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Dynamic Modeling of A Class of Soft Growing

Robots Using Euler-Lagrange Formalism

Haitham El-Hussieny∗1, Sang-Goo Jeong2†and Jee-Hwan Ryu2

1Electrical Engineering Department, Faculty of Engineering (Shoubra), Benha University, Egypt

(E-mail: haitham.elhussieny@feng.bu.edu.eg)

2School of Mechanical Engineering, Korea University of Technology and Education, Cheonan-si, Republic of Korea

(E-mail: jsg1215z@kut.ac.kr, jhryu@kut.ac.kr)

Abstract: Due to their morphological adaptation and tip-extension capabilities, soft growing robots have shown an in-

creased potential for many different scenarios, including the inspection and navigation of conﬁned and challenging envi-

ronments. However, evaluating the performance of such robots iteratively over a variety of design aspects or environmental

conditions could be challenging due to their increased lengths that could require long manufacturing time. Thus, in this

paper, utilizing the Euler-Lagrange equation, a computationally efﬁcient dynamics model is derived for the “vine robot”,

a class of soft growing robots that can extend its length up to tens of meters long. Simulation experiments have been

performed to validate the derived dynamics model of the vine robot that could in turns facilitate model-based feedback

control design and simulation-based performance evaluation in the future.

Keywords: continuum robot; growing robots; dynamics model; euler-lagrange

1. INTRODUCTION

Inspired by the morphological adaptation capability

shown by snakes, elephant trunks, and octopus tentacles,

continuum robots have proved the potential to enable ma-

neuvering in tight and conﬁned environments [1]. As

compared to rigid robots, continuum robots have curvi-

linear structures with continually bending backbones that

make them highly adaptable to the surroundings [2].

However, continuum robots are commonly found to have

small lengths which could restrain their applicability in

the navigation of distant environments [3].

Investigating the growing process exhibited by plants,

new mobility by growth approach has been recently pro-

posed to come up with growing robots. This kind of

robots emulates biological growth by incrementally ex-

panding either their lengths, volumes or knowledge [4].

Soft growing robots with expanding lengths have the abil-

ity to reach narrow spaces searching for victims or can

serve as conduits to transfer air or water for them in dis-

aster scenarios [5].

Previous studies have reported the embodiment of long

ﬂexible robots in congested environments. For instance,

Tsukagoshi et al. [5] have proposed multiple degrees of

freedom growing robot, called “Active Hose,” that was

used for rescue and searching scenarios. This robot was

designed to be ﬂexible with the capability of expanding

its length by connecting small ﬂexible units of two de-

grees of freedom in series. An active ﬂexible long cable

with ciliary vibration mechanism was developed by Isaki

et al. [6] to achieve navigation in narrow spaces. Ex-

pandable soft robots have also proposed such as “Slime

Scope” [7] that was a pneumatically driven expendable

arm with a camera attached to its tip used for search and

rescue people in the rubble environments. Tsukagoshi et

al. [8] have developed a ﬂexible hose-like robot that was

able to steer in narrow environments by manual control.

†Sang-Goo Jeong is the presenter of this paper.

Recently, vine-like growing robots, which imitate the

growing process exhibited by plants, have proved superb

performance towards tackling inspection and rescue pur-

poses such as [9, 10]. Hawkes et al. [11] have developed

a novel growing robot using the concept of tip eversion

[12]. This robot made of thin-walled polyethylene tub-

ing that can expand up to several tens of meters to nav-

igate challenging environments either through teleopera-

tion [13] or guided by obstacles [14]. A steerable version

has developed by Greer et al. [15] by inﬂating multi-

ple series pneumatic artiﬁcial muscles placed around the

robot’s backbone. The increased length-to-diameter ra-

tios, the lengthening capability, and the ﬂexible structures

allow vine-like robots to penetrate into cluttered environ-

ments as evaluated in [16].

