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Angular Kinematics

Learning Objectives:• Define angular kinematics• Understand & compute absolute & relative angles,

angular displacements, velocities & accelerations• Estimate instantaneous angular velocity & accel.• Be able to identify phases of movement & infer

sources of propulsion and braking• Be able to use the laws of constant angular accel.• Grasp the applications to human movement

Questions to Think About

• Why would tight calf muscles restrict the ability to run more than the ability to walk?

• Which muscles are used to speed up the extension of the elbow during a jump shot in basketball?

• As the knee flexes after landing from a jump, why are the knee extensors active?

• If you are trying to increase a baseball player’s bat speed at impact, what kinematic variables should you consider?

Angular KinematicsKinematics• The description of motion as a function of space

and time• Forces causing the motion are not considered

Angular Motion (Rotation)• All points in an object move in a circle about a

single, fixed axis of rotation – All points move through the same angle in the

same time

Angular Kinematics• The kinematics of particles, objects, or systems

undergoing angular motion

Angular Kinematics & Motion

• Volitional movement performed through rotation of the body segments

• The body is often analyzed as a collection of rigid, rotating segments linked at the joint centers

• This is a rough approximation

ANKLE

KNEE

HIP

ELBOW

SHOULDER

NECK

LUMBAR

2

Measuring Angles

0, 2π

π/2

π

3π/2

π = 3.14159

0, 360

90

180

270

Degrees: Radians:

1 radian57.3°

1 radian = 57.3°

1 revolution = 360° = 2π radians

Positive vs. Negative Angles

0,+360°

+90°

+180°

+270°

Positive: Negative:

+57.3°

Typical convention:

• Positive angles Counterclockwise rotation

• Negative angles Clockwise rotation

0,-360°

-270°

-180°

-90°-57.3°

Absolute Angle (or Inclination Angle)

• Orientation of a line segment with respect to a fixed line of reference

• Use absolute angles for equations relating torques to motion

θ

Trunk angle from vertical

Trunk angle from horizontal

θ

θ

X’

Computing Absolute Angles in 2-D• Use trigonometry to compute absolute angles from

(x, y) coordinates of two points

(xD, yD)

(xP, yP)

y P–

y D

xP – xD

X

Y

−−=

DP

DP

xx

yyatanθ

3

Relative Angle• Angle between two line segments

• Can compute relative angle by subtracting absolute angles of segments:

θ2/1 = θ2 – θ1

axis of rotation

θ2/1

θ2

θ1

segment 1segment 2

Joint Angles• Joint angles are relative angles between adjacent

body segments

θankle

θelbow

θknee

θshoulder

θhip

• Can think of as:– Rotation of distal

segment relative to proximal

and/or– Rotation of

proximal segment relative to distal

• Joint angle of zero = anatomical position

θelbow

θknee

Joint Angles in 2-D• Flexion & Abduction : θ between longitudinal axes

• External rotation: θ between AP or ML axes

Sagittal View Frontal View

APPELVIS

Transverse View

MLPELVIS

θflexionθabduct

KNEE

SHOULDER APTHIGH

MLTHIGH

HIP θexternal

Measuring Joint Angles• Devices for directly measuring joint angles:

– Goniometer

– Electrogoniometer V

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Computing Joint Angles• Involves subtracting absolute angles of segments

• Exact formula and order of subtraction depends on the joint and the convention chosen

θknee ANKLE

HIP

KNEE

θthigh=25°

θleg=70°

θtrunk=60°

θhip

θhip = θtrunk– θthigh

θknee = θleg– θthigh

If facing left and flexion > 0:

Range of Motion

Hip ROM

Flexion ROM

Extension ROM

• Can measure for person or task as:

– Maximum joint angle

– Difference between maximum and minimum joint angles

• Restrictions in range of motion can impair performance

• Exceeding functional range of motion can result in injury

• Excessive or restricted range of motion can indicate injury or other disorder

Angular Displacement (∆θ)• Change in the absolute or relative angle of an

object between two instants in time

• Doesn’t depend on the path between orientations

• Has angular units (e.g. degrees, radians)

angular displacement

axis of rotation

final orientation

initial orientation

Computing Angular Displacement• Compute angular displacement (∆θ) by subtracting

initial from final orientation angle:

∆θ = θfinal – θinitial

axis of rotation

∆θ

θfinal

θinitial

initial orientation

final orientation

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Computing Displacement• When computing displacement, θ must continuously

increase or decrease over the range of motion– # of full rotations thus included in displacement

ω

θi = 320°

θf = 390° (not 30°)

∆θ = θf – θi

= 70°

ω

θf = -40° (not 320°)

θi = 30°

∆θ = -70°

crossing the x-axis in the + direction

crossing the x-axis in the – direction

Angular Velocity (ω)• Rate of change of the angle of an object

ω =∆θ∆t

angular velocity

change in angle

change in time

angular displacement

change in time=

θfinal – θinitial

tfinal – tinitial=

• Can compute for an absolute or a relative angle

• Symbolic notation:

• Has units of (angular units)/time (e.g. radians/s, °/s)

=

gives average angular velocity from tinitial to tfinal

Example Problem #1At stride foot contact of a baseball pitch, a pitcher’s

shoulder is in 88° of external rotationIn the arm cocking phase, the shoulder externally

rotates through a displacement of 86°.When the ball is released 37 ms later, at the end of

the acceleration phase, the shoulder is in 64° of external rotation

What was a) the shoulder angle at the end of the arm cocking phase? b) the average shoulder angular velocity during the acceleration phase?

