Abstract:
The present invention relates to a system and method for deploying devices, such as routers ( 107 ) and mobile nodes ( 102 ) or tags, that communicate in a wireless multihopping ad-hoc communication network ( 100 ), that enables a monitoring system such as a dispatcher ( 212 ) to create a reference system to identify the location of each device. The system and method thus allows for prompt identification of the locations of persons or items, particularly firefighters and other persons operating in hazardous environments, to expedite rescue efforts.

Description:
[0001]     This application claims the benefit of U.S. Provisional Application No. 60/598,833, filed Aug. 5, 2004, the entire content being incorporated herein by reference.  
       CROSS-REFERENCE TO RELATED APPLICATIONS  
       [0002]     Related subject matter is described in copending U.S. patent application of Pertti  0 . Alapuranen and John M. Belcea entitled “Bandwidth Efficient System and Method for Ranging Nodes in a Wireless Communications Network”, Docket No. P2209/0167 (Mesh-107), filed concurrently herewith and incorporated herein by reference. 
     
    
     FIELD OF THE INVENTION  
       [0003]     The present invention relates to an autonomous reference system and method for monitoring the location and movement of objects. More particularly, the present invention relates to a system and method for deploying wireless routers and mobile nodes operating in a wireless multihopping ad-hoc network to create a reference system that allows identification of the locations of persons or items, particularly firefighters and other persons operating in hazardous environments.  
       BACKGROUND  
       [0004]     A Rapid Intervention Crew (RIC) is attached to each firefighting unit dealing with a fire incident. While other personnel are fighting the fire, this team stays on the side waiting in case somebody needs to be rescued. If any firefighter or group asks for help or does not answer when called, the RIC enter the action and proceed to the rescue operation. First, they have to find out the location of the firefighters to be rescued, then they proceed with the rescue. The procedure currently in use requires that RIC proceed first to the last known location of the firefighters in need, from where they start searching. Because of heat, low visibility and other factors, the firefighters may become confused and could report incorrect positions. In such cases the search may be conducted in inappropriate places delaying the rescue process, sometimes with fatal consequence. 
     
    
     BRIEF DESCRIPTION OF THE FIGURES  
       [0005]     The accompanying figures, where like reference numerals refer to identical or functionally similar elements throughout the separate views and which together with the detailed description below are incorporated in and form part of the specification, serve to further illustrate various embodiments and to explain various principles and advantages all in accordance with the present invention.  
         [0006]      FIG. 1  is a block diagram of an example ad-hoc wireless communications network including a plurality of nodes employing a system and method in accordance with an embodiment of the present invention;  
         [0007]      FIG. 2  is a block diagram illustrating an example of a node employed in the network shown in  FIG. 1 ; and  
         [0008]      FIG. 3  is a block diagram of an example of a network as shown in  FIG. 1  deployed to create a reference system in accordance with an embodiment of the present invention. 
     
