Roll Pitch and Yaw 3 D rotation matrices

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 3 x 3 rotation matrices • Limitation of single origin • Matlab: Modelling Rhino waist, shoulder • 3 D 4 x 4 Homogeneous matrices for translation • First Rhino Lab specs & protocols

Remember remember the dot of the vectors

Right-handed coordinate system

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

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

Looking down at rotation of θ around the z axis, in terms of unit vectors. j 0 j 1 θ i 0

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

Rotate φ around the y-axis

Rotate ζ around the x-axis

3 -D rotation matrix multiplication not commutative Rx*Ry*Rz*Po ≠ Rz*Ry*Rx*Po http: //www. lightandmatter. com/html_books/0 sn/ch 04/figs/book. png

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

An eensy weensy problem for Rhino • Basic concatenating of 3 D 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 3 D 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/instfig 1. gif

Matlab demo of rotation matrix sequence order • • exer. Rot 3. m lines 11, 16, 15 P_orig = [0 1 0]’ After rot. Z of 45º then rot. X 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; 0] ; %[ 0 1 0 ]' % [ 0. 70711; [P_dest, seq_stp] = Th 3 D_Rot_Seq(P_orig, rot_ang_seq); disp(P_dest) row_sze = size(seq_stp, 2); plot 3(seq_stp(1, : ), seq_stp(2, : ), seq_stp(3, : ), 'r*-') 0. 70711; 1 ] ; %

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

4 x 4 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 exer. Rot 4. m for demo of X-axis rotation mimicking Rhino forearm rotation

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• First Rhino Lab, “Limits” • on engin 1930 z. pbworks. com

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 interrelated 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.