Locating system and method for determining positions of objects

• US 6,946,956 B2
• Filed: 07/30/2002
• Issued: 09/20/2005
• Est. Priority Date: 07/29/2002
• Status: Active Grant

• Current Assignee: Nippon Telegraph and Telephone Corporation
• Original Assignee: Nippon Telegraph and Telephone Corporation
• Inventors: Shibuya, Akinori, Hayashi, Hitoshi, Shimizu, Masashi, Suda, Hirohito
• Primary Examiner(s): La, Anh V.
• Application Number:
  US10/207,021
• Publication Number:
  US 20040021566A1
• Time in Patent Office: 1,148 Days
• Field of Search: 340/539.13, 340/993, 340/540, 340/573.1, 340/686.1, 342/357.07, 342/450, 342/457, 702/150, 700/302
• US Class Current: 340/539.13
• CPC Class Codes: F41G 3/2633 using a TV type screen, e.g. a CRT, displaying a simulated target
• CPC Class Codes: G01S 11/06 using intensity measurements
• CPC Class Codes: G01S 11/12 using electromagnetic waves other than radio waves
• CPC Class Codes: G01S 11/14 using ultrasonic, sonic, or infrasonic waves

Abstract

A locating system for determining a position of an object comprises a transmitting station that transmits a first ID signal containing a first identifier of this transmitting station; a receiving station that receives the first ID signal, measures the intensity of the first ID signal, and extracts the first identifier; a data management unit that stores and manages the intensity in association with the first identifier, the intensity and the identifier being supplied from the receiving station; and a positioning computer that estimates the position of the transmitting station using a first correcting formula defining a relation between the intensity and the distance, based on the data stored in the data management unit.

Claims

1. A system for determining a position of an object, comprising:

a transmitting station configured to transmit a first ID signal containing a first identifier in a periodic manner;

a receiving station configured to receive the first ID signal, measure an intensity of the first ID signal, and extract the first identifier;

a data management unit configured to store and manage the intensity in association with the first identifier; and

a positioning computer configured to determine a first correcting formula defining a relation between the intensity measured at the receiving station and a distance between the transmitting station and the receiving station using a correcting coefficient, and to estimate a position of an unknown transmitting station using the first correcting formula and known position information stored in the data management unit.

2. The system according to claim 1, wherein the receiving station comprises:

an activation signal generator configured to generate an activation signal for causing the transmitting station to generate a second ID signal; and

a transmitter configured to transmit the activation signal to the transmitting station, and wherein the receiving station has an ID signal generator configured to generate the second ID signal containing a second identifier in response to the activation signal.

3. The system according to claim 1, wherein the transmitting station receives an activation signal supplied from a remote base station, and has an ID signal generator configured to generate a second ID signal containing a second identifier in response to the activation signal.

4. The system according to claim 1, wherein the transmitting station has a sensor for sensing a change due to an external factor, and an ID signal generator configured to generate a third ID signal containing a third identifier when sensing the change.

5. The system according to claim 4, wherein the ID signal generator generates different types of said third identifiers depending on types of changes.

6. The system according to claim 1, wherein the first correcting formula is

e_ij=f₀(d_ij)=S₁×

log₁₀(d_ij)+S₂

or

e_ij=f₀(d_ij)=S₁×

log₁₀(d_ij+S₂)

where e_ij is the intensity of a signal transmitted from an i^th transmitting station at (xi,yi) and measured at a j^th receiving station at (uj, vj), d_ij is the distance from the i^th transmitting station to the j^th receiving station expressed as

d_ij=√

{square root over ((x_i−

u_j)²+(y_i−

v_j)²)}{square root over ((x_i−

u_j)²+(y_i−

v_j)²)}

and S1 and S2 are correcting coefficients, wherein the positioning computer determines the correcting coefficients S1 and S2 using the known position information to estimate the position of the unknown transmitting station.

