Life for old maps

• US 5,581,259 A
• Filed: 11/03/1994
• Issued: 12/03/1996
• Est. Priority Date: 11/03/1994
• Status: Expired due to Fees

• Current Assignee: Trimble Navigation Limited (Trimble Inc.)
• Original Assignee: Trimble Navigation Limited (Trimble Inc.)
• Inventors: Schipper, John F.
• Primary Examiner(s): Issing, Gregory C.
• Application Number:
  US08/333,855
• Time in Patent Office: 761 Days
• Field of Search: 342/357, 342/451, 342/458, 364/449, 364/731
• US Class Current: 342/451
• CPC Class Codes: G01C 21/20 Instruments for performing navigational calculations G01C21/24, G01C21/26 take precedence
• CPC Class Codes: G01C 21/28 with correlation of data from several navigational instruments
• CPC Class Codes: G09B 29/106 using electronic means

Abstract

Methods for determining the present location coordinates of a user moving in a two-dimensional or three-dimensional space, with reference to an old map that may be inaccurate. Location coordinates (xi'"'"',yi'"'"') of two or three non-collinear landmarks in two dimensions), or coordinates (xi'"'"',yi'"'"',zi'"'"') of three or four non-coplanar landmarks (in three-dimensions), are indicated on the old map. Corresponding location coordinates (xi,yi) or (xi,yi,zi) are also determined or made available using a location determination system (LDS), such as GPS, GLONASS, Loran, Tacan, Decca, Omega, VOR, DME, JTIDS, PLRS or an FM subcarder system. Location coordinates (xi,yi) (or (xi,yi,zi)) for two or more physical landmarks can be combined into coordinates for a landmark representative. The user'"'"'s location coordinates (xu,yu) (or (xu,yu,zu)) are also determined or provided using the LDS. In a first class of embodiments, a transformation T (linear or not necessarily linear) is found that maps the LDS location coordinates (xi,yi) (or (xi,yi,zi)) onto the corresponding location coordinates (xi'"'"',yi'"'"') (or (xi'"'"',yi'"'"',zi'"'"')) on the old map. The image of the location coordinate pair (xu,yu) (or triple (xu,yu,zu)) under the transformation T determines the user location coordinates on the old map. In a second class of embodiments, the range Ri from the user location to landmark number i (i=1, . . . , N; N=2,3) is determined in the LDS. Two or more locations are computed on the old map for which the range Ri'"'"' from landmark number i to this old map computed location is also equal to Ri, and one of these computed locations is designated as the user location on the old map. Several metrics are developed to measure the relative acceptability of an old map user location, as determined above.

Claims

1. A method for determining the location coordinates of a user on a two-dimensional surface, with reference to a map that may be substantially inaccurate, the method comprising the steps of:

(1) providing a user with a location determination (LD) unit having an LD antenna and an LD receiver/processor that can determine the location coordinates of the user in an LD coordinate system and can determine or provide the location coordinates, relative to the user, of a chosen location that is spaced apart from the user'"'"'s location;

(2) providing a selected map of a region that includes the user'"'"'s location and that has at least first and second landmark representatives with first and second map location coordinate pairs (xr1'"'"',yr1'"'"') and (xr2'"'"',yr2'"'"'), respectively;

(3) using the LD unit to determine user LD location coordinates (xu,yu) of the user'"'"'s location;

(4) providing first and second LD location coordinate pairs (xr1,yr1) and (xr2,yr2) for the first and second landmark representatives, respectively, in the LD coordinate system;

(5) using a computer to determine a coordinate transformation T of the LD location coordinate pairs (xr1,yr1) and (xr2,yr2) onto the respective map location coordinate pairs (xr1'"'"',yr1'"'"') and (xr2'"'"',yr2'"'"');

(6) using the computer to express a user map location coordinate pair (xu'"'"',yu'"'"') on the selected map as the image under the transformation T of the user LD location coordinate pair (xu,yu); and

(7) displaying the user map location coordinate pair (xu'"'"',yu'"'"') in at least one of a visually perceptible form and an audibly perceptible form.

2. The method of claim 1, further comprising the step of visually indicating a location on said LD unit corresponding to said user map location coordinate pair (xu'"'"',yu'"'"').

3. The method of claim 1, further comprising the steps of:

expressing said LD-determined user location as a vector pu that is a linear combination of two linearly independent vectors I and J in said two-dimensional surface, in the form

space="preserve" listing-type="equation">pu=xu I+yu J,

where xu and yu are said user LD location coordinates;

expressing the locations of said first and second landmark representatives, as determined by said LD unit, as the respective vectors

space="preserve" listing-type="equation">p1=xr1 I+yr1 J,

space="preserve" listing-type="equation">p2=xr2 I+yr2 J;

expressing the locations of said first and second landmark representatives, as determined on said selected map, as the respective vectors

space="preserve" listing-type="equation">p1=xr1'"'"'I+yr1'"'"'J,

space="preserve" listing-type="equation">p2=xr2'"'"'I+yr2'"'"'J;

determining two vectors pu'"'"'=xu'"'"'I+yu'"'"'J and pu"=xu"I+yu" J in said selected map that satisfy the relations

space="preserve" listing-type="equation">|pu'"'"'-p1'"'"'|=|pu"-p1'"'"'|=|pu-p1.vertline.,

space="preserve" listing-type="equation">|pu'"'"'-p2'"'"'|=|pu|-p2'"'"'|=|pu-p2|; and

selecting one of the vectors pu'"'"' and pu" as said user location in said selected map.

4. The method of claim 1, wherein said map location coordinate pairs (xr1'"'"',yr1'"'"') and (xr2'"'"',yr2'"'"') are the location coordinate pairs of physical landmarks L1'"'"' and L2'"'"' on said selected map, and said LD location coordinate pairs (xr1,yr1) and (xr2,yr2) are the location coordinate pairs of physical landmarks L1 and L2 in said LD coordinate system corresponding to the respective landmarks L1'"'"' and L2'"'"'.

5. The method of claim 1, further comprising the step of choosing said LD unit from the class of satellite-based and ground-based LD units consisting of GPS, GLONASS, Loran, Tacan, Decca, Omega, VOR, DME, JTIDS, PLRS and an FM subcarrier system.

6. The method of claim 1, wherein steps (5)-(6) comprise the steps of:

expressing said LD-determined user location as a vector pu that is a linear combination of two linearly independent vectors I and J in said two-dimensional surface, in the form

space="preserve" listing-type="equation">pu=xu I+yu J,

where xu and yu are said user location coordinates;

expressing the locations of said first and second landmark representatives in said LD unit as the respective vectors

space="preserve" listing-type="equation">p1=xr1 I+yr1 J,

space="preserve" listing-type="equation">p2=xr2 I+yr2 J,

where xr1, yr1, xr2 and yr2 are said location coordinates in said LD coordinate system, and p1 and p2 are linearly independent;

expressing the locations of said first and second landmark representatives on said selected map as the respective vectors

space="preserve" listing-type="equation">p1'"'"'=xr1'"'"'I+yr1'"'"'J,

space="preserve" listing-type="equation">p2'"'"'=xr2'"'"'I+yr2'"'"'J,

where xr1'"'"', yr1'"'"', xr2'"'"' and yr2'"'"' are said location coordinates on said selected map; and

expressing said user location on said selected map as a vector

space="preserve" listing-type="equation">pu'"'"'=xu'"'"'I +yu'"'"'J,

where

space="preserve" listing-type="equation">xu'"'"'={(yr2 xr1'"'"'-yr1 xr2'"'"') xu+(xr1 xr2 '"'"'-xr2 xr1'"'"') yu}/D

space="preserve" listing-type="equation">yu'"'"'={(yr2 yr1'"'"'-yr1 yr2'"'"') xu+(xr1 yr2'"'"'-xr2 yr1'"'"') yu}/D,

space="preserve" listing-type="equation">D=xr1 yr2-xr2 yr1.

