Abstract:
Apparatus for determining the distance, accurate to within an error of no more than 1 mm, from a first position to a second position. The apparatus includes a distance marker that includes a first strip having a sequence of uniformly spaced light-reflecting (or light-transmitting) regions in one embodiment and includes a second strip having a sequence of uniformly spaced light-reflecting (or light-transmitting) regions, the two strips being positioned parallel and adjacent to each other with the positions of the light-reflecting (or light-transmitting) regions of the first strip being translated relative to the positions of the light-reflecting (or light-transmitting) regions of the second strip by a fixed translation distance. The apparatus also includes a light sensor positioned adjacent to the strip to sense each such light-reflecting region (or light-transmitting region) and/or each non-reflecting region (or non-transmitting region) as that region moves past the light sensor. When the strip is extended to measure the distance from the first position to the second position, the number of light-reflecting (or light-transmitting) regions and/or non-reflecting (or non-transmitting) regions moving past the light sensor is counted to determine the distance. When the extended portion of the distance marker is reduced in length, the counting process is reversed to determine a new distance. The number of strips on the distance marker can be increased to N≧3 for improved distance measurement accuracy.

Description:
FIELD OF THE INVENTION 
     This invention relates to measurement of distances by electronic instruments. 
     BACKGROUND OF THE INVENTION 
     In many fields of activity, such as surveying, construction and navigation, accurate measurement of short and medium size distances, generally no more than 100 meters is needed. The required accuracy will vary with the activity. In some activities such as global surveying and navigation, distance measurement that is accurate to within a fraction of a millimeter (mm) is required for purposes of initial positioning. 
     Several non-contact distance measurement schemes have been proposed by workers in this field. One approach is frequency modulation/phase detection, which can be used over distances as large as 100 meters with a distance uncertainty, about 3,000 parts per million over the full measurement range. The accuracy obtainable is often limited by phase shift due to the dynamic range of the signals received. The full measurement range is often limited to 1 meter in order to obtain measurements accurate to within 1 mm. Schiek et al, in U.S. Pat. No. 4,238,795, disclose a microwave range measuring device in which microwaves are emitted at a first frequency, then at a second frequency, and are reflected from the object whose distance is to be determined. The reflected waves return to and are sensed by the device to determined the distance. A medium distance measuring device, disclosed by Budde in U.S. Pat. No. 4,829,305, also uses return of reflected microwaves from a target to determine the distance from the device to the target. 
     Optical triangulation of position is another attractive approach, possibly allowing distance measurements that are accurate to within a fraction of a millimeter over a modest distance range. This approach requires emission of a collimated beam from a reference position and determination of the distance by use of geometrical calculations based upon measurements of angles of reflection of the beam at the object. Sawabe et al, in U.S. Pat. No. 4,908,648, disclose use of radiation transmitted toward and reflected by the target, using two radiation signals that can overlap in space or in time. Resolution is very much dependent upon the distance measured and becomes monotonically worse as the distance increases. Further, the emitted beam would need to be modulated to insure immunity from changes in ambient conditions. 
     What is needed is apparatus for distance measurements that is good to within a millimeter or a fraction thereof, that does not require use of complex, movement-sensitive optical and electronics components, that provides approximately the same accuracy over the entire measurement range, that takes account of changing ambient temperature, that requires no recalibration at subsequent times, and that is compact and rugged enough for field work. 
     SUMMARY OF THE INVENTION 
     These needs are met by the invention, which provides an electronic tape measurement system that may be constructed with an inaccuracy as small as a fraction of a millimeter over measured distances as great as tens of meters. In one embodiment, the system includes a distance marker, such as an optically reflecting tape, that extends in a longitudinal direction with two parallel strips, each containing a sequence of length indicia. The length indicia consist of light reflecting regions alternating with regions that reflect little or no light. The alternating light-reflecting and non-reflecting regions in each sequence are spaced apart by a uniform distance d, such as 2 mm, and the positions of the light-reflecting regions in one sequence partly, but not completely, overlap the light-reflecting regions in the other sequence. This distance marker is extended from an initial position to determine the distance between two positions. As the tape is extended, the tape moves through a light sensing instrument that directs light at a small region on each of the two strips on the tape and senses and counts the number of length indicia from each of the two sequences that pass through the scanning instrument as the marker is extended (or contracted). 
     In one embodiment, signals produced and issued by the optical scanning instrument for the two sequences are received by two peak detector-comparator combinations that determine whether the light sensor signal produced by each strip is presently high or low. The number of light-reflecting and/or non-reflecting regions that pass the light sensor as the distance marker moves is counted for each of the two sequences, and the counts are compared to determine how far the distance marker has been extended or contracted, with a distance error of no more than d/2. The counting of light-reflecting regions that pass the light sensor can be done in hardware or in software, and an embodiment for each of these is disclosed. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 is a top view of a portion of a distance marker constructed according to a first embodiment of the invention. 
     FIGS. 2A, 2B and 2C illustrate other shapes, namely triangles, trapezoids and ovals, that can be used for the light-reflecting regions and non-reflecting regions according to the invention. 
     FIGS. 3A and 3B are perspective views illustrating use of the distance marker together with an associated electronic sensing module in reflection and transmission modes, respectively. 
     FIG. 4 is a schematic view of an electronic signal processing module or circuit, constructed according to the invention, to sense the length indicia, and count the number thereof, on the distance marker of FIG. 1 or FIG. 12 that pass a given point. 
     FIGS. 5 and 6 are schematic views of alternative arrangements of the peak detector portion of the invention shown in FIG. 4. 
     FIGS. 7A and 7B are graphical views of the signals V A  (t) and V B  (t) that would be processed by the module in FIG. 4 if the length of the distance marker is being extended. 
     FIGS. 7C and 7D are graphical views illustrating the signals V u  (t) and V d  (t), respectively, that would be generated by the module in FIG. 4 if the length of the distance marker is increasing and decreasing, respectively. 
     FIGS. 8, 9 and 10 are schematic views of the quadrature decoder shown in FIG. 4 for use of two adjacent strips, implemented in hardware, hardware and software, respectively, according to the invention. 
     FIGS. 11 and 12 are graphical views illustrating the signals V A  (t) and V B  (t) for two other embodiments of the invention. 
     FIG. 13 is a schematic view of a three-strip distance marker for another embodiment of the invention. 
     FIGS. 14A, 14B and 14C are graphical views illustrating an arrangement of three voltage signals V A  (t), V B  (t) and V C  (t) that is suitable for practicing the invention of FIG. 12. 
     FIG. 15 is a schematic view of a quadrature decoder module, implemented in hardware, corresponding to the three-strip distance marker shown in FIG. 13. 
    
