Locomotion

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Locomotion
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Dinosaur Locomotion

• We would like to estimate how fast dinosaurs
  could move.
• Solving the problem is straightforward if we
  assume that dinosaurs are like other terrestrial
  animals with erect gaits.
• First, we observe there is a relationship between
  how fast you walk or run and how long your stride
  is.

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Dinosaur Locomotion

• Increasing your speed results in increased stride
  length.
• We can estimate the relationship via regression
  for any animal. Of course, all dinosaurs are
  dead, so we will have to do it indirectly.

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Dinosaur Locomotion

• Begin by estimating the relative stride length of
  a dinosaur.
• Sr Stride Length / Leg Length

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Dinosaur Locomotion

• We expect animals of different sizes to have
  equal relative stride lengths when running at
  equivalent speeds.
• That is, if we compare a human adult with a human
  child, their relative stride lengths should be
  the same when they are running at equivalent
  speeds. However, 5mph for an adult is not the
  same as 5mph for a child.

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Dinosaur Locomotion

• What is the equivalent speed? We call it the
  dimensionless speed. This comes from
  shipbuilding, since it is not practical to make
  full size models to test various hull designs.
• Sd S / (Hull Length g) ½

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Dinosaur Locomotion

• In this equation, Sd is dimensionless speed, S is
  speed, Hull length is self explanatory, and g is
  gravitational acceleration (9.81m/s/s).
• Imagine we wish to build a ship with a hull
  length of 300m, and we expect to operate the ship
  at 15m/s. We build a model which is 5m long.

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Dinosaur Locomotion

• At what speed should the model be tested to give
  results equivalent to those of the actual ship?
• 15 / (300 10)½ 0.27
• The dimensionless speed is 0.27.

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Dinosaur Locomotion

• For our model, we have
• x / (5 10) ½ 0.27
• and solving for x yields an equivalent speed of
  1.9m/s.

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Dinosaur Locomotion

• This is called dynamic similarity, and is used by
  the motion picture industry when they explode
  model buildings or cars. The same general
  approach applies to animals. In our case, we
  use
• Sd S / (Leg Length g) ½

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Dinosaur Locomotion

• Now, we assume that animals move in the most
  efficient or most economical way possible, and
  thus we expect dynamic similarity.
• Therefore, we expect them to use equal relative
  stride lengths.

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Dinosaur Locomotion

• If we compute relative stride lengths for a large
  variety of animals ranging in size from dogs to
  elephants, and regress this against dimensionless
  speed, we get a very tight straight line fit, as
  per the figure.

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Dinosaur Locomotion

• If we look at a large variety of articulated
  dinosaur skeletons, and compare leg length with
  foot length, we find that leg length is about 4
  times the foot length.
• Now, assume we have a dinosaur with a foot length
  of 0.64m. We then expect leg length to be 4 x
  0.64 2.56m.

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Dinosaur Locomotion

• We measure stride length (the distance from the
  impression of one foot print to the next
  impression of the same foot) and determine it to
  be 3.31m.
• Relative stride length is then
  3.31 2.56 1.3

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Dinosaur Locomotion

• Now, using the graph, we see that a relative
  stride length of 1.3 should result in a
  dimensionless speed of about 0.4.

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Dinosaur Locomotion

• For our dinosaur with a leg length of 2.56m, we
  get
• x / (2.56 10) ½ 0.4
• and solving for x yields a speed of about 2m/s.

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Dinosaur Locomotion

• Compare this with a human with much shorter legs,
  who reaches 2m/s (4.5mph) easily in a brisk walk.
• Some dinosaur data are given below. These data
  show that dinosaurs for the most part, were
  pretty slow. This is in stark contrast to views
  held by some that dinosaurs were rapid,
  aggressive, and fleet. It appears that only the
  smaller forms could run.

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Dinosaur Locomotion

• Track Leg L Speed Gait
• Large Theropod 2.0 2.2 4.9 walk
• Large Theropod 2.6 2.0 4.5 walk
• Small Theropod 0.13 3.0 6.7 run
• Small Theropod 0.22 3.5 7.8 run
• Small Theropod 1.0 3.6 8.1 run
• Large Sauropod 3.0 1.0 2.2 walk
• Small Sauropod 1.5 1.1 2.5 walk
• Ornithopod .14 4.3 9.6 run
• Ornithopod 1.6 4.8 10.7 run

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Dinosaur Locomotion

• There are some tracks which show running and
  indicate speeds of about 12m/s. However, the
  larger of these animals weighed about 0.6 metric
  tons. This is actually quite fast. Humans run
  at about 10m/s, race horses at about 17m/s, and
  various antelopes run at 14m/s. However, note
  that both of these theropods are not particularly
  large.

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Dinosaur Mass

• How can we estimate the weight of a dinosaur?
• This is actually very simple. We know that mass
  increases as a cube, while the strength of the
  appendages increases only as the square.
• If we plot total circumference of the limbs
  against body mass in metric tons for a variety of
  animals, we get the following

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Dinosaur Mass

• You can see that the relationship is actually
  guite tight. In fact, the regression is
• Kg 0.000084 (total circumference in mm)2.73

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Dinosaur Mass

• Masses (tonnes) of dinosaurs
• Theropods
• Tyrannosaurus rex 4.5 - 7.7
• Allosaurus fragilis 1.4 - 2.3
• Sauropods
• Apatosaurus louisae 33.5 - 37.5
• Brachiosaurus brancai 31.6 - 87.0

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Dinosaur Mass

• Ornithopods
• Iguanodon bernissartensis 5.0 - 5.4
• Anatosaurus copei 3.4 - 4.0
• Stegosaurs
• Stegosaurus ungulatus 2.0 - 3.1
• Ceratopians
• Triceratops prorsus 6.1 - 9.4

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Turtle Locomotion
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Turtle Locomotion

• Stride Frequency of strides / unit time.
  This refers to a particular foot.
• Stride Length Distance traveled in a single
  stride (distance from footprint to footprint made
  by the same foot).

