Prediction of kinetics and kinematics of running animals using an analytical approximation to the planar spring-mass system
1 Structure and Motion Laboratory, The Royal Veterinary College, North
Mymms, Hatfield, Hertfordshire AL9 7TA, UK
2 Centre for Human Performance, University College London, Brockley Hill,
Stanmore, Middlesex, HA7 4LP,
* Author for correspondence (e-mail: awilson{at}rvc.ac.uk)
Accepted 28 September 2005
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Summary |
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Key words: spring-mass, running, locomotion, force
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Introduction |
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The spring-mass model has been shown to be an accurate tool for modelling
trotting, running and hopping over a range of species differing dramatically
in terms of body size and shape, including cockroaches, quail, dogs, humans,
horses and kangaroos (McMahon and Cheng,
1990; Alexander,
1992
; Blickhan and Full,
1993
; Bullimore and Burn,
2002
). Although the simplicity of the spring-mass system suggests
that the mathematics of its mechanics would be straightforward, this is not
the case: the motion is complex and cannot be solved using a simple analytical
solution. Horizontal and vertical COM deflections, velocities and
accelerations interact to make analytical solutions very difficult to
describe. Indeed, the situation is formally classed as a `non-integrative
Hamilton equation', which indicates that an explicit analytical solution does
not exist (Whittackler, 1904
;
Schwind and Koditschek,
2000
).
A number of approximations to the spring-mass system have been made.
Schwind and Koditschek conducted a thorough mathematical investigation of this
system and produced complex estimates of COM position in a two degrees of
freedom monopod. Simpler analyses included consideration of specific points in
the stance phase (Farley et al.,
1993; Ferris et al.,
1998
), small angle sweep assumptions
(Geyer et al., 2005
) and
numerical iterations that search for a symmetrical solution in terms of motion
of the stance phase (Blickhan,
1989
) or mapping of successive maximum heights
(Seyfarth et al., 2002
).
Numerical step-by-step solutions, although computer-intensive, do generate
accurate answers. However, they do not provide an intuitive relationship
between the five input variables. The advantage of an analytical solution is
that it is fast and intuitive; it also allows the inter-relationships between
variables to be immediately identified.
A relatively simple solution that can elegantly predict the COM trajectory and limb kinetics produced by the spring-mass model is desirable. Arguably an exact answer is not required, as in the real world limbs are obviously not mass-less, perfectly elastic springs of fixed stiffness even in steady, level locomotion, muscles are required to provide power to replace energy dissipated because of hysteresis in tendons or other loaded structures. The leg is not necessarily the same effective length (i.e. the length between the hip and the toe) at foot-on to foot-off due to plantar flexion at the end of stance. Therefore, we are interested in determining whether a simple spring-mass model and related mathematical approximations are adequate in modelling running in real animals.
In steady state bouncing gaits such as running and trotting, the motion of
the COM can be described in two components. In the vertical component, the
downward vertical velocity peaks at foot-on and gradually reaches zero by
mid-stance. In the second half of stance the velocity is reversed and peaks
again at foot-off. The differential of this motion, acceleration, would
approximate to a half sine wave. As acceleration is directly proportional to
force, then vertical ground reaction force (GRF) can be approximated to a half
sine wave (Cavagna et al.,
1964,
1977
;
Alexander et al., 1979
;
Full et al., 1991
;
Farley et al., 1993
;
Witte et al., 2004
). In the
horizontal component, the COM first decelerates from foot-on to a minimum
velocity at midstance before being accelerated forward in the second half of
stance. It is therefore reasonable to consider the horizontal acceleration
curve of the mass during the stance phase as a full negative sine wave, since
this approximates the shape of the horizontal forces measured using force
plates.
We hypothesize that an analytical approximation to the spring-mass model based on sine wave assumptions matches both numerical solutions of the model across the biological range and data collected from real animals.
In this paper, the accuracy of a number of analytical solutions based on
sine wave approximations is assessed in three ways. Firstly, they are compared
to a numerical solution produced by a computer simulation. Secondly, they are
compared to data collected in this study from a Standardbred racehorse
trotting over a range of speeds. Thirdly, a comparison is made with previously
published data of a man running, dog trotting and a kangaroo hopping from
McMahon and Cheng (1990) based
on the data of Cavagna (1988
).