    
     DETAILED DESCRIPTION 
     FIG. 1 illustrates a top view of a first embodiment of a portion of a distance marker 11 constructed according to the invention. The distance marker 11 includes a first and second parallel strips 13 and 15, separated by a thin strip 17 (optional) of material that is not light reflecting, in order to help isolate the two strips 13 and 15. The thickness of the thin strip is not critical, and it may be narrow or wide relative to the widths of the two parallel strips 13 and 15. The first strip 13 includes a first sequence 13-1, 13-3, 13-5, 13-7, etc. of rectangles or other suitable polygonal or ovular shapes, such as triangles, quadrilaterals and ovals, (all referred to as &#34;Polygons&#34; herein) that are non-reflecting for light incident on the marker 11 from the top or the bottom thereof. The first strip 13 also includes second sequence 13-2, 13-4, 13-6, etc. of Polygons that are highly reflecting for light incident from the top or the bottom of the marker 11. The second strip 15 includes a third sequence 15-1, 15-3, 15-5, 15-7, etc. of non-reflecting Polygons and a fourth sequence 15-2, 15-4, 15-6, etc. of light-reflecting Polygons. The light-reflecting Polygons have approximately the same area as, and preferably but not necessarily the same shape as, the non-reflecting Polygons. The non-reflecting Polygons and the light-reflecting Polygons are used together to determine the distance of extension of the distance marker 11. FIGS. 2A, 2B and 2C illustrate some other shapes of light-reflecting Polygons and non-reflecting Polygons that can be used with the invention. 
     A Polygon, such as 15-3, in the third sequence in the second strip 15 is positioned so that it &#34;overlaps&#34;, in a sense illustrated in FIG. 1, a nearest Polygon 13-3 from the first sequence of the first strip 13 and overlaps another nearest Polygon 13-2 from the second sequence of the first strip 13. Preferably, the amount of overlap of the Polygon 15-3 with each of the two Polygons 13-2 and 13-3 is equal in this embodiment. In a similar manner, a Polygon, such as 15-4, in the fourth sequence of the second strip 15 is positioned so that this Polygon overlaps a nearest Polygon 13-4 from the second sequence of the first strip 13 and overlaps another nearest Polygon 13-3 from the first sequence of the first strip 13, preferably by equal amounts. Each Polygon 13-i is assumed to be spaced apart from the Polygons 13-(i-2) and 13-(i+2) by a first center-to-center distance (i=3, 4, 5, . . . ); and each Polygon 15-i is assumed to be spaced apart from the Polygons 15-(i-2) and 15-(i+2) by a second center-to-center distance. The Polygons themselves may be any size, consistent with the requirements for sensing of reflected light from the light reflecting Polygons on each strip 13 and 15. For convenient reference, in this first embodiment the Polygons in the first and second strips 13 and 15 are assumed to have the same diameter d, measured in the direction of extension of the distance marker. 
     Assume that a first narrow slit of light of width w&lt;d/2 is directed at the first strip 13 and a second narrow slit of light of width w&lt;d/2 is aligned with the first slit and directed at the second strip 15. If the light produced by these slits is received at the surface of the distance marker 11, either (1) light from the first slit will be reflected and/or light from the second slit will not be reflected, or (2) light from the first slit will not be reflected and/or light from the second slit will be reflected, depending upon where the first and second slits are positioned along the distance marker 11. The two slits of light may be combined into a single slit extending across the two strips 13 and 15, if desired. Thus, if the distance marker 11 is moved past a two-slit light scanner or other suitable light sensing instrument to measure a distance from a first position to a second position, the scanner will count the number of transitions from reflecting to non-reflecting (&#34;light-to-dark&#34;), and from non-reflecting to reflecting (&#34;dark-to-light&#34;) that occur in each of the first and second strips 13 and 15. At any given position along the distance marker, a first light sensor and a second light sensor will independently sense the presence or absence of light reflected from the first strip 13 and the second strip 15, respectively. Comparing the number of transitions sensed and the presence or absence of reflected light at the scanner for each of the two slits, the position of the distance marker can be determined within a distance Δx≦d/2, because of the overlap of the Polygons in the first and second strips. The two light-producing slits need not be aligned with one another; the slits may be positioned a distance Δx=2Nd apart along the distance marker 11, where N is an integer, in the embodiment of FIG. 1. If the light reflecting Polygons and non-reflecting Polygons on one (but not both) of the strips are exchanged with one another, the distance Δx becomes Δx=(2N-1)d. 
     FIG. 3A illustrates use of the distance measuring unit in the embodiment of FIG. 1, using reflection of light from light reflecting Polygons of the distance marker 11, such as 13-4 and 13-6. As the distance marker 11 is extended in the direction of the arrow, the measured distance x increases, and the change in this distance x is monitored by a light sensing means 21 that receives light produced by a light source 23 and reflected by the light-reflecting Polygons on each strip of the distance marker 11. The electrical signals produced at the light sensor 21 for each of the strips 13 and 15, as the Polygons move past the sensor, are passed to a signal processing module 31 that is illustrated in more detail in FIG. 4. The embodiment shown in FIG. 3A uses reflected light to alternatingly activate and not activate certain phototransistors in certain optical switches shown in FIG. 4. In the embodiment shown in FIG. 3A, the light sensor 21 and light source 23 are positioned on the same side of the distance marker 11. 
     In an alternative embodiment, illustrated in FIG. 3B, the two adjacent strips 13 and 15 of the distance marker 11 are replaced by two adjacent strips 14 and 16, respectively; of a distance marker 12, the light-reflecting Polygons 13-2, 13-4, 13-6, etc. are replaced by cut-out Polygons or apertures 14-2, 14-4, 14-6, etc.; and the light-reflecting Polygons 15-2, 15-4, 15-6, etc. are replaced by cut-out Polygons or apertures 16-2, 16-4, 16-6, etc. A light sensor 25 is positioned on one side of the distance marker 12 and a light source 27 is positioned on a second side of the distance marker 12, both devices being adjacent to the strip 14. A light sensor and light source (not shown in FIG. 2B) are similarly positioned adjacent to the strip 16. Light emitted by the source 27 passes through a cut-out Polygon, such as 14-4, and is received by the light sensor 25 when that cut-out Polygon lies between the light source 27 and the light source 25. When a non-cut-out Polygon, such as 14-5, lies between the light source 27 and the light sensor 25, no light from the source is received by the light sensor 25, assuming that the light source 27 is contained in an appropriate housing (not shown in the interests of clarity) that absorbs or shields most of the light and allows light to exit from the housing only through an aperture in the distance marker 12. 