7. The system according to claim 6, wherein when using

e_ij=f₀(d_ij)=S₁×

log₁₀(d_ij)+S₂

as the first correcting formula, the positioning computer derives a distance ${\mathrm{md}}_{\mathrm{ij}}={10}^{\left(e_{\mathrm{ij}}-S_2\right)/S_1}$

from the measured intensity, and determines an estimation function $h_{\mathrm{ij}}=\underset{j=1}{\overset{\mathrm{rn}}{\sum <br><br>}}<br><br>\text{\hspace{1em}<br><br>}<br><br>{\left({10}^{\left(e_{\mathrm{ij}}-S_2\right)/S_1}-d_{\mathrm{ij}}\right)}^2,$

and when using

e_ij=f₀(d_ij)=S₁×

log₁₀(d_ij+S₂)

as the first correcting formula, the positioning computer derives a distance ${\mathrm{md}}_{\mathrm{ij}}={10}^{\left(e_{\mathrm{ij}}\right)/\left(S_1\right)}-S_2$

from the measured intensity, and determines an estimation function $h_{\mathrm{ij}}=\sum <br><br>{\left({10}^{\left(e_{\mathrm{ij}}\right)/\left(S_1\right)}-S_2-d_{\mathrm{ij}}\right)}^2$

to weight with h_ij/md_ij to estimate the position of the unknown transmitting station.

8. The system according to claim 1 wherein the first correcting formula is

e_ij=f₀(d_ij)=S₁×

log₁₀(d_ij)+S₂−

K_rj

or

e_ij=f₀(d_ij)=S₁×

log₁₀(d_ij+S₂)−

K_rj

where e_ij is the intensity of a signal transmitted from an i^th transmitting station at (xi,yi) and received at a j^th receiving station at (uj, vj), d_ij is the distance from the i^th transmitting station to the j^th receiving station expressed as

d_ij=√

{square root over ((x_i−

u_j)²+(y_i−

v_j)²)}{square root over ((x_i−

u_j)²+(y_i−

v_j)²)},

S1 and S2 are correcting coefficients, and K_rj is an environmental coefficient for the receiving station, wherein the positioning computer determines the coefficients S1, S2 and K_rj using the known position information to estimate the position of the unknown transmitting station.

9. The system according to claim 8, wherein when using

e_ij=f₀(d_ij)=S₁×

log₁₀(d_ij)+S₂−

K_rj

as the first correcting formula, the positioning computer derives a distance $\begin{array}{cc}{\mathrm{md}}_{\mathrm{ij}}={10}^{\left(e_{\mathrm{ij}}-S_2+K_{\mathrm{rj}}\right)/S_1}& {\left(4\right)}^{\mathrm{\prime <br><br>\prime <br><br>\prime <br><br>}}\end{array}$

from the measured intensity, and determines an estimation function $h_{\mathrm{ij}}=\sum <br><br>{\left({10}^{\left(e_{\mathrm{ij}}-S_2+K_{\mathrm{rj}}\right)/S_1}-d_{\mathrm{ij}}\right)}^2,$

and when using

e_ij=f₀(d_ij)=S₁×

log₁₀(d_ij+S₂)−

K_rj

as the first correcting formula, the positioning computer derives a distance ${\mathrm{md}}_{\mathrm{ij}}={10}^{\left(e_{\mathrm{ij}}+K_{\mathrm{rj}}\right)/\left(S_1\right)}-S_2$

from the measured intensity, and determines an estimation function $h_{\mathrm{ij}}=\sum <br><br>{\left({10}^{\left(e_{\mathrm{ij}}+K_{\mathrm{rj}}\right)/\left(S_1\right)}-S_2-d_{\mathrm{ij}}\right)}^2$

to weight with h_ij/md_ij to estimate the position of the unknown transmitting station.

10. The system according to claim 1 wherein the first correcting formula is

e_ij=f₀(d_ij)=S₁×

log₁₀(d_ij)+S₂−

K_ti

or

e_ij=f₀(d_ij)=S₁×

log₁₀(d_ij+S₂)−

K_ti

where e_ij is the intensity of a signal transmitted from an i^th transmitting station at (xi,yi) and received at a j^th receiving station at (uj, vj), d_ij is the distance from the i^th transmitting station to the j^th receiving station expressed as

d_ij=√

{square root over ((x_i−

u_j)²+(y_i−

v_j)²)}{square root over ((x_i−

u_j)²+(y_i−

v_j)²)},

S1 and S2 are correcting coefficients, and K_ti is an environmental coefficient for the transmitting station, wherein the positioning computer determines the coefficients S1, S2 and K_ti using the known position information to estimate the position of the unknown transmitting station.

