(2) BIOMECHANICS of TERRESTRIAL LOCOMOTION - …€¦ · (2) BIOMECHANICS of TERRESTRIAL LOCOMOTION...

(2) BIOMECHANICS of TERRESTRIAL LOCOMOTION

Questions:

- How does size influence the mode and speed of locomotion ?

- What determines the energy cost oflocomotion ?

- Why do humans walk and run the way we do ?

- What determines how high an animal can jump ?

Mechanics:

- Linear motion.- Position, velocity, acceleration.- Force, mass, momentum, energy

(kinetic and potential), power.

- Rotational motion.- Angle, angular velocity, angular

acceleration, moment of inertia, angular momentum.

Biomechanics of Terrestrial Locomotion 2-1

2.1 LEGGED MOTION

Terrestrial legged motion consists of repeated cycles of leg movement.

Principal parameters:

- Size - Leg length, l (m) (Hip height while standing).- Mass, m (kg) (supported by each pair of legs).

- Velocity of locomotion, v (m/s) (mean value over a complete cycle).

- Stride length, λ (m).

- Stride frequency f = v / λ (Hz).

- Duty factor of each foot, β, (fraction of duration of stride in which a foot is on ground).

Gaits:

- Gaits are distinguished by relative phases of the feet.

- Compare the gaits of animals of different size using the dimensionless terms- Relative stride length, λ / l- Froude number, v2 / g l

- Different gaits used at different Froude numbers.Biomechanics of Terrestrial Locomotion 2-2

Two-legged locomotion consist of two gait (walk, hop or run).

Four-legged locomotion consist of three or more gaits (walk, trot, canter, gallop).


2.2 WALKING

Slow symmetrical gait with at least one foot (or pair) in contact with the ground at all times.

- Duty factor (β > 0.5).

Effect of Leg Length on Walking Speed

Quadrupedal walking = two bipeds in tandem.

“Gravitational walking”:- Gravity exerts a swinging torque on the leg.- In recovery phase the leg swings like a

pendulum.- Body and legs oscillate at natural frequency

(not forced) with no muscular effort.∴ No muscular effort.∴ Economical transport.

walking velocity = (stride length).(stride frequency)

fλv .=

lg

πθl

21

).(≈

lv∝Biomechanics of Terrestrial Locomotion 2-4

SteppingFrequency(Hz)

Shoulder height (∝ leg length) (m)

Result:Longer legs → Faster walking speed (compare the speed of a horse and a dog).

Assumptions of model:

- Constant (small) angle of swing, θ (only a small correction is required for large amplitude swings).

- All of the leg mass is concentrated in the feet.


Physical pendulum model (a more general model of the leg).

Stride frequency =

where m ≡ mass of legd ≡ distance from c.g. to axis of rotationI ≡ moment of inertia of leg

about axis of rotation (I = ∫ r2 dm)

Moment of inertia is a measure of the resistance a body offers to have its rotational momentum changed by a given torque.

For mass on string I = ml2, d = l

for uniform rod I = (1/3)ml2, d = (1/2)l

Same result:walking velocity

Imgd

π21

lgf

π21

=∴

lgf

23

21π

=∴

l∝


2.2 RUNNING

Gait features periods of time where all feet are off the ground ∴ lengthen stride (duty factor β < 0.5).

Different gaits can be attributed to variations in the relative phases of the feet.- Symmetrical gaits (pace, trot)- Asymmetrical gaits (bound, pronk, transverse gallop

rotary gallop).

Effect of Leg Length on Maximum Running Speed

Hip muscles exert a torque to accelerate and decelerate the leg.

In geometrically similar animals:running velocity = (stride length).(stride frequency)

v = λ . fwhere λ ∝ (length)and it may be shown that for a forced pendulum

f ∝ (length)-1

v ∝ (length) . (length)-1

∝ (length)0Result:

Maximum running speed is independent of leg length (l).


Advantage of long legs (in geometrically similar animals):- Walk faster- Same maximum running speed


The result f ∝ (length)-1 was obtained using the relationship between torque, τ, angular acceleration α ( = d 2θ / dt2) and moment of inertia, I

τ = I α

Where I ∝ (length)2 . (mass)∝ (length)2 . (length)3

∝ (length)5

The maximum torque exerted by the hip:max torque = (max muscle force) . (moment arm)

τ ∝ (cross-sectional area) . (length)∝ (length)2 . (length)∝ (length)3

α ∝ (length)3 / (length)5

∝ (length)-2

By integrating α twice, the time taken to accelerate the leg through a fixed angle is proportional to the length.i.e. stride period ∝ (length)

∴ f ∝ (length)-1


To increase running speed ( v = λ / f ):- Increase stride frequency, f, by reducing moment of inertia of legs:

- Flex legs during recovery phase.- Evolve large muscles in upper body only.

