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Abstract—This paper presents a new lizard-inspired robot that

can maintain its direction of movement via S-shaped lateral body

motions during its high-speed bipedal running. For this purpose,

a dynamic model and a control scheme for the proposed robot

were derived to simulate the S-shaped lateral body motions of a

real lizard. Based on simulations using the dynamic model with the

proposed control scheme, a prototype of lizard-inspired robot was

built to reproduce a bipedal running gait as well as the lateral body

motions of a real lizard. The simple robot body consisted of three

links connected by two rotational joints and its hind legs were

optimally designed upon a 4-bar linkage combined with an ankle

mechanism. The experimental tests using the proposed robot were

performed by confining its motion in the horizontal plane, which

demonstrated that the proposed robot can successfully maintain

its direction of movement via the S-shaped lateral motion adjusted

by the yaw angle of its posterior body during its bipedal running

gait similar to the gait of a real lizard.

Index Terms— 4-bar linkage, bipedal running gait, lizard-

inspired robot, S-shaped lateral body motion

I. INTRODUCTION

EGGED animals have naturally evolved to attain the ability

to overcome various types of obstacles encountered in their

surrounding environments. Therefore, for a long time, many

investigations have been carried out to focus on developing

legged and wheel-legged robots to traverse rough terrains stably

by mimicking the locomotive mechanisms of legged animals

with excellent obstacle-climbing abilities [1]-[7].

Because some legged robots are fully or redundantly actuated,

their poses, directions of movement, and moving speeds can be

controlled simultaneously and rapidly and as a result, they have

excellent locomotive capabilities. For example, humanoids can

walk on rough terrains naturally using two feet like humans [8]-

[10]. Furthermore, the MIT Cheetah can jump over obstacles up

to 40 cm in height at the running speed of 2.4 m/s [11] and the

cheetah developed by Boston Dynamics is reported to run at a

speed of 12.6 m/s. In general, these fully or redundantly

This work was supported by Basic Science Research Program through the

National Research Foundation of Korea grants funded by the Ministry of Education (NRF-2016R1D1A1B03935516 and NRF-2019R1A2C1008163)

and by the GRRC program of Gyeonggi province [GRRC2019-B02, Research

on Innovative Intelligent Manufacturing System]. (Co-corresponding authors: TaeWon Seo and Hwa Soo Kim)

J. Kim, H. Kim, Y. Kim, and J. Kim are with School of Mechanical and Aerospace Engineering, Seoul National University, 1 Gwanak-ro, Gwanak-gu, Seoul 08826, South Korea (e-mails: {jrkim, hmkim, yskim}

@rodel.snu.ac.kr, [email protected]).

actuated legged robots are large, heavy and complicated due to

their many actuators and sensors.

On the contrary, some legged robots are under-actuated, but

they can achieve performances comparable to those of fully or

redundantly actuated ones without implementing all the degrees

of freedom required to mimic the locomotive mechanisms of

legged animals. For example, iSprawl is the six-legged robot

inspired by a cockroach that can run at a speed of 2.3 m/s by

converting the rotational motions from two motors into periodic

thrusting motions. Also, flexible push/pull cables are adopted

to rapidly swing six legs [12]. Similarly, VelociRoACH is the

six-legged robot that can run at a speed of 2.7 m/s by combining

two motors with a lightweight linkage mechanism. Moreover,

it has the ability to steer its direction of movement easily by

using an aerodynamic stabilizer [13], [14]. It is noteworthy that

iSprawl and VelociRoACH are hexapedal runner so that more

than three legs always have good contact with the ground so

that their locomotion is always stable. Since their body consists

of one rigid section, their directions of movements are mainly

controlled by their leg motions.

Typically, an under-actuated bipedal robot cannot maintain

its posture as well as its direction of movement simultaneously

during bipedal running due to the inherent lack of degrees of

freedom (DOFs) of its motion. A possible solution to such lack

of DOFs caused by under-actuation may be additional motions

of the body. The investigations have been carried out on legged

robots that can keep their postures stable by utilizing some parts

of their bodies besides their legs or wheels. For example, an

active tail has been widely used due to its structural simplicity:

the motion stability of six-legged water-running robot in the

yaw- and pitch-direction is improved by using a 2 DOF active

tail [15]. The six-legged robot TAYLRoACH with a 1 DOF

active tail can make sudden 90° turns at a constant rotational

speed of 360°/s in the horizontal plane [16]. Also, the lizard-

sized wheeled robot Tailbot can keep its pitch angle constant

via the swing of its 1 DOF active tail in a manner to redirect the

angular momentum from its body to its tail [17]. Similarly, the

J. Park is with the Department of Transdisciplinary Studies, Seoul National

University, 1 Gwanak-ro, Gwanak-gu, Seoul 08826, South Korea (e-mail: [email protected])

T. Seo is with the School of Mechanical Engineering, Hanyang University, Seoul 04763, Republic of Korea (e-mail: [email protected])

H. S. Kim is with the Department of Mechanical System Engineering,

Kyonggi University, Suwon, Gyeonggi-do 16227, South Korea (e-mail: [email protected]).

A New Lizard-inspired Robot with S-shaped

Lateral Body Motions

Jeongryul Kim, Hongmin Kim, Youngsoo Kim, Jaeheung Park, Member, IEEE, TaeWon Seo,

Member, IEEE, Hwa Soo Kim, Member, IEEE, and Jongwon Kim

L

mailto:yskim%7d%[email protected]

mailto:yskim%7d%[email protected]

mailto:[email protected]

mailto:[email protected]




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bipedal robot MSU Tailbot can maneuver in midair for safe

landings by utilizing a 1-DOF active tail [18]. It is noted that

TAYLRoACH, Tailbot and MSU tailbot can control their poses

stably by using the active pendulum unlike iSprawl without a

tail or VelociRoACH with a fixed tail. The design, scaling and

control of active tails for planar aerial reorientation is detailedly

discussed in [31].