7. The method of claim 1, wherein said selected map has N physical landmarks (N≧

3) for which location coordinates (xi'"'"',yi'"'"') (i=1, 2, . . ., N) are provided on said selected map and for which N corresponding landmarks with location coordinates (xi,yi) are provided in said LD coordinate system, and wherein steps (2) and (4) comprise the steps of;

providing a first sequence of real numbers cj (j=1, . . . , N1) and a second sequence of real numbers dj (j=N1+1, . . . , N1+N2), where N1+N2=N, Ni≧

1 and N2≧

1, and where the real numbers cj and dj satisfy the constraints 0≦

cj ≦

1, 0≦

dj≦

1, and ##EQU26## selecting a first set G1 of N1 adjacent landmarks and associated location coordinates and a second set G2 of N2 adjacent landmarks and associated location coordinates, from among the N landmarks in said LD coordinate system, where the first set and the second set are mutually exclusive, where a landmark in the first set G1 has location coordinates (xj,yj) (j=1, . . . , N1) and a landmark in the second set G2 has location coordinates (xj,yj) (j=N1+1, . . . , N1+N2);

selecting a first set G1'"'"' of N1 landmarks and associated location coordinates and a second set G2'"'"' of N2 landmarks and associated location coordinates, from among the N landmarks on said selected map, where a landmark in the first set G1'"'"' has location coordinates (xj'"'"',yj'"'"') (j-1, . . . , N1) and corresponds to a landmark in the first set G1, and a landmark in the second set G2'"'"' has location coordinates (xj'"'"',yj'"'"') (j=N1+1, . . . , N1+N2) and corresponds to a landmark in the second set G2;

determining said location coordinates (xr1,yr1) and (xr2,yr2) for said first and second landmark representative locations by said said LD unit by the relations ##EQU27## determining said location coordinates (xr1'"'"',yr1'"'"') and (xr2'"'"',yr2'"'"') for said first and second landmark representative locations on said selected map by the relations ##EQU28##

8. The method of claim 7, further comprising the step of choosing at least one of said sequences of real numbers cj (j=1, . . . , N1) and dj (j=N1+1, . . . , N1+N2) according to the relations

9. The method of claim 7, further comprising the step of choosing at least one of said sequences of real numbers cj (j=1, . . . , N1) and dj (j=N1+1, . . . , N1+N2) according to the relations

10. The method of claim 1, further comprising the step of providing a third landmark representative on said selected map with a third map location coordinate pair (xr3'"'"',yr3'"'"') and a corresponding third landmark representative in said LD coordinate system with an LD location coordinate pair (xr3,yr3), and wherein steps (5)-(6) comprise the steps of:

providing a coordinate transformation from a location with location coordinates (x,y) in said LD coordinate system, to a location with location coordinates (x'"'"',y'"'"') on said selected map, where the transformation has the form

space="preserve" listing-type="equation">x'"'"'-x0=a11(x-x0)+a12(y-y0),

space="preserve" listing-type="equation">y'"'"'-y0=a21 (x-x0)+a22(y-y0),

where a11, a12, a21, a22, x0 and y0 are determined by the requirement that, the coordinate transformation carries the location coordinate pairs (xr1,yr1), (xr2,yr2) and (xr3,yr3) onto the respective location coordinate pairs (xr1'"'"',yr1'"'"'), (xr2'"'"',yr2'"'"') and (xr3'"'"',yr3'"'"'); and

designating the image of said user location coordinate pair (xu,yu) under the transformation as said user location coordinate pair (xu'"'"',yu'"'"') on said selected map.

11. The method of claim 1, wherein said selected map has N physical landmarks (N≧

4) for which location coordinates (xi'"'"',yi'"'"') (i=1, 2, . . . , N) are provided on said selected map and for which N corresponding landmarks with location coordinates (xi,yi) are provided in said LD coordinate system, and wherein steps (2) and (4) comprise the steps of;

providing a first sequence of real numbers cj (j=1, . . . , N1), a second sequence of real numbers dj (j=N1+1, . . . , N1+N2) and a third sequence of real numbers ej (j=N1+N2+1, . . . , N1+N2+N3), where N1+N2+N3=N, N1≧

1, N2≧

1 and N3>

1, and where the real numbers cj, dj and ej satisfy the constraints 0≦

cj≦

1,0≦

dj≦

1,0≦

ej≦

1 and ##EQU29## selecting a first set G1 of N1 landmarks and associated location coordinates, a second set G2 of N2 landmarks and associated location coordinates, and a third set G3 of landmarks and associated location coordinates, from among the N landmarks in said LD coordinate system, where the first set, second set and third set are mutually exclusive, where a landmark in the first set G1 has location coordinates (xj,yj) (j=1, . . . , N1), a landmark in the second set G2 has location coordinates (xj,yj) (j=N1+1, . . . , N2), and a landmark in the third set G3 has location coordinates (xj,yj) (j=N1 +N2+1, . . . , N1+N2+N3 );

selecting a first set G1'"'"' of N1 landmarks and associated location coordinates, a second set G2'"'"' of N2 landmarks and associated location coordinates, and a third set G3'"'"' of N3 landmarks, from among the N landmarks on said selected map, where a landmark in the first set G1'"'"' has location coordinates (xj'"'"',yj'"'"') (j=1, . . . , N1) and corresponds to a landmark in the first set G1, a landmark in the second set G2'"'"' has location coordinates (xj'"'"',yj'"'"') (j=N1+1, . . . , N1+N2) and corresponds to a landmark in the set G2, and a landmark in the set G3'"'"' has location coordinates (xj'"'"',yj'"'"') (j=N1+N2+1, . . . , N1+N2+N3) and corresponds to a landmark in the set G3;

determining said location coordinates (xr1,yr1), (xr2,yr2) and a location coordinate pair (xr3,yr3) for first, second and third landmark representative locations in said LD coordinate system by the relations ##EQU30## determining said location coordinates (xr1'"'"',yr1'"'"'), (xr2'"'"',yr2'"'"') and a location coordinate pair (xr3'"'"',yr3'"'"') for first, second and third landmark representative locations on said selected map by the relations ##EQU31##

12. The method of claim 11, further comprising the step of choosing at least one of said sequences of real numbers cj (j=1, . . . , N1 ), dj (j=N1 +1, . . . , N1+N2) and ej (j=N1+N2+1, . . . , N1+N2+N3) according to the relations

13. The method of claim 11, further comprising the step of choosing at least one of said sequences of real numbers cj (j=1, . . . , N1), dj (j=N1+1, . . . , N1+N2) and ej (j=N1+N2+1, . . . , N1+N2+N3) according to the relations

14. The method of claim 1, wherein said selected map has N landmark representatives (N≧

3) for which location coordinates (xi'"'"',yi'"'"') (i=1, 2, . . . , N) are provided on said selected map and for which N corresponding landmark representatives with location coordinates (xi,yi) are provided by said LD unit, and wherein steps (5)-(6) comprise the steps of;

choosing said linear transformation T to be of the form

space="preserve" listing-type="equation">x'"'"'=a11 x+a12 y,

space="preserve" listing-type="equation">y'"'"'=a21 x+a22 y,

where a11, a12, a21 and a22 are parameters to be determined, x'"'"' and y'"'"' are location coordinates on said selected map, and x and y are location coordinates in said LD coordinate system;

choosing the parameters a11, a12, a21 and a22 to minimize an error term defined by

space="preserve" listing-type="equation">ε

=Σ

ε

xi (a11 xi+a12 yi-xi'"'"').sup.2 +Σ

ε

yi (a21 xi+a22 yi-yi'"'"').sup.2,

where ε

xi and ε

yi are selected non-negative coefficients and the sums are over the index i=1, . . . , N; and

expressing said user location coordinates (xu'"'"',yu'"'"') on said selected map by the relations

space="preserve" listing-type="equation">xu'"'"'=a11 xu+a12 yu,

space="preserve" listing-type="equation">yu'"'"'=a21 xu+a22 yu.