11. The system according to claim 10, wherein when using

e_ij=f₀(d_ij)=S₁×

log₁₀(d_ij)+S₂−

K_ti

as the first correcting formula, the positioning computer derives a distance ${\mathrm{md}}_{\mathrm{ij}}={10}^{\left(e_{\mathrm{ij}}-S_2+K_{\mathrm{ti}}\right)/S_1}$

from the measured intensity, and determines an estimation function $h_{\mathrm{ij}}=\sum <br><br>{\left({10}^{\left(e_{\mathrm{ij}}-S_2+K_{\mathrm{ti}}\right)/S_1}-d_{\mathrm{ij}}\right)}^2,$

and when using

e_ij=f₀(d_ij)=S₁×

log₁₀(d_ij+S₂)−

K_ti

as the first correcting formula, the positioning computer derives a distance ${\mathrm{md}}_{\mathrm{ij}}={10}^{\left(e_{\mathrm{ij}}+K_{\mathrm{ti}}\right)/\left(S_1\right)}-S_2$

from the measured intensity, and determines an estimation function $h_{\mathrm{ij}}=\sum <br><br>{\left({10}^{\left(e_{\mathrm{ij}}+K_{\mathrm{ti}}\right)/\left(S_1\right)}-S_2-d_{\mathrm{ij}}\right)}^2$

to weight with h_ij/md_ij to estimate the position of the unknown transmitting station.

12. The system according to claim 1, wherein the receiving station comprises:

an activation signal generator configured to generate an activation signal for causing the transmitting station to transmit a second ID signal containing a second identifier;

a transmitter configured to transmit the activation signal to the transmitting station; and

a time computation unit configured to measure a transmission time required to acquire the second identifier from the transmitting station in response to the activation signal.

13. The system according to claim 12, wherein the positioning computer determines a second correcting formula defining a relation between a propagation time of a signal through the air and a distance between the transmitting station and the receiving station, and estimates a position of an unknown transmitting station using the second correcting formula and known position information.

14. The system according to claim 13, wherein the second correcting formula is expressed as

p_ij=f₁(t_ij,e_ij)=t_ij−

B−

g exp(−

h×

e_ij)=Kd_ij=K{square root over (√

(u_i−

u_j)²+(v_i−

v_j)²)}{square root over (√

(u_i−

u_j)²+(v_i−

v_j)²)}

where e_ij is the intensity of the second ID signal transmitted from an i^th transmitting station at (xi,yi) and measured at a j^th receiving station at (uj, vj), t_ij is the transmission time, p_ij is the propagation time, d_ij is the distance from the i^th transmitting station to the j^th receiving station expressed as

d_ij=√

{square root over ((x_i−

u_j)²+(y_i−

v_j)²)}{square root over ((x_i−

u_j)²+(y_i−

v_j)²)},

B, g, and h are correcting coefficients, and K is a proportional constant, and wherein the positioning computer determines the correcting coefficients B, g, and h, and the proportional constant K from the known position information to estimate the position of the unknown transmitting station.

15. The system according to claim 14, wherein the positioning computer derives a distance ${\mathrm{nd}}_i$

 

⁢

j=pi⁢

 

⁢

j/K={ti⁢

 

⁢

j-B-g⁢

 

⁢

exp⁡

(-h×

ei⁢

 

⁢

j)}/K

from the propagation time through the air, and determines an estimation function ${\mathrm{hh}}_{i<br><br>\text{\hspace{1em}<br><br>}<br><br>j}=\sum <br><br>{\left(\left\{t_{i<br><br>\text{\hspace{1em}<br><br>}<br><br>j}-B-g<br><br>\text{\hspace{1em}<br><br>}<br><br>\exp <br><br>\left(-h\times <br><br>e_{i<br><br>\text{\hspace{1em}<br><br>}<br><br>j}\right)\right\}/K-d_{i<br><br>\text{\hspace{1em}<br><br>}<br><br>j}\right)}^2$

using the derived distance to weight with hh_ij/nd_ij to estimate the position of the unknown transmitting station.

