A Sextant Calibrator

13 02 2011
Previous posts in this category are : Jesse Ramsden and his dividing engine, and Backlash and micrometer errors. Click on the figures to obtain larger images and use the back arrow to return to the text.

In Chapter 16 of my book, The Nautical Sextant, I explain some of the various ways of calibrating a sextant. For the most part, they employ methods and equipment not accessible to the amateur, but sometimes I let my amateur enthusiasms run away with me. Over a period of years I have gradually accumulated for relatively small sums versatile precision measuring devices that, with some loving attention, could be given a second lease of life. For measuring small angles with great accuracy and precision, the autocollimator is almost indispensable. About ten  years ago I spotted a “Job lot of five Hilger and Watts telescopes” and thought I could see in the picture two Hilger and Watts Microptic autocollimators, less their bases, the tube of another, and an Angle Dekkor (which need not concern us further). As the price was £100, I snapped them up. I made bases for the two autocollimators and overhauled them, while the third tube now forms part of a horizontal collimator for calibrating bubble sextants. Most people will perhaps need the terms “collimator” and “autocollimator” to be explained. It comes from the Latin colineare, to make straight, but a medieval scribe probably miscopied it as collimare and so it has stayed.

Collimator

This is simply a good quality lens held rigidly in a tube with, at its focus, a set of cross wires. The wires are illuminated by a lamp behind them (Figure 1). Since the wires are at the focus of the lens, the rays that make up their image as they leave the lens are parallel, and so the wires appear as if they were at infinity. It is as if a very distant object had been imported into the workshop. If the wires are made to be adjustable so that the centre line of the tube coincides  with the optical axis of the lens-crosswire combination,  the addition of a sensitive level to the tube gives us a means of importing a horizontal line of sight.

Figure 1: Diagram of Collimator

Autocollimator

Imagine now that the wires are illuminated indirectly by having the light reflected on to them by a semi-reflective mirror (or ” beam splitter”) set at 45 degrees and that the emerging parallel rays (full red in Figure 2) strike a mirror (full black) that is exactly at right angles to the axis of the instrument. It is obvious that the rays will be reflected back to where they came from and through the semi-reflector. Suppose now that the mirror is tilted through a small angle, α, shown in broken black in Figure 2. The rays will be deflected through twice this angle (broken red). If an eyepiece, with a measuring scale or other device at its focus is used to examine the displacement of the reflected image of the cross wires, it will  be possible to deduce this angle if the focal length, f,  of the lens is known. In fact, the shift of the image will be 2f tan α. Note that the distance of the mirror from the instrument does not appear in the equation.

Figure 2: Diagram of autocollimator

The Hilger and Watts Microptic autocollimator has a pair of moveable wires for measuring the deflection. The wires are moved by means of a micrometer drum that is calibrated directly in minutes and seconds of an arc, with the least division being 0.2 seconds and the total range being 10 minutes. Figure 3 shows a view through the eyepiece with the parallel setting wires deliberately offset from the reflected image by 15 seconds. The parallel wires can be centred with great precision around the image.  It is easy to see an error of as little as 2 seconds, and the experienced user is said to do rather better.

Figure 3: View through autocollimator eyepiece

The Calibrator

At the heart of the calibrator is a Soviet SNO-T sextant stripped of shades, mirrors and telescope mounting and mounted on three stout feet which have been ground and lapped so that the axis of the index arm is at right angles to the surface on which the sextant is placed. I did not of course sacrifice one of these fine sextants; I was able to obtain the part-instrument for US$45 from a source in India.

In place of the index mirror I mounted a 6 mm-thick steel sub-plate (Figure 4) ,which has a flat, a conical hole and a vee groove in which sit the ball-ended adjusting screws of the top plate (Figure 5). The random holes betray its source as a scrap heap.

