![](https://csdnimg.cn/release/download_crawler_static/11816724/bge.jpg)
14
1. MOTION OF A RIGID BODY
The Lie group SO(3) is the appropriate mathematical model of the orientations of a rigid
body. Also the angular velocities of rigid bodies must get a good mathematical model.
This is given by the so-called Lie algebra of SO(3), which is denoted with so(3). A Lie
algebra describes the local (“first order”) properties of its parent Lie group (irrespective
of what the Lie group represent), while maintaining the geometric properties.
From the theory on left and right invariant vector fields, Eq. (B.5) and Eq. (B.6), we can
see that elements of the algebra should look like T
L
= R
−1
˙
R, or like T
R
=
˙
RR
−1
, for any
curve R(t) ∈ SO(3). Because R
−1
= R
T
, differentiating the equation R
T
R = I yields:
˙
R
T
R + R
T
˙
R =0⇒ R
T
˙
R = −(R
T
˙
R)
T
,
which means that the matrix T
L
is skew-symmetric. Similarly, T
R
is also skew-symmetric,
so that so(3) is the vector space of skew-symmetric matrices. Note that due to the struc-
ture of the matrix commutator of the algebra, the commutation of skew-symmetric ma-
trices is still skew-symmetric.
There is a bijective relation between 3 × 3 skew-symmetric matrices and 3 vectors as it
was shown in Eq. (1.1).
It is straightforward to see that the algebra matrix commutator of so(3) corresponds to
the vector product of the corresponding three vectors:
[˜ω
x
, ˜ω
y
]=
ω
x
∧ ω
y
∀ω
x
,ω
y
∈ R
3
.
The elements of so(3) correspond to the angular velocity of the frames whose relative
change of coordinates is represented by the matrix R
2
1
∈ SO(3). We will analyse in more
detail the general case, i.e., angular and translational velocities. A lot of coordinate rep-
resentations of rotations exists, besides the above-mentioned skew-symmetric matrix
representation of SO(3); for example, the unit quaternions, which are isomorphic to an-
other Lie group, namely SU(2) (the Special Unitary group. They are both double covers
of SO(3) (i.e., each rotation in SO(3) corresponds to two elements in the double cover
group). In contrast to SO(3), they are both topologically speaking simply connected.
SO(3) is instead not contractible and it is a typical example of a non simply connected
manifold “without holes.” This means that if we define a continuous closed curve in
SO(3) (a ring), there is not always a way to continuously make this ring smaller and
smaller in order to let it become a point.
Remark 4 (Vector product) It is a misconception to interpret the vector product in three
space as an extra structure on E
∗
(the scalar product being the other structure). This
is NOT the case, because the vector product is a consequence of the fact that for E the
Lie group of rotations SO(3) can be intrinsically defined. As we have just seen, the Lie
algebra so(3) has a commutator defined on it and therefore, after a choice of orientation
of space, the intrinsic bijection between so(3) and E
∗
carries the commutation operation
defined in so(3) over to the vector product in E
∗
.
It is possible to find two bijections in the following way. A vector of E
∗
is characterized by
a direction d, an orientation v and a module m. An element of so(3) represents an angular
velocity. It is possible to see that any angular motion leaves a line d
invariant. This is
the first step in this bijection: the vector of E
∗
which we associate to a vector in so(3) will
have as direction the line left invariant by this rotation (d = d
). We still need a module
and an orientation. It is possible to show that there exists a positive definite metric on
so(3), called the Killing form (Selig 1996), which defines the magnitude of a rotation in a
unique, coordinate-independent way. We can therefore choose as m the module of this
angular velocity vector calculated using the metric on so(3). The only choice we are left