Floor patterns for navigation corrections

Abstract

For a vehicle using dead reckoning or some other type of navigation which accumulates error as the vehicle moves, this invention provides a simple, single-sensor, low-cost, highly-accurate system for correcting navigation errors. The system uses a marker structure with optical density which is formed from one or more periodic patterns. The vehicle's navigation computer records the density, measured by the sensor, as the sensor moves on a line over a marker at a known location, then it processes the recorded density function to get the correct navigation parameters. If the vehicle's usual path passes over a marker, that path can be used without change for acquiring navigation corrections.

Description

OTHER PUBLICATIONS

Odin, a robot for odometry, http://philohome.com/odin/odin.htm

Bong-Su Cho, Woo-sung Moon, Woo-Jin Seo and Kwang-Ryul Baek,

• • “A dead reckoning localization system for mobile robots using inertial sensors and wheel revolution encoding,” Journal of Mechanical Science and Technology 25 (11) (2011) 2907 2917, Springer

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

Not applicable

REFERENCE TO A SEQUENCE LISTING, A TABLE, OR A COMPUTER PROGRAM LISTING COMPACT DISC APPENDIX

Not applicable

STATEMENT REGARDING PRIOR DISCLOSURE BY THE INVENTOR

There have been no prior disclosures by the inventor.

BACKGROUND OF THE INVENTION

Field of the Invention

This invention relates generally to methods for correcting dead-reckoning navigation errors in vehicles which can move past an area having a patterned position marker. In particular, the area can be on a horizontal floor, such as a warehouse floor or a table top; or it can be on a wall, a ceiling, or some other surface. The present invention describes a class of patterns to be used for markers, and methods that an autonomous robot or other vehicle can use to analyze the patterns for determining coordinates to be used for navigational corrections.

The purpose of this invention is providing a simple, single-sensor, low-cost, highly-accurate system for correcting navigation errors as a vehicle moves without changing direction along a dynamically variable path which crosses one or more two-dimensional patterned markers.

The Prior Art

Mobile robots commonly use odometry for position estimation. From wheel rotation measurements a robot can estimate its direction and travel distance along its path, and use those estimates to approximate its position. It is well known that this dead-reckoning navigation suffers from accumulation of errors, so the use of dead-reckoning for extended travel needs a method for correcting the estimated position. Many methods are known.

For example, a U.S. Pat. No. 4,541,049 by S. H. N. Ahlbom, et al. derives corrections using signals from a vehicle-mounted set of optical sensors arranged in a line as the vehicle crosses known-position lines on the floor. The present invention is simpler in that it uses a single optical sensor.

The robot known as “Odin” (reference: “Odin, a robot for odometry”) uses a related method: the robot optically senses one leg of a known-position L-shaped pattern as it moves into the angle between the two legs, then it turns to cross and sense the other leg; it uses the positions sensed to determine its location. The present invention does not need to make large path direction changes to get the data necessary for navigation corrections.

U.S. Pat. No. 4,847,773 by C. C. van Helsdingen, et al. describes a grid of “passive” markers over which a sensor-equipped vehicle moves, but it does not describe markers with patterns similar to those of the present invention.

U.S. Pat. No. 5,111,401 by H. R. Everett, et al teaches the use of a stripe with position markers at various points; the robot moves until it encounters the stripe, then follows the stripe until it encounters a position marker, which it uses to establish its location by referring to predetermined coordinates of the position markers. The position markers are wider places on the stripe. The patent mentions that the “simplistic marker pattern” can be replaced with a more complex pattern providing unique identification of the marker, but it does not provide a detailed description of any such complex pattern. The present invention does not need stripe following, and it has complete descriptions of position markers.

BRIEF SUMMARY OF THE INVENTION

Overview

The key concept for this invention is the fact that Fourier analysis can reveal features of an object that is the superposition of periodic structures. The prototype for the patterned position marker is a finite, two-dimensional region, such as a circular disk, for which the optical density varies with position in a particular way. It will be apparent to persons familiar with the appropriate art that optical density is not the only characteristic which is appropriate, but using it here simplifies the description. Generally, the optical density will vary periodically in various directions; more precise descriptions of this are given below. For simplicity, in the following the term marker will mean “patterned position marker.”

A vehicle using this invention carries a sensor which measures an average optical density in a small region. The exact nature of the averaging is not important, and the size of the region is not important as long as it is small compared with the sizes of features of the marker pattern, but large enough to reduce signal noise to an acceptable level. As the vehicle moves the sensor reports the measured density to the navigation computer (henceforth, computer) of the vehicle. The computer stores a portion of the measured density as a function f(s) of distance s along the sensor's path, and uses that function to determine corrections of navigation errors. In the following, this function f is called the path-density function, and the term sensor refers to the density sensor just described.

Using Simple Patterns

Some Definitions

FIG. 1 shows a simple-pattern marker 1 with part of the path 2 of the sensor as the vehicle moves past the marker, the dashed parts indicating that the path is incomplete.

Capital letters are used for quantities related to markers, and lower case letters are used for quantities related to the sensor path. Boldface sans-serif letters are used for vectors and points. Thus, points on sensor path 2 are lower-case letters p, p₁, p₂, etc., whereas the pattern reference point C₀=<X₀, Y₀> on pattern axis 3 is upper case. For descriptions of specific notations, see TABLE 1: LIST OF SYMBOLS USED. Terms specific to this invention are italicized when defined, and shown in TABLE 2: LIST OF TERMS.