Recently, many investigations of lizard behavior have been

carried out due to their excellent mobility and stability on rough

terrains. For example, the effect of periodical lateral motion of

a real lizard during running is studied in [20], [21]. Moreover,

the relation between the pace angulation and the pelvic rotation

is uncovered [22] and also, to reveal the role of muscles in real

lizards, the investigations using the electromyography (EMG)

have been carried out [23]-[25]. Based on these observations,

some research has verified the principle of body motion of a

real lizard via lizard-inspired robotic platforms. For example, it

is shown that the tail of a lizard plays an important role in

keeping its motion stable [15], [17] or the flexibility of lizard

body is strongly related with the efficiency of its crawling gait

[26]. However, since their locomotion is based on multiple legs

or wheels, their motions have nothing to do with the bipedal

running of a lizard. On the other hand, Salamander Robotica is

known for the similarity of its gait controlled by the central

pattern generator (CPG) to that of a real salamander [19], [32].

It can control the periodic yaw angle of its body by changing

the curvature of body consisting of eight flexible spines. Since

the gait of Salamander Robotica is based on a quadruped gait,

its duty factor (defined as the ratio of contact time of one leg

over one gait cycle using both legs) is larger than that of bipedal

running gait of lizard so that its walking on the ground is quite

stable but relatively slow. So far as is known, the effect of S-

shaped lateral bending of body along the yaw direction during

bipedal running has not been studied in previous investigations

on bio-inspired robots comprehensively.

The final goal of our study is to develop a perfect bipedal

running robot that can keep its direction of movement via the

S-shaped lateral body motions similar to those of real lizard

Callisaurus draconoides. In this work, as a stepping stone to a

perfect bipedal running robot, we have focused on verifying

that the direction of movement of a bipedal running robot can

be managed by adjusting its body motion in the 2-D horizontal

plane. To this end, we have first derived a 2-D dynamic model

of a prototype of bipedal running robot to generate a bipedal

running gait without slip as well as a S-shaped body motion

adjusted by the error between the yaw angle of body and the

direction of movement. Second, for the experimental validation,

we have constructed a prototype of bipedal running robot

consisting of three links, whose hind legs are optimally

designed upon a 4-bar linkage to reproduce the tip trajectory

very similar to that of C. draconoides. Also, for sufficient

friction, the ankle linkage is added to the leg to ensure the

constant contact angle between the tip and a ground. Third, we

have adopted the high- and low-level PD controllers in series

driven by the yaw error of robot body in order to achieve a S-

shaped motion of the prototype of bipedal running robot and we

have experimentally verified that it can keep its direction of

movement during its bipedal running. It is noteworthy that in

order to consider only the effect of lateral body motion of the

prototype of robot on the direction of movement, the effect of

balancing by its hind legs is minimized by confining the motion

of prototype of robot within the 2-D horizontal plane. However,

different from previous investigations inspired by lizards, this

work is the only case to handle the direction of movement of

the bipedal running robot by swinging its body properly in the

horizontal plane without the force control of its legs, i.e., its legs

just run for locomotion. In summary, dynamic modeling, design

of the prototype robot and two-level feedback control scheme

have been successfully incorporated to verify the effect of

lateral body undulation on the direction of movement.

The rest of paper is organized as follows: Section II presents

a dynamic model of the lizard-inspired robot and a control

framework for gait planning of the proposed robot. The design

of prototype of the new lizard-inspired robot is in details shown

in Section III. In order to investigate the ability of the proposed

robot to maintain its direction of movement even during bipedal

running gait, the experiments using the prototype of robot are

carried out on a treadmill to constrain its motions in the 2-D

horizontal plane, and the results are discussed in Section IV.

II. DYNAMIC MODELING OF A LIZARD-INSPIRED ROBOT

A. Dynamic Modeling

It is noteworthy that the main goal of this study was to build

a lizard-inspired robot that could maintain its direction of

movement by mimicking the S-shaped lateral motions of C.

draconoides. To this end, some attributes of dynamic model of

the lizard-inspired robot were simplified to focus on the goal.

First, the degree-of-freedom (DOF) of the dynamic model is

assumed to be three so that the possible motions of proposed

robot would be two translations on the XY plane and rotation

along the Z-axis, as shown in Fig. 1. Next, the dynamic model

is assumed to move at a constant speed during its bipedal

running gait (similar to C. draconoides). Third, the time periods

for the motions of body, tail and hind legs are assumed to be

constant. Finally, the leg motion of the dynamic model is

controlled by the open-loop control so that its legs repeated the

same motion in every period and touched the ground. It is

Fig. 1. Schematics of (a) symbols and (b) coordinate frames and joint variables

for the lizard-inspired robot.




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noteworthy that unlike a snake whose sideways motion slipping

on the ground plays an essential role in forward propulsion, the

lateral body motion of the proposed robot had little effect on its

forward propulsion because there is a negligible slip between

its body and the ground.

Fig. 1(a) shows the schematics of coordinate frames and joint

variables for the proposed robot, where y is the yaw angle of

the posterior body section, and s

l and b

d are the stroke of the

tip of the hind leg and the distance from the posterior body

section to the hind leg. i

l and i

w are the length and the width

of the anterior and posterior body sections and tail ( i = 1, 2, 3),

with values from the length and width of C. draconoides, as

shown in Table I [20]. Because the weight of the front leg of C.

draconoides is just 2% of its weight, the front legs is neglected

in the dynamic modeling, but the hind legs are regarded as slide

joints. This is because, while the hind leg of C. draconoides

stays in contact with the ground, its tip trajectory moving from

front to back can be approximated by a linear reciprocating

motion on the XY plane [27], [28].

To express the position and yaw angle of the dynamic model

in the XY plane, three virtual variables (1q ,

2q , and 3q ) are

used as shown in Fig. 1(b). While 4q and

5q are the angles of

the waist and tail joints, 6q and

7q are the displacements of the

slide joints, respectively. With 1 7[ ]Tq q q= , the equation of

motion is derived as

( , )( ) T

c cA q b q q Jq f+ + = (1)

where 7 7( )A q R and 7 1)( ,b q q R are the joint space inertia

matrix and the Coriolis and centrifugal force vector. 3 7

cJ R ,

3 1

cf R , and 7 1R are the Jacobian matrix, the reaction

force and moment vector and the torque vector, respectively.