15. The method of claim 1, wherein said selected map has N landmark representatives (N≧

4) for which location coordinates (xi'"'"',yi'"'"') (i=1, 2, . . . , N) are provided on said selected map and for which N corresponding landmark representatives with location coordinates (xi,yi) are provided in said LD coordinate system, and wherein steps (5)-(6) comprise the steps of;

choosing said linear transformation T to be of the form

space="preserve" listing-type="equation">x'"'"'-x0=a11 (x-x0)+a12 (y-y0),

space="preserve" listing-type="equation">y'"'"'-y0=a21 (x-x0)+a22 (y-y0),

where a11, a12, a21, a22, x0 and y0 are parameters to be determined, x'"'"' and y'"'"' are location coordinates on said selected map, and x and y are location coordinates in said LD coordinate system;

choosing the parameters a11, a12, a21, a22, x0 and y0 to minimize an error term ε

defined by

space="preserve" listing-type="equation">ε

=Σ

ε

xi (a11 (xi-x0)+a12 (yi-y0)-xi'"'"'+x0).sup.2 ++Σ

ε

yi (a21 (xi-x0)+a22 (yi-y0)-yi'"'"'+y0).sup.2,

where ε

xi and ε

yi are selected non-negative coefficients and the sums are over the index i=1, . . . , N; and

expressing said user location coordinates (xu'"'"',yu'"'"') on said selected map by the relations

space="preserve" listing-type="equation">xu'"'"'=a11 xu+a12 yu-(a11-1) x0-a12 y0,

space="preserve" listing-type="equation">yu'"'"'=a21 xu+a22 yu-a21 x0-(a22-1) y0.

16.

16. The method of claim 1, wherein steps (5)-(6) comprise the steps of:

expressing said LD-determined user location as a vector pu that is a linear combination of two linearly independent vectors I and J in said two-dimensional surface, in the form

space="preserve" listing-type="equation">pu=xu I+yu J,

where xu and yu are said user location coordinates in said LD coordinate system;

expressing the locations of said first and second landmark representatives in said LD coordinate system as the respective vectors

space="preserve" listing-type="equation">p1=xr1 I+yr1 J,

space="preserve" listing-type="equation">p2=xr2 I+yr2 J,

where xr1, yr1, xr2 and yr2 are said location coordinates in said LD coordinate system;

expressing the locations of said first and second landmark representatives on said selected map as the respective vectors

space="preserve" listing-type="equation">p1'"'"'=xr1'"'"'I+yr1'"'"'J,

space="preserve" listing-type="equation">p2'"'"'=xr2'"'"'I+yr2'"'"'J,

where xr1'"'"', yr1'"'"', xr2'"'"' and yr2'"'"' are said location coordinates on said selected map;

expressing the vector pu as a linear combination

space="preserve" listing-type="equation">pu=a1 p1+a2 p2,

where a1 and a2 are selected coefficients;

expressing the vectors p1'"'"' and p2'"'"' as linear combinations

space="preserve" listing-type="equation">p1'"'"'=b11 p1+b12 p2,

space="preserve" listing-type="equation">p2'"'"'=b21 p1+b22 p2,

where b11, b12, b21 and b22 are selected coefficients; and

expressing said user location on said selected map as a vector ##EQU32##

17. The method of claim 1, further comprising the step of providing a landmark representative on said selected map that has a third location coordinate pair (xr3'"'"',yr3'"'"') and providing a corresponding landmark representative in said LD coordinate system that has a third location coordinate pair (xr3,yr3), wherein steps (5)-(6) comprise the steps of:

expressing the LD-determined user location as a vector pu that is a linear combination of two linearly independent vectors I and J in said two-dimensional surface, in the form

space="preserve" listing-type="equation">pu=xu I+yu J,

where xu and yu are said user location coordinates in said LD coordinate system;

expressing the locations of said first and second landmark representatives and the third landmark representative in said LD coordinate system as the respective vectors

space="preserve" listing-type="equation">p1=xr1 I+yr1 J,

space="preserve" listing-type="equation">p2=xr2 I+yr2 J,

space="preserve" listing-type="equation">p3=xr3 I+yr3 J,

where xr1, yr1, xr2, yr2, xr3 and yr3 are said location coordinates in said LD coordinate system;

expressing the locations of said first and second landmark representatives and the third landmark representative on said selected map as the respective vectors

space="preserve" listing-type="equation">p1'"'"'=xr1'"'"'I+yr1'"'"'J,

space="preserve" listing-type="equation">p2'"'"'=xr2'"'"'I+yr2'"'"'J,

p3'"'"'=xr3'"'"'I+yr3'"'"'J,

where xr1'"'"', yr1'"'"', xr2'"'"', yr2'"'"', xr3'"'"' and yr3'"'"' are said location coordinates on said selected map;

expressing the vector pu as a linear combination

space="preserve" listing-type="equation">pu=a1 p1+a2 p2+(1-a1-a2) p3,

where a1'"'"' and a2'"'"' are selected coefficients;

expressing the vectors p1'"'"', p2'"'"' and p3'"'"' as linear combinations

space="preserve" listing-type="equation">p1'"'"'=b11 p1+b12 p2+(1-b11-b12) p3,

space="preserve" listing-type="equation">p2'"'"'=b21 p1 +b22 p2+(1-b21-b22) p3,

space="preserve" listing-type="equation">p3'"'"'=b31 p1+b32 p2+(1-b31-b32) p3,

where b11, b12, b21, b22, b31 and b32 are selected coefficients; and

expressing said user location on said selected map as a vector pu'"'"' with components xu'"'"' and yu'"'"' given by ##EQU33##

18. The method of claim 1, further comprising the step of providing a landmark representative on said selected map that has a third location coordinate pair (xr3'"'"',yr3'"'"') and providing a corresponding landmark representative in said LD coordinate system that has a third location coordinate pair (xr3,yr3), wherein steps (5)-(6) comprise the steps of:

expressing the LD-determined user location as a vector pu that is a linear combination of two linearly independent vectors I and J in said two-dimensional surface, in the form

space="preserve" listing-type="equation">pu=xu I+yu J,

where xu and yu are said user location coordinates in said LD coordinate system;

expressing the locations of said first and second landmark representatives and the third landmark representative in said LD coordinate system as the respective vectors

space="preserve" listing-type="equation">p1=xr1 I+yr1 J,

space="preserve" listing-type="equation">p2=xr21+yr2 J,

space="preserve" listing-type="equation">p3=xr31+yr3 J,

where xr1, yr1, xr2, yr2, xr3 and yr3 are said location coordinates in said LD coordinate system;

expressing the locations of said first and second landmark representatives and the third landmark representative on said selected map as the respective vectors

space="preserve" listing-type="equation">p1'"'"'=xr1'"'"'I+yr1'"'"' J,

space="preserve" listing-type="equation">p2'"'"'=xr2'"'"'I+yr2'"'"' J,

space="preserve" listing-type="equation">p3'"'"'=xr3'"'"'I+yr3'"'"' J,

where xr1'"'"', yr1'"'"', xr2'"'"', yr2'"'"', xr3'"'"' and yr3'"'"' are said location coordinates on said selected map;

expressing the vector pu as a linear combination

space="preserve" listing-type="equation">pu=a1'"'"'p1'"'"'+a2'"'"'p22'"'"'+(1-a1'"'"'-a2'"'"') p3'"'"',

where a1'"'"' and a2'"'"' are selected coefficients;

expressing the vectors p1'"'"', p2'"'"' and p3'"'"' as linear combinations

space="preserve" listing-type="equation">p1=b11'"'"'p1'"'"'+b12'"'"'p2'"'"'+(1-b11'"'"'-b12'"'"') p3'"'"',

space="preserve" listing-type="equation">p2=b21'"'"'p1'"'"'+b22'"'"'p2+(1-b21'"'"'-b22'"'"') p3'"'"',

space="preserve" listing-type="equation">p3=b31'"'"'p1'"'"'+b32'"'"'p2'"'"'+(1-b31'"'"'-b32'"'"') p3'"'"',

where b11'"'"', b12'"'"', b21'"'"', b22'"'"', b31'"'"' and b32'"'"' are selected coefficients; and

expressing said user location on said selected map as a vector pu'"'"' with components xu'"'"' and yu'"'"' given by ##EQU34##

