Estimating dinosaur maximum running speeds
using evolutionary robotics
William Irvin Sellers
and Phillip Lars Manning
Lecturer in Integrative Vertebrate Biology, Faculty of Life Sciences, University of Manchester,
Jackson’s Mill, PO Box 88, Sackville Street, Manchester M60 1QD, UK
Lecturer in Palaeontology, School of Earth, Atmospheric and Environmental Sciences,
University of Manchester, Manchester M13 9PL, UK
Maximum running speed is an important locomotor parameter for many animals—predators as well as
prey—and is thus of interest to palaeobiologists wishing to reconstruct the behavioural ecology of extinct
species. A variety of approaches have been tried in the past including anatomical comparisons, bone scaling
and strength, safety factors and ground reaction force analyses. However, these approaches are all indirect
and an alternative approach is to create a musculoskeletal model of the animal and see how fast it can run.
The major advantage of this approach is that all assumptions about the animal’s morphology and
physiology are directly addressed, whereas the exact same assumptions are hidden in the indirect
approaches. In this paper, we present simple musculoskeletal models of three extant and ﬁve extinct
bipedal species. The models predict top speed in the extant species with reasonably good agreement with
accepted values, so we conclude that the values presented for the ﬁve extinct species are reasonable
predictions given the modelling assumptions made. Improved musculoskeletal models and better estimates
of soft tissue parameters will produce more accurate values. Limited sensitivity analysis is performed on key
muscle parameters but there is considerable scope for extending this in the future.
Keywords: locomotion; computer simulation; bipedalism; running
Chasing down prey is a vital factor in the lives of extant
predators, as is the avoidance of being captured for prey
animals. It is therefore of little surprise that speed
estimation is of such interest to palaeobiologists who
study dinosaurs. The range of predicted speeds is as
variable as the methods chosen with some authors
favouring high speeds (Paul 1988, 1998) while others
prefer moderate (Farlow et al. 1995) or low speeds
(Alexander 1989; Hutchinson & Garcia 2002). Current
analysis techniques are based on anatomical comparisons,
bone scaling and strength, risk factors and ground reaction
forces, and a recent review (Hutchinson & Gatesy 2006)
summarizes the current state of the art and concludes,
among other things, that a ‘rigorous dynamic simulation
of a moving dinosaur, one encompassing all motions and
forces, cannot yet plausibly be done’. However, such an
approach is clearly the best option because it both
explicitly requires a complete set of modelling assump-
tions and is conceptually simple. It requires a musculos-
keletal model to be constructed necessitating assumptions
about skeletal geometry, body mass and mass distribution,
together with muscle and tendon properties. These
properties all have a considerable effect on locomotor
performance and all that happens in any quantitative
technique that does not make explicit assumptions about
their values is that they will be implicitly scaled from
whatever reference species are employed. This is likely to
be suboptimal given that some of these values can often be
inferred from fossil evidence directly, and unknown values
need to be clearly identiﬁed so that their importance to the
ﬁnal result can be assessed.
It is now possible to produce highly detailed muscu-
loskeletal models and these can be used for fossil
locomotor reconstruction ( Nagano et al. 2005) and such
could be constructed for dinosaur species. However, such
complex models require a great deal of work to construct
and good predictive results can be obtained from rather
simpler models ( Ya m a z a k i et al. 1996; Ogihara &
Yamazaki 2001; Sellers et al. 2003, 2004, 2005). These
latter models have the great advantage of requiring no
knowledge of locomotor kinematics and allow the
generation of a range of gaits de novo, which maximize
or minimize some global optimization parameter such as
speed or energy consumption. These models use an evolu-
tionary robotics approach where so-called evolutionary
algorithms are used to ﬁnd muscle activation patterns
(Nolﬁ & Floreano 2000). However, such models are still
computationally demanding. It is perfectly possible to
solve the mechanical constraints of the system (internal
forces generated by muscles and the spring recoil of
tendons, segment movement constraints imposed by
joints, external forces generated by gravity and ground
reaction through contacts with the environment) in more
or less real time on a modern computer using freely
available software. However, ﬁnding the activation pattern
of the muscles that produces high-quality gait is extremely
challenging. Even a simple model such as ours with 12
muscles and 5 activation levels per step (half a gait cycle)
leads to 61 dimensions in the search space, which is
Proc. R. Soc. B (2007) 274, 2711–2716
Published online 21 August 2007
Electronic supplementary material is available at http://dx.doi.org/10.