    
       [0009]     Skilled artisans will appreciate that elements in the figures are illustrated for simplicity and clarity and have not necessarily been drawn to scale. For example, the dimensions of some of the elements in the figures may be exaggerated relative to other elements to help to improve understanding of embodiments of the present invention.  
       DETAILED DESCRIPTION  
       [0010]     Before describing in detail embodiments that are in accordance with the present invention, it should be observed that the embodiments reside primarily in combinations of method steps and apparatus components related to an autonomous reference system and method for monitoring the location and movement of objects. Accordingly, the apparatus components and method steps have been represented where appropriate by conventional symbols in the drawings, showing only those specific details that are pertinent to understanding the embodiments of the present invention so as not to obscure the disclosure with details that will be readily apparent to those of ordinary skill in the art having the benefit of the description herein.  
         [0011]     In this document, relational terms such as first and second, top and bottom, and the like may be used solely to distinguish one entity or action from another entity or action without necessarily requiring or implying any actual such relationship or order between such entities or actions. The terms “comprises,” “comprising,” or any other variation thereof, are intended to cover a non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements does not include only those elements but may include other elements not expressly listed or inherent to such process, method, article, or apparatus. An element proceeded by “comprises . . . a” does not, without more constraints, preclude the existence of additional identical elements in the process, method, article, or apparatus that comprises the element.  
         [0012]     It will be appreciated that embodiments of the invention described herein may be comprised of one or more conventional processors and unique stored program instructions that control the one or more processors to implement, in conjunction with certain non-processor circuits, some, most, or all of the functions of an autonomous reference system and method for monitoring the location and movement of objects described herein. The non-processor circuits may include, but are not limited to, a radio receiver, a radio transmitter, signal drivers, clock circuits, power source circuits, and user input devices. As such, these functions may be interpreted as steps of a method to perform operations for monitoring the location and movement of objects. Alternatively, some or all functions could be implemented by a state machine that has no stored program instructions, or in one or more application specific integrated circuits (ASICs), in which each function or some combinations of certain of the functions are implemented as custom logic. Of course, a combination of the two approaches could be used. Thus, methods and means for these functions have been described herein. Further, it is expected that one of ordinary skill, notwithstanding possibly significant effort and many design choices motivated by, for example, available time, current technology, and economic considerations, when guided by the concepts and principles disclosed herein will be readily capable of generating such software instructions and programs and ICs with minimal experimentation.  
         [0013]     As described in more detail below, the present invention provides an autonomous system and method that is capably of monitoring the location of a person or item, to thus expedite a search and rescue process, in particular, a search and rescue of a firefighter. The system comprises a number of routers and mobile nodes or “tags” that organize themselves as a network as they are deployed therefore they create an ad-hoc multihopping network. Each device has double functionality: collecting range data (TOF) and providing multihopping network services for transporting data to dispatcher. The range data is processed in order to locate the object to which any mobile nodes or tags are attached. The location of the routers can also be identified in this manner.  
         [0014]     Turning now to the figures,  FIG. 1  is a block diagram illustrating an example of an ad-hoc packet-switched wireless communications network  100  employing an embodiment of the present invention. Specifically, the network  100  includes a plurality of mobile wireless user terminals  102 - 1  through  102 - n  (referred to generally as nodes  102  or mobile nodes  102 ), and can, but is not required to, include a fixed network  104  having a plurality of access points  106 - 1 ,  106 - 2 , . . .  106 - n  (referred to generally as nodes  106  or access points  106 ), for providing nodes  102  with access to the fixed network  104 . The fixed network  104  can include, for example, a core local access network (LAN), and a plurality of servers and gateway routers to provide network nodes with access to other networks, such as other ad-hoc networks, the public switched telephone network (PSTN) and the Internet. The network  100  further can include a plurality of fixed routers  107 - 1  through  107 - n  (referred to generally as nodes  107  or fixed routers  107 ) for routing data packets between other nodes  102 ,  106  or  107 . It is noted that for purposes of this discussion, the nodes discussed above can be collectively referred to as “nodes  102 ,  106  and  107 ”, or simply “nodes” or “terminals”.  
         [0015]     As can be appreciated by one skilled in the art, the nodes  102 ,  106  and  107  are capable of communicating with each other directly, or via one or more other nodes  102 ,  106  or  107  operating as a router or routers for packets being sent between nodes, as described in U.S. patent application Ser. No. 09/897,790 and U.S. Pat. Nos. 6,807,165 and 6,873,839 referenced above.  
         [0016]     As shown in  FIG. 2 , each node  102 ,  106  and  107  includes a transceiver, or modem  108 , which is coupled to an antenna  110  and is capable of receiving and transmitting signals, such as packetized signals, to and from the node  102 ,  106  or  107 , under the control of a controller  112 . The packetized data signals can include, for example, voice, data or multimedia information, and packetized control signals, including node update information.  
         [0017]     Each node  102 ,  106  and  107  further includes a memory  114 , such as a random access memory (RAM) that is capable of storing, among other things, routing information pertaining to itself and other nodes in the network  100 . As further shown in  FIG. 2 , certain nodes, especially mobile nodes  102 , can include a host  116  which may consist of any number of devices, such as a notebook computer terminal, mobile telephone unit, mobile data unit, or any other suitable device. Each node  102 ,  106  and  107  also includes the appropriate hardware and software to perform Internet Protocol (IP) and Address Resolution Protocol (ARP), the purposes of which can be readily appreciated by one skilled in the art. The appropriate hardware and software to perform transmission control protocol (TCP) and user datagram protocol (UDP) may also be included.  
         [0018]      FIG. 3  illustrates a rescue operation in progress utilizing the present invention with a Rapid Intervention Crew (RIC) operating on a single building floor  200 . While proceeding towards the last known position of the firefighter  202 , the RIC deploys data routers  204  that provide data transfer over data links  208  between the RIC and a monitoring device, which can be referred to as a dispatcher  212 . These routers  204  can be deployed on a flat area before entering the building and on several floors while the RIC advances upward into the building.  
         [0019]     It should also be noted that the location references  206  and data routers  204  represent identical or similar equipment. The difference between them is that the data routers  204  are generally too far from the scene to receive radio signals from the firefighter  202 . The location references  206  are closer and can exchange signals with the firefighter  202 . The exchanges of signals have as primary purpose of collecting the time of flight (TOF) data between all routers and firefighter&#39;s wireless movable device located on the same floor. The location references  206  collect TOF data between each other and between each location reference  206  and a firefighter  202  that is equipped with a movable device, such as a mobile node  102  shown in  FIGS. 1 and 2 , that can be referred to as a “tag”. The firefighter  202  also collects the TOF between other firefighter  202  and each location reference  206 . Any device, such as the MEA™ WMC 6300 manufactured by Motorola, Inc., that can (1) measure distances and (2) provide transport of data to a processing location, can be used as a data router  204 , location reference  206 , mobile node  102  or tag. Examples of techniques for collecting TOF are presented in U.S. Pat. Nos. 6,453,168, 6,486,831, 6,539,231, 6,600,927, and 6,665,333. These patents and all other documents cited herein are incorporated herein by reference in their entirety.  
         [0020]     The exchanges of messages that have the primary purpose of collecting TOF are presented in  FIG. 3  as reference links  210 . The collected TOF values in this example are transferred by data links  208  to the dispatcher  212  for processing. On the scene, the location references  206  can provide multi-hopping services for transferring TOF data on these data links  208 . When the signal leaves the scene, the data routers  204  provide the transfer using the multi-hopping capability to the dispatcher  212 . In summary, the reference links  210  are mainly used for collecting TOF data, while the data links  208  are used for transporting collected data to the dispatcher  212 .  
         [0000]     RIC Team Components  
         [0021]     The team has one leader that directs the team search and a suitable number of members, for example, at least four members, that are involved in search and rescue operation. The leader remains outside the operating field and directs the team based on information provided by the dispatcher  212 . The dispatcher  212  can include, for example, a software component installed in a laptop computer with a MEA™ WMC 6300™ interface. The leader is in continuous radio connection with its team.  
         [0000]     Router Deployment  
         [0022]     When the team turns on a router (such as router  107  as shown in  FIG. 1 ), the active router can be deployed as a location reference  206  and is shown on the screen of the dispatcher  212 . When the router is deployed on a floor  200  as a location reference  206 , the leader enters the floor number where the device is deployed. When entering the firefighter&#39;s floor, the RIC deploys one router in front of the elevator or the stairwell door. It is called “Point Zero.” The dispatcher  212  shows the “Point Zero” router in the middle of the searching scene screen, for example. The RIC deploys a second router  107  named “East” or “3 o&#39;clock” at 5 (five) to 25 (twenty-five) meters distance from Point Zero. The dispatcher shows the East router on the right side of the Point Zero at a distance corresponding to the TOF measured between Point Zero and the East router. On the left or right side of the direction from Point Zero to the East router, the RIC deploys a third router called “North” or “12 o&#39;clock.” It should be at 5 (five) to 25 (twenty-five) meter distance from “point Zero”.  
         [0023]     The three deployed routers  107  thus create a triangle, where the angle between the direction from Point Zero to North and the direction from Point Zero to East is preferably 90 (ninety) degrees or about 90 (ninety) degrees. Although the system works for any size of the angle, the highest precision of computation is achieved when the angle is close to 90 (ninety) degrees, while almost collinear placement of routers causes the worst computation of precision. The routers  107  can be identified by their own names, but, the search procedure does not require such identification. With the deployment of these three routers  107 , RIC has defined the horizontal system of coordinates with the routers  107  operation a location references  206 . The dispatcher displays on the screen the relative positions of the three location references  206 , the position of the firefighter  202 , and the position of each member of the RIC, within the bounded area representing the floor of the building, in a manner similar to the block diagram shown in  FIG. 3 , for example. When the search for a firefighter  202  is executed in a building or other type of structure with a variation in elevation, the first three routers  107  must be on the same or about the same plane (e.g., on the same floor), which is a horizontal plane, or substantially or relatively horizontal plane, while a fourth router  107  must be placed on a different plane from the three previously deployed routers  107 .  
         [0000]     Coordinates Computation  
         [0024]     The computation of the coordinates depends on the number of routers  107  (location references  206 ) deployed at every moment. When only Point Zero was deployed, the location of a firefighter  202  could be anywhere in a sphere around Point Zero at a radius of the distance computed from the TOF between Point Zero and the firefighter. As more routers  107  are deployed, the location of a firefighter  202  is computed more accurately.  
         [0025]     In this location system, the information about all n stationary devices is processed at the same time. Some of the stationary devices could be routers providing location references  206 , and some could be firefighters&#39; tags or mobile nodes  102 . The distance between devices is computed from TOF, which is a measured quantity and therefore is affected by random errors. 
 
