Phase shift between joint rotation and actuation reflects dominant forces and predicts muscle activation patterns

Abstract

During behavior, the work done by actuators on the body can be resisted by the body's inertia, elastic forces, gravity, or viscosity. The dominant forces that resist actuation have major consequences on the control of that behavior. In the literature, features and actuation of locomotion, for example, have been successfully predicted by nondimensional numbers (e.g. Froude number and Reynolds number) that generally express the ratio between two of these forces (gravitational, inertial, elastic, and viscous). However, animals of different sizes or motions at different speeds may not share the same dominant forces within a behavior, making ratios of just two of these forces less useful. Thus, for a broad comparison of behavior across many orders of magnitude of limb length and cycle period, a dimensionless number that includes gravitational, inertial, elastic, and viscous forces is needed. This study proposes a nondimensional number that relates these four forces: the phase shift (ϕ) between the displacement of the limb and the actuator force that moves it. Using allometric scaling laws, ϕ for terrestrial walking is expressed as a function of the limb length and the cycle period at which the limb steps. Scale-dependent values of ϕ are used to explain and predict the electromyographic (EMG) patterns employed by different animals as they walk.

Full text

Phase shift between joint rotation and actuation reects
dominant forces and predicts muscle activation patterns
G. P. Sutton
a,
*
,1
, N. S. Szczecinski
b,1
, R. D. Quinn
c
and H. J. Chiel
d,e,f
a
School of Life Sciences, University of Lincoln, Lincoln LN6 7TS, UK
b
Department of Mechanical and Aerospace Engineering, West Virginia University, Morgantown, WV 26506-6106, USA
c
Department of Mechanical Engineering, Case Western Reserve University, Cleveland, OH 44106, USA
d
Department of Biology, Case Western Reserve University, Cleveland, OH 44106, USA
e
Department of Neuroscience, Case Western Reserve University, Cleveland, OH 44106, USA
f
Department of Biomedical Engineering, Case Western Reserve University, Cleveland, OH 44106, USA
*To whom correspondence should be addressed: Email: RScealai@gmail.com
1
G.P.S. and N.S.S. contributed equally to this work.
Edited By: Krzysztof Zur
Abstract
During behavior, the work done by actuators on the body can be resisted by the body’s inertia, elastic forces, gravity, or viscosity. The
dominant forces that resist actuation have major consequences on the control of that behavior. In the literature, features and
actuation of locomotion, for example, have been successfully predicted by nondimensional numbers (e.g. Froude number and
Reynolds number) that generally express the ratio between two of these forces (gravitational, inertial, elastic, and viscous). However,
animals of different sizes or motions at different speeds may not share the same dominant forces within a behavior, making ratios of
just two of these forces less useful. Thus, for a broad comparison of behavior across many orders of magnitude of limb length and
cycle period, a dimensionless number that includes gravitational, inertial, elastic, and viscous forces is needed. This study proposes a
nondimensional number that relates these four forces: the phase shift (ϕ) between the displacement of the limb and the actuator
force that moves it. Using allometric scaling laws, ϕ for terrestrial walking is expressed as a function of the limb length and the cycle
period at which the limb steps. Scale-dependent values of ϕ are used to explain and predict the electromyographic (EMG) patterns
employed by different animals as they walk.
Signicance Statement
There have been many discussions about how scaling in locomotion changes the relationship between inertial, gravitational, viscous,
and elastic forces, with inertial forces governing motion in large limbs and elastic forces governing small limbs. We show that regimes
of differing force dominance require the nervous system to solve differing control problems and show that the dominant regime of a
movement can be determined by measuring a single parameter: the phase relationship between muscle force and joint angle. This
work shows locomotion of large fast creatures (e.g. horses) and small slow creatures (e.g. snails) exists upon a continuum that can
be evaluated and represented by this single dimensionless parameter. This will greatly inform and aid locomotion researchers in
the study of locomotion.
Competing Interest: The authors declare no competing interest.
Received: February 28, 2023. Accepted: August 29, 2023
© The Author(s) 2023. Published by Oxford University Press on behalf of National Academy of Sciences. This is an Open Access article
distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0/), which permits
unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.
Introduction
Brains are embodied (1). Consequently, the mechanics of the body
provide both opportunities for and constraints on the nervous sys-
tem. Thus, while it is natural to consider how evolutionary pres-
sures shape neural structures that control behavior, such
pressures can only be fully understood within the context of an
animal’s mechanics. Muscles apply forces within the body; how
do those forces interact with the mechanical properties of the
limb (inertia, gravity, elasticity, and viscosity) as a function of
size and speed?
To better understand how muscle force interacts with limb
mechanical properties, studies often employ dimensionless
numbers that express the ratio of two forces within a particular
motion. For example, the Froude number, which is the ratio of
centripetal inertial force to gravitational force acting on the
body of a locomoting animal, has been shown to predict the gait
(2, 3), duty factor, and stride length (4) of many large animals as
they walk and run. As another example, the Reynolds number,
which is the ratio between the inertial force and viscous force of
uid acting on an animal as it moves, predicts whether the animal
should swim most efciently through that uid via a corkscrew or
paddling motion (5). These dimensionless numbers, although ex-
tremely useful, are subject to certain assumptions and limitations
that prevent their application to all systems. Specically, each is
best applied to a system whose motion is dominated by the two
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forces compared by the dimensionless number. For example, if
viscous forces are much larger than both centripetal and gravita-
tional forces, Froude numbers will not reect major portions of
the movement. Likewise, if gravitational forces are much larger
than inertial or viscous forces, Reynolds numbers will not reect
useful insights about a movement.
