a channel, see gure 2.6c. These cilinders are pressed to the walls by a tensioned strip. Now the cilinders are able to roll within the slit with minimal slip and hysteresis. Connecting the top of the rocker with body A and one cilinder with body B a 1 over 2 ratio is obtained when the radius of the rocker is 3 times higher than the radii of the cilinders To choose between the two proposed options of synchronization, the question rises: is it manufacturable? The rocker needs some adjustments at the connection with the main body and the tension of them is also not easy. Notice that the rocker still have some shortening in those connections, depending on the type of the connection with the intermediate body, gure 2.6b or c. This can be minimized using another shape of the rocker instead of a circular one. Then the best way of making the rocker would be by using wire EDM and make it out of one piece of material. But now the lever and the rocker are both equal in diffculty and on top of that the lever has the advantage of a hole in the middle where the measurement beams are able to pass through together whereas the rocker will be larger to enable the same. This is due to the dierence in distance between pivot point and the middle of the measurement beams, the rocker makes a larger rotation. The choice for the 'simple' lever is based on the manufacturing diculty, the feasibility of the dierent connections including the connection to the xed world: the -axis. Final geometry is shown in gure 2.9b and the dimensions in appendix C.

Figure 2.6: Connections between rocker and double parallellogram, in (a) the link using tensioned strings to connect the rocker with the main body. In (b) the rocker is held using two folded plate springs that are connected to the intermediate body B and a loaded cilinder at the bottom to x the vertical position. (c) Represents the guidance of the rocker using two rolling cilinders within a slit. The cilinders are pushed against the side walls with a strip under tension and the rocker is connected to the lower one. It is also possible that the rocker itself is one of the cilinders.

2.4 The double parallellogram assembly

The above discussed elements: double parallellogram, synchronization and actuator areassembled into one nal design. An overview is shown in gure 2.9, the synchronizationlever (b) is placed partially in the double parallellogram (c). The beams are surroundedby a rectangular tube that can hold the optics (mirror and focal lens) to the right and,at the top, connects the lever to the main body of the double parallellogram (d). Aroundthe tube the synchronization lever is placed with one connection rod at the top and twofor the intermediate body, this is done to minimize the rotation of the lever around the vertical axis. Also the connectors between the lever and the intermediate body are shown,the connector between the tube and the main body could be a plate at the left end and arod between tube and body to totally constrain it to the main body. This connection is notfully designed yet, and needs some further attention if this setup turned out to be betterthan the six folded plate-spring guidance of chapter 3. The actuators (e) t exact on bothsides inside the double parallellogram and are connected to the world on both sides. Thisis also the reason why this rectangular setup is preferred above the one using A-shapedplate springs of gure 2.2d. The maximal width of the plate springs is reached alreadyand using plate springs with an A-shape of the same width will result in a decrease ofthe stiness in y-direction (use equation 2.11). The coils are connected to the main bodyusing a coil-holder that is xed at the side of the body and two rods at the bottom tominimize the motion of the coils in y-direction; this mode is now at 1500 Hz (see gureC.5c). Other specic dimensions can be found in appendix C. 

The mode of the guidance (the moving mass is 56 g) round the z-axis is approximately1030 Hz. This value is obtained using a static force in y-direction to obtain the stinessof the double parallellogram and the following equation:

The maximal stresses that occur in the thin parts of the thickened plate springs at maximalstroke is approximately 100 MPa, mainly based on the choice to get a low overall stinessin the guidance and have long plate springs to have sucient room for the synchronizationand the actuators.A disadvantage of this setup is the need of a connection to the  -axis of the measurementmachine at the top, bottom and sides of the assembly to connect the double parallellogram,the synchronization lever and the actuators.

To calculate the total dissipated power during a normal measurement, rst this 'normal'measurement has to be dened. Propose an averaged freeform the motion of the measuringhead, and thus the guidance, would come close to a sinusoidal motion of 10 Hz with 1mm amplitude. The maximal acceleration during this motion is !2= (2f)2= 3:95m/s2. The dynamic force needed to accelerate 56 g is 0.22 N, the static stiness of doubleparallellogram and synchronization lever is 470 N/m, so the maximal static force is 0.47N. The absolute mean force needed for this motion is:

            (2.12)

and equals 0.6 N. Using the best obtained lineair actuator constant  = 0:41 W/N2thedissipated power reads 0.15 W. If the guidance is for example in the lower position, thedirection of the static force is opposite to the direction of the dynamic force. This isneglected in the analysis above.