For the part of the path shown, the motion starts at point p₁, then goes through points p₂, p₃, p₄, and p₅, to p₆. The part of the path between p₂ and p₅ is a straight line aligned with unit direction-of-motion vector v. The point p₂ is the reference point for distance s (i.e., s=0 at p₂) on the straight segment of the sensor's path.

Simple-pattern marker 1 (see FIG. 1) is a circular disk of radius R₀ centered at C₀. Its optical density varies in the direction of unit orientation vector I₀, but is independent of position in the direction of unit vector J₀ orthogonal to I₀. The counterclockwise angles in radians from the x-axis to I₀ and J₀ are α₀ and α₀+π/2, respectively.

Within the marker, the optical density is given by the marker-density function F, which will appear as F(x, y) when it is necessary to show the x and y coordinates. At times it will be convenient to use the notation F(p), where p is the point with coordinates x and y. For a simple-pattern marker, F is a sinusoidal function of position in the direction of I₀. The distance between adjacent density peaks measured in the direction of I₀ is the wavelength Λ₀. The function is independent of position in the direction of J₀.

The function F has not been defined outside the circular disk of the marker, but the discussion which follows is simplified if F is extended to the entire x,y-plane. Therefore, let F⁺ denote the function which, throughout the x,y-plane, is constant on lines parallel to J₀, and periodic in the direction of I₀; and equals F inside disk 1. The maxima of F⁺ are on certain lines parallel to J₀.

Points p₃ and p₄ are the points at which path 2 intersects the boundary of 1. Line 3 is the pattern axis; it passes through C₀ and is parallel to the orientation vector I₀. The counterclockwise angle in radians between I₀ and v is θ₀ satisfying 0≦θ₀<2π.

As the sensor moves along the straight-line path across the marker it will report a density which is sinusoidal with wavelength λ₀. If cos(θ₀)≠0, λ₀ satisfies |cos(θ₀)|λ₀=Λ₀. The following discussion uses angular frequencies ω₀=2π/λ₀ and Ω₀=2π/Λ₀. Thusω₀=|cos(θ₀)|Ω₀.  (1)

In FIG. 1, there are 12 cycles of the density across the disk, so Λ₀=R₀/6, and Ω₀=12π/R₀. Values of Ω₀ of about this size will be used in the figures, but practical applications will have much higher values of Ω₀.

The density reported by the sensor is a function of x and y, but since it is measured on the sensor's path, the computer stores it as periodic samples of the path-density function f(s). On the part of the path intersecting disk 1f(s)=a cos(ω₀s+ψ₀)+b,  (2)where a, b, and ψ₀ are constants with b≧|a| and 0≦ψ₀<2π.

The function F(x, y) is anchored to the x, y-axes and C₀, but f(s) is not. Instead, v and p₂ determine its relationship to F(x, y), hence to the x, y-axes and C₀. Therefore, f(s) carries some information about where the path is located. The computer has approximate values {tilde over (p)}₂ and {tilde over (θ)}₂ for p₂, and θ₀, respectively. The angle {tilde over (θ)}₀ determines a unit vector {tilde over (v)} which is an approximation of v. The goal is to use f(s) to get {tilde over (θ)}₀ and δ₀, the latter being the component of the displacement δ={tilde over (p)}₂−p₂ in the direction of I₀. The method for this involves mathematical concepts described in the following.

Mathematics for Getting Position and Direction from Phase and Frequency

An expression for F(p) on the path between p₂ and p₅ will be obtained, then that will be related to f(s).

For a point p (e.g., on the sensor's path), let S(p) denote the signed distance (positive toward I₀) from C₀ to the orthogonal projection P of p onto line 3. The reader is reminded that since I₀ is a unit vector, the dot product (p−C₀)−I₀ is the (signed) length of the projection of the line segment from C₀ to p onto line 3, and δ₀=δ·I₀, soP=C₀+((p−C₀)·I₀)I₀,andS(p)=(p−C₀)·I₀.  (3)The value of the function F⁺(p) at p is determined by S(p). Let φ₀ be the phase at C₀, soF⁺(p)=A cos(Ω₀S(p)+φ₀)+B,  (4)where A and B are constants associated with the pattern. Usually, B≧|A| so that F(p) is always nonnegative.

Between p₂ and p₅ on the sensor's path, p is a function of s given by p(s)=p₂+sv, so from equations (2) and (4),a cos(ω₀s+ψ₀)+b=f(s)=F⁺(p(s))=A cos(Ω₀S(p(s))+φ₀)+B,  (5)which is true while p(s) is between p₃ and p₄.

An early part of the computer processing of f is removal of the DC component b by any well-known technique (high-pass filtering, finding the average value b and subtracting it from f, etc.). Thus, in the following, without loss of generality, b, and B will be taken to be zero. Also, it is obvious that the cosine amplitudes must be equal: a=A. Therefore, the cosine factors in equation (5) are equal:cos(ω₀s+ψ₀)=cos(Ω₀S(p(s))+φ₀) for p(s) between p₃ and p₄.Since this is true over an extended interval of s, the arguments of the cosines are equal except for an additive multiple of 2π to account for the periodicity of the cosines. Thus, from equation (3),ω₀s+ψ₀=Ω₀(p(s)−C₀)·I₀+φ₀−m2π,  (6)where m is an integer. In particular, since p(0)=p₂, for s=0 equation (6) isψ₀=Ω₀(p₂−C₀)·I₀+φ₀−m2π.  (7)This can be rearranged and applied to the definition of δ₀ to getδ₀=({tilde over (p)}₂−p₂)·I₀=({tilde over (p)}₂−C₀)·I₀+(φ₀−ψ₀−2πm)/Ω₀.  (8)

Therefore, if ψ₀ and m can be found, any position can be partially corrected by subtracting δ₀I₀ from it. In particular, the partially-corrected {tilde over (p)}₂ is {tilde over (p)}₂−δ₀I₀.

