Roll, Pitch and Yaw: 3-D rotation matrices

1. Roll, Pitch and Yaw: 3-D rotation matrices Right handed coordinate systems Follow development in Wolovich chpt 2 Start with common origin for 2 frames Rotate w.r.t. each axis Three 3x3 rotation matrices Limitation of single origin Matlab: Modelling Rhino waist, shoulder 3D 4x4 Homogeneous matrices for translation First Rhino Lab specs & protocols

2. Remember remember the dot of the vectors

3. Right-handed coordinate system

4. Two coordinates systems rotated at the origin http://personal.uncc.edu/jamiller/coordinates/rotate.gif

5. Expressing one coordinate system Point in terms of another coordinate system Point

6. Looking down at rotation ofθ around the z axis, in termsof unit vectors. j0 θ j1 i1 θ i0

7. Rotate θaround the z-axis • We’ve already done that, for the 2-D x-y rotation matrix

8. Rotate φaround the y-axis

9. Rotate ζaround the x-axis

10. 3-D rotation matrix multiplication not commutative Rx*Ry*Rz*Po ≠ Rz*Ry*Rx*Po http://www.lightandmatter.com/html_books/0sn/ch04/figs/book.png

11. Roll, Pitch and Yaw Roll around the x-axis Pitch around the y-axis Yaw around the z-axis http://ultimatepointer.com/images/YawPitchRoll.jpg

12. An eensy weensy problem for Rhino • Basic concatenating of 3D rotation matrices to find where a point ends up assumes the rolling, pitching and yawing all take place around a single origin of one cartesian coordinate system. • Example of 3D rotation: human shoulder socket • But Rhino shoulder and elbow can have their coordinates systems moved w.r.t. waist http://www.hopkinsortho.org/orthopedicsurgery/images/instfig1.gif

13. Matlab demo of rotation matrix sequence order • exerRot3.m • lines 11, 16, 15 • P_orig = [0 1 0]’ • After rotZ of 45º then rotX of 45º correct P_dest = [ .5 .5 .707]’ rot_ang_seq = [ [3*ones(4,1) (pi/16)*ones(4,1)]; [1*ones(4,1) (pi/16)*ones(4,1)] ]; % shoulder "extend" rotation around x, then waist to "left" 4 clicks. % start at y = 1; 4 steps around x ; 4 steps around z % looking rot_ang_seq = [3*ones(4,1) (pi/16)*ones(4,1)] ; % rot_ang_seq = [ [1*ones(4,1) (pi/16)*ones(4,1)]; [3*ones(4,1) (pi/16)*ones(4,1)] ]; % wrong: z rot first, losing arm coordinate system P_orig = [ 0; 1; 0] % [ .707; .707; 0] ; %[ 0 1 0 ]' % [ 0.70711; 0.70711; 1 ] ; % [P_dest, seq_stp] = Th3D_Rot_Seq(P_orig, rot_ang_seq); disp(P_dest) row_sze = size(seq_stp, 2); plot3(seq_stp(1, :), seq_stp(2, :), seq_stp(3, :), 'r*-')

14. Homogeneous matrix to represent translation by multiplication scaling rotation translation example, from http://www.riemers.net/eng/ExtraReading/homogenous_matrices.php

15. 4x4 matrix idea: • translate elbow joint location to [0 0 0]’ • rotate elbow points (and distal) as required • translate rotated elbow element back to correct joint location. • run exerRot4.m for demo of X-axis rotation mimicking Rhino forearm rotation

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17. First Rhino Lab, “Limits” • on engin1930z.pbworks.com

18. Rhino Lab 1: Limits of rotation • Waist, Shoulder, Elbow and Wrist Flex all have limits of movement. • Waist can rotate about 340º independent of other joints. • Shoulder, Elbow and Wrist have inter-related angles at the limits of rotation: too far back, too far down. • Your challenge: keep track of clicks for each joint and stop movement near limits.