RESEARCH ARTICLE
Elastic energy savings and active energy cost
in a simple model of running
Ryan T. SchroederID1*, Arthur D. Kuo1,2
1 Faculty of Kinesiology, University of Calgary, Alberta, Canada, 2 Biomedical Engineering Program,
University of Calgary, Alberta, Canada
* ryan.schroeder@ucalgary.ca
Abstract
The energetic economy of running benefits from tendon and other tissues that store and
return elastic energy, thus saving muscles from costly mechanical work. The classic
“Spring-mass” computational model successfully explains the forces, displacements and
mechanical power of running, as the outcome of dynamical interactions between the body
center of mass and a purely elastic spring for the leg. However, the Spring-mass model
does not include active muscles and cannot explain the metabolic energy cost of running,
whether on level ground or on a slope. Here we add explicit actuation and dissipation to the
Spring-mass model, and show how they explain substantial active (and thus costly) work
during human running, and much of the associated energetic cost. Dissipation is modeled
as modest energy losses (5% of total mechanical energy for running at 3 m s-1) from hyster-
esis and foot-ground collisions, that must be restored by active work each step. Even with
substantial elastic energy return (59% of positive work, comparable to empirical observa-
tions), the active work could account for most of the metabolic cost of human running (about
68%, assuming human-like muscle efficiency). We also introduce a previously unappreci-
ated energetic cost for rapid production of force, that helps explain the relatively smooth
ground reaction forces of running, and why muscles might also actively perform negative
work. With both work and rapid force costs, the model reproduces the energetics of human
running at a range of speeds on level ground and on slopes. Although elastic return is key to
energy savings, there are still losses that require restorative muscle work, which can cost
substantial energy during running.
Author summary
Running is an energetically economical gait whereby the legs bounce like pogo sticks. Leg
tendons act elastically to store and return energy to the body, thus saving the muscles
from costly work with each running step. Although elasticity is known to save energy, it
does not explain why running still requires considerable effort, and why the muscles still
do substantial work. We use a simple computational model to demonstrate two possible
reasons why. One is that small amounts of energy are lost when the leg collides with the
ground and when the tendons are stretched, and muscles must restore that energy during
PLOS COMPUTATIONAL BIOLOGY
PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1009608
November 23, 2021
1 / 22
a1111111111
a1111111111
a1111111111
a1111111111
a1111111111
OPEN ACCESS
Citation: Schroeder RT, Kuo AD (2021) Elastic
energy savings and active energy cost in a simple
model of running. PLoS Comput Biol 17(11):
e1009608. https://doi.org/10.1371/journal.
pcbi.1009608
Editor: Barbara Webb, The University of Edinburgh,
UNITED KINGDOM
Received: May 10, 2021
Accepted: November 2, 2021
Published: November 23, 2021
Peer Review History: PLOS recognizes the
benefits of transparency in the peer review
process; therefore, we enable the publication of
all of the content of peer review and author
responses alongside final, published articles. The
editorial history of this article is available here:
https://doi.org/10.1371/journal.pcbi.1009608
Copyright: © 2021 Schroeder, Kuo. This is an open
access article distributed under the terms of the
Creative Commons Attribution License, which
permits unrestricted use, distribution, and
reproduction in any medium, provided the original
author and source are credited.
Data Availability Statement: The source code and
data used to produce the results and analyses
presented in this manuscript are available from the
Bitbucket Git repository: https://bitbucket.org/hbcl/
runoptsol/src/main/.
steady running. A second reason is that muscles may perform work to avoid turning on
and off rapidly, which may be even more energetically costly. The resulting muscle work,
while small in quantity, may still explain most of the energetic cost of running. Economy
may be gained from elasticity, but running nonetheless requires muscles to do active
work.
Introduction
Running is distinguished by the spring-like, energy-saving behavior of the stance limb [1–4],
analogous to a pogo stick (Fig 1). It is modeled well by a classic analogy, the Spring-mass
model, where the limb acts elastically to support and redirect the body center of mass (CoM)
between flight phases, and all mechanical energy is conserved throughout each step. This sim-
ple model can reproduce the motion and forces of running remarkably well and explains how
series elastic tissues such as tendon can improve running economy. It applies to bipeds and
even polypeds, making it one of the most universal and elegant models for running. However,
it does not include muscles that actively contract against series elasticity, and it fails to explain
Human
Spring-
mass
B.  Force
C.  Power
D. V. work loop
V. disp. (cm)
V. GRF (Mg)
Time (s)
0
0.1
0.2
0.3
0
2
4
Power (kW)
0
0.1
0.2
0.3
-2
2
0
Time (s)
V. acc. (g)
-6
-1
0
1
2
0
3
-3
Ground Slope
Cost of Transport
-1.2
E.  Cost vs. slope
-0.4
0
0.4
0.8
1.2
0
0.4
0.25
Human
Spring-
mass
Touchdown
Take-off
A. Trajectory
Stance
Flight
Point-mass
body
Spring-like
stance leg
Foot-ground
collision
Fig 1. Human running and the Spring-mass model. (A) Human stance phases resemble motion of a spring-mass system with no
energy loss, alternating with parabolic Flight phases. Body mass is lumped into a single point center of mass (CoM). Traces of (B)
vertical ground reaction forces (V. GRF) vs. Time, (C) leg Power vs. Time, (D) and vertical acceleration vs. displacement (V. acc. vs. V.
disp.; termed vertical work loop curve) are all shown for both human data (gray lines) and the Spring-mass model (dark solid lines). (E)
The energetic Cost of Transport (cost per unit weight and distance) for humans running on slopes (after [11]) is not explained by the
Spring-mass model, which only operates at zero ground slope and zero energetic cost. The spring-like behaviors (B-D) should be
regarded as pseudo-elastic, because humans and other animals experience dissipation such as in foot-ground collision, and thus, require
active muscle actuation.
https://doi.org/10.1371/journal.pcbi.1009608.g001
PLOS COMPUTATIONAL BIOLOGY
A simple optimization model of running
PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1009608
November 23, 2021
2 / 22
Funding: This work was funded by the Dr. Benno
Nigg Research Chair (ADK) and the National
Sciences and Engineering Research Council of
Canada (https://www.nserc-crsng.gc.ca/; NSERC
CRC, Tier 1 to ADK; NSERC Discovery to ADK). The
funders had no role in study design, data collection
and analysis, decision to publish, or preparation of
the manuscript.
Competing interests: The authors have declared
that no competing interests exist.
the substantial metabolic cost measured during running. An extension of the Spring-mass
model to include active actuation may help explain organismal running energetics and how
best to exploit series elasticity for economy.
The Spring-mass model agrees well with a wide body of experimental evidence [4]. It repro-
duces mechanical characteristics such as the body’s trajectory in space (Fig 1A), ground reac-
tion forces (Fig 1B), leg mechanical power (Fig 1C), and even the leg’s vertical work loop curve
(vertical acceleration vs. vertical displacement similar to an elastic spring’s work loop, Fig 1E;
[4]). The spring can passively store and return mechanical energy to the CoM, reducing the
active work otherwise required of active muscle, and thus, improve running economy. Some
have therefore proposed that more compliance or longer tendons are key to running economy
[5,6]. For example, the energetic cost of human running is less than half of that expected if
muscles alone performed work on the CoM [7]. In turkeys, tendon contributes over 60% of
the shortening work performed by the lateral gastrocnemius [8]. Although the simple Spring-
mass model (Fig 2A) applies mainly to bipedal running or polypedal trotting, multiple leg
springs can reproduce galloping, and indeed, practically all of the running gaits observed in
nature [9,10]. Few other models reproduce so many behaviors with such simplicity.
There are also important aspects of running not captured by the Spring-mass model. A crit-
ical feature is metabolic energy expenditure by muscle [12], considered important for selecting
gait and speed [13,14], and more generally for a variety of animal behaviors [15]. The Spring-
mass model is conservative of mechanical energy and predicts no such expenditure. Lacking
muscle actuation, it is incapable of accelerating from rest or running on sloped ground. Steeper
slopes in particular have energetic costs approaching that expected from muscles performing
positive and negative work against gravity at their respective efficiencies (Fig 1E; [11]). Even
steady running on the level entails substantial muscle shortening work, as shown in turkeys
(e.g., 40% of muscle-tendon work; [8]), and in human running [16,17]. Some of that work is
fundamentally necessary because of dissipation, for example by tendons with hysteresis (26%
loss per cycle in Achilles tendon during hopping; [18]), by the heel pad [19] and other soft tis-
sues that deform (33% per step of human running at 3 m s-1, [20]). Restoration of those losses
alone could account for up to 29% of the energetic cost of human running [20], and the overall
active work of muscle for as much as 76% of the energetic cost of human running [21]. The
spring-like mechanics of running (Fig 1B–1D) should therefore be regarded as pseudo-elastic,
as opposed to purely elastic. Beyond the conceptual illustration of energy savings, the Spring-
mass does not account for dissipation and is not predictive of actual energy costs observed in
nature.
Other simple models of running have included elements other than springs. Perhaps the
simplest of these has only an active, extending actuator (Actuator-only model, Fig 2B; [14]).
Minimization of its work alone is sufficient for both walking and running to emerge as optimal
gaits, with running more economical at faster speeds and exhibiting a pseudo-elastic, bounc-
ing-like stance phase despite no passive elasticity [14]. In addition to mechanical work as a
cost, we have proposed that muscles also expend energy for a force-rate cost, associated with
rapid production of force [22–25], which helps to explain human-like ground reaction forces
[26]. Others have optimized the actuation of robots, including both elasticity and dissipative
elements, and have shown a variety of running gaits to emerge [27,28]. For organisms, similar
models have been used to explore stability [29] and economical strategies [30] of terrain navi-
gation. Still, while dissipation has been characterized empirically (e.g., [31]), most running
energetics models have not included such dissipation with series elasticity and actuation to
explain energy cost.
Here we propose a running model that combines series elasticity with active actuation and
passive dissipation (Fig 2). The classic Spring-mass (Fig 2A) and pure Actuator-only (Fig 2B)
PLOS COMPUTATIONAL BIOLOGY
A simple optimization model of running
PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1009608
November 23, 2021
3 / 22
models serve as opposing reference points that can produce running mechanics (Fig 1A–1D)
with and without elasticity. We propose to combine the spring and actuator in series, along
with dissipation, in an Actuated Spring-mass model (Fig 2C) that may be more representative
of running in organisms. We expect that such a model will leverage series elasticity to perform
minimal work, as needed to restore dissipative losses. We test whether such a model is suffi-
cient to explain both the mechanics and metabolic cost of running (Fig 1E), with work mini-
mization as the sole objective, or work plus the proposed force-rate cost. Such a model may
help determine whether more compliant tendon is indeed economical [6], and provide insight
on the energy expended by muscle. We use human data for running at different speeds and
ground slopes as experimental comparison, but the principles revealed by the model are
intended to help explain running across a range of animal species.
Methods
We used dynamic optimization to determine optimal actuation strategies for the proposed
Actuator-Spring-mass running model. The model extends the classic Spring-mass model by
adding an active series actuator and dissipative losses (Fig 2C). The actuator can perform posi-
tive and/or negative work, in part to compensate for two modes of passive energy dissipation:
collision loss associated with foot-ground contact and hysteresis of the tendon spring. The
model was optimized for energy economy, as defined by the energetic costs of that active actu-
ator work, briefly summarized here.
Two of the most basic elements of the model are the mass and elastic spring. The point
mass M was supported by a spring with stiffness k, which was varied as a free parameter to pro-
duce a wide spectrum of gaits. These included low stiffnesses ranging from grounded running
with no flight phase, to more impulsive running with a brief stance period and a relatively long
flight phase. The limiting case of impulsive running has infinitesimal stance and an infinitely
stiff spring, where the body’s motion is almost entirely described by its parabolic trajectory
during flight.
M
F
g
Actuator-only
B.
M
g
k
m
LlLl
LlLl
C. Actuated Spring-mass
Spring-mass
A.
Actuator
Hysteresis
M
c
k
Fm
Lm
Lt
g
Collision
Fig 2. Simple running models with and without elastic spring, active actuator, and dissipative elements. (A) The
Spring-mass model comprises a point-mass body and a massless spring for a leg. (B) The Actuator-only model replaces
the spring with a massless, active actuator producing extension forces in the leg [14]. (C) The proposed Actuated
Spring-mass model combines an actuator and a spring (analogous to a muscle-tendon unit), along with two passive,
dissipative elements: a damper in parallel with the spring to model tendon hysteresis, and collision loss to model
dissipation of kinetic energy at touchdown. In the models, g is gravitational acceleration and M is body mass. Ll(t),
Lt(t), and Lm(t) are time-varying lengths of the leg, spring and actuator, respectively. Parameters k and c are spring
stiffness and damping coefficient. Fm(t) is the active actuator force in the leg’s extension direction.
https://doi.org/10.1371/journal.pcbi.1009608.g002
PLOS COMPUTATIONAL BIOLOGY
A simple optimization model of running
PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1009608
November 23, 2021
4 / 22
Two types of dissipation were included in the model, representing losses from collisions
and hysteresis. Collisions model the kinetic energy dissipated when the body impacts the
ground. For example, humans lose momentum associated with 2.6–7.8% body mass [31]. We
modeled this as a simple discontinuity in the CoM velocity vector magnitude at touchdown,
defining collision fraction (CF) as the fraction of momentum lost in the collision. A nominal
collision fraction of 3% resulted in a 5.9% loss in kinetic energy (see S1 Text for details).
Hysteresis was included to model the imperfect energy return of tendons and other series
elastic tissues. Hysteretic energy losses of 10–35% per stretch-shortening cycle have been esti-
mated in vivo for tissues such as the human Achilles tendon [32]. Estimates of soft tissue defor-
mation suggest that much of the actual dissipation occurs during the first half of stance [20],
modeled with a viscous damper (in parallel with the spring) only dissipating energy during
spring loading (Fig 2C). Damping was parameterized by damping ratio, z ¼ c=
ffiffiffiffiffiffiffiffiffi
4Mk
p
, where c
is the damping coefficient. A nominal damping value of z = 0.1 was selected to yield 26% hys-
teresis, roughly within the estimated range of humans.
Each stance phase was computed with dynamic optimization for energy economy. The
main control variable was the time-varying actuator length during the leg’s stance (specifically,
its third derivative L
. . .
mðtÞ was used to allow for calculations of force rate during implementa-
tion), treating the stance and swing phases as periodic and symmetric between legs. The objec-
tive function to be minimized was the energy cost per step E,
E ¼ EW þ ER;
ð1Þ
as the sum of a cost EW for work, and another cost ER for force rate. The work cost depends on
positive and negative efficiencies for muscle, η+ and η- respectively (25% and -120% from
[11]), defined as work divided by metabolic energy (superscript + or − for positive and nega-
tive work, respectively). The energetic cost was therefore defined by actuator work (Wm per
step, and power Pm for work per time, superscripts for positive and negative work),
EW ¼ R T
0
Pþ
m
Zþ the leg’s posture at touchdown and did not allow slipping of that contact. Model states
included the position (xb, yb) and velocity (_xb, _yb) of the point-mass body and the first and sec-
ond derivatives of the actuator’s length ( _Lm, €Lm) to facilitate inclusion of work and force-rate
in the objective function.
All optimizations were conducted with the MATLAB software GPOPS-II [42] and the
resulting nonlinear problem was solved using SNOPT [43]. All variables and equations were
non-dimensionalized with parameter combinations (L = 0.90 m, M = 70 kg and g = 9.81 m s2)
during optimization and outputs were subsequently re-dimensionalized as indicated in figures.
Further details regarding the model and implementation of the optimization problem can be
found in S1 Text.
Running parameters were chosen to represent human-like gait. Speed v was varied over a
range of 2–4 m s-1, with empirical preferred step frequency given by f = 0.26v + 2.17 [44]. Step
length s was defined as the distance travelled over one periodic step, and step frequency f as the
inverse of time duration, T, per step (i.e. T = 1/f), such that v = s f. Gaits were produced while
varying parameters such as tendon stiffness k (4.93–122 kN m-1 or equivalently, 6.46–160
MgL-1) and force-rate coefficient ε (0–210−2 M-1g-1.5L1.5). Furthermore, a single set of nominal
parameter values (force-rate coefficient ε, spring stiffness k and negative and positive work
efficiency η- and η+) were selected for comparisons with human ground reaction forces and
metabolic data (see Table 1).
For comparison with our model, we included representative human data to qualitatively
illustrate well-established patterns for ground reaction forces and other trajectories. The data
consist of one representative subject from a separate published study [45]: a male (25 years,
body mass = 75.3 kg, leg length = 0.79 m) running on an instrumented force treadmill at 3.9 m
s-1. An average step was determined from 20 s of steady-state ground reaction force data and
used in plots including CoM power [46] and vertical acceleration vs. displacement [2,4], as
comparison against the model. These plots reproduce patterns from accepted literature, and so
no statistical analysis was performed.
Results
Optimization results are presented in two parts, first examining the effects of individual model
components (Fig 2), and then combining them into a single, unified model. Part I presents
Table 1. Model parameters and values.
Symbol
Description
Range
Nominal
Units
η-
negative work efficiency
-1.05
η+
positive work efficiency
0.32
CF
collision fraction
0–0.06
0.03
z
damping ratio
0–0.2
0.1
k
spring stiffness
6.46–160
46.7
MgL-1
ε
force-rate coefficient
0–210−2
510−4
M-1g-1.5L1.5
v
running speed
0.67–1.35
1.01
g0.5L0.5
M
body mass
70
kg
g
gravitational acceleration
9.81
m s-2
L
maximum leg length
0.9
m
Parameters used in the Actuated Spring-mass model, along with the ranges of values examined for parameter sensitivity analysis, and nominal values for comparison
with human. Range is left empty if the parameter was not varied in optimizations. Units are left empty if the parameter has no units. The model was implemented in
normalized units, with M, g and L as base units (nominal human values shown).
https://doi.org/10.1371/journal.pcbi.1009608.t001
PLOS COMPUTATIONAL BIOLOGY
A simple optimization model of running
PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1009608
November 23, 2021
6 / 22
individual sensitivity studies, beginning with the Spring-mass and Actuator-only models,
which have been examined in prior literature: e.g. [2,4] and [26,36,40], respectively. Next, the
Actuated Spring-mass model is evaluated as an alternative, since it still uses a spring but also
requires an actuator to account for passive energy dissipation occurring with each step. Ini-
tially, this model is evaluated with the cost of work only (i.e. zero force-rate coefficient ε) over
varying speeds and spring stiffnesses. Next, non-zero force-rate coefficients ε are introduced
so changes in actuation strategies may be independently evaluated. Finally, in Part II, a single,
unified set of parameters is applied to the Actuated Spring-mass model, which is then used to
simulate gait over a range of running speeds, spring stiffnesses and ground slopes to assess its
utility in predicting locomotion energetics and optimal spring-actuator coordination patterns.
Part I: Individual model components and their contributions to running
behavior
Spring-mass and Actuator-only models produce similar pseudo-elastic running gaits.
Both models (Fig 3) can produce similar gaits, ranging from very flat to very bouncy. For the
Spring-mass model at a given speed and step length, the spring stiffness determines the gait
trajectory, as described by CoM trajectories, vertical ground reaction force profiles, mechanical
power profiles, and vertical work loop curves illustrating the spring-like leg behavior [4]. As
reported by others [3], the Spring-mass model can run with a vast range of stiffnesses k (Fig 3,
top). A less stiff (or more compliant) spring can produce grounded running, in which the
stance phase occupies the entire step (blue curves in Fig 3). With greater stiffness comes a flight
phase, yielding gaits that resemble more typical human running, where both stance and flight
phases are finite in duration (redder curves, Fig 3). These gaits generally include a single-
peaked ground reaction force profile, with higher peak forces and powers with increasing
Actuator-only
Spring-mass
V. acc. (g)
-5
V. disp. (cm)
0
2
4
6
8
0
5
V. disp. (cm)
-5
0
5
0
2
4
6
8
V. acc. (g)
V. GRF (Mg)
Time (s)
0
0.1
0.2
0.3
0
2
4
6
8
0
0.1
0.2
0.3
Time (s)
0
2
4
6
8
V. GRF (Mg)
Increasing
stiffness k
k=∞
low k
Power (kW)
0
0.1
0.2
0.3
-5
5
0
Time (s)
Power (kW)
0
0.1
0.2
0.3
-5
5
0
Time (s)
ε=0
high ε
min work &
force-rate (EW+ER)
A. Trajectories
B.  Force
C.  Power
D. V. work loop
Increasing
k
Increasing
k
Increasing
ε
Increasing
ε
Increasing
ε
one step
one step
Fig 3. Running gaits from the Spring-mass (top) and Actuator-only (bottom) models. They are illustrated by (A) center of mass (CoM) trajectory, (B)
vertical ground reaction forces vs. time, (C) leg power performed on the CoM vs. time, and (D) vertical acceleration vs. vertical displacement of the body
(or vertical work loop curve). A range of running gaits are shown, varying stiffness k in the Spring-mass model, and the force-rate cost coefficient ε in the
Actuator-only model, for a single running speed v (3.5 m s-1) and step frequency f (3 Hz). In the limiting case of infinite spring stiffness or zero force-rate
cost, touchdown forces become perfectly impulsive (red arrows).
https://doi.org/10.1371/journal.pcbi.1009608.g003
PLOS COMPUTATIONAL BIOLOGY
A simple optimization model of running
PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1009608
November 23, 2021
7 / 22
stiffness, as well as steeper vertical work loop curves during stance. In the limit toward infinite
stiffness, the model produces impulsive running [14], where flight takes up nearly the entire
step and stance occurs as an instantaneous impulse (red arrows, Fig 3). In all cases, the Spring-
mass model is purely elastic and has no actuator and no losses. Thus, no single spring stiffness,
and no single spring-like gait, can be considered beneficial over another in terms of energy
cost.
The Actuator-only model can produce a very similar range of running gaits, despite the
complete lack of an elastic spring (Fig 3, bottom). The optimization produces pseudo-elastic
behavior resembling a spring, and tuned by a single parameter: the force-rate coefficient ε.
With a coefficient of zero (i.e. work cost only), impulsive running is optimal, because least
work is performed with least displacement, albeit with infinite force [14]. A greater force-rate
cost results in increasing stance time and shorter flight time, more similar to humans. Increas-
ing that cost further eventually causes the flight phase to disappear, producing grounded run-
ning similar to a very compliant spring. For any non-zero force-rate cost, the model
consistently produces an approximately linear vertical work loop curve, similar to the Spring-
mass model. However, this behavior is purely active and requires substantial positive and neg-
ative work. As a result, the Actuator-only model does incur an energy cost for work but has no
passive elasticity to reduce that work.
We thus find that the two diametrically opposed models can reproduce the pseudo-elastic
behaviors similar to humans, whether or not there is true passive elasticity. There is certainly
strong evidence that elasticity is important for running in humans and other animals, but
spring-like ground reaction forces and vertical work loop curves are not necessarily indicative
that elasticity is the dominant mechanism in running. If it were, the energetic cost of running
might be expected to be close to zero. Conversely, the pure Actuator-only model also obviously
cannot demonstrate that humans are purely inelastic. The work performed by humans, if there
were no elasticity, would result in unrealistically high muscle efficiencies of at least 45% [7]. It
is more realistic to regard the human as having some combination of series elasticity and active
actuation, both contributing to the actual energetic cost of human running.
Dissipative energy losses require compensatory, active positive mechanical
work
We next consider the effect of passive energy dissipation in the Actuated Spring-mass model
(Fig 4), optimizing for the cost of work alone (with zero force-rate coefficient). Again, gaits
roughly similar to those of humans are produced, for either stiff or compliant springs. How-
ever, the actuator must perform positive work to restore the lost energy. With a stiff spring, it
is optimal to produce a relatively “bouncy” CoM trajectory where the body spends more time
in the air and thus, reaches greater heights above the ground (Fig 4A). The optimum also
favors a spikier vertical force and power over shorter stance durations (Fig 4B and 4C). Con-
versely, a compliant spring makes it optimal to produce a “flatter” CoM trajectory with briefer
flight time (Fig 4A), with lower peak vertical force, longer stance duration, and less stiff vertical
work loop curves (Fig 4D). The accompanying leg angle at touchdown also varies, with a more
vertical orientation for increasing spring stiffness.
With zero force-rate coefficient ε, it is generally optimal to perform only positive actuator
work. For steady motion, the energy lost to hysteresis and collision must be restored with an
equal magnitude of positive work, to yield zero net work per step. The optimization reveals
this is performed most economically in the second half of stance (Fig 4C, blue shaded areas),
in concert with elastic energy return from the spring. It is also generally economical to
completely avoid active negative work, which would also require an additional amount of
PLOS COMPUTATIONAL BIOLOGY
A simple optimization model of running
PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1009608
November 23, 2021
8 / 22
positive work to be performed. Thus, actuator work is minimized when it is performed only to
restore dissipative losses.
Dissipative losses are minimized by increasing spring stiffness with
running speed
We next evaluate a range of spring stiffnesses and running speeds to determine how the cost of
actuator work may be minimized (Fig 5). Here we find that optimal spring stiffness increases
with running speed (Fig 5A), while generally preserving the timing of positive work within the
second half of stance (Fig 5B). At lower speed v (2.5 m s-1; Fig 5A left), both hysteresis and col-
lision losses are reduced with less spring stiffness. However, at higher speed (3.5 m s-1; Fig 5A
right), hysteresis losses are reduced with greater stiffness whereas collision losses are relatively
unchanged. For a moderate speed (3.0 m s-1; Fig 5A middle) both hysteresis and collision trade
off over stiffness, and an intermediate stiffness is optimal. Overall, hysteresis losses change
mainly with spring stiffness, and collision losses increase mainly with running speed, so that
losses are generally minimized by a stiffness increasing with speed.
The collision and hysteresis losses have distinct dependencies on running speed and/or
stiffness. The model’s collision loss increases with touchdown velocity and thus, running
speed, but is relatively insensitive to spring stiffness. Stiffness does affect the CoM trajectory
and the distribution between horizontal and vertical velocity components but has relatively lit-
tle effect on the vector magnitude. Overall, collision losses increase with speed but are rela-
tively unaffected by spring stiffness (Fig 5).
In contrast, hysteresis loss occurs as a fraction of elastic strain energy, which is largely deter-
mined by the angle of the leg during stance. For example, a vertical leg posture is used in con-
junction with a stiff spring (Fig 4), and this results in greater strain (and hysteresis losses) to
redirect vertical CoM velocity of the bouncier gait. Alternatively, a less vertical leg posture is
used with a compliant spring and results in greater strain to redirect horizontal velocity of the
body. As such, stiff springs allow for efficient gait at higher speeds, since the vertical leg is effec-
tive at mitigating excessive strain to redirect high horizontal CoM velocity. At lower speeds,
compliant springs are better since a less vertical leg is better at mitigating higher vertical veloc-
ity associated with bouncy running at these speeds.
The overall effects of dissipation are as follows. At low speeds, both collision and hysteresis
losses are reduced with relatively low spring stiffness and a shallower leg touchdown angle,
A. Actuated Spring-mass
V. GRF (Mg)
Time (s)
0
0.1
0.2
0.3
0
2
4
6
8
B. Force
Power (kW)
Time (s)
0
0.1
0.2
0.3
-5
5
0
Actuator
Collision loss
Leg
C. Power
V. disp. (cm)
V. acc. (g)
-5
0
2
4
6
0
5
D. V. work loop
min work (EW),
zero force rate
Fig 4. Effects of stiff vs. compliant springs on Actuated Spring-mass model minimizing cost of work (with zero force-rate cost). Optimal
running gaits (speed v of 3.0 m s-1) are shown for the Actuated Spring-mass model, including (A) CoM trajectory, (B) vertical ground reaction
forces (V. GRF) vs. time, (C) leg power vs. time, and (D) vertical acceleration vs. vertical displacement (V. acc. vs. V. disp.). The model
includes passive dissipation (hysteresis and collision), optimized for two spring stiffnesses k (13.7 kN m-1 for compliant and 109.5 kN m-1 for
stiff). The stiffer spring yields a more vertical leg, shorter stance time and bouncier gait, with higher peak forces and leg power. Net actuator
work is similar in both cases.
https://doi.org/10.1371/journal.pcbi.1009608.g004
PLOS COMPUTATIONAL BIOLOGY
A simple optimization model of running
PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1009608
November 23, 2021
9 / 22
thereby also reducing actuator work. But at higher speeds, hysteresis losses are actually
reduced by greater spring stiffness and steeper leg touchdown angle, so that actuator work is
minimized with relatively high stiffness. These effects together cause the optimal spring stiff-
ness for minimizing actuator work to increase with running speed.
An added force-rate cost favors active actuator dissipation
We have thus far found that the model with zero force-rate cost avoids active negative work,
whereas some animal muscles are observed to perform non-negligible negative work during
running [5]. It may seem uneconomical to perform any amount of active negative work,
because it only adds to the costly positive work needed to restore the losses. Perhaps there is
some indirect energetic advantage to active negative work, not explained by a cost for work
alone. In fact, the addition of a force-rate cost with coefficient ε (Fig 6) makes it favorable for
the actuator to perform both negative and positive work. This distributes ground reaction
forces over a longer stance duration with lower peak forces and reduced force rate. However, it
also comes at the expense of additional actuator work, which is made worthwhile by its capac-
ity to reduce force rate (Fig 6). Overall, the force-rate cost yields less impulsive forces and
smoother CoM trajectories, at the expense of active dissipation and increased work.
Model Work per Step (x10-2)
101
102
0
2
4
6
8
10
3.0 m s-1
101
102
0
2
4
6
8
10
3.5 m s-1 
101
102
0
2
4
6
8
10
2.5 m s-1
10 J
Power P(t)
5 kW
Leg
Spring
Actuator
Collision
Hysteresis
Actuator
0.1 s
More work
Less work
Running Speed
Collision
Collision
Hysteresis
Actuator
Time
101
102
101
102
101
102
Spring Stiffness k (kN m-1)
Spring Stiffness k (kN m-1)
Spring Stiffness k (kN m-1)
A.
B.
min
work
Fig 5. Actuator work and power as a function of spring stiffness and running speed in the Actuated Spring-mass model, minimizing cost of work
(with zero force-rate cost). (A) Work vs. stiffness for speeds of 2.5–3.5 m s-1. Shown are active Actuator work (black), Collision work magnitude (red), and
Hysteresis work magnitude (blue). Spring diagrams (inset) illustrate touchdown angles for each stiffness. (B) Power vs. time for very compliant and very stiff
springs, for each running speed. Shown are net Leg power (black lines), Spring power (orange shaded area), Actuator power (blue shaded area). Results are
for spring stiffness ranging 13.7–109.5 kN m-1.
https://doi.org/10.1371/journal.pcbi.1009608.g005
PLOS COMPUTATIONAL BIOLOGY
A simple optimization model of running
PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1009608
November 23, 2021
10 / 22
The added force-rate cost yields a shift in timing of positive work so that it is optimally per-
formed late in stance. Experiments have shown that the triceps surae muscles undergo sub-
stantial shortening throughout stance, but particularly late in stance, during human running
[16,17]. On the other hand, negative work is optimally performed early in stance as ground
reaction forces rise [5,47].
Part II: Unified actuated spring-mass model of human-like running and
energy expenditure
We next apply the Actuated Spring-mass model with a single, unified set of parameter values
selected to produce human-like running in a variety of conditions. The full model therefore includes
an elastic spring, both hysteresis and collision losses, and an objective to minimize both work and
force rate costs. A single force-rate coefficient is selected (ε of 0.510−3) to approximately match the
model’s output to human data, along with stiffness k of 35.6 kN m-1, positive work efficiency η+ of
32%, and negative work efficiency η- of -105%. The resulting model, with parameters thus fixed, is
then applied to three comparisons with human data: mechanics of a nominal gait, energetic cost as
a function of running speed, and energetic cost as a function of ground slope.
Unified model produces human-like gait mechanics
The resulting model qualitatively matches the human CoM trajectory (Fig 7A), vertical ground
reaction forces (Fig 7B), leg power vs. time (Fig 7C), and vertical acceleration vs. displacement
10-2
Force-Rate Coefficient, ε     (M
–1g
–1.5L
1.5)
Energy Cost per Step
Power
V. GRF
V. disp.
10-3
10-4
10 cm
2 Mg
2 kW
0.1 s
Leg
Spring
Actuator
Work cost, EW
Collision
Total cost, EW+ER
2 J kg-1
Force-rate
cost, ER
Unified ε*
min work &
force-rate
(EW+ER)
Fig 6. Effect of work and force-rate costs on running using Actuated Spring-mass model. (top:) Vertical CoM displacement vs.
time, vertical GRF vs. time, and leg Power vs. time, for varying force-rate cost coefficient ε. (bottom:) Actuator work cost EW (thin
blue line), Total cost EW + ER (work and force rate, solid black line), and Force-rate cost ER (difference between lines) vary with
the coefficient. Impulsive actions (red arrows, V. GRF and leg Power) occur at Collision, and overall leg power (solid black line)
includes contributions from the Spring and Actuator. All solutions are shown for v of 3 m s-1 and f of 2.94 Hz.
https://doi.org/10.1371/journal.pcbi.1009608.g006
PLOS COMPUTATIONAL BIOLOGY
A simple optimization model of running
PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1009608
November 23, 2021
11 / 22
(Fig 7D). The model also reproduces some features that the classical Spring-mass model can-
not, such as the brief initial peak at touchdown (from collision [31], Fig 7B) and a more grad-
ual decrease in force than the increase (temporal asymmetry [30], Fig 7B). The model collision
produces a transient burst of negative power [46] followed by elastic energy storage and return
(about 53%; Fig 7C), as well as slightly non-linear vertical work loop curves (as in [4], Fig 7D),
qualitatively similar to human. Whereas the Spring-mass model produces a nearly linear curve
that retraces itself almost perfectly (Fig 1D), the human curve has an initial transient and self-
intersecting profile resembling a tilted figure-eight. The shape indicates hysteresis with a dissi-
pative counter-clockwise loop, followed by a (positive work) clockwise loop. The present
model crudely reproduces these broad features, even if imperfect in detail.
Unified model has increasing energy cost with speed for level running
The model’s energetic cost per time (Fig 8A) increases with running speed at a rate similar to
human data [48]. Here, the model’s step frequency was constrained to the empirical human
preferred step frequency, but other parameters were kept fixed. The increasing overall cost
with speed may be explained by the constituent force-rate and work costs (Fig 8B), evaluated
as a function of spring stiffness and speed. The work cost is considerably greater in magnitude
than the force-rate cost (e.g., 68% vs. 32%, respectively at 3 m s-1) and increases more as a func-
tion of speed, primarily for restoring collision losses. Thus, most of the model’s overall cost for
running at higher speeds is due to increased actuator work, which is not included in the
Spring-mass model.
Nevertheless, the force-rate cost has a large influence on the model’s gait as a function of
spring stiffness (Fig 8B). Greater stiffness is associated with more impulsive ground reaction
forces and briefer stance durations (as in Fig 4), thus resulting in higher force-rate cost. Fur-
thermore, the actuator performs additional negative (and therefore also positive) work with
greater stiffness as a trade-off against even higher force-rate costs (like in Fig 6). Overall, the
Leg
Actuator
Human
V. disp. (cm)
V. GRF (Mg)
Time (s)
0
0.1
0.2
0.3
0
2
4
Power (kW)
0
0.1
0.2
0.3
-2
2
0
Time (s)
V. acc. (g)
-1
0
1
2
0
3
-3
V. disp. (cm)
V. GRF (Mg)
Time (s)
0
0.1
0.2
0.3
0
2
4
Power (kW)
0
0.1
0.2
0.3
-2
2
0
Time (s)
V. acc. (g)
-1
0
1
2
0
3
-3
Collision
Collision
A.  Unified model
B.  Force
C.  Power
D.  V. work loop
53%
Elastic
return
min work & 
force-rate
(EW+ER)
Fig 7. Comparison of Unified Actuated Spring-mass model (top) including work and force-rate costs against human data (bottom). Shown are (A)
CoM trajectories, (B) vertical ground reaction forces, (C) leg mechanical power, and (D) vertical acceleration vs. displacement. Initial force transients are
highlighted (red impulse arrow for model, red line for human). In (C), spring (orange shaded area) and actuator work (blue shaded area) contributions
are shown. Gait parameters v and f are 3.9 m s-1 and 3 Hz, respectively. Stiffness and force-rate coefficient in the model are selected to approximately
match stance time duration: k is 35.6 kN m-1 and ε is 0.510−3.
https://doi.org/10.1371/journal.pcbi.1009608.g007
PLOS COMPUTATIONAL BIOLOGY
A simple optimization model of running
PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1009608
November 23, 2021
12 / 22
presence of a force-rate cost makes particularly stiff springs more costly to the model and may
indicate benefits of some compliance when running.
Unified model explains energetic cost of running on inclines
The model may also be applied to uphill and downhill running. In humans, metabolic cost
asymptotically converges toward the costs of muscle performing positive and negative work
(about 25% and -120%, respectively; [11]). At intermediate slopes, the cost smoothly transi-
tions between these two extremes, passing through the cost for level running. The unified
model produces a similar cost curve (Fig 9A), with similar asymptotes. However, the force-
rate cost adds to work costs in such a way that the human asymptotes are actually achieved
with slightly different positive and negative actuator efficiencies (32% and -105%), though con-
sistent with estimates on cross-bridge efficiency [49].
The model’s energetic cost is dominated by positive and negative work at steeper upward
and downward slopes, respectively (Fig 9B). Of course, increasing work is required of steeper
slopes, but force-rate becomes less costly at those extremes. Additionally, for slopes surround-
ing zero, the force-rate cost contributes to the relatively smooth transition from positive to
negative efficiency tangent lines identified by Margaria [11]. The minimum of the cost curve
occurs approximately where passive energy dissipation approaches the net negative mechani-
cal work of descending the ground slope (about -0.08 slope) and is consistent with simple colli-
sion models indicating optimal running slopes [50]. The force-rate cost is relatively high for
shallow slopes and level ground, because it favors more impulsive forces that take advantage of
passive dissipation to reduce active negative work. While even more passive dissipation at
steeper negative slopes could reduce work costs further, this would come at a higher force-rate
cost. In fact, it is less costly overall to actively dissipate energy at steeper slopes to avoid a high
force-rate, but at increased work cost.
A.
Running Speed (m s-1)
2
3
4
0
4
8
12
16
20
Energy Cost per Time (W kg-1)
Model
Human
101
102
0
4
8
12
16
20
Energy Cost per Time (W kg-1)
2.5 m s-1
101
102
Spring Stiffness k  (kN m-1)
0
4
8
12
16
20
3.0 m s-1
101
102
0
4
8
12
16
20
3.5 m s-1
B.
Running Speed
Force
rate
Work
cost
Off-
set
k*
68%
32%
Total
cost
Total
cost
min work & 
force-rate
(EW+ER)
Fig 8. Unified model energy cost vs. speed and spring stiffness, including force-rate cost. (A) Energetic cost per time versus speed is shown for the Actuated
Spring-mass model (k of 35.6 kN m-1, ε of 0.510−3; red curve) and for empirical metabolic data of human subjects running on a treadmill (mean ± standard
deviation; [48]). Model cost includes costs for work and force-rate, plus a constant offset associated with human resting metabolism (dashed horizontal line). (B)
The model’s energetic cost is shown for three speeds v (2.5–3.5 m s-1) and with parameter variation of spring stiffnesses k (13.7–109.5 kN m-1), with total cost
(black lines), force-rate cost (difference between offset and magenta lines), and actuator work cost (difference between total and magenta lines). The unified
model’s spring nominal stiffness k is indicated (red line in A, red symbol in B).
https://doi.org/10.1371/journal.pcbi.1009608.g008
PLOS COMPUTATIONAL BIOLOGY
A simple optimization model of running
PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1009608
November 23, 2021
13 / 22
Parameter variation was used to assess the model’s cost sensitivity to spring stiffness k (Fig
9C), force-rate coefficient ε (Fig 9D), damping ratio z (Fig 9E) and collision fraction CF (Fig
9F). Costs generally increase with each of these parameters, particularly for shallow and level
slopes. For example, increasing spring stiffness resulted in greater energy cost, largely due to
increased force-rate cost associated with more impulsive forces. Greater force-rate coefficient
was also more costly, since total cost is proportional to the coefficient (Fig 6). Increasing either
dissipation parameter resulted in relatively modest increases in cost, mainly because more dis-
sipation must be offset by more active work. But the overall result is that, even for extreme
parameter variations, the cost of running up or down steeper slopes still tends to asymptote
toward positive and negative work efficiencies. On shallow and level slopes, each parameter
contributed toward a non-zero cost, resulting in an overall cost comparable to human
(Fig 9A).
Discussion
We have proposed several additions to the Spring-mass model that help to explain the energy
expenditure of running. The Actuated Spring-mass model includes passive energy dissipation,
active work, and an additional energetic cost related to force-rate. In combination, these ele-
ments show how running can still cost substantial energy, even though series elasticity acts
Ground Slope
Cost of Transport
-0.4
-0.2
0
0.2
0.4
Ground Slope
-0.4
-0.2
0
0.2
0.4
Force rate
Increasing
CF
Increasing
damping, ζ
Increasing
ε
Cost of Transport
Cos  t con irt bu it ons
   .
B
Energy cost
   .
A
Increasing
k
Ground Slope
 .F Co ill sion
D  . Force r ate
ness
ffit
C  . S
E. Hysteresis
Ground Slope
Ground Slope
Ground Slope
Model
Positive work
Negative
work
-120%
25%
Human
Model
min work & 
force-rate
(EW+ER)
Offset
Cost of Transport
0.2
0.2
min work & 
force-rate
(EW+ER)
Fig 9. Energetic cost of running vs. ground slope for unified Actuated Spring-mass model. (A) Model Cost of Transport (solid
line) compared to humans (circles; [11]). Also shown are asymptotes (thin lines) for muscle efficiency of positive and negative
mechanical work (25% and -120%, respectively). (B) Contributors to model Cost of Transport: positive work cost, negative work
cost, force-rate cost ER, and a constant offset. Parameter sensitivities are included for varying (C) stiffness k, (D) force-rate
coefficient ε, (E) hysteresis (damping ratio z), and (F) collision fraction CF. Each trace indicates variation from lowest to highest
parameter values: k ranging 13.7 kN m-1 –1, ε ranging 0–210−3, z ranging 0–0.2, CF ranging 0–0.06. All model results are for
nominal running at speed of 3 m s-1 and step frequency of 2.94 Hz.
https://doi.org/10.1371/journal.pcbi.1009608.g009
PLOS COMPUTATIONAL BIOLOGY
A simple optimization model of running
PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1009608
November 23, 2021
14 / 22
conservatively to reduce the active work required of muscle. We next re-examine each of the
elements individually, to consider both the justification and the contribution of each to an
overall model of running.
Active work dominates energy expenditure despite elastic return
The primary energetic cost observed in the models considered here was for active work.
Empirical estimates based on work during human running, along with assumed 50% elastic
energy return, suggest that active work could account for 76% of the energetic cost [21]. Our
model also had substantial elastic energy return, for example 53% at a speed of 3.9 m s-1 (Fig
7), yet mechanical work still accounted for 66–70% of the overall energetic cost over the range
of speeds considered.
The unified model actively performed both positive and negative work (Fig 7). Empirical
measurements reveal modest active lengthening (and thus negative power) during early stance
in humans (vastus lateralis; [47]) and in turkeys (lateral gastrocnemius; [5]). The perspective
provided by our model is that such active negative work should generally be avoided if work
were the only energetic cost. But active dissipation may be justified by opposing costs such as
for force rate (Fig 8), which justify performing active negative work and extending stance
durations (e.g. Fig 7C), but at the cost of yet more positive work to offset the active dissipation.
The performance of active positive and negative work on level ground also explains the smooth
transition in cost between uphill and downhill slopes. A leg that actively performs negative
work on the level should simply perform more such dissipation for downward slopes, and less
for upward slopes, and similarly for positive work, with cost asymptotes defined by work per-
formed against gravity alone [11]. Even if passive elasticity performs most of the work of run-
ning, the remaining active work by muscles [16,17,47] could still explain much of the overall
energetic cost.
The model also reproduces empirical correlations with energy cost. Kram and Taylor [51]
observed energy costs increasing with speed, and proportional to body weight divided by
ground contact time. The present unified model also yields similar correlations (cost propor-
tional to inverse contact time, and inverse contact time to speed, R2 = 0.95 and R2 = 0.98
respectively), but as an outcome of optimizing costs for work and force rate. In fact, a simple
analysis demonstrates that mechanical work of a series actuator can explain this proportional-
ity explicitly (in supplementary material of [21]). We do not consider contact time to be a
direct determinant of cost, because running (with constant body weight) with an especially flat
CoM trajectory results in both greater contact time and greater energy cost [52]. Rather, a
mechanistic energy cost from actuator work and force can potentially explain why quantities
such as contact time can appear correlated (or not) with cost.
Passive dissipation is a major determinant of mechanics and energetics of
running
There are several features of running that are reproduced by the inclusion of dissipation. Dissi-
pative losses are the primary driver of active mechanical work, which is needed to offset losses
and obtain the zero net work of a periodic gait cycle. Even a relatively small amount of dissipa-
tion can be costly. For example, the unified model passively dissipated less than 5% of the
body’s mechanical energy and passively returned 59% of positive shortening work at 3 m s-1
(compared to 60% in turkey; [8]), yet the remaining active positive work can still explain 61%
of the net metabolic cost of equivalent human running. We also found dissipation (particularly
collision loss) to increase substantially with running speed, and therefore contribute to greater
energy expenditure rate (Fig 8).
PLOS COMPUTATIONAL BIOLOGY
A simple optimization model of running
PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1009608
November 23, 2021
15 / 22
Passive dissipation also helps to explain some time-asymmetries in running. For example,
in humans the vertical ground reaction force increases faster early in stance than it decreases
later in stance (Fig 7B). Such asymmetries have previously been attributed to the muscle force-
velocity relationship [53] and to dissipation [54]. The present model predicts such asymmetry
to be energetically optimal (Fig 7B). Given the leg extends after mid-stance when forces are
already decreasing, the leg must undergo net extension by the time of take-off (Fig 7D). The
model also predicts a more vertical leg orientation during touchdown versus at take-off to
reduce dissipation similar to humans [54]. Other models have also demonstrated the economy
of asymmetrical trajectories in bird running [30] but do not include a brief transient in ground
reaction force and work at touchdown (Fig 7B and 7C), observed prominently in human foot
contact.
Asymmetrical trajectories are not observed in the models without passive dissipation. The
Spring-mass model produces more time-symmetric trajectories (Fig 3) lacking initial tran-
sients and predicts zero active work. The Actuator-only model also produces symmetric trajec-
tories since it has no passive dissipation. However, it lacks passive elasticity resulting in cost
over twice that of equivalent human running (Fig 3, bottom column). The time asymmetries
in force and work profiles might seem like minor details, but here they are emergent from
energy optimality. Passive dissipation helps to explain these asymmetries and is a major contri-
bution to the energetic cost modeled here.
Elastic tendon is critical to energy economy but not the pseudo-elastic
mechanics of running
We next re-examine whether long, compliant tendons are helpful for locomotion economy
[5,6,55]. If the total work per step from muscle and tendon were fixed, a more compliant ten-
don could indeed perform more of the work passively, and thus reduce the energetic cost. But
in running, the total work need not be fixed, and changing the series stiffness also yields a dif-
ferent optimal trajectory (Fig 4), and a different amount of total work for a given speed and
step length. In our model with zero force rate cost coefficient ε, a stiffer spring is actually more
economical at higher speeds (3.5 m s-1, Fig 5). The optimal trajectory yields lower spring dis-
placement and hysteresis loss, and thus, less active work.
Another potential factor for compliance is the proposed force-rate cost. Avoidance of that
cost can favor more active work (Fig 8), and overall cost may indeed be reduced with more
compliance. More complex running models have also suggested that the optimal compliance
may actually be different for each muscle [56]. Of course, there may also be other benefits to
tendon compliance beyond economy [57]. But for running economy alone, there is no general
prescription that favors greater tendon length or compliance.
Tendon compliance has long been implicated in the spring-like mechanics of running. This
is manifested in CoM trajectories, ground reaction forces, power, and vertical work loop
curves (Fig 3), which are all suggestive of elastic behavior. But all the models considered here,
including those with dissipation and actuation, and even those with no elasticity at all (Actua-
tor-only Model, Fig 3) also exhibit similar pseudo-elastic behavior. In models, springs are also
not critical to the economical advantage of running over walking at high speeds [14], nor to
the general cost trends for running on slopes (Fig 9C). This is not to diminish the importance
of elasticity, which allows leg muscles to operate at lower and more efficient shortening veloci-
ties [5], and store and return substantial energy. But the basic resemblance of running to elastic
bouncing, and the associated pseudo-elastic mechanics (e.g., Fig 3), should not be regarded as
evidence of true elasticity in running.
PLOS COMPUTATIONAL BIOLOGY
A simple optimization model of running
PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1009608
November 23, 2021
16 / 22
Force-rate cost contributes to mechanics and energetics of running
In addition to the cost of active work, our model also includes an energetic cost for force rate.
Such a cost has been experimentally observed in tasks such as cyclic muscle contractions
[24,35], and was included as an energetic penalty for rapid transients in force production. We
found that work as the only energetic cost tended to favor overly impulsive running motions
(Fig 6). The force-rate cost acts as a trade-off against work, resulting in reduced peak ground
reaction forces and longer stance durations, more similar to human data (Fig 7). The trade-off
also makes it economical to perform active negative work, and to favor more compliant series
elasticity, contributing to more human-like mechanics (Fig 7) and energetics (Fig 8). Work
alone also reveals a particularly economical strategy for running downhill, where only a mini-
mal amount of active negative work is performed to dissipate gravitational potential energy.
However, humans expend more than the predicted amount of energy, perhaps because the
force-rate makes such a strategy more costly, resulting in a smoother transition toward the
negative work asymptote observed experimentally [11]. The force-rate cost produces these
effects despite a relatively modest energetic cost. In the unified model, force-rate accounts for
only 32% of energetic cost, compared to 68% for work (Fig 8). And across slopes, work alone
predicts too low an energetic cost for running (Fig 9D). A cost such as force-rate is thus helpful
for explaining human-like energetics.
Implications for legged robots
The present study may be relevant to running robots that employ spring-like behavior. In the
SLIP (spring-loaded inverted pendulum) paradigm, a controller causes the overall leg to
behave like a spring, despite internal dissipation. Robotic dissipation includes actuator heat
and transmission losses (analogous to actuator work efficiency and hysteresis in our model) as
well as from interactions with the environment (e.g. collisions) [58]. With SLIP control, the
stiffness is sometimes selected to resemble animal gait [59], just as our model resembles
human. Our results suggest that SLIP may actually be reasonably economical, because our
model, similar to others modeling dissipation with more detail [27,28], yield optimal force pro-
files that are still remarkably spring-like. However, closer examination also reveals that better
economy is achieved with small but significant differences such as force asymmetry (Fig 7).
We also note that matching a gait to human or animal is not necessarily the best option for
economy, which might favor a quite different stiffness (Fig 8B). There are also potential bene-
fits to including passive dissipation, which can yield improved stability [29] and velocity esti-
mation, which is considered important for robust control [60,61]. Robotic controllers can take
advantage of passive dissipation by modeling it explicitly (e.g., [62]) and will respond differ-
ently when optimized for economy.
Limitations
The running model proposed here has a number of limitations, with regard to passive dissipa-
tion, actuator costs and running dynamics. For dissipation, we modeled hysteresis during leg
compression alone to reproduce empirically estimated energy losses, without a detailed model
of hysteresis mechanics. Similarly, we modeled collision losses with a simple reduction in
momentum at touchdown, without considering the direction and mechanical properties of the
body mass experiencing impact. To our knowledge, most models of running do not include
explicit dissipation. We consider the present model to be an indicator that dissipation is
important, but also in need of better-informed dissipation mechanics. The same is true for our
model of energetic cost, intended to extend Spring-mass models with a highly simplified
dependence on active mechanical work. Our model also stands to be improved with other
PLOS COMPUTATIONAL BIOLOGY
A simple optimization model of running
PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1009608
November 23, 2021
17 / 22
features potentially relevant to running, such as muscle moment arms and force-velocity rela-
tionships (e.g., [63]).
There are also limitations of the proposed energetic cost for force rate, which is not
included in most other running models. This cost is intermediate in complexity between
abstract “effort” costs such as force squared [64,65], and more detailed models of muscle ener-
getics [66,67], while also being supported by empirical data (e.g. [23–25,34,35]). But the spe-
cific dynamics of such a cost in muscle are not well understood, with considerable uncertainty
in the appropriate formulation. For example, we have found different derivative order and
exponent (e.g., 2 and 1, respectively for Eq 3) to yield similar forces [26] and agree well with
empirical energy costs [23]. We therefore regard the force-rate cost as a provisional implemen-
tation that stands for considerable improvement.
Similar to the Spring-mass model, the present model is also a gross simplification of human
body dynamics. It is intended only to model basic features of running, such as pseudo-elastic
mechanics and overall energy expenditure. More complexity would be required to address
motion of multiple-jointed models, intersegmental dynamics, and activation or co-contraction
of multiple muscles. For example, the model neglects a swing leg, whose active motion may
also cost energy [22], and be exploited to modulate collision losses via active leg retraction just
before touchdown [68,69]. We also fixed step length and frequency with respect to running
speed, whereas these could also be included in optimization for preferred gait parameters (e.g.,
[38]). The present model only provides basic suggestions, that dissipation and energetic costs
for work and force rate may be important for running. These suggestions are intended to
apply to more complex models of running, but this remains to be tested.
Conclusions
The energetic cost of running is not addressed by the classic Spring-mass model of running.
Although elastic tissues store and return energy during stance, there is still some dissipation
due to touchdown collision and hysteresis. For steady gait, these losses must be restored by
active, positive work from muscle. The present Actuated Spring-mass model shows that even a
relatively small amount of work can still incur a substantial energetic cost. It is also particularly
costly to perform active negative work, because the associated losses must be restored by addi-
tional positive work. Muscles may also expend energy for high rates of force development that
make it economical to perform some active negative work, ultimately helping to explain the
energetics of running at different speeds and slopes. Spring-like forces and other mechanics
emerge from an actively controlled model optimized for economy, even if there were no elas-
ticity. Series elasticity may be critical to saving energy, but active work and passive dissipation
appear important for determining the energetic cost of running.
Supporting information
S1 Text. Dynamic optimization model details.
(DOCX)
Acknowledgments
The authors would like to thank Arash Khassetarash (University of Calgary) for sharing exper-
imental data.
Author Contributions
Conceptualization: Ryan T. Schroeder, Arthur D. Kuo.
PLOS COMPUTATIONAL BIOLOGY
A simple optimization model of running
PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1009608
November 23, 2021
18 / 22
Data curation: Ryan T. Schroeder.
Formal analysis: Ryan T. Schroeder.
Funding acquisition: Arthur D. Kuo.
Investigation: Ryan T. Schroeder, Arthur D. Kuo.
Methodology: Ryan T. Schroeder, Arthur D. Kuo.
Project administration: Ryan T. Schroeder, Arthur D. Kuo.
Resources: Arthur D. Kuo.
Software: Ryan T. Schroeder.
Supervision: Arthur D. Kuo.
Validation: Ryan T. Schroeder, Arthur D. Kuo.
Visualization: Ryan T. Schroeder, Arthur D. Kuo.
Writing – original draft: Ryan T. Schroeder, Arthur D. Kuo.
Writing – review & editing: Ryan T. Schroeder, Arthur D. Kuo.
References
1.
Alexander RM. A model of bipedal locomotion on compliant legs. Philosophical Transactions of the
Royal Society of London Series B: Biological Sciences. 1992; 338(1284):189–98. https://doi.org/10.
1098/rstb.1992.0138 PMID: 1360684
2.
Blickhan R. The spring-mass model for running and hopping. Journal of Biomechanics. 1989 Jan; 22
(11–12):1217–27. https://doi.org/10.1016/0021-9290(89)90224-8 PMID: 2625422
3.
Geyer H, Seyfarth A, Blickhan R. Compliant leg behaviour explains basic dynamics of walking and run-
ning. Proceedings of the Royal Society B-Biological Sciences. 2006 Nov 22; 273(1603):2861–7. https://
doi.org/10.1098/rspb.2006.3637 PMID: 17015312
4.
McMahon TA, Cheng GC. The mechanics of running: How does stiffness couple with speed? Journal of
Biomechanics. 1990;23, Supplement 1:65–78. https://doi.org/10.1016/0021-9290(90)90042-2 PMID:
2081746
5.
Biewener AA, Roberts TJ. Muscle and tendon contributions to force, work, and elastic energy savings:
a comparative perspective. Exerc Sport Sci Rev. 2000 Jul; 28(3):99–107. PMID: 10916700
6.
Bramble DM, Lieberman DE. Endurance running and the evolution of Homo. Nature. 2004 Nov; 432
(7015):345–52. https://doi.org/10.1038/nature03052 PMID: 15549097
7.
Cavagna GA, Kaneko M. Mechanical work and efficiency in level walking and running. J Physiol. 1977
Jun 1; 268(2):467–81. https://doi.org/10.1113/jphysiol.1977.sp011866 PMID: 874922
8.
Roberts TJ, Marsh RL, Weyand PG, Taylor CR. Muscular Force in Running Turkeys: The Economy of
Minimizing Work. Science. 1997 Feb 21; 275(5303):1113–5. https://doi.org/10.1126/science.275.5303.
1113 PMID: 9027309
9.
Gan Z, Remy CD. A passive dynamic quadruped that moves in a large variety of gaits. In: 2014 IEEE/
RSJ International Conference on Intelligent Robots and Systems. 2014. p. 4876–81.
10.
Gan Z, Yesilevskiy Y, Zaytsev P, Remy CD. All common bipedal gaits emerge from a single passive
model. Journal of The Royal Society Interface. 2018 Sep 30; 15(146):20180455. https://doi.org/10.
1098/rsif.2018.0455 PMID: 30257925
11.
Margaria R. Positive and negative work performances and their efficiencies in human locomotion. Int Z
Angew Physiol. 1968 May 28; 25(4):339–51. https://doi.org/10.1007/BF00699624 PMID: 5658204
12.
Taylor CR, Heglund NC, Maloiy GM. Energetics and mechanics of terrestrial locomotion. I. Metabolic
energy consumption as a function of speed and body size in birds and mammals. J Exp Biol. 1982 Apr
1; 97(1):1–21. PMID: 7086334
13.
Long LL, Srinivasan M. Walking, running, and resting under time, distance, and average speed con-
straints: optimality of walk–run–rest mixtures. Journal of The Royal Society Interface. 2013 Apr 6; 10
(81):20120980.
PLOS COMPUTATIONAL BIOLOGY
A simple optimization model of running
PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1009608
November 23, 2021
19 / 22
14.
Srinivasan M, Ruina A. Computer optimization of a minimal biped model discovers walking and running.
Nature. 2006; 439(7072):72–5. https://doi.org/10.1038/nature04113 PMID: 16155564
15.
Alexander RM. Optima for Animals [Internet]. Princeton, NJ: Princeton University Press; 1996 [cited
2020 Sep 5]. Available from: https://press.princeton.edu/books/paperback/9780691027982/optima-for-
animals
16.
Lai A, Lichtwark GA, Schache AG, Lin Y-C, Brown NAT, Pandy MG. In vivo behavior of the human
soleus muscle with increasing walking and running speeds. Journal of Applied Physiology. 2015 Mar
26; 118(10):1266–75. https://doi.org/10.1152/japplphysiol.00128.2015 PMID: 25814636
17.
Lichtwark GA, Bougoulias K, Wilson AM. Muscle fascicle and series elastic element length changes
along the length of the human gastrocnemius during walking and running. Journal of Biomechanics.
2007 Jan 1; 40(1):157–64. https://doi.org/10.1016/j.jbiomech.2005.10.035 PMID: 16364330
18.
Lichtwark GA, Wilson AM. In vivo mechanical properties of the human Achilles tendon during one-leg-
ged hopping. Journal of Experimental Biology. 2005 Dec 15; 208(24):4715–25. https://doi.org/10.1242/
jeb.01950 PMID: 16326953
19.
Ker RF. The time-dependent mechanical properties of the human heel pad in the context of locomotion.
J Exp Biol. 1996; 199:1501–8. PMID: 8699155
20.
Riddick RC, Kuo AD. Soft tissues store and return mechanical energy in human running. Journal of Bio-
mechanics. 2016 Feb 8; 49(3):436–41. https://doi.org/10.1016/j.jbiomech.2016.01.001 PMID:
26806689
21.
Riddick RC, Kuo AD. Mechanical Work Accounts for Most of the Energetic Cost in Human Running.
bioRxiv. 2020;
22.
Doke J, Donelan JM, Kuo AD. Mechanics and energetics of swinging the human leg. The Journal of
experimental biology. 2005; 208(3):439–45. https://doi.org/10.1242/jeb.01408 PMID: 15671332
23.
Doke J, Kuo AD. Energetic cost of producing cyclic muscle force, rather than work, to swing the human
leg. Journal of Experimental Biology. 2007; 210(13):2390–8. https://doi.org/10.1242/jeb.02782 PMID:
17575044
24.
van der Zee TJ, Kuo AD. The high energetic cost of rapid force development in cyclic muscle contrac-
tion. Journal of Experimental Biology [Internet]. 2020 [cited 2020 Sep 8]; Available from: https://www.
biorxiv.org/content/10.1101/2020.08.25.266965v1
25.
Wong JD, Cluff T, Kuo AD. The energetic basis for smooth human arm movements [Internet]. Neurosci-
ence; 2020 Dec [cited 2021 Mar 17]. Available from: http://biorxiv.org/lookup/doi/10.1101/2020.12.28.
424067
26.
Rebula JR, Kuo AD. The Cost of Leg Forces in Bipedal Locomotion: A Simple Optimization Study.
PLOS ONE. 2015 Feb 23; 10(2):e0117384. https://doi.org/10.1371/journal.pone.0117384 PMID:
25707000
27.
Xi W, Remy CD. Optimal gaits and motions for legged robots. In: 2014 IEEE/RSJ International Confer-
ence on Intelligent Robots and Systems. 2014. p. 3259–65.
28.
Xi W, Yesilevskiy Y, Remy CD. Selecting gaits for economical locomotion of legged robots. The Interna-
tional Journal of Robotics Research. 2016 Aug 1; 35(9):1140–54.
29.
Heim S, Millard M, Le Mouel C, Badri-Spro¨witz A. A little damping goes a long way: a simulation study
of how damping influences task-level stability in running. Biol Lett. 2020 Sep; 16(9):20200467. https://
doi.org/10.1098/rsbl.2020.0467 PMID: 32961093
30.
Birn-Jeffery AV, Hubicki CM, Blum Y, Renjewski D, Hurst JW, Daley MA. Don’t break a leg: running
birds from quail to ostrich prioritise leg safety and economy on uneven terrain. Journal of Experimental
Biology. 2014 Nov 1; 217(21):3786–96. https://doi.org/10.1242/jeb.102640 PMID: 25355848
31.
Chi KJ, Schmitt D. Mechanical energy and effective foot mass during impact loading of walking and run-
ning. J Biomech. 2005; 38:1387–95. https://doi.org/10.1016/j.jbiomech.2004.06.020 PMID: 15922749
32.
Finni T, Peltonen J, Stenroth L, Cronin NJ. Viewpoint: On the hysteresis in the human Achilles tendon.
Journal of Applied Physiology. 2013 Feb 15; 114(4):515–7. https://doi.org/10.1152/japplphysiol.01005.
2012 PMID: 23085961
33.
Bergstrom M, Hultman E. Energy cost and fatigue during intermittent electrical stimulation of human
skeletal muscle. Journal of Applied Physiology. 1988; 65:1500–5. https://doi.org/10.1152/jappl.1988.
65.4.1500 PMID: 3182513
34.
Chasiotis D, Bergstro¨m M, Hultman E. ATP utilization and force during intermittent and continuous mus-
cle contractions. J Appl Physiol. 1987 Jul; 63(1):167–74. https://doi.org/10.1152/jappl.1987.63.1.167
PMID: 3624122
35.
Hogan MC, Ingham E, Kurdak SS. Contraction duration affects metabolic energy cost and fatigue in
skeletal muscle. Am J Physiol. 1998; 274:E397–402. https://doi.org/10.1152/ajpendo.1998.274.3.E397
PMID: 9530120
PLOS COMPUTATIONAL BIOLOGY
A simple optimization model of running
PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1009608
November 23, 2021
20 / 22
36.
Croft JL, Schroeder RT, Bertram JE. Determinants of optimal leg use strategy: horizontal to vertical
transition in the parkour wall climb. Journal of Experimental Biology. 2019; 222(1):jeb190983. https://
doi.org/10.1242/jeb.190983 PMID: 30446542
37.
Handford ML, Srinivasan M. Robotic lower limb prosthesis design through simultaneous computer opti-
mizations of human and prosthesis costs. Sci Rep. 2016 Apr; 6(1):19983. https://doi.org/10.1038/
srep19983 PMID: 26857747
38.
Kuo AD. A simple model of bipedal walking predicts the preferred speed-step length relationship. J Bio-
mech Eng. 2001 Jun; 123(3):264–9. https://doi.org/10.1115/1.1372322 PMID: 11476370
39.
Polet DT, Bertram JEA. An inelastic quadrupedal model discovers four-beat walking, two-beat running,
and pseudo-elastic actuation as energetically optimal. Donelan M, editor. PLoS Comput Biol. 2019 Nov
21; 15(11):e1007444. https://doi.org/10.1371/journal.pcbi.1007444 PMID: 31751339
40.
Schroeder RT, Bertram JE. Minimally Actuated Walking: Identifying Core Challenges to Economical
Legged Locomotion Reveals Novel Solutions. Front Robot AI. 2018 May 22; 5(58):58. https://doi.org/
10.3389/frobt.2018.00058 PMID: 33644120
41.
Schroeder RT, Bertram JEA, Son Nguyen V, Vinh Hac V, Croft JL. Load carrying with flexible bamboo
poles: optimization of a coupled oscillator system. J Exp Biol. 2019 Dec 1; 222(23):jeb203760. https://
doi.org/10.1242/jeb.203760 PMID: 31801848
42.
Patterson MA, Rao AV. GPOPS-II: A MATLAB Software for Solving Multiple-Phase Optimal Control
Problems Using hp-Adaptive Gaussian Quadrature Collocation Methods and Sparse Nonlinear Pro-
gramming. ACM Trans Math Softw. 2014 Oct 27; 41(1):1–37.
43.
Gill PE, Murray W, Saunders MA. SNOPT: An SQP Algorithm for Large-Scale Constrained Optimiza-
tion. SIAM Rev. 2005 Jan; 47(1):99–131.
44.
Gutmann AK. Constrained optimization in human running. Journal of Experimental Biology. 2006 Feb
15; 209(4):622–32. https://doi.org/10.1242/jeb.02010 PMID: 16449557
45.
Khassetarash A, Vernillo G, Kruger RL, Edwards WB, Millet GY. Neuromuscular, biomechanical, and
energetics adjustments following repeated bouts of downhill running. J Sport Health Sci. 2021;
46.
Arampatzis A, Knicker A, Metzler V, Bru¨ggemann G-P. Mechanical power in running: a comparison of
different approaches. Journal of Biomechanics. 2000 Apr; 33(4):457–63. https://doi.org/10.1016/s0021-
9290(99)00187-6 PMID: 10768394
47.
Bohm S, Marzilger R, Mersmann F, Santuz A, Arampatzis A. Operating length and velocity of human
vastus lateralis muscle during walking and running. Scientific Reports. 2018 Mar 22; 8(1):5066. https://
doi.org/10.1038/s41598-018-23376-5 PMID: 29567999
48.
Kipp S, Grabowski AM, Kram R. What determines the metabolic cost of human running across a wide
range of velocities? Journal of Experimental Biology [Internet]. 2018 Sep 15 [cited 2020 Jul 9]; 221(18).
Available from: https://doi.org/10.1242/jeb.184218 PMID: 30065039
49.
Barclay CJ. Chapter 6—Efficiency of Skeletal Muscle. In: Zoladz JA, editor. Muscle and Exercise Physi-
ology [Internet]. Academic Press; 2019 [cited 2021 Feb 3]. p. 111–27. Available from: http://www.
sciencedirect.com/science/article/pii/B9780128145937000062
50.
Dewolf AH, Willems PA. Running on a slope: A collision-based analysis to assess the optimal slope.
Journal of Biomechanics. 2019 Jan; 83:298–304. https://doi.org/10.1016/j.jbiomech.2018.12.024
PMID: 30611540
51.
Kram R, Taylor CR. Energetics of running: a new perspective. Nature. 1990 Jul 19; 346(6281):265–7.
https://doi.org/10.1038/346265a0 PMID: 2374590
52.
McMahon TA, Valiant G, Frederick EC. Groucho running. Journal of Applied Physiology. 1987 Jun 1; 62
(6):2326–37. https://doi.org/10.1152/jappl.1987.62.6.2326 PMID: 3610929
53.
Cavagna GA. The landing–take-off asymmetry in human running. Journal of Experimental Biology.
2006 Oct 15; 209(20):4051–60. https://doi.org/10.1242/jeb.02344 PMID: 17023599
54.
Dewolf AH, Willems PA. A collision-based analysis of the landing-takeoff asymmetry during running.
Computer Methods in Biomechanics and Biomedical Engineering. 2017 Oct 30; 20(sup1):S65–6.
https://doi.org/10.1080/10255842.2017.1382863 PMID: 29088651
55.
Kielty M, Radke H. Radiolab: Man Against Horse [Internet]. [cited 2021 Mar 16]. Available from: https://
www.wnycstudios.org/podcasts/radiolab/articles/man-against-horse
56.
Uchida TK, Hicks JL, Dembia CL, Delp SL. Stretching Your Energetic Budget: How Tendon Compliance
Affects the Metabolic Cost of Running. Zadpoor AA, editor. PLoS ONE. 2016 Mar 1; 11(3):e0150378.
https://doi.org/10.1371/journal.pone.0150378 PMID: 26930416
57.
Daley MA, Usherwood JR. Two explanations for the compliant running paradox: reduced work of bounc-
ing viscera and increased stability in uneven terrain. Biol Lett. 2010 Jun 23; 6(3):418–21. https://doi.org/
10.1098/rsbl.2010.0175 PMID: 20335198
PLOS COMPUTATIONAL BIOLOGY
A simple optimization model of running
PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1009608
November 23, 2021
21 / 22
58.
Seok S, Wang A, Meng Yee Chuah, Otten D, Lang J, Kim S. Design principles for highly efficient quad-
rupeds and implementation on the MIT Cheetah robot. In: 2013 IEEE International Conference on
Robotics and Automation [Internet]. Karlsruhe, Germany: IEEE; 2013 [cited 2021 Aug 26]. p. 3307–12.
Available from: http://ieeexplore.ieee.org/document/6631038/
59.
Zhang C, Zou W, Ma L, Wang Z. Biologically inspired jumping robots: A comprehensive review. Robot-
ics and Autonomous Systems. 2020 Feb; 124:103362.
60.
Johnson M, Shrewsbury B, Bertrand S, Wu T, Duran D, Floyd M, Abeles P, Stephen D, Mertins N, Les-
man A, Carff J, Rifenburgh W, Kaveti P, Straatman W, Smith J, Griffioen M, Layton B, de Boer T, Koolen
T, Neuhaus P, Pratt J. Team IHMC’s Lessons Learned from the DARPA Robotics Challenge Trials:
Team IHMC’s Lessons Learned from the DARPA Robotics Challenge Trials. J Field Robotics. 2015
Mar; 32(2):192–208.
61.
Xinjilefu X, Feng S, Atkeson CG. A distributed MEMS gyro network for joint velocity estimation. In: 2016
IEEE International Conference on Robotics and Automation (ICRA) [Internet]. Stockholm, Sweden:
IEEE; 2016 [cited 2021 Aug 26]. p. 1879–84. Available from: http://ieeexplore.ieee.org/document/
7487334/
62.
Hutter M, Remy CD, Hopflinger MA, Siegwart R. SLIP running with an articulated robotic leg. In: 2010
IEEE/RSJ International Conference on Intelligent Robots and Systems [Internet]. Taipei: IEEE; 2010
[cited 2021 Mar 16]. p. 4934–9. Available from: http://ieeexplore.ieee.org/document/5651461/
63.
McMahon TA, Greene PR. The influence of track compliance on running. Journal of Biomechanics.
1979; 12(12):893–904. https://doi.org/10.1016/0021-9290(79)90057-5 PMID: 528547
64.
Ackermann M, van den Bogert AJ. Optimality principles for model-based prediction of human gait. Jour-
nal of Biomechanics. 2010 Apr; 43(6):1055–60. https://doi.org/10.1016/j.jbiomech.2009.12.012 PMID:
20074736
65.
Martin AE, Schmiedeler JP. Predicting human walking gaits with a simple planar model. Journal of Bio-
mechanics. 2014 Apr; 47(6):1416–21. https://doi.org/10.1016/j.jbiomech.2014.01.035 PMID: 24565183
66.
Ma SP, Zahalak GI. A distribution-moment model of energetics in skeletal muscle. J Biomech. 1991; 24
(1):21–35. https://doi.org/10.1016/0021-9290(91)90323-f PMID: 2026631
67.
Zahalak GI, Motabarzadeh I. A re-examination of calcium activation in the Huxley cross-bridge model. J
Biomech Eng. 1997 Feb; 119(1):20–9. https://doi.org/10.1115/1.2796060 PMID: 9083845
68.
Haberland M, Karssen JGD, Kim S, Wisse M. The effect of swing leg retraction on running energy effi-
ciency. In: 2011 IEEE/RSJ International Conference on Intelligent Robots and Systems [Internet]. San
Francisco, CA: IEEE; 2011 [cited 2021 Mar 16]. p. 3957–62. Available from: http://ieeexplore.ieee.org/
document/6094627/
69.
Hasaneini SJ, Bertram JEA, Macnab CJB. Energy-optimal relative timing of stance-leg push-off and
swing-leg retraction in walking. Robotica. 2017 Mar; 35(3):654–86.
PLOS COMPUTATIONAL BIOLOGY
A simple optimization model of running
PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1009608
November 23, 2021
22 / 22

Elastic energy savings and active energy cost in a simple model of running.
Schroeder, Ryan T,Kuo, Arthur D