![](https://csdnimg.cn/release/download_crawler_static/9003691/bg4.jpg)
(Eq. 1.2)
It is now easy to notice that DCM
B
= (DCM
G
)
T
or
DCM
G
= (DCM
B
)
T
, in other words the two
matrices are translates of each other, we’ll use this important property later on.
Also notice that DCM
B
. DCM
G
= (DCM
G
)
T
.DCM
G
= DCM
B
. (DCM
B
)
T
= I
3
, where I
3
is the 3x3
identity matrix. In other words the DCM matrices are orthogonal.
This can be proven by simply expanding the matrix multiplication in block matrix form:
(Eq. 1.3)
To prove this we use such properties as for example: i
GT
. i
G
= | i
G
|| i
G
|cos(0) = 1 and i
GT
. j
G
= 0 because (i and j are orthogonal) and so forth.
The DCM matrix (also often called the rotation matrix) has a great importance in
orientation kinematics since it defines the rotation of one frame relative to another. It can
also be used to determine the global coordinates of an arbitrary vector if we know its
coordinates in the body frame (and vice versa).
Let’s consider such a vector with body coordinates:
r
B
= { r
x
B
, r
y
B
, r
z
B
}
T
and let’s try to determine its coordinates in the global frame, by using a
known rotation matrix DCM
G
.
We start by doing following notation:
r
G
= { r
x
G
, r
y
G
, r
z
G
}
T
.
Now let’s tackle the first coordinate r
x
G
: