Improved Inertial/Odometry/GPS Positioning of Wheeled Robots Even in GPS-Denied Environments - Global Navigation Satellite Systems: Signal,Theory and Applications - page 266

Using Equations 25 and 26, Equation 24 can be rewritten as:

⎡
⎣

⎤
⎦

⎛
⎝

⎞
⎠

δ V e

V e V n V u

ω z

δ V n

δθ = − 1

f x + V f ω z
g cos

2

V f ( g cos ρ )

−

δ V u

ρ

V f

−

1

2 δω z f x + V f ω z

g cos

g cos ρ

−

ρ

f g 2 ⎡

⎤
⎦

δ f x

( )

f x + V f ω z sin ρ

1

1
g cos

1

⎣

−

1 (27)

ρ

f x + V f ω z
g cos

2

δ

f y g 2 cos 2 ρ −

−

ρ

Using Equations 22 and 27 the equation for the attitude components of the velocity error
can be re-arranged to accommodate the error terms belonging to . Similar terms will

be grouped to produce a set of equations that describe the attitude errors within

ρ and θ

˙

V l .

δ The

˙

V att will be used to describe the attitude-portion of the velocity errors states. Once the
equation for the components of velocity errors due to attitude errors is described it can be
easily combined with the other terms in the velocity error states.

term δ

⎡
⎣

⎤
⎦

−

f u V f

1

f x + V f ω z
g cos

2

⎡
⎣

⎤
⎦

⎡
⎣

⎤
⎦ =

⎡
⎣

⎤
⎦ δψ +

−

g cos ρ

0 f u

−

f n δρ

−

f n
f e
0

ρ

˙

−

f u 0 f e δθ 0

f e V f

V att =

δ δω z

f n

−

f e 0 δψ

1

f x + V f ω z
g cos

2

−

g cos ρ

ρ

⎡
⎣

⎤
⎦

( )

−

f x + − f u V f ω z sin ρ

f u

1

f g 2 1

1

fx + V f ω z
g cos

2 f x + V f ω z

g cos

2

g cos ρ − g 2 cos 2 ρ − −

ρ ρ

⎡
⎣

⎤
⎦

δ f x

− f u

0

g 1

f g 2

+

−

δ f y

( )

f e f x + V f ω z sin ρ

f e f n

2 +

1

f g 2 1

g 1

1

f x + V f ω z
g cos

2 f x + V f ω z

g cos

f g 2

−

g cos ρ − g 2 cos 2 ρ − −

ρ ρ

⎡
⎣

⎤
⎦

2
V f

σ

0

+ W

(

t

)

(28)

V f

σ

Experimental data shows that the forward speed originating from the encoders does not suffer
from stochastic errors such as stochastic scale factor. This is due to the fact that the robot's