Although the potential of vine-like growing robots in

unstructured and congested environments, there is still

a notable paucity towards investigating dynamics mod-

els for this kind of robots. For a ﬁxed length contin-

uum robot, the dynamics model has been already de-

rived, such as in [17-19]. However, due to their ex-

panding lengths and subsequently complexity, deriving

a simple and effective dynamics model for vine robots

could be challenging. Having a mathematical model for

growing robots in general and vine robots, in particu-

lar, could in turns reduces the time required for manu-

facturing and subsequently facilitates their performance

evaluation over a variety of design or environmental as-

pects. In [13], a growing robot simulation is developed

using Unity3D game engine to evaluate the robot perfor-

mance in unstructured scenarios. However, a mathemat-

ical model could enrich the development of model-based

control schemes with ease to control the growth of grow-

ing robots. Thus, in this paper, a computationally efﬁ-

cient spatial dynamics model is derived, based on Euler-

Lagrange formalism, for the vine growing robot proposed

by Coad et al. [16].

Proceedings of the SICE Annual Conference 2019

September 10-13, 2019, Hiroshima, Japan

978-4-907764-66-1 PR0001/19 ¥400 © 2019 SICE 453

PP

PP

P1

camera

sPAMs

Growing

direction

(a) (b) (c)

P

P

Fig. 1 Working principal of the growing vine robot.

2. KINEMATICS MODEL

In this research, the “vine robot” developed in [16] is

under discussion, where this kind of robots can elongate

their tips up to tens of meters via eversion mechanism

[15]. Air pressure is applied to its core tube as depicted

in Figure 1 to facilitate tip extension while steering is

achieved by applying air pressure through one or two of

the thsee rerial Pneumatic Actuator Muscles (sPAM) that

are placed around the robot circumference. A camera or

other sensing device could be added to its tip to facilitate

the navigation capability of the robot.

2.1. Direct Kinematics

The constant-curvature model [20] that is commonly

applied in modeling continuum-like robots is assumed

here to ﬁnd the forward kinematics of the vine growing

robot. The distal tip transformation matrix Tb

rwith re-

spect to its base is derived in terms of the robot conﬁg-

uration parameters q∈IR3including its length s, the

bending angle θand the plane angle φas shown in Figure

2. Thus, Tb

ris obtained as

Tb

r=

cos2φ(cos θ−1) + 1 sin φcos φ(cos θ−1)

sin φcos φ(cos θ−1) cos2φ(1 −cos θ) + cos θ

cos φsin θsin φsin θ

0 0

−cos φsin θscos φ(cos θ1)

θ

−sin φsin θssin φ(cos θ1)

θ

cos θssin θ

θ

0 1

(1)

As noted in the last column in (1) that represents the

robot Cartesian tip position, the denominators are di-

vided by θthat could create mathematical singularities

in case of the robot is axially stretched, i.e. θ= 0.

Hence, to avoid such singularity, the tip Cartesian posi-

tion x= [x, y, z]∈IR3is approximated by the ﬁfth

order Taylor expansion as follows,

x=−scos(φ)θ5

a1

−θ3

a2

+θ

a3,

y=−ssin(φ)θ5

a1

−θ3

a2

+θ

a3,

z=s1 + θ4

a4

−θ2

a5

(2)

θ

s

φ

D

β

d

p

li

x

y

z

Fig. 2 Schematic of vine growing robot with its conﬁgu-

ration parameters.

where a1= 720, a2= 24, a3= 2, a4= 120, a5= 6

are the series coofﬁencts.

The actuator lengths l∈IR4including the core tube

length l0and the sPAM lengths could be expressed in

terms of the robot conﬁguration qas follows,

l0=s,

li=s−1

2D θ cos(φ+βi)i= 1,2,3; (3)

where liis the length of sPAM actuator i,Dis the core

diameter and βi= [0,2

3π, −2

3π]as higlighted in Figure

2.

2.2. Differential Kinematics

The growing robot tip velocity, ˙

x∈R3, is related to

the time derivatives of the robot conﬁguration parameters

˙q as follows,

˙

x=Jq(q)˙

q(4)

where the Jacobian matrix, Jq(q)∈R3×3, is computed

analytically as follows,

Jq(q) = ∂x

∂q=∂x

∂(s, θ, φ)(5)

where xis the robot tip Cartesian position mentioned in

Eq. (2).

The time derivative of the sPAM actuator lengths, ˙

l,

could also be stated in terms of ˙q as follows,

˙

l=Jl(q)˙

q(6)

where Jl(q)∈R4×3, is computed analytically as fol-

lows,

Jl(q) = ∂l

∂q=∂l

∂(s, θ, φ)(7)

where lis the set of actuator lengths in Eq. (3).

