Transcript Document 7738624

Quaternions
A quaternion vector can represent a rotation about a unit vector:
e0   cos(  / 2) 
 e  sin(  / 2) u 
x
e   1  
e 2  sin(  / 2) u y 
 e  sin(  / 2) u 
 3 
z
e  e0  e1i  e2 j  e3k
e02  e12  e 22  e32  1
The components are also called Euler parameters
Quaternions
A quaternion completely describes the rotation of an object
Quaternions can be converted to / from Euler angles and
rotation matrices
Quaternions are similar to Euler angles, but have the
advantage of no singularities
Quaternions are also better for computation and interpolation
Quaternion Integration
The quaternion of a rigid body can be calculated given the
body frame rotation rate from the equation of motion
e 0 
  e1
 e 
e
1
 1   0
e 2  2  e3
 e 
 e
 3
 2
 e2
 e3
e0
e1
 e3 
e 0 
e 
e2 
ω  k  1 
 e1 
e 2 
e 
e0 
 3
  1  (e02  e12  e 22  e32 )
   Mi  ω  (Iω) , Mi  ri  Fi
Iω
An alternative to the correction term is to re-normalize
the quaternion periodically