where ϕ is the angle of rotation and u ¯ is the axis of rotation. Please note that the angle ϕ is not unique and the direction of u ¯ can be opposite. We can further define the Euler parameters, c0 c1 c2 c3 or define the quater = cos ϕ 2
1
1.1
Instataneous Axis
Euler’s Theorem on Rotation
Any displacement of a rigid body with a fixed point is equivalent to a rotation about a fixed axis throught the fixed point. Let the rotation matrix be R11 R12 R13 R = R21 R22 R23 , R31 R32 R33 we can define cosϕ = u ¯ = 1 (R11 + R22 + R33 − 1) 2 ) 1 ( R − RT 2sinϕ
ϕ 2 ϕ = uy sin 2 ϕ = uz sin 2 = ux sin
c0 [ ] [ ] c1 c0 c0 c= = = ¯ . c2 d u ¯sin ϕ 2 c3 ¯¯T ¯ˆ R = I (2c2 0 − 1) + w c, then we can figure out the rotation matrix
Example: Let the rotation be a combination of two consecutive rotations about the axes, i.e. R = Rot(z, θ)Rot(x, β ) Cθ −Sθ 0 1 0 = Sθ Cθ 0 0 Cβ 0 0 1 0 Sβ Cθ −SθCθ SθSβ = Sθ CθCβ −CθSβ 0 Sβ Cβ 1 0 −Sβ Cβ

3 4 1 2
0
If we write the axis in vector form, then we have √ √3 4 u ¯= √ 3 7 Therefore, from u ¯sin ϕ = 2
√ √3 7 1 √ √7 √3 7 3 3 4 3 4 √ 3 3 4
Please note that we can choose the opposite sign of c0 too. From here we can easily obtain √ √ 63 2 sinϕ = 1 − cos ϕ = 8 √ √ ϕ ϕ 7 2 sin = 1 − cos = 2 2 4 which leads to ) 1 ( R − RT 2sinϕ 1 2·
Then we can calculate cosϕ = 1 (Cθ + CθCβ + Cβ − 1) 2
If we know that θ = 60◦ , β = 60◦ , then cosϕ = therefore, we can have ϕ c0 = cos = 2 √ cosϕ + 1 = 2 √ 9/8 3 = 2 4 1 1 1 1 1 1 ( + · + − 1) = 2 2 2 2 2 8
3
√ 63 8
u ¯ =
=
0
1 2 √ 3 4
−

0
√ 3 3 4 −3 4
=
4 √ 3 7
√ −343 √ 3 3 4
3 4 1 4 √ 3 4
√
−
3 4 √


3 4 1 2
− −
1 2 √
3 4
0
3 4√ −343

0
√ 3 4 1 4 √ − 43
0 √
=
√
7 1 √ √7 √3 7


√ 7 = · 4
√ 3 4 1 4 √ 3 4

2
we can get the quarternian
c=
1 8 √ 3 4 1 4 √ 3 4
.
Charles’ Theorem
The most general rigid body motion can be produced by a translation along a line followed or preceedeed by a rotation about that line. This line is often described as a finite screw motion. The line rotated about is called screw axis. If we consider the infinitesimal motion, or velocity, then we can always find an axis such that ω ¯ //v ¯. The axis is called Instantaneous screw axis (ISA). In [ ]T planar case, ω ¯= 0 0 ω , the axis degrades to the instantaneous center. Generally, given a rigid body with ω ¯ and v ¯o (where o is the point of the rigid body that happend to be at the origin of the inertial coordinate system at the instant), we can write this motion as [ ] ω ¯ T = . v ¯o It is called a velocity twist. Then we can find an ISA such that the rigid body will rotate about it with angular velocity ω ¯ and translate along the axis with linear velocity u ¯.