RESEARCH ARTICLE
Effects of body movement on yaw motion in
bipedal running lizard by dynamic simulation
Jeongryul Kim1, Hongmin Kim2¤, Jaeheung Park3,4*, Hwa Soo KimID5*, TaeWon Seo6
1 Center for Healthcare Robotics, Korea Institute of Science and Technology, Seoul, South Korea, 2 School
of Mechanical and Aerospace Engineering, Seoul National University, Seoul, South Korea, 3 Department of
Intelligence and Information, Seoul National University, Seoul, South Korea, 4 Advanced Institutes of
Convergence Technology (AICT), Suwon, Gyeonggi-do, South Korea, 5 Department of Mechanical System
Engineering, Kyonggi University, Suwon-si, South Korea, 6 School of Mechanical Engineering, Hanyang
University, Seoul, South Korea
¤ Current address: Institute of Advanced Machines and Design (IAMD), Seoul National University, Seoul,
South Korea
* park73@snu.ac.kr (JP); hskim94@kgu.ac.kr (HSK)
Abstract
Lizards run quickly and stably in a bipedal gait, with their bodies exhibiting a lateral S-shaped
undulation. We investigate the relationship between a lizard’s bipedal running and its body
movement with the help of a dynamic simulation. In this study, a dynamic theoretical model
of lizard is assumed as a three-link consisting of an anterior and posterior bodies, and a tail,
with morphometrics based on Callisaurus draconoides. When a lizard runs straight in a sta-
ble bipedal gait, its pelvic rotation is periodically synchronized with its gait. This study shows
that the S-shaped body undulation with the yaw motion is generated by minimizing the
square of joint torque. Furthermore, we performed the biomechanical simulation to figure
out the relationship between the lizard’s lateral body undulation and the bipedal running
locomotion. In the biomechanical simulation, all joint torques significantly vary by the waist
and tail’ motions at the same locomotion. Besides, when the waist and tail joint angles
increase, the stride length and duration of the model also increase, and the stride frequency
decreases at the same running speed. It means that the lizard’s undulatory body move-
ments increase its stride and help it run faster. In this study, we found the benefits of the
lizard’s undulatory body movement and figured out the relationship between the body move-
ment and the locomotion by analyzing the dynamics. In the future works, we will analyze
body movements under different environments with various simulators.
1. Introduction
The running patterns of vertebrates vary depending on the species. The development of these
running patterns is known to be an evolutionary result of adapting to the environment. Biolo-
gists use several methods, such as comparative approaches, to study the principles of observed
running patterns. In the field of biomechanics, the dynamics analysis technique is used to ver-
ify the principles of running patterns by simulation. In this study, we explain the principle of
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OPEN ACCESS
Citation: Kim J, Kim H, Park J, Kim HS, Seo T
(2020) Effects of body movement on yaw motion
in bipedal running lizard by dynamic simulation.
PLoS ONE 15(12): e0243798. https://doi.org/
10.1371/journal.pone.0243798
Editor: Marc H.E. de Lussanet, University of
Mu¨nster, GERMANY
Received: June 1, 2020
Accepted: November 26, 2020
Published: December 31, 2020
Copyright: © 2020 Kim et al. 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: All relevant data are
available at https://github.com/imationzzz/
LizardSimulation.
Funding: H. S. Kim was supported by the GRRC
program of Gyeonggi province [GRRC KGU 2020-
B02, Research on Innovative Intelligent
Manufacturing Systems]. 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 running pattern of a fast bipedal running lizard. More specifically, we investigate the effect
of lateral body undulation on bipedal running at high speed by simulating a theoretical
mechanical model of a running lizard.
A lizard can run in a straight line with a very stable posture even at a high speed. While run-
ning, it has been observed that a lizard bends its body and tail in the horizontal plane, which is
referred to as a lateral body undulation, in the same manner as snakes and fish. Because this
movement is quite large and dynamic, we hypothesize that such a body movement plays an
important role in the bipedal lizard’s locomotion.
In previous studies, the lateral undulation of lizards has been observed. Ritter observed
the unique S-shaped lateral body undulation of a lizard with its various running speeds [1].
Whether such a lateral undulation is a result of active movement or passive movement has also
been investigated. Ritter analyzed the electromyography and kinematic data of the epaxial
muscle of a running lizard and concluded that the lizard actively bends its waist to stabilize its
trunk [2,3]. Similarly, by analyzing the electromyography of the epaxial muscle, Bennett et al.
confirmed that at a low speed, the epaxial muscle resists the body’s torsional force, whereas at
a high speed, the muscle is used to bend the waist [4]. These studies showed that the lateral
body undulation is not a result of a passive external force, but of active internal moments.
However, these studies did not explain the actual purpose of such a lateral body undulation or
its effect on locomotion.
The effect of a lizard’s tail was also studied. Jusufi et al. and Libby et al. studied the balanc-
ing effect of the tail when a lizard free-falls or jumps [5,6]. Gillis et al. showed that after tail
loss, a lizard could no longer control its body when it jumps [7]. Kim et al. improved the
motion stability of a six-legged water-running robot in the yaw and pitch direction using a two
degrees-of-freedom (DOF) active tail [8]. MSU Tailbot can control body angle for a safe land-
ing with swing tail in the air [9]. Such studies have focused on the tail’s role in the lizards’ pitch
and roll. On the other hand, TAYLRoACH can change its running direction as 90˚ turns at a
constant rotational speed of 360˚/s using a 1 DOF active tail in the horizontal plane [10]. Sala-
mandra Robotica controlled by the Central Pattern Generator (CPG) is similar to that of a real
salamander in the horizontal plane [11]. These robots studied the yaw movement of the body,
but the effect of the periodic lateral bending of the body during running was not investigated.
The contributions of this paper are to 1) find the benefits of the lateral undulations of the
lizard body during bipedal running, and 2) confirm the relationship between the lizard’s undu-
latory body movement and the bipedal running locomotion. The target of this study, Calli-
saurus draconoides, is a creature that can run very fast compared to its size, and the body of the
lizard dynamically moves in the yaw direction. We analyze the effect of the lateral body undu-
lation of a steady-state bipedal running lizard on its yaw motion via a dynamic theoretical sim-
ulation. Besides, we simulate the effect of the undulatory body motion on bipedal running
locomotion through a dynamic simulation using a lizard biomechanical model with the legs.
The biomechanical model derived in this study has 33 DOFs, based on the morphometrics of
Callisaurus draconoides. The extensive simulations verify that the undulatory body movements
dominantly affects its bipedal running locomotion.
2. Analysis for the benefit of the lizard’s undulatory body
movement
When a lizard runs straightly with a bipedal gait, its posterior body swings symmetrically and
periodically in the horizontal plane. Furthermore, the lateral undulations of a lizard generate a
periodical S-shaped movement. When the lizard’s running is stable, the angle between the
trunk and the tail is very close to zero [12]. It means that if the lizard ideally balances the body,
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the undulatory body movement will occur only in the horizontal plane. Therefore, we assumed
an ideal situation that the lizard was stably balanced, and modeled the lizard’s undulatory
body movement only in the horizontal plane. In addition, we studied the relationship between
the theoretical model’s running performance and the movement of the lizard body using the
optimization method and assuming the data obtained from the previous studies.
A. Three-link theoretical model of a lizard
After observing the Callisaurus draconoides, a theoretical model comprising a three-link sys-
tem with an anterior and posterior bodies, and a tail is devised as shown in Fig 1. The joints
are located such that each link can rotate in the horizontal plane relative to its neighboring
links. Animal muscles have both rigid and flexible properties so that when generating force
actively, they are rigid but when absorbing forces like springs, they become passive and flexi-
ble. This study modeled a lizard’s joint as the rotating joint to mimic the active muscle and
extract joint torque. The forelegs are attached to the anterior body and the hind legs to the pos-
terior body. The moment of inertia for the segment of the body is calculated by assuming the
segment as homogenous cylinder. The morphometrics of Callisaurus draconoides obtained
from the real measured data in [13]. For such lizards as Anolis sagrei and Anolis carolinensis,
which are similar to the Callisaurus draconoides and run with a bipedal gait, Legreneur et al.
found that the mass percentages of the forelegs relative to the total body mass were 2.05% and
1.8%, and those for the hind limbs were 6.2% and 4.25%, respectively [14]. Thus, the effect of
the leg motion is assumed to be negligible, and the leg masses are included in the anterior and
posterior bodies. The morphometrics of Callisaurus draconoides used for the three-link theo-
retical model are summarized in Table 1, and the details of the derivation of the three-link the-
oretical model are presented in Appendix A1.
B. Kinematic analysis of a lizard’s bipedal running
Fig 2 shows the forces and moments acting on each body segment of the lizard when it runs
straightly with a bipedal gait. The roles of the ground reaction force (GRF), ground reaction
moment (GRM), and lateral undulation of the waist and tail of a lizard must be uncovered for
the analysis. When a lizard runs, its legs exert a force on the ground with every stride, and as a
result, a fore-aft ground reaction force GRFx (fx) and lateral force GRFy (fy) are produced at its
feet. Besides, GRFz (fz) occurs in the height direction of the lizard’s foot. GRFz is closely related
to the roll and pitch motion of the lizard body. However, the body of the lizard is assumed to
have relatively little movement in the roll and pitch directions compared to the yaw direction
from the previous studies [12,15]. Therefore, the effect of GRFz on the lizard’s undulatory
body movement was judged to be small, and the force analysis of the lizard was performed in
Fig 1. Three-link theoretical model of a lizard, Callisaurus draconoides, in the horizontal plane. The moment of
inertia for the segment of the body is assumed as a homogenous cylinder. Since the mass percentages of the forelegs
and the hind limbs are low relative to the total body mass, the leg motion’s effect is assumed to be negligible.
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the horizontal plane, excluding GRFz, as shown in Fig 2. If we consider that the right leg
swings first while the left leg is in the stance stage, the GRF produces both a translational force
and a moment Mf, which acts on the posterior body. In the case of the swinging of the left leg,
while the right leg is in the stance stage, the lizard must bring its left leg into the appropriate
position by moving it by an amount equal to the distance from the position of the left leg at the
first swing, which is called the stride length.
In order to implement the stride length for the left leg, the posterior body is required to
rotate clockwise. Hence, a moment Mswing, which acts on the posterior body, is required to
realize the second swing. The posterior body is unable to rotate by itself, which implies that
Mswing is required to be produced relative to the posterior body. Because Mswing is not equal to
Mf, the GRM (Mz) generated by the twisting of the foot relative to the ground and the body tor-
que (Tbw, Tbt) generated by the joint movement of the bodies relative to the posterior body are
used to generate Mswing. A lizard is able to run forward by repeating these steps.
C. Dynamics analysis for the lizard’s undulatory body movement with
optimization method
The dynamics analysis of the lizard theoretical model is performed according to the process in
Fig 3(a). We set the initial desired value of the lizard body theoretical model. In the theoretical
Fig 2. Moments and forces periodically acting on the lizard in a period in the horizontal plane. When a lizard runs, its legs exert a force
on the ground with every stride, a fore-aft ground reaction force GRFx (fx), lateral force GRFy (fy), and GRM are produced at its feet. Also,
the body torques (Tbw, Tbt) are generated by the bodies’ joint movement. A lizard controls these forces, moments, and torques to run
forward by repeating the gait.