The Jacobian matrix cJ is defined as

c cx J q= with the contact

vector 3 1

cx R in Fig. 1(b). By replacing q with

1( )c c c

q J x J q−= − , (1) becomes

1 1( , )( )

Tcc

c c c c c c c c

J

J A q fx b q q J J Aq

− − =+ − + (2)

where c and

c denote the inertia matrix and the Coriolis

and centrifugal force vector at contact point, and cJ is the

dynamically consistent inverse matrix of cJ [29]. In this study,

to ensure no slip of foot at the contact point, the acceleration of

the foot is supposed to satisfy the following constraint [30]:

22 0c c cx x x + + = (3)

where and are the positive Baumgarte parameters. By

combining (3) with (2), the reaction force and moment vector

cf can be obtained as

2( 2 )T

c c c c c cf J x x = − − − − (4)

Substituting (4) into (1) derives the forward dynamics of the

proposed robot in the contact condition as

1 2( ) ( ), () 2T

c c c c cxq A q b q q J x − − − − −= −− (5)

In the non-contact condition (3 1cf = 0 ), the forward dynamics

is given by

1( ) ( , )q A q b q q− −= (6)

More information about the parameters and forward kinematics

of the proposed robot are given in details in the Appendix.

B. Locomotion Simulation

It is noted that the aim of dynamic modeling was to simulate

a stable bipedal running gait for the proposed lizard-inspired

robot to maintain its direction of movement via proper angular

movement of waist and tail. In this study, the direction of

movement of the robot is defined as the median value of the

yaw angle of its posterior body section. Note that even when a

lizard runs straight, the yaw angle of its posterior body section

may change continuously but its median value remains constant

toward its direction of movement.

In this study, the duty factor chosen for the bipedal running

gait of the dynamic model is set to be 0.25 (similar to that of C.

draconoides) so that the duration that one leg is in touch with

the ground is equal to the duration that both legs are in the air,

as shown in Fig. 2(a). For a S-shaped body motion, the dynamic

model is supposed to move like a sinusoidal wave so that the

directions of the anterior body section and tail were exactly the

same, but opposite to that of the posterior body section, as

shown in Fig. 2(b). For the simplicity of simulation, the rotating

angle of the tail is equal to that of the anterior body section.

The basic principle of maintaining the direction of movement

via angular motions of the dynamic model is based upon the

conservation of angular momentum. The dynamic model may

be placed in two situations: either one foot is in contact with the

ground or all feet are in the air. When one foot is in contact with

the ground, the direction of movement of the dynamic remains

constant due to friction for any angular motions of the anterior

body section and tail, as shown in phases [I] and [III] of Fig.

2(b). In contrast, when all feet are in the air, the yaw angle of

posterior body section rotates in the opposite direction to the

rotating angles of waist and tail joints by the principle of

conservation of angular momentum, as shown in phases [II] and

[IV] of Fig. 2(b). Thus, during time period T , the direction of

movement of dynamic model is changed by the difference

between counter-clockwise and clockwise rotating angles 4

ccwq

and 4

cwq in Fig. 2(b), where the direction of movement of the

dynamic model is denoted by the black dotted arrow.

TABLE I

SPECIFICATIONS OF DYNAMIC MODEL OF LIZARD-INSPIRED ROBOT

Length Width Weight Inertia

Anterior link 182 mm 55 mm 0.89 kg 0.0026 kg m2

Posterior link 115 mm 55 mm 1.46 kg 0.0019 kg m2

Tail link 231 mm 20 mm 0.19 kg 0.0008 kg m2

Foot 30 mm 30 mm 0.18 kg 0.000027 kg m2

TABLE I

SPECIFICATIONS OF DYNAMIC MODEL OF LIZARD-INSPIRED ROBOT

Length Width Weight Inertia

Anterior link 182 mm 55 mm 0.89 kg 0.0026 kg m2

Posterior link 115 mm 55 mm 1.46 kg 0.0019 kg m2

Tail link 231 mm 20 mm 0.19 kg 0.0008 kg m2

Foot 30 mm 30 mm 0.18 kg 0.000027 kg m2




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It is noteworthy that for the TAYLRoACH to change its

direction of movement, it swings its tail in the desired direction

of movement all at one time. However, because its tail is as

heavy as its main body, it is likely to suffer from slip of its feet

and as a result, its center of mass (CoM) may be shifted, or its

capability to change its direction of movement may be degraded

by the limited rotating angle [16]. In contrast, the principle of

motion used for the dynamic model allows the waist and tail to

come back to their original positions, as shown in phases [II]

and [IV] of Fig. 2(b). This enables the dynamic model to rotate

over a sufficiently large angle with a relatively stable CoM

during a bipedal running gait.

Fig. 3 shows a block diagram of the control scheme for the

locomotion simulation. The yaw angle error e is defined by

the difference between the desired moving angle d and the

real yaw angle of the posterior body section re . From this yaw

angle error e , the desired waist and tail joint angles 4,dq and

5,dq are calculated by (7):

( )4,

5, 4,

)s

)

( ) 2(

( )

in /

(

y y

p

d

dd

d

q t k k e t

q q

Te

t t

= +

=

− (7)

where y

pk and y

dk denote the positive control gains of high-

level PD controller. It is noted that, because the magnitudes of

waist and tail joints angles are the same, but their signs are

opposite, the S-shaped lateral motion of the dynamic model can

be efficiently implemented. Recall that during the locomotion

simulation, the desired slide joints 6,dq and 7,dq for the hind

legs are computed by (7) irrespective of the yaw angle error:

4

2 4

4 5

3 6 4

6,

7, 6, 2

if 0( )

if

( ) ( )

s

s

s

sl l T

T

l ld

d

T

T

d

t tq t

t t T

q t q t

− =

− +

= +

(8)

where T and sl are the time period of one gait cycle and the

stroke of the tip of the hind leg when it remains in contact with

the ground, respectively. Note that the motions of the left and

right hind legs have the time difference of 0.5T and that,

according to the motion in (8), the speed to move the hind leg

from back to front is three times slower than the speed to move

the hind leg from front to back. Also, note that the real yaw

angle of the posterior body section is checked every half period

of 0.5T s when the whole body is straight, for example, it is

checked at the times 2t and

4t in Fig. 2(b). Based on the joint

error iqe between the desired and real joint angle

,i dq and ,i req ,

the required torque i , i = 1, , 7 is generated by:

, , (( ) , )( )i ii q

t t

i p q i dA q k e k e b q q ++ = (9)

where ,

t

i pk and ,

t

i dk are the positive control gains of low-level

PD controller. Note that i = 0 holds because

iq , i = 1, 2, 3 is

the virtual joint variable used to express the position and the

orientation of the dynamic model. Therefore, If the foot of

dynamic model stays in contact with the ground, the 2nd order

derivative of real joint variable ,i req of dynamic model is

obtained from (5) by using cf and

i given by (5) and (9),

respectively. Otherwise, ,i req is obtained by (6) with

3 1cf = 0

and i given by (9). Then, the real joint variable ,i req can be

calculated by using the 4th order Runge-Kutta method. The

positive gains of high- and low-level PD controllers for the

dynamic model are chosen by the trial and error method to

ensure the stable motion of the dynamic model.

