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Dynamic Modeling of A Class of Soft Growing Robots Using Euler-Lagrange Formalism

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Due to their morphological adaptation and tip-extension capabilities, soft growing robots have shown an increased potential for many different scenarios, including the inspection and navigation of confined and challenging environments. 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.
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Dynamic Modeling of A Class of Soft Growing
Robots Using Euler-Lagrange Formalism
Haitham El-Hussieny1, Sang-Goo Jeong2and Jee-Hwan Ryu2
1Electrical Engineering Department, Faculty of Engineering (Shoubra), Benha University, Egypt
2School of Mechanical Engineering, Korea University of Technology and Education, Cheonan-si, Republic of Korea
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
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
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(a) (b) (c)
Fig. 1 Working principal of the growing vine robot.
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 qIR3including its length s, the
bending angle θand the plane angle φas shown in Figure
2. Thus, Tb
ris obtained as
cos2φ(cos θ1) + 1 sin φcos φ(cos θ1)
sin φcos φ(cos θ1) cos2φ(1 cos θ) + cos θ
cos φsin θsin φsin θ
0 0
cos φsin θscos φ(cos θ1)
sin φsin θssin φ(cos θ1)
cos θssin θ
0 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,
z=s1 + θ4
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 lIR4including the core tube
length l0and the sPAM lengths could be expressed in
terms of the robot configuration qas follows,
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. Differential Kinematics
The growing robot tip velocity, ˙
xR3, is related to
the time derivatives of the robot configuration parameters
˙q as follows,
where the Jacobian matrix, Jq(q)R3×3, is computed
analytically as follows,
Jq(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, ˙
could also be stated in terms of ˙q as follows,
where Jl(q)R4×3, is computed analytically as fol-
Jl(q) = l
(s, θ, φ)(7)
where lis the set of actuator lengths in Eq. (3).
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
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.
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= 120around 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
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
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].
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= [(PsYe)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(lileq
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
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
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
where M(q)IR3×3is the inertia matrix that is com-
puted using the kinetic energy Tin (9) as follow
Mij =
∂qi,{qi, qj} ∈ q(18)
while C(q,˙
q)IR3×3is the Coriolis matrix computed
Ckj =
∂qk˙qi,{qj, qk} ∈ q
and G(q)IR3is the gravitational term obtained from
the computed potential energy in (10) as follow
, qiq(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
x4= ˙s(t)
Therefore, by introducing the state variables, the equation
of motion can be written as follows
where IIR3×3is 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. x40.
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
Time (s)
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.
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
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
0 5 10
Time (s)
Input force (N)
0 5 10
Robot length (m)
Fig. 5 Response of the vine robot length due to step input
force of 5 N.
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 uIR4is 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)
sPAM tension (N)
sPAM 1
sPAM 2
0 5 10
(t) (rad)
0 5 10
(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 eIR3is the state error vec-
tor that is defined as e=qdq,qdIR3repre-
sents the desired state, Kp,KdIR3×3are 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-
Time (s)
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.
hibited by the vine robot, where once it has grown to a
certain length, it can not be retracted to lower values.
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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We present the design and development of a plant-inspired robot, named Plantoid, with sensorized robotic roots. Natural roots have a multi-sensing capability and show a soft bending behaviour to follow or escape from various environmental parameters (i.e., tropisms). Analogously, we implement soft bending capabilities in our robotic roots by designing and integrating soft spring-based actuation (SSBA) systems using helical springs to transmit the motor power in a compliant manner. Each robotic tip integrates four different sensors, including customised flexible touch and innovative humidity sensors together with commercial gravity and temperature sensors. We show how the embedded sensing capabilities together with a root-inspired control algorithm lead to the implementation of tropic behaviours. Future applications for such plant-inspired technologies include soil monitoring and exploration, useful for agriculture and environmental fields.
Soft continuum robots exhibit access and manipulation capabilities in constrained and cluttered environments not achievable by traditional robots. However, navigation of these robots can be difficult due to the kinematics of these devices. Here we describe the design, modeling, and control of a soft continuum robot with a tip extension degree of freedom. This design enables extremely simple navigation of the robot through decoupled steering and forward movement. To navigate to a destination, the robot is steered to point at the destination and the extension degree of freedom is used to reach it. Movement of the tip is always in the direction tangent to the end of the robot's backbone, independent of the shape of the rest of the backbone. Steering occurs by inflating multiple series pneumatic artificial muscles arranged radially around the backbone and extending along the robot's whole length, while extension is implemented using pneumatically driven tip eversion. We present models and experimentally verify the growing robot kinematics. Control of the growing robot is demonstrated using an eye-in-hand visual servo control law that enables growth and steering of the robot to designated locations.
Across kingdoms and length scales, certain cells and organisms navigate their environments not through locomotion but through growth. This pattern of movement is found in fungal hyphae, developing neurons, and trailing plants, and is characterized by extension from the tip of the body, length change of hundreds of percent, and active control of growth direction. This results in the abilities to move through tightly constrained environments and form useful three-dimensional structures from the body. We report a class of soft pneumatic robot that is capable of a basic form of this behavior, growing substantially in length from the tip while actively controlling direction using onboard sensing of environmental stimuli; further, the peak rate of lengthening is comparable to rates of animal and robot locomotion. This is enabled by two principles: Pressurization of an inverted thin-walled vessel allows rapid and substantial lengthening of the tip of the robot body, and controlled asymmetric lengthening of the tip allows directional control. Further, we demonstrate the abilities to lengthen through constrained environments by exploiting passive deformations and form three-dimensional structures by lengthening the body of the robot along a path. Our study helps lay the foundation for engineered systems that grow to navigate the environment.
Conference Paper
Artificial apical extension in a soft robot, inspired by biological systems from plant cells to neurons, offers an interesting alternative to movement forms found traditionally in robots. Apically extending systems can move effectively in some environments that impede traditional locomotion. Artificial apical extension has been realized using a continuous stream of surface material, thin-walled, flexible plastic, which is everted at the tip by internal pressure. Understanding artificial apical extension as a form of movement requires a model to describe and predict the capabilities of the system. Unlike many other forms of movement, the model includes components that are dependent on the previous path in addition to path-independent terms associated with actuation. The model draws inspiration from biological models of apical extension and mechanical models of compliant Bowden cable actuation, and is verified though a series of tests on physical systems that isolate each term of the model.
The goal of this work is to develop a soft-robotic manipulation system that is capable of autonomous, dynamic, and safe interactions with humans and its environment. First, we develop a dynamic model for a multi-body fluidic elastomer manipulator that is composed entirely from soft rubber and subject to the self-loading effects of gravity. Then, we present a strategy for independently identifying all of the unknown components of the system; these are the soft manipulator, its distributed fluidic elastomer actuators, as well as the drive cylinders that supply fluid energy. Next, using this model and trajectory-optimization techniques we find locally-optimal open-loop policies that allow the system to perform dynamic maneuvers we call grabs. In 37 experimental trials with a physical prototype, we successfully perform a grab 92% of the time. Last, we introduce the idea of static bracing for a soft elastomer arm and discuss how forming environmental braces might be an effective manipulation strategy for this class of robots. By studying such an extreme example of a soft robot, we can begin to solve hard problems inhibiting the mainstream use of soft machines.