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Table 1. Morphometrics of Callisaurus draconoides used for the three-link theoretical model.
Callisaurus draconoides
Total length (mm)
192
Anterior body length (mm)
66
Posterior body length (mm)
42
Tail length (mm)
84
Anterior body width (mm)
20
Posterior body width (mm)
20
Tail width (mm)
6
Total mass (g)
10
Anterior body mass (g) (with fore leg added)
4
Posterior body mass (g) (with hind leg added)
4
Tail mass (g)
2
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simulation, torque is generated in a proportional-derivative (PD) controller to follow the body
joint’s initial desired angle. We extracted the lizard joint torque function over one cycle in the
simulation. Next, we have defined a function that integrates the square of this torque value
over one period. As a result, jd_minimized of the body joint is derived when the function is mini-
mized by changing the desired value of the lizard body joint through iteration.
After both the desired waist and tail movements are generated, the theoretical simulation
can be formulated as shown in Fig 3(b). This type of periodic angular motion is assumed to be
a sine function in their study on the design of a machine with fish-like biomimetic locomotion
in a liquid environment [16]. The waist and tail movements of a lizard can be realized via the
use of a sine function similar to the previous research method. These assumptions are not per-
fectly suited to mimic target animals but have the advantage of being able to reveal the effects
and relationships of animal movements through simple calculations. The absolute angle of the
posterior body is controlled by the GRM, but Tbw and Tbt generated by the waist and tail move-
ments affect the magnitude of the GRM and the model’s running dynamics. GRM (Mz), Tbw,
and Tbt are described in Fig 3(c).
According to Kubo et al. in [17], the pelvic rotation angle varies with the species of lizard
and ranges from 11.7˚ to 25.8˚. Reilly et al. observed the pelvic rotation to comprise a periodic
sinusoidal movement [18]. The objective of our study is not to obtain a numerically perfect
simulation of a running lizard, but to verify the effect of its lateral body undulation. Accord-
ingly, the pelvic rotation is approximated as a sinusoidal movement of 20 sin(2πt/Tp) ˚ (Tp =
0.092 s).
The magnitude of the GRF is mathematically derived while referring to related studies.
GRFx, which acts in the forward-aft direction, was modeled as a sine function by Aerts et al. in
[19]. The same mathematical derivation of GRFx is used in the simulation presented herein. In
the case of GRFy, the direction of the force is always towards the midline of the body. The
maximum value of GRFy is close to that of GRFx, whereas the impulse is much larger [20]. For
the simulation, the direction, relative magnitude, and impact of GRFy are modeled according
to the studies in [20,21], and mathematically formulated as a sine function. The GRF depends
on the running speed of the lizard model, and thus, in the steady-state running, the GRF is
periodically repeated in every stride. Therefore, the GRF was approximated as a sine function
similar to the previous studies, as shown in Fig 4.
The position of the foot relative to the posterior body for every stride is determined from
the leg length and angle data as presented in [13]. The location of the right foot is initially set
as 38 mm away and at −45˚ from the center of mass of the posterior body, while the left foot is
38 mm away and at +45˚. Without the loss of generality, the feet are modeled in a no-slip state
Fig 3. (a) Overall architecture of the dynamics analysis for benefits of the undulatory body movement, (b) simulation for
one cycle of the gait of the lizard model with angular body movement (waist, tail, and leg controller are a proportional-
derivative (PD) controller) and (c) forces and moments in the lizard model.
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during contact with the ground. The foot position is fixed, and the center of mass moves rela-
tive to the feet according to the GRF.
The movement of the waist and tail is minimized by the optimization, as shown in Fig 3(a).
The desired angular movements of the waist and tail, which are assumed to be represented by
sine functions, can be formulated as shown in Eq (1).
Jd;i¼1;2 ¼ ai  sinð2p
TP
t þ biÞ
ð1Þ
The angular body movement is achieved by combining the amplitude (ai) and phase differ-
ence (bi). As this study focuses on the effect of the body motion at high velocities, the fre-
quency during running is fixed The angular movements of the waist and tail vary depending
on the constants ai and bi in Eq (1). The values of the ai and bi determine the sinusoidal profile
of the desired angular movement over time; then, the PD controller generates the waist and
tail joints torques Tbw and Tbt to follow the desired profiles, as shown in Fig 3(b). Therefore,
we determined ai and bi as the design parameters for the optimization. Also, we derived the
objective function with the appropriate assumptions, as shown in Eq (2).
Minimize FðTbwðtÞ; TbtðtÞÞ ¼
Ð Tp
0 ½T2
bwðtÞ þ T2
btðtÞdt
ð2Þ
Subject to GRM  Const1 and q2 ¼ 20  sin
2p
Tp t


.
The objective function to be minimized is set as (Tbw
2 + Tbt
2). The term (Tbw
2 + Tbt
2) can be
expanded as ((Tbw + Tbt)2 + (Tbw − Tbt)2)/2, and thus, minimizing ðT2
bw þ Tbt
2Þ implies that the
magnitude (Tbw + Tbt) is minimized along with the uniform distribution of the moments to
the joints via the term (Tbw − Tbt). The moment generated at either the waist or tail joint acts
as a burden on the active joint because it has to generate a relatively large moment. The uni-
form distribution of the moments over the joints reduces the load, and because a real lizard is
presumed to act in the same manner, (Tbw
2 + Tbt
2) is selected as the objective function.