- Increase angle of leg swing θ (i.e. increase stride length λ = l θ ).- Evolve relatively long legs (deviate from geometrical similarity).

Advantage of running on 4 legs:- hind limbs have principal running muscles (fore limbs act as props)- increase stride length, bending and extending the back (hind limbs land

just in front of fore limbs).

∴ increase speed of runninghuman 8 - 12 m/s (bipedal)greyhound 15 - 16 m/shorse 16 - 17 m/s

- energy stored in tendons of lower back reduces metabolic energy cost of locomotion.


Gait change:- Mammals change gait at equal Froude numbers.

Walk → Trot ≈ 0.5Trot → Gallop ≈ 2.5

- Gait changes to minimize the energy cost locomotion.

Walk Trot Gallop


Dynamically similar locomotion:

- Each leg swings ~60° while foot on ground.- Takes longer strides at higher speeds.- Same relative stride length at equal Froude

numbers (±30%)

( ) 3.023.2≈ glvlλ

Describes walking and running of terrestrial mammals (humans, dogs, horses, rodents, kangaroos), flightless birds and dinosaurs.


Leg Straightness and Size

Animals do not quite move in dynamically similar fashion over the whole range of sizes.

- Small mammals → bent legs (higher metabolic cost, but animal is ready to jump or accelerate)

- Largest mammals → straighter legs (avoid large muscular stress)

Explanation for straight legs:

- For animals that move in dynamically similar fashion, all forces are scaled in same proportion.

Muscle must exert force ∝ gravitational force∝ mass

- For geometrically similar animals: Max. muscle force ∝ Cross-sectional area∝ (length)2

∝ (mass)2/3

Max. stress in muscle ∝ (force) / (cross sectional area)∝ (mass) / (mass)2/3

∝ (mass)1/3

∝ length not (length)0

- i.e., Muscles work closer to the limit of strength in large animals than in smaller animals.- Large animals must use straighter legged locomotion to reduce muscular stresses.


Cat Elephant


Elastic Mechanisms in Terrestrial Locomotion

Use of springs during running:

1) Replace muscle:

The force exerted by muscle increases and decreases as leg muscles lengthen and shorten.

∴ Replace muscle by an elastic tendon (spring)- Same mechanical effect.- No metabolic energy cost.

2) Store and return energy.

Store KE and PE as elastic strain energy (in tendons, ligaments, or muscle).- Partially returned as elastic recoil.- Reduces amplitude of fluctuations in total mechanical energy.

∴ Reduce metabolic power required.

3) Reduce length change of muscle.

∴ use more efficient muscle fibres.

(short fibres, or lower maximum shortening speed).


Tendons: - Achilles tendon stores significant amount of elastic energy during running.- Highly elastic (93% of stored energy is returned in recoil only 7% dissipated as

heat).

Ligaments: - Ligaments of foot store some elastic energy during running.- Less elastic than tendons.

Muscles: - Only some muscles store significant elastic energy, and only in some gaits (not effective in walking).

Plantaris tendon of kangaroo Achilles tendon and ligaments of human



2.4 ENERGETICS of TERRESTRIAL LOCOMOTION

Metabolic energy required for

- Activation of muscles (activity of crossbridges, i.e. generating force).- Performing work (moving limbs)

- Concentric contraction (+ve work).- Eccentric contraction (-ve work).- Isometric contraction (no work).

Energy interactions during each stride in animal running:

- Body and body parts accelerate/decelerate (inertia)∴ KE of animal changes.

- COG of body and body parts rise/fall (gravity)∴ Gravitational PE of animal changes.

- Tendons stretch and recoil∴ Store and release elastic strain energy.

Also work required to:- Overcome antagonist muscle groups.- Overcome aerodynamic drag (always small, only 3% of total energy cost in human sprinting)- Overcome joint friction (negligible).

Total mechanical energy (KE + PE + elastic) of animal

- Fluctuates during each stride.- Energy supplied and removed (converted to heat) by muscle action.

In unaccelerated locomotion over level ground:- Net work ≈ 0 (+ve work ≈ -ve work), the mechanical energy of animal is the same at

corresponding points in successive strides.

- High metabolic energy consumption rate due to metabolic energy cost of- Exerting force (∝ force)- Performing work (+ve, -ve)

Cost of performing 1 J of work in humans:

+ve work, requires 4 J metabolic energy-ve work, requires 0.8 J metabolic energy

To minimize metabolic energy consumption during locomotion:

- keep leg joints straight- keep ground reaction force in line with the leg

∴ Minimize moments at joints∴ Minimize forces required by muscles.