ρ i,j =√{square root over (( x   i   −x   j ) 2 +( y   i   −y   j ) 2 +( z   i   −z   j ) 2 )}, i,j=1, 2,  Equation 1 
 
         [0026]     In this equation, the indexes i and j identify two routers with coordinates (x i , y i , z i ) and (x j , y j , z j ). The correct distance between any two routers is p i,j . This is an unknown distance as the correct positions of all routers are initially unknown. The same distance is also estimated by the measured range r j,i  from the TOF between router j and router i. Because these values are measured, r i,j ≠r j,i  because of measuring errors, while ρ i,j =ρ j,i . For any set of coordinates of considered routers, the differences between measured and computed distances are: 
 
ε i,j =ρ i,j   −r   i,j , i,j=1, 2, . . . n  Equation 2 
 
         [0027]     In Equation 2 each error ε i,j  is a function of six variables: x i , y i , z i , x j , y j , and z j . The error function E shows the precision of the current estimation:  
       E   =       ∑   i     ⁢       ∑     j   ,     j   ≠   i         ⁢     ɛ     i   ,   j     2             
 
         [0028]     Equation 3  
         [0029]     Although not explicitly shown, E is a function of 3n variables (x i , y i , z i ) i=1, 2, . . . n, most of them unknown. The most probable coordinates of routers are those values that minimize the error function E:  
                         {                 ∂   E       ∂     x   k         =         ∑   i     ⁢       ∑     j   ≠   i       ⁢       ∂     ɛ     i   ,   j     2         ∂     x   k             =       2   ⁢           ⁢       ∑   i     ⁢       ∑     j   ≠   i       ⁢       ɛ     i   ,   j       ⁢       ∂     ɛ     i   ,   j           ∂     x   k                 =   0                       ∂   E       ∂     y   k         =         ∑   i     ⁢       ∑     j   ≠   i       ⁢       ∂     ɛ     i   ,   j     2         ∂     y   k             =       2   ⁢           ⁢       ∑   i     ⁢       ∑     j   ≠   i       ⁢       ɛ     i   ,   j       ⁢       ∂     ɛ     i   ,   j           ∂     y   k                 =   0                       ∂   E       ∂     z   k         =         ∑   i     ⁢       ∑     j   ≠   i       ⁢       ∂     ɛ     i   ,   j     2         ∂     z   k             =       2   ⁢           ⁢       ∑   i     ⁢       ∑     j   ≠   i       ⁢       ɛ     i   ,   j       ⁢       ∂     ɛ     i   ,   j           ∂     z   k                 =   0               ;                   k   =   1     ,   2   ,     …   ⁢           ⁢   n                   Equation   ⁢           ⁢   4               
 
         [0030]     From Equation 2 the derivative of δ i,j  can be computed as follows:  
                       ∂     ɛ     i   ,   j           ∂     x   k         =       ∂     ρ     i   ,   j           ∂     x   k                             ⁢     =       ∂           (       x   i     -     x   j       )     2     +       (       y   i     -     y   j       )     2     +       (       z   i     -     z   j       )     2             ∂     x   k                               ⁢     =       1   2     ⁢       2   ⁢     (       x   i     -     x   j       )     ⁢     (         ∂     x   i         ∂     x   k         -       ∂     x   j         ∂     x   k           )               (       x   i     -     x   j       )     2     +       (       y   i     -     y   j       )     2     +       (       z   i     -     z   j       )     2                                   ⁢     =         (       x   ij     -     x   i       )     ⁢     (       δ   k   i     -     δ   k   j       )         ρ     i   ,   j                           ∂     ɛ     i   ,   j           ∂     y   k         =         (       y   i     -     y   j       )     ⁢     (       δ   k   i     -     δ   k   j       )         ρ     i   ,   j                         ∂     ɛ     i   ,   j           ∂     z   k         =         (       z   i     -     z   j       )     ⁢     (       δ   k   i     -     δ   k   j       )         ρ     i   ,   j                       Equation   ⁢           ⁢   5             
 
         [0031]     In Equation 5 the symbol δ k   i  is equal to one when i=k and zero in all other cases. The symbol is known as “Kronecker delta symbol” and is a particularization of the Dirac delta function. The complement of Kronecker symbol {overscore (δ k   i )} which is equal to zero is used in a case when i=k or in any other case that the complement is suitable as can be appreciated by one skilled in the art.  
                       (       x   i     -     x   j       )     ⁢     (       δ   k   i     -     δ   k   j       )       =     {             i   ≠     k   ⋀   j     ≠   k     ⇒   0               i   =         k   ⋀   j     ≠   k     ⇒       x   k     -     x   j                       i   ≠     k   ⋀   j       =     k   ⇒       x   k     -     x   i                     i   =       k   ⋀   j     =     k   ⇒   0                                 (       y   i     -     y   j       )     ⁢     (       δ   k   i     -     δ   k   j       )       =     {             i   ≠     k   ⋀   j     ≠   k     ⇒   0               i   =         k   ⋀   j     ≠   k     ⇒       y   k     -     y   j                       i   ≠     k   ⋀   j       =     k   ⇒       y   k     -     y   i                     i   =       k   ⋀   j     =     k   ⇒   0                                 (       z   i     -     z   j       )     ⁢     (       δ   k   i     -     δ   k   j       )       =     {             i   ≠     k   ⋀   j     ≠   k     ⇒   0               i   =         k   ⋀   j     ≠   k     ⇒       z   k     -     z   j                       i   ≠     k   ⋀   j       =     k   ⇒       z   k     -     z   i                     i   =       k   ⋀   j     =     k   ⇒   0                               Equation   ⁢           ⁢   6             
 
         [0032]     Replacing the derivative of ε i,i  and using {overscore (δ j   i )} for avoiding addition of ε i,i  in Equation 4, the equations become:  
                   {                     ∑   i     ⁢       ∑   j     ⁢         δ   j   i     _     ⁢           ⁢     ɛ     i   ,   j       ⁢         (       x   i     -     x   j       )     ⁢     (       δ   k   i     -     δ   k   j       )         ρ     i   ,   j               =       ⁢         ∑   i     ⁢       ∑   j     ⁢         δ   j   i     _     ⁢           ⁢       δ   k   i     ⁡     (       ρ     i   ,   j       -     r     i   ,   j         )       ⁢         x   i     -     x   j         ρ     i   ,   j               +       ∑   i     ⁢       ∑   j     ⁢         δ   j   i     _     ⁢           ⁢       δ   k   j     ⁡     (       ρ     i   ,   j       -     r     i   ,   j         )       ⁢           ⁢         x   j     -     x   i         ρ     i   ,   j                             =       ⁢         ∑   j     ⁢         δ   j   k     _     ⁢           ⁢     (       ρ     k   ,   j       -     r     k   ,   j         )     ⁢         x   k     -     x   j         ρ     k   ,   j             +       ∑   i     ⁢         δ   k   i     _     ⁢           ⁢     (       ρ     i   ,   k       -     r     i   ,   k         )     ⁢           ⁢         x   k     -     x   i         ρ     i   ,   k                           =       ⁢         ∑   i     ⁢           δ   k   i     _     ⁡     (     1   -       r     k   ,   j         ρ     i   ,   k           )       ⁢     (       x   k     -     x   i       )         +       ∑   i     ⁢         δ   k   i     _     ⁢           ⁢     (     1   -       r     i   ,   k         ρ     i   ,   k           )     ⁢           ⁢     (       x   k     -     x   i       )                       =       ⁢       ∑   i     ⁢         δ   k   i     _     ⁢           ⁢     (     2   -         r     i   ,   k       +     r     k   ,   j           ρ     i   ,   k           )     ⁢           ⁢     (       x   k     -     x   i       )                     =       ⁢   0                               ∑   i     ⁢       ∑   j     ⁢         δ   j   i     _     ⁢           ⁢     ɛ     i   ,   j       ⁢         (       y   i     -     y   j       )     ⁢     (       δ   k   i     -     δ   k   j       )         ρ     i   ,   j               =       ⁢       ∑   i     ⁢           δ   k   i     _     ⁡     (     2   -         r     i   ,   k       +     r     k   ,   j           ρ     i   ,   k           )       ⁢     (       y   k     -     y   i       )                     =       ⁢   0                               ∑   i     ⁢       ∑   j     ⁢         δ   j   i     _     ⁢           ⁢     ɛ     i   ,   j       ⁢         (       z   i     -     z   j       )     ⁢     (       δ   k   i     -     δ   k   j       )         ρ     i   ,   j               =       ⁢       ∑   i     ⁢           δ   k   i     _     ⁡     (     2   -         r     i   ,   k       +     r     k   ,   i           ρ     i   ,   k           )       ⁢     (       z   k     -     z   i       )                     =       ⁢   0                               k   =   1     ,   2   ,     …   ⁢           ⁢   n                   Equation   ⁢           ⁢   7             
 