16. The system according to claim 1, wherein the receiving station includes a fixed-position receiving station and a moving receiving station, and wherein the positioning computer carries out the steps of:

(a) determining a first correcting formula defining a relation between the intensity of a received signal and a distance using known position information supplied from the fixed-position receiving station;

(b) estimating the position of the moving receiving station using the first correcting formula, together with signal information transmitted from a known or position-estimated transmitting station and position information about said known or position-estimated transmitting station; and

(c) estimating a position of an unknown transmitting station based on signal information transmitted from said unknown transmitting station to the fixed-position receiving station or the moving receiving station at an estimated position, and position information about the fixed-position receiving station and said estimated position of the moving receiving station, and wherein the positioning computer repeats the steps (b) and (c) to successively estimate positions of multiple unknown transmitting stations as the moving receiving station travels.

17. The system according to claim 12, wherein the receiving station includes a fixed-position receiving station and a moving receiving station, and wherein the positioning computer carries out the steps of:

(a) determining a second correcting formula defining a relation between a signal propagation time through the air and a distance using the transmission time and the intensity measured at the fixed-position receiving station and known position information supplied from the fixed-position receiving station;

(b) estimating a current position of the moving receiving station using the second correcting formula, together with signal information transmitted from a known or position-estimated transmitting station to the moving station, position information about said known or position-estimated transmitting station, and a transmission time measured at the moving receiving station; and

(c) estimating a position of an unknown transmitting station based on signal information transmitted from said unknown transmitting station to the fixed-position receiving station or the moving receiving station at an estimated position, position information about the fixed-position receiving station and said estimated position of the moving receiving station, and a transmission time measured at the fixed-position receiving station or the moving receiving station at the estimated position, and wherein the positioning computer repeats the steps (b) and (c) to successively estimate positions of multiple unknown transmitting stations as the moving receiving station travels.

18. The system according to claim 1, wherein the receiving station includes a single moving receiving station, and wherein the positioning computer carries out the steps of:

(a) determining a first correcting formula defining a relation between the intensity of a received signal and a distance using signal information transmitted from a transmitting station whose position is known to the moving receiving station at an unknown position, and position information of said transmitting station;

(b) estimating a current position of the moving receiving station using the first correcting formula, based on signal information transmitted from a known transmitting station or a position-estimated transmitting station and position information about said known transmitting station or the position-estimated transmitting station; and

(c) estimating a position of an unknown transmitting station based on signal information transmitted from said unknown transmitting station to the moving receiving station at the estimated current position, and position information about the estimated current position, and wherein the positioning computer repeats the steps (b) and (c) to successively estimate positions of multiple unknown transmitting stations as the moving receiving station travels.

19. The system according to claim 12, wherein the receiving station includes a single moving receiving station, and wherein the positioning computer carries out the steps of:

(a) determining a second correcting formula defining a relation between a signal propagation time through the air and a distance using signal information transmitted from a transmitting station whose position is known to the moving receiving station at an unknown position, position information of said transmitting station, and the transmission time measured at the moving receiving station;

(b) estimating a current position of the moving station using the second correcting formula, based on signal information transmitted from the known transmitting station or a position-estimated transmitting station, position information about said known transmitting station or the position-estimated transmitting station, and the transmission time measured by the moving receiving station; and

(c) estimating a position of an unknown transmitting station based on signal information transmitted from said unknown transmitting station to the moving receiving station at the estimated current position, position information about the estimated current position, and the transmission time measured by the moving receiving station at the estimated current position, and wherein the positioning computer repeats the steps (b) and (c) to successively estimate positions of multiple unknown transmitting stations as the moving receiving station travels.

20. The system according to claim 1, wherein the first ID signal is transmitted using electromagnetic waves or sound waves.

21. The system according to claim 2, wherein the first and second ID signals are transmitted using electromagnetic waves or sound waves, and the activation signal is transmitted using electromagnetic waves or sound waves.