GRM (Mz) is a constant that can be varied to produce different outcomes. Previously, we
assume that the GRM (Mz) produced by the lizard is very small. Thus, in the optimization pro-
cess, the GRM (Mz) is set to a minimum value. Here, q2 is the required absolute angle for the
posterior body when the lizard is running at 4 m/s, as determined via a simple calculation
Fig 4. Profiles of (a) GRFx and (b) GRFy approximated as a sine function similar to the previous studies [20,21].
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comprising the stride length and time. We performed the optimization by sequential quadratic
programming (SQP) using fmincon in MATLAB1.
D. Result of the dynamic analysis for a lateral undulation of a bipedal
running lizard
For various lizard theoretical models of different lengths and mass ratios, their lateral body
undulations are generated in the simulations. We chose the size and weight of the lizard model
based on the real lizard. It is worthwhile to note that the optimized motions may be affected by
the size and weight of lizard model. Therefore, in this study, lizard models of different sizes
and weights were used for simulations. The dynamic theoretical simulation is implemented in
MATLAB1 with the numerical integration by the fourth-order Runge–Kutta method. The ini-
tial velocity of the model is set as V0 = 4 m/s. The model movement resulting from the optimi-
zation is presented in Fig 5. As expected, to minimize ðT2
bw þ Tbt
2Þ, the waist and tail joints
are required to generate moments that oppose each other, which results in a natural S-shaped
movement of the body.
We have obtained the joint angles of the actual lizard’s waist and tail by capturing the
images from a video file related with its running motion, as shown in Fig 6(a) from [22]. Then,
we normalized them for comparison with the simulation. Since the shape of the lizard’s undu-
latory body movement is determined by the phases of the waist and tail movement angles, we
compared the phase difference between the actual lizard’s and the simulated waist and tail
joint angles as shown in Fig 6(b). The phase difference of the waist was 0.002T (T is the
period), and the phase difference of the tail was 0.067T, indicating that the dynamics theoreti-
cal model and the real lizard’s waist and tail movements were very similar. In conclusion, we
found that real lizards can minimize the sum of squares of torque by moving their body in an
S-shape.
3. Lizard biomechanical modeling: For relationship between the
undulatory body movement and the locomotion
In the previous section, we observe that the body movement of the lizard’s theoretical model
when the joint torques are minimized. In addition, we have studied the relationship between
Fig 5. Movements of the lizard models calculated by the dynamics analysis when the joint torques are minimized. (
indicates the point at which the foot contacts the ground). (a) l1 = 66 mm, l2 = 42 mm, l3 = 84 mm, m1 = 4.0 g, m2 = 4.0 g, m3
= 2.0 g, (b) l1 = 60 mm, l2 = 40 mm, l3 = 92 mm, m1 = 4.0 g, m2 = 4.0 g, m3 = 2.0 g, (c) l1 = 72 mm, l2 = 42 mm, l3 = 78 mm, m1
= 4.0 g, m2 = 4.0 g, m3 = 2.0 g, (d) l1 = 66 mm, l2 = 42 mm, l3 = 84 mm, m1 = 4.2 g, m2 = 4.0 g, m3 = 1.8 g, (e) l1 = 66 mm, l2 =
42 mm, l3 = 84 mm, m1 = 3.8 g, m2 = 4.0 g, m3 = 2.2 g.
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the body movement and the joint torques of the theoretical model. In this section, to confirm
the relationship between the lizard’s undulatory body movement and bipedal running locomo-
tion, we established the lizard biomechanical model including its legs. To clearly identify the
effect of the body movement, we have constructed a biomechanical simulation to generate the
motion of the biomechanical model without control and to extract the joint torque through
inverse dynamics, to exclude the effect of control performance.
A. Bipedal running lizard biomechanical model
The biomechanical model of a bipedal running lizard is established by morphometrics of the
Callisaurus draconoides based on [13], as in the previous section. The kinematics model is pre-
sented in Fig 7. The body of the model consists of three links referred to as body1, body2, and
body3. In addition, the fore and hind legs of the biomechanical model comprise three links.
Fig 6. (a) Captured image of the running lizard for the joint angles of waist and tail from [22] and (b) comparison of the phase differences of the
waist and tail joint angles between the real lizard and the dynamic model.
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Fig 7. Kinematic biomechanical model of the lizard to confirm the relationship between the undulatory body movement and
bipedal running locomotion: (a) bodies, (b) joints, and (c) lengths. The number of links used in the model is fifteen, and the
number of joints is 33.
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The total number of the links used in the kinematics model of the lizard is fifteen, as shown in
Fig 7(a).
The joints used for describing the angles of each body are set as rotational joints with one
DOF. The fore and hind legs comprise shoulder, knee, and ankle joints. The shoulder and
ankle joints are spherical joints with three DOFs. The knee joint is a rotational joint with one
DOF. We add two linear joints as virtual joints to calculate the position of the biomechanical
model in the fixed coordinate frame. The number of joints in the kinematics model is 33, as
shown in Fig 7(b). The length and mass of the links of the biomechanical model, based on
[13], are listed in Table 2.
The center of the mass of each link is in the middle of the link. The positions of the center
of the mass are explained in the Appendix A2 using the lengths and the joint angles of the liz-
ard biomechanical model.