The validity of the sine wave assumptions across the biologically relevant
parameter space are investigated.
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Materials and methods |
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The vertical orientation
During the gait cycle in steady state locomotion, the impulse due to
gravity on the centre of mass (i.e. the product of body mass m,
gravitational constant g and stride time) is balanced over the
stride by the vertical impulse generated by the leg during stance. If the
vertical GRF of animals is modelled as a half sine wave
(Alexander et al., 1979), the
vertical force experienced by the COM can be calculated. The sine wave that
would represent the vertical GRF is of the form Fy
=Fmaxsin(at), where Fy is the
vertical force, Fmax is the peak force, t is time
and 2
/a is the period of the wave. The area under this curve, or
the vertical impulse produced by the leg during contact time,
Tc, is therefore:
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The integral of velocity is position, so the vertical position is:
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The horizontal orientation
The slowing of the mass during the first half of stance in the horizontal
direction means that an analysis based on a constant horizontal velocity will
underestimate stance time and the vertical impulse generated. The horizontal
acceleration of the mass can be shown to approximate to a full negative sine
wave. The horizontal velocity of the animal decreases from foot-on until a
minimum velocity is reached at midstance before increasing throughout the
second half of the stance phase. As velocity is the integral of acceleration,
this is consistent with a full negative sine wave for acceleration. Since
force is directly proportional to acceleration, and the horizontal force trace
measured by a force plate during steady state locomotion resembles a full
negative sine wave with a total impulse of zero, this also supports the
approximation of the horizontal acceleration of the COM to a full negative
sine wave.
So, in our analytical approximation, the horizontal acceleration trace in
terms of time Ax(t) follows the curve:
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In addition to this, the area under the horizontal velocitytime
trace during stance phase will be equal to the total horizontal distance
travelled:
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Horizontal forces Fx can be calculated from horizontal
acceleration using Newton's second law of motion, F=ma:
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Production of numerical solutions to the planar spring-mass model
We created simulations using a computer software model (MSC
visualNastran4D, MSC.Software, California, USA run through Matlab via
Simulink, The MathWorks Inc, MA, USA). The simulation consisted of a mass
above a perfect linear spring tethered to the ground at coordinates (0,0). The
simulation starts with the same initial conditions as for the above
mathematics and stops when the tension in the spring returns to zero at the
end of stance phase. Adjustments were made to spring stiffness or horizontal
velocity until the simulation produced a symmetrical trajectory for the mass.
The resulting stance time was used as the input for the mathematical
calculations. All simulations were run at an integration rate of step size
0.0002 s. Halving the step size to 0.0001 s did not change the required spring
stiffness or horizontal velocity, expressed accurate to 4 decimal places.
Comparison of the analytical and numerical solutions
Absolute and relative differences for all kinematic variables (horizontal
and vertical position, velocity and acceleration) as well as resultant leg
force and mechanical energy were compared between 17 numerical simulation
solutions and the corresponding analytical solutions. For the computer
simulations, and those obtained from the biological data, the same initial leg
length, mass, leg stiffness and initial vertical velocity were used. Initial
leg angle was varied from 2.5° to 60° and the required initial
horizontal velocity was found so that a symmetric solution was produced. The
inputs for the analytical solutions were the constant input parameters stated
above and initial horizontal velocity and contact times that were obtained
from the output of the computer simulations.
To compare the results of the analytical and numerical solutions across the biologically relevant limb angles (1530°), each variable was plotted with respect to stance time, where zero represents midstance.
Percentage differences for each variable were calculated using [1(variableanalytical/variablenumerical)]x100, where the values for each variable were either the maximum values (cases Vy, Ax, Ay, F) or minimum values (cases y, Vx). In the case of x position and total mechanical energy, this value was calculated for every time point and the maximum value was taken.
