사용자:Kanie/연습장
위키백과 ― 우리 모두의 백과사전.
- What is the quaternion q1 that represents the rotation of 180 degree about the x-axis?
- What is the quaternion q2 that represents the rotation of 180 degree about the z-axis?
- What rotation is represented by composite quaternion q = q1q2?

- rotation of 180 degree about the y-axis
- Let
be a point and let
be a quaternion whose scalar part is zero and whose vector part is equal to
. Show that if
is a unit quaternion, the product qXq − 1 is a purely imaginary quaternion and the vector part of qXq − 1 satisfies:
-
- Show that q and -q represent same rotation using the result of Exercise 4.


- Therefore q and − q represents the same rotation.
- Compare the number of additions and multiplications needed to perform the following operations:
- Compose two rotation matrices.
- given n × n matrices A and B,

- this requires at least (n-1) additions and n multiplications per single element
- 3 × 3 matrix multiplication requires (2 additions + 3 multiplication) * 9 elements = 18 additions + 27 multiplications)
- given n × n matrices A and B,
- Compose two quaternions

- this requires 12 additions + 16 multiplications
- Apply a rotation matrix to a vector

- (2 additions + 3 multiplications) * 3 elements = 6 additions + 9 multiplications
- Apply a quaternion to a vector (as in Exercise 4)
: 3 additions + 4 multiplications
: 2 additions + 3 multiplications- total : (4 additions + 5 multiplications) * 3 + 5 additions + 7 multiplications
- = 17 additions + 22 multiplications
- (if times 2 is counted as multiplication, then 6 more multiplications) : 17 additions + 28 multiplications
- Compose two rotation matrices.
- Show that a rigid body rotating at angular velocity
can be represented by the quaternion differential equations

Hint: Recall that the angular velocity
indicates that the body is instantaneously rotating about the ω axis with magnitude
. Suppose that a body were to rotate with a constant angular velocity
. Then the rotation of the body after a period of time
is represented by the quaternion

At times
(for small
), the orientation of the body is (to within the first order)

compute
by differentiating the above equation





![\begin{array}{lcl} \dot q (t) & = & \lim \limits_{\Delta t \to 0} \frac{q(t + \Delta t) - q(t)}{\Delta t}\\ & = & \lim \limits_{\Delta t \to 0} \left [ \left ( \cos \frac{\left|\left| \omega (t) \right|\right| \Delta t}{2} , \frac{\omega (t)}{\left|\left| \omega (t) \right|\right|} \sin \frac{\left|\left| \omega (t) \right|\right| \Delta t}{2} \right ) - 1 \right ] \frac{q(t)}{\Delta t} \\ & = & \lim \limits_{\Delta t \to 0} \left [ \left ( \cos \frac{\left|\left| \omega (t) \right|\right| \Delta t}{2} - 1 \right ) / \Delta t , \frac{\omega (t)}{\left|\left| \omega (t) \right|\right|} \sin \frac{\left|\left| \omega (t) \right|\right| \Delta t}{2} / \Delta t \right ] q(t) \\ & = & \left ( \frac {\partial}{\partial \Delta t} \cos \frac{\left|\left| \omega (t) \right|\right| \Delta t}{2} , \frac{\omega (t)}{\left|\left| \omega (t) \right|\right|} \frac {\partial}{\partial \Delta t} \sin \frac{\left|\left| \omega (t) \right|\right| \Delta t}{2} \right ) q(t) \,\,\,\,\,\, \mbox{(by lHospitals rule)}\\ & = & \left ( 0, \frac{\omega (t)}{\left|\left| \omega (t) \right|\right|} \frac{\left|\left| \omega (t) \right|\right|}{2} \right ) q(t) \\ & = & \frac{1}{2} \left ( 0, \omega (t) \right ) q(t) \end{array}](../../../math/5/8/d/58d4f98d7acb8d644b01b22bcf4d8a30.png)