Figure 4: Calibrator sub-plate

 

Figure 5: Calibrator top plate

The top plate in its turn carries an assortment of conical holes, a moveable hardened steel vee groove, and flats to accommodate the legs of the sextant being calibrated. This method of mounting prevents movement in any direction except upwards, while avoiding redundancy of constraints that might introduce stresses and strains; and if the mountings are accidentally disturbed it is easy to relocate them accurately.

Finally, a counterweight slides along an arm attached to the bottom plate in order to reduce asymmetry of loading of the index arm bearing. It counters the weights of the plates themselves together with the weight of the sextant being calibrated. There is a scale on the arm to cover sextants weighing between 0.8 and 1.6 kg. Before the calibrator can be put into service it must itself be calibrated.

Calibrating the calibrator

This uses the method described by Bouillon, Delisle and Pichard in the Canadian Journal of Physics for May 1, 1976 (Vol 54, No 9, pp 917 to 927). While it is an extremely tedious method in which it is very easy to displace instruments accidentally, it does have the merit of comparing the sextant with a fundamental standard of angle, the straight line. This of course always subtends exactly 180 degrees. It requires two autocollimators (though one plus a plain collimator would do), an accurately plane surface like a large surface table, a good quality mounted auxiliary mirror and an auxiliary base for the sextant. Though the authors do not mention it, the better the quality of the index mirror, the sharper is the reflected image as seen in the autocollimator and the easier it is to make accurate settings.

          Setting the standard

Before embarking on this, it is as well to overhaul the index arm bearing and micrometer mechanism to ensure that everything operates with silky smoothness and without any trace of backlash between the micrometer screw and rack. The latter is adjusted by means of a screw whose head lies under a blob of yellow paint at the rear of the mechanism.

Figure 6: Setting axes parallel to reference plane, 1

The axes of the two autocollimators have first to be set facing each other and parallel to the table surface, and for this a good quality mirror mounted on an adjustable  stand is needed. Its face should be set approximately perpendicular to the table and then it is placed between the autocollimators (Figure 7). It is viewed first through one instrument and the angle a of the reflected image of the horizontal wire on the vertical scale is noted. It is then rotated through 180 degrees and the vertical scale reading b obtained on the second instrument. If this instrument is now adjusted by an amount (a – b)/2 its axis will then be horizontal. If the other autocollimator is also brought to the same reading, then their axes will be horizontal and coincide with each other in the horizontal plane. Note this reading. Figure 7 may help to make this clearer.

Figure 7: Setting axes parallel to reference plane, 2 (afterBouillon et al.)

 Once this has been done, the autocollimators are adjusted to align the images of the vertical wires with each other. In practice, the setting wires are offset somewhat from the midline and set to embrace the image of the other’s wire, so that the direct image does not obscure the reflected images. It is convenient to set the micrometer of one to, say, 4′ 30″ and the other to 5 ′ 30″.  The axes of the autocollimators are then aligned in a straight line in both the vertical and horizontal plane and neither micrometer is changed until the end of the calibration process. Blobs of plasticine help to prevent accidental movement of the micrometers. These are visible in Figure 8, below.

          Setting axis and mirror

The upper plate is used as an auxiliary base for this process in which the plane of the sextant is brought parallel to the reference surface or, more exactly, that the index arm axis is brought vertical to the surface. In an undamaged sextant of good quality the two should mean practically the same thing. A dial indicator is used to bring each end and the middle of the arc into the same plane. After this, when an autocollimator is directed at the index mirror with the index at each end and at the middle of the arc, the readings of the horizontal wire on the vertical scale should be closely the same and if they are not, the base adjusting screw at the index mirror corner is adjusted until they are. The index mirror, temporarily borrowed from another sextant,  can then be adjusted to make the direct and reflected images of the horizontal wire coincide, so that the mirror is then aligned with the axis of rotation. Note also that sextant legs have been borrowed too.

          Dividing the standard

The sextant and one autocollimator, designated the moving autocollimator are now used as an optical divider to divide the standard of 180 degrees into segments, which must be factors of 180. Seven and a half degrees is the smallest practicable factor for my instruments, representing 15 degrees on the sextant scale, as the bases interfere with each other at smaller values. The lamp of the fixed autocollimator can be temporarily switched off, to avoid confusion of images.