         [0033]     This is a nonlinear system of 3 n equations that is solved iteratively using the first degree terms of Taylor series. The previous three functions can be decomposed as:  
             {                   F   ⁢           ⁢     x   k   0       +       ∑   l     ⁢     (       a   ⁢           ⁢     x     k   ,   l         ⁢     |   0     ⁢           ⁢     δ   ⁢           ⁢     x   l         )       +       ∑   l     ⁢     (       a   ⁢           ⁢     y     k   ,   l         ⁢     |   0     ⁢           ⁢     δ   ⁢           ⁢     y   l         )       +       ∑   l     ⁢     (       a   ⁢           ⁢     z     k   ,   l         ⁢     |   0     ⁢           ⁢     δ   ⁢           ⁢     z   l         )         =   0                   F   ⁢           ⁢     y   k   0       +       ∑   l     ⁢     (       b   ⁢           ⁢     x     k   ,   l         ⁢     |   0     ⁢           ⁢     δ   ⁢           ⁢     x   l         )       +       ∑   l     ⁢     (       b   ⁢           ⁢     y     k   ,   l         ⁢     |   0     ⁢           ⁢     δ   ⁢           ⁢     y   l         )       +       ∑   l     ⁢     (       b   ⁢           ⁢     z     k   ,   l         ⁢     |   0     ⁢           ⁢     δ   ⁢           ⁢     z   l         )         =   0                   F   ⁢           ⁢     z   k   0       +       ∑   l     ⁢     (       c   ⁢           ⁢     x     k   ,   l         ⁢     |   0     ⁢           ⁢     δ   ⁢           ⁢     x   l         )       +       ∑   l     ⁢     (       c   ⁢           ⁢     y     k   ,   l         ⁢     |   0     ⁢           ⁢     δ   ⁢           ⁢     y   l         )       +       ∑   l     ⁢     (       c   ⁢           ⁢     z     k   ,   l         ⁢     |   0     ⁢           ⁢     δ   ⁢           ⁢     z   l         )         =   0           ;     k   =   1       ,   2   ,     …   ⁢           ⁢   n               Equation   ⁢           ⁢   8             
 
         [0034]     The iterative process starts with selecting some initial values (x i   0 , y i   0 , z i   0 ); i=1, 2, . . . n for the unknown variables. With Equation 8 are computed the corrections (δx i , δy i , δz i ); i=1, 2, . . . n that are then used for finding improved values of the coordinates:  
             {                 x   i     =       x   i   0     +     δ   ⁢           ⁢     x   i                       y   i     =       y   i   0     +     δ   ⁢           ⁢     y   i                       z   i     =       z   i   0     +     δ   ⁢           ⁢     z   i                 ;     i   =   1       ,   2   ,     …   ⁢           ⁢   n               Equation   ⁢           ⁢   9             
 
 The first term in each line of Equation 8 is:  
                     F   ⁢           ⁢     x   k   0       =       ∑   i     ⁢           δ   k   i     _     (     2   -         r     j   ,   k     0     +     r     k   ,   j     0         ρ     i   ,   k           )     ⁢     (       x   k   0     -     x   j   0       )                       F   ⁢           ⁢     y   k   0       =       ∑   i     ⁢           δ   k   i     _     (     2   -         r     j   ,   k     0     +     r     k   ,   j     0         ρ     i   ,   k           )     ⁢     (       y   k   0     -     y   j   0       )                       F   ⁢           ⁢     z   k   0       =       ∑   i     ⁢           δ   k   i     _     (     2   -         r     j   ,   k     0     +     r     k   ,   j     0         ρ     i   ,   k           )     ⁢     (       z   k   0     -     z   j   0       )                       Equation   ⁢           ⁢   10             
 