454

θ

s

m

φ

Kθ

Dθ

T1

T2

Fig. 3 A vine-like growing robot is modeled as point

mass on its tip, a damper Dsacting on its length and

a torsional spring and damper Kθ, Dθacting on its

curvature.

3. DYNAMICS MODELING

In order to derive a computationally efﬁcient dynam-

ics model for the growing vine robot under discussion,

the distributed mass of the robot is assumed to be concen-

trated on the distal end of the robot. Hence, the position

of this mass with respect to the robot base is continuously

moving while the vine robot is growing. This assump-

tion could be reasonable since the mass of the robot body

could be neglected compared to the mass of the sensory

system that is commonly attached to the robot’s tip. The

dynamics equations of the vine robot are obtained using

the Euler-Lagrange equations along each generalized co-

ordinate qi, i.e. the robot length s, the bending angle θ

and the angle of curvature φ.

d

dt

∂L

∂˙qi

−∂L

∂qi

=τi(8)

where L=T − U is the Lagrangian that denotes the

difference between the kinetic energy T(q,˙

q)and the

potential energy U(q), while τiis the non-conservative

generalized force that includes the external and the dissi-

pative forces exerted on qi.

3.1. General Model Description

Neglecting the masses of the robot body and the sPAM

actuators, the mass mof the attached sensory system is

located at a position xfrom the robot’s base (Figure 3).

Axial damping Dsis added along the length sto rep-

resent the damping action along the longitudinal motion

while torsional damping Dθis added to represent the

damping along the bending movement. A torsional stiff-

ness Kθis added in the direction of θto represent the

robot tendency towards being straight (θ= 0). Three

identical sPAM actuators with diameter dare placed with

an angle β= 120◦around the robot core tube that has a

main diameter D.

3.2. Kinetic Energy

The kinetic energy of the vine robot with mass mis

the sum of the translational and rotational kinetic ener-

gies. Since rotational energies in continuum-like robots

are much smaller than the corresponding translational en-

ergies [18], the total kinetic energy is

T=1

2˙

qT[mJq(q)TJq(q)˙

q(9)

where Jq(q)is the Jacobian obtained in Eq. (4).

3.3. Potential Energy

The potential energy Uincludes both the gravitational

potential Ugof the mass mand the spring potential Uθ

due to the torsional stiffness

U=Ug+Uθ(10)

where

Ug=−mgTr(11)

and

Uθ=1

2Kθθ2(12)

3.4. Generalized Forces

The generalized forces τin (8) are the set of forces

acting on the selected generalized coordinates q. These

forces include the set of actuation forces τa, used for both

lengthening and steering the robot, and the dissipative

forces τddue to damping. [12].

τ=τa+τd(13)

3.4.1. Actuation forces

The actuation forces acting on the vine robot are re-

sulting from the input force Psdue to the air pressure ap-

plied to its core tube to lengthen the robot beside the ten-

sion forces Ti= [T1, T2, T3]Tapplied by the three sPAM

actuators to steer the vine robot. Since these tensions are

not directly acting on the generalized coordinates q, they

are represented as external forces that are subsequently

projected into qcoordinates as follows

τa= [(Ps−Ye)T1T2T3]Jl(q)(14)

where Yeis the yield force below which no extension oc-

curs [12], while Jl(q)∈IR4×3is the Jacobian that relates

the actuator coordinates lto the generalized coordinates

qcomputed in Eq. (6). The tension forces Tiexerted by

sPAM actuators are directly proportional to the inside air

pressure pias follows

T i =kipi(li−leq

i)for sPAM i= 1,2,3(15)

where liis the sPAM length, leq

iis the equilibrium length,

piis the air pressure, and kiis the proportional constant

[15].