The starting position is with the sextant index and micrometer set to 15 degrees exactly, using a magnifying glass to ensure accurate setting of the micrometer (Figure 8). The sextant on its base is then moved so that the reflected image of the vertical crosswire is precisely between the setting wires. Taking great care not to move the sextant, it is then set to zero, which of course then resets the mirror at an angle of 7½ degrees to its previous position. The auto collimator this time  is moved to intercept the beam and to bring the image of the wire back between the setting wires.

Figure 8: Starting position for calibration

Then the sextant is re-set to 15 degrees and rotated to  bring the vertical wire back again between the setting wires, after which it is again set to zero and the autocollimator moved. After 24 movements of the sextant and 24 of the autocollimator, the latter will be alongside the fixed autocollimator at a nominal angle of 7½ degrees (Figure 9). To avoid confusion, its lamp is then switched off and the fixed autocollimator switched on.

Figure 9: Final position

If there is no error of the sextant segment 0 to 15 degrees, when it is brought finally to zero the index mirror should be at right angles to the axis of the fixed autocollimator. If there is an error, Ε, in the segment, plus or minus, it will have been applied to each movement. The total can then be measured by bringing the vertical wire between the setting wires of the fixed autocollimator and noting the reading, ε. But this error applies to 24 movements (180/7.5) and the actual error Ε = ε/24.

There will of course have been setting errors of the sextant, which with care can be reduced to well under 3 seconds, thanks to vernier acuity aided by a magnifying glass and the superb workmanship of the SNO-T sextant (Figure 10), and there will have been reading errors of the autocollimators, which should also have been well under 3 seconds. However, these errors will have been averaged over 24 movements, and if the errors are random rather than systematic, will tend to cancel out, improving accuracy by a factor of √24, or very nearly five.

In a separate investigation of setting errors in which I tried to determine the errors involved in repeatedly re-setting the sextant micrometer to zero, always approaching from the same direction, I found the standard deviation of thirty readings to be 0.24″. This is a measure of dispersion of results around the mean and implies that 95 percent of results are likely to be within ± 0.47 seconds, say 0.5 seconds.

Figure 10: SNO-T sextant micrometer drum
E could then of course simply be applied to each reading 15 to 30°, 30 to 45°, and so on, leaving the autocollimators where they lie at a nominal 7½° while moving the sextant, but this would not average out setting errors. Instead, I elected to measure the error for each 15 degree segment by going through the full dividing procedure, to take advantage of the averaging effect. This was very tedious, but by adopting a zombie-like state and moving in an unhurried way, I believe I have avoided gross errors, and I had to do it only once. The results are given in Table 1.   The last-but-one column gives the error, plus or minus, for each 15 degrees segment and the final column gives the cumulative error from zero. These are truly remarkably small errors and a great tribute to Russian workmanship, though the results for the two other SNO-T sextants I have calibrated showed errors of between 0.7 and 14 seconds at various points on the arc.

Table 1: Calibrator errors

 
Calibrating a sextant
The   borrowed index mirror and legs had now to be replaced by the sub-plate and short legs mentioned in the first paragraph, and the top plate replaced on top of the sub-plate with its ball-ended feet resting in their respective places. After checking that the legs of the instrument to be calibrated are tight, it is placed on the top plate, one leg in the vee groove, one in a conical hole and one on a flat. The position of the vee groove can be altered and, within reason, new conical holes drilled to accommodate different sizes of sextant.     The autocollimators have to be raised higher off the surface table so that their axes are roughly at the same height as the index mirror of the sextant. Scrap pieces of large diameter steel water pipe with the ends faced in the lathe  make excellent raising blocks for this purpose. The axes of the autocollimators are then brought parallel to the reference surface as described above under Setting the standard following which one autocollimator can be returned to its box. Bouillon et al.’s paper describes a method of setting the index error to zero using the two autocollimators, but as this should be checked each time the sextant is used, I have omitted it here.
Next the sextant frame is set parallel to the plane and the index mirror adjusted as described above under Setting axis and mirror. The sextant is then set to zero and the calibrator to 120 degrees (Figure 11), using a magnifying glass to aid acuity and approaching the setting by rotating the micrometers always in the same direction, to avoid the effects of any backlash in either instrument. If you overshoot, it is necessary to go back half a turn or so and try again. The remaining autocollimator micrometer is set to zero and then directed at the index mirror to intercept the reflected image of the vertical wire between the setting wires. The autocollimator is not moved from this position and can be guarded against minor knocks by anchoring the base with plasticine.