 Other coefficients on Equation 8 are computed as follows:  
               a   ⁢           ⁢     x     k   ,   l         =       ∂     (       ∑   i     ⁢           δ   k   i     _     ⁡     (     2   -         r     i   ,   k       +     r     k   ,   j           ρ     i   ,   k           )       ⁢     (       x   k     -     x   i       )         )         ∂     x   l                 Equation   ⁢           ⁢   11                 a   ⁢           ⁢     y     k   ,   l         =       ∂     (       ∑   i     ⁢           δ   k   i     _     ⁡     (     2   -         r     i   ,   k       +     r     k   ,   j           ρ     i   ,   k           )       ⁢     (       x   k     -     x   i       )         )         ∂     y   l                                   a   ⁢           ⁢     z     k   ,   l         =       ∂     (       ∑   i     ⁢           δ   k   i     _     ⁡     (     2   -         r     i   ,   k       +     r     k   ,   j           ρ     i   ,   k           )       ⁢     (       x   k     -     x   i       )         )         ∂     z   l                                   b   ⁢           ⁢     x     k   ,   l         =       ∂     (       ∑   i     ⁢           δ   k   i     _     ⁡     (     2   -         r     i   ,   k       +     r     k   ,   j           ρ     i   ,   k           )       ⁢     (       y   k     -     y   i       )         )         ∂     x   l                 Equation   ⁢           ⁢   12                 b   ⁢           ⁢     y     k   ,   l         =       ∂     (       ∑   i     ⁢           δ   k   i     _     ⁡     (     2   -         r     i   ,   k       +     r     k   ,   j           ρ     i   ,   k           )       ⁢     (       y   k     -     y   i       )         )         ∂     y   l                                   b   ⁢           ⁢     z     k   ,   l         =       ∂     (       ∑   i     ⁢           δ   k   i     _     ⁡     (     2   -         r     i   ,   k       +     r     k   ,   j           ρ     i   ,   k           )       ⁢     (       y   k     -     y   i       )         )         ∂     z   l                                   c   ⁢           ⁢     x     k   ,   l         =       ∂     (       ∑   i     ⁢           δ   k   i     _     ⁡     (     2   -         r     i   ,   k       +     r     k   ,   j           ρ     i   ,   k           )       ⁢     (       z   k     -     z   i       )         )         ∂     x   l                 Equation   ⁢           ⁢   13                 c   ⁢           ⁢     y     k   ,   l         =       ∂     (       ∑   i     ⁢           δ   k   i     _     ⁡     (     2   -         r     i   ,   k       +     r     k   ,   j           ρ     i   ,   k           )       ⁢     (       z   k     -     z   i       )         )         ∂     y   l                                   c   ⁢           ⁢     z     k   ,   l         =       ∂     (       ∑   i     ⁢           δ   k   i     _     ⁡     (     2   -         r     i   ,   k       +     r     k   ,   j           ρ     i   ,   k           )       ⁢     (       z   k     -     z   i       )         )         ∂     z   l                                   a   ⁢           ⁢     x     k   ,   l         =       ∂       ∑   i     ⁢           δ   k   i     _     ⁡     (     2   -         r     i   ,   k       +     r     k   ,   i           ρ     i   ,   k           )       ⁢     (       x   k     -     x   i       )             ∂     x   l                 Equation   ⁢           ⁢   14                       ⁢     =       ∑   i     ⁢         δ   k   i     _     ⁢           ⁢         ∂     (     2   -         r     i   ,   k       +     r     k   ,   i           ρ     i   ,   k           )       ⁢     (       x   k     -     x   i       )         ∂     x   l                                               ⁢     =         ∑   i     ⁢         δ   k   i     _     ⁢           ⁢       ∂     (     2   -         r     i   ,   k       +     r     k   ,   i           ρ     i   ,   k           )         ∂     x   l         ⁢     (       x   k     -     x   i       )         +                                       ⁢       ∑   i     ⁢         δ   k   i     _     ⁢       ∂     (       x   k     -     x   i       )         ∂     x   l         ⁢     (     2   -         r     i   ,   k       +     r     k   ,   i           ρ     i   ,   k           )                                         ⁢     =         ∑   i     ⁢       -         δ   k   i     _     ⁡     (       r     i   ,   k       +     r     k   ,   i         )         ⁢     (       x   k     -     x   i       )     ⁢       ∂     (     1     ρ     i   ,   k         )         ∂     x   l             +                                       ⁢       ∑   i     ⁢         δ   k   i     _     ⁢           ⁢       ∂     (       x   k     -     x   i       )         ∂     x   l         ⁢     (     2   -         r     i   ,   k       +     r     k   ,   i           ρ     i   ,   k           )                                         ⁢     =         ∑   i     ⁢         δ   k   i     _     ⁢         (       r     i   ,   k       +     r     k   ,   i         )     ⁢     (       x   k     -     x   i       )         ρ     i   ,   k     2       ⁢       ∂     ρ     i   ,   k           ∂     x   l             +                                       ⁢       ∑   i     ⁢           δ   k   i     _     ⁡     (       δ   l   k     -     δ   l   i       )       ⁢     (     2   -         r     i   ,   k       +     r     k   ,   i           ρ     i   ,   k           )                                         ⁢     =         ∑   i     ⁢         δ   k   i     _     ⁢         (       r     i   ,   k       +     r     k   ,   i         )     ⁢     (       x   k     -     x   i       )         ρ     i   ,   k     2       ⁢         (       x   i     -     x   k       )     ⁢     (       δ   l   i     -     δ   l   k       )         ρ     i   ,   k             +                                       ⁢       ∑   i     ⁢           δ   k   i     _     ⁡     (       δ   l   k     -     δ   l   i       )       ⁢     (     2   -         r     i   ,   k       +     r     k   ,   i           ρ     i   ,   k           )                                         ⁢     =       -       ∑   i     ⁢         δ   k   i     _     ⁢         (       r     i   ,   k       +     r     k   ,   i         )     ⁢       (       x   k     -     x   i       )     2         ρ     i   ,   k     2       ⁢     δ   l   i           +                                       ⁢         δ   l   k     ⁢       ∑   i     ⁢         δ   k   i     _     ⁢           ⁢         (       r     i   ,   k       +     r     k   ,   i         )     ⁢       (       x   k     -     x   i       )     2         ρ     i   ,   k     2             +                                     ⁢         δ   l   k     ⁢           ⁢       ∑   i     ⁢         δ   k   i     _     ⁡     (     2   -         r     i   ,   k       +     r     k   ,   i           ρ     i   ,   k           )           -                                     ⁢       ∑   i     ⁢         δ   k   i     _     ⁢           ⁢       δ   l   i     ⁡     (     2   -         r     i   ,   k       +     r     k   ,   i           ρ     i   ,   k           )                                           ⁢     =         -       δ   k   i     _       ⁢         (       r     l   ,   k       +     r     k   ,   l         )     ⁢       (       x   k     -     x   l       )     2         ρ     l   ,   k     3         +                                       ⁢         δ   l   k     ⁢           ⁢       ∑   i     ⁢         δ   k   i     _     (           ⁢     2   -           ⁢         r     i   ,   k       +     r     k   ,   i           ρ     i   ,   k         +           ⁢         (       r     i   ,   k       +     r     k   ,   i         )     ⁢       (       x   k     -     x   i       )     2         ρ     i   ,   k     3         )         ⁢           -                                     ⁢         δ   k   l     _     ⁡     (     2   -         r     l   ,   k       +     r     k   ,   l           ρ     l   ,   k           )                                       ⁢     =         δ   l   k     ⁢       ∑   i     ⁢         δ   k   i     _     ⁡     (     2   -           r     i   ,   k       +     r     k   ,   i           ρ     i   ,   k         ⁢     (     1   -         (       x   k     -     x   i       )     2       ρ     i   ,   k     2         )         )           -                                       ⁢         δ   k   l     _     ⁡     (     2   -           r     l   ,   k       +     r     k   ,   l           ρ     l   ,   k         ⁢     (     1   -         (       x   k     -     x   l       )     2       ρ     l   ,   k     2         )         )                                 a   ⁢           ⁢     y     k   ,   l         =       ∂       ∑   i     ⁢           δ   k   i     _     ⁡     (     2   -         r     i   ,   k       +     r     k   ,   i           ρ     i   ,   k           )       ⁢     (       x   k     -     x   i       )             ∂     y   l                 Equation   ⁢           ⁢   15                       ⁢     =       ∑   i     ⁢         δ   k   i     _     ⁢         ∂     (     2   -         r     i   ,   k       +     r     k   ,   i           ρ     i   ,   k           )       ⁢     (       x   k     -     x   i       )         ∂     y   l                                               ⁢     =       ∑   i     ⁢         δ   k   i     _     ⁢       ∂     (     2   -         r     i   ,   k       +     r     k   ,   i           ρ     i   ,   k           )         ∂     y   l         ⁢     (       x   