A single simple-pattern marker does not, by itself, provide enough information to determine m but there is a condition under which m can be computed. Rearrange equation (8) to get (since Ω₀=2π/Λ₀)

$\begin{array}{cc}m=\frac{1}{{\Lambda }_0}\left({\tilde{p}}_2-C_0\right)·I_0+\frac{1}{2\pi }\left({\phi }_0-{\psi }_0\right)-\frac{{\delta }_0}{{\Lambda }_0}.& \left(9\right)\end{array}$ If|δ₀|<Λ₀/2,  (10)then m is the nearest integer to the right side of (9) with δ₀=0; i.e.,

$\begin{array}{cc}m=\mathrm{Round}\left(\frac{1}{{\Lambda }_0}\left({\tilde{p}}_2-C_0\right)·I_0+\frac{1}{2\pi }\left({\phi }_0-{\psi }_0\right)\right).& \left(11\right)\end{array}$ The restriction on δ₀ is too severe for most applications. Markers comprising superposed patterns, described below, have much weaker restrictions.

The computer will have the following information about the marker: the coordinates X₀ and Y₀ of the marker reference point C₀, the orientation vector I₀, the angular frequency Ω₀, and the phase φ₀ at C₀. All of these values, collectively called the marker parameters, are determined at the time of installation of the position marker, then delivered to the computer.

Under condition (10), after ψ₀ has been found, the computer can use equation (11) to get m, then use m in equation (8) to get δ₀. Then the corrected value of {tilde over (p)}₂ is {tilde over (p)}₂−δ₀I₀.

After ω₀ has been found, the computer can use it in equation 1 to put restrictions on the approximation {tilde over (θ)}₀ to θ₀. This does not work well when |d cos({tilde over (θ)}₀)/d{tilde over (θ)}₀| is close to zero (i.e., when {tilde over (θ)}₀ is close to 0 or π).

A given value of ω₀ determines several values of {tilde over (θ)}₀, however: if {tilde over (θ)}₀ is a solution satisfying 0<{tilde over (θ)}₀<π, then each of π+{tilde over (θ)}₀, π−{tilde over (θ)}₀, and 2π−{tilde over (θ)}₀ is a solution. Nevertheless, if one of these values is close enough to the current value of θ₀ it can be used for the revised value of {tilde over (θ)}₀. Usually, this is the case if θ₀ and {tilde over (θ)}₀ are reasonably close to the same one of π/4, 3π/4, 5π/4, and 7π/4. Therefore, it is best if the path crosses the marker with θ₀ close to one of these values.

Getting the Frequency and Phase

The computer uses odometry to measure s and to estimate the sensor's position {tilde over (p)}and direction of motion {tilde over (v)} (at angle {tilde over (θ)}₀). When it starts on the straight segment from p₂ to p₃ it stores {tilde over (p)} as {tilde over (p)}₂, an approximation to p₂.

After the sensor has passed over the marker, the computer determines ω₀ and ψ₀ by some means. In principle, the frequency ω₀ and phase ψ₀ can be determined by finding the midlevel points of f, and doing some simple calculations. This is a poor method because it is relatively sensitive to random errors (e.g., optical noise from dirt or breaks in the density pattern).

Consider, instead, using the Fourier transform {circumflex over (f)}(ω) of the stored path-density function f. Since the complex representation of the cosine is

$\cos \left(\gamma \right)=\frac{1}{2}\left(e^{\mathrm{j\gamma }}+e^{-\mathrm{j\gamma }}\right),$ where j is the imaginary unit (j²=−1), f(s) can be written (with b=0; see equation (2))

$\begin{array}{cc}f\left(s\right)=\frac{a}{2}\left(e^{j\left({\omega }_{0}s+{\psi }_0\right)}+e^{-j\left({\omega }_{0}s+{\psi }_0\right)}\right),& \left(12\right)\end{array}$ so {circumflex over (f)}(ω) is zero except at two points:{circumflex over (f)}₊=cae^jψ0 at ω=ω₀,and{circumflex over (f)}₋=cae^−jψ0 at ω=−ω₀,  (13)where c is a positive constant determined by the form of the Fourier transform used.

Therefore, to get both ω₀ and ψ₀, the computer can examine the values of {circumflex over (f)}(ω) for ω>0 to find the nonzero value at ω₀, then evaluateψ₀=arc tan(Im({circumflex over (f)}₊(ω₀))/Re({circumflex over (f)}₊(ω₀))),  (14)where Re(•) and Im(•) extract the real and imaginary parts, respectively.

It will be apparent to persons familiar with the appropriate art that since the computer stores discrete samples it should use the Discrete Fourier Transform (DFT) instead of the continuous-domain Fourier transform appearing in the discussion above. Application of the DFT is illustrated in FIGS. 2 and 3, which, for purposes of illustration, show functions with continuous lines instead of discrete points.

FIG. 2 shows path-related functions, and FIG. 3 shows features of corresponding DFTs. The function ω is the window function which multiplies the extended density function f⁺ to produce the path-density function f that the sensor reports. FIG. 2 shows the wavelength λ₀ expressed in terms of the corresponding angular frequency ω₀.