Fig. 4 shows the locomotion simulation results of the yaw

angle of the posterior body section of the dynamic model, where

the desired moving angle is set to zero and denoted by the red

dotted line. While Fig. 4(a) corresponds to the case that the

locomotion of the dynamic model was done by open-loop leg

propulsion without angular motions of waist and tail, Fig. 4(b)

corresponds to the case that the locomotion was done by open-

loop leg propulsion with angular motions of the waist and tail

according to the control framework in Fig. 3. Because the mean

value of the yaw angle of the posterior body section in Fig. 4(a)

increases gradually with respect to the reference line, the

resulting direction of movement of the posterior body section

of dynamic model will eventually be upward. In contrast, the

direction of movement of the dynamic model in Fig. 4(b) is kept

relatively constant with the help of periodic S-shaped lateral

motions of waist and tail computed by (7) and (8) (i.e., the mean

Fig. 3. Block diagram of control framework for locomotion simulation of

proposed dynamic model.

Fig. 2. (a) Motion plan of left and right hind legs of dynamic model with duty

ratio of 0.25 during a time period and (b) corresponding phases of change in its direction of movement via conservation of angular momentum.




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value of yaw angle remains close to zero while the yaw angle is

within the range from -9.4° to 8.6°).

Additional simulations were performed to investigate how

the mass of each body affects the lateral body motion of the

proposed robot in that the mass of the proposed robot (about 2.5

kg) is larger than that of real lizard (about 10 g). During the

simulations, the masses of the anterior and posterior body

sections and the tail, vary from 0.8 kg to 1.2 kg and from 0.1 kg

to 0.3 kg with intervals of 0.02 kg and 0.01 kg, respectively.

Under options of different body masses, the waist and tail joints

rotate from 0° to 20° at time 0.25t T= , return to their starting

positions at time 0.5t T= , and remain stationary until the end

of time t T= . Then, the angle of the direction of movement of

the dynamic model is measured. In additional simulations,

under constant masses of anterior body section, posterior body

section, and tail (1, 1, and 0.2 kg, respectively), the rotating

angles of waist and tail joints vary from 10° to 30° with

intervals of 1°, with similar motion scheduling of the dynamic

model. Fig. 4(c) shows the simulated direction of movement of

the posterior body section in the dynamic model under various

masses and rotating angles. As shown in Fig. 4(c), the direction

of movement of the dynamic model is within the range ±5° even

for varying body masses, so that the effect of mass of each body

on the moving direction seems much smaller than the effect of

rotating the waist and tail joints. Specifically, the effect of the

mass of the posterior body section is negligible, which implies

that the rotation of the waist and tail joints plays more important

roles in achieving the desired lateral motion of the proposed

robot. However, because the mass of a body is related to the

moving speed of the proposed robot, each body option of the

proposed robot is designed to minimize its mass, even though it

could not be reduced to the mass of a real lizard owing to heavy

components such as motors and drives.

III. A LIZARD-INSPIRED ROBOT

A. Prototype of Lizard-inspired Robot

From the locomotion simulations in Section II by using the

dynamic model of the lizard-inspired robot, it is successfully

shown that the S-shaped lateral bending of waist and tail allows

the dynamic model of the proposed robot to maintain its

direction of movement. In this section, the design of a new

lizard-inspired robot is presented, which consists of anterior and

posterior body sections, a tail, and two hind legs.

The schematics of the prototype of the lizard-inspired robot

is given in Fig. 5. The dimension of the prototype robot is

233×567×185 mm3 (width×length×height), which is about 2.75

times larger than that of C. draconoides. As shown in Fig. 5, the

joints of the waist and tail of the robot are composed of a motor

(Maxon EC-max 22) and one axle connected by gears with the

ratio of 128:1. The Arduino board and the EPOS2 boards are

located on the anterior body section as shown in Fig. 5. The

double wishbone compliant structure is combined with the leg

to reduce impacts that may occur whenever a foot contacts the

ground.

In this study, a 4-bar linkage is used as the basic mechanism

for the hind leg of the proposed robot as shown in Fig. 5. This

choice is made because the entire leg motion can easily be

controlled by one motor so that its structure is quite simple.

Please note that the main goal of this study is not to imitate the

motions of a real lizard but build a lizard-inspired robot that can

maintain its direction of movement via lateral body motion.

From this point of view, the essential functions of the robot such

as lateral swinging of the waist and the tail and the forward

propulsion by the hind legs, are constructed in the simplest way

possible. In order to reduce the weight of the legs, each leg of

tail joint

28 mm207 mm

125 mm

235 mm

waist joint

waist jointtail joint

125 mm

Arduino board

22

3 m

m

Motors for legs

18

5 m

m

EPOS2 boards

4 bar-based leg

mechanism

207 mm235 mm

Wishbone mechanism

Motors for waist and tail

Fig. 5. Schematics of prototype of lizard-inspired robot (top and side view).

Fig. 4. Locomotion simulation results: yaw angle of posterior body section of dynamic model while its locomotion is done by (a) only leg propulsion, (b)

combination of leg propulsion and angular motions of waist and tail and (c)

direction of movement of posterior body section with various link mass and joint angles.




6

the proposed robot is made of ABS plastic.

