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Angular Kinematics D. Gordon E. Robertson, PhD, FCSB School of Human Kinetics University of Ottawa Slide 2 4/13/2015 Biomechanics Lab, University of Ottawa 2 Angular Kinematics Differences vs. Linear Kinematics Three acceptable SI units of measure revolutions (abbreviated r) degrees (deg or , 360 = 1 r) radians (rad, 2 rad = 1 r, 1 rad 57.3 deg) Angles are discontinuous after one cycle Common to use both absolute and relative frames of reference In three dimensions angular displacements are not vectors because they do not add commutatively (i.e., a + b b + a ) Slide 3 4/13/2015 Biomechanics Lab, University of Ottawa 3 Slide 4 4/13/2015 Biomechanics Lab, University of Ottawa 4 Absolute or Segment Angles Uses Newtonian or inertial frame of reference Used to define angles of segments Frame of reference is stationary with respect to the ground, i.e., fixed, not moving In two-dimensional analyses, zero is a right, horizontal axis from the proximal end Positive direction follows right-hand rule Magnitudes range from 0 to 360 or 0 to +/180 (preferably 0 to +/180) deg Slide 5 4/13/2015 Biomechanics Lab, University of Ottawa 5 Angle of Foot Slide 6 4/13/2015 Biomechanics Lab, University of Ottawa 6 Angle of Leg Slide 7 4/13/2015 Biomechanics Lab, University of Ottawa 7 Relative or Joint Angles Uses Cardinal or anatomical frame of reference Used to define angles of joints, therefore easy to visualize and functional Requires three or four markers or two absolute angles Frame of reference is nonstationary, i.e., can be moving Origin is arbitrary depends on system used, i.e., zero can mean neutral position (medical) or closed joint (biomechanical) Slide 8 4/13/2015 Biomechanics Lab, University of Ottawa 8 Angle of Ankle Slide 9 4/13/2015 Biomechanics Lab, University of Ottawa 9 Angle of Knee Slide 10 4/13/2015 Biomechanics Lab, University of Ottawa 10 Absolute vs. Relative knee angle = [thigh angle leg angle] 180 =[60(120)]180 = 120 Slide 11 4/13/2015 Biomechanics Lab, University of Ottawa 11 Joint Angles in 2D or 3D = cos 1 [(ab)/ab] a and b are vectors representing two segments ab = product of segment lengths ab= dot product Slide 12 4/13/2015 Biomechanics Lab, University of Ottawa 12 Angular Kinematics Finite Difference Calculus Assuming the data have been smoothed, finite differences may be taken to determine velocity and acceleration. I.e., Angular velocity omega i = i = ( q i+1 q i-1 ) / (2 t) where t = time between adjacent samples Angular acceleration: alpha i = a i = ( i+1 i-1 ) / t = ( q i+2 2 q i + q i-2 ) / 4(t) 2 or i = ( q i+1 2 q i + q i-1 ) / (t) 2 Slide 13 4/13/2015 Biomechanics Lab, University of Ottawa 13 3D Angles Euler Angles Ordered set of rotations: a, b, g Start with x, y, z axes rotate about z () to N rotate about N () to Z rotate about Z () to X Finishes as X, Y, Z axes Slide 14 4/13/2015 Biomechanics Lab, University of Ottawa 14 Visual3D Angles Segment Angles Segment angle is angle of a segment relative to the laboratory coordinate system x, y, z vs. X, Y, Z z-axis: longitudinal axis y-axis: perpendicular to plane of joint markers (red) x-axis: orthogonal to y-z plane Slide 15 4/13/2015 Biomechanics Lab, University of Ottawa 15 Visual3D Angles Joint Cardan Angles Joint angle is the angle of a segment relative to a second segment x1, y1, z1 vs. x2, y2, z2 order is x, y, z x-axis: is flexion/extension y-axis: is varus/valgus, abduction/adduction z-axis: is internal/external rotation Slide 16 4/13/2015 Biomechanics Lab, University of Ottawa 16 Computerize the Process Visual3D, MATLAB, Vicon, or SIMI etc.