B. Joint angles of the biomechanical model for the bipedal running
locomotion
We simulate the locomotion and joint movement of the lizard’s biomechanical model. In this
simulation, the joint angles of the robot model are simplified as a sine function according to
the actual lizard’s joint angle. The sine functions of the joint motion are defined as follows:
qi ¼ ai  sin
2p
period t þ bi


þ ci; i ¼ 3;    ; 33
ð3Þ
where ai is the magnitude, bi is the phase difference, and ci is the offset of the sine function. As
we change these variables, the joint angle of the model changes dramatically.
The movement of the biomechanical model is laterally symmetric because it is assumed
that the model runs in a straight line without any deviation. To realize symmetric movement
in the lateral direction, the body joint angle has two variables of magnitude ai and phase differ-
ence bi without an offset ci. In addition, the left and right hind legs perform the same symmet-
ric movement with a 180˚ phase difference. The joints of the forelegs are fixed at a constant
angle to mimic a real lizard’s posture. We summarized the motion of each joint in Table 3.
The ankle joint of the lizard model is set with the foot always facing forward. This setting is
determined by observing that the foot of a real lizard faces forward when the lizard’s foot
touches on the ground. Therefore, we set the ankle joint angle to x, z = 0 and fix the y-axis
value in the fixed coordinate frame, such that lizard’s foot is facing forward when it touches
the ground at any time.
Table 2. Size and mass of a lizard biomechanical model.
Lizard model links
Length (mm)
Width (mm)
Mass (g)
Body1
66
16.5
3.6
Body2
42
13
2.9
Body3
84
4.5
2.0
Body4, 7
16
4
0.11
Body5, 8
12
3
0.06
Body6, 9
15
3
0.03
Body10, 13
19
7.5
0.36
Body11, 14
21
4.5
0.18
Body12, 15
13
3.5
0.08
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In the biomechanical simulation, it is assumed that the body of the model is maintained at a
constant distance from the ground. The lizard’s body is observed to have a small up and down
movement [12,15,23]. Thus, in order to simplify the simulation and focus on the lateral body
undulation of the biomechanical model, we assume that the body of the biomechanical model
only moves in the horizontal plane.
The distance between the body of the biomechanical model and the ground is determined
based on the trajectory of the hind legs. Based on the body position, the z-direction position of
the ground should be between the minimum of the foot end and the minimum of the hind
ankle. If the ground position is lower than the minimum position of the foot end of the hind
foot, the hind foot of the model cannot reach the ground. On the other hand, if the ground
position is higher than the minimum position of the hind ankle, then the model’s ankle
touches the ground, which is different from the actual lizard’s movement. Therefore, in this
study, we set the ground position of the body from the center of the foot at the minimum foot
position in the z-direction.
Before the hind foot touches the ground, the posture of the foot is fixed, as described in the
previous section and as shown in Fig 8(a). When the position of the foot end of the hind foot is
lower than the position of the ground, the angle of the ankle is changed such that the position
of the foot end is the same as the position of the ground, as shown in Fig 8(b). If the position of
the foot end of the back foot is higher than the position of the ground, the hind foot returns to
its original posture as shown in Fig 8(c). Therefore, when the foot touches the ground, the
position of the foot is fixed, and after the foot leaves the ground, it returns to its original pos-
ture. While in contact with the ground, the movement of the model is implemented based on
the fixed position of the foot end of the hind leg.
When the model is in the air without contacting the ground, we assume that the center of
the mass of the biomechanical model moves linearly. Since the biomechanical model is not
Table 3. Joint motion of the lizard biomechanical model.
Joint’s name
Symbol
Joint motion
Waist, tail joint
q4, q5
qi ¼ ai  sin
2p
period t þ bi


(4)
Shoulder X, Z axis & Knee of Left hind leg
q20, q21 q22
qi ¼ ai  sin
2p
period t þ bi


þ ci
(5)
Shoulder X, Z axis & Knee of Right hind leg
q27, q28 q29
qi ¼ aiforced in the air, the velocity of the model does not change. The velocity and the direction of
the model in the air are calculated from the previous velocity and direction before falling off
the ground.
C. Inverse dynamics of the lizard biomechanical model
The number of joints in the biomechanical model is k = 30 and with three virtual joints, the
total number of joints is n = k+3 = 33. When the biomechanical model does not touch the
ground as shown in Fig 9(a), the equation of motion is derived as follows.
AðqÞ€q þ bðq; _qÞ þ gðqÞ ¼ G
ð10Þ
where q 2 R33×1 is the joint vector and Γ 2 R33×1 is the torque vector. A(q) 2 R33×33 is the mass
and inertia matrix and bðq; _qÞ 2 R331 is the Coriolis and centrifugal vector.g(q) 2 R33×1 is the
gravity vector. When the foot of the lizard biomechanical model contacts the ground, as
shown in Fig 9(b), the reaction forces and moments are taken into consideration in the equa-
tion of motion from the ground as follows.
AðqÞ€q þ bðq; _qÞ þ gðqÞ þ JT
c fc ¼ G
ð11Þ
JT
c is the transpose of the Jacobian of the position and the angle at the foot on the ground. In
addition, fc is the reaction force and moment vector.Jc must satisfy Eq (12).
_xc ¼ Jc _q
ð12Þ
where xc is the position vector of the foot that contacts the ground and q is the joint vector.
Then, by replacing €q with €q ¼ Jwhere,
Lc ¼ ðJcAwith a 180˚ phase difference. The phase difference of the waist joint angle is different from that
of the tail joint angle. In Fig 11, it appears that the waist joint moves first, followed by the tail
joint. For the readers, the MATLAB code used for the simulation in Fig 11 can be accessed
with the instruction file (https://github.com/imationzzz/LizardSimulation).
Fig 12 shows the trajectory of the left hind tiptoe of the lizard biomechanical model running
at 4 m/s in place and compares it with that of the actual lizard captured from its running loco-
motion [22]. As shown in Fig 12(a) and 12(b), in the stance mode, both the lizard biomechani-
cal model and the real lizard move their tiptoes back in a straight line but bring them forward
in the swing mode. The trajectory length of the tiptoe of the lizard biomechanical model in the
top view is 58.5 mm in the horizontal direction, and its maximum width is 25.8 mm in the ver-
tical direction. On the other hand, the trajectory length of the real lizard in the top view is 199
mm in the horizontal direction, and its maximum width is 49 mm in the vertical direction. As
shown in Fig 12(c) and 12(d), both the lizard biomechanical model and the real lizard raise
Fig 10. Squares of the torques of all joints obtained by varying the body movement. The magnitude of the waist
joint motion changes in the vertical direction, and the magnitude of the tail joint motion changes in the horizontal
direction.
https://doi.org/10.1371/journal.pone.0243798.g010
Table 4. Design parameters for optimization.
Order
Description
Variable
1
Magnitude of the waist joint angle
a4
2
Phase difference of the waist joint angle
b4
3
Magnitude of the tail joint angle
a5
4
Phase difference of the tail joint angle
b5
5
Magnitude of the shoulder z joint angle
a20
6
Offset of the shoulder z joint angle
b20
7
Magnitude of the shoulder x joint angle
a21
8
Magnitude of the shoulder y joint angle
a22
9
Magnitude of the knee joint angle
a23
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their tiptoes upward in swing mode. The trajectory length of the lizard biomechanical model
in the side view is 58.5 mm in the horizontal direction, and its maximum height is 8.7 mm in
the vertical direction. The trajectory length of the real lizard in the side view is 199 mm in the
horizontal direction, and its maximum height is 56 mm in the vertical direction. Through this
simulation, we found that when the lizard biomechanical model ran in the bipedal locomotion
with minimizing the sum of squares of the torque, the resulting trajectory of tiptoe is similar to
that of the actual lizard even though their sizes are different.
We performed a simulation to confirm the bipedal running locomotion of the lizard bio-
mechanical model by gradually increasing the size of the waist and tail joint angle, as shown in
Fig 13(a). The body movement is a sine function, and the waist and tail joints are set to be
same. Excluding the waist and tail joints, the legs’ movements were derived to minimize the
square of the torque, as in the previous simulation. As a result, the characteristics of locomo-
tion were obtained, as shown in Fig 13(b)–13(d). When the lizard biomechanical model runs
at the same speed, Fig 13(b) shows that the stride length increases from 97.8 mm to 135.2 mm
as the body movement increases from 2 degrees to 40 degrees. The stride duration also
Fig 11. Snapshot of a moving video clip of a simulated lizard when the square of the joint torque is minimized.
The lizard biomechanical model has bipedal running locomotion at a speed of 4 m/s.
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increases from 24.3 ms to 34.0 ms (see Fig 13(c)). The stride frequency is the reciprocal num-
ber of the stride duration and decreases from 41.2Hz to 29.4Hz. Therefore, as the body move-
ment increases, the stride length increases, so the lizard can achieve a high running speed with
fewer moves. On the other hand, Fig 13(d) shows that the duty factor of the lizard maintains
its value, about 25.7%, independent of the body’s movement.
When the joint angle of the lizard’s waist and the tail are 40 degrees, the characteristics of
simulated locomotion are compared with those of actual lizard obtained from [13] in Table 5.
The stride length of the lizard biomechanical model is smaller than that of the real lizard. It is
guessed that the difference between the simulation model and the real lizard may stem from
the modeling error. On the other hand, since other values are similar between the biomechani-
cal model and the lizard, the modeling error may be negligible.
5. Conclusion
This study shows that the steady-state bipedal straight running of a lizard is highly related to
the symmetric and periodical angular movement of its posterior body. It is hypothesized that
such pelvic rotation is produced owing to the GRM and internal torque. The dynamic theoreti-
cal simulation is used to discover the relationship between the GRM and internal torque. In
Fig 12. Comparison of the tiptoe’s trajectory between the dynamic biomechanical model and the real lizard. (a)
The trajectory of the tiptoe of the lizard biomechanical model in the top view has a length of 58.5 mm in the horizontal
direction, and 25.8 mm in the vertical direction. (b) The trajectory of the tiptoe of the real lizard in the top view has a
length of 199 mm in the horizontal direction, and 49 mm in the vertical direction. (c) The trajectory of the tiptoe of the
lizard biomechanical model in the side view has a length of 58.5 mm in the horizontal direction, and 8.7 mm in the
vertical direction. (b) The trajectory of the tiptoe of the real lizard in the side view has a length of 199 mm in the
horizontal direction, and 56 mm in the vertical direction.
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the lizard theoretical model of bipedal and steady-state running, the GRF associated with the
running speed is assumed to be identical for every stride based on previous studies. When the
GRM, torque distribution, and torque magnitude are minimized, an S-shaped lateral body
undulation similar to that of a real lizard is observed in the model.