22. The system according to claim 1, wherein if the intensity of the first ID signal transmitted form an i^th transmitting station is measured at a j^th receiving station, but is not measured at an m^th receiving station, then the positioning computer determines the first correcting formula adding an restrictive condition

d_ij<

d_im

where d_ij is a distance between the i^th transmitting station and the j^th receiving station, and d_im is a distance between the i^th transmitting station and the m^th receiving station.

23. A method for determining a position of an object comprises the steps of:

receiving at a receiving station a first ID signal containing a first identifier transmitted from a transmitting station;

measuring an intensity of the first ID signal received at the receiving station;

determining a first correcting formula defining a relation between the intensity measured at the receiving station and a distance between the transmitting station and the receiving station using a correcting coefficient; and

estimating a position of an unknown transmitting station using the first correcting formula and known position information.

24. The method according to claim 23, wherein the first correcting formula is expressed as

e_ij=f₀(d_ij)=S₁×

log₁₀(d_ij)+S₂

or

e_ij=f₀(d_ij)=S₁×

log₁₀(d_ij+S₂)

where e_ij is the intensity of a signal transmitted from an i^th transmitting station at (xi,yi) and measured at a j^th receiving station at (uj, vj), d_ij is the distance from the i^th transmitting station to the j^th receiving station expressed as

d_ij=√

{square root over ((x_i−

u_j)²+(y_i−

v_j)²)}{square root over ((x_i−

u_j)²+(y_i−

v_j)²)},

and S1 and S2 are correcting coefficients.

25. The method according to claim 24, further comprising the step of deriving a distance ${\mathrm{md}}_i$

 

⁢

j=10(ei⁢

 

⁢

j-S2)/S1

from the measured intensity if using

e_ij=f₀(d_ij)=S₁×

log₁₀(d_ij)+S₂

as the first correcting formula and determining an estimation function $h_{i<br><br>\text{\hspace{1em}<br><br>}<br><br>j}=\underset{j=1}{\overset{\mathrm{rn}}{\sum <br><br>}}<br><br>{\left({10}^{\left(e_{i<br><br>\text{\hspace{1em}<br><br>}<br><br>j}-S_2\right)/S_1}-d_{i<br><br>\text{\hspace{1em}<br><br>}<br><br>j}\right)}^2$

to weight with h_ij/md_ij to estimate the position of the unknown transmitting station.

26. The method according to claim 24, further comprising the step of deriving a distance ${\mathrm{md}}_i$

 

⁢

j=10(ei⁢

 

⁢

j)/(S1)-S2

from the measured intensity if using

e_ij=f₀(d_ij)=S₁×

log₁₀(d_ij+S₂)

as the first correcting formula and determining an estimation function $h_{i<br><br>\text{\hspace{1em}<br><br>}<br><br>j}=\sum <br><br>{\left({10}^{\left(e_{i<br><br>\text{\hspace{1em}<br><br>}<br><br>j}\right)/\left(S_1\right)}-S_2-d_{i<br><br>\text{\hspace{1em}<br><br>}<br><br>j}\right)}^2$

to weight with h_ij/md_ij to estimate the position of the unknown transmitting station.

27. The method according to claim 23, further comprising the steps of:

transmitting an activation signal from the receiving station to the transmitting station;

transmitting a second ID signal containing a second identifier from the transmitting station to the receiving station in response to the activation signal;

measuring a transmission time required to acquire the second ID signal in response to the activation signal at the receiving station;

determining a second correcting formula defining a relation between a signal propagation time through the air and a distance based on the measured transmission time; and

estimating a position of an unknown transmitting station using the second correcting formula and known position information.

28. The method according to claim 27, wherein the second correcting formula is

p_ij=f₁(t_ij,e_ij)=t_ij−

B−

g exp(−

h×

e_ij )=Kd_ij=K√

{square root over ((u_i−

u_j)²+(v_i−

v_j)²)}{square root over ((u_i−

u_j)²+(v_i−

v_j)²)}

where e_ij is the intensity of the second ID signal transmitted from an i^th transmitting station at (xi,yi) and measured at a j^th receiving station at (uj, vj), t_ij is the transmission time, p_ij is the propagation time, d_ij is the distance from the i^th transmitting station to the j^th receiving station expressed as

d_ij=√

{square root over ((x_i−

u_j)²+(y_i−

v_j)²)}{square root over ((x_i−

u_j)²+(y_i−

v_j)²)},

B, g, and h are correcting coefficients, and K is a proportional constant, and wherein the positioning computer determines the correcting coefficients B, g, and h, and the proportional constant K from the known position information to estimate the position of the unknown transmitting station.

