IEEE Robotics & Automation Magazine - September 2023 - 113

the dynamic simulations presented in this article, the reader
may refer to the supplementary video available at https://doi.
org/10.1109/MRA.2022.3202488.
TEST 4: ENERGY BALANCE IN SOROSIM DYNAMICS
Here, we investigate the energy balance in damped and
undamped cantilever beams released from rest with an initial
strain under gravity. We create a soft link with a length (L) of
0.5 m and a radius that linearly decreases from 2 to 1 cm.
The material properties of the link are
and t =1,000 kg/m3
E 1MPa=
, o 05= .,
. For the damped rod, we use an elastic
damping of 11.2 KPa. We enable all three angular modes of
deformation (torsion and rotations about the y- and z-axes)
with a cubic polynomial approximation for the strains. To
obtain a complex dynamic motion, the soft link is released
from rest with an initial bending of 1 rad/m about the y-axis.
We run the dynamic simulation for 5 s. Figure 7 describes the
motion of the damped and the undamped rod between time
t 0= and
t ..= 075s
The total energy of the beam at a given time is given by the sum
of kinetic, gravitational potential, and elastic potential energies:
EdXAg hdX
dX
tot
=+
+- -
2
1
=+ +qq qKq
2
1
..
T
Ug
2
1
T
where h is the screw velocity, Mr
##r
#
() ()
qM ()
00
hh
M
2
1
LLT
T
L ))
R
pp pp
t
(5)
is the cross-sectional screw
inertial matrix, A is the area of cross section, g is acceleration
due to gravity, h is the y coordinate of a cross section of the rod
(in the opposite direction of gravity), p is the screw strain vector,
p) is the reference screw strain, and R is the cross-sectional
screw stiffness matrix. Here, M and K are defined as in (4).
The first, second, and third terms on the right-hand side
of the equation represent kinetic, gravitational potential, and
elastic potential energies, respectively. We plot these for the
damped and the undamped cases in Figure 7(a). The total
energy of the damped system is decaying, and that of the
undamped system remains a nonzero positive constant corresponding
to the initial strain energy. The results highlight the
robustness of the SoRoSim dynamics, as any deviation in the
energy conservation is solely attributed to errors in numerical
integration (in time and space).
For the undamped case, the system's total energy should
remain the same as the initial elastic potential energy, while
for the damped case, the total energy should decrease monotonically.
Figure 7(a) conveys the change in the values of various
forms of energies of the system. The total energy of the
damped system is decaying, and that of the undamped system
remains a nonzero positive constant, as expected. This
example also emphasizes that apart from dynamic and static
simulations, we can use the toolbox for postprocessing. The
methods of the SorosimLinkage class allow the user to compute
quantities, such as the configuration (the position and orientations
of cross sections), velocities, and generalized mass
.. .
matrices, as functions of q and
q .
.
The user can compute these
values in a separate MATLAB code for postprocessing, such
as the energy computations in this example.
MODELING APPLICATIONS
This section presents examples of the static equilibrium and
dynamic simulations relevant to the soft robotics field that
the toolbox can solve. We demonstrate the applicability of
the toolbox to different systems and the use of custom functions
to model various external forces, including contact and
fluid interactions.
HYBRID MANIPULATORS
Serial robotic arms are widely used to automate manufacturing
processes and any task requiring object gripping and
manipulation. We show the toolbox's ability to model and
analyze hybrid manipulators for two different cases.
CASE 1: STATIC EQUILIBRIUM
This example demonstrates the static equilibrium analysis
of a hybrid (rigid-continuum) robotic arm with soft grippers
under a wide range of loading conditions, including gravity
(distributed load), a point force (F), rigid joint actuation, and
cable actuation of soft links for gripping. We use the toolbox
to investigate the static equilibrium analysis of the
manipulator and demonstrate its ability in modeling systems
with multiple types of links, joints, and forces within
the same framework.
The system consists of 11 links that form open-, branched-,
and closed-chain components, as illustrated in Figure 8(a).
The first link is a rigid link that is connected to the ground
via a planar joint, which allows displacement in the xy-plane
and rotation about the z-axis. The second and third links are
rigid links connected using revolute joints, which rotate
about the y- and z-axes, respectively. These three rigid links
0.1
-0.1
-0.2
(0.7, -0.025, -0.025)
(0.682, -0.009, -0.153)
0.1298 m
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
X (m)
(a)
C: Copy of Static Structural
Total Deformation
Type: Total Deformation
Unit: m
Time: 1 s
2/10/2022 4:24 PM
0.13265 Max
0.11791
0.10317
0.088435
0.073696
0.058956
0.044217
0.029478
0.014739
0 Min
0.075
0.15
0.225
(b)
FIGURE 5. (a) The SoRoSim results showing the reference and
deformed shape of
the linkage. (b) The total deformation and
deformed shape obtained from ANSYS.
SEPTEMBER 2023 IEEE ROBOTICS & AUTOMATION MAGAZINE
113
0.3 (m)
z
y
x
Z (m)
https://doi
IEEE Robotics & Automation Magazine - September 2023

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