TULIP Trilateration Algorithm

minRTT = propagation delay + extra delay (due to extra circular routes , congestion and router delays)

delta(T) measured= delta(t) + delta(t0)  

(Pseudo-distance)

PD = delta(T) measured . alpha

(Actual distance)

D = delta(T) . alpha

PD = (delta(T) + delta(T0)). alpha

PD = D + delta(T0) .c  .................................(1)

D = actual distance from the landmark.

C = speed of light

alpha = X(c) i.e. Speed of digital info in fiber optic cable

X = factor of c with which digital info travels in fiber optic cable.

delta(T) = actual propagation delay along the greater circle router/paths.

delta(T0) = the extra delay causing overestimation.

PD = pseudo distance

Graphically,

 
 H: host
L1: Landmark 1
L2: landmark 2
L3: landmark 3
 
Using distance farmula:
D1= sqrt(XL1- Xh)^2 + (YL1 - YH)^2    ............................................. (2)
FROM (1) & (2)

PD1= (sqrt(XL1 - XH )^2 + (YL1 - YH)^2) + delta(T0) . alpha       .................... (A)
Similarly for other 2 landmarks:
PD2= (sqrt(XL2 - XH )^2 + (YL2 - YH)^2) + delta(t0) . alpha     .................... (B)
PD3= (sqrt(XL3 - XH )^2 + (YL3 - YH)^2) + delta(t0) . alpha    .....................(C)
We need to linearize (A), (B) & (C) to solve them

 Using Taylor Series:
 

                        f(x0) . ( x- x0)        f ' (x0) . (x - x0)

  f( x ) = f (x0) + ----------  + --------------------

                              1!                             2!

 Considering the simplified part first

 f ( x ) = f (x0) + f(x0)(x - x0)

 put (x-x0) =  delta(x) 

f ( x ) = f (x0) + f' (x0) delta ( x ) .......................(3)

Hence to compute the original value of X an arbitrary value x0 is required, this is done by simple trilateration  

 We know that:

 Hx = X est + delta ( x )

 Hy = Y est + delta ( y )

 Also

 Est Di =  (Lhi - Xest) + (Hy - Yest) .......................(4)

 From (3) and (4)

                       d(EstDi) . delta ( x )          d(EstDi) . delta( y )
 

 PDi = Est Di + ----------      +         ----------------- 

                              dX                                 dY

 After carrying out partial differentiation

                           (Xest - Xli)                   (Yest - Yi)

  ODi = Est Di +  --------. delta ( x )  + ---------------- delta ( y) + c. delta(To)
                             dX                                  dY

 Now we need to solve ( x , y , delta(to))

Solution from (4) is put in eq(D) to get new estimations.

Hx,Hy becomes the new estimated position

Reference

http://www.ece.cmu.edu/research/publications/2003/CMU-ECE-2003-038.pdf