19. The method of claim 1, further comprising the steps of:

using the computer to compute a dimensionless ratio ρ

, whose numerator increases monotonically with the magnitude of the vector difference |pu-pu'"'"'|, where pu and pu'"'"' are user location vectors as determined by said LD unit and on said selected map, respectively, and are given by

space="preserve" listing-type="equation">pu=xu I+yu J,

space="preserve" listing-type="equation">pu'"'"'=xu'"'"'I+yu'"'"'J,

where I and J are linearly independent vectors in said two-dimensional surface, and whose denominator increases monotonically with at least one of the following magnitudes of vector differences;

(1) min {|pu-p1|, |pu-p2|}, where p1=xr1 I+yr1 J and p2=xr2 I +yr2 J are vectors indicating said locations of said first and second landmark representatives as determined by said LD unit;

(2) min{|pu'"'"'-p1'"'"'|, |pu'"'"'-p2'"'"'|}, where p1'"'"'=xr1'"'"'I+yr1'"'"'J and p2'"'"'=xr2'"'"'I+yr2'"'"'J are vectors indicating said locations of said first and second landmark representatives on said selected map;

(3) min_bε

|pb-pu|, where pb is a location on a selected boundary B that is adjacent to the user location vector pu in said LD coordinate system; and

(4) min_bε

B |pb-pu'"'"'|, where pb is a location on a selected boundary B that is adjacent to the user location vector pu'"'"' on said selected map;

accepting said user location coordinates (xu'"'"',yu'"'"') on said selected map when the ratio 13 satisfies ρ

≦

F1, where F1 is a selected positive real number that is no greater than 1.0; and

rejecting said user location coordinates (xu'"'"',yu'"'"') on said selected map when the ratio ρ

satisfies ρ

≧

F2, where F2 is a selected positive real number that is greater than F1.

20. The method of claim 10, wherein steps (5)-(6) further comprise the steps of:

determining said parameters a11, a12, a21 and a22 by the relations ##EQU35## D123=(x1-x3)(y2-y3)-(x2-x3)(y1-y3); and

determining said parameters x0 and y0 by the relations ##EQU36##

21. A method for determining the present location coordinates of a user moving on a two-dimensional surface, with reference to a map that may be substantially inaccurate, the method comprising the steps of:

providing a user with a location determination (LD) unit that can determine the location coordinates of the user and can determine or provide the location coordinates, relative to the user, of a chosen location that is spaced apart from the user'"'"'s location;

using the LD unit to determine user LD location coordinates (xu,yu);

using the LD unit to determine first and second ranges R1 and R2 from the user'"'"'s location to selected first and second landmark locations, respectively, that are spaced apart from the user'"'"'s location;

providing a selected map of a region that includes selected first and second landmark locations and that indicates location coordinates (x1'"'"',y1'"'"') and (x2'"'"',y2'"'"') for the selected first and second landmark locations on the selected map;

using a computer to determine computed location coordinates (xu'"'"',yu'"'"') on the selected map of at least one location that has the respective ranges R1 and R2 from the first and second selected landmark locations, as indicated on the selected map; and

designating the computed location coordinates (xu'"'"',yu'"'"') as the user location coordinates on the selected map.

22. A method for determining the present location coordinates of a user moving in a three-dimensional region, with reference to a map that may be substantially inaccurate, the method comprising the steps of:

providing a user with a location determination LD unit that can determine the location coordinates of the user in an LD coordinate system and can determine the location coordinates, relative to the user, of a chosen location that is spaced apart from the user'"'"'s location;

using the LD unit to determine user LD coordinates (xu,yu,zu);

using the LD Unit to determine first, second and third ranges R1, R2 and R3 from the user'"'"'s location to selected first, second and third landmark locations, respectively, that are spaced apart from the user'"'"'s location;

providing a selected map of a region that includes the selected first, second and third landmark locations and that indicates location coordinates (x1'"'"',y1'"'"',z1'"'"'), (x2'"'"',y2'"'"',z2'"'"') and (x3'"'"',y3'"'"',z3'"'"') for the selected first, second and third landmarks on the selected map;

using a computer to determine computed location coordinates (xu'"'"',yu'"'"',zu'"'"') on the selected map of at least one location that has the respective ranges R1, R2 and R3 from the first, second and third selected landmark locations, as indicated on the selected map; and

designating the computed location coordinates (xu'"'"',yu'"'"',zu'"'"') as the user location coordinates on the selected map.

23. A method for determining the location coordinates of a user in a three-dimensional region, with reference to a map that may be substantially inaccurate, the method comprising the steps of:

(1) providing a user with a location determination (LD) unit having an LD antenna and an LD receiver/processor that can determine the location coordinates of the user in an LD coordinate system and can determine or provide the location coordinates, relative to the user, of a chosen location that is spaced apart from the user'"'"'s location;

(2) providing a selected map of a region that includes the user'"'"'s location and that has at least first, second and third landmark representatives with first, second and third map location coordinate triples (xr1'"'"',yr1'"'"',zr1'"'"'), (xr2'"'"',yr2'"'"',zr2'"'"') and (xr3'"'"',yr3'"'"',zr3'"'"') , respectively;

(3) using the LD unit to determine user LD location coordinates (xu,yu,zu) of the user'"'"'s location in the LD coordinate system;

(4) providing first, second and third LD location coordinate triples (xr1,yr1,zr1), (xr2,yr2,zr2) and (xr3,yr3,zr3) for the first, second and third landmark representatives, respectively, in the LD coordinate system;

(5) using a computer to determine a coordinate transformation T of the LD location coordinate triples (xr1,yr1,zr1), (xr2,yr2,zr2)and (xr3,yr3,zr3) onto the respective map location coordinate triples (xr1'"'"',yr1'"'"',zr1'"'"'), (xr2'"'"',yr2'"'"',zr2'"'"') and (xr3'"'"',yr3'"'"',zr3'"'"');

(6) using the computer to express a user map location coordinate triple (xu'"'"',yu'"'"',zu '"'"') on the selected map as the image under the transformation T of the user LD location coordinate triple (xu,yu,zu); and

(7) displaying the user map location coordinate triple (xu'"'"',yu'"'"',zu'"'"') in at least one of a visually perceptible form and an audibly perceptible form.

24. The method of claim 23, further comprising the step of visually indicating a location on said LD unit corresponding to said location coordinate triple (xu'"'"',yu'"'"',zu'"'"').

25. The method of claim 23, further comprising the steps of:

expressing said LD-determined user location as a vector pu that is a linear combination of three linearly independent vectors I, J and K in said three-dimensional region, in the form

space="preserve" listing-type="equation">pu=xu I+yu J+zu K,

where xu, yu and zu are said user LD location coordinates;

expressing the locations of said first and second landmark representatives, as determined by said LD unit, as the respective vectors

space="preserve" listing-type="equation">p1=xr1 I+yr1 J+zr1 K,

space="preserve" listing-type="equation">p2=xr2 I+yr2 J+zr2 K,

space="preserve" listing-type="equation">p3=xr3 I+yr2 J+zr3 K;

expressing the locations of said first, second and third landmark representatives, as determined on said selected map, as the respective vectors

space="preserve" listing-type="equation">p1=xr1'"'"'I+yr1'"'"'J+zr1'"'"'K,

space="preserve" listing-type="equation">p2'"'"'=xr2'"'"'I+yr2'"'"'J+zr2'"'"'K,

space="preserve" listing-type="equation">p3'"'"'=xr3'"'"'I+yr2'"'"'J+zr3'"'"'K;

determining two vectors pu'"'"'=xu'"'"'+yu'"'"'J+zu'"'"'K and pu"=xu"+yu"J +zu"K in said selected map that satisfy the relations

space="preserve" listing-type="equation">|pu'"'"'-p1'"'"'|=|pu"-p1'"'"'|=|pu-p1.vertline.,

space="preserve" listing-type="equation">|pu'"'"'-p2'"'"'|=|pu"-p2'"'"'|=|pu-p2.vertline., and

space="preserve" listing-type="equation">|pu'"'"'-p3'"'"'|=|pu"-p3'"'"'|=|pu-p3.vertline.; and

selecting one of the vectors pu'"'"' and pu" as said user location in said selected map.

