# Angular Kinematics D. Gordon E. Robertson, PhD, FCSB

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

Angular Kinematics Differences vs. Linear Kinematics
Three acceptable SI units of measure revolutions (abbreviated r) degrees (deg or º, 360º = 1 r) radians (rad, 2p 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) 4/10/2017 Biomechanics Lab, University of Ottawa

Biomechanics Lab, University of Ottawa
4/10/2017 Biomechanics Lab, University of Ottawa

Biomechanics Lab, University of Ottawa
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 4/10/2017 Biomechanics Lab, University of Ottawa

Biomechanics Lab, University of Ottawa
Angle of Foot 4/10/2017 Biomechanics Lab, University of Ottawa

Biomechanics Lab, University of Ottawa
Angle of Leg 4/10/2017 Biomechanics Lab, University of Ottawa

Biomechanics Lab, University of Ottawa
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) 4/10/2017 Biomechanics Lab, University of Ottawa

Biomechanics Lab, University of Ottawa
Angle of Ankle 4/10/2017 Biomechanics Lab, University of Ottawa

Biomechanics Lab, University of Ottawa
Angle of Knee 4/10/2017 Biomechanics Lab, University of Ottawa

Biomechanics Lab, University of Ottawa
Absolute vs. Relative knee angle = [thigh angle – leg angle] –180 =[–60–(–120)]–180 = –120 4/10/2017 Biomechanics Lab, University of Ottawa

Biomechanics Lab, University of Ottawa
Joint Angles in 2D or 3D q = cos–1[(a∙b)/ab] a and b are vectors representing two segments ab = product of segment lengths a∙b= dot product 4/10/2017 Biomechanics Lab, University of Ottawa

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 omegai = wi = (qi+1 – qi-1) / (2 Dt) where Dt = time between adjacent samples Angular acceleration: alphai = ai = (wi+1 – wi-1) / Dt = (qi+2 –2qi +qi-2) / 4(Dt)2 or ai = (qi+1 –2qi +qi-1) / (Dt)2 4/10/2017 Biomechanics Lab, University of Ottawa

Biomechanics Lab, University of Ottawa
3D Angles Euler Angles Ordered set of rotations: a, b, g Start with x, y, z axes rotate about z (a) to N rotate about N (b) to Z rotate about Z (g) to X Finishes as X, Y, Z axes 4/10/2017 Biomechanics Lab, University of Ottawa

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 4/10/2017 Biomechanics Lab, University of Ottawa

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 4/10/2017 Biomechanics Lab, University of Ottawa

Computerize the Process
Visual3D, MATLAB, Vicon, or SIMI etc. 4/10/2017 Biomechanics Lab, University of Ottawa

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