Chapter 3

Guidance with six folded platesprings

Figure 3.1: Example of a guidance made of a monolith with six folded plate springs. (see [Ein03,chapter 1])

Another way to elastically guide a body along a (nearly) straight line is by using six foldedplate springs: three on either end of a moving tube oriented around in 120 degrees. Thissetup is shown in gure 3.1. Using six springs the -direction is once overconstrained,but using ve springs the guidance isn't symmetrical anymore. However there is still thepossibility after manufacturing to cut one plate spring if the 'click clack' eect is disruptingthe measurement or the straightness of the guidance. A challenge is now to place the platesprings within the design space. Figure 3.2 shows some dierent possible solutions thatenables the three measurement beams passing through one tube. An improvement canbe made by changing the rectangular plate springs into A-shaped ones (gure 3.2b). Thestiness in radial direction will nearly be the same but the stiness in z-direction willdecrease from $2500 to 1700 N/m.

As shown above it is more dicult to place the three folded plate springs in the designspace. To simplify the design also for this setup a Matlab-script is written to predictthe stresses and required actuator forces at dierent thicknesses and lengths of the springs (used formulas are explained in appendix B). A good starting point are folded plate springswith a length of 17 mm for a spring that makes the whole stroke (horizontally placed) and8 mm for the spring that has to compensate for the shortening of the longer ones. Thiscombination has the potential to t within the design space and requires an actuator thatalso ts. The minimal plate length as function of the thickness of the thin plate parts andthe total static force are shown in gure 3.3. Using a thickness of 0.09 mm, the maximalstresses in this setup are 390 MPa and the stiness is 2460 N/m. The static force neededto make the 2.5 mm translation is 6.2 N (as also shown in gure 3.3). To optimize thedesign A-form shaped plate springs are used where the stiness linearly decreases withthe width of the thin parts of the springs. A nal stiness of 1700 N/m is obtained.

 

Figure 3.2: Dierent setups of six folded plate springs within the design space. Within (a) alsothree linear actuators can be used, but they are much less effcient than a ring shaped-actuatorthat ts within all designs.

Normally the guidances of this type are made monolithically out of a simple piece ofmaterial (see gure 3.1), to minimize errors in the production proces and assembly. In thisdesign however, the choice is made to assemble the guidance. Three cilinders are separatelyturned on a lathe to make a circular tting and are provided with pins to x the rotationcompared to each other. Then that assembly is manufactured further. Out of the two outercilinders the folded plate springs are made, the inner cilinder is the basis of a ring-shapedactuator. The folded plate springs are made in the following steps: rst some larger piecesare cut out using normal EDM, to make a raw A-shape. Then the folded plate springswill be made using wire EDM. The 'legs' of the A-shaped springs can be milled before thewire EDM or eroded afterwards with normal EDM.

 

3.1 Ring-shaped actuator

To dene what type of ring-shaped actuator is the best, some dierent magnet locationsare investigated and shown in gure 3.4. The comparison is made using the analysis ofsection 2.3. The setup of (a) is used by the company BEI1and gives the best result at alow actuator diameter. The best thermal option is shown in 3.4b, where the soft magneticmaterial do not introduce stresses in the magnet in uctuating thermal conditions. Withinthis setup however the height of the actuator is a more critical design parameter, and the best actuator with the lowest height is shown in 3.4d. As displayed in the picture theactuator has a chamfer in the middle to allow a cone to connect the coil with a centraltube. The specications of this actuator are given in table 3.1. It is possible to makethe magnet in for example four segments to reduce stresses due to dierences in thermalexpansion coeffcients.

 

Figure 3.3: The minimal length L of a plate spring as function of the thickness and below it thetotal static force needed to de ect the six folded plate springs (L = 17 mm) over a distance of 2.5mm as function of the thickness.

Table 3.1: Specications of the ring-shaped actuator.