Figure 11: Sextant about to be calibrated

Calibration can now begin. The sextant is set to 15 degrees while the calibrator is set to 105 degrees. If there is no error, this should bring the vertical wire back between the setting wires, and if there is an error, this can be measured by re-setting the wires and reading the autocollimator micrometer, also applying the calibrator error with due regard to its sign, plus or minus. The sextant is next set to 30 degrees and the calibrator to 90 degrees, and so on. For a quintant, the calibrator is set at 135 degrees when the sextant is at zero. The calibration procedure is so rapid, once the sextant has been set up, that it is easy to do a set of three observations and average the results. the results for a SNO-T sextant are given in Table 2 .

Table 2 : Errors for SNO-T sextant

For most purposes of course, such precision is not required and results could be given to the nearest 5 or 10 seconds, or, as C Plath, Cassens and Plath and Weems and Plath did, simply state that the instrument is free of error for practical use. Nowadays, probably only “Lunartics”, those insterested in making lunar distance observations, would wish to know the actual value of the errors. However, I hope that this illustration of what is possible will be of interest to a few people and should be grateful if they would point out (kindly of course)  important errors or omissions.

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Backlash and Micrometer Errors

12 03 2010

Lately on NavList there has been much written about accurate determination of index error, some of it relating to the instruments used and some relating to the observers’ physiology. Recently, Richard Pisko mentioned the subject of backlash, in the context of surveying instruments and I see there was quite extensive (though not always well-informed) discussion of this on the list in 2005. Not for the time being having any new sextants to pull apart, I have devoted a little time to considering aspects of accuracy of a few micrometer sextants of various ages. All are second hand but in good condition and have been completely overhauled.

 Resetting accuracy

 I did a prelimary determination that each sextant could be reset to a given reading within the precision with which I can read the testing instrument, by taking a series of thirty readings, each time resetting the sextant to the initial reading, from the same direction, to avoid backlash. Give or take a second or two,  all met this criterion.

 Backlash or lost motion

 Backlash due to lost axial motion in the micrometer mechanism was dealt with in three ways by different manufacturer.

 C Plath seems to have got things correct from the beginning (in about 1909). Manufacturers, like W Ludolph, Tamaya and Astra who imitated his design principle did well to do so. In this design, a small leaf spring (labelled”pre-load spring” in Figure 1) bears on one end of the worm shaft to hold a shoulder on the shaft against some sort of thrust bearing, and axial clearance is automatically taken up.

Figure 1: General arrangement of Plath micrometer mechanism

 Heath and Co, with their Patent Automatic Clamp arranged for the micrometer shaft to be held between two adjustable, hardened, conical centres (Figure 2). In principal, careful adjustment could remove for practical purposes all backlash while still allowing the shaft to rotate. It also allows for wear to be taken up if necessary. In theory, differential expansion of the shaft and the frame in which it is held might lead to end play developing.

Figure 2 : Micrometer shaft held between centres (Heath and Co)

 The third group, which includes Freibergers, SNO-Ts and pre-WWII Husuns, attempted to eliminate backlash by careful construction at the manufacturing stage and made no provision for user or automatic adjustment for wear, though one would not expect much wear in slow moving surfaces.