Furthermore, we established the lizard biomechanical model to figure out the relationship
between the lizard’s undulatory body movement and the bipedal running locomotion. The
biomechanical model consists of 15 bodies and 33 DOFs. When the biomechanical model per-
forms running in bipedal locomotion, the square of the joint torque is changed by the waist
and tail joints. It means that the lizard’s undulatory body movement significantly affects all
joint torques when the lizard performs bipedal running locomotion. When the square of the
joint torque was minimized, we observed that the undulatory body movement are quite similar
to those of a lizard. The trajectory of the lizard’s biomechanical model tiptoe is small than a
real lizard’s trajectory; however, the trajectory shape is similar to a real lizard’s tiptoe trajec-
tory. We performed a simulation to confirm the bipedal running locomotion of the lizard bio-
mechanical model by gradually increasing the size of the waist and tail joint angle. As a result,
when the lizard biomechanical model runs at the same speed, the stride length and stride
Fig 13. The locomotion characteristics of the lizard biomechanical model changed by increasing the undulatory body movement. (a) Concept of the
increase of the lizard’s undulatory body movement. The waist and tail joint angles are set to the same angle. (b) The stride length and (c) duration are
increased by increasing the lizard’s undulatory body movement. (c) The duty factor does not be changed by the undulatory body movement of the lizard
biomechanical model.
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Table 5. Comparison of locomotion specification between the biomechanical model and the real lizard.
Variable
Lizard model with 40˚ of body joint
Callisaurus draconoides [13]
Speed (m/s)
4.0
4.0±0.1
Stride length (mm)
135.2
319±11
Stride width (mm)
40.9
51±4
Stride duration (ms)
34.0
80±3
Duty factor (%)
25.7
24±1
Hip height (mm)
30.3
28±1
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duration increase, and the stride frequency decrease. Therefore, as the body movement
increases, the lizard can achieve a high running speed with fewer moves. It means that the liz-
ard’s undulatory body movements increase its stride and help it run faster.
As a result, we identify the benefits of the lizard-body movements that minimize the square
of the joint torque. Besides, we confirmed that the undulatory body movements of lizard affect
the lizard’s locomotion. To the best of our knowledge, there is no research to figure out the
principle of the running lizard’s undulatory body movement and relationship with the loco-
motion. In future work, we will analyze body movements in various ways through various
environments and simulators.
6. Appendix: A1
As shown in Fig 14, for dynamic equation, the vectors for position, angle, mass and moment
of inertia are defined in R3×1. The horizontal position vector of center of mass of the link is
defined as x = [x1, x2, x3]T and the lateral position vector is defined as y = [y1, y2, y3]T. The
angle vector of the link in global frame is q = [q1, q2, q3]T, the joint angle vector between each
links is j = [j1, j2]T. The mass vector is defined as m = [m1, m2, m3]T. For numerical calculation,
M = diag(m) is defined. The moment of the inertia vector is defined by i = [i1, i2, i3]T and in
the same manner, I = diag(i). The sine and cosine vectors are defined as sinq = [sinq1, sinq2,
sinq3]T, Sinq = diag(sinq), cosq = [cosq1, cosq2, cosq3]T cosq = diag(cosq).
The model has a boundary as shown in Fig 14 ([24,25])
xiþ1 C and L are defined in Eqs (26) and (27).
C ¼
From the assumption that the reaction force exists only in hind leg, the vectors of the reac-
tion force are Fext,x = [0, fext,x, 0]T, Fext,v = [0, fext,v, 0]T. The inner force vector are defined as
Fx = [fx1, fx2]T and Fv = [fv1, fv2]T.
Eqs (32) and (33) are derived by differentiating Eqs (24) and (25) twice:
C€x ¼ X4 ¼
q1
q2
0
2
64
3
75 þ Rot q3; z
ð
Þ 
l2
2 þ shoulderX
X10
¼
q1
q2
0
2
6664
3
7775 þ Rot q3; z
ð
Þ 
l1
0
0
2
6664
3
7775 þ Rot q3; z
ð
Þ  Rot q4; z
ð
Þ 
l2
2 þ pelvicX
X15
¼ X14 þ Rot q3; z
ð
Þ  Rot q4; z
ð
Þ  Rot q27; z
ð
Þ  Rot q28; x
ð
Þ  Rot q29; y
ð
Þ  Rot q30; z
ð
Þ 
l14
2
0
0
2
66664
3
77775
þRot q31; z
ð
Þ  Rot q32; x
ð
Þ  Rot q33; y
ð
Þ 
l15
2
0
0
2
66664
3
77775
ð51Þ
where Rot(qi, axis) means that {ith} joint rotates in the axis direction. ShoulderX and shoulderY
are the distances from the center of the mass of body1. PelvicX and pelvicY are the distances in
the x and y directions from the center of the mass of body2.
Supporting information
S1 File.
(TXT)
S1 Video.
(WMV)
Author Contributions
Investigation: Jeongryul Kim, Hongmin Kim, Jaeheung Park, TaeWon Seo.
Supervision: Hwa Soo Kim.
Writing – original draft: Jeongryul Kim.
Writing – review & editing: Jaeheung Park, Hwa Soo Kim, TaeWon Seo.
References
1.
RITTER D, Ritter R. Lateral bending during lizard locomotion. J Exp Biol. 1992; 173: 1–10.
2.
Ritter D. Epaxial muscle function during locomotion in a lizard (Varanus salvator) and the proposal of a
key innovation in the vertebrate axial musculoskeletal system. J Exp Biol. 1995; 198: 2477–2490.
PMID: 9320404
3.
Ritter D. Axial muscle function during lizard locomotion. J Exp Biol. 1996; 199: 2499–2510. PMID:
9320426
4.
Bennett WO, Simons RS, Brainerd EL. Twisting and bending: the functional role of salamander lateral
hypaxial musculature during locomotion. J Exp Biol. 2001; 204: 1979–1989. PMID: 11441039
5.
Jusufi A, Goldman DI, Revzen S, Full RJ. Active tails enhance arboreal acrobatics in geckos. Proc Natl
Acad Sci. 2008; 105: 4215–4219. https://doi.org/10.1073/pnas.0711944105 PMID: 18347344
6.