Which forces dominate a given behavior, however, is governed
by the scale and speed of the movement. For example, because
mass scales with the cube of length, mass-dependent forces dom-
inate in large animals (6, 7), whereas elastic forces, which scale
with the square of length, dominate in small animals (8–10).
Similarly, the timing of a movement also affects the forces that
dominate. In an oscillatory movement like walking, because elas-
tic forces are a function of position, they are thus insensitive to
movement speed, whereas inertial forces are proportional to ac-
celeration, and consequently the faster the oscillation, the greater
the inertial forces will be relative to the elastic forces. This has
major consequences for the neural control of these movements:
for large animals, neural output is required to react to gravitation-
al and inertial forces, whereas, for small animals, gravity and in-
ertial forces can be disregarded (10, 11), with the nervous
system instead needing to react to the increased role of elastic
forces. Changes in the dominant force in turn change how best
to quantify and control movement in animals of differing sizes
and speeds, and these relationships have been quantied in
some models (e.g. insect walking versus horse walking) (10–12).
Similarly, internal viscous forces of a joint (damping) would also
affect movement (10, 13), so that the behavior of smaller and fast-
er animals is more dominated by damping than that of larger and
slower animals.
Consequently, a broad comparison of movement across many
orders of magnitude of limb length cannot be done with dimen-
sionless numbers that specify the ratio of only two forces.
Comparing movements across a wide range of sizes and speeds re-
quires a dimensionless number that includes gravitational, iner-
tial, elastic, and viscous forces, allowing comparison of the
relative magnitudes of all these forces to determine which best
quanties a behavior. In this study, we propose such a dimension-
less number that relates gravitational forces, inertial forces, elas-
tic forces, and viscous forces: the phase shift, ϕ, between actuator
force and limb displacement.
We quantify how forces are partitioned among gravitational,
elastic, viscous, and inertial forces during simulations of legged
locomotion (including stance and swing), nding that the relation-
ships between these forces can be reected by a single measurable
nondimensional number: the phase shift (ϕ) between actuator
force and limb displacement (Fig. 1). Using allometric scaling
laws, we expressed ϕ in terms of two quantities: limb length and
cycle period. As a result, we identied three “regions” of limb
length and cycle period in which actuator force is primarily re-
sisted by gravitational forces, inertial forces, elastic forces, or vis-
cous forces. We will use swing and stance of simulated
locomotion to show that, while the dominant force in a given mo-
tion is dependent on the size and frequency of the movement, the
relationship between phase shift (ϕ) and the dominant force re-
mains the same in both swing and stance, demonstrating that
phase shift can be used to quantify the dominant force in a
behavior.
Moreover, phase shift (ϕ) can also predict two more aspects of
the movement. First, the phase shift can predict the limb’s re-
sponse to perturbation; and second, the phase shift can predict
the timing of electromyography (EMG) during a limb movement.
Since the nervous system must activate muscles with an
appropriate timing relative to limb position to generate a move-
ment, the EMG must also be shifted by ϕ. We test this prediction
by demonstrating how the phase shift (ϕ) can predict the EMG re-
cordings of locomotion (swing and stance) at two different speeds
in two very differently sized animals: horse and stick insect.
Results
To develop a dimensionless number for limb movement, we cre-
ated a model that represents the simplied geometry and dynam-
ics of a limb segment in both swing and stance, e.g. a horse’s foreleg
rotating about the shoulder or an insect’s leg rotating about its
thoraco-coxal joint (Fig. 1A). We model the leg of a walking animal
as a rigid pendulum in swing and an inverted pendulum in stance
(14–17). The limb is moved by an antagonistic pair of actuators, e.g.
a shoulder protractor (exor) and retractor (extensor). Together,
the actuators have total inherent elastic stiffness k
elas
and viscous
damping c, resulting in elastic and viscous moments about the
shoulder. The model is concerned with which forces resist actuator
work so the actuators themselves do not include muscle dynamics,
e.g. force–velocity limits (see supplementary materials for further
justication of this simplication), although muscle properties lim-
it what motions an animal can execute volitionally (18). The dy-
namics of the leg–body system depend on whether the leg is in
swing, during which the leg is moved anteriorly (i.e. protracted)
and does not support the body (Fig. 1B), or stance, during which
the leg is moved posteriorly (i.e. retracted) while supporting and
propelling the body (Fig. 1C). In both cases, the limb is assumed
to have length L with mass mL and moment of inertia about the
hip JL. The limb is assumed to operate in a gravitational eld with
acceleration g, and the rotation of the limb relative to the direction
of gravity is measured by θ. As in Hooper and Alexander (11, 15), we
neglected aerodynamic drag.
To calculate the ratios of inertial, elastic, gravitational, and vis-
cous forces within the limb, we applied allometric scaling rela-
tionships to express the inertia, gravitational forces, elastic
forces, and viscous damping in terms of limb length (Fig. 1G)
and limb movement speed. For example, a limb’s mass very near-
ly scales with its volume, that is, its length cubed (19, 20). Similar
scaling laws describe how spring stiffness scales proportional to
length (21, 22). Because we were unaware of an established allo-
metric scaling relationship for joint damping as a function of leg
length, we developed one using previously published data from
studies in human, stick insect, and cockroach joints (8, 13, 23–
25). We found that joint damping, like spring stiffness, scales pro-
portional to length and predicts the values for joint damping re-
ported for limb lengths spanning two orders of magnitude
(Fig. 1H). These relationships were extended to account for the ro-
tational motion of a joint by the principle of virtual work (26) (see
the supplementary materials).
