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Appendix I / Utilities / Axis Transforms
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22.4 Axis Transformations
Consider the two Cartesian frames (A) and (B) shown in Figure 15-2 with a
common origin at point (a), and X B passing through point (b). The position
of (b) with respect to (a), expressed in frame A, is denoted by vector ( A
P ). b
Y B
a
2
Φ
Z A
Z A
2
Θ
2
φ
1
Ψ
1
Θ
Y A
22.4-1
X A
Figure 22-6 : YPR and RPY Euler Triplet Definition
Euler triplets defining the orientation of frame (B) with respect to frame (A),
YPR
RPY
B
A
1
B
A
E ≡ E (roll angle redundant)
B
A
2
B
A
E ≡ E (yaw angle redundant)
b
X B
Equation 22.4-1
Equation 22.4-2
Since the YPR Euler triplet is the most commonly used its prefix has been
dropped when the meaning of an equation is clear in context.
22.4.1 Cartesian to Euler YP Transformation
XYZ_TO_YP takes the position of point (b) with respect to point (a)
expressed in frame (A), and returns the YPR Euler triplet representing the
orientation of frame (B) with respect to frame (A).
XYZ_TO_YP
A B
E A
( P , E ) ≡ ϕ ( P )
∆
b
: =
A
10
−15
X
b
Equation 22.4-3
Equation 22.4-4