k     -     x   i       )                                           ⁢     =       ∑   i     ⁢       -         δ   k   i     _     ⁡     (       r     i   ,   k       +     r     k   ,   i         )         ⁢     (       x   k     -     x   i       )     ⁢           ⁢       ∂     (     1     ρ     i   ,   k         )         ∂     y   l                                               ⁢     =       ∑   i     ⁢         δ   k   i     _     ⁢         (       r     i   ,   k       +     r     k   ,   i         )     ⁢     (       x   k     -     x   i       )         ρ     i   ,   k     2       ⁢       ∂     ρ     i   ,   k           ∂     y   l                                               ⁢     =       ∑   i     ⁢         δ   k   i     _     ⁢         (       r     i   ,   k       +     r     k   ,   i         )     ⁢     (       x   k     -     x   i       )         ρ     i   ,   k     2       ⁢         (       y   i     -     y   k       )     ⁢     (       δ   l   i     -     δ   l   k       )         ρ     i   ,   k                                               ⁢     =         ∑   i     ⁢         δ   k   i     _     ⁢         (       r     i   ,   k       +     r     k   ,   i         )     ⁢     (       x   k     -     x   i       )     ⁢     (       y   i     -     y   k       )         ρ     i   ,   k     3       ⁢     δ   l   i         -                                       ⁢       δ   l   k     ⁢       ∑   i     ⁢         δ   k   i     _     ⁢         (       r     i   ,   k       +     r     k   ,   i         )     ⁢     (       x   k     -     x   i       )     ⁢     (       y   i     -     y   k       )         ρ     i   ,   k     3                                             ⁢     =         δ   l   k     ⁢       ∑   i     ⁢         δ   k   i     _     (         (       r     i   ,   k       +     r     k   ,   i         )     ⁢     (       x   k     -     x   i       )     ⁢     (       y   k     -     y   j       )         ρ     i   ,   k     3       )         -                                       ⁢         δ   k   l     _     ⁢         (       r     l   ,   k       +     r     k   ,   l         )     ⁢     (       x   k     -     x   l       )     ⁢     (       y   k     -     y   l       )         ρ     l   ,   k     3                                   a   ⁢           ⁢     z     k   ,   l         =       ∂       ∑   i     ⁢           δ   k   i     _     ⁡     (     2   -         r     i   ,   k       +     r     k   ,   i           ρ     i   ,   k           )       ⁢     (       x   k     -     x   i       )             ∂     z   l                 Equation   ⁢           ⁢   16                       ⁢     =         δ   l   k     ⁢           ⁢       ∑   i     ⁢         δ   k   i     _     (         (       r     i   ,   k       +     r     k   ,   i         )     ⁢     (       x   k     -     x   i       )     ⁢     (       z   k     -     z   j       )         ρ     i   ,   k     3       )         -                                       ⁢         δ   k   l     _     ⁢         (       r     l   ,   k       +     r     k   ,   l         )     ⁢     (       x   k     -     x   l       )     ⁢     (       z   k     -     z   l       )         ρ     l   ,   k     3                                   b   ⁢           ⁢     x     k   ,   l         =       ∂       ∑   i     ⁢           δ   k   i     _     ⁡     (     2   -         r     i   ,   k       +     r     k   ,   i           ρ     i   ,   k           )       ⁢     (       y   k     -     y   i       )             ∂     x   l                 Equation   ⁢           ⁢   17                       ⁢     =         δ   l   k     ⁢       ∑   i     ⁢         δ   k   i     _     (         (       r     i   ,   k       +     r     k   ,   i         )     ⁢     (       y   k     -     y   i       )     ⁢     (       x   k     -     x   j       )         ρ     i   ,   k     3       )         -                                       ⁢         δ   k   l     _     ⁢         (       r     l   ,   k       +     r     k   ,   l         )     ⁢     (       y   k     -     y   l       )     ⁢     (       x   k     -     x   l       )         ρ     l   ,   k     3                                   b   ⁢           ⁢     y     k   ,   l         =       ∂       ∑   i     ⁢           δ   k   i     _     ⁡     (     2   -         r     i   ,   k       +     r     k   ,   i           ρ     i   ,   k           )       ⁢     (       y   k     -     y   i       )             ∂     y   l                 Equation   ⁢           ⁢   18                       ⁢     =       ∑   i     ⁢         δ   k   i     _     ⁢         ∂     (     2   -         r     i   ,   k       +     r     k   ,   i           ρ     i   ,   k           )       ⁢     (       x   k     -     x   i       )         ∂     x   l                                               ⁢     =         δ   l   k     ⁢       ∑   i     ⁢         δ   k   i     _     ⁡     (     2   -           r     i   ,   k       +     r     k   ,   i           ρ     i   ,   k         ⁢     (     1   -         (       y   k     -     y   i       )     2       ρ     i   ,   k     2         )         )           -                                       ⁢         δ   k   l     _     ⁡     (     2   -           r     l   ,   k       +     r     k   ,   l           ρ     l   ,   k         ⁢     (     1   -         (       y   k     -     y   l       )     2       ρ     l   ,   k     2         )         )                                 b   ⁢           ⁢     z     k   ,   l         =       ∂       ∑   i     ⁢           δ   k   i     _     ⁡     (     2   -         r     i   ,   k       +     r     k   ,   i           ρ     i   ,   k           )       ⁢     (       y   k     -     y   i       )             ∂     z   l                 Equation   ⁢           ⁢   19                       ⁢     =         δ   l   k     ⁢       ∑   i     ⁢         δ   k   i     _     (         (       r     i   ,   k       +     r     k   ,   i         )     ⁢     (       y   k     -     y   i       )     ⁢     (       z   k     -     z   j       )         ρ     i   ,   k     3       )         -                                       ⁢         δ   k   l     _     ⁢         (       r     l   ,   k       +     r     k   ,   l         )     ⁢     (       y   k     -     y   l       )     ⁢     (       z   k     -     z   l       )         ρ     l   ,   k     3                                   c   ⁢           ⁢     x     k   ,   l         =       ∂       ∑   i     ⁢           δ   k   i     _     ⁡     (     2   -         r     i   ,   k       +     r     k   ,   i           ρ     i   ,   k           )       ⁢     (       z   k     -     z   i       )             ∂     x   l                 Equation   ⁢           ⁢   20                       ⁢     =         δ   l   k     ⁢       ∑   i     ⁢         δ   k   i     _     (         (       r     i   ,   k       +     r     k   ,   i         )     ⁢     (       z   k     -     z   i       )     ⁢     (       x   k     -     x   j       )         ρ     i   ,   k     3       )         -                                       ⁢         δ   k   l     _     ⁢         (       r     l   ,   k       +     r     k   ,   l         )     ⁢     (       z   k     -     z   l       )     ⁢     (       x   k     -     z   l       )         ρ     l   ,   k     3                                   c   ⁢           ⁢     y     k   ,   l         =       ∂       ∑   i     ⁢           δ   k   i     _     ⁡     (     2   -         r     i   ,   k       +     r     k   ,   i           ρ     i   ,   k           )       ⁢     (       z   k     -     z   i       )             ∂     y   l                 Equation   ⁢           ⁢   21                       ⁢     =         δ   l   k     ⁢       ∑   i     ⁢         δ   k   i     _     (         (       r     i   ,   k       +     r     k   ,   i         )     ⁢     (       z   k     -     z   i       )     ⁢     (       y   k     -     y   j       )         ρ     i   ,   k     3       )         -                                       ⁢         δ   k   l     _     ⁢         (       r     l   ,   k       +     r     k   ,   l         )     ⁢     (       z   k     -     z   l       )     ⁢     (       y   k     -     y   l       )         ρ     l   ,   k     3                                   c   ⁢           ⁢     z     k   ,   l         =       ∂       ∑   i     ⁢           δ   k   i     _     ⁡     (     2   -         r     i   ,   k       +     r     k   ,   i           ρ     i   ,   k           )       ⁢     (       z   k     -     z   i       )             ∂     z   l                 Equation   ⁢           ⁢   22                       ⁢     =       ∑   i     ⁢         δ   k   i     _     ⁢         ∂     (     2   -         r     i   ,   k       +     r     k   ,   i           ρ     i   ,   k           )       ⁢     (       z   k     -     z   i       )         ∂     z   l                                               ⁢     =         δ   l   k     ⁢           ⁢       ∑   i     ⁢         δ   k   l     _     ⁡     (     2   -           r     i   ,   k       +     r     k   ,   i           ρ     i   ,   k         ⁢     (     1   -         (       z   k     -     z   i       )     2       ρ     i   ,   k     2         )         )           -                                       ⁢         δ   k   l     _     ⁡     (     2   -           r     l   ,   k       +     r     k   ,   l           ρ     l   ,   k         ⁢     (     1   -         (       z   k     -     z   l       )     2       ρ     l   ,   k     2         )         )                             
 