In swing, the phase shift (ϕ) between force and movement is de-
termined by limb length and cycle period. In turn, the phase shift
quanties the ratio of the inertial, gravitational, potential, and vis-
cous forces, as shown in Fig. 2A. Although the phase shift varies
continuously with limb length and cycle time, there are regions
in which large changes in the phase shift occur over small changes
in cycle time or length, and these determine distinct “regions.” In
region I (yellow), the cycle period is so short relative to the limb’s
natural period of oscillation that the actuator force is almost en-
tirely out of phase with the motion, resulting in a phase shift (ϕ)
of 180°. This is indicated by the work loops (27) shown in Fig. 2B
and F, which plot the actuator force versus the limb angle. We
call this region “kinetic” because most of the actuator work is
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resisted by inertial forces and is thus converted into kinetic energy
(represented by the yellow shaded area). In region II (red), the
cycle period is longer than the limb’s natural period, and actuator
torque is almost entirely in phase with the limb angle, as indicated
by the positive-slope work loops shown in Fig. 2C and D. The phase
shift (ϕ) in this region is 0°. We call this region “quasi-static” be-
cause the static forces of gravity and elasticity dominate (28),
and most actuator work is converted into potential energy
(shaded red). Finally, in region III (orange), cycle period is short
relative to the resonant frequency of the limb, but the limb has
very little mass, so most actuator energy is dissipated due to
viscous forces within the joint (orange shading in Fig. 2E). The
phase shift in this region is intermediate but usually near 90°.
For swing at all sizes and cycle times, the phase shift (ϕ) indicates
whether the motion is dominated by inertia, gravity and elasticity,
or viscosity.
In stance, the relationship between limb length, cycle period,
and phase shift changes dramatically compared to swing, as indi-
cated by Fig. 2G. The primary difference is that movements that
previously resided in region II now reside in region I (the kinetic
region), meaning that the phase shift changes substantially be-
tween swing and stance in some cases. Specically, for relatively
A
D E
F G H
B C
Fig. 1. Formulation of the model. A) Horses, stick insects, and many animals possess rotary joints actuated by antagonistic muscle groups. B) In swing, we
model the leg as a pendulum rotating about a xed point O and actuated by a force F. The joint possesses intrinsic viscoelasticity, and gravity pulls the
leg’s center of mass (white and black circle) downward. C) In stance, we model the leg as an inverted pendulum (14–16) anchored at point P with the same
mass and viscoelastic joint properties, plus the mass of the body concentrated at point O. Gravity pulls the leg’s center of mass and the body’s center of
mass downward. D) The equations of motion in swing and stance can be written in terms of x, the actuator position. The rotary pendular dynamics
is computed from the torque, i.e. the cross product of r with the equations of motion. Although rotating the joint will change the moment arm vector, this
effect is negligible for joint motions less than 30° in either direction (Fig. S2). Each term is color coded: inertial forces and moments are yellow (in
greyscale: light grey), viscous forces and moments are orange (in greyscale: medium grey), and elastic forces and moments are red (in greyscale: dark
grey). Gravitational moment is shaded red in swing and yellow in stance to indicate that it acts in phase with motion (like elastic force) in swing and out of
phase with motion (like inertial force) in stance. This color scheme will be used throughout the manuscript. E) When the joint angle θ(t)=sin(ωt), the
angular velocity leads the displacement by 90° of the cycle period and the angular acceleration leads the displacement by 180° (i.e. out of phase). F) The
phase shift ϕ between the actuator moment M and the joint angle θ can be represented graphically. G) Allometric scaling laws enable phase shift ϕ to be
expressed in terms of the limb length L and the cycle period T. H) Our allometric scaling law for joint damping is consistent with literature references cited
in the gure (see also Table S1). See Tables S1 and S3 for parameters and Figs. S1 to S6 for more details and parameter sensitivity analysis.
Sutton et al. | 3
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G
- Phase shift
φ
’s dependence on T and L
kinetic
-I
viscous
-III
quasi-static
-II
J
- Behaviour 3
-0.5 0 0.5
Angle, (rad)
0
I
- Behaviour 2
-0.5 0 0.5
0
Moment, M (Nm)
H
- Behaviour 1
-0.5 0 0.5
0
Moment, M (Nm)
K
- Behaviour 4
-0.5 0 0.5
Angle, (rad)
0
L
- Behaviour 5
-0.5 0 0.5
Angle, (rad)
0
2
4
6
Moment, M (Nm)
A
- Phase shift
φ
’s dependence on T and L
kinetic
-I
viscous
-III
quasi-static
-II
D
- Behaviour 3
-0.5 0 0.5
Angle, (rad)
0
1
2
C
- Behaviour 2
-0.5 0 0.5
0
50
100
150
Moment, M (Nm)
B
- Behaviour 1
-0.5 0 0.5
0
100
200
Moment, M (Nm)
E
- Behaviour 4
-0.5 0 0.5
Angle, (rad)
0
410
-6
F
- Behaviour 5
-0.5 0 0.5
Angle, (rad)
0
0.1
0.2
Moment, M (Nm)
Swing work loops
Stance work loops
Phase angle,
φ
(deg)
Limb length, L (m)
Cycle period, T (s)
2
10
-3
10
-2
10
-1
10
0
10
1
10
2
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
10
1
0
45
90
135
180
1
3
5
4
underdamped
overdamped
quasi-
static - II
viscous - III
kinetic - I
Phase angle,
φ
(deg)
Limb length, L (m)
Cycle period, T (s)
10
-3
10
-2
10
-1
10
0
10
1
10
2
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
10
1
0
45
90
135
180
2
1
3
5
4
unstable
stable
φ=5
φ=85
φ=95
φ=175
2
φ=5
φ=85
φ=95
φ=175
210
-6
10
-7
10
-7
210
-6
410
-6
110
-7
.5 10
-7
510
3
110
4
110
4
210
4
quasi-
static - II
viscous - III
kinetic - I
Fig. 2. Muscle forces are resisted by inertial (kinetic zone), elastic (quasi-static), or viscous forces depending on limb length and cycle period of stepping.