B. Optimal Design of Hind Leg for Lizard-inspired Robot

To optimize the 4-bar mechanism for the hind legs of the

proposed robot, the real trajectory of a foot of C. draconoides

is captured from a video of its bipedal running in Fig. 6(a). The

captured end-point is denoted by the blue circle. During one

cycle of propulsive motion of a hind leg, the number of captured

end-points is 23 as shown in Fig. 6(b). In order to streamline the

optimization procedure, the captured trajectory ( i

reX , i

reY ) is

approximated by ( i

appX , i

appY ) which are combinations of cosine

and sine functions along the X- and Y-axes as follows:

( ) ( ) ( )

( ) ( ) ( )

1 2 1 2 1 22 4 2

1 2 1 2 1 22 4 2

cos cos sin

cos cos sin

i i i

i i i

t t t

app

t t ti

pp

i

a

X c d

Y

a a b b c

e e ff g g h

= + +

+

+

+

+

=

+ +

+ + ++ (10)

where i =1, , 23, and a1, a2, , g2 and h can be obtained by

minimizing the sum of squared differences between the real and

captured end-points along each axis via the “fmincon” function

in Matlab. The resulting minimized differences between the real

and captured points were 0.991 and 0.989 along the X- and Y-

axes with the coefficients in Table II.

The schematic of the 4-bar linkage used for the hind legs is

shown in Fig. 7(a) with the parameters linkX = }{ , , ,i x yL c c :

iL , i =1, , 4, is the length of each Link i of the four bars and

is the angle between the horizontal line and the line

connecting the input and output points. The end-point of the

hind leg is determined by xc and

yc with respect to the local

frame {xy} on the joint 12J .

The required constraints on the 4-bar linkage are given in Fig.

7(b). As the crank-rocker mechanism, Link 3 should rotate

through a limited range of angles under full rotation of Link 1.

In combination with the GRSHOF condition, the following

conditions can be derived:

1 4 2 3

1

1

2

2

3 4

3 4

L L L L

L L L L

L L L L

+ +

+ + + +

(11)

With regard to locations of input/output points, the following

condition is derived considering the size of the proposed robot:

0 120 mm, 0 50 mmyxp p (12)

Based on the approximated trajectory of the foot in (10), the

following must hold:

1 2130 mm, 15 mm, 20 mmy yH C C= = = (13)

It is noteworthy that, as discussed in Section II.A, all motions

of the proposed robot are assumed to be within the horizontal

XY-plane. This implies that its weight is ignored so that an

additional method is required to ensure good contact with the

ground. To this end, as given in (13), the end-point of the hind

leg is designed to penetrate the ground while contacting it,

which generates a normal force to provide friction between the

foot of the robot and the ground. Also, for the given constant

rotating speed of Link 1 (= 120 RPM), the difference between

entry and release velocities 1xv and

2xv in Fig. 7(b) should be

within a certain range:

2 1 0.2x xv v− m/s (14)

The desired trajectories of the foot in the air and on the ground

are denoted as air

desT and o

des

grT , and the trajectories of the end-

point of the 4-bar linkage to be optimized in the air and on the

ground are denoted as air

barT and o

des

grT , respectively. Then, the

cost functions are defined as follow:

Fig. 6. (a) Captured image of C. draconoides in bipedal running mode and (b)

real and approximated trajectories of end-point of C. draconoides.

TABLE II

COEFFICIENTS FOR APPROXIMATION OF CAPTURED TRAJECTORY

a1 282.3 f1 21.20 a2 0.367 f2 0.855

b1 58.96 g1 13.93 b2 0.042 g2 0.514

c1 67.90 d 1661 c2 0.965 h 1910

e1 132.1 e2 2.368

Fig. 7. (a) Schematic of the 4-bar linkage used for the hind legs of the lizard-

inspired robot and (b) required constraints on the 4-bar linkage.




7

,1

1

2 2

, , ,1( ) ( )

mgro gro air

des i bar i des i b

m air

t m ar iiraj iV T T Ta T b

= =−+= − (15)

,1

1

2 2

, , ,1( ) ( )

mgro gro air

des i b

m air

ve bar i des i il m i ariV a v v v vb

= =+= − − (16)

where ( ) ( )

( ), ( ), 1

1

( )

( ),

gro air gro airdes bar i des bar i

i i

gr To air

des b

T

ar i t tv

−

−

−

−= . The subscript i denotes the ith

sample of a trajectory or velocity, a and b are the weighting

coefficients for a trajectory or velocity on the ground and in the

air, and m is the total number of sample points. Then, the

optimization of the 4-bar linkage is done in two steps. First, the

design parameters are searched to minimize (16) under the

given constraints as follow: * arg min

link

velX

linkX V= (17)

subject to (11), (12), (13) and (14)

It is noted that in the 1st step optimization, the design parameters *

linkX = * * * *{ , , , }i x yL c c are selected to make the shape of the

trajectory determined by them as similar to that of the desired

one as possible, irrespective of its overall size. Then, a scaling

factor s is searched in the 2nd step optimization to make the

overall size of the trajectory determined in the 1st optimization,

as close as possible to that of the desired one, which is described

by

* * * *, }

*

{ , ,

arg mini x y

trajsL sc sc

s V

= (18)

Note that in (18), the size of the trajectory of the 4-bar linkage

is optimized to minimize (15) so that the angle * obtained in

the 1st optimization is used. In the optimizations, the weighting

factors a = 3 and b = 1 are used in that the trajectory on the

ground is more important in the bipedal gait than that in the airs.

The optimal parameters are summarized in Table III and the

corresponding trajectory of a 4-bar linkage is shown in Fig. 8(a).

The red and blue lines denote the optimized and desired

trajectories of a hind leg of the proposed robot. Note that due to

a higher weighting factor a , the optimized trajectory on the

ground matches the desired one well while the optimized one in

the air shows some discrepancies with the desired one in the air.

Fig. 8(b) describes the CAD design of the 4-bar-based hind leg

of the proposed robot, where additional links denoted by the

yellow lines are combined with the optimized 4-bar linkage to

ensure a constant contact angle between the foot of the hind leg

and the ground for sufficient friction. As a result, the contact

angle between the foot of the 4-bar-based hind leg and the

ground remained almost constant within the range −2.28°

(when the foot leaves the ground) to 0.84° (when the foot

contacts with the ground) as shown in Fig. 8(c).