The functions {circumflex over (f)}⁺ and {circumflex over (f)}, which are the DFTs of f⁺ and f, respectively, are complex functions of ω. For simplicity, FIG. 3 shows |{circumflex over (f)}⁺| and |{circumflex over (f)}|, the absolute values of these functions. The graphs of the functions and the discussion that follows have been simplified by ignoring the fact that the DFT might not be computed over the entire span from p₃ to p₄, or over an integral number of cycles of the function f⁺.

Multiplication of functions implies convolution of their DFTs. In particular, {circumflex over (f)} is the convolution of {circumflex over (f)}⁺ and {circumflex over (ω)}, the DFT of ω. The latter is a multiple of a sinc function, so |{circumflex over (f)}| is a multiple of the absolute value of a sinc function translated so its center is at ω₀, as shown in FIG. 4. The width of the central peak is inversely proportional to the distance between p₃ and p₄.

The computer can use |{circumflex over (f)}| to determine ω₀, either from the position of the peak, or by determining the centroid of the function. The latter should be more accurate if certain precautions are taken, such as using only that part of the function in the central peak. After getting ω₀, the computer can determine ψ₀ as indicated in equation (14).

Some details that will be apparent to persons familiar with the appropriate art have been omitted. The graphs in FIG. 4 are drawn as if the DC components of f⁺ and f have been removed.

Since the method for determining the frequency and phase integrates measured values (by summation, through the DFT), the accuracy is better than that of methods that use single point measures. It is also much less sensitive to noise.

Using Multiple Simple Patterns

The method just described fails if the vector v is nearly orthogonal to I₀. One way to provide for complete correction is to have the sensor path pass across several markers. This is illustrated in FIG. 4, in which the sensor path 8 passes sequentially across markers 5, 6, and 7 as the sensor moves from p₁ to p₆. The patterns are rotated from each other, as indicated by the orientation vectors I₁, I₂, and I₃. Each of markers 5 and 7 provide coordinate corrections in the directions of their orientation vectors.

Marker 6 cannot be used for corrections because I₂ is nearly orthogonal to the direction of motion. It is easy to see that the corresponding path wavelength λ₂ is very large, and that only two full cycles of the density sinusoid are crossed, so the determination of λ₂ would not be accurate. Better rotations of the three markers can be found.

A main disadvantage of the arrangement of FIG. 4 is that the sensor path is restricted. As will be shown below, it is possible to construct a single marker by superposing several patterns of the type being used in FIG. 1 and FIG. 4.

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS

FIG. 1 shows a patterned position marker and the path of the density sensor passing over it.

FIG. 2 displays the path-density function and some related functions.

FIG. 3 displays discrete Fourier transforms (DFTs) of some functions in FIG. 3.

FIG. 4 shows the sensor path crossing three position markers with different orientations.

FIG. 5 has two patterns with different wavelengths, and a marker made by superposing them, with the path of a density sensor passing over the marker.

FIG. 6 shows the path-density function of a sensor passing over a marker comprising two superposed patterns of different wavelengths.

FIG. 7 displays the discrete Fourier transform (DFT) of the path-density function in FIG. 6.

FIG. 8 exhibits the dependency of the frequencies on the direction of motion.

FIG. 9 shows the path-density function of a sensor passing over a marker comprising two superposed patterns of different wavelengths and different orientations.

FIG. 10 displays the DFT of the path-density function in FIG. 9.

FIG. 11 exhibits the dependency of the path-density-function frequencies on the direction of motion for a marker of two patterns with different wavelengths and different orientations.

FIG. 12 displays the geometry of a marker comprising two patterns with different orientations.

DETAILED DESCRIPTION OF THE INVENTION

This description is divided into several sections and it uses technical abbreviations, so the following lists are provided as an aid to the reader. Other than in Table 2, below, the first time an abbreviation is used, it is put in parentheses after its definition.