A) Swing phase: Each contour of constant phase shift ϕ (in 10° increments) represents identical distributions of inertial, elastic, and viscous forces
during swing. Three regions (I, II, and III) appear in this plot corresponding to the plateaus in the gure. B) Work loop plotting the moment applied to the
joint by one antagonistic actuator versus the joint angle. Areas below the curve represent energy added to the body. Areas within the loop represent
energy dissipated due to viscosity. In each plot, the joint sweeps a range of 1 radian (approximately 60°) symmetrically about 0. In behavior 1, most
actuator work is converted into kinetic energy, indicated by the large yellow region. C) In behavior 2, most actuator work is stored as potential energy,
indicated by the large red region. D) In behavior 3, most actuator work is stored as potential energy and a noticeable amount is dissipated by viscosity,
indicated by the orange region. E) In behavior 4, most actuator work is dissipated by viscosity. F) In behavior 5, most actuator work is converted into
kinetic energy. G) Stance phase: Each contour of constant phase shift ϕ (in 10° increments) represents identical distributions of muscle work into
kinetic, viscous, and potential energy during stance. H) In behavior 1, most actuator work is converted into kinetic energy, similar to what is observed in
swing. I) In behavior 2, most actuator work is converted to kinetic energy, in contrast to the energy distribution in swing (C). J) In behavior 3, most
actuator work is converted to potential energy. K) In behavior 4, most actuator work is dissipated due to viscosity. L) In behavior 5, most actuator work is
converted into kinetic energy.
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slow motions in relatively large animals, swing would be quasi-
static but stance would be kinetic. This change occurs because
gravity dominates at this limb length and cycle period, and al-
though it stabilizes the leg in swing, it destabilizes the body’s pos-
ture during stance, changing the phase shift of the gravity term by
180°. This transition is evident when comparing Fig. 2C and I. In
our simulated behaviors, slow movements for large animals
have a phase shift of 0° in swing (Fig. 2A, behavior 2) and a phase
shift of 180° in stance (Fig. 2G, behavior 2). For both stance and
swing, however, the relationship between phase shift and the
dominant forces within the movement remains the same. There
are still three regions (inertia and gravity dominated, quasi-static,
and viscous dominated), with a 180° phase shift indicating iner-
tially or gravitationally dominated movements, a 0° phase shift in-
dicating elastically dominated movements and a 90° phase shift
indicating viscously dominated movements.
The relationship between phase shift and dominant force can
be illustrated more thoroughly by looking at a cut through of the
phase shift plot across time for both swing and stance. For a cycle
time of 1 s, the swing phase of limbs shorter than 10
−2
m is domi-
nated by the elastic and viscous forces within the limb, while the
swing phase of limbs longer than 10
−2
m is dominated by the
gravitational and inertial forces within the limb (Fig. 3A).
Likewise, for limbs shorter than 10
−2
m, the phase shift between
actuator force and limb position is close to 0°, while for limbs lon-
ger than 10
−2
m, the phase shift is 180°. For stance, at the same
cycle time, the shift from elastically dominated forces to inertially
dominated forces occurs at a length of 2 × 10
−1
m (Fig. 3D), with
the change in phase shift occurring at that length as well.
The same relationship between phase shift and dominant force
in the limb is also seen when looking at a cut across length for both
simulations. In swing, for a limb length of 5 × 10
−3
m, a movement
with a cycle time of 10
−2
s is dominated by viscous forces within
the limb, with a transition to elastic force dominance when the
cycle time becomes longer than 1 s (Fig. 3C). As the limb move-
ment is more and more dominated by elastic forces, the phase
shift changes from near 90° to near 0°. In stance, at the same
length scale (Fig. 3F), the relationship between cycle time and
the dominant limb force is very different than for swing, with in-
ertial forces being dominant at cycle times of less than 10
−2
s and
elastic forces becoming dominant at cycle times greater than
10
0.5
s (Fig. 3F). Despite the difference in which forces are present,
the relationship between phase shift and the dominant force is the
same in stance as for swing, with a phase shift of 180° reecting
the dominance of inertial forces, a phase shift of 0° reecting
the dominance of elastic forces, and a phase shift of 90° reecting
the dominance of viscous forces. The phase shift between actu-
ator force and limb angle thus illustrates which forces are domin-
ant within the limb, for both stance and swing, at a wide range of
length and time scales. For all simulations, a phase shift of 180°
indicates dominant inertial forces (the kinetic region), a phase
shift of 0° indicates dominant elastic forces (the quasi-static re-
gion), and a phase shift of 90° indicates dominant viscous forces.
The only difference between these movements (i.e. stance and
swing) is the direction of gravity; if gravity stabilizes the motion,
its phase shift is 0°; if gravity destabilizes the motion, its phase
shift is 180°. If the phase shift can be precisely measured, inter-
mediate values of phase shift can also indicate relative magni-
tudes of these forces: for example, in stance at a length of 5 ×
10
−3
m (Fig. 3F), at a cycle period of 7 × 10
−1
s, the viscous forces
and elastic forces are equal in magnitude, with a commensurate
phase shift of 45°, exactly halfway between a phase of 0° (elastic
force dominance) and 90° (viscous force dominance).
Recognizing the phase shift and the scale dependence of iner-
tia, gravity, elasticity, and damping also has major consequences
for how a simulated limb movement reacts to a perturbation.
Regions I and II can be divided between two areas, one in which
the limb is mechanically underdamped (examples seen in behav-
ior 1 [kinetic underdamped] and 2 [quasi-static underdamped]),
and one in which the limb is mechanically overdamped (example
seen in behavior 5 [kinetic overdamped] and behavior 3 [quasi-
static overdamped]). Depending on the region of a given move-
ment, responses may be stable overdamped (a perturbation is
quickly removed from the system through damping), stable
underdamped (a perturbation is eventually removed from the sys-
tem, but there are multiple oscillations), or unstable (perturba-
tions lead to uncontrolled movements). In our swing
simulations, all responses are stable, with large limbs being
underdamped and small limbs being overdamped. For large limbs
in region I, perturbation causes lasting alterations to the ongoing
motion unless excess kinetic energy is absorbed by the actuator
(Fig. 4B and C). However, for small limbs, the damping parameter
(Fig. 1H) is large enough that the joint will rapidly dissipate excess
kinetic energy (Fig. 4E and F). Large amounts of kinetic energy can
only be built up, however, when inertial forces are dominant (i.e.
when the phase shift is 180°). When the phase shift is 90° or less,
such as for motions with long cycle periods (i.e. quasi-static mo-
tions), energy is dissipated quickly relative to the cycle period, im-
plying that perturbations may not have a noticeable effect on the
limb’s motion (Fig. 4D).
In stance, the kinetic region, where the phase shift is 180°, is
quite large, and in most of this area, the posture is unstable.
The model predicts that locomotion with limbs longer than about
1 cm is unstable, implying that the locomotion of animals larger
than 1 cm will destabilize in response to perturbation if no feed-
back control is used. This results in the limb angle “exploding”
(i.e. the animal falls down) if it is perturbed (Fig. 4H, I, and L).
Interestingly, the posture of animals with limbs shorter than
about 1 cm is expected to be passively stable, implying that an
animal could stand up without any active muscle contraction as
long as their feet do not slip on the substrate. Animals with short
limbs could also passively reject perturbations during stance
(Fig. 4J and K).
Phase shift can be used to predict EMG patterns of locomoting
animals. Figure 5A and B overlay the reported swing and stance
durations of multiple animals’ walking on the plot of the phase
shift (horse (29), human (30), cat (31), rat (32), stick insect (33),
mouse (34), American cockroach (35), and fruit y (36)). Due to
the unique energetics and stability of motion within each region,
all these walking motions should arise from widely varying motor
output. To better understand how motor output should vary, the
torques required to actuate the hip of two model organisms, horse
(29) and the stick insect (37), were calculated via inverse dynam-
ics. The calculation was performed twice: once using the full set
of parameter values J,cr,kr,elas , and kr,grav and again with cr=0
and kr,elas =0 to determine roles of damping and elastic forces
within this behavior.
EMG patterns were approximated by assuming joint torque in
the retractor direction was applied by the retractor muscle and
torque in the protractor direction was applied by the protractor.
EMG patterns were advanced relative to the calculated torque to
mimic the approximately 50-ms lag between EMG activity and
muscle force production (38) (see also the supplementary
materials and Fig. S7). Our approximated EMG patterns cannot ac-
count for cocontraction of antagonist muscles and primarily re-
ect changes in muscle activity throughout the stepping cycle.
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The model predicts unique EMG recordings of hip muscle ac-
tivity driving the same joint motion in two animals, the horse
and stick insect. In the horse (which exists in the kinetic region,
phase shift 180°), gravity dominates the stance phase, so the mo-
ment due to gravity about the foot should be counteracted by the
hip retractor muscles at the beginning of stance (Fig. 5Ci) and
the hip protractor muscles at the end of stance (Fig. 5Cii). The
predicted activations are observed in EMG recordings from walk-
ing horses (Harrison et al. (29) and Fig. 5D). During swing, the
protractor muscles initially accelerate the leg with an impulse
(Fig. 5Ciii), and then the retractor muscles decelerate the leg
with an opposing impulse (Fig. 5Civ). As in stance, the model pre-
diction and experimental data conrm these activation pat-
terns. Because horses are large, removing elastic and viscous
10
-3
10
-2
10
-1
10
0
10
1
10
2
Cycle period, T (s)
0
0.5
1
relative
magnitude
elas. grav. visc. iner.
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
10
1
Limb length, L (m)
0
90
180
Phase shift
(deg)
0
101
100
10-1
45
10-2
Limb length, L (m)
10-3
10
2
10
1
10-4
Cycle period, T (s)
90
10
0
Phase angle,
φ
(deg)
10
-1
10-5
10
-2
10
-3
135
180
10
-3
10
-2
10
-1
10
0
10
1
10
2
Cycle period, T (s)
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
10
1
Limb length, L (m)
0
45
90
135
180
101
100
10-1
10-2
Limb length, L (m)
10-3
10
2
10
1
10-4
Cycle period, T (s)
10
0
Phase angle,
φ
(deg)
10
-1
10-5
10
-2
10
-3
A
Relative force magnitude and phase shift
in swing, T = 1.0 s
D
Relative force magnitude and phase shift
in stance, T = 1.0 s
B
Phase shift  versus cycle period and leg
length in swing
E
Phase shift  versus cycle period and leg
length in stance
0
0.5
1
relative
magnitude
0
90
180
Phase shift
(deg)
C
Relative force magnitude and phase shift
in swing, L = 0.005 m
F
Relative force magnitude and phase shift
in stance, L = 0.005 m
elas. grav. visc. iner.
elas. grav. visc. iner. elas. grav. visc. iner.
Fig. 3. The relationship between phase shift and dominant force can be illustrated by looking at cut throughs of the phase shift as either time or length is
held constant and the other is varied. A) At a constant cycle period of 1 s under swing phase conditions, the motion of small limbs (L < 10
−1
m) is
dominated by elastic and viscous forces, but the motion of large limbs (L > 1 m) is dominated by inertial force. This is reected by the corresponding plot of
phase shift ϕ, which is near 45° for small limbs and near 180° for large limbs at this cycle period. B) Plot of the full ϕ versus T and L landscape for swing. The
color shading and contours are the same as in Fig. 2A. Cut throughs are indicated by dotted (constant period T = 1 s) or solid (constant length L = 0.005 m)
black lines. C) For a limb of length 0.005 m, rapid oscillations (T < 10
−1
s) are dominated by viscous force and slow oscillations (T > 10 s) are dominated by
elastic force. This is reected by the corresponding plot of phase shift ϕ, which is near 90° for rapid oscillations and near 0° for slow oscillations. D) Same
analysis as in A, but for stance phase conditions, mass is much greater and gravity is shifted 180° relative to the joint angle (indicated by change from red
to yellow dashed line). Because of gravity’s phase shift, ϕ approaches 180° at shorter limb lengths than in swing. E) Plot of the full ϕ versus T and L
landscape for stance. The color shading and contours are the same as in Fig. 2G. Cut throughs are indicated by dotted (constant period T = 1 s) or solid
(constant length L = 0.005 m) black lines. F) Same analysis as in C, but for stance phase conditions. Inertial force dominates extremely rapid oscillations
(T < 10
−2
s), viscous force dominates intermediate-speed oscillation (10
−2
s < T < 1 s), and elastic force dominates slow oscillations (T > 1 s). Note that at
slow oscillations, the contribution of stabilizing elastic forces is greater than that from destabilizing gravitational forces.
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AB
C
D
EF
G
L K J
H
I
Behavior 1
Behavior 1
Behavior 2
Behavior 3
Behavior 4
Behavior 5
Behavior 2
Behavior 3
Behavior 4
Behavior 5
Fig. 4. Muscle forces are resisted by differing amounts of inertial, viscous, elastic, and gravitational forces depending on limb length and cycle period of
stepping. A) Swing phase: The plot from Figs. 2A and 3B, “attened” into two dimensions (see also Fig. S5). Each contour of constant phase shift ϕ (in 10°
increments) represents identical distributions of muscle work into kinetic, viscous, and potential energy during swing. Three regions appear in this plot
corresponding to the regions in Fig. 2. B) To test the response to perturbation, we applied a perturbation equal to 20% of the magnitude of the steady-state
actuation torque for one half of a cycle period (black bar on time axis). We then plot the subsequent joint angle versus time. To track how the amplitude
varies from cycle to cycle, the maximum angle reached during each cycle is traced with a spline (solid black line in each plot). In behavior 1, a small
perturbation alters the motion in a highly erratic way for many subsequent cycles because the energy cannot be easily dissipated. C) In behavior 2, energy
is dissipated rapidly compared to the natural period of oscillation and the system returns to its previous oscillatory pattern. D) In behavior 3, the system
does not oscillate because it is overdamped. E) In behavior 4, a perturbation does not cause erratic oscillation as in B, but it does alter the mean angle of
the ongoing motion. F) In behavior 5, a perturbation substantially alters the mean angle of the ongoing motion but does not cause erratic oscillation as in B
because it is overdamped. G) Stance phase: The plot from Figs. 2G and 3E, “attened” into two dimensions (see also Fig. S5). Each contour of constant
phase shift ϕ (in 10° increments) represents identical distributions of muscle work into kinetic, viscous, and potential energy during stance. In behaviors 1
(H), 2 (I), and 5 (L), unlike in swing, static posture is unstable. J) In regions 3 (J) and 4 (K), static posture is predicted to be stable because elastic torques at
the hip are greater than those from gravity.
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A
C D E
F G H
B
Cycle period (s) Cycle period (s)
Limb length (m)
* Stance and swing phase
were normalized separately
to their mean values.
Protractor muscle activation
Fig. 5. Our model predicts the disparate EMG patterns observed in animals of very different sizes. A) The phase shift during swing is plotted (0° in red, 90° in
orange, and 180° in yellow), along with the duration of swing and leg length of several species (horse (29), human (30), cat (31), rat (32), stick insect (33), mouse
(34), American cockroach (35), and fruit y (36)) Numerical values are presented in Table S2. Animals should require region-specic motor patterns to
accomplish the same motion. B) The phase shift during stance is plotted, along with the reported range of stance duration and leg length of the same species as
in A. This shows that for the same length and cycle period, stance and swing can have different phase shifts. C) For horse, the model predicts decreasing
retractor activity during the rst half of stance (inset: stance leg black, swing leg gray, retractor muscle blue, and protractor muscle red; gray background
stance and white background swing), bimodal protractor activity straddling the stance-swing transition, and reactivation of retractor muscles at the end of
swing. D) Averaged EMG recordings from three thoroughbred horses walking with mean stance phase duration 0.74 s and mean swing phase duration 0.44 s
(29). Harrison et al. (29) classify the triceps brachii’s long head and deltoideus muscles as shoulder exors that retract the foot and the biceps brachii and
supraspinatus as shoulder extensors that protract the foot. The scale of each muscle’s EMG was normalized to the maximum reading during a canter gait by
Harrison et al. (29). E) The model’s prediction does not visibly differ from a model in which no elastic or viscous forces are present, which is consistent with a
horse’s large size. F) For an animal on the scale of a stick insect, the model predicts decreasing retractor muscle activity throughout stance, a brief burst of
retractor muscle activity at the end of stance, and almost exclusively protractor muscle activity during swing (inset: stance leg black, swing leg gray, retractor
muscle blue, protractor muscle red, gray background stance, and white background swing). G) Averaged EMG recordings from 174 steps from ve stick insects
walking unsupported at their preferred speed (steps at 1 Hz, 70% in stance phase) (37). The coxal retractor (blue) and coxal protractor (red) actuate the
shoulder-like thorax-coxa joint during walking. Note that this EMG pattern differs substantially from the horse recording in Fig. 4D. The increase of EMG
activity in the retractor coxae muscle (blue) observed at the end of stance (gray background) is successfully predicted by our model and is a consequence of
the phase shift of stick insect locomotion representing a dominance of elastic forces. H) When the size-dependent effects of elastic and viscous forces are not
considered, the predicted EMG patterns are dramatically different from the recordings, emphasizing the predictive capability of our modeling framework.
Our modeling framework explains interspecies differences in the EMG patterns of walking animals and makes testable predictions for future experiments.
This is, of course, with the caveat that EMG patterns can be suboptimal reections of the forces within the system (38, 39). For more details, example joint
torques for horse and stick insect are presented in Fig. S7.
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parameters does not noticeably affect the predicted EMG
(Fig. 5E).
We also used the model to predict EMG of stick insect walking,
which, in contrast to the horse, exists in the quasi-static region
(phase shift 0°). Due to the small size and slow movement of the
stick insect, the model’s prediction of its EMG pattern is dramatic-
ally different from that for horse. Figure 5B predicts stance to be
nearly kinetic (inertia dominated) for the stick insect, meaning
that as in the horse, stance begins with hip retractor (i.e. the
thoraco-coxal retractor) activation (Fig. 5Fi). However, due to its
small size, the stick insect experiences relatively large viscous
moments at the shoulder, which modify the relative phase of
muscle activation. Furthermore, it experiences relatively large
elastic forces at the shoulder, which act opposite to gravity.
Thus, near the end of the stance phase, the gravitational forces
are counteracted almost entirely by elastic forces, resulting in
nearly no muscle activation of the coxal retractor or protractor
(Fig. 5Fii). As the swing phase begins, the foot is lifted from the
substrate, and the gravitational force that had counteracted elas-
tic force disappears, requiring the retractor to greatly increase its
activation to prevent the leg from “snapping” forward like a
mousetrap (Fig. 5Fiii). This unexpected feature is also observed
in kinematics and experimental EMG recordings from freely walk-
ing stick insects (37) (Fig. 5G) and is a consequence of the domin-
ance of elastic forces in stick insect locomotion. As the swing
phase continues, the protractor (i.e. the thoraco-coxal protractor)
activates to overcome the viscous forces that resist the swing
phase motion (Fig. 5Fiv), as predicted by the phase shift during
swing (Fig. 5A). Since elastic and viscous forces dominate the loco-
motion of small animals, removing the elastic and viscous ele-
ments from the model greatly reduces the model’s prediction
accuracy (Fig. 5H).
In both models of horse and stick insect locomotion, the phase
shift (ϕ) between force and limb angle indicates which forces are
most dominant during the movement and helps to explain experi-
mental EMG patterns seen in the literature.
Discussion
To quantify the dominant force (gravitational, elastic, viscous,
and/or inertial) during a behavior, we have described a nondimen-
sional number: the phase shift (ϕ) between actuator force and limb
displacement (Fig. 1). Using allometric scaling laws, we expressed
the phase shift (ϕ) in terms of two quantities: limb length and cycle
period. By modeling both swing and stance for a wide range of dif-
fering limb sizes and limb cycle times, we identied “regions” of
limb length and cycle period in which actuator force is primarily
resisted by the body’s inertia, gravitational force, elastic force,
or viscous force (Fig. 2). In each region, movement has very differ-
ent responses to perturbation (Fig. 4) and is driven by dramatically
different patterns of force over time. However, for both swing and
stance, the relationship between phase shift (ϕ) and the dominant
force within the behavior was the same, with inertially dominated
behaviors having a phase shift of 180°, quasi-static (elastically do-
minated) behaviors having a phase shift of 0°, and viscous domi-
nated behaviors having a phase shift of 90° (Fig. 4). We then
showed how this phase shift can be used to predict the differing
EMG patterns observed for a large locomoting animal (horse)
and a small locomoting animal (stick insect, Fig. 5).
Despite the value of the phase shift for understanding broad
trends in the control of movement, this framework has limitations
that can be addressed in future work. Because this study was
based on allometric scaling of mechanical properties like mass,
viscous damping, and joint stiffness, it cannot account for
species-specic variations. This framework is not meant to be a
replacement for species-specic investigations of the mechanical
properties of animal legs (9, 10, 13, 24, 25, 40, 41). Instead, it is in-
tended to facilitate a comparison of dynamics in legged locomo-
tion across many different scales. Furthermore, while this
framework should be broadly applicable to other periodic mo-
tions, e.g. insect apping-wing ight (42, 43) or soft-bodied feeding
(44), the parameters within this model were tuned with leg joints
in mind and the model may not accurately describe these motions
without some retuning of parameters. Finally, this framework
treats the leg as a single rigid link (a commonly used simplication
(45, 46)), despite legs utilizing many joints with coupled dynamics
(47, 48). We anticipate that the basic framework presented here
will lead to future studies that rene its predictions and extend
its applicability to more systems.