IV. EXPERIMENTAL RESULTS AND DISCUSSIONS

A. Lizard-inspired Robot and Experimental Set-up

The prototype of the lizard-inspired robot and experimental

set-up are shown in Fig. 9(a) and 9(b), respectively. As shown

in Fig. 9(a), the main chassis of the robot is made of aluminum

and, its hind legs and tail are made of ABS plastic. Because all

the motors of the robot are located on the posterior body section,

the masses of the anterior and posterior bodies and tail are 0.89,

1.46, and 0.19 kg, respectively. Thus, the total weight of the

proposed robot is same as that in the dynamic model in Section

II. Fig. 9(b) shows the experimental set-up for the lizard-

inspired robot, which is located on a commercial treadmill

equipped with the linear motion (LM) guides and the rotational

axis to allow only the translations along the X- and Y-axes and

the rotation along the yaw direction. It is noted that during the

experiments, the motion of proposed robot is confined within

the 2-dimensional horizontal plane in order to minimize the

effects of other factors like balancing and locomotion by its

hind legs on the direction of movement undesirably. As shown

in Fig. 9(b), the IMU sensor (RTxQ, Ruva Tech, Inc.) is

installed at the rotational axis connected to the robot to measure

the real yaw angle of the proposed robot, and the sampling rate

and resolution are 125 Hz and 0.1°, respectively. All

communications between the PC and the high- and low-level

controllers are done at the sampling time of 15 ms through the

RS232. The time period T in the experiment is set to be 2.4 s

(the same as in the simulation). The gains of the high- and low-

level PD controllers in (7) and (9) for the new robot are tuned

based on those gains for the dynamic model, which are given in

Table IV of the Appendix.

First, in order to compute the duty factor of the new robot, its

bipedal running gait on the treadmill is examined using a high-

speed camera to capture a motion at the speed of 480 frames per

second (fps). Snapshots of the robot running in a bipedal gait

are given in Fig. 10(a), where the running speed of lizard-

inspired robot is set to be 0.472 m/s and the rotational joint of

Fig. 8. (a) Comparison of optimized and desired trajectories of 4-bar linkage,

(b) CAD design of 4-bar-based hind leg of the proposed robot and (c)

simulated contact angles between a hind foot and the ground.

TABLE III OPTIMAL DESIGN PARAMETERS FOR HIND LEG OF LIZARD-INSPIRED ROBOT

*

1L 58.9 mm *

2L 104 mm

*

3L 98.1 mm *

4L 107 mm

*

xc 104 mm *

yc 38.3 mm

* 0.0734 rad *s 1.27




8

LM guide is fixed. As shown in Fig. 10(a), the right and left

hind legs of the robot remained in contact with the ground in

the snapshots (s1)–(s3), (s12), and (s6)–(s9), respectively. On

the other hand, no leg remains in contact with the ground in the

snapshots (s4), (s5), (s10), and (s11). The measured time

duration of the snapshots in Fig. 10(a) is shown in Fig. 10(b).

Because the duty factor of the lizard-inspired robot is defined

by dividing the contact time (0.297 s) of one leg by the contact

time and air-borne time (1.203 s), the computed duty factor of

the proposed robot is about 0.246 (almost the same as for C.

draconoides: 0.25). Therefore, it is shown that the 4-bar-based

hind leg of the proposed robot successfully performs the desired

bipedal running gait similar to C. draconoides.

B. Experiments and Discussions

First, the S-shaped lateral motion in the dynamic model is

compared with those of a real lizard. To this end, the motions

of the waist and tail of a real lizard are obtained from a video

file [28] by assuming it by three links as shown in Fig. 11(a).

The motions are approximated by a combination of sinusoidal

functions in a manner similar to that used in Section III.B. For

fair comparison, the approximated phase value of each joint is

normalized because the size and weight of the dynamic model

are quite different from those of a real lizard. As shown in Fig.

11(b), the phase difference between the waist angles of a real

lizard and the dynamic model is just 0.2% and the phase

difference between the tail angles of a real lizard and the

dynamic model is 6.7%, which implies that the S-shaped lateral

motions of the proposed dynamic model show the similar trend

to those of a real lizard.

Two types of experiments are carried out to investigate the

ability of the robot to maintain its direction of movement via S-

shaped lateral motions of waist and tail. While the locomotion

of the robot in the first experiment is accompanied by the

angular motions of the waist and tail combined with the open-

loop leg propulsion, the locomotion in the second experiment is

done only by open-loop leg propulsion. Recall that the angular

motions of waist and tail are controlled by the error between the

desired and real yaw angles of the posterior body section of

robot. The time duration of both experiments is 30 s and the

proposed robot runs forward at the same speed as in Fig. 10(a).

It is noteworthy that the fact that the direction of movement of

the robot remains constant can be verified if the two body

sections and the tail of robot are parallel to the center line of the

treadmill regardless of the position of the robot on the treadmill.

Snapshots of the first and second experiments are shown in

Figs. 12 and 13, where the center line of the treadmill is denoted

Fig. 9. Photographs of (a) Lizard-inspired robot and (b) Experimental set-up.

(s1) (s2) (s3)

(s4) (s5) (s6)

(s7) (s8) (s9)

(s10) (s11) (s12)

(a)

Fig. 10. (a) Snapshots of bipedal running robot on the treadmill and (b)

measured durations of bipedal running gait.

Joint 1Waist jointTail joint

(a)Anterior

body

Posterior

bodyTail

Fig. 11. (a) Captured image of real lizard and approximated joint angles and

(b) phase differences of waist angle (top) and tail angle (bottom) between real lizard and dynamic model.




9

by the blue dotted line. The time step between each snapshot is

0.33 s. During the entire time duration in Fig. 12, the direction

of movement of the proposed robot seems to remain almost

constant by virtue of S-shaped angular motions of the waist and

tail. Even though the position of the robot is not perfectly at the

center of the treadmill, it still remains close to the center line of

the treadmill. It is observed that from (s5) to (s7), and from (s13)

to (s14) in Fig. 12, the robot fulfills the function to correct its

direction of movement when it is likely to be far away from the

center line of the treadmill. To the contrary, the yaw angle of

the robot driven only by the open-loop leg propulsion becomes

rapidly inclined toward the upper side and cannot be restored to

zero as shown in Fig. 13. Recall that the ability of the robot to

maintain its direction of movement is examined at its maximum

bipedal running speed of 0.472 m/s.