1. Tables

Table of Contents

1. TABLES

• • TABLE OF CONTENTS
  • TABLE 1: LIST OF SYMBOLS USED
  • TABLE 2: LIST OF TERMS

2. SUPERPOSING ALIGNED PATTERNS

3. SUPERPOSING UNALIGNED PATTERNS

4. USING HIGH-CONTRAST MEDIA

5. OTHER CONSIDERATIONS

TABLE 1
LIST OF SYMBOLS USED
In this list k is the numerical index of a patterned position marker;
it can be blank; in some cases it can be a pair of indices.
Symbol                Meaning
Mathematical Notation
<x, y>                The point (or vector) with coordinate components x and y; e.g., δ = <δx, δy>.
┌·┐                   The ceiling function: ┌r┐ is the smallest integer ≧ r.
Round                 Round to the nearest integer, ties rounding down: Round(u) = ┌u − ½┐.
U · V                 For any vectors U and V, the dot product U · V = UxVx + UyVy.
k ∈ { . . . }         k is one of the members of the set { . . . }; e.g., k ∈ {1, 2, 3} means k is
                      1,2, or 3
Symbols for the Patent
αk                    The angle between the positive x axis and the pattern orientation vector Ik.
δk                    Offset in the direction Ik of the estimated coordinates from the true coordinates.
θk                    The counterclockwise angle in radians between Ik and v with 0 ≦ θk < 2π.
{tilde over (θ)}k     The computer's estimate of θk.
λk                    Wavelength of the path-density function fk.
Λk                    Wavelength of the pattern-density function Fk(x, y).
φk                    The phase of Fk+ at Ck: Fk+ (Ck) = A cos(φk) + Bk.
ψk                    Phase of fk(s) at s = 0.
ωk                    The angular frequency corresponding to λk (thus, ωk = 2π/λk).
Ωk                    The angular frequency corresponding to Λk (thus, Ωk = 2π/Λk).
f                     The path-density function.
f+                    The periodic extension of f to the entire s axis.
{circumflex over (f)} The discrete Fourier transform (DFT) of f.
s                     The distance along the sensor path from p2.
v                     The direction-of-motion vector for the density sensor on the segment from
                      p2 to p5.
{tilde over (v)}      The computer's estimate of v.
Ck                    The pattern reference point <Xk,Yk> of pattern k.
Ik                    The orientation vector of pattern k.
Jk                    The vector orthogonal to and on the left side of Ik.
Fk                    The pattern density function of pattern k.
Fk+                   The periodic extension of Fk to the entire x, y-plane.
Rk                    Radius of the position marker pattern (when circular).
δ                     The displacement vector of p2: δ = {tilde over (p)}2 − p2.
p                     A point on the sensor path, especially the sensor location.
{tilde over (p)}      The computer's estimate of p.
p1                    Starting point of the sensor path in the figures.
p2                    Starting point of the straight path segment crossing a position marker;
                      used as a reference point for s (s = 0 at p2).
p4                    Last point at which the straight path segment meets the position marker;
p5                    Ending point of the straight path segment crossing the position marker;
p6                    Ending point of the sensor path in the figures.
Pk                    The projection of p on pattern axis Ik
{tilde over (p)}2     The computer's approximation of p2.
S(p)                  Distance from C0 to P: S(p) = (p − C0) · I0

TABLE 2
LIST OF TERMS
		Meaning, Example,
Paragraph	Term	Symbol, or Comment
8	marker	patterned position marker
9	sensor	optical density sensor
9	path-density function	f
9	computer	navigation computer
11	pattern reference point	Ck
11	pattern axis	line 3 in FIG. 1;
		lines 33, and 34 in FIG. 12
12	direction-of-motion vector	v
13	orientation vector	Ik
14	pattern-density function	Fk
20	displacement	δ
27	marker parameters	Xk, Yk, Ik, Ωk, and φk
58	underlying navigation	e.g., dead reckoning
60	superposing	F3 = A F1 + B F2 + C

2. Superposing Aligned Patterns

No matter what form of navigation (called underlying navigation in the following), dead-reckoning or some other, is used to estimate the instantaneous position of the vehicle, operation of the vehicle must provide for the sensor to cross a position marker properly. Thus, the line from p₂ to p₅ must pass reasonably close to the center of the marker; within one-half of the radius of the marker may be sufficient.

As discussed above, for the geometry shown in FIG. 1, avoiding phase ambiguity requires the underlying navigation to insure that the orthogonal projection of {tilde over (p)}₂ on line 3 is within Λ₀/2 of the corresponding projection of p₂; i.e., that |δ|<Λ₀/2. This reqirement is too severe in many applications.

The |δ|<Λ₀/2 restriction can be relieved if there is a second marker pattern which provides for phase identification over a wider range. This can be done by superposing two or more patterns. The term superposing means combining the patterns by weighted addition of their density functions and adjustment of the DC component to produce the final pattern.

This is illustrated in FIG. 5, in which the pattern density function F₃ of marker 11 is the weighted sum of the density functions F₁ and F₂ of patterns 12 and 13, respectively. Specifically, for FIG. 5F₃(x, y)=A(F₁(x, y)+0.75 F₂(x, y))+B,  (15)where A and B are constants. The orientation vectors, I_k for k=1, 2, and 3, are all rotated 15 degrees from the x-axis (so I₁=I₂=I₃). The direction-of-motion vector v is rotated 20 degrees from the orientation vectors, so the two path frequencies are ω₁=cos(20°)Ω₁ and ω₂=cos(20°)Ω₂ (see equaion (1)). Points p₂, p₃, p₄, and p₅, are the same as in FIG. 1. FIG. 5 shows the wavelengths Λ₁, Λ₂, and Λ₃ corresponding to Ω₁, Ω₂, and Ω₃, respectively; Ω₃=(Ω₁−Ω₂)/2.

The path-density function f of FIG. 5 and the absolute value |{circumflex over (f)}| of its DFT are shown in FIG. 6 and FIG. 7. From the two peaks in FIG. 7 the computer can determine the frequencies ω₁ and ω₂, and the phases ψ₁ and ψ₂ of the two components of the path-density function.

This can be generalized to the superposition of L≧2 patterns with angular frequencies Ω_l, and orientation vectors I_l for l=1, 2, . . . , L. Here it will be assumed that the I_l are all equal.