The relationship between phase shift and dominant force re-
sults from physics and is explained by classical mechanics
(Fig. 1and supplementary material). For any linear second-order
system, these relationships will hold. Even if the values of the
damping, elasticity, and inertia coefcients differ (i.e. allometric
scaling laws must be adjusted), the relationship between phase
shift (ϕ) and the dominant force within a behavior will still hold.
Thus, we believe this framework could readily be extended to en-
compass different environmental media within which behavior
occurs. An example of such a scenario would be legged locomo-
tion through water. The water would increase the inertia and
damping of the leg, but that increase in inertia and damping
would in equal measure increase the phase shift between the ac-
tuator force and limb position. Similarly, at the speed at which
ies ap their wings, the nonstationary dynamics of air become
important (49). Once again, the framework could readily be ad-
justed to incorporate these environmental features, by incorpor-
ating the uid forces that the wings must overcome, as a
function of the kinematics. We believe the phase shift analysis
we present could be applied to different neuromechanical sys-
tems and environments as long as the mass, stiffness, and damp-
ing parameters are adjusted to reect that neuromechanical
system and environment.
Analysis of phase shift links multiple nondimensional numbers
that describe locomotion at particular scales. The interfaces be-
tween our named regions of force dominance (i.e. kinetic, viscous,
and quasi-static) are not only level curves of ϕ; they are also level
curves of other nondimensional numbers. For example, the
Froude number is the ratio between inertial centripetal force
and gravitational force, Fr =v2
gL. Level curves of the Froude number
run parallel to the boundary between kinetic and quasi-static re-
gions, for example, between regions I and II in Fig. 2A. As another
example, the Reynolds number is the ratio between inertial force
and viscous force in a owing uid, Re =ρvL
μ. Level curves of the
Reynolds number run parallel to the boundary between kinetic
and viscous regions, for example, between regions I and III in
Fig. 2A. A third example is within the quasi-static region (region
II), in which the ratio between gravitational and elastic forces
(quantied by “specic modulus”) is the important dimensionless
number. The phase shift between force and position thus illus-
trates which dimensionless quantity is most important for a given
motion—showing, for example, that when the phase shift is 180°,
Froude number is very relevant for a behavior (such as horse loco-
motion), whereas when the phase shift is close to 0, Froude num-
ber is not very relevant for a behavior (such as stick insect
locomotion). Furthermore, because ϕ varies continuously over
the entire space of limb lengths and cycle periods, it may facilitate
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meaningful comparison between apparently similar motions in
which different forces dominate.
What needs to be measured experimentally to predict neural
control patterns? Measuring ϕ between force and displacement
while moving the limb in a cyclic pattern with frequency ω would
enable an experimentalist to quantify the dominant force within a
limb motion. Because this framework does not directly rely on
muscle contraction properties, the limb can be moved by any
sort of force in such an experiment, including muscle force, iner-
tia, a mechanical manipulator, or an applied magnetic eld (50,
51). Direct measurement of ϕ is important because although this
study employed allometric scaling to predict ϕ at different scales,
allometric scaling is approximate and does not explain all vari-
ability in mechanical parameters between species (though the
overall predictions of the model are robust to variations in key
parameter values; Fig. S6). Moreover, correlations between differ-
ent forces and kinematics can be used to indirectly infer the distri-
bution of muscle work into elastic potential, gravitational
potential, kinetic, and viscous (dissipated) energy (12). To create
a more detailed model of a limb segment, an experimentalist
may approximate the model parameters kr,elas ,kr,grav, and cr by
measuring the limb’s mass, its length, and ϕ in response to two dif-
ferent forcing periods, T.
Analysis of phase shift also informs the construction of more
accurate neurorobotic models of animals. Because our analysis
does not consider muscle dynamics, robotic motions can also
be classied by ϕ. Typical robot construction methods, in which
massive segments are actuated by electric gearmotors, produce
robots dominated by inertial and gravitational forces, much
like large animals. Such a robot would likely exhibit values of ϕ
near 0° during slow motions and 180° during rapid motions,
which would be a poor model of a small arthropod, whose ϕ value
should be between 0° and 90° in all contexts (Fig. 5A and B). As
pointed out by Hooper (11), to make meaningful comparisons be-
tween an animal and a robotic model, it is important for the robot
to match the fundamental relative physics of its inspiration
(relative inertial, elastic, and viscous forces). In the future, engi-
neers may construct more accurate and useful neurorobotic
models of insects by altering the robot’s mechanics (e.g. by add-
ing springs that resist motor output (52)) and slowing its speed of
operation to ensure that its ϕ values match the model animal’s.
Such alterations would guarantee the same energy allocation
(although at different magnitude) between robot and animal.
Matching energy allocation will both improve robots as models
for animals and allow animal neural control patterns to be
used more effectively in robots.
Acknowledgments
The authors would like to thank four anonymous reviewers,
whose comments signicantly improved an earlier draft of the
paper.
Supplementary material
Supplementary material is available at PNAS Nexus online.
Funding
G.P.S. was funded by the UK MRC (MR/T046619/1), and N.S.S.,
R.D.Q., and H.J.C. were funded by NSF DBI 2015317, both as part
of the NSF/CIHR/DFG/FRQ/UKRI-MRC Next Generation Networks
for Neuroscience Program. G.P.S. was also funded by the Royal
Society (UK) (UF120507) and the US Army Research Ofce
(W911NF-15-038).
Author contributions
G.S. and N.S. designed and performed the research, analyzed the
data, and wrote the paper. R.Q. and H.C. designed the research
and wrote the paper.
Data availability
The data set and models used for this work will be made available
online. The data can also be taken from the individual references
cited. Simulations and parameters for this work can be found on
this GitHub site: https://github.com/nicksz12/dynamicScaling.
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