26. The method of claim 23, wherein said map location coordinate triples (xr1'"'"',yr1'"'"',zr1'"'"'), (xr2'"'"',yr2'"'"',zr2'"'"') and (xr3'"'"',yr3'"'"',zr3'"'"') are the location coordinate triples of physical landmarks L1'"'"', L2'"'"' and L3'"'"' on said selected map, and said LD location coordinate triples (xr1,yr1,zr1), (xr2,yr2,zr2) and (xr3,yr3,zr3) are the location coordinate triples of physical landmarks L1, L2 and L3 in said LD coordinate system, corresponding to the respective landmarks L1'"'"', L2'"'"' and L3'"'"'.

27. The method of claim 23, further comprising the step of choosing said LD unit from the class of satellite-based and ground-based LD units consisting of GPS, GLONASS, Loran, Tacan, Decca, Omega, VOR, DME, JTIDS, PLRS and an FM subcarrier system.

28. The method of claim 23, wherein steps (5)-(6) comprise the steps of:

expressing said LD-determined user location as a vector PU that is a linear combination of three linearly independent vectors I, J and K in said three-dimensional region, in the form

space="preserve" listing-type="equation">pu=xu I+yu J+zu K,

where xu, yu and zu are said user location coordinates;

expressing the locations of said first, second and third landmark representatives in said LD coordinate system as the respective vectors p1=xr1 I+yr1 J+zr1 K,

space="preserve" listing-type="equation">p2=xr2 I+yr2 J+zr2 K,

space="preserve" listing-type="equation">p3=xr3 I+yr3 J+zr3 K,

where xr1, yr1, zr1, xr2, yr2, zr2, xr3, yr3 and zr3 are said location coordinates in said LD coordinate system and p1, p2 and p3 are linearly independent;

expressing the locations of said first, second and third landmark representatives on said selected map as the respective vectors

space="preserve" listing-type="equation">p1'"'"'=xr1'"'"'I+yr1'"'"'J+zr1'"'"'K,

space="preserve" listing-type="equation">p2'"'"'=xr2'"'"'I+yr2'"'"'J+zr2'"'"'K,

space="preserve" listing-type="equation">p3'"'"'=xr3'"'"'I+yr3'"'"'J+zr3'"'"'K,

where xr1'"'"', yr1'"'"', zr1'"'"', xr2'"'"', yr2'"'"', zr2'"'"', xr3'"'"', yr3'"'"' and zr3'"'"' are said location coordinates on said selected map;

forming a first 3×

3 matrix ##EQU37## having as its entries the LD location coordinates of said first, second and third landmark representatives;

forming a second 3×

3 matrix ##EQU38## having as its entries the map location coordinates of said first, second and third landmark representatives;

determining a 3×

3 matrix A=M(X'"'"'1,2,3) M(X1,2,3)^-1 with entries aij (i=1,2,3;

j=1,2,3), where A represents said coordinate transformation T; and

expressing said user location coordinates xu'"'"', yu'"'"' and zu'"'"' on said selected map by the relations

space="preserve" listing-type="equation">xu'"'"'=a11 xu+a12 yu+a13 zu.

space="preserve" listing-type="equation">yu'"'"'=a21 xu+a22 yu+a23 zu.

space="preserve" listing-type="equation">zu'"'"'=a31 xu+a32 yu+a33 zu.

29. The method of claim 23, wherein said selected map has N physical landmarks (N≧

4), for which location coordinates (xi'"'"',yi'"'"',zi'"'"') (i=1, 2, . . . , N) are provided on said selected map and for which N corresponding landmarks with location coordinates (xi,yi,zi) are provided in said LD coordinate system, and wherein steps (2) and (4) comprise the steps of;

providing a first sequence of real numbers cj (j=1, . . . , N1), a second sequence of real numbers dj (j=N1+1, . . . , N1+N2), and a third sequence of real numbers ej (j=N1+N2+1, . . . , N), where N1+N2+N3=N, N1≧

1, N2≧

1, N3≧

1, and where the real numbers cj, dj and ej satisfy the constraints 0≦

cj≦

1, 0≦

dj≦

1,0≦

ej≦

1, and ##EQU39## selecting a first set G1 of N1 landmarks and associated location coordinates, a second set G2 of N2 landmarks and associated location coordinates, and a third set G3 of N3 landmarks and associated location coordinates, from among the N landmarks in said LD coordinate system, where the first set, the second set and the third set are mutually exclusive, where a landmark in the first set G1 has location coordinates (xj,yj,zj) (j=1, . . ., N1), a landmark in the second set G2 has location coordinates (xj,yj,zj) (j=N1+1, . . . , N1+N2) and a landmark in the third set G3 has location coordinates (xj,yj,zj) (j=N1+N2+1, . . . , N1+N2+N3);

selecting a first set G1'"'"' of N1 landmarks and associated location coordinates, a second set G2'"'"' of N2 landmarks and associated location coordinates, and a third set G3'"'"' of N3 landmarks and associated location coordinates, from among the N landmarks on said selected map, where a landmark in the first set G1'"'"' has location coordinates (xj'"'"',yj'"'"',zj'"'"')(j=1, . . . , N1) and corresponds to a landmark in the first set G1, a landmark in the second set G2'"'"' has location coordinates (xj'"'"',yj'"'"',zj'"'"') (j=N1+1, . . . , N1+N2) and corresponds to a landmark in the second set G2, and a landmark in the third set G3'"'"' has location coordinates (xj'"'"',yj'"'"',zj'"'"') (j=N1+N2+1, . . . , N1+N2+N3) and corresponds to a landmark in the third set G3;

determining said location coordinate triples (xr1,yr1,zr1), (xr2,yr2,zr2) and (xr3,yr3,zr3) for said first, second and third landmark representative locations in said LD coordinate system by the relations ##EQU40## determining said location coordinate triples (xr1'"'"',yr1'"'"',zr1'"'"'), (xr2'"'"',yr2'"'"',zr2'"'"') and (xr3'"'"',yr3'"'"',zr3'"'"') for said first, second and third landmark representative locations on said selected map by the relations ##EQU41##

30. The method of claim 29, further comprising the step of choosing at least one of said sequences of real numbers cj (j=1, . . . , N1), dj (j=N1+1, . . . , N1+N2) and ej (j=N1+N2+1, . . . , N1+N2+N3) according to the relations

31. The method of claim 29, further comprising the step of choosing at least one of said sequences of real numbers cj (j=1, . . . , N1), dj (j=N1+1, . . . , N1+N2) and ej (j=N1+N2+1, . . . , N1+N2+N3) according to the relations