There are L equations, one for each pattern, corresponding to equation (8):δ₁=({tilde over (p)}₂−C_l)·I_l+(φ_l−ψ_l−2πm_l)/Ω_l, for l∈{1,2, . . . , L},  (16)where 0≦ψ_l<2π, and m_l is an integer (there is only one δ₁ because all I_l=I₁; all I_l are shown, however). Combine these as follows. For l∈{1, 2, . . . , L} let n_l be an integer. Multiply equation (16) by n_lΩ_l, then add the equations together and rearrange the sum to get

$\begin{array}{cc}{\delta }_1=\frac{1}{\sum _{l=1}^L{n}_l{\Omega }_l}\left(\sum _{l=1}^L\left(n_l\left({\Omega }_l\left({\tilde{p}}_2-C_l\right)·I_l+\left({\phi }_l-{\psi }_l\right)\right)\right)-2\pi \sum _{l=1}^L{n}_l{m}_l\right),& \left(17\right)\end{array}$ which can be rearranged into

$\begin{array}{cc}\sum _{l=1}^L{n}_l{m}_l=\sum _{l=1}^L\left(n_l\left(\frac{1}{{\Lambda }_l}\left({\tilde{p}}_2-C_l\right)·I_l+\frac{1}{2\pi }\left({\phi }_l-{\psi }_l\right)\right)\right)-{\delta }_1\sum _{l=1}^L\frac{n_l}{{\Lambda }_l}.& \left(18\right)\end{array}$ The left side of this is an integer, so if

$\begin{array}{cc}{\delta }_1<\frac{1}{2}{\sum _{l=1}^L\frac{n_l}{{\Lambda }_l}}^{-1}& \left(19\right)\end{array}$ then

$\begin{array}{cc}\sum _{l=1}^L{n}_l{m}_l=\mathrm{Round}\left(\sum _{l=1}^L{n}_l\left(\frac{1}{{\Lambda }_l}\left({\tilde{p}}_2-C_l\right)·I_l+\frac{1}{2\pi }\left({\phi }_l-{\psi }_l\right)\right)\right).& \left(20\right)\end{array}$

As before (see the text following equation (11)), after ψ_l has been found for l∈{1, . . . L}, if condition (19) is satisfied, the computer can get Σ_l=1^L n_lm_l, then δ₁, and partially correct {tilde over (p)}₂ to {tilde over (p)}₂−δ₁I₀.

Condition (19) is much better than the one (|δ₀|<Λ₀/2) for a single pattern. For example, with L=2, Λ₁=Λ₀, and Λ₂=Λ₁/0.8, the values for FIGS. 5, 6, and 7, with n₁=1, if n₂=−1, the right hand side of (19) is 2.5 Λ₀, which is 5 times larger than the other bound. With other choices of the frequencies it can be much larger.

The text after equation (11) applies mutatis mutandis to ψ_l, ω_l, etc.

FIG. 8 shows how the path frequencies ω₁ and ω₂ change as the direction-of-motion vector v changes. Since ω₁ and ω₂ are periodic, the figure shows only one period. Near θ=20° there is enough variation of the path frequencies that equation (1) with ω₀ replaced by either ω₁ or ω₂ can be used for accurate correction of θ₀, but that is not true at θ=0° where dω₁/dθ=dω₂/dθ=0. Another extreme is near θ=±90°, where ω₁ and ω₂ are so small that there is no useful information about θ₀ (as is the case with position marker 6 in FIG. 4). This kind of graph will be seen again in the following.

3. Superposing Unaligned Patterns

Consider a marker which is the same as that of FIG. 5 except that patterns 12 and 13 are rotated so that, say, α₁=15° (i.e., 12 is unchanged) and α₂=60°. Then the two patterns provide information about coordinate errors in two different directions, so the coordinates can be corrected completely.

FIG. 9 and FIG. 10 show the path-density function f and the absolute value |{circumflex over (f)}| of its DFT, respectively, for this arrangement. The peaks of |{circumflex over (f)}| are somewhat farther apart than those of FIG. 7. FIG. 11 shows how the wavelengths ω₁ and ω₂ change as the angle-of-motion θ changes.

Consider a marker with L≧2 pattern axes. For each l∈{1, . . . , L}, the displacement vector δ={tilde over (p)}₂−p₂ has a component δ_l=δ·I_l along I_l. Given δ_l for all l∈1, . . . , L, δ can be found by taking δ=Σ_l=1^L a_lI_l and solving the simultaneous linear equationsδ_l=δ·I_l, for l∈{1, . . . , L}.  (21)for a₁, . . . , a_L.

FIG. 12 shows the geometrical relationships for two pattern axes, but omits the marker pattern, which is represented by its boundary circle 31. Line 32 is the sensor path. The pattern axes are line 33 for the axis associated with I_l, and line 34 for the axis associated with I_i. The orientation vectors I_l and I_i are at angles α_l and α_i from the positive x-axis, respectively. The angles from I_l and I_i to the direction-of-motion vector v are θ_l and θ_i, respectively; for purposes of illustration, θ_i is shown as negative, but all calculations assume that both θ_l and θ_i are in [0, 2π).

For FIGS. 9, 10, and 11 the pattern for each of I₁ and I₂ is sinusoidal with a single frequency. As was noted earlier, this severely restricts the size of the displacement.

Instead, in the following, each pattern axis I_l has K_l≧1 associated patterns. For each k∈{1, . . . , K_i} the pattern has refernce point C_lk, angular frequency Ω_lk, phase ψ_lk at C_lk, and corresponding path frequency ω_lk; and for each of these patterns there is an equation like equation (16) for δ_l=({tilde over (p)}₂−p₂)·I_l, the displacement in the direction of I_l. Let J_l be the unit vector orthogonal to I_l and pointing to the left of I_l. Then there are numbera a_li and b_li such that I_i=a_liI_l+b_liJ_l. Note that a_ll=1 and b_ll=0. FIG. 12 shows vectors J_i and J_l.

