Xsens MTi 10 Series, User Manual

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Document MT0605P.E
© Xsens Technologies B.V. MTi User Manual
34
norm is always equal to one (1) and a rotation R
LS
followed by the inverse rotation R
SL
naturally yields
the identity matrix I
3
.
|𝑅| = 1
𝑅
𝐿𝑆
𝑅
𝑆𝐿
= 𝐈
3
The rotation matrix, R
LS
, can be interpreted in terms of quaternions;
𝑅
𝐿𝑆
= [
𝑞
0
2
+ 𝑞
1
2
− 𝑞
2
2
− 𝑞
3
2
2𝑞
1
𝑞
2
− 2𝑞
0
𝑞
3
2𝑞
0
𝑞
2
+ 2𝑞
1
𝑞
3
2𝑞
0
𝑞
3
+ 2𝑞
1
𝑞
2
𝑞
0
2
− 𝑞
1
2
+ 𝑞
2
2
− 𝑞
3
2
2𝑞
2
𝑞
3
− 2𝑞
0
𝑞
1
2𝑞
1
𝑞
3
− 2𝑞
0
𝑞
2
2𝑞
2
𝑞
3
+ 2𝑞
0
𝑞
1
𝑞
0
2
− 𝑞
1
2
− 𝑞
2
2
+ 𝑞
3
2
]
= [
2𝑞
0
2
+ 2𝑞
1
2
− 1 2𝑞
1
𝑞
2
− 2𝑞
0
𝑞
3
2𝑞
1
𝑞
3
+ 2𝑞
0
𝑞
2
2𝑞
1
𝑞
2
+ 2𝑞
0
𝑞
3
2𝑞
0
2
+ 2𝑞
2
2
− 1 2𝑞
2
𝑞
3
− 2𝑞
0
𝑞
1
2𝑞
1
𝑞
3
− 2𝑞
0
𝑞
2
2𝑞
2
𝑞
3
+ 2𝑞
0
𝑞
1
2𝑞
0
2
+ 2𝑞
3
2
− 1
]
or in terms of Euler-angles;
𝑅
𝐿𝑆
= 𝑅
ѱ
𝑍
𝑅
𝜃
𝑌
𝑅
𝜑
𝑋
= [
cos ѱ − sin ѱ 0
sin ѱ cos ѱ 0
0 0 1
] [
cos 𝜃 0 sin 𝜃
0 1 0
− sin 𝜃 0 cos 𝜃
] [
1 0 0
0 cos 𝜑 − sin 𝜑
0 sin 𝜑
cos 𝜑
]
= [
cos θ cos ѱ sin 𝜑 sin 𝜃 cos ѱ − 𝑐𝑜𝑠𝜑 sin ѱ cos 𝜑 sin 𝜃 cos ѱ + sin 𝜑 sin ѱ
cos θ sin ѱ sin 𝜑 sin 𝜃 sin ѱ + 𝑐𝑜𝑠𝜑 cos ѱ cos 𝜑 sin 𝜃 sin ѱ − sin 𝜑 cos ѱ
− sin 𝜃 sin 𝜑 cos 𝜃 cos 𝜑 cos 𝜃
]
As defined here R
LS
, rotates a vector in the sensor co-ordinate system ( S ) to the global reference
system ( L ):
𝒙
𝐿
= 𝑅
𝐿𝑆
𝒙
𝑠
= (𝑅
𝑆𝐿
) 𝒙
𝑠
𝑇
It follows naturally that, R
SL
rotates a vector in the global reference co-ordinate system ( L ) to the
sensor co-ordinate system ( S ).
For the rotation matrix (DCM) output mode it is defined that:
𝑅
𝐿𝑆
= [
𝑎 𝑑 𝑔
𝑏 𝑒 ℎ
𝑐 𝑓 𝑖
] = [
𝑅
11
𝑅
12
𝑅
13
𝑅
21
𝑅
22
𝑅
23
𝑅
31
𝑅
32
𝑅
33
]
𝑅
𝑆𝐿
= [
𝑎 𝑏 𝑐
𝑑 𝑒 𝑓
𝑔 ℎ 𝑖
] = [
𝑅
11
𝑅
12
𝑅
13
𝑅
21
𝑅
22
𝑅
23
𝑅
31
𝑅
32
𝑅
33
]