For the detailed analysis, the yaw angles of the posterior body

section of the robot in both experiments are measured and

compared in Fig. 14(a), where the blue and red lines denote the

yaw angles in the first and second experiments, respectively. As

shown in Fig. 14(a), the yaw angle of the posterior body section

of the robot in the second experiment drastically increases in

less than 3 s and eventually deviates from the treadmill. On the

other hand, the yaw angle of the posterior body section of the

robot in the first experiment periodically varies within the range

from −18.2° to 21.5° during the entire time. Although the yaw

angle of the posterior section adjusted by the S-shaped lateral

motions of the waist and tail is not perfectly symmetric with

respect to 0°, its mean value seems well maintained around 5.5°,

as shown in Fig. 14(b). In comparison with the simulated yaw

angle, the measured yaw angle of the posterior body section of

the robot is slightly shifted so that some discrepancies between

their magnitudes are observed. However, both yaw angles in

Fig. 14(b) show the same phase tendency to keep the direction

of movement of the proposed robot constant. It is noted that the

discrepancy in their magnitudes is effectively controlled within

the bound of 10° during the bipedal running of the robot as

confirmed in Fig. 14(b) (for more detailed information, refer to

the attached video file of bipedal running lizard-inspired robot).

V. CONCLUSIONS

In this study, we have carried out the investigation to keep

the direction of movement of lizard-inspired robot via its S-

shaped lateral body motions. Through the simulation using the

simplified dynamic model of lizard-inspired robot together with

the high- and low-level PD controllers, it is verified that the

mean value of yaw angle of the dynamic model remains close

to zero via the periodic swing of its body ranging from -9.4° to

8.6°. Based on the dynamic model, the prototype of lizard-

inspired robot is simply constructed by using only three links

combined with two rotational joints. When the proposed robot

runs in the bipedal gait at the maximum speed of 0.472 m/s, the

one cycle of its gait and the resulting duty ratio are measured to

be 1.2 s and 0.246, almost same as that of real lizard (=0.25).

Also, the S-shaped lateral motions of dynamic model are shown

quite similar to those of real lizard in that the phase differences

of waist and tail angles between them are 0.2% and 6.7%,

respectively. During bipedal running of the proposed robot on

the treadmill, the yaw angle of its posterior body section

periodically varies within a range from -18.2° to 21.5° to

achieve the S-shaped lateral body motion so that its direction of

movement is kept almost constant in that its body is maintained

parallel to the center line of the treadmill. It is noted that the

previous investigations of lizard-inspired robots have been

mainly related with maintaining the pitch angle of the robot or

making rapid turns using only a tail when their locomotion is

based on wheels or multiple legs. On the contrary, this study

(s1) (s2) (s3)

(s4) (s5) (s6)

(s7) (s8) (s9)

(s10) (s11) (s12)

(s13) (s14) (s15)

Fig. 12. Snapshots of bipedal running lizard-inspired robot controlled by the

angular motions of its waist and tail.

Fig. 14. (a) Yaw angles of real lizard-inspired robot with/without angular motions of waist and tail, (b) simulated and real yaw angles of lizard-inspired

robot controlled by angular motions of waist and tail.

(s1) (s2) (s3)

(s4) (s5) (s6)

(s7) (s8) (s9)

Fig. 13. Snapshots of bipedal running lizard-inspired robot driven by only the

open-loop leg propulsion without angular motions of waist and tail.




10

has focused on managing the yaw angle of the lizard-inspired

robot by using whole-body motion during the bipedal running

gait. As our future work, the bipedal running ability of the

proposed robot will be improved by reducing its size as well as

weight and utilizing the force control of leg motion so that it

can run fast on its own feet. Moreover, the S-shaped motion of

the robot to manage the yaw angle will be refined by

constructing the posterior body section and tail with multiple

flexible links so that it can change and maintain its direction of

movement more effectively like a real lizard.

APPENDIX

A. Forward Kinematics of the Proposed Robot

For simplicity, cos i iq c= , sin i iq s= , cos( )i j ijq q c+ = , and

sin( )i j ijq q s+ = are used below. For the center of mass (CoM)

of each body iO , i = 1, 2, 5 in Fig. 1(a), its position with

respect to the frame {XYZ} is given by

1 2

3

4 5

1 1 2

2 2 2

1 2 3 5

3

1 3 1 3 34

1 3 1 3 34

1 3 34 34

1 3 34 34

1 3 6 34 34 1

3

3 7 34

1 3 6 3

2 2 5

1 1

4 342

0.5 c 0.5 c, ,

0.5 s s 0.5 s

c c 0.5 c

s s 0.5

c c s c c s

s s cf

O

f f

O

O

O O

q l c q l lP P

q l q l l

q l l lP

q l l l s

q l a y q l a yP P

q l a y

+ + + = =

+ + +

+ + + =

+ + +

+ + + + + − = =

+ + −

4

1 3 7 342 34s s cfq l a y

+ + +

where fx and fy are the distances from the CoM 2O of the

posterior body section to the foot along the 2X - and

2Y -axes in

Fig. 1(b) and 20.5i ifa x l q= − + + , i =6, 7. It is noted that in Fig.

1(b), the yaw angles of anterior and posterior body section

sections and tail are 3q ,

3 4q q+ , and 53 4q qq+ + while the yaw

angle of a left/right hind leg is 3 4q q+ . The Jacobian matrix

3 7[ ]i iw

T T T

i vJ J J R = , i = 1, 5 is derived as

1 3 1 3 34 34

1 3 1 3 34 3

2

4

1 3 34 3 345 34 3 345 3 345

1 3 34 3

2 2

1 2 2 2

2 2

3 342 5

1 0 0.5 s 0 0 0 0 1 0 s 0.5 s 0.5 s 0 0 0

0 1 0.5 c 0 0 0 0 , 0 1 c 0.5 c 0.5 c 0 0 0

0 0 1 0 0 0 0 0 0 1 1

s 0.5 0.s 5 0

0 0 0

1 0 s s s s 0 0

0 1 c .c

.5

0 5 c

l l l l

J l J l l l

l l l

J

l l l

ll l l

− − − −

= = +

− − −

= +

− − −

+ 34 3 345 3 345

1 3 6 34 34 6 34 34 34

1 3 6 34 34 6 34 34 34

1 3 34 34 34 34 34

1 3 34 34 34 3

4

7 7

5 7 7

c c 0.5 c 0 0

0 0 1 1 1 0 0

1 0 s s c s c 0 c 0

0 1 c c s s s 0 s 0

0 0 1 1 0 0 0

1 0 s s c s c 0 c 0

0 1 c

0

c s

.

