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Angular Kinematics D. Gordon E. Robertson, PhD, FCSB School of Human Kinetics University of Ottawa

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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 º, 360º = 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 )

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4/13/2015 Biomechanics Lab, University of Ottawa 3

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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

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4/13/2015 Biomechanics Lab, University of Ottawa 5 Angle of Foot

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4/13/2015 Biomechanics Lab, University of Ottawa 6 Angle of Leg

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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)

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4/13/2015 Biomechanics Lab, University of Ottawa 8 Angle of Ankle

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4/13/2015 Biomechanics Lab, University of Ottawa 9 Angle of Knee

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4/13/2015 Biomechanics Lab, University of Ottawa 10 Absolute vs. Relative knee angle = [thigh angle – leg angle] –180 =[–60–(–120)]–180 = –120

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4/13/2015 Biomechanics Lab, University of Ottawa 11 Joint Angles in 2D or 3D = cos –1 [(a∙b)/ab] a and b are vectors representing two segments ab = product of segment lengths a∙b= dot product

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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

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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

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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

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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

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4/13/2015 Biomechanics Lab, University of Ottawa 16 Computerize the Process Visual3D, MATLAB, Vicon, or SIMI etc.

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