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:
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 
     Related subject matter is described in copending U.S. patent application Ser. No. 11/197,950 of Pertti O. Alapuranen and John M. Belcea entitled “Bandwidth Efficient System and Method for Ranging Nodes in a Wireless Communications Network”, filed concurrently herewith and incorporated herein by reference. 
     FIELD OF THE INVENTION 
     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 
     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 
       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. 
         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; 
         FIG. 2  is a block diagram illustrating an example of a node employed in the network shown in  FIG. 1 ; and 
         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. 
     
    
    
     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 
     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. 
     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. 
     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. 
     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. 
     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”. 
     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. 
     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. 
     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. 
       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. 
     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. 
     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 . 
     RIC Team Components 
     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. 
     Router Deployment 
     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”. 
     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 . 
     Coordinates Computation 
     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. 
     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 )}{square root over (( x   i   −x   j ) 2 +( y   i   −y   j ) 2 +( z   i   −z   j ) 2 )}{square root over (( x   i   −x   j ) 2 +( y   i   −y   j ) 2 +( z   i   −z   j ) 2 )},  i,j= 1, 2,  Equation 1
 
     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
 
     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: 
     
       
         
           
             
               
                 
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     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  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. 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
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                                         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 
                 
               
             
           
         
       
     
     Replacing the derivative of ε i,i  and using  δ j   i    for avoiding addition of ε i,i  in Equation 4, the equations become: 
     
       
         
           
             
                 
             
             ⁢ 
             
                 
               
                 
                   Equation  7 
                 
                 ⁢ 
                 
                   
 
                 
                 ⁢ 
                 
                   { 
                   
                     
                       
                         
                           
                             
                               
                                 
                                   
                                     ∑ 
                                     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 
                     
                   
                 
               
             
           
         
       
     
     This is a nonlinear system of 3n equations that is solved iteratively using the first degree terms of Taylor series. The previous three functions can be decomposed as: 
     
       
         
           
             
                 
             
             ⁢ 
             
               Equation  8 
             
           
         
       
       
         
           
             { 
             
               
                 
                   
                     
                       
                         
                           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 
                       
                     
                   
                   
                     
                         
                     
                   
                 
               
             
           
         
       
     
     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           ;                   Equation   ⁢           ⁢   9                 i   =   1     ,   2   ,     …   ⁢           ⁢   n                             
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 
                               
                               3 
                             
                           
                           ⁢ 
                           
                             δ 
                             l 
                             i 
                           
                         
                       
                     
                     + 
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         δ 
                         l 
                         k 
                       
                       ⁢ 
                       
                         
                           ∑ 
                           i 
                         
                         ⁢ 
                         
                           
                             
                               δ 
                               k 
                               i 
                             
                             _ 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               
                                 ( 
                                 
                                   
                                     r 
                                     
                                       i 
                                       , 
                                       k 
                                     
                                   
                                   + 
                                   
                                     r 
                                     
                                       k 
                                       , 
                                       i 
                                     
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 
                                   ( 
                                   
                                     
                                       x 
                                       k 
                                     
                                     - 
                                     
                                       x 
                                       i 
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                             
                             
                               ρ 
                               
                                 i 
                                 , 
                                 k 
                               
                               3 
                             
                           
                         
                       
                     
                     + 
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         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 
                                 
                               
                             
                             ) 
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                     
                 
               
             
           
         
       
     
     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 
                           
                         
                         ) 
                       
                     
                   
                   + 
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   23 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       ∑ 
                       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 
                 
               
               
                 
                     
                 
               
             
           
         
       
     
     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. 
     Those skilled in the will easily recognize that the error function E could be also minimized by other methods. 
     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. 
           
         
       
    
     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. 
     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 
                     
                   
                 
               
             
             ⁢ 
             
                 
             
           
         
       
     
     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. 
             6 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. 
             7 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. 
           
         
       
    
     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.