     The embodiments of FIGS. 3A and 3B rely upon light reflection and upon light transmission, respectively, at the distance marker. In the embodiment of FIGS. 3A or 3B, the amount of light received by a light sensor from a light source undulates between minimum and maximum values as the distance marker 11 or 12 is extended in a first longitudinal direction in the direction of the arrow in FIG. 3A (coordinate x increasing) or is contracted in a second longitudinal direction that is opposite to the first longitudinal direction (coordinate x decreasing), as indicated by the arrow direction in FIG. 3B. 
     FIG. 4 illustrates an embodiment 30 of an electronic circuit according to the invention that is suitable for sensing and counting the number of light-to-dark and dark-to-light transitions in both channels that occur as the distance marker is extended to measure a distance. The light-dark pulses from these transitions are produced at first, second and third substantially identical optical switches 31A, 31B and 31C, including the respective light sources 32A, 32B and 32C and the respective phototransistors 33A, 33B and 33C, each light source having a predetermined current source, denoted I cc . The amount of charge flowing at a particular instant of time in each of the phototransistors 33A, 33B and 33C increases with the amount of light incident upon the device. A resistor 34A connected between ground voltage and an emitter of the phototransistor 33A converts the output signal from the phototransistor 33A to a voltage signal, and this voltage is amplified by an amplifier 35A. The amplified voltage signal from the amplifier 35A is received by a peak detector 37A that detects and stores the most recently received peak voltage value, until this peak value is replaced by a subsequently received peak value. Optionally, the peak detector 37A can be preset to an initial value, at the time of system power-up, by a presetting module 39A. The amplified voltage signal issued by the amplifier 35A is also passed through a second resistor 41A to the non-inverting input terminal of an operational amplifier or comparator 43A. The comparator is arranged in a positive feedback loop, with the output terminal thereof being connected to the non-inverting terminal through a third resistor 45A to provide hysteresis. 
     The peak detector 37A issues an output signal, representing the most recently received peak voltage value, and this peak output signal is received by an input terminal of a potentiometer 47A, then passed from an output terminal of the potentiometer through a fourth resistor 49A to ground. In a first alternative, the fourth resistor 49A may be deleted and the potentiometer 47A may be set to produce a selected voltage issued at a third terminal of the potentiometer and received at the inverting terminal of the comparator 43A. The potentiometer 47A and the fourth resistor 49A form a voltage ladder that can be used to adjust the voltage appearing at a node 48A between the potentiometer 47A and the fourth resistor 49A. In a second alternative, the potentiometer 47A can be replaced by a fifth resistor or other resistive means, if no adjustment of the voltage appearing at the node 48A is required. The voltage signal appearing at the node 48A is received at an inverting terminal of the comparator 43A. A comparator output signal, corresponding to the voltage difference between the signals received at the non-inverting and inverting terminals of the comparator 43A, is issued at the output terminal of the comparator 43A and is received by a first input terminal 51A of a quadrature detector 53. This completes the processing of an input signal by channel A, which includes the phototransistor 33A, the first resistor 34A, the amplifier 35A, the peak detector 37A, the preset module 39A, the second resistor 41A, the comparator 43A, the third resistor 45A, the potentiometer (or other resistive means) 47A, the fourth resistor 49A and the first input terminal 51A. 
     In a similar manner channel B includes an optical switch 31B (light source 32B and phototransistor 33B), a first resistor 34B, an amplifier 35B, a peak detector 37B, a preset module 39B, a second resistor 41B, a comparator 43B, a third resistor 45B, a potentiometer (or other resistive means) 47B, a fourth resistor 49B and a second input terminal 51B, which operate in the same manner as the optical switch 31A, the first resistor 34A, the amplifier 35A, the peak detector 37A, the preset module 39A, the second resistor 41A, the comparator 43A, the third resistor 45A, the potentiometer (or other resistive means) 47A, the fourth resistor 49A and the input terminal 51A, respectively, of channel A. 
     The C channel, includes an optical switch 31C (light source 32C and phototransistor 33C), a first resistor 34C, an amplifier 35C, a peak detector 37C, a preset module 39C, a second resistor 41C, a comparator 43C, a third resistor 45C, a potentiometer (or other resistive means) 47C, a fourth resistor 49C and a second input terminal 51C, is not present or operative in the embodiment corresponding to the distance marker shown in FIG. 1 but is present in the embodiment corresponding to the distance marker shown in FIG. 13, discussed below. 
     Voltage signals V A  (t) and V B  (t) are received more or less continuously by a quadrature decoder 53 at the first and second input terminals 51A and 51B, respectively. The quadrature decoder 53 compares the transition signals being received at each input terminals 51A and 51B and determines if the extension distance of the marker is increasing or decreasing, by a hardware approach or by a software approach, as discussed below in more detail. The extension distance information (increase/decrease) is passed from the quadrature decoder 53 to an up/down counter 55 that counts the number of transitions and passes this information to a central processing unit or CPU 57 of a computer. The present position of the distance marker is then determined by the CPU 57 and is passed to a visual monitor or other visually perceptible display means for display of the present distance. The quadrature decoder 53 also issues an error signal on a separate line 61, and this error signal is received by the CPU 57 at a second input terminal, to advise the CPU that the quadrature decoder 53 has detected a non-valid transition in signals received from the two phototransistors 33A and 33B. Generation of this error signal is discussed below in more detail. Optionally, the CPU 57 may receive ambient temperature information from a temperature sensor module 63, in order to determine the temperature-corrected length L c  of a given section of the distance marker 11 in FIG. 1 whose measured length L u  is determined by the CPU 53 without reference to any change in temperature from a reference temperature T ref . The temperature-corrected length L c  is determined by the relation 
     
