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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, 2 p 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) 5/23/2017 Biomechanics Lab, University of Ottawa 2 absolute angles for segments 5/23/2017 relative angles for joints Biomechanics Lab, University of Ottawa 3 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 5/23/2017 Biomechanics Lab, University of Ottawa 4 Angle of Foot right horizontal axis from proximal end angle of foot is –60 deg or 300 deg 5/23/2017 Biomechanics Lab, University of Ottawa 5 Angle of Leg right horizontal axis from proximal end angle of leg is –75 deg or 285 deg 5/23/2017 Biomechanics Lab, University of Ottawa 6 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) 5/23/2017 Biomechanics Lab, University of Ottawa 7 Angle of Foot ankle angle is +110 deg 5/23/2017 Biomechanics Lab, University of Ottawa 8 Angle of Knee knee angle is –120 deg 5/23/2017 Biomechanics Lab, University of Ottawa 9 Absolute vs. Relative knee angle = (thigh angle – leg angle) –180 = –60–(–120) – 180 = –120 5/23/2017 angle of thigh is –60 deg knee angle is –120 deg Biomechanics Lab, University of Ottawa angle of leg is –120 deg 10 Joint Angles in 2D or 3D q = cos–1[(a.b)/ab] a & b are vectors representing two segments b knee angle is –120 deg ab = product of segment lengths a a∙b= dot product 5/23/2017 Biomechanics Lab, University of Ottawa 11 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 5/23/2017 Biomechanics Lab, University of Ottawa 12 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 5/23/2017 Biomechanics Lab, University of Ottawa 13 Visual3D Angles Segment Angles Segment angle is angle of a segment relative to the lab coordinate system (LCS) x, y, z vs X, Y, Z z-axis: longitudinal axis y-axis: perpendicular to plane of joint markers (red points) x-axis: orthogonal to y-z plane (crossproduct) 5/23/2017 Biomechanics Lab, University of Ottawa 14 Visual3D Angles Joint Cardan Angles Joint angle is the angle of a segment relative to another segment x1, y1, z1 vs x2, y2, z2 order is x, y, z x-axis: is flexion/extension y-axis: is abduction/ adduction x-axis: is internal/external rotation 5/23/2017 Biomechanics Lab, University of Ottawa 15 Visual3D Angles 3 or 4 point angles calculates angle between two vectors with (3point) or without (4point) a common point can be an angle projected onto a plane (XY, XZ or YZ) or a 3D angle limited to ranges of motion of less than 180 degrees 5/23/2017 Biomechanics Lab, University of Ottawa 16 Electrogoniometry Sensors potentiometer polarized light optical fibre (e.g., Measurand) strain gauge (e.g., Biometrics) videography (e.g., Visual3D, Polygon) 5/23/2017 Biomechanics Lab, University of Ottawa 17 Electrogoniometry Potentiometry can measure absolute or relative angles usually use one-turn “pots” for human motions essentially a variable resistor with dc-power input a “wiper” changes output voltage depending on its angular position 5/23/2017 Biomechanics Lab, University of Ottawa 18 Electrogoniometry Potentiometry simple circuit, has dc-input and one or more outputs signal condition changes gain and offset 5/23/2017 Biomechanics Lab, University of Ottawa 19 Electrogoniometry Types single-axis and torsional (e.g., ShapeSensor, Biometrics) single-axis with four-bar linkage twin-axis (e.g., Biometrics) triaxial (e.g., CARS-UBC, ShapeTape) 5/23/2017 Biomechanics Lab, University of Ottawa 20 Electrogoniometry Four-bar linkage permits linear translation of one arm of the goniometer without causing rotation of the potentiometer potentiometer four-bar linkage 5/23/2017 Biomechanics Lab, University of Ottawa hinges 21 Electrogoniometry Triaxial linkages requires 4x4 matrix (Chao, et al. J Biomech, 3:459-71, 1970) transformation to estimate internal joint motion and test jig for calibration 5/23/2017 Biomechanics Lab, University of Ottawa 22