 In the Freiberger, the worm shaft runs in plain parallel bearings.  A 1 mm-thick bronze washer, shown in Figures 3 and 4), separates two thrust surfaces at one end of the shaft and if wear develops the solution is to fit a thicker washer. “Thicker” means an increase as little as 2 or 3 thousandths of a millimetre. Six arc seconds was considered the maximum acceptable amount of backlash in its first cousin, the SNO-T.

Figure 3 : Freiberger thrust washer in situ

Figure 4: Freiberger thrust washer exposed

 Husun fitted two opposed conical surfaces on the shaft to matching surfaces in a split bearing. Careful fitting here could again remove backlash for practical purposes and if play did develop, then it could be taken up by closing up the bearing.

 The micrometer mechanism is mounted on a frame to allow the worm to swing out of engagement with the rack. I have chosen to call it the “swing arm chassis” (Figure 1), since noone else seems to have given it a name . The swing arm chassis rotates around a bearing (Figure 1) that is attached to the expanded lower end of the index arm. Here, it is radial, rather than axial, play that can contribute to backlash.

 In the original C Plath, this bearing was a tapered one, similar to the index arm bearing, with provision to reduce clearances by closing up the bearing, but by the second half of the twentieth century, most manufacturers that adopted Plath’s pattern settled for well-fitting plain parallel bearings without provision for adjustment. Provided that it was well made in the first place, this seems to have been satisfactory and little wear was to be expected.

 Heath and later Kelvin and Hughes sextants mounted the swing arm chassis between adjustable centres (one of which is labelled “cone-pointed screw”  in Figure 2).

 Freibergers are altogether more complex. The bearings which carry the worm shaft rotate in a bearing machined in the casting carried on the end of the index arm. This bearing is eccentric, so that as it rotates against spring pressure in the indeex arm casting, the worm swings out of engagement with the rack. Axial backlash of this bearing can be taken up by tightening a nut which has a radial locking screw.

 All micrometer sextants take up lost motion between the worm and the rack by spring loading the contact between the two in various ways, so that the only clearance between the two is occupied by the lubricating oil film. The spring is labelled “swing arm spring” in Figure 1 and a leaf spring lies between the swing arm and the index arm in Figure 2.  Backlash here in a well made, undamaged sextant can arise if the oil is too thick, so that the thickness of the oil film varies with loading, or if it is too thin or absent, when “stick-slip” occurs to give irregular readings. Damage to the teeth of the rack or to the thread of the worm, while a cause of irregular movement and readings, should not in itself cause backlash, as contact is maintained by the spring loading. Rise or fall of the lower end of the index arm from the face of the limb with change in direction of rotation of the micrometer could in theory give rise to backlash, but in practice the keepers that hold the index arm close to the limb are usuallyadequate for their task.

 Finally, incorrect adjustment or wear in the index arm bearing can cause backlash. Most bearings are tapered ones with provision to adjust the clearance by moving the conical bearing surfaces axially against each other by means of a slender screw. If too slack, there is lost movement and if too tight, stick-slip, sometimes called “stiction”.

 This is perhaps a good place to warn inveterate fiddlers about overtightening the screw. The washer beneath the head of the screw usually has a square hole in it that fits on to a square on the shaft, or there is some other means of preventing the washer from rotating on the shaft. This ensures that as the shaft rotates, no rotational forces get transmitted to the screw head. The purpose of the screw is simply to move the shaft axially until there is a little drag indicating that clearance has been taken up. The screw may feel quite slack at this point. If you “firm it up” as one does with most screws, you may introduce stick-slip or, worse, twist its head off. Freiberger, SNO-T and some later C Plath sextants used well-fitting plain parallel index arm bearings that have no provision for adjustment.