         L.sub.c =L.sub.u [1+α(T-T.sub.ref)],                 (1) 
    
     where α is the thermal linear expansion coefficient of the material used for the distance marker 11 in FIG. 1. Optionally, chosen values of the parameters α and/or T ref  can be provided at two input terminals of the CPU 57. 
     In a preferred mode of the invention, the first, second, third and fourth resistors 34A, 41A, 45A and 49A of channel A (and, similarly, the corresponding resistors 34B, 41B, 45B and 49B of channel B and the corresponding resistors 34C, 41C, 45C and 49C of channel C) have the respective resistance values 4.64, 5.11, 511 and 348 kilo-ohms and the power supply voltage V cc  for the comparators 43A, 43B and 43C is 5 volts to be compatible with circuitry in the quadrature decoder 53. 
     The circuit 31 shown in FIG. 4 includes a processing module that performs quadrature decoding and pulse counting. The phototransistor modules 33A and 33B convert a light beam reflected by the respective strips 13 and 15 of the distance marker 11 into signal currents. The optimum distance from a strip 13 or 15 to the corresponding light receiving surface of the phototransistor 33A or 33B is 0.8 mm in this embodiment, but the distance used in operation of the invention may be greater or less than the optimum distance. Preferably, the light receiving surfaces of the phototransistors that receive the reflected light beams from the Polygons that form the strips 13 and 15 are oriented for maximum sensitivity relative to these Polygons. 
     The currents produced by the phototransistors 33A and 33B are ultimately converted into voltage signals for the respective comparators 43A and 43B. If, say, the light receiving surface of the phototransistor 33A is directly over a &#34;light&#34; Polygon such as 13-2 in FIG. 1, the voltage produced at the non-inverting input terminal of the comparator 43A will exceed the threshold voltage V thr  produced at node 48A, and the output voltage signal of the comparator 43A will pull high. At the same time, the light receiving surface of the phototransistor 33B may be directly over a &#34;dark&#34; Polygon, such as 15-3 or directly over a &#34;light&#34; Polygon, such as 15-2, producing a low output voltage or a high output voltage, respectively, at the output terminal of the comparator 43B. As the distance marker 11 shown in FIG. 1 moves past the light receiving surfaces of the two phototransistors 33A and 33B in FIG. 4, the output voltage signals from the two comparators 43A and 43B will alternatingly pull high and pull low, being out of synchronism with one another by a time π=d/2w, where w is the rate of change of distance extension of the distance marker 11. The distance moved by the distance marker may thus be determined to within an error of no more than d/2. Increasing the resistance values of the first resistors 34A and 34B of channels A and B would increase the effective gain of the respective amplifiers 35A and 35B but would decrease the bandwidths of the phototransistors 33A and 33B. For the resistance values chosen in the preferred mode for the first resistors 34A and 34B, the bandwidths of the phototransistors 33A and 33B are each around 5 KHz. If the Polygon widths d on the distance marker (FIG. 1) are chosen as 2 mm, a 5 KHz signal corresponds to movement of the distance marker past the phototransistors at a theoretical maximum rate of 20 M/sec, which is much faster than any velocity contemplated for the distance marker. Thus, the available bandwidth of the phototransistors should not be a problem. 
     FIG. 5 illustrates an alternative arrangement of the peak detector portion of Channel A in the embodiment shown in FIG. 4. The peak detector 37A in FIG. 4 is replaced by a maximum peak detector 36A and a minimum peak detector 38A, both of which are optionally preset by a preset module 40A as shown in FIG. 5. The maximum peak detector 36A detects and holds the most recently received maximum value of a sequence of maximum peaks, and the minimum peak detector 38A detects and holds the most recently received minimum value of a sequence of minimum peaks. Normally, each two consecutive maximum peaks are separated by a minimum peak, and each two consecutive minimum peaks are separated by a maximum peak. The output signal from the maximum peak detector 36A and the output signal from the minimum peak detector 38A are received at a first terminal of a potentiometer 44A and at a first terminal of a potentiometer 46A, respectively, with a second terminal of each potentiometer 44A and 46A and an inverting input terminal of a comparator 43A being connected together, as illustrated in FIG. 5. The peak detector portion(s) of channel B (and of channel C) in FIG. 4 may similarly be replaced by a maximum peak detector and a minimum peak detector, arranged in parallel as in FIG. 5. For each channel A and B, a sequence of maximum voltage peaks and a sequence of minimum voltage peaks is generated. Two consecutive maximum (or minimum) voltage peaks of channel A are separated by a maximum (or minimum) voltage peak of channel B and two consecutive maximum (or minimum) voltage peaks of channel B are separated by a maximum (or minimum) voltage peak of channel A. Thus, four sets of voltage cycles occur, a maximum sequence for each of channels A and B and a minimum sequence for each of channels A and B. 