1098/rspb.2007.0846 or via http://www.journals.royalsoc.ac.uk.
* Author for correspondence (firstname.lastname@example.org).
Received 23 June 2007
Accepted 24 July 2007
2711 This journal is q 2007 The Royal Society
consequently far too large to search exhaustively. For-
tunately, using a parallel implementation of a suitable non-
exhaustive, evolutionary search procedure such as a
genetic algorithm means that even this is a tractable
problem that can be solved in days or weeks on a modern
It is important that any model attempting to predict the
behaviour of extinct species be tested on extant animals.
Thus, a model attempting to predict the top speeds of a
range of fossil bipedal dinosaurs should also be tested using
equivalent data from living bipeds. However, there is
remarkably little high-quality top-speed data for any living
species other than those actively involved in racing, such as
humans, horses and greyhounds (Alexander 1989) and
even elephants (Hutchinson et al. 2006). The values used
for comparison in this paper (Alexander et al. 1979; Patak &
Baldwin 1998) are based on anecdotal observations and
need to be treated with some caution since there is no way of
knowing whether these animals were actually running as
fast as they could. The anatomy, posture and gait of bipedal
dinosaurs is unique, with no equivalent modern analogue
available for the comparison of locomotor abilities. The
successful testing and validation of the computational
approach using extant species indicates that it is possible to
generate a robust model applicable to extinct species.
2. MATERIAL AND METHODS
Our previous models (Sellers et al. 2003, 2004, 2005) used the
YNAMECHS simulation library ( McMillan et al. 1995).
However, this system has very limited support for contacts
between the simulation and the environment so, for this
model, we switched to using the open dynamics engine (ODE;
http://www.ode.org) to provide the physics simulation. This is
again a CCC library so we were able to modify our existing
AITSYM code to use the new simulator. The models
themselves were speciﬁed in a custom XML format that
deﬁned the necessary segments, joints, muscles, tendons and
contacts. To perform the optimization, we modiﬁed our
existing distributed genetic algorithm system to support the
new format. While it is difﬁcult to compare the results using the
two systems, it is certainly our impression that ODE is both
faster and more numerically stable than D
YNAMECHS in this
application. Communication between the optimization code
and the simulation code was performed using sockets via the
PTypes abstraction library (http://www.melikyan.com/ptypes)
to allow easy portability over the variety of computer systems
available to us, and to allow the code to take advantage of
multiple, remote computer clusters, using over 300 processor
cores when available. Graphical display and user interaction
used the GLUI OpenGL widget library (http://glui.
sourceforge.net/) to again allow easy portability. The muscle
model was derived from Minetti & Alexander (1997) to
generate both eccentric and concentric velocity-dependent
force characteristics extended with the addition of Hill-style
linear serial and parallel elastic elements (Hill 1938). The
combined musculotendinous unit is solved analytically and
exported as a standard C function using M
www.wolfram.com) that greatly aids numerical stability. A
cylindrical wrapping operator was used where necessary to
maintain the muscle and tendon path around joints.
A set of three extant and ﬁve extinct bipeds were modelled.
These are two-dimensional models with a rigid trunk, and left
and right thigh, shank and composite foot segments. These
segments are linked using three hinge joints per limb. The
segment properties are based on published data (Hutchinson
2004a,b) with the single composite foot segment combining
the metatarsus and foot with all ﬂoor contact occurring at the
distal end of the metatarsus. The species were chosen to cover a
reasonable size range and are listed in table 1. The published
dataset does not include moments of inertia so these were
calculated by modelling the segments as geometric shapes
chosen to match the published lengths, masses and centres of
mass. Conic segments were used for the limb segments and
back-to-back circular cones were used for the trunk. Muscle
ﬁbre lengths, physiological cross-section areas (PCSAs) and
moment arms were available for limb extensors (Hutchinson
2004a,b), and these were used to create appropriate muscle
paths on the model. Cylindrical wrapping operators were used
for the knee and ankle extensors. Limb ﬂexors were assumed to
have the same properties, but only 59% of the mass was based
on human proportions ( Pierrynowski 1995) but not dissimilar
from the values found in other species (Alexander et al. 1979;
Maloiy et al. 1979; Bennett 1996).