Libby T, Moore TY, Chang-Siu E, Li D, Cohen DJ, Jusufi A, et al. Tail-assisted pitch control in lizards,
robots and dinosaurs. Nature. 2012; 481: 181–184. https://doi.org/10.1038/nature10710 PMID:
22217942
7.
Gillis GB, Bonvini LA, Irschick DJ. Losing stability: tail loss and jumping in the arboreal lizard Anolis car-
olinensis. J Exp Biol. 2009; 212: 604–609. https://doi.org/10.1242/jeb.024349 PMID: 19218510
8.
Kim H, Sitti M, Seo T. Tail-Assisted Mobility and Stability Enhancement in Yaw and Pitch Motions of a
Water-Running Robot. IEEE/ASME Trans Mechatronics. 2017; 22: 1207–1217.
9.
Zhao J, Zhao T, Xi N, Mutka MW, Xiao L. MSU tailbot: Controlling aerial maneuver of a miniature-tailed
jumping robot. IEEE/ASME Trans Mechatronics. 2015; 20: 2903–2914.
PLOS ONE
Effects of body movement on yaw motion in bipedal running lizard
PLOS ONE | https://doi.org/10.1371/journal.pone.0243798
December 31, 2020
22 / 23
10.
Kohut NJ, Pullin AO, Haldane DW, Zarrouk D, Fearing RS. Precise dynamic turning of a 10 cm legged
robot on a low friction surface using a tail. 2013 IEEE International Conference on Robotics and Auto-
mation. IEEE; 2013. pp. 3299–3306.
11.
Ijspeert AJ, Crespi A, Ryczko D, Cabelguen J-M. From swimming to walking with a salamander robot
driven by a spinal cord model. Science (80-). 2007; 315: 1416–1420. https://doi.org/10.1126/science.
1138353 PMID: 17347441
12.
Irschick DJ, Jayne BC. Effects of incline on speed, acceleration, body posture and hindlimb kinematics
in two species of lizard Callisaurus draconoides and Uma scoparia. J Exp Biol. 1998; 201: 273–287.
PMID: 9405318
13.
Irschick DJ, Jayne BC. Comparative three-dimensional kinematics of the hindlimb for high-speed
bipedal and quadrupedal locomotion of lizards. J Exp Biol. 1999; 202: 1047–1065. PMID: 10101105
14.
Legreneur P, Homberger DG, Bels V. Assessment of the mass, length, center of mass, and principal
moment of inertia of body segments in adult males of the brown anole (Anolis sagrei) and green, or
Carolina, anole (Anolis carolinensis). J Morphol. 2012; 273: 765–775. https://doi.org/10.1002/jmor.
20022 PMID: 22461036
15.
Druelle F, Goyens J, Vasilopoulou-Kampitsi M, Aerts P. Compliant legs enable lizards to maintain high
running speeds on complex terrains. J Exp Biol. 2019;222. https://doi.org/10.1242/jeb.195511 PMID:
30796100
16.
y Alvarado PV, Youcef-Toumi K. Performance of machines with flexible bodies designed for biomimetic
locomotion in liquid environments. Proceedings of the 2005 IEEE International Conference on Robotics
and Automation. IEEE; 2005. pp. 3324–3329.
17.
Kubo T, Ozaki M. Does pace angulation correlate with limb posture? Palaeogeogr Palaeoclimatol
Palaeoecol. 2009; 275: 54–58. https://doi.org/10.1016/j.palaeo.2009.02.001
18.
Reilly S, Delancey M. Sprawling locomotion in the lizard Sceloporus clarkii: quantitative kinematics of a
walking trot. J Exp Biol. 1997; 200: 753–765. PMID: 9318518
19.
Aerts P, Van Damme R, D’Aouˆt K, Van Hooydonck B. Bipedalism in lizards: whole–body modelling
reveals a possible spandrel. Philos Trans R Soc London Ser B Biol Sci. 2003; 358: 1525–1533. https://
doi.org/10.1098/rstb.2003.1342 PMID: 14561343
20.
McElroy EJ, Wilson R, Biknevicius AR, Reilly SM. A comparative study of single-leg ground reaction
forces in running lizards. J Exp Biol. 2014; 217: 735–742. https://doi.org/10.1242/jeb.095620 PMID:
24198262
21.
McElroy EJ, Hickey KL, Reilly SM. The correlated evolution of biomechanics, gait and foraging mode in
lizards. J Exp Biol. 2008; 211: 1029–1040. https://doi.org/10.1242/jeb.015503 PMID: 18344476
22.
https://www.youtube.com/watch?v=ExyMxKDxT9M.
23.
Fieler CL, Jayne BC. Effects of speed on the hindlimb kinematics of the lizard Dipsosaurus dorsalis. J
Exp Biol. 1998; 201: 609–622. PMID: 9438835
24.
Liljeback P, Pettersen KY, Stavdahl O. Modelling and control of obstacle-aided snake robot locomotion
based on jam resolution. 2009 IEEE International Conference on Robotics and Automation. IEEE;
2009. pp. 3807–3814.
25.
Liljeback P, Pettersen KY, Stavdahl Ø, Gravdahl JT. Controllability and stability analysis of planar
snake robot locomotion. IEEE Trans Automat Contr. 2010; 56: 1365–1380.
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Effects of body movement on yaw motion in bipedal running lizard by dynamic simulation.
Kim, Jeongryul,Kim, Hongmin,Park, Jaeheung,Kim, Hwa Soo,Seo, TaeWon