December 2003 MassMind newsletter

Yes, I know I skipped September, it is still in the works but having digital camera issues. I can't seem to get good closeups. Does anyone know how to adapt a macro lens to an HP Photosmart 435?

Position

Finding your position seems to have only one answer these days: GPS. And it is excellent. New systems are becomming available to increase the resolution of a reciever within a limited area. For example, one research group was able to control the position of a small "car" to within a few centimeters inside a large building. This is done by adding "pseudolites" which act as local area positioning beacons for the GPS system. Also, GPS recievers continue to decrease in size and price. It will not be long before linear encoders will be replaced with GPS units!

David Beals [DBeals at transdyn,com] has sent along some very nice SX code to read a GPS unit and display the info on an LCD.

Optical Encoders

One of the things that pushes up the price of encoders is the need for fine resolution and the tiny sensors and disks that this implies. There is a way to use larger components and achieve fine resolution. I'm not exactly sure why it isn't used more, but I think the reason is that, although it is very accurate and fine in the long run, it does not produce a perfect position signal for each "click." The idea is to use a "vernear" arrangement of sensors just like the marks on a vernier caliper as invented by Pierre Vernier in 1631. What! You don't know what that is?

Vernier Caliper:

"An instrument for accurately measuring length. It has two graduated scales, a main scale like a ruler and a second scale, the vernier, that slides parallel to the main scale and enables readings to be made to a fraction of a division on the main scale."
simulations: HTML Java

Now, it turns out that this vernier trick accounts for some of our visual abilities:

• http://www.bradford.ac.uk/acad/lifesci/optometry/research/projects/Localisation.htm "Spatial Localisation"

  Tasks that require one thing to be positioned with respect to another are referred to as spatial localisation tasks. These tasks are of interest because humans are generally 10 times better at this than at other positioning tasks. One example is vernier acuity. The precision with which vernier tasks can be performed by the visual system is amazing. As this paper shows, vernier thresholds are considerably smaller than the foveal cone diameter!
   

• http://www.bradford.ac.uk/acad/lifesci/optometry/PDF/VR-39-3681.pdf "Vernier acuity with compound gratings: the whole is equal to the better of its parts"

Vernier Simulation

Absolute Position Encoders

This type of encoder also uses several sensors, but each sensor follows its own track, reading one ring of the encoder disk. Each ring is printed with an encoding of a single binary digit; the LSB on the outside track and the MSB on the innermost track. This order takes advantage of the increased space towards the edge. By reading all the sensors, the rotational position of the disk can be known as a binary value, to the limit of resolution dictated by the width of the marks on the LSB track.

This disadvantages of this system are that it requires more sensors and so more IO pins, its accuracy is limited by the width of the LSB track, and it must be large enough to separate each track / sensor.

The advantage is, of course, absolute position sensing, which means it can be used with a very slow processor, as there is no worry about missing a pulse.

Psuedo random pattern encoders

This type of encoder has the unique property of letting you determine the absolute position (within the entire range) with less than the expected minimum number of bits required for complete absolute resolution. It does this at the expensive of you needing to actually move the encoder several steps in order to "see" the pattern in the changing bits. When enough steps in the encoder position has passed by, the resultant pattern then is unique to one and only one spot within the full combination of the code. In essence, after a minimum number of increments, the absolute position can then be easily determined.

The deal is that you need a cyclic (repeating) code sequence N bits long in which any given string of M bits is unique. You then draw this sequence around your code disk, and you need to move only M bit positions to read M bits of code at startup; you then know where you are in the whole N-bit sequence.

The figure shows, as an example, a 31-bit term Psuedo Random Binary Sequence: 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0,1, 0, 0, 1, generated by a 5-bit shift register. If you imagine a window, 5 bits wide, which can be pieced together by reading 5 bits in turn with a single photocell with another photocell following a standard linear pattern which acts as a "clock", each window position is unique and tells your position to within 1 bit.

