lecture 4. terrestrial locomotion iii: mechanical analysis of gaits...
TRANSCRIPT
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Lecture 4. Terrestrial locomotion III: mechanical analysis of gaits and jumpiness.• Recap: gaits and ballistic walking
• When the Froude Number (V2/g L) is greater than 1, simple ballistic walking is no longer possible.
• More aspects of jumping (energy and force)
• The jumper model accounts for an airborne phase of movement.
• Calculating optimal gaits for energy expenditure
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Lecture 4. Terrestrial locomotion III: mechanical analysis of gaits and jumpiness.
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Review of walking
Matthis & Fajen, 2013
How much energy input is needed for walking?
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Energy tradeoffs in walking and running
What’s different?
KE
PE
KE
PE
elastic potential energy
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A model for running:the spring-loaded inverted pendulum (SLIP)
BBC
tendons are springy!
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The jumper: for speeds greater than Fr = 1, gait must change with an airborne phase
Spring Loaded Inverted Pendulum
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Goal: compute the power (rate of energy expenditure) as a function of size (M,m) and gait (jumpiness)
M
m
h
d
VAssume constant Vno air resistanceno skidding
j = (1-β)/β
Maynard-Smith Mathematical Ideas in Biology
L
βL (1-β)L
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Goal: compute the power (rate of energy expenditure) as a function of size (M,m) and gait (jumpiness)
M
m
h
d
j = (1-β)/β
Maynard-Smith Mathematical Ideas in Biology
L
βL (1-β)L
V
Energy to launch? = (M +m)g (h+d) Energy to break the fall? Energy to move the foot?
γ= (M +m)g (h+d) {-1 < γ < 1}= mV2/2
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= (1+γ )(M +m) + mV2/2 g2 L2(1-β) 8V2
Goal: compute the power (rate of energy expenditure) as a function of size (M,m) and gait (jumpiness)
M
m
h
d
j = (1-β)/β
Maynard-Smith Mathematical Ideas in Biology
L
βL (1-β)L
V
Total Energy
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= (1+γ )(M +m) + mV3/ (2 L) g2 L(1-β) 8V
Goal: compute the power (rate of energy expenditure) as a function of size (M,m) and gait (jumpiness)
M
m
h
d
j = (1-β)/β
Maynard-Smith Mathematical Ideas in Biology
L
βL (1-β)L
V
Total Power
= (1+γ )(M +m) + mV3/ (2 Lβ(1+j)) g2 Lβj 8V
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How does Power vary with foot mass (m)?body mass (M)?body velocity (V)?jumpiness (j)?
P
0 inf.
= (1+γ )(M +m) + mV3/ (2 Lβ(1+j)) g2 Lβj 8VTotal Power
dP/dj = 0 —> Max, Min, inflection….