The three equations
can be divided into two overlapping groups. Group I contains ds2 = ds1 + k1,
ds3 = ds2 + k2; while group II contains ds3 = ds2 + k2, ds4 = ds3 + k3. Each
group can be used to compute S??™s coordinates in the next step independently.
4.3 Step 3: Location Computation
From Equations (9)(10), dSA = dSB+k1, dSC = dSB+k2, we get the following
three equations with three unknowns x, y and dSB based on trilateration:
(x ??’ xb)2 + (y ??’ yb)2 = d2
sb, (11)
(x ??’ xa)2 + (y ??’ ya)2 = (dsb + k1)2, (12)
(x ??’ xc)2 + (y ??’ yc)2 = (dsb + k2)2. (13)
As proposed in [8], we can solve these equations in two steps: First, transform
the coordinates into a system where A, B, C reside at (x1,0), (0,0) and (x2, y2),
respectively; Second, solve the equations with the e?±cient method proposed
in [8]. Since the positions at the original coordinate system can always be
obtained through rotation and translation, the solution provided by [8] can
be treated as a general one:
x =
??’2k1dsb ??’ k2
1 + x2
1
2x1
, (14)
y =
(2k1x2 ??’ 2k2x1)dsb
2x1y2
+
k2
1x2 ??’ k2
2x1 + x2
2x1 + y2
2x1 ??’ x2
1x2
2x1y2
, (15)
where dsb is the root of ?®d2
sb + ??dsb + ?° = 0, with
?® = 4[k2
1y2
2 + (k1x2 ??’ k2x1)2 ??’ x2
1y2
2], (16)
?? = 4[k1(k2
1 ??’ x2
1)y2
2 +
(k1x2 ??’ k2x1)(k2
1x2 ??’ k2
2x1 + x2
2x1 + y2
2x1 ??’ x2
1x2)], (17)
?° = (k2
1 ??’ x2
1)2y2
2 + (k2
1x2 ??’ k2
2x1 + x2
2x1 + y2
2x1 ??’ x2
1x2)2.
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