The extensor muscle mass was limited to 5% of the body
mass per joint (Hutchinson 2004b) and all muscles were
considered to act over a single joint, with each joint having a
single ﬂexor and extensor. Muscle volume was calculated
using the standard value of 1056 kg m
for muscle density
(Winter 1990) and PCSA was calculated by dividing this
volume by the ﬁbre length. Force per unit area was chosen to
be 300 000 N m
(Hutchinson 2004b), but there are other
values in the literature: Umberger et al. (2003) use
250 000 N m
, and Alexander (2003) reports an in vitro
maximum value of 360 000 N m
for frog and
330 000 N m
for cat for parallel-ﬁbred leg muscles.
Zheng et al. (1998) recommend a value of 400 000 N m
for human quadriceps and Pierrynowski (1995) suggests
350 000 N m
. There is a similarly large range for
maximum contraction speed. Winter (1990) suggests values
from 6 to 10 times the muscle’s resting length per second for
humans. This value is clearly highly dependent on both the
Table 1. Data for the species used in the study and key simulator outputs.
mass (kg) speed (m s
) cycle time (s) stride length (m) leg length (m) l/hu
Dromaius 27.2 13.3 0.441 5.865 0.864 6.79 20.89
Struthio 65.3 15.4 0.370 5.688 1.086 5.24 22.23
Homo 71 7.9 0.475 3.732 0.994 3.75 6.33
Compsognathus 3 17.8 0.106 1.894 0.179 10.58 180.43
Velociraptor 20 10.8 0.284 3.058 0.489 6.25 24.18
Dilophosaurus 430 10.5 0.579 6.092 1.350 4.51 8.36
Allosaurus 1400 9.4 0.615 5.795 1.750 3.31 5.17
Tyrannosaurus 6000 8.0 1.199 9.559 3.089 3.09 2.10
2712 W. I. Sellers & P. L. Manning Dinosaur running
Proc. R. Soc. B (2007)
ﬁbre-type composition of the muscle and the temperature.
Westneat (2003) reports a range of values for ﬁshes from 3 to
for different ﬁbre types and Umberger et al. (2003)
recommend values of 12 s
for fast twitch and 4.8 s
slow twitch. A value of 8 s
was chosen to represent a mixed-
ﬁbred muscle. Joint limits were set to extreme values for all
species since these limits are not generally available and the
range of motion permitted by the muscle and tendon lengths
speciﬁed should be sufﬁcient to limit joint excursion. Serial
and parallel elastic constants were set so that the serial
element strain was 6% at the maximum isometric contraction
and parallel element strain was 60% for the same force as
used in a previous model (Sellers et al. 2005), and the lengths
of the tendons were scaled based on average human
1995). Muscle attachment points were arranged so that the
muscle plus tendon was at its resting length in the published
mid-stance positions (Hutchinson 2004a,b), so that the
moment arm was approximately correct. This procedure
was performed automaticallyusingacustomM
program (http://www.mathworks.com/). The full speciﬁ-
cation for each of the models is included as human-readable
XML ﬁles as electronic supplementary material.
Gait is generated using a distributed genetic algorithm
optimization system using a genome that represents the gait
cycle duration and the muscle activation levels at 10 time
periods through the gait cycle. The genome contains 61
parameters representing the cycle time and the 5 activation
levels for 12 muscles for half a gait cycle. The left–right
activation levels are simply swapped for the other half of the
gait cycle. This is implemented as a client–server architecture
with the simulators running on multiple client machines and a
central server that gathers the results from the multiple
simulations. The starting conditions used published mid-
stance positions with a trunk forward velocity giving a Froude
number of approximately 1.5 (which equates to a medium
running speed) using the formula velocityZ1.5!(9.81!
(thigh lengthCshank lengthCmetatarsus length)) following
Alexander (2003) but using leg length as a proxy for hip
height. Forward velocities for the support leg segments
decreased to zero depending on the height of the segment
from the ground, and the velocities of the swing-leg segments
increased to double the trunk forward velocity again
depending on height. The ﬁtness criterion used for the
genetic algorithm optimization was the maximum forward
distance achieved in a ﬁxed time (3 s for most species but 5 s
for the Allosaurus and Tyrannosaurus to allow a reasonable
number of complete gait cycles). Thus, runs where the animal
fell over scored very badly and the runs with the highest
average speed would score the highest. The population was
1000 and up to 1000 generations were allowed in each run
unless a steady maximum average forward velocity was
achieved earlier. This procedure was repeated at least ﬁve
times until a good quality run was obtained. Runs were
judged to be good quality when the animal did not fall over
within the time limit and managed at least 15 m forward
( f )
2.5 3.0 3.5 4.0 4.5 5.0
3.0 3.5 4.0 4.5
1.0 1.5 2.0 2.5 3.0
4 6 8 10121416
Figure 1. Overlay images of the simulated gaits. All scales are in metres and 10 images are generated per gait cycle. (a) Dromaius;
(b) Struthio ;(c) Homo;(d ) Compsognathus;(e) Velociraptor;(f ) Dilophosaurus; ( g) Allosaurus;(h) Tyrannosaurus.