There are integers m_ik such that equation (7) mutatis mutandis becomesΩ_ik(p₂−C_ik)·I_i=−φ_ik+ψ_ik+2πm_ik for i∈{1,2, . . . , L} and k∈{1,2, . . . , K_i}.  (22)Since(p₂−C_ik)·I_i=(p₂−{tilde over (p)}₂+{tilde over (p)}₂−C_ik)·I_i=−δ·(a_liI_l+b_liJ_l)+({tilde over (p)}₂−C_ik)·I_i=−a_liδ_l−b_liδ·J_l+({tilde over (p)}₂−C_ik)·I_iequation (22) can be rewritten as

$\begin{array}{cc}{\delta }_l\frac{{\Omega }_{\mathrm{ik}}{a}_{\mathrm{li}}}{2\pi }=\frac{{\Omega }_{\mathrm{ik}}}{2\pi }\left(\left({\tilde{p}}_2-C_{\mathrm{ik}}\right)·I_i-b_{\mathrm{li}}\delta ·J_l\right)+\frac{1}{2\pi }\left({\phi }_{\mathrm{ik}}-{\psi }_{\mathrm{ik}}\right)-m_{\mathrm{ik}}\mathrm{for}i\in \left\{1,2,\dots ,L\right\},k\in \left\{1,2,\dots ,K_i\right\}.l\in \left\{1,2,\dots ,L\right\}\mathrm{and}{a}_{\mathrm{li}}\ne 0& \left(23\right)\end{array}$ (omit any of these equations with a_li=0).

Combine all of these equations to get a single equation for δ_l: choose integers n_lik, multiply the equation by them, and sum on i and k to get

$\begin{array}{cc}{\delta }_l\sum _{\underset{a_{\mathrm{li}}\ne 0}{i=1}}^L\sum _{k=1}^{K_i}\frac{n_{\mathrm{lik}}{\Omega }_{\mathrm{ik}}{a}_{\mathrm{li}}}{2\pi }=\sum _{\underset{a_{\mathrm{li}}\ne 0}{i=1}}^L\sum _{k=1}^{K_i}\frac{n_{\mathrm{lik}}{\Omega }_{\mathrm{ik}}}{2\pi }\left(\left({\tilde{p}}_2-C_{\mathrm{ik}}\right)·I_i-b_{\mathrm{li}}\delta ·J_l\right)+\sum _{\underset{a_{\mathrm{li}}\ne 0}{i=1}}^L\sum _{k=1}^{K_i}\frac{n_{\mathrm{lik}}}{2\pi }\left({\phi }_{\mathrm{ik}}-{\psi }_{\mathrm{ik}}\right)-\sum _{\underset{a_{\mathrm{li}}\ne 0}{i=1}}^L\sum _{k=1}^{K_i}{n}_{\mathrm{lik}}{m}_{\mathrm{ik}}\mathrm{for}l\in \left\{1,2,\dots ,L\right\}.& \left(24\right)\end{array}$ If the last term can be computed, this can be solved for δ_l. As before, since the last term is an integer, for each l∈{1, 2, . . . , L}

$\begin{array}{cc}\sum _{\underset{a_{\mathrm{li}}\ne 0}{i=1}}^L\sum _{k=1}^{K_i}{n}_{\mathrm{lik}}{m}_{\mathrm{ik}}=\mathrm{Roun}d\left(\sum _{\underset{a_{\mathrm{li}}\ne 0}{i=1}}^L\sum _{k=1}^{K_i}\frac{n_{\mathrm{lik}}{\Omega }_{\mathrm{ik}}}{2\pi }\left(\left({\tilde{p}}_2-C_{\mathrm{ik}}\right)·I_i+\frac{n_{\mathrm{lik}}}{2\pi }\left({\phi }_{\mathrm{ik}}-{\psi }_{\mathrm{ik}}\right)\right)\right),& \left(25\right)\end{array}$ provided

$\begin{array}{cc}{\delta }_l\sum _{\underset{a_{\mathrm{li}}\ne 0}{i=1}}^L\sum _{k=1}^{K_i}\frac{n_{\mathrm{lik}}{\Omega }_{\mathrm{ik}}{a}_{\mathrm{li}}}{2\pi }+\delta ·J_l\sum _{\underset{a_{\mathrm{li}}\ne 0}{i=1}}^L\sum _{k=1}^{K_i}\frac{n_{\mathrm{lik}}{\Omega }_{\mathrm{ik}}}{2\pi }{b}_{\mathrm{li}}<\frac{1}{2}.& \left(26\right)\end{array}$ It is sufficient to make

${\delta }_l<\frac{\pi }{2}{\sum _{\underset{a_{\mathrm{li}}\ne 0}{i=1}}^L\sum _{k=1}^{K_i}{n}_{\mathrm{lik}}{\Omega }_{\mathrm{ik}}{a}_{\mathrm{li}}}^{-1}\mathrm{and}\delta <\frac{\pi }{2}{\sum _{\underset{a_{\mathrm{li}}\ne 0}{i=1}}^L\sum _{k=1}^{K_i}{n}_{\mathrm{lik}}{\Omega }_{\mathrm{ik}}{b}_{\mathrm{li}}}^{-1}.$

In summary, under the assumption in equation (26), for each the computer can use equation (25) to get

$\sum _{i=1}^L\sum _{k=1}^{K_i}{n}_{\mathrm{lik}}{m}_{\mathrm{lk}},$ use that in equation (24) to get δ_l, use δ_l in equation (21) to get the vector displacement δ, then subtract δ from {tilde over (p)}₂ to get a new estimate of p₂.