32. The method of claim 23, further comprising the step of providing a fourth landmark representative on said selected map with a map location coordinate triple (xr4'"'"',yr4'"'"',zr4'"'"') and a corresponding fourth landmark representative in said LD coordinate system with an LD location coordinate triple (xr4,yr4,zr4) that is not coplanar with said LD location coordinate triples (xrk,yrk,zrk) (k=1,2,3);

wherein steps (5)-(6) comprise the steps of;

providing a coordinate transformation from a location with location coordinates (x,y,z) in said LD coordinate system to a location with location coordinates (x'"'"',y'"'"',z'"'"') on said selected map, where the transformation has the form

space="preserve" listing-type="equation">x'"'"'-x0=a11(x-x0)+a12(y-y0)+a13(z-z0),

space="preserve" listing-type="equation">y'"'"'-y0=a21(x-x0)+a22(y-y0)+a23(z-z0),

space="preserve" listing-type="equation">z'"'"'-z0=a31 (x-x0)+a32(y-y0)+a33(z-z0),

and where the transformation carries said LD location coordinate triples (xr1,yr1,zr1), (xr2,yr2,zr2), (xr3,yr3,zr3) and (xr4,yr4,zr4) onto said respective map location coordinate triples (xr1'"'"',yr1'"'"',zr1'"'"'), (xr2'"'"',yr2'"'"',zr2'"'"'), (xr3'"'"',yr3'"'"',zr3'"'"') and (xr4'"'"',yr4'"'"',zr4'"'"'), where the coefficients of this transformation are determined by the following steps;

forming a first 3×

3 matrix ##EQU42## having as its elements differences between said LD location coordinates xrk, yrk and zrk (k=2, 3,

     4) and said corresponding LD location coordinates xr1, yr1 and zr1;

forming a second 3×

3 matrix ##EQU43## having as its elements differences between said map location coordinates xrk'"'"', yrk'"'"' and zrk'"'"' (k=2,3,4) and said corresponding map location coordinates xr1'"'"', yr1'"'"' and zr1'"'"';

determining a 3×

3 matrix

A=M(Δ

X'"'"'1;

2,3,4) M(Δ

X1;

2,3,4)^-1 with entries aij (i=1,2,3;

j=1,2,3);

determining a vector X0=^tr, whose entries are the translation coordinates x0, y0 and z0, and which is related to the vectors X1=^tr and X1'"'"'=^tr by the relation

X0=(A-I)^-1 (A X1-XI'"'"'); and

expressing said user location coordinates xu'"'"', vu'"'"' and zu'"'"' on said selected map by the relations

space="preserve" listing-type="equation">xu'"'"'=a11 xu+a12 yu+a13 zu-(a11-1) x0-a12 y0-a13 z0,

space="preserve" listing-type="equation">yu'"'"'=a21 xu+a22 yu+a23 zu-a21 x0-(a22-1) y0-a23 z0,

space="preserve" listing-type="equation">zu'"'"'=a31 xu+a32 yu+a33 zu-a31 x0-a32 y0-(a33-1)z0.

33.

33. The method of claim 23, wherein said selected map has N landmark representatives (N≧

5), for which location coordinates (xi'"'"',yi'"'"',zi'"'"') (i=1, 2, . . . , N) are provided on said selected map and for which N corresponding landmark representatives with location coordinates (xi,yi,zi) are provided in said LD coordinate system;

wherein steps (2) and (4) comprise the steps of;

providing a first sequence of real numbers cj (j=1, . . . , N1), a second sequence of real numbers dj (j=N1+1, . . . , N1+N2), a third sequence of real numbers ej (j=N1+N2+1, . . . , N1+N2+N3) and a fourth sequence of real numbers fj (j=N1+N2+N3+1, . . . , N1+N2+N3+N4), where N1+N2+N3+N4 =N, N1≧

1, N2≧

1, N3≧

1 and N4≧

1, and where the real numbers cj, dj, ej and fj satisfy the constraints 0≦

cj≦

1, 0≦

dj≦

1,0≦

ej≦

1, 0≦

fj≦

1 and ##EQU44## selecting a first set G1 of N1 landmarks and associated location coordinates, a second set G2 of N2 landmarks and associated location coordinates, a third set G3 of landmarks and associated location coordinates, and a fourth set G4 of landmarks and associated location coordinates, from among the N landmarks in said LD coordinate system, where the first set, second set, third set and fourth set are mutually exclusive, where a landmark in the first set G1 has location coordinates (xj,yj,zj) (j=1, . . . , N1), a landmark in the second set G2 has location coordinates (xj,yj,zj) (j=N1+1, . . . , N1+N2), a landmark in the third set G3 has location coordinates (xj,yj,zj) (j=N1+N2+1, . . . , N1+N2+N3) and a landmark in the fourth set G4 has location coordinates (xj,yj,zj) (j=N1+N2+N3+1, . . . , N1+N2+N3+N4);

selecting a first set G1'"'"' of N1 landmarks and associated location coordinates, a second set G2'"'"' of N2 landmarks and associated location coordinates, a third set G3'"'"' of N3 landmarks and associated location coordinates, and a fourth set G4'"'"' of landmarks, from among the N landmarks on said selected map, where a landmark in the first set G1'"'"' has location coordinates (xj'"'"',yj'"'"',zj'"'"') (j =1, . . . , N1) and corresponds to a landmark in the first set G1, a landmark in the second set G2'"'"' has location coordinates (xj'"'"',yj'"'"',zj'"'"') (j=N1+1, . . . , N1+N2) and corresponds to a landmark in the set G2, a landmark in the set G3'"'"' has location coordinates (xj'"'"',yj'"'"',zj'"'"') (j=N1+N2+1, . . . , N1+N2+N3) and corresponds to a landmark in the set G3, and a landmark in the fourth set G4'"'"' has location coordinates (xj'"'"',yj'"'"',zj'"'"') (j=N1+N2+N3+1, . . . , N1+N2+N3+N4);

determining said LD location coordinates (xr1,yr1,zr1), (xr2,yr2,zr2), (xr2,yr3,zr3) and an LD location coordinate triple (xr4,yr4,zr4) for said first, second, third and fourth landmark representative locations in said LD coordinate system by the relations ##EQU45## determining said location coordinates (xr1'"'"',yr1'"'"',zr1'"'"'), (xr2'"'"',yr2'"'"',zr2'"'"'), (xr3'"'"',yr3'"'"',zr3'"'"') and a map location coordinate triple (xr4'"'"',yr4'"'"',zr4'"'"') for said first, second, third and fourth landmark representative locations on said selected map by the relations ##EQU46##

34. The method of claim 33, further comprising the step of choosing at least one of said sequences of real numbers cj (j=1, . . . , N1), dj (j=N1+1, . . . , N1+N2), ej (j=N1+N2+1, . . . , N1+N2+N3) and fj (j=N1+N2+N3+1, . . . , N1+N2+N3+N4) according to the relations

35. The method of claim 33, further comprising the step of choosing at least one of said sequences of real numbers cj (j=1, . . . , N1), dj (j=N1+1, . . . , N1+N2), ej (j=N1+N2+1, . . . , N1+N2+N3) and fj (j=N1+N2+N3+1, . . . , N1+N2+N3+N4) according to the relations

36. The method of claim 23, wherein said selected map has N landmark representatives (N≧

4), for which location coordinates (xi'"'"',yi'"'"',zi'"'"') (i=1, 2, . . . , N) are provided on said selected map and for which N corresponding landmarks with location coordinates (xi,yi,zi) are provided in said LD coordinate system;

wherein steps (5)-(6) comprise the steps of;

choosing said coordinate transformation T to be of the form

space="preserve" listing-type="equation">x'"'"'-a11 x+a12 y+a13 z,

space="preserve" listing-type="equation">y'"'"'-a21 x+a22 y+a23 z,

space="preserve" listing-type="equation">z'"'"'-a31 x+a32 y+a33 z,

where a11, a12, a13, a21, a22, a23, a31, a32 and a33 are parameters to be determined, x'"'"', y'"'"' and z'"'"' are location coordinates on said selected map, and x, y and z are location coordinates in said LD coordinate system;

choosing the parameters a11, a12, a13, a21, a22, a23, a31, a32 and a33 to minimize an error term ε

defined by ##EQU47## where ε

xi, ε

yi and ε

zi are selected non-negative coefficients and the sums are over the index i=1, . . . , N; and

expressing said user location coordinates (xu'"'"',yu'"'"',zu'"'"') on said selected map by the relations

space="preserve" listing-type="equation">xu'"'"'=a11 xu+a12 yu+a13 zu,

space="preserve" listing-type="equation">yu'"'"'=a21 xu+a22 yu+a23 zu,

space="preserve" listing-type="equation">zu'"'"'=a31 xu+a32 yu+a33 zu.

