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) 9/19/2018 Biomechanics Lab, University of Ottawa

Biomechanics Lab, University of Ottawa 9/19/2018 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 9/19/2018 Biomechanics Lab, University of Ottawa

Biomechanics Lab, University of Ottawa Angle of Foot 9/19/2018 Biomechanics Lab, University of Ottawa

Biomechanics Lab, University of Ottawa Angle of Leg 9/19/2018 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) 9/19/2018 Biomechanics Lab, University of Ottawa

Biomechanics Lab, University of Ottawa Angle of Foot 9/19/2018 Biomechanics Lab, University of Ottawa

Biomechanics Lab, University of Ottawa Angle of Knee 9/19/2018 Biomechanics Lab, University of Ottawa

Biomechanics Lab, University of Ottawa Absolute vs. Relative knee angle = (thigh angle – leg angle) –180 = –60–(–120) – 180 = –120 9/19/2018 Biomechanics Lab, University of Ottawa

Biomechanics Lab, University of Ottawa Joint Angles in 2D or 3D q = cos–1[(a.b)/ab] a & b are vectors representing two segments ab = product of segment lengths a∙b= dot product 9/19/2018 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 9/19/2018 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 9/19/2018 Biomechanics Lab, University of Ottawa

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 (cross-product) 9/19/2018 Biomechanics Lab, University of Ottawa

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 9/19/2018 Biomechanics Lab, University of Ottawa

Visual3D Angles 3 or 4 point angles calculates angle between two vectors with (3-point) or without (4-point) 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 9/19/2018 Biomechanics Lab, University of Ottawa

Electrogoniometry Sensors potentiometer polarized light optical fibre (e.g., Measurand) strain gauge (e.g., Biometrics) videography (e.g., Visual3D, Polygon) 9/19/2018 Biomechanics Lab, University of Ottawa

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 9/19/2018 Biomechanics Lab, University of Ottawa

Electrogoniometry Potentiometry simple circuit, has dc-input and one or more outputs signal condition changes gain and offset 9/19/2018 Biomechanics Lab, University of Ottawa

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) 9/19/2018 Biomechanics Lab, University of Ottawa

Electrogoniometry Four-bar linkage permits linear translation of one arm of the goniometer without causing rotation of the potentiometer potentiometer four-bar linkage hinges 9/19/2018 Biomechanics Lab, University of Ottawa

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 9/19/2018 Biomechanics Lab, University of Ottawa