Usually, the number Σ_i=1^L K_i of patterns is greater than two, and the peaks of the function |{circumflex over (f)}| must be associated with the corresponding pattern. This can be done if the patterns have unique amplitudes so the computer can associate each pattern with the peak of corresponding size.

Markers for particular applications can be designed by varying the number of pattern axes, the number of patterns for each axis, the pattern frequencies, and the directions of the pattern orientation vectors. The integers n_lik can be chosen to provide the best range for δ.

4. Using High-Contrast Media

The discussion above assumed that the marker density can take on all possible values over some range. This allowed the marker-density function F to be the sum of several cosine functions. That is not possible for high-contrast media which have only few density values.

Consider a medium that can only be black or white. In this case, patterns can be represented by marker-density functions F, F₁, F₂, . . . having only the values 0 and 1, say 0 for white, and 1 for black. The function F cannot be formed by weighted addition of the pattern density functions F₁, F₂, . . . , but it can be formed by multiplying them. Periodic functions of this type can be represented by Fourier series. Multiplication of the functions produece complicated combinations of the sinusoidal components, but analysis of the path-density function can still provide phase and frequency information needed for correction of navigation parameters. For two patterns, if

$\begin{array}{cc}{F}_k\left(s\right)=\frac{1}{2}+\sum _{\underset{m\ne 0}{m=-\infty }}^{\infty }{c}_{\mathrm{km}}{e}^{{\mathrm{j\omega }}_k\mathrm{ms}},& \left(27\right)\end{array}$ where c_km is a complex constant, then

$\begin{array}{cc}{F}_1\left(s\right)F_2\left(s\right)=\frac{1}{4}+\frac{1}{2}\left(c_{1,-1}{e}^{-{\mathrm{j\omega }}_{1}s}+c_{11}{e}^{{\mathrm{j\omega }}_{1}s}+c_{2,-1}{e}^{-{\mathrm{j\omega }}_{s}s}+c_{21}{e}^{{\mathrm{j\omega }}_{2}s}\right)+\mathrm{terms}\mathrm{with}\mathrm{higher}\mathrm{fequencies},& \left(28\right)\end{array}$ so the low frequency terms are separated and can be found using the DFT. This can be extended to more than two patterns. The functions on the left of this equation can be raised to different powers to weight the functions.

5. Other Considerations

Pattern density functions are periodic, but they need not be sinusoidal. Since periodic functions can be represented as a Fourier series like that of equation (27), the technique described for high-contrast media can be adapted for patterns which are not sinusoidal.

It is well known that window functions ω other than the one shown in FIG. 2 can be used to reduce the level of |{circumflex over (f)}| outside the central peaks. While such windows can be used to taper the edge of a marker, it is better and more flexible to have the computer apply windows in the processing of the path-density function f.

Persons knowledgeable of the relevant art will recognize that markers need not be circular disks, so that other regions can be used.

Although it is not clear how one could construct markers for three (or higher) dimensional applications, equations (23) through (26), which were derived for two-dimensional markers, also apply to markers of higher dimensions comprising superposed periodic patterns.

Claims

1. An automatic system for determining a position and direction-of-motion of a vehicle, which system comprises set of position markers fixed in known positions,a density sensor mounted on the vehicle and positioned so it determines density along a line as the vehicle moves,a computer receiving density values determined by the density sensor, and maintaining an estimate of the position and direction-of-motion of the vehicle, anda computer control system that directs the computer to perform computations for estimating the position and direction-of-motion of the vehicle from the density values as the sensor passes across any one of the position markers;each of the position markers comprising one or more density patterns in a finite two-dimensional region;each of the density patterns filling a two-dimensional region of a position marker;each of the density patterns having a known orientation vector;in the two-dimensional region, a density of each density pattern being a known periodic function of position in a direction of its orientation vector, and independent of position in a direction orthogonal to its orientation vector;at each point of the two-dimensional region the position marker having a density which is a combination of its density-pattern densities at the same point;the computer storing the density values it receives as a path density function until it determines that it has a quantity of them that is sufficient for performing the computations needed to estimate a current position and direction-of-motion of the vehicle;the computer then performing mathematical operations necessary to determine frequency, phase, and amplitude values of various sinusoidal components of the path density function;the computer then using so-determined amplitude value of each sinusoidal component of the path density function to determine which density pattern corresponds to that sinusoidal component;the computer then performing mathematical operations necessary to use so-determined frequency and phase values of sinusoidal components of the path density function and spatial frequency and phase values of corresponding known density patterns of position markers to determine the vehicle's direction-of-motion and an offset of a current estimate of the vehicle's position from a vehicle's actual position, and then using that offset to correct the current estimate of the vehicle's position.

2. A system as in claim 1 for which a density-determining periodic function of a density pattern is a sinusoid.

3. A system as in claim 1 for which at each point of the two-dimensional region the position marker has a density which is a constant plus a weighted sum of its density-pattern densities at the same point.

4. A system as in claim 1 for which the density of each of the density patterns is between zero and one;and at each point of the two-dimensional region the position marker has a density which is a multiplicative product of the densities of the density patterns at the same point,with adjustment of a range of densities.

5. A system as in claim 1 for which a set of position markers are positioned so the density sensor passes over them sequentially.

6. A system as in claim 1 for which the frequencies, phases, and amplitudes of the sinusoidal components of the computer-stored path-density function are determined by using the Discrete Fourier Transform.