3.4.2. Dissipative forces

Due to the damping Dsand Dθin the longitudinal and

the bending directions, energy is lost from the generalized

coordinates, sand θwhen the robot moves. Thus, the

dissipative force Fdcan be stated as

τd=−

Ds˙s

Dθ˙

θ

0

(16)

455

3.5. Resulting Dynamics Model

By plugging the Lagrangian and its derivatives and the

generalized forces in Eq. (13) into the Euler-Lagrange

equation in Eq. (8), three coupled second-order differ-

ential equations are obtained representing the vine robot

equation of motion in the form

M(q)¨

q+C(q,˙

q)˙

q+G(q)=τ,(17)

where M(q)∈IR3×3is the inertia matrix that is com-

puted using the kinetic energy Tin (9) as follow

Mij =∂

∂qj∂T

∂qi,{qi, qj} ∈ q(18)

while C(q,˙

q)∈IR3×3is the Coriolis matrix computed

as

Ckj =

3

X

i=1

1

2∂Mkj

∂qi

+∂Mki

∂qj

−∂Mij

∂qk˙qi,{qj, qk} ∈ q

(19)

and G(q)∈IR3is the gravitational term obtained from

the computed potential energy in (10) as follow

Gi=∂U

∂qi

, qi∈q(20)

The obtained dynamics equations are written in such

a way to show the input-output behavior of the growing

vine robot. Thus, six state variables are selected to rep-

resent the robot conﬁguration and their rate of change as

follows

x1=s(t)

x2=θ(t)

x3=φ(t)

x4= ˙s(t)

x5=˙

θ(t)

x6=˙

φ(t)

(21)

Therefore, by introducing the state variables, the equation

of motion can be written as follows

˙x1

˙x2

˙x3

=I

x4

x5

x6

,

˙x4

˙x5

˙x6

=D−1

τ−C

x4

x5

x6

−G

(22)

where I∈IR3×3is an identity matrix. Then, Eq. (22)

be written in the general form

˙x(t) = f(x, t) + g(x, t)u(t)(23)

where xrepresents state variables, f(.)and g(.)are non-

linear functions, and u(t)∈IR4represents the command

vector composed of the input force Psused to grow the

robot length besides the tension forces Ti, i = 1,2,3ap-

plied by the sPAM actuator to steer the robot. To account

for the irrevisable growth exhibited by vine robot, where

once the robot has grown to a certain length it cannot re-

tract to a smaller value, a constraint is imposed on the

time derivative of the robot length ˙s, i.e. x4≥0.

Table 1 Parameters of the selected growing robot

Parameter Description Value

mSensory system mass 0.1 Kg

KθTorsional stiffness 20 N.m/rad.

DsAxial damping 1.5 N.s/m

DθTorsional damping 1.1 N.m.s/rad.

YeYield force 4 N

DRobot diameter 0.05 m

gGravity constant 9.81 m/s2

0246810

Time (s)

-0.05

0

0.05

0.1

(t)

s = 4 m

s = 2 m

Fig. 4 Static analysis of the vine robot dynamics model

with zero actuation under initial condition θ= 0.1

and φ= 0 with two different robot lengths.

4. SIMULATION EXPERIMENTS

To simulate and verify the derived vine robot dynamics

model, a set of simulation experiments have conducted

using MATLAB with ode45 solver with reference to the

selected robot parameters in Table 1.

4.1. Static Analysis

The simulation of the vine robot static analysis con-

siders the robot is under static equilibrium with zero ac-

tuation, either in the core tube pressure or the sPAMs ten-

sions. The robot is simulated under its initial condition

θ0= 0.1and φ0= 0 at different robot lengths s01 = 2 m

and s02 = 4 m. The dynamic response of the robot bend-

ing angle is shown in Figure 4. As expected, the bend-

ing angle shows damped oscillations due to the torsional

stiffness Kθand the torsional damping Dθadded to the

model. These oscillations are in proportion to the robot

length since the moment caused by the mass mis greater

at s= 4 mthan that of length s= 2 m.

4.2. Open Loop Dynamics

In this set of experiments, we present the simulation

results of the vine robot response under open loop actua-

tion. First, the robot length sis plotted in Figure 5 over

time under step input of Ps= 5 N. This value is selected

to be higher than the yield force of Ye= 4 N to ensure the

growth of the robot. As highlighted, the robot length in-

creases linearly after the application of the constant input

force.

In the second simulation, we applied a ramp tension

on the ﬁrst sPAM actuator while keeping the other two

sPAMs inactive to investigate the response of the robot

bending angle. As shown in Figure 6, the bending an-

gle increases gradually with the input tension that op-

poses the weight of the sensory system attached to the

robot’s tip. The observed bending angle oscillations are

signiﬁcant with increased robot length as mentioned in

456

0 5 10

Time (s)

0

2

4

6

Input force (N)

0 5 10

0

5

10

Robot length (m)

Fig. 5 Response of the vine robot length due to step input

force of 5 N.