s s

5

f f

f f

f f

f f

l a y a y

J l a y a y

l a y a y

J l a y

l

y

l

a

− − −

= + −

− − + − +

=

−

+ + −

+

− −

−

+ 4 340 s 0

0 0 1 1 0 0 0

where 2 7

ivJ R and 1 7

iwJ R are related to the linear and

angular velocities of iOP . It is noted that if all feet of inverse

dynamic model are in the air, 3 7cJ = 0 and

3 1cx = 0 , but if its

left (or right) foot contacts the ground, 4cJ J= and

4cx x=

hold (or5cJ J= and

5cx x= hold). In this study, the structures

of anterior and posterior body sections and tail are assumed as

cylinders while those of the left and right hind legs were

assumed as rectangles. Thus, the moments of inertia of

anterior/posterior body sections and tail ( i =1, 2, 3) and

left/right hind legs ( i = 4, 5) were calculated as

( )

( )

2 23

2

1

2 4

2

1

1

12

1, 2, 3

4, 5

i i i i

i i i i

I m l i

I m l i

w

w

= =

= =

+

+

where il and

iw are the length and the width of link i . Then,

the joint space inertia matrix ( )A q and the Coriolis matrix

( , )C q q are calculated by 7 ( ) ( )( )1

5

11 2)( (( ) , , )ij jki

j i

k

k

A q A qA qT

i i i kq q qi kA q J M J qq qC

= == + +=

where diag , ,i i i iM m m I= with inertia matrix iI , i = 1, 2,

5. The Coriolis and centrifugal force vector is derived by

)( ), ( ,q C q q qb q = .

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TABLE IV

PD GAINS OF HIGH-LEVEL AND LOW-LEVEL CONTROLLERS

Type Gain Simulation

(Experiment) Gain

Simulation

(Experiment)

High-level y

pk 0.1 (0.08)

y

dk 0.9(1.03)

Low-level

2,

t

pk 257.8 (218.5) 3,

t

pk 65.02 (53.3)

4,

t

pk 221.25 (265.5) 5,

t

pk 230 (201.5)

2,

t

dk 26.68 (22.7) 3,

t

dk 69.05 (59.01)

4,

t

dk 63.10 (69.4) 5,

t

dk 50 (42.2)




11

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[26] X. Gu, Z. Guo, Y. Peng, G. Chen, and H. Yu, “Effects of compliant and

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[27] https://www.youtube.com/watch?v=ExyMxKDxT9M

[28] S. Reilly and M. Delancey, “Sprawling locomotion in the lizard Sceloporus clarkia: quantitative kinematics of a walking trot, Journal of

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[32] A. J. Ijspeert et al., “From swimming to walking with a salamander robot

driven by a spinal cord model,” Science, vol. 315, pp. 1416-1420, 2007.

Jeongryul Kim received the B.S., M.S., and

Ph.D. degrees in mechanical engineering from

Seoul Nat’l University, South Korea in 2009,

2011 and 2015, respectively. From 2015 to

2018, he worked at Samsung Electronics. He is

currently a Post-Doctoral Researcher at KIST,

South Korea. His research interests include

bio-inspired robotics, robot mechanism design and medical

robotics.

Hongmin Kim received the B.S. degree in

mechanical engineering from KAIST in 2013,

the M.S. degree in mechanical engineering

from Seoul Nat’l University, South Korea in

2015. He is currently pursuing his Ph.D.

degree at Seoul Nat’l University.

Youngsoo Kim received the B.S. and Ph.D.

degrees in mechanical and aerospace

engineering from Seoul Nat’l University,

South Korea in 2013 and 2019, respectively.

He is currently a Post-Doctoral Researcher at

Seoul Nat’l University. His research interest

includes design of mobile platforms.

Jaeheung Park (M’10) received the B.S. and

M.S. degrees in aerospace engineering from

Seoul Nat’l University, South Korea, in 1995

and 1999, respectively, and the Ph.D. degree

in aeronautics and astronautics from Stanford

University, CA, USA in 2006. He is now an

Associate Professor in the Department of

Intelligent Convergence Systems at Seoul Nat’l University,

South Korea. His research interests are in areas of robot–

environment interaction, contact force control, robust haptic

teleoperation.

TaeWon Seo (M’10) received the B.S. and

Ph.D. degrees from school of mechanical and

aerospace Engineering, Seoul Nat’l University.

He is currently an Associate professor at the

School of Mechanical Engineering, Hanyang

University, South Korea. His research

interests include robot design, analysis,

control, optimization, and planning. Dr. Seo received the Best

Paper Award of the IEEE/ASME Transaction on Mechatronics

in 2014, and currently works as a Technical Editor of

IEEE/ASME Trans. on Mechatronics, Associate Editor of IEEE

Robotics and Automation Letters and Intelligent Service

Robots.

Hwa Soo Kim (M’15) received the B.S. and

Ph.D. degrees in mechanical engineering from

Seoul Nat’l University, in 2000 and 2006,

respectively. He is currently an Associate

professor at the Department of Mechanical

System Engineering, Kyonggi University,

South Korea. His research interests include

design, modeling and control of various mobile platforms.

Jongwon Kim received the B.S. degree in

mechanical engineering from Seoul Nat’l

University in 1978, the M.S. degree in

mechanical and aerospace engineering from

the KAIST in 1980, and the Ph.D. degree in

mechanical engineering from the University of

Wisconsin-Madison, USA in 1987. He is now

a Professor in the School of Mechanical Engineering, Seoul

Nat’l University, South Korea. His research interests include

parallel mechanisms, Taguchi methodology, and field robots.

https://www.youtube.com/watch?v=ExyMxKDxT9M

Documents

A New Lizard-inspired Robot with S-shaped Lateral Body Motionsdyros.snu.ac.kr/wp-content/uploads/2019/11/20191119_072531.pdf · motions of a real lizard. The simple robot body consisted