29. A method for determining a position of an object comprising the steps of:

(a) determining a first approximate function

e_ij=f₀(d_ij)=f₀(√

{square root over ((u_i−

x_j)²+(v_i−

y_j)²)}{square root over ((u_i−

x_j)²+(v_i−

y_j)²)})

as a first correcting formula, using a first intensity e_ij of a first signal transmitted from at least one known transmitting station “

i”

located at a known position (ui,vi) and measured at a receiving station “

j”

located at a first unknown position (xj,yj), as well as a distance d_ij from the transmitting station “

i”

to the receiving station “

j”

, to derive a position (uj, vj) of the first unknown position of the receiving station “

j”

;

(b) defining a second approximate function

e_ij=f₀(√

{square root over ((x_i−

u_j)²+(y_i−

v_j)²)}{square root over ((x_i−

u_j)²+(y_i−

v_j)²)})

based on the first correcting formula, using a second intensity e_ij of a second signal transmitted from an unknown transmitting station “

i”

located at an unknown position (xi, yi) and measured at a known or position-estimated receiving station “

j”

, as well as position information (uj, vj) of said known or position-estimated receiving station “

j”

, to derive a position (ui, vi) of said unknown transmitting station “

i”

; and

(c) defining a third approximate function

e_ij=f₀(√

{square root over ((u_i−

x_j)²+(v_i−

y_j)²)}{square root over ((u_i−

x_j)²+(v_i−

y_j)²)})

based on the first correcting formula, using a third intensity e_ij of a third signal transmitted from a known or position-estimated transmitting station “

i”

located at (ui,vi) and measured at the receiving station “

j”

at a second unknown position, as well as position information of said known or position-estimated transmitting station “

i”

, to derive a position (uj, vj) of the second unknown position of the receiving station “

j”

, the steps (b) and (c) being repeated to successively estimate positions of an unknown receiving station and an unknown transmitting station.

30. A method for determining a position of an object comprising the steps of:

(a) determining a first approximate function

e_ij=f₀(d_ij)=f₀(√

{square root over ((x_i−

u_j)²+(y_i−

v_j)²)}{square root over ((x_i−

u_j)²+(y_i−

v_j)²)})

as a first correcting formula, using a first intensity e_ij of a first signal transmitted from a first unknown transmitting station “

i”

located at a first known position (xi,yi) and measured at one or more known receiving stations “

j”

located at known positions (uj,vj), as well as a distance d_ij from the transmitting station “

i”

to the receiving station “

j”

, to derive a position (ui, vi) of the first unknown transmitting station “

i”

;

(b) defining a second approximate function

e_ij=f₀(√

{square root over ((u_i−

x_j)²+(v_i−

y_j)²)}{square root over ((u_i−

x_j)²+(v_i−

y_j)²)})

based on the first correcting formula, using a second intensity e_ij of a second signal transmitted from a known or position-estimated transmitting station “

i”

located at (ui, vi) and measured at an unknown receiving station “

j”

located at an unknown position (xj, yj), as well as position information (ui, vi) of said known or position-estimated transmitting station “

i”

, to derive a position (uj, vj) of said unknown receiving station “

j”

; and

(c) defining a third approximate function

e_ij=f₀(√

{square root over ((x_i−

u_j)²+(y_i−

v_j)²)}{square root over ((x_i−

u_j)²+(y_i−

v_j)²)})

based on the first correcting formula, using a third intensity e_ij of a third signal transmitted from a second unknown transmitting station “

i”

located at a second unknown position (xi,yi) and measured at a known or position-estimated receiving station “

j”

at (uj, vj), as well as position information of said known or position-estimated receiving station “

j”

, to derive a position (ui, vi) of the second unknown position of the second unknown transmitting station “

i”

, the steps (b) and (c) being repeated to successively estimate positions of an unknown transmitting station and an unknown receiving station.

