IEEE Robotics & Automation Magazine - September 2023 - 111

The Twist class specifies the active DoF of lumped joints
and the deformation modes and corresponding strain orders
of soft link divisions. For a soft link division, the class
allows the user to enable the required modes among six
deformation modes: torsion about the x-axis, bending about
the y-axis, bending about the z-axis, elongation along the
x-axis, shear along the y-axis, and shear along the z-axis.
The order of a particular mode corresponds to the polynomial
that is used to estimate the strain values. The Twist
class also allows the user to define a reference strain value
for the soft division corresponding to its rest configuration.
The inset in Figure 3(b) shows a sample GUI with which
the user creates the Twist class for a soft link division. The
GVS model allows the definition of the strain base of the
soft links as a continuous or discontinuous function of X.
The user may change the default polynomial basis of soft
links once the SorosimLinkage is defined.
The SorosimLinkage class allows the user to assemble
previously defined links into various single-, branched-,
open-, and closed-chain systems. SorosimLinkage calls the
Twist class to determine each link's DoF. The user can also
add closed-loop joints by selecting appropriate joint types.
The class allows the definition of various external forces and
actuation inputs. The class automatically precomputes and
saves constant properties of the linkage, such as the generalized
stiffness matrix (K) and the generalized damping matrix
(D). It also allows the users to program custom external and
actuation forces. An overview of the SorosimLinkage creation
is given in Figure 3(b).
We pack the SorosimLinkage class with " methods " that
facilitate the analysis of multibody systems and the postprocessing
of results (static equilibrium configuration and
dynamic video output). The methods for analysis include
functions to solve the GVS model for static and dynamic analyses.
The SorosimLinkage methods can also compute values
of system parameters, such as the Jacobian (J), generalized
mass (M), and Coriolis (C) matrices for a given value of q
and
q .
.
Users can utilize these methods for problem-specific
analyses. The reader may refer to the toolbox manual [15] for
a detailed description of all the properties and methods of
SoRoSim classes.
TOOLBOX VALIDATION
To validate the toolbox, we conduct several numerical tests
and comparisons with verified and published data. We present
these tests in this section.
TEST 1: FIXED-FREE BEAM WITH
A FOLLOWER TIP FORCE
We first simulate the bending behavior of a cantilever beam
with a follower force applied at the tip. Many authors have
considered this problem and solved it using numerical
approaches, such as the finite-strain rod method [21] and
other geometrically exact models [17]. Using the toolbox, we
construct a 100-m-long cylindrical beam with Young's modulus
E
.675GPa and a diameter of 57 cm. We set a fourth=
order
bending strain about the y-axis to model the deflection
of the rod when subjected to the follower tip force. The force
is varied between 0 N and 130 kN. We use 15 Gauss quadrature
points, specified during the SorosimLink creation process,
for the computation of integrals (in this case, K).
Figure 4(a) illustrates the deflection of the link as well as the
horizontal and vertical displacement of the tip as modeled by
the toolbox. The results obtained match those obtained in
[21], as detailed in Figure 4(b). The total DoF of the rod
modeled using SoRoSim is 5 DoF, while Simo et al. [21]
used a 1D finite-element mesh consisting of five elements
with quadratic shape functions corresponding to 10 DoF.
This demonstrates the GVS approach's ability to recreate
accurate results by using fewer DoF. Additionally, the reported
computation time is 16.4 s per loading step in [17], while
it took 42 ms for the same in the SoRoSim toolbox.
We also use this example to highlight the scaling process
the toolbox uses. The toolbox performs internal computations
on a soft division after normalizing its length into one unit
(here, there are 100 m to one unit). Consequently, physical
quantities with length dimensions are scaled using the original
length of the division. Once the simulation is complete, the
toolbox scales back the resulting values of joint coordinates
into their actual dimensions. The normalization of soft divisions
avoids poorly scaled matrices, such as the generalized
stiffness matrix. This allows faster static solutions and more
stable dynamic simulations.
TEST 2: FEM STUDY OF A
FIXED-FIXED L-SHAPED BEAM
To test the performance of the toolbox with respect to commonly
used methods of modeling, we compare the results
obtained when a soft linkage is subjected to a distributed
load (gravity) with those obtained through FEM simulation.
Two 0.7-m-long soft links with a 55-cm#
square
cross section (aspect ratio: 14:1) are connected to form an
L-shaped linkage clamped at each end. We use a fixed
closed-loop joint to fix the rear end of the second link with
the ground. All six deformation modes are enabled for this
simulation; quadratic and linear polynomials are used to
estimate the rotational and translational modes, respectively.
Hence, there are 15 DoF for a link (30 in total). For the
material, E 10 MPa;=
t =1,200 kg/m3
Poisson's ratio, o 05= .; and density,
, are used.
We compare the static equilibrium results obtained from
the toolbox with those of ANSYS Workbench. A linkage with the
same geometry and material properties is created in ANSYS.
A total of 152 quadratic 3D elements with 1,062 nodes (~3,000
DoF) are used for the simulation. Figure 5(a) describes the
toolbox result, while Figure 5(b) conveys that of ANSYS. The
maximum deformation obtained from the toolbox result is
12.98 cm, whereas the FEM simulation gives a deformation of
13.27 cm. Hence, with fewer DoF (: ),
1 100
the GVS method
can estimate the deformed shape of the L-shaped beam. The
simulation results also suggest that the toolbox can effectively
handle closed-loop problems.
SEPTEMBER 2023 IEEE ROBOTICS & AUTOMATION MAGAZINE
111
IEEE Robotics & Automation Magazine - September 2023

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