37. The method of claim 23, wherein steps (5)-(6) comprise the steps of:

expressing the locations of said first, second and third landmark representatives in said LD coordinate system as the respective vectors

space="preserve" listing-type="equation">p1=xr1 I+yr1 J+zr1 K,

space="preserve" listing-type="equation">p2=xr2 I+yr2 J+zr2 K,

space="preserve" listing-type="equation">p3=xr3 I+yr3 J+zr3 K,

where xr1, yr1, zr1, xr2, yr2,zr2, xr3, yr3 and zr3 are said location coordinates in said LD coordinate system and I, J and K are three linearly independent vectors in said three-dimensional region;

expressing the locations of said first, second and third landmark representatives on said selected map as the respective vectors

space="preserve" listing-type="equation">p1'"'"'=xr1'"'"'+I+yr1'"'"'J+zr1 K,

space="preserve" listing-type="equation">p2'"'"'=xr2'"'"'I+yr2 J+zr2'"'"'K,

space="preserve" listing-type="equation">p3'"'"'=xr3'"'"'I+yr3 J+zr3'"'"'K,

where xr1'"'"', yr1'"'"', zr1'"'"', xr2'"'"', yr2'"'"',zr2'"'"', xr3'"'"', yr3'"'"' and zr3'"'"' are said location coordinates on said selected map;

expressing said LD-determined user location as a vector pu that is a linear combination of vectors p1, p2 and p3 representing said first, second and third landmark representatives

space="preserve" listing-type="equation">pu=a1 p1+a2 p2+a3 p3,

where a1, a2 and a3 are selected coefficients;

expressing the vectors p1'"'"', p2'"'"' and p3'"'"' as linear combinations

space="preserve" listing-type="equation">p1'"'"'=b11 p1+b12 p2+b13 p3,

space="preserve" listing-type="equation">p2'"'"'=b21 p1+b22 p2+b23 p3,

space="preserve" listing-type="equation">p3'"'"'=b31 p1+b32 p2+b33 p3,

where b11, b12, b13, b21, b22, b23, b31, b32 and b33 are selected coefficients;

and

expressing said user location on said selected map as a vector ##EQU48##

38. The method of claim 23, further comprising the step of providing a landmark representative on said selected map that has a fourth map location coordinate triple (xr4'"'"',yr4'"'"',zr4'"'"') and providing a corresponding landmark representative in said LD coordinate system that has a fourth LD location coordinate triple (xr4,yr4,zr4), wherein steps (5)-(6) comprise the steps of:

expressing the locations of said first, second and third landmark representatives and of the fourth landmark representative in said LD coordinate system as the respective vectors

space="preserve" listing-type="equation">p1=xr1 I+yr1 J+zr1 K,

space="preserve" listing-type="equation">p2=xr2 I+yr2 J+zr2 K,

space="preserve" listing-type="equation">p3=xr3 I+yr3 J+zr3 K,

space="preserve" listing-type="equation">p4=xr4 I+yr4 J+zr4 K,

where xr1, yr1, zr1, xr2, yr2, zr2, xr3, yr3, zr3, xr4, yr4 and zr4 are said location coordinates in said LDS and I, J and K are linearly independent vectors in said three-dimensional region;

expressing the locations of said first, second and third landmark representatives and of the fourth landmark representative on said selected map as the respective vectors

space="preserve" listing-type="equation">p1'"'"'=xr1'"'"'I+yr1'"'"'J+zr1'"'"'K,

space="preserve" listing-type="equation">p2'"'"'=xr2'"'"'I+yr2'"'"'J+zr2'"'"'K,

space="preserve" listing-type="equation">p3'"'"'=xr3'"'"'I+yr3'"'"'J+zr3'"'"'K,

space="preserve" listing-type="equation">p4'"'"'=xr4'"'"'I+yr4'"'"'J+zr4'"'"'K,

where xr1'"'"', yr1'"'"', zr1'"'"', xr2'"'"', yr2'"'"', zr2'"'"', xr3'"'"', yr3'"'"', zr3'"'"', xr4'"'"', yr4'"'"' and zr4'"'"' are said location coordinates on said selected map;

expressing said LD-determined user location as a vector pu that is a linear combination

space="preserve" listing-type="equation">pu=a1 p1+a2 p2+a3 p3+(1-a1-a2-a3) p4,

where a1, a2 and a3 are selected coefficients;

expressing the vectors p1'"'"', p2'"'"', and p3'"'"' and p4'"'"' as linear combinations

space="preserve" listing-type="equation">p1'"'"'=b11 p1+b12 p2+b13 p3, +(1-b11-b12-b13) p4

space="preserve" listing-type="equation">p2'"'"'=b21 p1+b22 p2+b23 p3, +(1-b21-b22-b23) p4,

space="preserve" listing-type="equation">p3'"'"'=b31 p1+b32 p2+b33 p3, (1-b31-b32-b33) p4)

space="preserve" listing-type="equation">p4'"'"'=b41 p1+b42 p2+b43 p3+(1-b41-b42-b43) p4,

where b11, b12, b31, b21, b22, b23, b31, b32 and b33 are selected coefficients;

and

expressing said user location on said selected map as a vector pu'"'"' with components xu'"'"' and yu'"'"' given by ##EQU49##

39. The method of claim 23, further comprising the step of providing a landmark representative on said selected map that has a fourth map location coordinate triple (xr4'"'"',yr4'"'"',zr4'"'"') and providing a corresponding landmark representative in said LD coordinate system that has a fourth LD location coordinate triple (xr4,yr4,zr4), wherein steps (5)-(6) comprise the steps of:

expressing the LDS-determined user location as a vector pu that is a linear combination of linearly independent vectors I, J and K in said three-dimensional region, in the form

space="preserve" listing-type="equation">pu=xu I+yu J+zu K,

where xu, yu and zu are said user location coordinates in said LDS;

expressing the locations of said first, second and third landmark representatives and of the fourth landmark representative in said LD coordinate system as the respective vectors

space="preserve" listing-type="equation">p1=xr1 I+yr1 J+zr1 K,

space="preserve" listing-type="equation">p2=xr2 I+yr2 J+zr2 K,

space="preserve" listing-type="equation">p3=xr3 I+yr3 J+zr3 K,

space="preserve" listing-type="equation">p4=xr4 I+yr4 J+zr4 K,

where xr1, yr1, zr1, xr2, yr2, zr2, xr3, yr3, zr3, xr4, yr4 and zr4 are said location coordinates in said LD coordinate system;

expressing the locations of said first, second and third landmark representatives and of the fourth landmark representative on said selected map as the respective vectors

space="preserve" listing-type="equation">p1'"'"'=xr1'"'"'I+yr1'"'"'J+zr1'"'"'K,

space="preserve" listing-type="equation">p2'"'"'=xr2'"'"'I+yr2'"'"'J+zr2'"'"'K,

space="preserve" listing-type="equation">p3'"'"'=xr3'"'"'I+yr3'"'"'J+zr3'"'"'K,

space="preserve" listing-type="equation">p4'"'"'=xr4'"'"'I+yr4'"'"'J+zr4'"'"'K,

where xr1'"'"', yr1'"'"', zr1'"'"', xr2'"'"', yr2'"'"', zr2'"'"', xr3'"'"', yr3'"'"', zr3'"'"', xr4'"'"', yr4'"'"' and zr4'"'"' are said location coordinates on said selected map;

