1 / 8

Understanding Rotation Principles with Euler Angles and Quaternions

Learn about rotating objects using Euler angles and quaternions, understanding fixed angles and axis-angle rotations. Explore the equivalence of fixed angles and Euler angles. Discover how to represent orientations with quaternions.

phillipl
Download Presentation

Understanding Rotation Principles with Euler Angles and Quaternions

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Orientation Rotation about principle axes - fixed angles Rotation about object’s axes - Euler angles Axis-angle rotation Quaternion

  2. Orientation 1 0 0 0 cos(q) -sin(q) 0 sin(q) cos(q) We know how to rotate about the global axes cos(q) 0 sin(q) 0 1 0 -sin(q) 0 cos(q) cos(q) -sin(q) 0 sin(q) cos(q) 0 0 0 1

  3. y x z Fixed angles Rotate about global axes in a fixed order Rotating about global axes is what the rotation matrices do Can use most any triple of axes Rotate about x, then y, then z (10, 90, -45)

  4. y x z Gimbal lock From some orientations, can’t do some rotations (0,90,0) Can’t rotate around x-axis

  5. y x z Euler angles Rotate about axes of object Can use most any triple of axes Roll, Pitch, Yaw (10, 90, -45)

  6. Equivalence of Fixed angles and Euler angles Ru(a) = Rx(a) Rv (b)Ru (a) = Rx(a)Ry (b)Rx (- a)Rx (a) = Rx (a)Ry (b) Rw (g) Rv (b)Ru (a) = Rx (a)Ry (b) Rz (g)

  7. Quaternions Keep axis-angle orientation as 4-tuple Q = (s, x, y, z) = (s,v) Q1*Q2 = (s1,v1)*(s2,v2) = (s1*s2+v1*v2 , s1*v2 + s2*v1 + v1xv2) Q1+Q2 = (s1,v1)+(s2,v2) = (s1+s2, v1+v2)

  8. Quaternions Keep axis-angle orientation as 4-tuple (sin(t/2), cos(t/2)*(x,y,z))

More Related