22_KINE426_Angular_Kinematics

1 Andrew Nordin, PhD Assistant Professor, Division of Kinesiology KINE 426 Exercise Biomechanics Angular Kinematics Chapter 9 Kinematics Linear Angular Scalar Vector Distance Speed Acceleration Displacement Velocity !"##$ = !"#$%&'() !"#$% !"#$% !"# = !"#$%&'()*)+, !"#$% !""#$#%!&'( ! = !"#$%&'() !"#$% • Spatial & temporal components of motion • Ignores forces causing motion Magnitude Magnitude & direction Scalar Vector Distance Speed Acceleration Displacement Velocity Magnitude Magnitude & direction ∆ ! = ! ! − ! ! ! = ! ! − ! ! ! ! − ! ! ! = ! ! − ! ! ! ! − ! ! Kinematics

2 Angular motion • Angular motion – All parts of a body segment move through the same angle, but different linear displacement • Axis of rotation • Human movement – Rotation & translation r Δθ axis line vertex angle line Rectilinear translation Curvilinear translation Translation with rotation Angular motion

4 Barroso Rosa et al., 2019, Orthopaedic surgery Units of measurement • Degrees – 360° in a circle • Revolutions – # rotations around a circle • Radians (rad) – Arc length ( L or s ) – Radius ( r ) – Considered unitless – 2π rad in a circle r s axis θ ! = ! ! = ! !"# = 180° → 180 ° ! !"# = 57 . 3 ° !"# !"#$""% 57 . 3 ° → !"# !"#$%&'(#()$( !" !"#!$% = 2 ! ! = 1 rad = 57.3 degrees s = r

5 r Δθ 1 axis Angular kinematics Scalar Vector Distance Speed Acceleration Displacement Velocity Magnitude Magnitude & direction ∆ ! = ! ! − ! ! ! = ! ! − ! ! ! ! − ! ! ! = ! ! − ! ! ! ! − ! ! Δθ 2 Δθ 3 Angular distance: ∆ ! = ! ! + ! ! Angular displacement: ∆ ! = ! ! θ : theta ω : omega α : alpha Δθ Angular kinematics θ f θ i Δθ Angle types • Absolute angle – Segment inclination – Fixed reference – Counterclockwise from right horizontal – Coordinate system @ distal segment end – Trigonometric tangent relationship • Relative angle (Intersegmental angle) – Angle between longitudinal axes of segments – Law of cosines !"# ! = !""!#$%& !"#!$%&' ! ! = ! ! + ! ! − 2 !" !"#$ ! ! = ! ! + ! ! − 2 !" !"#$ ! ! = ! ! + ! ! − 2 !" !"#$ opposite adjacent hypotenuse θ

7 Absolute angle !"# ! = !""!#$%& !"#!$%&' opposite adjacent hypotenuse θ proximal distal proximal distal tan ! !"# = ! ! tan ! !"# = − ! − ! tan ! !"# = − ! ! tan ! !"# = ! − ! proximal distal distal proximal θ θ θ θ + 180° + 180° + 360° Relative to right horizontal tan ࠵? !"#$"%& = ࠵? ’()*+$,- − ࠵? .+!,- ࠵? ’()*+$,- − ࠵? .+!&,- Absolute angle !"# ! = !""!#$%& !"#!$%&' opposite adjacent hypotenuse θ Distal to reference coordinate tan ࠵? &1+#1 = 0.80 − 0.51 1.14 − 1.22 = 0.29 −0.08 = −3.635 ࠵? &1+#1 = tan /0 (− 3.635) = −74.58° tan ࠵? &1+#1 = ࠵? 1+’ − ࠵? 2%"" ࠵? 1+’ − ࠵? 2%"" tan ࠵? !"#$"%& = ࠵? ’()*+$,- − ࠵? .+!,- ࠵? ’()*+$,- − ࠵? .+!&,- −࠵?࠵?. ࠵?࠵?° + ࠵?࠵?࠵?° = ࠵?࠵?࠵?. ࠵?° from right horizontal

8 Relative angle ! ! = ! ! + ! ! − 2 !" !"#$ ! ! = ! ! + ! ! − 2 !" !"#$ ! ! = ! ! + ! ! − 2 !" !"#$ a b c A B C Law of cosines No right angle Segment lengths ! ! = ! ! + ! ! − 2 !" !"#$ !"#$ = − ! ! + ! ! + ! ! 2 !" ! !" ! = 146 . 4 ° Thigh Leg !"#$% ℎ = ! ! − ! ! ! + ( ! ! − ! ! ) ! ! = ( ! ! !" − ! !"#$% ) ! + ( ! ! !" − ! !"#$% ) ! ! = ( 1 . 14 − 1 . 09 ) ! + ( 0 . 80 − 0 . 09 ) ! = 0 . 71 ! = ( ! ! !" − ! !"## ) ! + ( ! ! !" − ! !"## ) ! ! = ( 1 . 14 − 1 . 22 ) ! + ( 0 . 80 − 0 . 51 ) ! = 0 . 30 ! = ( ! !"## − ! !"#$% ) ! + ( ! !"## − ! !"#$% ) ! ! = ( 1 . 22 − 1 . 09 ) ! + ( 0 . 51 − 0 . 09 ) ! = 0 . 44 Relative knee angle • Absolute angles – Used in biomechanics in other calculations – Relative angles • Provide less information regarding body position in space – Multiple body positions → same relative angle • More common in practice (clinical meaning) • Converting from absolute to relative angles ! !"#$%&'" = ! !"#$%&'( !"#$%&'( − ! !"#$%&'( !"#$%& Relative vs. Absolute angles