         [0035]     Replacing all terms in Equation 8 with terms from Equation 14 to Equation 22 produces the final equations:  
                     ∑   i     ⁢           δ   k   i     _     (     2   -         r     j   ,   k     0     +     r     k   ,   j     0         ρ     i   ,   k           )     ⁢     (       x   k   0     -     x   j   0       )         +       ∑   l     ⁢       (         δ   l   k     ⁢       ∑   i     ⁢         δ   k   i     _     (     2   -           r     i   ,   k       +     r     k   ,   i           ρ     i   ,   k         ⁢     (     1   -         (       x   k     -     x   i       )     2       ρ     i   ,   k     2         )         )         -         δ   k   l     _     (     2   -           r     l   ,   k       +     r     k   ,   l           ρ     l   ,   k         ⁢     (     1   -         (       x   k     -     x   l       )     2       ρ     l   ,   k     2         )         )       )     ⁢           ⁢   δ   ⁢           ⁢     x   l         +       ∑   l     ⁢       (         δ   l   k     ⁢       ∑   i     ⁢         δ   k   i     _     (         (       r     i   ,   k       +     r     k   ,   i         )     ⁢     (       x   k     -     x   i       )     ⁢     (       y   k     -     y   j       )         ρ     i   ,   k     3       )         -         δ   k   l     _     ⁢         (       r     l   ,   k       +     r     k   ,   l         )     ⁢     (       x   k     -     x   l       )     ⁢     (       y   k     -     y   l       )         ρ     l   ,   k     3           )     ⁢           ⁢   δ   ⁢           ⁢     y   l         +       ∑   l     ⁢       (         δ   l   k     ⁢       ∑   i     ⁢         δ   k   i     _     (         (       r     i   ,   k       +     r     k   ,   i         )     ⁢     (       x   k     -     x   i       )     ⁢     (       z   k     -     z   j       )         ρ     i   ,   k     3       )         -         δ   k   l     _     ⁢         (       r     l   ,   k       +     r     k   ,   l         )     ⁢     (       x   k     -     x   l       )     ⁢     (       z   k     -     z   l       )         ρ     l   ,   k     3           )     ⁢           ⁢   δ   ⁢           ⁢     z   l           =   0     ⁢     
     ⁢           ∑   i     ⁢           δ   k   i     _     (     2   -         r     j   ,   k     0     +     r     k   ,   j     0         ρ     i   ,   k           )     ⁢     (       y   k   0     -     y   j   0       )         +       ∑   l     ⁢       (         δ   l   k     ⁢       ∑   i     ⁢         δ   k   i     _     (         (       r     i   ,   k       +     r     k   ,   i         )     ⁢     (       y   k     -     y   i       )     ⁢     (       x   k     -     x   j       )         ρ     i   ,   k     3       )         -         δ   k   l     _     ⁢         (       r     l   ,   k       +     r     k   ,   l         )     ⁢     (       y   k     -     y   l       )     ⁢     (       x   k     -     x   l       )         ρ     l   ,   k     3           )     ⁢           ⁢   δ   ⁢           ⁢     x   l         +       ∑   l     ⁢       (         δ   l   k     ⁢       ∑   i     ⁢         δ   k   i     _     (     2   -           r     i   ,   k       +     r     k   ,   i           ρ     i   ,   k         ⁢     (     1   -         (       y   k     -     y   i       )     2       ρ     i   ,   k     2         )         )         -         δ   k   l     _     (     2   -           r     l   ,   k       +     r     k   ,   l           ρ     l   ,   k         ⁢     (     1   -         (       y   k     -     y   l       )     2       ρ     l   ,   k     2         )         )       )     ⁢           ⁢   δ   ⁢           ⁢     y   l         +       ∑   l     ⁢       (         δ   l   k     ⁢       ∑   i     ⁢         δ   k   i     _     (         (       r     i   ,   k       +     r     k   ,   i         )     ⁢     (       y   k     -     y   i       )     ⁢     (       z   k     -     z   j       )         ρ     i   ,   k     3       )         -         δ   k   l     _     ⁢         (       r     l   ,   k       +     r     k   ,   l         )     ⁢     (       y   k     -     y   l       )     ⁢     (       z   k     -     z   l       )         ρ     l   ,   k     3           )     ⁢           ⁢   δ   ⁢           ⁢     z   l           =   0     ⁢     
     ⁢           ∑   i     ⁢           δ   k   i     _     (     2   -         r     j   ,   k     0     +     r     k   ,   j     0         ρ     i   ,   k           )     ⁢     (       z   k   0     -     z   j   0       )         +       ∑   l     ⁢       (         δ   l   k     ⁢       ∑   i     ⁢         δ   k   i     _     (         (       r     i   ,   k       +     r     k   ,   i         )     ⁢     (       z   k     -     z   i       )     ⁢     (       x   k     -     x   j       )         ρ     i   ,   k     3       )         -         δ   k   l     _     ⁢         (       r     l   ,   k       +     r     k   ,   l         )     ⁢     (       z   k     -     z   l       )     ⁢     (       x   k     -     z   l       )         ρ     l   ,   k     3           )     ⁢           ⁢   δ   ⁢           ⁢     x   l         +       ∑   l     ⁢       (         δ   l   k     ⁢       ∑   i     ⁢         δ   k   i     _     (         (       r     i   ,   k       +     r     k   ,   i         )     ⁢     (       z   k     -     z   i       )     ⁢     (       y   k     -     y   j       )         ρ     i   ,   k     3       )         -         δ   k   l     _     ⁢         (       r     l   ,   k       +     r     k   ,   l         )     ⁢     (       z   k     -     z   l       )     ⁢     (       y   k     -     y   l       )         ρ     l   ,   k     3           )     ⁢           ⁢   δ   ⁢           ⁢     y   l         +       ∑   l     ⁢       (         δ   l   k     ⁢       ∑   i     ⁢         δ   k   i     _     (     2   -           r     i   ,   k       +     r     k   ,   i           ρ     i   ,   k         ⁢     (     1   -         (       z   k     -     z   i       )     2       ρ     i   ,   k     2         )         )         -         δ   k   l     _     (     2   -           r     l   ,   k       +     r     k   ,   l           ρ     l   ,   k         ⁢     (     1   -         (       z   k     -     z   l       )     2       ρ     l   ,   k     2         )         )       )     ⁢           ⁢   δ   ⁢           ⁢     z   l           =   0             Equatio   ⁢           ⁢   n   ⁢           ⁢   23             
 
