Optimal Estimation of Dynamic Systems, 2nd Edition

610 Optimal Estimation of Dynamic Systems

2

ˆ

r

3

ˆ

r

2

ˆ

r

3

ˆ

r

1

ˆ

r

rotation

3

ˆ

r

1

ˆ

r

3

ˆ

r

1

ˆ

r

2

ˆ

r

rotation

1

ˆ

r

2

ˆ

r

1

ˆ

b

2

ˆ

b

3

ˆ

r

rotation

Figure A.5: Euler Angles for a 1-2-3 Rotation Sequence

body vector (b) through a rotation about the

ˆ

r

1

vector (the 1-axis rotation) first, with

r



=

⎡

⎣

10 0

0cos

φ

sin

φ

0 −sin

φ

cos

φ

⎤

⎦

r (A.163)

Then, a rotation about the

ˆ

r



2

vector is performed (the 2-axis rotation), with

r



=

⎡

⎣

cos

θ

0 −sin

θ

01 0

sin

θ

0cos

θ

⎤

⎦

r



(A.164)

Finally a rotation about the

ˆ

r



3

vector is performed (the 3-axis rotation), with

b =

⎡

⎣

cos

ψ

sin

ψ

0

−sin

ψ

cos

ψ

0

001

⎤

⎦

r



(A.165)

Substituting Equation (A.163) into Equation (A.164) and substituting the resulting

equation into Equation (A.165) leads t ...