Dinosaur running W. I. Sellers & P. L. Manning 2713
Proc. R. Soc. B (2007)
movement. The best run was then used as the basis for a gait
morphing procedure (Sellers et al. 2004), where the best
results for a previous run were used to generate the starting
conditions for subsequent runs. This procedure was repeated
at least 20 times to obtain the highest speed estimate for each
species. An individual simulation ran in approximately real
time but at least 1 000 000 repeats were needed to generate
the optimized running gait for each species.
All the models generated high-quality running gaits that
were stable over the whole simulation period as shown in
ﬁgure 1. This ﬁgure shows a series of snapshots of the
generated gaits at 1/10 cycle time intervals. The lines
connect the joint centres, except for the foot line that is
drawn between the ankle joint and the contact point on
the metatarsal head and the trunk line that is drawn from
the hip joint to the position of the centre of mass of the
The top speeds achieved for each species are shown in
table 1 and there is reasonably good correspondence for
the extant species between the speeds generated by the
model and those cited in the literature: Dromaius 14 m s
(Patak & Baldwin 1998); Struthio 17 m s
et al. 1979); and Homo 10 m s
(Alexander 1989). To
investigate whether mass alone is a good predictor of top
speed, we plotted the top speed against body mass in
ﬁgure 2, which shows a monotonic reduction in speed for
the extinct bipeds but not for the extant species. The
extinct species have a consistent 15% of body mass as
muscle mass for each limb, whereas the value for Dromaius
is 15.5%, for Struthio is 12.2% and for Homo is 10.6%, so
this alone does not explain the speed differences seen in
the extant species.
Similarly, there is a well-known empirical relationship
between speed, leg length and stride length based on
Froude numbers that is often used for trackway analysis:
, where l is the stride length; h is a
characteristic length (in this case, leg length); u is the
forward velocity and g is the acceleration due to gravity
(Alexander 1976). This information is shown in table 1
where the characteristic length used is leg length (the sum
of thigh, shank and metatarsus lengths). Figure 3 shows a
plot of l/h against u
/gh for the models to compare the
simulations against the empirical relationship demonstrat-
ing extremely good agreement.
A key area of uncertainty in our model is the choice of
muscle parameters, so a simple sensitivity analysis was
performed. The maximum force available per muscle is
directionally proportional to the mass of muscle (F
dM/rl, where d is the force per unit area; M is the muscle
mass; r is the muscle density and l is the muscle ﬁbre
length). Of these the greatest uncertainty is likely to be the
value of muscle mass (Hutchinson & Garcia 2002); hence,
this was varied over a range of 2.5–7.5% per joint, which
should encompass the probable variation. Similarly, V
known to vary widely and it may scale negatively with body
mass (Medler 2002), although there is little information
about contraction velocities in large animals. Values
between 4 and 12 s
were tested to both encompass
the probable range and give the same degree of variation
(G50%) as used for muscle mass. The results of the
sensitivity analysis on the Tyrannosaurus rex model are
shown in ﬁgure 4. Both parameters have an approximately
linear effect over this range, and the effect of muscle mass is
approximately double the effect of contraction velocity.
However, the model was unable to sustain locomotion
when the muscle mass was reduced to 2.5% per joint.
Multibody dynamic simulations using evolutionary
robotic optimization approaches appear to provide reliable
estimates for the maximum running speeds of extant
animals. Multiple simulations with small changes in both
starting conditions and muscle activation patterns pro-
duced highly consistent estimates. Maximum running
speed is a highly variable character and difﬁcult to estimate
in any animal. The observed speed is always a lower bound
estimate of maximum speed since the animal might not be
running as fast as it can. In addition, many quoted values
for running speed are based on observations made in less
than ideal conditions, which may lead to considerable
body mass (kg)
eed (m s
Figure 2. Graph showing the body mass and top speeds of the
1 10 100 1000
Figure 3. Graph showing the Froude number, stride length
relationship for the simulations and also the empirical
relationship (Alexander 1976).
