Angular Kinematics

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Angular Kinematics

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

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Angular KinematicsDifferences 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)

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Absolute or Segment AnglesUses 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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Angle of Foot
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Angle of Leg
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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)

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Angle of Foot
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Angle of Knee
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Absolute vs. Relative

• knee angle
• thigh angle
• leg angle 180
• 60(120)180
• 120

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Joint Angles in 2D or 3D

• q cos1(ab)/ab
• a and b are vectors representing two segments
• ab product of segment lengths
• ab dot product

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Angular KinematicsFinite Difference Calculus

• Assuming the data have been smoothed, finite
  differences may be taken to determine velocity
  and acceleration. I.e.,
• Angular velocity
• omegai wi (qi1 qi-1) / (2 Dt)
• where Dt time between adjacent samples
• Angular acceleration
• alphai ai (wi1 wi-1) / Dt (qi2 2qi
  qi-2) / 4(Dt)2
• or ai (qi1 2qi qi-1) / (Dt)2

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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

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Visual3D AnglesSegment Angles

• Segment angle is angle of a segment relative to
  the lab. 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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Computerize the Process

• Visual3D, MATLAB, Vicon or SIMI etc.

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Visual3D AnglesJoint Cardan Angles

• Joint angle is the angle of a segment relative to
  the lab. reference system
• x1, y1, z1 vs. x2, y2, z2
• order is x, y, z
• x-axis is flexion/extension
• y-axis is varus/valgus, abduction/adduction
• x-axis is internal/external rotation