Fundamentals of Positioning

Chapter 6

Fundamentals of Positioning

• Accuracy and precision

• Yield and consistency

• Overhead

• Power consumption

• Latency

• Roll-out and operating costs

6.1 Classification of Positioning Infrastructures

6.1.1 Integrated and Stand-alone Infrastructures

6.1.2 Network and Terminal-based Positioning

6.1.3 Satellites, Cellular, and Indoor Infrastructures

6.1.3.1 Satellite Infrastructures 6.1.3.2 Cellular Infrastructures 6.1.3.3 Indoor Infrastructures

6.2 Basic Positioning Methods

6.2.1 Proximity Sensing

6.2.2 Lateration

6.2.2.1 Circular Lateration

2

2 ( )

)

(X x Y y

r_i = _i − + _i − ) 1 . 6 (

2 2

2 ( ) ( )

)

(X x Y y Z z

r_i = _i − + _i − + _i − (6.2)

ε +

= _i

i p

r (6.3)

~ )

~ ,

~ , (

) (

) (

) (

) , ,

( ^{2 2 2}

z z

y y

x x

p

z Z y

Y x

X z

y x p

i

i i

i i

∆ +

∆ +

∆ +

=

− +

− +

−

= (6.5)

z z y p

y x p

x z p

y x p

z z

y y

x x

p

i i

i i

i

∂ ∆ + ∂

∂ ∆ + ∂

∂ ∆ + ∂

=

∆ +

∆ +

∆ +

~ ~ ) ~

,~ ,~ (~

~ )

~ ,

~ , (

) 6 . 6 (

i i

i i

i i

i i

i i

i i

i i

i i

i i

c z

Z y

Y x

X

z Z z

p

b z

Z y

Y x

X

y Y y

p

a z

Z y

Y x

X

x X x

p

− = +

− +

−

+

= −

∂

∂

− = +

− +

−

+

= −

∂

∂

− = +

− +

−

+

= −

∂

∂

2 2

2

2 2

2

2 2

2

~) (

~) (

~) (

~

~

~) (

~) (

~) (

~

~

~) (

~) (

~) (

~

~

) 7 . 6 (

∑

+

−

= ⁿ

i

n i

i

a x R a

i x a x f

f

0 ) (

) , ( )

! ( ) ) (

( (6.4)

z c y b x a

p_i = _i∆ + _i∆ + _i∆

∆

Ax b =

















∆

∆

∆

=

















=

















∆

∆

∆

=

z y x

c b a

c b a

c b a

p p p

n n n n

x A

b , ^{2 2 2} ,

1 1 1

2 1









b A x = ⁻¹

) 9 . 6 (

) 10 . 6 (

) 11 . 6 (

) 12 . 6 ( z

c y b x a z

y x p

z z

y y

x x

p

i i

i i

i

∆ +

∆ +

∆ +

=

∆ +

∆ +

∆ +

~)

~,

~, (

~ )

~ ,

~ ,

( (6.8)

2 2

min b−A~x

~ 0 2

2 + =

− A^Tb A^TAx

b A A A x

b A x

A A

T T

T T

) 1

~ (

~

= −

⇔

= (6.16)

) 15 . 6 (

) 14 . 6 ( x

A A x b A x b

b

x A b x A b

~

~ 2~

~) (

~)

2 (

2

T T T

T T

T Tr

r r

+

−

=

−

−

=

= (6.13)

6.2.2.2 Hyperbolic Lateration

2 2

2 2

2

2 ( ) ( ) ( ) ( ) ( )

)

(X x Y y Z z X x Y y Z z

r r d

j j

j i

i i

j i ij

− +

− +

−

−

− +

− +

−

=

−

= (6.17)

i j

j j

i j

j j

i j

j j

r c z Z r

z Z z

d

r b y Y r

y Y y

d

r a x X

r x X x

d

+ =

− − +

= −

∂

∂

+ =

− − +

= −

∂

∂

+ =

− − +

= −

∂

∂

~

~

~

~

~

~

~

~

~

~

~

~

~

~

~

1 1 1

1 1 1

1 1 1

2 2

2 ( ~) ( ~)

~)

~ (X x Y y Z z

r_i = _i − + _i − + _i −

) 18 . 6 (

) 19 . 6 (

b Q A A Q A

x ( ¹ ) ^{1 1}

~ = ^{T − − T −} (6.20)

6.2.3 Angulation



 





−

= −

x X

y Y

i i i arctan α

~ )

~ , ( arctan )

, , (

y y

x x

x X

y z Y

y x

i

i i i

∆ +

∆ +

=



 





−

= −

φ φ

ε φ α = +

i i

i i

i i

i i

r b y

r a x

=

−

∂ =

=

∂ =

φ φ

φ φ

cos

~

sin

~

) 21 . 6 (

) 22 . 6 (

) 23 . 6 (

) 24 . 6 (

6.2.4 Dead Reckoning

α

α, sin

cos _{1 0}

0

1 x L y y L

x = + = +

t v L = ∆

) 25 . 6 (

) 26 . 6 (

6.2.5 Pattern Matching 6.2.6 Hybrid Approaches

6.3 Range Measurements

6.3.1 Time Measurements

6.3.1.1 Measurement Methods

• Pulse ranging

• Carrier phase ranging

• Code phase ranging

6.3.1.2 Clocks

6.3.2 Received Signal Strength

6.4 Accuracy and Precision

n x x

n

i

t i D

∑

−

= ¹

2

1

) (

σ _(6.27)

E N

E N

D

σ σ

σ σ

σ

4142 .

1 4142

. 1

2

2 ^{2 2}

2

=

=

=

= (6.28) σ_D2 = 0.5(σ_N +σ_E) (6.29)

6.5 Error Sources

• Clocks

• Ionospheric and tropospheric refraction

• Non line of sight

• Multipath propagation

• Medium access

• Base station coordinates

• Bad geometry

6.6 Conclusion