2714 W. I. Sellers & P. L. Manning Dinosaur running
Proc. R. Soc. B (2007)
errors (Garland 1983; Alexander 1989). Even for humans,
the situation is not straightforward: while 200 m sprint
averages are in excess of 10 m s
, the peak speed reached
can be in excess of 12 m s
(Brown et al. 2004), and these
are values for elite athletes who have considerably greater
leg muscle mass than the average values used in our
simulations. Tests on more general female athletes from
other sports give typical speeds of approximately 6 m s
with short bursts of less than 8 m s
(Brown et al. 2004).
Our estimates are broadly in line with other biomechanical
estimation techniques, which predict 18 m s
and 13 m s
for emu (Blanco & Jones 2005). It is self-
evident (and has been demonstrated in various models;
Hutchinson & Garcia 2002; Sellers & Paul 2005) that
changes in muscle mass will affect maximum speed and
this is a major source of uncertainty in these predictions.
However, we would propose that limits can be set using
functional bracketing to set minimum and maximum values.
The 15% value used here is similar to that found in large
extant bipeds and as our sensitivity analysis shows, an
appreciably smaller value does not even allow the animal
to walk. An upper limit would be harder to estimate but
one approach that is possible (even if computationally
expensive) is to calculate the bone loading during high-
speed locomotion and compare the safety factors with
those known for extant animals (Alexander 1997). For
these models, the general decrease in top speed as body
size increases is certainly in line with predictions made
elsewhere (Hutchinson 2004b) and the predicted top
speeds, with the possible exception of 17.8 m s
Compsognathus, are not exceptional. However, we would
expect that the actual top speeds are likely to be somewhat
higher than those given here since it is probable that
improvements could be made by altering the distribution
of the leg muscle and optimizing the ﬁbre and tendon
lengths. In addition, all the models are relatively simplistic
and lack multiple-joint muscles and accessory elastic
storage structures, which when combined should increase
the maximum speed. Our initial sensitivity analysis shows
that changing our assumptions about the muscles has a
considerable effect on our estimates. A 50% increase in
muscle mass leads to a 60% increase in top speed and a
50% decrease stops the model working at all. Similarly, a
50% increase in maximum contraction velocity leads to
a 30% increase in top speed with a 50% decrease leading
to a 20% decrease. However, there is considerable scope
for further sensitivity analyses such as variation in muscle
attachment points (Sellers & Crompton 2004) and the
effects of perturbation ( Wilson et al. in press).
The kinematics generated by the models show a wide
range of variation. It is very hard to judge the probable
accuracy of these values without subject-speciﬁc kinematic
matching, and in any case with such simple models it may
be optimistic to expect high-quality kinematics. In
particular, we would expect better estimates of muscle
and tendon lengths (and to a lesser extent joint limits) to
have a considerable effect on kinematics. However, as
shown in ﬁgure 3, the mechanical accuracy of the
simulations is high. The models represent a wide range
of Froude numbers and follow the predicted stride length
relationship very closely indeed. Overall, simulations such
as ours illustrate how an animal could have moved given its
physiological and morphological constraints, and perhaps
also indicate probable movement patterns, but we are still
some way off saying that this is how it must have moved.
There are a number of possible future developments in
animal gait simulation. Our models already allow us to
calculate the energetic costs of gaits in some detail. This
includes estimates of metabolic and mechanical power,
including active contraction and passive elasticity contri-
butions on a muscle by muscle basis. However, these
predictions need better validation and this can be achieved
using a combination of species-speciﬁc modelling and
in vivo physiological measurement. As computers get
faster and cheaper, it would be useful to increase the
bioﬁdelity of the models by adding more muscles,
including the forelimb in the simulation, and having
considerably greater detail in the feet and the interactions
with the substrate. In addition, reconstruction improve-
ments such as new estimates of inertial parameters
(Hutchinson et al. 2007) and better comparative ana-
tomical and physiological data will certainly help. In
particular, very little comparative data are available on the
elastic properties of muscles. Similarly, the addition of a
feedback-based control system will increase the range of
locomotor activities that can be encompassed and allow
this technology to look at non-continuous activities, such
as starting, stopping, cornering and complex movements
such as sitting and standing up.
We would like to thank Prof. Robin Crompton for allowing
access to his Beowulf Cluster to perform the required
simulations; Cliff Addison for arranging access to the
NW-Grid; funding from BBSRC, NERC and the Leverhulme
Trust; and two anonymous referees for their helpful
comments on the original draft of the manuscript.
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Figure 4. Graph showing the effects of altering the maximum
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