Average vs. Instantaneous Velocity• Previous formula gives average velocity between an

initial time and a final time

• Instantaneous angular velocity = angular velocity at a single instant in time

• Instantaneous angular velocity often more important

• Estimate using the central difference method:

ω at t1 = [θ at (t1 + ∆t)] – [θ at (t1 – ∆t)]

(t1 + ∆t) – (t1 – ∆t)

where ∆t is a very small change in time

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Relative Angular Velocity• Rate of change of the angle between two segments

• If segment 2 is rotating at velocity ω2/1 relative to segment 1, and segment 1 is rotating at velocity ω1, the angular velocity of segment 2 is:

ω2/1

θ1

θ2/1

Angular velocity of segment 2 relative to segment 1

Segment 1

Segment 2

ω1

ω2 = ω1 + ω2/1

ω2

Example Problem #2

During a forehand tennis stroke, a player is rotating her pelvis towards the ball at 200°/s, horizontally adducting her shoulder at 540°/s, and extending her wrist at 150°/s.

What are the absolute angular velocities of:

• Her pelvis-and-torso?

• Her upper limb?

• Her hand-and-racquet?

Angular Velocity as a Slope• Given a graph of angular position vs. time:

slope = average ωfrom t1 to t2

time (s)

θ(d

egre

es)

t2t1

∆t

slope = instantaneousω at t1

∆t(2–1)

∆θ(2

–1)

• Can estimate ω vs. time from slope, as done previously

Angular Acceleration• Rate of change of angular velocity

α =∆ω∆t

angular acceleration

change in angular velocity

change in time

ω1 – ω0

t1 – t0=

• Symbolic notation:

• Has units of (angular units)/time2 (e.g. rad/s2, °/s2)

=

gives average angular accel. from t0 to t1

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Instantaneous Angular Accel.• Previous formula gives average angular acceleration

between an initial time and a final time

• Instantaneous angular acceleration = angular acceleration at a single instant in time

• Estimate using the central difference method:

α at t1 = [ω at (t1 + ∆t)] – [ω at (t1 – ∆t)]

(t1 + ∆t) – (t1 – ∆t)

where ∆t is a very small change in time

Effects of Angular Acceleration

Velocity Acceleration Change in Velocity

(+) (+)

(+) (–)

(–) (–)

(–) (+)

• Velocity ω and acceleration α …

– In same direction: velocity increases magnitude

– Opposite directions: velocity decreases magnitude

• Larger accel. magnitude faster change in velocity

Increase in + dir.

Decrease in + dir.

Increase in – dir.

Decrease in – dir.

Angular Acceleration as a Slope• Given a graph of angular velocity vs. time:

time (s)

ω(d

eg/s

)

t2t1

∆t∆t(2–1)

∆ω(2

–1)

slope = instantaneousα at t1

slope = average αfrom t1 to t2

• Can estimate α vs. time from slope, as done previously

Example Problem #3

A volleyball player spikes the ball

Starting with her shoulder flexed, she begins to extend her shoulder to bring her arm forward

She contacts the ball 120 ms later, with her shoulder extending at 700°/s

After another 100 ms, at the end of follow-through, her shoulder stops extending

What was the average acceleration at the shoulder before ball contact and after ball contact?

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Propulsive & Braking PhasesPropulsive Phase• Magnitude of velocity ω increases

• Velocity ω and acceleration α in same direction• Propulsion produced by:

– Agonist muscles (concentric contraction)– External torques in direction of motion

Braking Phase• Magnitude of velocity ω decreases

• Velocity ω and acceleration α in opposite directions• Braking produced by:

– Antagonist muscles (eccentric contraction)– External torques opposite to direction of motion

0

40

80

120

0 0.5 1 1.5 2 2.5 3

Elb

ow A

ngle

(de

g)

Example: Biceps Curl

-800

-400

0

400

800

0 0.5 1 1.5 2 2.5 3

Acc

eler

atio

n (d

eg/s

2 )

-250

-150

-50

50

150

250

0 0.5 1 1.5 2 2.5 3

Ve

loci

ty (d

eg

/s)

FW

θ

-60

-40

-20

0

20

40

60

80

0 0.2 0.4 0.6 0.8 1

Time (s)

Stic

k an

gle

(deg

)

Example Problem #4aPictured is the absolute angle of a hockey stick during

a slap shot. Sketch the angular velocity and angular acceleration during the shot and identify its phases.

θ

How would the solution differ if the player pauses at the end of the backswing?

How and why might this affect the speed of the shot?

-60

-40

-20

0

20

40

60

80

0 0.2 0.4 0.6 0.8 1 1.2

Stic

k A

ngle

(deg

)Example Problem #4b

-200

-100

0

100

200

300

400

0 0.2 0.4 0.6 0.8 1 1.2

Stic

k V

eloc

ity (d

eg/s

)

P B P B

P B P B-2000

-1000

0

1000

2000

0 0.2 0.4 0.6 0.8 1 1.2

Stic

k A

ccel

. (de

g/s2

)Time (s)

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Laws of Constant Angular Accel.

where:α = angular accelerationω0 = angular velocity at initial time t0ω1 = angular velocity at final time t1∆θ = angular displacement between t0 and t1∆t = change in time (= t1 – t0)

Use + values for + direction, – values for – direction

ω1 = ω0 + α ∆t

∆θ = ½ (ω0 + ω1) ∆t

∆θ = ω0 ∆t + (½) α (∆t)2

ω12 = ω0

2 + 2 α (∆θ)

• When angular acceleration is constant:Example Problem #5

A discus thrower stands facing the back of the circle and starts to spin.

He releases the discus 2 seconds later after spinning 540° (1.5 revolutions) to his left.

Assume that he accelerates at a constant rate.

What was his angular acceleration as he spun?

How fast was he spinning after the first 180°?

How fast was he spinning at the time of release?