Vertical stiffness, Kvert, was calculated by:
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Comparison to biological data
To determine how accurately the analytical and numerical solutions reflect
biology, data were collected from one trotting Standardbred racehorse. Data
collection took place during an exercise session on a training track, with the
horse pulling an exercise sulky driven by an experienced driver. Data were
collected across a range of trotting speeds. Footfall data (foot-on and
foot-off events) were measured using ±50 g solid-state capacitive
accelerometers (ADXL150, Analog Devices, Norwood, MA, USA) as described in
Witte et al. (2004). Each
accelerometer was encased with epoxy-impregnated Kevlar fibres (total mass 2
g) and attached to the dorsal midline of the hoof of each forelimb using hot
glue (Bostik Findley Inc., Stafford, UK) so that the sensitive axis was in the
proximo-distal direction. Each accelerometer was attached to a telemetry
transmitter and battery via a short fatigue-resistant cable. The
transmitter and battery were securely attached over a protective pad over the
lateral aspect of the third metacarpal bone using a custom modified exercise
bandage. Output signals were telemetered using custom programmed FM radio
telemetry devices (ST/SR500, Wood and Douglas Ltd., Tadley, Hampshire, UK) and
logged at 1000 Hz via a 12-bit A/D converter and PCMCIA card (DAQcard
700, National Instruments, Austin, TX, USA) into a laptop using custom-made
software (Matlab, Natick, MA, USA). Speed was measured using a WAAS-enabled
GPS receiver (G30-L, Laipac Technology, Richmond Hill, Ontario, Canada)
attached to the sulky and sampled at a frequency of 1 Hz. This system is
accurate to within 0.2 m s1 for 57% of all samples, as
described in Witte and Wilson
(2005
).
Forelimb leg length was measured using a tape measure and recorded accurate to the nearest cm. Limb length measurements were taken when the horse was standing square. Forelimb length was taken as the vertical distance between the ground and the approximate insertion of serratus ventralis. When standing the limb is loaded to approximately 30% body weight. This loading introduces a small error that is not significant.
Foot-on angle for each stride was calculated by assuming symmetry of the
stance phase about the vertical in terms of leg length (leg length) and angle,
such that foot-on angle, 0 is:
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Trot is a symmetrical `2 beat' gait where the diagonal limbs can be
considered to operate as a single virtual leg. A computer simulation was made
by using average input variables for the horse trotting at 7 m
s1. The input variables were Vxi,
Vyi, 0, mass and leg length. The spring
stiffness was adjusted using a custom made software (MSCvisualNastran4D run
through Matlab via Simulink) until a symmetrical stance phase was
produced in which the final force in the leg spring was 0 N and the vertical
height of the COM was equal at foot-on and foot-off. The resulting calculated
spring stiffness represents the combined leg stiffness of the fore and
hindlimb.
Comparison to previously published data
The mathematical results were compared to previously published data of
Cavagna et al. (1988) published
by McMahon and Cheng (1990
).
These consisted of a step cycle from a running man, a hopping kangaroo and a
trotting dog. Absolute and percentage differences between the numerical and
analytical solutions are reported for peak vertical force, peak h
(decrease in vertical height from the foot-on position) and k. The
analytical solutions were also superimposed over the original data of Cavagna
et al. Data from the analytical solutions were normalised using the same
method as McMahon and Cheng: vertical accelerations were normalised by
dividing by accelerations due to gravity (9.81 m s2) and
forces divided by body weight to obtain dimensionless horizontal force
(Fx/mg) and dimensionless vertical force
(Fy/mg). Dimensionless time was calculated
as
0t where
0=(vertical leg
stiffness/m)0.5.
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Results |
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Fig. 6 show that the simulations very closely matched the stance time and initial contact angle data obtained from the racehorse over the entire speed range.
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Fig. 8 compared the normalised vertical acceleration against displacement traces for the trotting dog and the hopping kangaroo. There are small differences between the methods but the results are very similar.
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Only an h value was obtained for comparison with the analytical and numerical solutions for the kangaroo. Values of 7.7 cm, 6.4 cm and 6.9 cm (difference from the published data of 1.3 cm and 0.8 cm; 13% and 10.5% difference, respectively) were obtained for the published data, the analytical solution and the numerical solution, respectively.
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Discussion |
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Sine wave approximations produce results similar to numerical simulations and biological data
This study has shown that a simple analytical solution, requiring a small
number of easily measurable input values, can predict the kinetics and
kinematics of a numerical solution to the planar spring-mass system over the
range of biologically relevant sweep angles and horizontal velocities. For
initial leg angles of between 17.5° and 30° the percentage differences
between the analytical solution and the numerical solution for the majority of
variables were less than 1% (total ME, COM position, total limb
compression and COM velocity) and a difference of less than 2% was obtained
for total limb force and vertical acceleration of the COM. Less accurate
results were obtained for horizontal acceleration, with up to 31% difference
between the analytical and numerical solutions.