     The peak detector 37A and preset module 39A in FIG. 4 may be replaced by a reference voltage source 42 whose output voltage is received by one terminal of the variable resistor 47A, as illustrated in FIG. 6. The output signal of the comparator 43A goes high or low, depending upon the sign of the voltage difference between the output signal from the amplifier 35A and an offset voltage signal at node 48A. This offset voltage is a controllable fraction of the reference voltage from the source 42. 
     The quadrature decoder 53 in FIG. 4 receives the voltage signals V A  (t) and V B  (t) and receives and stores the voltage signals V A  (t-Δt) and V B  (t-Δt), each of which is either high (=1) or low (=0), indicating that the corresponding light sensor is receiving or is not receiving light reflected from one of the Polygons on the corresponding strip 13 or 15 in the distance marker 11 in FIG. 1. The quadrature decoder 53 forms the logical variables V u  (t)=[V A  (t)⊕V B  (t-Δt)*]·[V A  (t-Δt)⊕V B  (t)] and V d  (t)=[V A  (t)⊕V B  (t-Δt)]·[V A  (t-Δt)*⊕V B  (t)], where the time interval length Δt is predetermined and may be as short as Δt=50 μsec. The length Δt of this time interval limits the maximum time rate of change of extension of the distance marker to at most d/2Δt, if this ratio is less than 2 Bd, where B is the smaller of the bandwidths of the phototransistors 33A and 33B. The logical variable V u  (t)=1 if the distance marker length is increasing and V u  (t)=0 if the distance marker length is not increasing. The logical variable V d  (t)=1 if the distance marker length is decreasing and V d  (t)=0 if the distance marker length is not decreasing. The only three &#34;legal&#34; possibilities at any time t are (V u  (t), V d  (t))=(0,0), (1,0) and (0,1). The state (1,1) is not a legal condition or possibility and presence of this state indicates that the system is operating erroneously. Illegal transitions for this configuration are (0,0) . . . (1,1) and (1,0) . . . (0,1). An optional error signal V e  (t)=[V A  (t)⊕V A  (t-Δt)]·[V B  (t)⊕V B  (t-Δt)] may be computed and passed from the quadrature decoder 53 to the CPU 57 to sense the presence of skipped pulses. 
     FIGS. 7A and 7B illustrate the time variation of the functions V A  (t) and V B  (t), where the length of the distance marker 11 in FIG. 1 is assumed to be increasing at a uniform rate u &lt;&lt;d/Δt for convenience of representation, and channels A and B correspond to the strips 13 and 15, respectively. The function V A  (t) lags the function V B  (t) by a time interval of length Δt BA  =t 1  -t 0  =t 2  -t 1  =t 3  -t 2  =t 4  -t 3 , and it is assumed that Δt BA  &gt;Δt. By a consideration of cases, one easily verifies that ##EQU1## When the length of the distance marker 11 in FIG. 1 is decreasing, the function V A  (t) leads the function V B  (t) by the same time interval Δt BA  =t 1  -t 0 . This may be formally achieved by exchanging V A  (t) and V B  (t) and also exchanging V A  (t-Δt). By making these exchanges in Eqs. (2) and (3), one verifies that ##EQU2## FIGS. 7C and 7D graphically illustrate the time variation of the respective functions V u  (t) and V d  (t) where the length of the distance marker 11 in FIG. 11 is increasing and decreasing, respectively. If the length of the distance marker 11 is increasing with increasing time, the function V d  (t) will be identically zero and the function V u  (t) will differ from zero only during certain time intervals t i  &lt;t&lt;t i  +Δt. In a similar manner, if the length of the distance marker 11 is decreasing with increasing time, the function V u  (t) is identically zero and the function V d  (t) differs from zero only during certain time intervals t i  &lt;t&lt;t i  +Δt. 
     FIG. 8 illustrates a hardware embodiment 71 of the quadrature encoder 53 shown in FIG. 4. Time varying signals V A  and V B  are received at the input terminals of first and second latch type flipflops 73A and 73B, respectively. These flipflops may be D type or any other type that allows latching of an input signal until a subsequent clock pulse CP is received. A sequence of periodic clock pulses CP(t), with period Δt, is received on a clock pulse line 72 and is passed through a first signal inverter 74, with the inverted clock output signal from the first inverter 74 being received at the clock input terminals of the flipflops 73A and 73B. The inverted clock output signal from the first inverter 74 is passed through a second signal inverter 76, and the output signal from the second inverter 76 is passed to the clock input terminals of third and fourth flipflops 75A and 75B. The input terminals of the flipflops 75A and 75B receive the non-inverted output signals V A  (t) and V B  (t) from the non-inverting output terminals of the first and second flipflops 73A and 73B and issue the non-inverted output signals V A  (t-Δt) and V B  (t-Δt) and the inverted output signals V A  (t-Δt)* and V B  (t-Δt)*. 