Compare the efficiency of “Groucho” running on bent legs, like apes.


Energy Cost of Transport

Metabolic energy cost of locomotion is determined from:

- Oxygen consumption rate when when running on a treadmill (respiratory gas exchange).

metabolic rate = total metabolic rate - resting (postural)during locomotion metabolic rate

Energy cost of transport:

- Metabolic energy required to transport unit mass of animal per unit distance [ J/(kg.m) ]

energy cost = total energy cost - energy cost ofof transport standing still

For walking/running, energy cost of transport is constant (independent of velocity)



As velocity increases (↑) (i.e. increasing Froude number v2 / gl ) the component of the energy cost of transport associated with:

- Internal kinetic energy ↑ (limbs accelerated to higher angular velocity).- Gravitational potential energy ↓ (duration of floating phase decreases).- Elastic strain energy ↑ (duty factor decreases at higher speeds).

∴ Higher forces act on feet.∴ More tendon stretch.∴ More energy stored in tendon.

Confirmed experimentally formammals 10-100 kg (smallermammals have a higher costof transport)


Metabolic Power Consumption vs Running Speed

Approximately linear relation between metabolic power consumption and running speed.P(v) ≈ P(0) + C.v

whereP(v) ≡ rate of metabolic energy consumption when running at speed v.P(o) ≡ rate of metabolic energy consumption when standing.C.v ≡ extra rate of energy use (power) for running at speed v.C ≡ extra rate of energy consumption per unit velocity (energy cost per unit distance).

Determined from many species of mammals running on treadmill (ignore effect of different gaits).

Speed (km/h)

Metabolic Power Consumption


Energy cost of transport(running mammals).

C ∝ (mass)0.68

C/m ∝ (mass)-0.32

Large animals aremore economical.

Explanation is uncertain: FLYING

- Possibly due to energy RUNNINGcost of exerting forceis more important thancost of performing work.

SWIMMING


2.5 HUMAN LOCOMOTION

Walking- Is a unique two legged style. Straightest legs of any animal, with an erect spine.- At least one foot in contact with the ground at all times (usually duty factor β = 0.55 – 0.70).

Walking at a constant speed: It would seem that we would require only:- Vertical forces to support body weight.- Horizontal force to overcome air resistance (usually negligible).

Actual walking technique is quite different. People do not walk with Fvert = body weight (no vertical acceleration, i.e with the center of gravity level)

Walking technique:

- Knee is almost straight when in contact with the ground.- COG moves in arcs of a circle (rises and falls ~ 35 mm during stride).- Fvert ≠ body weight (at all times).

- Moments of force about knee are small∴ Little muscle activation∴ Low metabolic energy cost

(High energy efficiency locomotion)



Additional factors:

- Pelvic rotation, pelvic tilt, stance, leg flexion, ankle flexion. ∴ Smooth the arc of the COG.∴ Does not require infinite force for change in direction of velocity of COG at midstance.

Limbs as pendulums:

- Motion of the legs as (passive) swinging pendulums.

- Leg-swing half-period:

T ≈ 0.35 sec

(leg fixed at hip, knee allowed to bend freely,allow for pelvic motion)

≈ observed swing period for fast walk, v ≈ 2.0 m/s.

- Economical transport(Little muscular activity in legs duringwalking from EMG studies).

- Adjust stride length with walking speedto maintain a passive leg swing.

Maximum Speed of Walking

Walking model:

- Assume all body mass located at hip.

- Body on straight leg behaves like an invertedpendulum (mass mounted on top of thependulum).

- Torque exerted at hip.

- Need to maintain contact with the groundwith at least one leg at all times (feet cannotpull down on the ground).

Requires that (centripetal force) ≤ (body weight)

During the motion of the c.g. along the arc of acircle to avoid flying off at tangent to arc.


Leg length l, walking speed v;

(Froude number)

Result:Longer legs → faster maximum walking speed.

Examples:

- Adult human: l = 0.9 m vmax = 3.0 m/s

- Child: l = 0.5 m vmax = 2.2 m/s

Child has slower maximum speed due to shorter legs.

∴ Starts running at a lower velocity than adult.

mgl

mv≤

2

1≤=2

glv

glv =∴ max


Running

Human running:- Duty factor usually β ≈ 0.3 - 0.4- Must run to attain speeds above the maximum walking speed- Abrupt change of gait from walk to run at a critical speed.

- Legs bent during support phase.- Ground reaction force in line with leg.

- Large muscular forces∴ high metabolic cost

- KE and PE - Low at midstance- High at midstride


Muscles behave like springs:

- Motion of legs not like pendulums, COG motion more like bouncing ball or pogo stick.- Muscles of knee (quadruceps) and ankle (gastrocnemius and soleus) behave like

springs.- Muscles activated when foot on ground.