The analytical solution was most accurate for leg contact angles of between
17.5° and 30°, which represent the range of angles used by running
animals including the racehorse in this study
(Fig. 6B). Running animals
typically have an initial leg angle of approximately 17.520° when
they make the transition from walking to running
(Blickhan, 1989;
Lee and Farley, 1998
), and a
typical contact angle of 3035° at their maximum running speed
(Farley et al., 1993
;
Alexander and Jayes, 1983
;
Gatesy and Biewener,
1991
).
The analytical and numerical solutions matched the data obtained from our
trotting racehorse across the whole horizontal speed range. The racehorse in
this study used contact angles ranging from 15° at 3 m
s1 to 32° at 11 m s1. Stance time and
duty factor data matched the predicted values. The value of combined limb
stiffness (i.e. the combined stiffness of the forelimb and hindlimb) required
for the numerical calculations was 80.9 kN m1. McGuigan and
Wilson found an overall forelimb leg stiffness of 55 kN m1
in horses of similar size to the one in this study
(McGuigan and Wilson, 2003).
If 57% of the mass of the horse were carried by the forelimb in a trot stance
phase (Witte et al., 2004
)
then the total forelimb and hindlimb stiffness would be 96 kN
m1, a value approximately 20% higher than the value used in
this study. However using the isometric scaling relationship of
kleg
Mb0.67, where
Mb is body mass
(Farley et al., 1993
), which
was based on number of species ranging from kangaroo rat (0.112 kg) to horse
(135 kg), the combined leg stiffness for the racehorse in his study would be
57.9 kN m1. This is approximately 30% less than the value
than estimated in this study; however, the horse used in this inter-species
scaling study was considerably smaller (135 kg vs 426 kg), had a
shorter leg length (0.75 m vs 1.43 m) and trotted at a slower speed
(approximately 3 m s1 against 7 m s1). Our
estimate of limb stiffness is therefore within the range of previously
published values.
Both analytical and numerical solutions predicted the changes in biological
vertical stiffness across the trotting speed range. A change in leg angle from
25° to 30° results in an increase in Kvert of
201%. This is of a similar magnitude to the 170% increase for a smaller
trotting horse reported by Farley et al.
(1993). Therefore the
analytical approximation not only matches the numerical solutions, but can
also represent the mechanics of a complex biological system the
trotting horse, and can be expected to be appropriate for other symmetrical
bouncing gaits.
The solutions were also similar to other previously published models and
kinetic and kinematic measurements. Peak limb forces predicted by Alexander et
al. (1979) were very similar to
the peak forces produced by the analytical and numerical solutions. Our
solutions have similar accuracy to the analytical solution reported by Geyer
et al. (2005
), who report a
less than 1% error for total leg compression and 0.6° difference for
angular motion compared to a numerical solution (based on a simulation with
input consistent with a running human). Our analytical solution produced
errors of less than 1% for total leg compression and a maximum difference in
leg angle less than 0.1° across the entire biological range; however, our
solution uses different assumptions and cannot be considered an equivalent
approach. Comparison of our analytical and numerical solutions with the
previously published results of McMahon and Cheng based on the biological data
of Cavagna et al. showed that the solutions were nearly indistinguishable for
the running man: maximum force and limb stiffness differed by less than 2% and
vertical height decrease (h) differed by 0.2%. The results for the
dog were similar: a difference of less than 5% in leg stiffness and a
difference of 0.1 cm for vertical compression (as measured from their figure)
were detected between the two methods and the published result. The kangaroo
data provided the poorest fit both for McMahon and Cheng's model and for the
analytical and numerical solution, and possible explanations are discussed
below.
While previous methods remain valuable, the advantages of our analytical
solution are twofold. Like, and indeed following, Alexander's sine wave
approximation, which just considers vertical GRF, it is very simple
(Alexander et al., 1979), but
our addition of the horizontal component allows for calculation of
fluctuations in velocity and mechanical energies.