     A first Exclusive OR (&#34;EX-OR&#34;) gate 77 receives the output signals V A  (t-Δt) and V B  (t) from the flipflops 75A and 73B at its two input terminals and forms and issues the output signal V 1  (t)=[V A  (t-Δt)⊕V B  (t)] at its output terminal. A second EX-OR gate 79 receives the output signals V A  (t-Δt)* and V B  (t) from the flipflops 75A and 73B at its two input terminals and forms and issues the output signal V 2  (t)=[V A  (t-Δt)*⊕V B  (t)] at its output terminal. A third EX-OR gate 81 receives the output signals V A  (t) and V B  (t-Δt) from the flipflops 73A and 75B and forms and issues the output signal V 3  (t)=[V A  (t)⊕V B  (t-Δt)]  at its output terminal. A fourth EX-OR gate 83 receives the output signals V A  (t) and V B  (t-Δt)* from the flipflops 73A and 75B at its output terminals and forms and issues the output signal V 4  (t)=[V A  (t)⊕V B  (t-Δt)*] at its output terminal. A first two-input AND gate 85 receives the signals V 1  (t) and V 4  (t) at its two input terminals and forms and issues at its output terminal the inverted product V 5  (t)=V 1  (t) V 4  (t)=V u  (t), which is the up-count signal. A second two-input AND gate 87 receives the two signals V 2  (t) and V 3  (t) at its two input terminals and forms and issues at its output terminal the output signal V 6  (t)=V 2  (t) V 3  (t)=V d  (t), which is the down-count signal. The signals V u  (t) and V d  (t) are carried by two signal lines 95 and 97, respectively, to the up-down counter 55 shown in FIG. 4. 
     Optionally, a fifth EX-OR gate 89 receives the signals V A  (t) and V A  (t-Δt) at its input terminals and forms and issues the output signal V 7  (t)=[V A  (t)⊕V A  (t-Δt)] at its output terminal; a sixth EX-OR gate 91 receives the signals V B  (t) and V B  (t-Δt) at its input terminals and forms and issues the output signal V 8  (t)=[V B  (t)⊕V B  (t-Δt)] at its output terminal; and a third two-input AND gate 93 receives the signals V 7  (t) and V 8  (t) and forms and issues at its output terminal the output signal V 9  (t)=V 7  (t) V 8  (t)=V e  (t), which is the error signal carried on the signal line 61 in FIG. 3. 
     FIG. 9 illustrates an alternative embodiment of the quadrature decoder module 53 and up-down counter 55 in hardware. Here, two signal inverters 97 and 99 receive the signals [V A  (t-Δt)⊕V B  (t)] and [V A  (t)⊕V B  (t-Δt)], respectively, and form and issue the output signals [V A  (t-Δt)⊕V B  (t)]* and [V A  (t)⊕V B  (t-Δt)]*, respectively. The signals [V A  (t-Δt)⊕V B  (t)] and [V A  (t)⊕V B  (t-Δt)]* are received at the two input terminals of the AND gate 85, and the product thereof is formed and issued as before. The signals [V A  (t-Δt)⊕V B  (t)]* and [V A  (t)⊕V B  (t-Δt)] are received at two input terminals of the AND gate 87, and the product thereof is formed and issued as before. 
     FIG. 10 illustrates a flow diagram used for an embodiment of the quadrature decoder 53 and the up-down counter 55 of FIG. 4 in software. The variables V A  (t-Δt) and V B  (t-Δt) and counter are initialized in step 113. The distance marker 11 in FIG. 1 begins to move, and the system begins to receive a stream of values V A  (t) and V B  (t) in step 115, which are assumed to be sampled with a time intervalΔt between consecutive samples. In step 117, the system forms the variable V u  (t)=[V A  (t)⊕V B  (t-Δt)*]·[V A  (t-Δt)⊕V B  (t)] and determines the present value of V u  (t), high or low. If V u  (t) is high, the system proceeds to step 119, the counter is incremented, and the system proceeds to step 129. If V u  (t) is not high, the system proceeds to step 121, forms the variable V d  (t)=[V A  (t)⊕V B  (t-Δt)]·[V A  (t-Δt)*⊕V B  (t)], and determines the present value of V d  (t), high or low. If V d  (t) is high, the system proceeds to step 123 and decrements the counter, and proceeds to step 129. If V d  (t) is low, so that neither V u  (t) nor V d  (t) is high, the system proceeds to step 125 and determines the present value of the error signal V e  (t)=[V A  (t)⊕V A  (t-Δt)]·[V B  (t)⊕V B  (t-Δt)]. If V e  (t) is high, a system error is present, and the system notifies the user of this in step 127. If V e  (t) is low, no system error is present, and the system proceeds to step 129, where the system determines if the distance marker is still moving and if the system should continue to process incoming values V A  (t) and V B  (t). 
     Optionally, the rate of change u of extension of the distance marker 11 in FIG. 1 may be computed as 
     