0 5 10

Time (s)

-100

-50

0

sPAM #1 tension (N)

0 5 10

-0.2

0

0.2

0.4

Bending (t) (rad)

s = 1 m

s = 5 m

Fig. 6 Bending angle responses at different robot lengths

after applying a ramp tension force at one of the

sPAM actuator while keeping the others inactive.

the statch analysis subsection.

The plane angle φplays a crucial role in moving the

vine robot in 3D space where it rotates the robot plane

around its vertical axis. Thus, in the third simulation,

we have applied tension forces on two sPAM actuators

subsequently as depicted in Figure 7 to investigate the re-

sponse of the angle φ. In the ﬁrst ﬁve seconds, sPAM 1 is

active with constant tension force T1=−20 N and sPAM

2 was inactive. During this period the bending angle θ

shows a constant value around 0.1 rad. while the angle

φwas zero. In the next ﬁve seconds, the actuation has

switched between sPAM 1 and 2. As shown, the bend-

ing angle starts to oscillate and has stabilized ﬁnally at

the equilibrium value of 0.1 rad, while the value of angle

φhas jumped and oscillated around 2 rad. This, in fact,

demonstrates the ability of the derived model to consider

the robot spatial movements.

4.3. Closed-Loop Feedback Control

The following Proportional-Derivative (PD) controller

with gravity compensation is proposed to ensure trajec-

tory following in the vine robot’s state space

u=Jl(q) [G(q) + Kpe+Kd˙e],(24)

where u∈IR4is the control action applied to the core

tube pressure and the tension of the three sPAM actu-

ators while Jl(q)represents the actuator jacobian ob-

0 5 10

Time (s)

-20

-10

0

sPAM tension (N)

sPAM 1

sPAM 2

0 5 10

0

0.2

0.4

(t) (rad)

0 5 10

0

2

4

(t) (rad)

Fig. 7 Responses of bending and plan angle, θand φ

respectively when applying tension forces by sPAM 1

then sPAM 2.

tianed in Eq. (6). The e∈IR3is the state error vec-

tor that is deﬁned as e=qd−q,qd∈IR3repre-

sents the desired state, Kp,Kd∈IR3×3are symmet-

ric positive deﬁnite diagonal matrices represent the pro-

portional and derivative gains respectively. The gravi-

tational term G(q)is added to ensure the state track-

ing globally and is obtained for the robot dynamics in

Eq. (20). The proposed control scheme is evaluated in

terms of tracking multiple desired states. The PD gains

are selected by trails until good tracking performance is

achieved with Kp=Diag(100,5×104,1000)and

Kp=Diag(10,1000,100). As depicted in Figure

8, the vine robot with the proposed control scheme has

proved reasonable tracking as was expected in the three

coordinates. However, the robot loses tracking when its

current length is higher than the desired one as noted.

This, in fact, is due to the irreversible growth process ex-

012345

0

2

4

6

s(t)

Reference

Measured

012345

0

1

2

3

(t)

012345

Time (s)

-2

0

2

(t)

Fig. 8 Dynamic response for s,θand φwith multiple

desired lengths. The robot loses tracking when the

desired length is lower than its current length.

457

hibited by the vine robot, where once it has grown to a

certain length, it can not be retracted to lower values.

5. CONCLUSIONS

In this paper, the Euler-Lagrange formalism is utilized

to derive a computationally efﬁcient dynamics model of

“vine robot”, a class of soft growing robots that is char-

acterized by its tip extension via eversion mechanism.

The model has derived with the assumption of constant-

curvature for the direct kinematics of the robot with an

assumed concentrated mass at the robot’s tip. The de-

rived model is simulated over different scenarios show-

ing the coupling between the robot’s length and the bend-

ing angle. In addition, a PD with gravity compensation

model-based closed loop controller is designed to ensure

tracking of the desired state. Hence, the derived dynamics

model can be used to evaluate vine robot design iterations

with no need to have a physical robot. For conﬁrming the

validity of the proposed dynamic model, comparison of

behaviors between the model and the actual vine robot is

planned for future work.

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