Limitations
Both the numerical and sine wave analyses are limited to gaits that are
similar to bouncing springs. In bipeds these are hopping and running (not
walking and probably not skipping). In quadrupeds these are trot, pace and
pronk (not walk and possibly not gallop). Both analyses are more accurate if
the limbs behave as simple compression springs of constant stiffness. In the
numerical solutions the leg stiffness (the gradient of the peak force against
peak compression curve shown in Fig.
9) remains constant for all simulations. For the analytical
solutions stiffness remains constant over the most of the solutions and the
range of angles used by animals and only increases between the two most
extreme initial conditions of 45° and 60° due to inaccuracies in peak
force estimation at these extreme contact angles
(Fig. 9).
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The analytical solution simplifies the motion of the spring mass model by considering the horizontal and vertical components of motion separately, thus the force vector does not have to remain in line with the virtual leg. However, the differences between the leg angles calculated using the numerical solution and those calculated using the analytical solutions were very similar across the biological range (Fig. 10A), with a maximum difference of less than 0.10°. A maximum difference in leg angle of less than 0.24° occurred between the two solutions for all simulations across the entire range of angles (Fig. 10B).
|
For instance, although the velocity amplitude through stance is greater in
the vertical than the horizontal components
(Fig. 4), the kinetic energy
fluctuations are much greater in the horizontal direction. This is because a
small change in velocity around a large mean velocity involves a large change
in kinetic energy as it is the difference in velocity squared. Therefore
horizontal components dominate changes in COM mechanical energy during
running. For an animal of a specific leg length and mass using a simple
bouncing gait the leg stiffness determines the relationship between speed and
initial leg angle. High stiffness legs use small contact angles. Conversely,
more compliant limbs require larger contact angles. High stiffness limbs are
beneficial since they result in small changes in mechanical energy and hence
low hysteresis losses, since biological springs do not return all energy
stored in them (the potential energy storage in the spring leg can be
calculated as 0.5kc2, and can also be calculated using our
method). Stiff legs, however, result in brief contact times and high forces.
These high forces require a stronger leg (which may be difficult to protract
quickly) or a reduced safety factor and hence an increased risk of injury.
There may also be a detrimental effect on locomotor and muscle efficiency due
to shorter stance times (Kram and Taylor,
1990). If maximum contact angle determines maximum running speed
then stiff legs will also enable a higher maximum speed. This trade-off
explains why, in systems where force is a minor issue, high-stiffness,
vertically orientated limbs are preferred, whereas in biological systems,
where there is a structural and energetic cost to `force generation', the
compromise solution tends towards a more compliant limb. Therefore our simple
analytical approximation allows us to explore intuitively the consequences of
observed and postulated strategies of locomotion. Compliant limbs may, of
course, also carry benefits in control of locomotion under variable
conditions.
Future directions
Apart from using these methods to determine forces and energies from the
measurements as outlined above, we plan to develop these approximations
further to provide insight into asymmetrical gaits, specifically skipping and
galloping. We foresee that these approaches will provide alternative but
convergent conclusions to the developing collision-based models
(Ruina et al., 2005). Although
understanding of galloping is improving quickly, the consequences and
desirability of this gait are certainly not yet fully understood. Simple
optimisations providing conclusions about gait parameters including footfall
sequences will have to take account of both force and energetic costs. As
discussed above, energy loss minimisation and peak force minimisation have
quite different requirements. While collision-based models may be effective in
determining energetic consequences of different gaits, spring-based models
will be required if force consequences are to be understood.
Conclusion
This study has shown that a simple analytical solution matches the
kinematics and kinetics of the motion of a spring-mass system, accurately
modelling the kinetics of running and trotting. Limb force, leg stiffness and
changes in mechanical energy can be determined for these gaits from a few
easily obtainable, morphological and kinematic observations. This will allow
key energetic consequences of observed locomotion to be determined in the
field and without the need for force plates. In addition this allows
consideration of the consequences of postulated morphologies and gait
strategies.
The authors wish to thank Henry Wilson and Thilo Pfau for their technical help, Jim Usherwood and Anna Wilson for help with manuscript preparation. J.R. is a BBSRC funded student and A.W. a BBSRC Research Fellow and holder of a Royal Society Wolfson Research Merit award.
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