         u=dΔ.sub.count /2k(Δt),                        (8) 
    
     where Δ count  is the difference in the count that occurs in a time interval k(Δt), where k is a positive integer. This is most easily implemented in software but may also be implemented in hardware. 
     The voltage signals V A  (t) and V B  (t) illustrated in FIGS. 7A and 7B have a temporal offset of Δt o  =Δt/4 relative to each other, although this is not required for practice of the invention. FIG. 11 illustrates a pair of voltage signals V A  (t) and V B  (t) with a more general temporal offset Δt o  relative to each other, where it is assumed that 
     
         0&lt;Δt.sub.o &lt;Δt/2.                              (9) 
    
     The spatial resolution R for the embodiment of the invention that incorporates the voltage signals shown in FIG. 11 is limited at the lower end by 
     
         R&lt;w·max[Δt/2-Δt.sub.o,Δt.sub.o ],(10) 
    
     where w is the present spatial rate of change of extension of the distance marker. The right side of this inequality achieves its minimum value at Δt o  =Δt/4. 
     FIGS. 7A, 7B and 11 assume that the duty cycles of the voltage signals V A  (t) and V B  (t) are 50 percents, which is not required for practice of the invention. FIG. 12 illustrates a pair of voltage signals V A  (t) and V B  (t) with general duty cycles τ 1  /(τ 1  +τ 2 ) and τ 3  /(τ 3  +τ 4 ), respectively, and with a general temporal offset Δt o , for which the constraints are 
     
         0&lt;Δt.sub.o &lt;τ.sub.3,                             (11) 
    
     
         τ.sub.3 &lt;τ.sub.1 +Δt.sub.o,                  (12) 
    
     
         Δt.sub.o +τ.sub.1 &lt;τ.sub.3 +τ.sub.4      (13) 
    
     
         τ.sub.1 +τ.sub.2 =τ.sub.3 +τ.sub.4=Δt.(14) 
    
     Here, τ 1  and τ 2  are the temporal lengths of the high and low voltage signals for the strip 13 and τ 3  and τ 4  are the temporal lengths of the high and low voltage signals for the strip 15 (FIG. 1). By setting τ 1  =τ 2  =τ 3  =τ 4 , the situation in FIG. 11 is implemented. By setting τ 1  =τ 3  and τ 2  =τ 4 , a new situation is implemented, with identical duty cycles τ 1  /τ 1  +τ 2 ) for the two strips 13 and 15 in FIG. 1. The spatial resolution R for the embodiment shown in FIG. 12 is limited at the lower end by 
     
         R≧wmax[Δt.sub.o,τ.sub.3 -Δt.sub.o,τ.sub.1 -τ.sub.3 +Δt.sub.o, τ.sub.2 -Δt.sub.o ],(15) 
    
     and the right side of this inequality achieves its minimum value for Δt o  =Δt/4, τ 1  =τ 2  =τ 3  =τ 4  =Δt/2. 
     FIG. 13 illustrates a distance marker including a combination of three strips 131, 133 and 135, each having an alternating sequence of light-reflecting (or light-transmitting) and non-reflecting (or non-transmitting) Polygons and separated by optional non-reflecting (or non-transmitting) strips 137 and 139, as indicated. The voltage signals V A  (t), V B  (t) and V C  (t) that would be received on three channels from light sensors that monitor the passing of the Polygons on the respective strips 131, 133 and 135 are illustrated in FIG. 14A, 14B and 14C, where the signal V C  (t) leads the signal V B  (t) by an amount Δt BC  and the signal V B  (t) leads the signal V A  (t) by an amount Δt AB , as shown. In a symmetric situation, Δt AB  =Δt BC  =Δt CA  =Δt/6 and the minimum spatial resolution R becomes 
     