The measurements

a) Backlash

 To measure backlash I used an autocollimator to project a light beam on to the index mirror of the sextant and to measure the deviation of the reflected beam after changing the micrometer reading (next photo). The least graduation of the autocollimator is 0.2 arc seconds, and people who used this particular type frequently were in ideal circumstances able to achieve an accuracy approaching 0.3 arcseconds. I find I can get reproducible readings to within 2 or 3 seconds.

Figure 5: Autocollimator set-up

 To measure backlash, I approached a zero micrometer reading alternately increasing or decreasing the reading to zero, and noted the differences between the two readings. I took the mean of ten pairs and calculated the standard deviation (SD). The latter is an estimate of the dispersion of the results around the mean. 1.96 SDs each side of the mean includes 95 percent of the data and a small SD implies that the results are tightly clustered about the mean (thirty readings might have been better, but there are limits to everyone’s patience…).

 b) Worm errors

 These are often neglected and much attention given to  calibration charts that says there is no error over the whole range of the sextant at 15 degree intervals, or that the instrument is “free from error for practical use”, generally meaning that the maximum error at these points does not exceed 6 or 12 seconds. But of course, this tells us nothing of the points between, nor of the errors within each degree

 I used the same  autocollimator set-up to estimate errors of the micrometer worm, by using the autocollimator to measure the error for each 5 minute step around a full rotation of the micrometer drum. Most people will be surprised at the size of some of these errors. It may be that the second-hand sextants that I used had been damaged, but I chose ones from my collection that I had carefully examined for damage during the course of their restoration or overhaul.

 Results

                                     Backlash                                 

 Tamaya 1977      Mean 0.9”      SD 1.1

 SNO-M 1966      Mean 1”         SD 0.8

 USN BuShips  

Mk II (Ajax Engineering)     Mean 1”          SD 0.8  

Freiberger  trommelsextant      Mean 6”          SD 0.7

Ditto, after  adjustment     Mean 2”          SD  2 

Hughes and Son 1938     Mean 6″       SD 2.5                           

Heath Navigational 1977     Mean 1’ 15”  SD 2.3      

                                            

Worm Error

 Discussion

a) Backlash

 The first three sextants all have leaf springs that oppose axial movements of the worm shaft, following the principle established by C Plath’s original design. It is easy to see how effective the method is in removing backlash, and the usual admonishment to always make a measurement turning the drum in the same direction can probably be ignored for this type of instrument.

 The Freiberger trommelsextant had backlash within its design parameters (assuming they were the same as for the SNO-T). However, I made up a washer a little thicker than 1 mm  (1.01 mm) and then hand lapped it to reduce it in thickness until the shaft just turned with a trace of drag. By then it was 0.99 mm thick and non-engineers may be surprised to know that the shaft would not turn at all when it was only 0.02 mm (less than a thousandth of an inch) thicker. The reduction in backlash was obvious.

 I could by no means reduce the backlash of the Heath sextant to less than 75”. The reason for it quite mystifies me and it may be something to do with the way the worm is skewed across the rack in this design of sextant. Making a new worm reduced the maximum worm error somewhat and certainly reduced the individual errors around the circumference. Truly, the heyday of British sextant making had long passed when this instrument was made.

 b) Worm error

Many people, I am sure, will be surprised at the size of some of the errors at first glance at the graphs. Bear in mind that the worst are those that deviate most from a horizontal straight line as, over a full rotation, there are always exactly 360 degrees in a circle and the errors have to sum to zero (in practice there is nearly always a deficit of plus or minus a handful of seconds).

 On this basis, the SNO-T, a 1938 Husun with a new worm, a 1938 Husun with the original worm, an Ajax Engineering US Navy MkII and a Freiberger all perform well, but before deciding that the Tamaya, Heath and SNO-M don’t cut the mustard, consider that they were second-hand instruments and that 10 seconds at the periphery of a radius of 150 mm represents a movement of about 4 thousandths of a millimetre. Invisible particles of dust and fibres and invisible nicks can thus easily lead to variations of this amount. Perhaps the moral of this story is to be careful not to drag the worm across the rack, to keep it well brushed and to oil it regularly.