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Turtle Locomotion

• Duty Factor for a foot, the fraction of time
  for which that foot is on the ground. Eg., a
  walking man has a duty factor of 0.6 for each
  foot, ?, 0.60.61.2, and ? 20 of the time, both
  feet are on the ground. A running man may have a
  duty factor of .3 for each foot, ? 0.30.30.6
  and so for 40 of time, both feet are off the
  ground.?

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Turtle Locomotion

• A gait with a duty factor gt .5 is called a walk,
  if duty factor lt .5 it is a run. (Trots,
  Canters, and Gallops are all running gaits).
• Relative Phase of a foot is time at which it is
  set down, expressed as a fraction of the stride.
  Eg., for a walking man, 1st foot start of
  stride and RP of second foot .5

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Turtle Locomotion

• Problem for a slow quadruped
• You can maintain perpetual equilibrium during a
  slow walk if you move only 1 foot at a time so
  that you always have 3 feet on the ground.
• ? average of Duty Factors must be at least 0.75.
• Also, it must move its feet in such an order that
  its center of mass is always over the triangle of
  support.

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Turtle Locomotion

• Tortoises have DF 0.8, so they should use gait
  B or C in the figure on the next slide.
• But, they actually use gait D, and at times
  only 2 feet are on the ground. ? they can not
  maintain perpetual equilibrium and they rise,
  fall, pitch, and roll as they walk.

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Turtle Locomotion

• Why not maintain perpetual equilibrium?
• Answer their muscles are too slow.
• If muscles are slow, what gaits are possible?
• Minimize range of ht. thru which center of mass
  rises and falls.
• Minimize range of ?s thru which shell pitches.
• Minimize range of ?s thru which shell rolls.

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Turtle Locomotion

• We need to understand how turtle feet work in
  terms of applying force to the ground.
• Let a foot be on the ground from time -T/2 to
  time T/2. At any time t while it is on the
  ground, let is exert a vertical force given by
• F A cos (? t/T) where A is a constant.

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Turtle Locomotion

• We can use this model to estimate a wide variety
  of gaits using different duty factors and
  relative phases.
• Using this model, only a very few gaits were
  possible. All others had unacceptable pitch,
  roll, or height fluctuation. The best possible
  gait was found to be that in E, which is not
  that different from D.

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Turtle Locomotion

• If we modify our model (make it more realistic)
  we get
• F A (cos (? t/T) r sin (2? t/T))
  where both A and r are constants.
• Using this model, we get an exact fit between
  tortoise gaits and prediction of the model (gait
  D).

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Energetics of Terrestrial Locomotion and Body Size

• Some general relationships
• metabolic power increases linearly with speed.
• Power is the rate at which work is done. So,
  metabolic power is the rate at which metabolic
  work is done.
• In other words, regardless of how fast an animal
  moves, oxygen consumption changes linearly with
  speed, not curvilinearly.

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Energetics of Locomotion Size

• Note that as mass increases, gram specific oxygen
  consumption decreases. That means, it costs less
  for a big animal to move than for a little animal
  to move.
• Note also the difference between lizards,
  mammals, and birds.
• There are 2 organisms that dont fit Humans and
  kangaroos.

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Energetics of Locomotion Size

• The human pattern is a consequence of the way
  bipeds walk. Our center of mass dips and rises
  with each stride. We have one optimal speed
  (1.1-1.7m/s) where cost of transport is minimal.
• Kangaroos use a neat trick with their ligaments.

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Energetics of Locomotion Size

• Cost of transport changes in a regular fashion
  with body size.
• We have already seen that metabolic power
  increases more rapidly in small animals than in
  large ones.
• Since metabolic power increases linearly with
  speed, the rate of increase (slope) of the
  relationship between between rate of oxygen
  consumption and speed is constant for each animal.

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Energetics of Locomotion Size

• This rate has the units J/kg m, and is the amount
  of energy required to move one unit of mass one
  unit of distance. The slope represents an
  incremental cost of transport.
• When these slopes (incremental costs) are plotted
  as a function of body mass on a logarithmic
  scale, we get a straight line.

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Energetics of Locomotion Size

• We can use this to generate a simple equation for
  predicting the incremental cost (slope) from body
  mass
• T 10.7 m -0.4 where m mass in kg and T is
  incremental cost in J / kg m.
• Notice that the relationship for lizards is not
  different from that for quadrupedal mammals.

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Energetics of Locomotion Size

• What happens when the animals speed 0?
• This is the resting metabolic power and the
  postural cost of locomotion (the cost of standing
  up).
• To calculate the total power input af an animal
  running at a given speed, we need both the slope
  and the y intercept. The y intercept is greater
  than resting metabolism. The difference is the
  postural cost.

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Energetics of Locomotion Size

• For mammals, resting metabolic power can be
  predicted via
• W 3.5 m -.25
• What is the relationship for lizards?
• W (0.013 m .80 ) 10 .3 T

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Energetics of Locomotion Size

• Can we predict power used for running from body
  mass and speed?
• SURE ! It is quite simple
• Metabolic power equals slope times speed plus the
  y-intercept.
• W run-4 (T ? V) 1.7 ? W std-4
• where V is speed in m/s. Using out previoius
  equations, we get the following

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Energetics of Locomotion Size

• W run-4 (10.7m-0.4 ? V) 1.7 ? 3.7m -0.4

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