         R(min)=w·Δ/6,                               (16) 
    
     which is an improvement on the spatial resolution R=w Δt/4 for the two-strip combination illustrated in FIG. 1. More generally, if N adjacent strips (N≧2) of alternating light-reflecting (or light-transmitting) and non-reflecting (or non-transmitting) Polygons are used in a configuration analogous to that shown in FIG. 1, the minimum spatial resolution R becomes 
     
         R(min)=w·Δt/2N.                             (17) 
    
     The up-count and down-count voltage signals for the number count transitions for the channels A and B, B and C, and C and A are 
     
         V.sub.uAB (t)=[V.sub.A (t)⊕V.sub.B (t-Δt)*]·[V.sub.B (t)⊕V.sub.A (t-Δt)],                            (18) 
    
     
         V.sub.uBC (t)=[V.sub.B (t)⊕V.sub.C (t-Δt)*]·[V.sub.C (t)⊕V.sub.B (t-Δt)],                            (19) 
    
     
         V.sub.uCA (t)=[V.sub.C (t-Δt)*]·[V.sub.A (t)⊕V.sub.C (t-Δt)],                                            (20) 
    
     
         V.sub.dAB (t)=[V.sub.A (t)⊕V.sub.B (t-Δt)]·[V.sub.B (t)⊕V.sub.A (t-Δt)*],                           (21) 
    
     
         V.sub.dBC (t)=[V.sub.B (t)⊕V.sub.C (t-Δt)]·[V.sub.C (t)⊕V.sub.B (t-Δt)*],                           (22) 
    
     
         V.sub.dCA (t)=[V.sub.C (t)⊕V.sub.A (t-Δt)]·[V.sub.A (t)⊕V.sub.C (t-Δt)*].                           (23) 
    
     The three up-count voltage signals V uxy  (t) (x, y=A, B or C) determine whether the distance marker length shown in FIG. 13 has increased within the last pulse increment of length Δt by a length τ 1  /3; and the three down-count voltage signals V dxy  (t) determine whether the distance marker length has decreased within the last pulse increment by a length τ 2  /3. Here τ 1  is the temporal length of a light-reflecting (or light-transmitting) Polygon in FIG. 13 and τ 1  +τ 2  =Δt. The up-count and down-count signals presented to the up-down counter for the three-strip embodiment are 
     
         V.sub.u (t)=V.sub.uAB (t)⊕V.sub.uBC (t) =V.sub.uBC (t)⊕V.sub.uCA (t)=V.sub.uCA (t)⊕V.sub.uAB (t),                      (24) 
    
     
         V.sub.d (t)=V.sub.dAB (t)⊕V.sub.dBC (t)=V.sub.dBC (t)⊕V.sub.dCA (t)=V.sub.dCA (t)⊕V.sub.dAB (t).                      (25) 
    
     Three error voltage signals V exy  (t) (x, y=A, B or C) are defined analogously, using Eq. (4), and the overall error signal becomes ##EQU3## The up-count and down-count signals presented to the up-down counter for an alternative three-strip embodiment is 
     
         V.sub.u (t)=V.sub.uAB (t)+V.sub.uBC (t)+V.sub.uCA (t),     (27) 
    
     
         V.sub.d (t)=V.sub.dAB (t)+V.sub.dBC (t)+V.sub.dCA (t).     (28) 
    
     The number of adjacent strips of alternating Polygons used may be any number N≧2, and the spatial resolution R for this configuration becomes 
     
         R=wmax[τ.sub.1 /2N, τ.sub.2 /2N].                  (29) 
    
     The electronic signal processing circuit used to receive and analyze the voltage signals V A  (t), V B  (t) and V C  (t) in the embodiment corresponding to the distance marker shown in FIG. 13 is constructed using the full configuration, including the third or C channel, being used in the configuration of FIG. 4. More generally, N such channels would be used in the configuration of FIG. 4, if the distance marker in FIG. 12 contains N adjacent strips of Polygon sequences. 
     A suitable quadrature decoder module, implemented in hardware, for the three-strip embodiment shown in FIG. 4 is illustrated in FIG. 15. A suitable quadrature decoder module, implemented in software, for the three-strip embodiment shown in FIG. 4 is illustrated in FIG. 10. The embodiment shown in FIGS. 15 operates analogously to the embodiments shown in FIGS. 8 or 9. For definiteness, the up-count signal [V uAB  (t)⊕V uAC  (t)] and the down-count signal [V dAC  (t)⊕V dBC  (t)] are used to drive the up-down counter 55 in FIG. 15. However, the up-count signal [V uAB  (t)⊕V uAC  (t)] may be replaced by either of the other two up-count signals [V uAC  (t)⊕V uBC  (t)] or [V uAB  (t)⊕V uBC  (t)], and the down count signal [V dAC  (t)⊕V dBC  (t)] may be replaced by either of the other two down-count signals [V dAB  (t)⊕V dAC  (t)] or [V dAB  (t)⊕V dBC  (t)], consistent with Eqs. (24) and (25), with no other changes required in the quadrature decoder module illustrated in FIG. 15.