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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)

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angular kinematics

Angular Kinematics

D. Gordon E. Robertson, PhD, FCSB

School of Human Kinetics

University of Ottawa

angular kinematics differences vs linear kinematics
Angular KinematicsDifferences vs. Linear Kinematics
  • Three acceptable SIunits 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)

Biomechanics Lab, University of Ottawa

absolute or segment angles uses newtonian or inertial frame of reference
Absolute or Segment AnglesUses Newtonian or inertial frame of reference
  • Used to define angles ofsegments
  • 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

Biomechanics Lab, University of Ottawa

angle of foot
Angle of Foot

Biomechanics Lab, University of Ottawa

angle of leg
Angle of Leg

Biomechanics Lab, University of Ottawa

relative or joint angles uses cardinal or anatomical frame of reference
Relative or Joint AnglesUses 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)

Biomechanics Lab, University of Ottawa

angle of foot1
Angle of Foot

Biomechanics Lab, University of Ottawa

angle of knee
Angle of Knee

Biomechanics Lab, University of Ottawa

absolute vs relative
Absolute vs. Relative
  • knee angle =

(thigh angle

– leg angle) –180

= –60–(–120) – 180

= –120

Biomechanics Lab, University of Ottawa

joint angles in 2d or 3d
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

Biomechanics Lab, University of Ottawa

angular kinematics finite difference calculus
Angular KinematicsFinite Difference Calculus
  • Assuming the data have beensmoothed, 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

Biomechanics Lab, University of Ottawa

3d angles euler angles
3D AnglesEuler 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

Biomechanics Lab, University of Ottawa

visual3d angles segment angles
Visual3D AnglesSegment 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)

Biomechanics Lab, University of Ottawa

visual3d angles joint cardan angles
Visual3D AnglesJoint 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

Biomechanics Lab, University of Ottawa

visual3d angles 3 or 4 point angles
Visual3D Angles3 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

Biomechanics Lab, University of Ottawa

electrogoniometry sensors
ElectrogoniometrySensors
  • potentiometer
  • polarized light
  • optical fibre (e.g., Measurand)
  • strain gauge (e.g., Biometrics)
  • videography (e.g., Visual3D, Polygon)

Biomechanics Lab, University of Ottawa

electrogoniometry potentiometry
ElectrogoniometryPotentiometry
  • 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

Biomechanics Lab, University of Ottawa

electrogoniometry potentiometry1
ElectrogoniometryPotentiometry
  • simple circuit, has dc-input and one or more outputs
  • signal condition changes gain and offset

Biomechanics Lab, University of Ottawa

electrogoniometry types
ElectrogoniometryTypes
  • 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)

Biomechanics Lab, University of Ottawa

electrogoniometry four bar linkage

potentiometer

four-bar linkage

hinges

ElectrogoniometryFour-bar linkage
  • permits linear translation of one arm of the goniometer without causing rotation of the potentiometer

Biomechanics Lab, University of Ottawa

electrogoniometry triaxial linkages
ElectrogoniometryTriaxial linkages
  • requires 4x4 matrix (Chao, et al. J Biomech, 3:459-71, 1970) transformation to estimate internal joint motion and test jig for calibration

Biomechanics Lab, University of Ottawa