Users Manual

13 Appendix
13.1 Pose formats
13.1.1 XYZABC format
The XYZABC format is used to express a pose by 6 values. 𝑋𝑌 𝑍 is the position in millimeters. 𝐴𝐵𝐶 are Euler
angles in degrees. The convention used for Euler angles is ZYX, i.e., 𝐴 rotates around the 𝑍 axis, 𝐵 rotates around
the 𝑌 axis, and 𝐶 rotates around the 𝑋 axis. The elements of the rotation matrix can be computed by using
𝑟
11
= cos 𝐵 cos 𝐴,
𝑟
12
= sin 𝐶 sin 𝐵 cos 𝐴 − cos 𝐶 sin 𝐴,
𝑟
13
= cos 𝐶 sin 𝐵 cos 𝐴 + sin 𝐶 sin 𝐴,
𝑟
21
= cos 𝐵 sin 𝐴,
𝑟
22
= sin 𝐶 sin 𝐵 sin 𝐴 + cos 𝐶 cos 𝐴,
𝑟
23
= cos 𝐶 sin 𝐵 sin 𝐴 − sin 𝐶 cos 𝐴,
𝑟
31
= − sin 𝐵,
𝑟
32
= sin 𝐶 cos 𝐵, and
𝑟
33
= cos 𝐶 cos 𝐵.
Note: The trigonometric functions sin and cos are assumed to accept values in degrees. The argument needs
to be multiplied by the factor
𝜋
180
if they expect their values in radians.
Using these values, the rotation matrix 𝑅 and translation vector 𝑇 are defined as
𝑅 =
⎛
⎝
𝑟
11
𝑟
12
𝑟
13
𝑟
21
𝑟
22
𝑟
23
𝑟
31
𝑟
32
𝑟
33
⎞
⎠
, 𝑇 =
⎛
⎝
𝑋
𝑌
𝑍
⎞
⎠
.
The transformation can be applied to a point 𝑃 by
𝑃
′
= 𝑅𝑃 + 𝑇.
13.1.2 XYZ+quaternion format
The XYZ+quaternion format is used to express a pose by a position and a unit quaternion. 𝑋𝑌 𝑍 is the position in
meters. The quaternion is a vector of length 1 that defines a rotation by four values, i.e., 𝑞 = (
𝑎 𝑏 𝑐 𝑤
)
𝑇
with ||𝑞|| = 1. The corresponding rotation matrix and translation vector are defined by
𝑅 = 2
⎛
⎝
1
2
− 𝑏
2
− 𝑐
2
𝑎𝑏 − 𝑐𝑤 𝑎𝑐 + 𝑏𝑤
𝑎𝑏 + 𝑐𝑤
1
2
− 𝑎
2
− 𝑐
2
𝑏𝑐 − 𝑎𝑤
𝑎𝑐 − 𝑏𝑤 𝑏𝑐 + 𝑎𝑤
1
2
− 𝑎
2
− 𝑏
2
⎞
⎠
, 𝑇 =
⎛
⎝
𝑋
𝑌
𝑍
⎞
⎠
.
The transformation can be applied to a point 𝑃 by
𝑃
′
= 𝑅𝑃 + 𝑇.
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