3.2 Assembly of the six folded plate spring guidance

The design is shown in gure 3.5 in a cross-section, the length of the thickened platesprings is 17 mm, the thin parts are 90 m thick. The base of the A-shape is 18 mmand the thickness of the thick parts of the springs is 6 mm. Overall outside diameterof the guidance is 58 mm, 3 mm larger than the target value but this will cause nofurther problems. Other nal dimensions are shown in appendix D. At the top in figure3.5 is a part that connects the guidance to the  -axis. There is on two sides some roomfor the measurement beams that measure the position of the axis (also shown in gure3.7a). Underneath it, the upper monolith with three folded plate springs is placed usinga cylindrical t. Below them a cone with a tube connects the upper and lower monolithwith each other and makes a connection with the coil. The outer upper ring is also thelimit for the maximal translation of 2.5 mm in the upward direction (so: away from thefreeform). The cone contacts the monolith directly under the outer connection of theplate springs, best seen in the upper left part of the picture. Below that, the limit ofthe downward direction is made between the upper part of the actuator and the cone.Below the actuator, the monolith with the lower three plate springs is placed, also usingcylindrical ts to position it. At the bottom, a cover is applied with a taper shape withan top angle of 125oto protect the plate springs. The individual parts are shown in gure3.6 to 3.8.

Figure 3.6: Monoliths with the three folded plate springs, showing in (a) the top view of the upperset springs that are close to the  -axis and in (b) a bottom view. In (c) and (d) the same for thelower set springs that are close to the freeform.

Figure 3.7: The connector with the  -axis in (a) where the three holes for the springs can bedistinguished and on the opposite side the two places that are left open for the  -axis measurementbeams. In (b) the lower housing with the taper shape of 125o.

Figure 3.8: The left gure shows the actuator with the circular soft magnetic core, the magnetwith the magnetization in diameter direction, the coil with a cross-section of 1.5¢6 mm and thecoil holder. At the right the cone shaped spacer that connects the coil holder to the upper andlower set of plate springs.

Analyzing the eigenfrequencies of the guidance (see gure D.2 for the mode-shapes) resultsin a rst disturbing frequency of 1924 Hz, this is the rotation of the tube round the z-axis.The displacement of the mean tube in radial direction is at 2225 Hz and the dierentmodes where the thickened parts of the plate springs are moving in z-direction occur at2311 Hz. Total mass of the assembly is approximately 400 gram, including the translatingmass of 50 g. The mode in radial direction is also calculated using a force in that directionand formula 2.11. The obtained stiness is 7700 k N/m, so the frequency is 2400 Hz (usingthe moving mass of 50 g). The advantage of this calculation is the possibility to usethe non-lineair solver in Unigraphics NX3 instead of the linear modal solver, but theresults are comparable as shown above. Using the stiness of 1700 N/m and the actuatoreciency of  = 0:056 W/N2the total dissipated power during a normal measurement canbe calculated in the same manner as in section 2.4. For this design the static force is 1.7N, the dynamic force 0.2 N, then the mean force is 1.65 N and thus the dissipated power is 0.15 W. By coincidence this value is the same as the power dissipated by the doubleparallellogram guidance so the selection has to be made based on machinability and thesimplicity of the assembly.

Chapter 4 Conclusion and recommendations

To summarize and be able to give some recommendations on which type of guidance tochoose, the main dierent properties and results are shown in table 4.1. As already men-tioned the dissipated heat during a normal measurement is the same for the two guidances.The rst disturbing eigenfrequency of the double parallellogram is approximately at 1030Hz. In this mode the whole moving mass of the guidance (= main body, coil with carrier,tube with optics and partially the lever and the intermediate body) is rotating around thez-axis and will in uence the optical measurement. This frequency is a lot lower than thelowest disturbing frequency of the six folded plate springs, 1924 Hz. This rotation modehas less aect on the optical measurement itself. The rst considerable disturbing motionis the movement of the guidance in x or y-direction at 2220 Hz.

Table 4.1: Specications of both the guidance types.

Looking at manufacturability of the components used in the both designs the lever of thedouble parallellogram and the three folded plate springs out of one piece are comparable indiculty. The advantage of the design with the six folded springs is the feasibility to easilyconnect the dierent parts using a circular tting made on a lathe. Also the connectionwith the  -axis is more simple, one connector at one end of the guidance is sucient fora proper mounting. The mounting of the double parallellogram is more dicult to realizebecause a xed world is needed at four sides of the guidance.

So a guidance using six folded plate springs is recommended as a good solution for designproblem stated the in the introduction. In the present layout there are still some improve-ments to be made in gap width of the ring shaped actuator. Now the air distance betweencoil and magnet is 1 mm, but optimizing the connection with the coil holder the actuatorcan be made even more ecient. Also the orientation of the folded plate springs mustbe investigated again. In the current design the folding line of the springs are inside, butplacing those to the outside (for example in the upper set), the rotation stiness can be considerably improved (>3000 Hz). The disadvantage is a higher stress in the connectorto the inner tube, but this perhaps can be solved using a good solid basis for the A-shape.