expressing the vector pu as a linear combination

space="preserve" listing-type="equation">pu=a1'"'"'p1'"'"'+a2'"'"'p2'"'"'+a3'"'"'p3'"'"'+(1-a1'"'"'-a2'"'"'-a3'"'"') p4'"'"',

where a1'"'"', a2'"'"' and a3'"'"' are selected coefficients;

expressing the vectors p1, p2, p3 and p4 as linear combinations

space="preserve" listing-type="equation">p1=b11'"'"'p1'"'"'+b12'"'"'p2'"'"'+b13'"'"'p3'"'"'+(1-b11'"'"'-b12'"'"'-b13'"'"') p4'"'"',

space="preserve" listing-type="equation">p2=b21'"'"'p1'"'"'+b22'"'"'p2'"'"'+b23'"'"'p3'"'"'+(1-b21'"'"'-b22'"'"'-b23'"'"') p4'"'"',

space="preserve" listing-type="equation">p3=b31'"'"'p1'"'"'+b32'"'"'p2'"'"'+b33'"'"'p3'"'"'+(1-b31'"'"'-b32'"'"'-b33'"'"') p4'"'"',

space="preserve" listing-type="equation">p4=b41'"'"'p1'"'"'+b42'"'"'p2'"'"'+b43'"'"'p3'"'"'+(1-b41'"'"'-b42'"'"'-b43'"'"') p4'"'"',

where b11'"'"', b12'"'"', b31'"'"', b21'"'"', b22'"'"', b23'"'"', b31'"'"', b32'"'"' and b33'"'"' are selected coefficients; and

expressing said user location on said selected map as a vector pu'"'"' with components xu'"'"' and yu'"'"' given by ##EQU50##

40. The method of claim 23, further comprising the steps of:

using the computer to compute a dimensionless ratio ρ

, whose numerator increases monotonically with the magnitude of the vector difference |pu-pu'"'"'|, where pu and pu'"'"' are user location vectors in said LD coordinate system and on said selected map, respectively, and are given by

space="preserve" listing-type="equation">pu=xu I+yu J+zu K,

space="preserve" listing-type="equation">pu'"'"'=xu'"'"'I+yu'"'"'J+zu'"'"'K,

space="preserve" listing-type="equation">pu-pu'"'"'=(xu-xu'"'"') I+(yu-yu'"'"') J+(zu-zu'"'"') K,

where I, J and K are linearly independent vectors in said three-dimensional region, and whose denominator increases monotonically with at least one of the following magnitudes of vector differences;

(1) min{|pu-p1|, |pu-p2|, |pu-p3|}, where p1=xr1 I+yr1 J+zr1 K, p2=xr2 I+yr2 J+zr2 K and p3=xr3 I+yr3 J+zr3 K are vectors indicating said locations of said first, second and third landmark representatives in said LD coordinate system;

(2) min{|pu'"'"'-p1'"'"',||pu'"'"'-p2'"'"'|, |pu=p3|}, where p1'"'"'=xr1'"'"'I+yr1'"'"'J+zr1'"'"'K, p2'"'"'=xr2'"'"'I+yr2'"'"'J+zr2'"'"'K and p3'"'"'=xr3'"'"'I+yr3'"'"'J+zr3'"'"'K are vectors indicating said locations of said first, second and third landmark representatives on said selected map;

(3) min_bε

B |pb-pu|,

where pb is a location on a selected boundary B that is adjacent to the user location vector pu in said LD coordinate system; and

(4) min_bε

B |pb-pu'"'"'|,

where pb is a location on a selected boundary B that is adjacent to the user location vector pu'"'"' on said selected map;

accepting said user location coordinates (xu'"'"',yu'"'"',zu'"'"') on said selected map when the ratio ρ

satisfies ρ

≦

F1, where F1 is a selected positive real number that is no greater than 1.0; and

rejecting said user location coordinates (xu'"'"',yu'"'"',zu'"'"') on said selected map when the ratio ρ

satisfies ρ

≧

F2, where F2 is a selected positive real number that is greater than F1.

41. Apparatus for determining the present location coordinates of a user moving in a two-dimensional region, with reference to a map that may be substantially inaccurate, the apparatus comprising:

a location determination (LD) unit, including an LD signal antenna and an LD signal receiver/processor electronically connected to the antenna, that can receive LD signals and can determine a user LD location coordinate pair (xu,yu) in an LD coordinate system of a user who transports the LD unit in a two-dimensional region, and that can determine or provide an LD location coordinate pair of a chosen location that is spaced apart from the user'"'"'s location;

where the user has a selected map of a region that includes M landmark representative locations (M≧

2) and a map location coordinate triple (xrk'"'"',yrk'"'"') (k=1, . . . , M) for each of the M landmark representative locations;

a computer, electronically connected with the LD unit and provided with the user LD location coordinate triple (xu,yu) and with M LD location coordinate triples (xrk,yrk) (k=1, . . . , M) representing locations in the LD coordinate system corresponding to the M landmark representatives on the selected map;

data entry means for entry of the M landmark representative location coordinate pairs on the selected map into the computer;

where the computer is programmed to determine a coordinate transformation T of the M landmark representative LD location coordinate pairs (xrk,yrk) onto the location coordinate pairs (xrk'"'"',yrk'"'"') for the M landmark representatives on the selected map, and to determine a location coordinate pair (xu'"'"',yu'"'"') on the selected map that is an image T(xu,yu) under the transformation T of the user LD location coordinate pair (xu,yu) representing the user'"'"'s location in the LD coordinate system; and

display means, associated with the computer, for displaying the user'"'"'s location coordinate triple for the image T(xu,yu) on the selected map, in at least one of a visually perceptible form and an audibly perceptible form.

42. The apparatus of claim 41, further comprising locator means, electronically connected to at least one of said LD unit and said computer, for determining an LD location coordinate pair for the location of a chosen location that is spaced apart from the LD unit, and for providing this location coordinate pair for at least one of said LD unit and said computer.

43. Apparatus for determining the present location coordinates of a user moving in a three-dimensional region, with reference to a map that may be substantially inaccurate, the apparatus comprising:

a location determination (LD) unit, including an LD signal antenna and an LD signal receiver/processor electronically connected to the antenna, that can receive LD signals and can determine a user LD location coordinate triple (xu,yu,zu) in an LD coordinate system of a user who transports the LD unit in a three-dimensional region, and that can determine or provide an LD location coordinate triple of a chosen location that is spaced apart from the user'"'"'s location;

where the user has a selected map of a region that includes M landmark representative locations (M≧

3) and a map location coordinate triple (xrk'"'"',yrk'"'"',zrk'"'"') (k=1, . . . , M) for each of the M landmark representative locations;

a computer, electronically connected with the LD unit and provided with the user LD location coordinate triple (xu,yu,zu) and with M LD location coordinate triples (xrk,yrk,zrk) (k=1, . . . , M) representing locations in the LD coordinate system corresponding to the M landmark representatives on the selected map;

data entry means for entry of the M landmark representative location coordinate triples on the selected map into the computer;

where the computer is programmed to determine a coordinate transformation T of the M landmark representative LD location coordinate triples (xrk,yrk,zrk) onto the location coordinate triples (xrk'"'"',yrk'"'"',zrk'"'"') for the M landmark representatives on the selected map, and to determine a location coordinate triple (xu'"'"',yu'"'"',zu'"'"') on the selected map that is an image T(xu,yu,zu) under the transformation T of the user LD location coordinate triple (xu,yu,zu) representing the user'"'"'s location in the LD coordinate system; and

display means, associated with the computer, for displaying the user'"'"'s location coordinate triple for the image T(xu,yu,zu) on the selected map, in at least one of a visually perceptible form and an audibly perceptible form.

44. The apparatus of claim 43, further comprising locator means, electronically connected to at least one of said LD unit and said computer, for determining an LD location coordinate triple for the location of a chosen location that is spaced apart from the LD unit, and for providing this location coordinate triple for at least one of said LD unit and said computer.