Now, it's obvious that the best you can possibly do is N=2^M, because by reading M bits you only have 2^M possible values. It may be that there are some codes that have this property. On the other hand it is very easy to find codes that do nearly as well, giving N=(2^M)-1. These are the "pseudo-random binary sequences" which can be generated using linear-feedback shift registers. There's a useful section on these in Horowitz and Hill, and no doubt many other places. The good news is that the idea is basically sound. Suppose M=10, then you can do a 1023-position encoder and get absolute position after moving only 10 positions, i.e. about four degrees. A lot better than forcing your milling machine, industrial robot or whatever all the way back to its "home" position on every startup. I think that the most useful application would be for long linear tracks such as would be found on gantry robots powered by linear motors.

The bad news is that the pattern you must print on the code disk is no longer simply periodic, and it is therefore impossible (I think - certainly very tricky) to use Moire or vernier techniques to get high resolution with only one encoder. So this might be something you would add to an existing low resolution absolute position encoder: Basically, two low res encoders, in a vernier arrangement, can produce a VERY high resolution encoder, with absolute position readout after a few steps.

The other problem is that finding a sequence in a list of 1023 bits will be slow at best. If you know a fast way to manage this, please let me in on it?

Index hole multiplication

I remember a Heidenhain rotary encoder with a 'normal' incremental code. As an extra feature it had multiple (32?) index holes. The distance between these index holes was not equal and therefore you could retrieve the absolute position by counting the number of encoder pulses between two index hole pulses.

Sort of like pulse width modulation.

In order to be useful, the lines have to be very thin. Otherwise you can't count high enough between two index holes to provide for a unique identification of the index holes or you can't have enough index holes to make it worth doing. Here is a perfect opportunity for vernier encoding to help reduce the cost of the system by allowing lower resolution sensors to count the lines.

See also:

• http://iwant2study.org/lookangejss/01_measurement/ejss_model_AAPTVernierCaliper/AAPTVernierCaliper_Simulation.xhtml And HTML5 Verier Caliper simulator. Drag the slide and sample object from the exact points shown to measure the sample, then enter your measurement at the top and press enter to see if you are right.
• http://www.phy.ntnu.edu.tw/java/ruler/vernier.html An excellent little Java Applett that shows how to read a Caliper. Here is a sample of it showing a reading of 3.7mm
  
• http://www.heidenhain.com/phaise2/posps.html

  When a scale with a line grating is moved relative to another grating with the same structure - the scanning reticle - the lines of two graduations alternately align. The resulting light-dark modulation is sensed by photovoltaic cells.
  
  The index grating with one scanning field of two interlaced phase gratings generates four images on the measuring standard, with each image phase shifted by one-quarter of the grating period. Since only one scanning field is used to generate all four scanning signals, fluctuations in the intensity of the light such as are caused, for example, by local contamination of the scale, have an equal effect on the four photocell signals. The sinusoidal output signals therefore retain a high quality even with a certain degree of contamination.

• https://www.youtube.com/watch?v=0QLZCfqUeg4 Using a linear position encoder strip from a printer with the arduino.+
• http://www.geology.smu.edu/~dpa-www/robo/Encoder/pitt_html/encoders.html
• https://hackaday.com/2018/03/07/0-05-mm-precision-thats-a-moire/ A parametric 3D printable caliper using moire interference.
• http://www.site.uottawa.ca/~petriu/PositionRecovery.pdf Sensors and Measurement Techniques for Position Recovery and Object Localization. Pseudo-Random Encoding. Includes a section on "3D Object Recognition Using Structured Light"
• http://www.gurley.com/Virtual_Absolute.pdf Commercially available Pseudorandom encoders. Very good summary of best application cases.
• http://openjscad.com/#https://gist.githubusercontent.com/JamesNewton/c8598878736442c440bbe41d086291ac/raw//encoderdisk.jscad A parametric quadrature optical encoder for laser cutting. (change Output to "Parts" and select DXF or SVG Generation).
• https://github.com/HaddingtonDynamics/Dexter/wiki/Encoders Analog optical readings of sensors 90' out of phase form a circle when plotted. Each circle results from the motion of a single slit. With a 12 bit A2D, there are 4096 possible values from full dark to full light, then another 4096 back to full dark, so 8192 values per slit. With up to 200 slots per disk, that gives more than 1 MILLION positions per revolution.

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