31. A method for determining a position of an object comprising the steps of:

(a) determining a first approximate function

p_ij=f₁(t_ij,e_ij)=Kd_ij=K√

{square root over ((u_i−

x_j)²+(v_i−

y_j)²)}{square root over ((u_i−

x_j)²+(v_i−

y_j)²)}

and a constant K as a second correcting formula, using a first intensity e_ij of a first signal transmitted from at least one known transmitting station “

i”

located at a known position (ui,vi) and measured at a receiving station “

j”

located at a first unknown position (xj,yj), as well as a first signal transmission time t_ij, a first signal propagation time p_ij through the air, and position information of said known transmitting station, to derive a position (uj, vj) of the first unknown position of the receiving station “

j”

;

(b) defining a second approximate function

 

f₁(t_ij,e_ij)=K√

{square root over ((x_i−

u_j)²+(y_i−

v_j)²)}{square root over ((x_i−

u_j)²+(y_i−

v_j)²)}

based on the second correcting formula, using a second intensity e_ij of a second signal transmitted from an unknown transmitting station “

i”

located at an unknown position (xi, yi) and measured at a known or position-estimated receiving station “

j”

, as well as a second signal transmission time (tij) and position information (uj, vj) of said known or position-estimated receiving station “

j”

, to derive a position (ui, vi) of said unknown transmitting station “

i”

; and

(c) defining a third approximate function

f₁(t_ij,e_ij)=K√

{square root over ((u_i−

x_j)²+(v_i−

y_j)²)}{square root over ((u_i−

x_j)²+(v_i−

y_j)²)}

based on the second correcting formula, using a third intensity e_ij of a third signal transmitted from a known or position-estimated transmitting station “

i”

located at (ui,vi) and measured at the receiving station “

j”

at a second unknown position, as well as a third signal transmission time (tij) and position information of said known or position-estimated transmitting station “

i”

, to derive a position (uj, vj) of the second unknown position of the receiving station “

j”

, the steps (b) and (c) being repeated to successively estimate positions of an unknown receiving station and an unknown transmitting station.

32. A method for determining a position of an object comprising the steps of:

(a) determining a first approximate function

p_ij=f₁(t_ij,e_ij)=Kd_ij=K√

{square root over ((x_i−

u_j)²+(y_i−

v_j)²)}{square root over ((x_i−

u_j)²+(y_i−

v_j)²)}

as a second correcting formula, using a first intensity e_ij of a first signal transmitted from a first unknown transmitting station “

i”

located at a first known position (xi,yi) and measured at one or more known receiving stations “

j”

located at known positions (uj,vj), as well as a first signal transmission time (tij), a first signal propagation time (pij) through the air, and position information of said known receiving station “

j”

, to derive a position (ui, vi) of the first unknown transmitting station “

i”

;

(b) defining a second approximate function

f₁(t_ij,e_ij)=K√

{square root over ((u_i−

x_j)²+(v_i−

y_j)²)}{square root over ((u_i−

x_j)²+(v_i−

y_j)²)}

based on the second correcting formula, using a second intensity e_ij of a second signal transmitted from a known or position-estimated transmitting station “

i”

located at (ui, vi) and measured at an unknown receiving station “

j”

located at an unknown position (xj, yj), as well as a second signal transmission time (tij) and position information (ui, vi) of said known or position-estimated transmitting station “

i”

, to derive a position (uj, vj) of said unknown receiving station “

j”

; and

(c) defining a third approximate function

f₁(t_ij,e_ij)=K√

{square root over ((x_i−

u_j)²+(y_i−

v_j)²)}{square root over ((x_i−

u_j)²+(y_i−

v_j)²)}

based on the second correcting formula, using a third intensity e_ij of a third signal transmitted from a second unknown transmitting station “

i”

located at a second unknown position (xi,yi) and measured at a known or position-estimated receiving station “

j”

at (uj, vj), as well as a third signal transmission time (tij) and position information of said known or position-estimated receiving station “

j”

, to derive a position (ui, vi) of the second unknown position of the second unknown transmitting station “

i”

, the steps (b) and (c) being repeated to successively estimate positions of an unknown transmitting station and an unknown receiving station.