(2) BIOMECHANICS of TERRESTRIAL LOCOMOTION
Questions:
-
How does size influence the mode
and speed of locomotion ?
-
What determines the energy cost of
locomotion ?
-
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
-
Duty factor of each foot, β, (fraction of duration of stride in which a foot is on ground).
= v / λ (Hz).
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
-
-
2/gl
Froude number, v
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).
Biomechanics of Terrestrial Locomotion 2-3
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)
v = λ. f
≈(lθ ).
1
2π
g
l
v∝ l
Biomechanics of Terrestrial Locomotion 2-4
Stepping
Frequency
(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.
Biomechanics of Terrestrial Locomotion 2-5
Physical pendulum model (a more general model of the leg).
Stride frequency =
where
m
d
I
1
2π
mgd
I
≡
mass of leg
≡
distance from c.g. to axis of rotation
≡
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.
∴f =
For mass on string
I = ml2, d = l
for uniform rod
I = (1/3)ml2, d = (1/2)l
∴f =
Same result:
walking velocity
1
2π
1
2π
g
l
3g
2l
∝ l
Biomechanics of Terrestrial Locomotion 2-6
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.f
λ ∝ (length)
where
and it may be shown that for a forced pendulum
f ∝ (length)-1
v ∝
∝
(length) . (length)-1
(length)0
Result:
Maximum running speed is independent of leg length (l).
Biomechanics of Terrestrial Locomotion 2-7
Advantage of long legs (in geometrically similar animals):
- Walk faster
- Same maximum running speed
Biomechanics of Terrestrial Locomotion 2-8
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 =Iα
I ∝
∝
∝
Where
(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)
∝
∝
α ∝
∝
(length)2 . (length)
(length)3
(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
Biomechanics of Terrestrial Locomotion 2-9
= λ / f ):
Increase stride frequency, f, by reducing moment of inertia of legs:
To increase running speed ( v
-
-
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 running
human
8 - 12 m/s
greyhound
15 - 16 m/s
horse
16 - 17 m/s
-
(bipedal)
energy stored in tendons of lower back reduces metabolic energy cost of
locomotion.
Biomechanics of Terrestrial Locomotion 2-10
Gait change:
- Mammals change gait at equal Froude numbers.
Walk → Trot
≈ 0.5
Trot → Gallop ≈ 2.5
-
Gait changes to minimize the energy cost locomotion.
Walk
Trot
Gallop
Biomechanics of Terrestrial Locomotion 2-11
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%)
λ
≈2.3 v 2 gl
l
(
)0.3
Describes walking and running of terrestrial mammals (humans, dogs, horses, rodents,
kangaroos), flightless birds and dinosaurs.
Biomechanics of Terrestrial Locomotion 2-12
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
-
For geometrically similar animals:
Max. muscle force
Max. stress in muscle
-
∝
∝
gravitational force
mass
∝ Cross-sectional area
∝ (length)2
∝ (mass)2/3
∝ (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.
Biomechanics of Terrestrial Locomotion 2-13
Cat
Elephant
Biomechanics of Terrestrial Locomotion 2-14
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).
Biomechanics of Terrestrial Locomotion 2-15
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
Biomechanics of Terrestrial Locomotion 2-16
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.
Biomechanics of Terrestrial Locomotion 2-17
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.
Biomechanics of Terrestrial Locomotion 2-18
Compare the efficiency of “Groucho” running on bent legs, like apes.
Biomechanics of Terrestrial Locomotion 2-19
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
during locomotion
=
total metabolic rate -
resting (postural)
metabolic rate
Energy cost of transport:
-
Metabolic energy required to transport unit mass of animal per unit distance [ J/(kg.m) ]
energy cost
of transport
= total energy cost
-
energy cost of
standing still
For walking/running, energy cost of transport is constant (independent of velocity)
Biomechanics of Terrestrial Locomotion 2-20
As velocity increases (↑) (i.e. increasing Froude number
cost of transport associated with:
-
v2 / gl ) the component of the energy
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 for
mammals 10-100 kg (smaller
mammals have a higher cost
of transport)
Biomechanics of Terrestrial Locomotion 2-21
Metabolic Power Consumption vs Running Speed
Approximately linear relation between metabolic power consumption and running speed.
P(v) ≈ P(0) + C.v
where
P(v)
P(o)
C.v
C
≡ rate of metabolic energy consumption when running at speed v.