         [0036]     Equation 23 is used for computing the coordinates of a system of stationary devices measuring the relative distance between them. The system has 3n equations and the same number of unknown variables δx i , δy i , δz i . The solution of the system of equations provides the value of δx i , δy i , δz i  that are the corrections to be applied to the supposed position of each router in the system. From the mathematical point of view, the system of equations does not have a unique solution, but an infinite number of “correct” solutions. The word “correct” means that in all configurations the computed positions of routers are in concert with the measured distances between them.  
         [0037]     Those skilled in the will easily recognize that the error function E could be also minimized by other methods.  
         [0038]     With regard to the error function E discussed above and the index i, assuming that the “Point Zero” router has i=1, the “East” router has i=2, the third deployed router has i=3 and the fourth deployed router has i=4, the system must verify the following set of constraints: 
        1 The following coordinates have fixed predefined values:  
             {             x   1     =   0                 y   1     =   0                 z   1     =   0                 y   2     =   0                 z   2     =   0                 z   3     =   0                   Equation   ⁢           ⁢   24             
    2 The sign of the Y coordinate of the third router corresponds to the deployment conditions.     3 The sign of the Z coordinates of the fourth router corresponds to the deployment conditions.        
 
         [0042]     These three constraints must be observed in order to assure that the computed configuration is not a rotation, translation or a reflection of the correct configuration. They assure that from the infinite number of possible solutions, is selected the solution that matches the reference system that has been created when Point Zero, East and North routers were deployed.  
         [0043]     After solving the linear system of equations, for selecting the solution matching the reference system, the following steps have to be executed at the end of each iteration: 
        1 After new coordinates are computed, is executed a translation that makes “Point Zero” the origin of the coordinates.  
       {                   ⁢       x   i   ′     =       x   i   0     -     x   1   0                         ⁢       y   i   ′     =       y   i   0     -     y   1   0                       z   i   ′     =       z   i   0     -     z   1   0               ;           ⁢     i   =   1       ,   2   ,     …   ⁢           ⁢   n           
       
 
         [0045]     After this translation, “Point Zero” has coordinates:  
       {             x   1     =   0                 y   1     =   0                 z   1     =   0                   
        2 On next step is executed a rotation around the OZ axis for bringing the “East” router on the of ZOX plane, which means to have y 2 =0. The rotation equations are:  
         sin   ⁢           ⁢   Ry     =       y   2           x   2   2     +     y   2   2               
         cos   ⁢           ⁢   Ry     =       x   2           x   2   2     +     y   2   2               
       {                     ⁢       x   i   ′     =         x   i     *   cos   ⁢           ⁢   Ry     +       y   i     *   sin   ⁢           ⁢   Ry                       y   i   ′     =         -     x   i       *   sin   ⁢           ⁢   Ry     +       y   i     *   cos   ⁢           ⁢   Ry                       ⁢       z   i   ′     =     z   i               ⁢           ⁢   i     =   2     ,   3   ,     …   ⁢           ⁢   n           
    3 On third step a rotation around the OY axis should bring the “East” router in the OX axis, which means to have y 2 =0 and z 2 =0.  
         sin   ⁢           ⁢   Rz     =       z   2           x   2   2     +     z   2   2               
         cos   ⁢           ⁢   Rz     =       x   2           x   2   2     +     z   2   2               
       {                     ⁢       x   i   ′     =         x   i     *   cos   ⁢           ⁢   Rz     +       z   i     *   sin   ⁢           ⁢   Ry                       y   i   ′     =     y   i                     ⁢       z   i   ′     =         x   i     *   sin   ⁢           ⁢   Rz     +       z   i     *   cos   ⁢           ⁢   Rz                 ⁢           ⁢   i     =   2     ,   3   ,     …   ⁢           ⁢   n           
    4 On next step the coordinates of routers are rotated around the OX axis to bring the third router in the OXY plane, which means to have z 3 =0.  
         sin   ⁢           ⁢   Rx     =       z   2           y   2   2     +     z   2   2               
         cos   ⁢           ⁢   Rx     =       y   2           y   2   2     +     z   2   2               
       {                     ⁢       x   i   ′     =     x   i   ′                     y   i   ′     =         y   i     *   cos   ⁢           ⁢   Rx     +       z   i     *   sin   ⁢           ⁢   Rx                       ⁢       z   i   ′     =         -     y   i       *   sin   ⁢           ⁢   Rx     +       z   i     *   cos   ⁢           ⁢   Rx                 ⁢           ⁢   i     =   2     ,   3   ,     …   ⁢           ⁢   n           
    5 On next step is checked the symmetry of the result against the ZOY plane. If the sign of x 2  is not the expected sign, all x i , for i=2, 3, . . . n need to have their sign changed, which assures that the “East” router has correct coordinates. On next step is checked the symmetry of the result against the XOZ plane. If y 3  does not have the expected sign, all y i , i=2, 3, . . . n, need to have the sign changed, which assures that the third deployed router has correct y 3  coordinate. In the final step is checked the symmetry of the result against the XOY plane. If the search is executed in a plane (one floor only), this check is not required because all routers must have z i =0. If the search is executed in three dimensional space, the sign of z 4  is verified. If the sign is incorrect, all z i  have their sign changed.        
 
         [0050]     In the foregoing specification, specific embodiments of the present invention have been described. However, one of ordinary skill in the art appreciates that various modifications and changes can be made without departing from the scope of the present invention as set forth in the claims below. Accordingly, the specification and figures are to be regarded in an illustrative rather than a restrictive sense, and all such modifications are intended to be included within the scope of present invention. The benefits, advantages, solutions to problems, and any element(s) that may cause any benefit, advantage, or solution to occur or become more pronounced are not to be construed as a critical, required, or essential features or elements of any or all the claims. The invention is defined solely by the appended claims including any amendments made during the pendency of this application and all equivalents of those claims as issued.