Energy cost of running is mainly due to the horizontal component of ground force.


Ground Reaction Forces

Recorded with a force platform. Some important features are:- Ground reaction force is in line with the leg and accelerates / retard body during stride.

Horizontal force:- Forward then backward force (retard, accelerate).- Represented by superposition of 2 sine terms.

Vertical force:- Greatly exceeds body weight when running.- Represented by the superposition of 2 cosine terms.


Impact peak:

- Damped oscillation superimposed on ground reaction force due to body mass on a leg spring.

- Provides impulse to halt motion of foot (mass ≈ 4 kg in 25 ms).

- Compliant foot pad moderates the impact force, improves "road holding" by preventing "chatter” (vibrations in which the foot repeatedly leaves and returns to the ground before settling).



Metabolic Energy Cost

Most economical speed for walking; v ≈ 1.3 m/s. Running more economical than walking at v ≈ 2.3 m/s.

- Humans change walk → run at ≈ 0.7 (Froude number)

i.e. when v ≈ 2.5 m/s for adult human.

Human walking: - More economical than walking for an animal of similar size.Human running: - Relatively uneconomical.

glv2

Energy storage in humans:

- Much of KE lost in running stride is stored as elastic strain energy in stretched tendons and ligaments.

Achilles tendon:

- Store 35 J of elastic strain energy (1/3 of KE and PE lost during running stride).- 93% elastic recoil (7% dissipated).- Thin tendon in proportion to strength of muscles (low stiffness, k)

∴ Large stretch.∴ Large energy storage.

- Calf muscles do not have to lengthen and shorten as much, or as fast.∴ Use more economical muscle type.

Ligaments in arch of foot:

- Store 17 J of elastic strain energy- 80% elastic recoil (20% dissipated).

Running shoes:

- Compress 10 mm- Store 7 J of elastic strain energy- 50-70% elastic recoil (30-50% dissipated)

2=,

21

=,=2

2

kF

EkxEkxF


2.6 JUMPING

Mode of locomotion used:- To capture prey.- Escape predators.- For locomotion in trees.

Peak height of jump:(conservation of energy)

Where

h = Increase in height of COG from take-off topeak of jump (m).

v = Velocity of COG at instant of take-off (m/s)m = Mass of animal (kg).

KEPE ∆=∆2

21

= mvmgh

gv

h2

=2


Work performed by animal during take-off:

(assuming velocity at start of take-off = 0)

where

F = Average take-off force (N).s = Take-off distance (m).

For geometrically similar animals:- Body mass, m ∝ (length)3

- Take-off distance, s ∝ length- Max take-off force, F ∝ Cross-sectional area

∝ (length)2

∴ Height of jump h ∝ -(length)∝ -(mass)1/3

- Jump height decreases with increasing size- Maximum size of jumping animal.

PEKEdymgF ∆=∆=∫ ).-(

mghmvsmgF =21

=)-( 2

smgFs

h -=∴


Ground reaction force generated in take-off depends on:- Moment of muscle force about joint- Shortening speed of muscle (less force exerted as shortening speed increases).

Stretch-shorten cycle:- Higher jumps when preceded by motion in opposite direction.- Advantage due to:

- Muscle pre-tensing- Enhanced force-velocity (stretching of muscle fibres).- Return of elastic energy stored in stretched tendon and muscle.


Jumping by Small Animals

For small animals:- The effect of aerodynamic drag is not negligible (work done against drag during flight

phase).- For a vertical jump:

Where Fdrag ∝ Frontal area Sf

For same take-off velocity- Smaller animals cannot jump as high.

(Fdrag / m ∝ Sf / m is larger)

∴More energy lost to aerodynamic drag).

dragFmgdtdv

m --=

m--= dragF

gdtdv



Jump by extending legs suddenly.- Body accelerates (uniformly) from rest to take-off velocity, v- Acceleration path, s ≤ length of legs, l

(Time taken to extend legs)

However, for small animals- Maximum contraction rate of muscles limits the jump height.

Flea jumping:- No known muscle can make an isolated contraction in a few milliseconds.- Fleas use muscles to slowly store elastic strain energy in rubber-like protein (resilin) at

base of hind leg.- Acts as a spring (built-in catapult).- Resilin recoils rapidly (release more power than by muscle contraction).

tvu

s2+

=

2≤

vtl

vl

t2

≤

V(ms)

h(m)

s(m)

t(s)

0.4 0.27

0.05

0.006

0.16

0.005

0.45

2.0

0.13

3.0

6.0

1.6

Human

Bushbaby

Flea

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