≡ rate of metabolic energy consumption when standing.
≡ extra rate of energy use (power) for running at speed v.
≡ 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
Biomechanics of Terrestrial Locomotion 2-22
Energy cost of transport
(running mammals).
C
∝
C/m ∝
(mass)0.68
(mass)-0.32
Large animals are
more economical.
Explanation is uncertain:
-
Possibly due to energy
cost of exerting force
is more important than
cost of performing work.
FLYING
RUNNING
SWIMMING
Biomechanics of Terrestrial Locomotion 2-23
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)
Biomechanics of Terrestrial Locomotion 2-24
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)
Biomechanics of Terrestrial Locomotion 2-25
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 during
walking from EMG studies).
-
Adjust stride length with walking speed
to maintain a passive leg swing.
Biomechanics of Terrestrial Locomotion 2-26
Maximum Speed of Walking
Walking model:
-
Assume all body mass located at hip.
-
Body on straight leg behaves like an inverted
pendulum (mass mounted on top of the
pendulum).
-
Torque exerted at hip.
-
Need to maintain contact with the ground
with at least one leg at all times (feet cannot
pull down on the ground).
Requires that (centripetal force) ≤ (body weight)
During the motion of the c.g. along the arc of a
circle to avoid flying off at tangent to arc.
Biomechanics of Terrestrial Locomotion 2-27
Leg length l, walking speed v;
mv 2
≤mg
l
v2
≤1
(Froude number) =
gl
∴ vmax = gl
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.
Biomechanics of Terrestrial Locomotion 2-28
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
Biomechanics of Terrestrial Locomotion 2-29
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.
Biomechanics of Terrestrial Locomotion 2-30
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.
Biomechanics of Terrestrial Locomotion 2-31
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).
Biomechanics of Terrestrial Locomotion 2-32
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
v2
gl
≈ 0.7
(Froude number)
i.e. when v ≈ 2.5 m/s for adult human.
Human walking: Human running: -
More economical than walking for an animal of similar size.
Relatively uneconomical.
Biomechanics of Terrestrial Locomotion 2-33
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.
2
1 2
F
F = kx, E = kx , E =
2
2k
-
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)
Biomechanics of Terrestrial Locomotion 2-34
2.6
JUMPING
Mode of locomotion used:
- To capture prey.
- Escape predators.
- For locomotion in trees.
Peak height of jump:
(conservation of energy)
∆PE = ∆KE
1
mgh = mv 2
22
v
h=
2g
Where
h
v
m
=
Increase in height of COG from take-off to
peak of jump (m).
= Velocity of COG at instant of take-off (m/s)
= Mass of animal (kg).
Biomechanics of Terrestrial Locomotion 2-35
Work performed by animal during take-off:
(assuming velocity at start of take-off = 0)
∫( F - mg ).dy = ∆KE = ∆PE
1
( F - mg ) s = mv 2 = mgh
2
Fs
∴ h=
-s
mg
where
F
s
=
Average take-off force (N).
=
Take-off distance (m).
For geometrically similar animals:
- Body mass,
-
Take-off distance,
-
Max take-off force,
∴
-
Height of jump
m ∝ (length)3
s ∝ length
F ∝ Cross-sectional area
∝ (length)2
h ∝ -(length)
∝ -(mass)1/3
Jump height decreases with increasing size
Maximum size of jumping animal.
Biomechanics of Terrestrial Locomotion 2-36
Biomechanics of Terrestrial Locomotion 2-37
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.
Biomechanics of Terrestrial Locomotion 2-38
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:
m
Where Fdrag
dv
= -mg - Fdrag
dt
Fdrag
dv
= -g m
dt
∝ 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).
Biomechanics of Terrestrial Locomotion 2-39
Jump by extending legs suddenly.
- Body accelerates (uniformly) from rest to take-off velocity, v
-
Acceleration path, s ≤ length of legs, l
u +v
t
2
vt
l≤
2
2l
t≤
v
s=
(Time taken to extend legs)
However, for small animals
- Maximum contraction rate of muscles limits the jump height.
V
h
s
t
(ms)
(m)
(m)
(s)
Human
3.0
0.45
0.4
0.27
Bushbaby
6.0
2.0
0.16
0.05
Flea
1.6
0.13
0.005
0.006
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).
Biomechanics of Terrestrial Locomotion 2-40