IEEE Robotics & Automation Magazine - September 2023 - 121

oscillatory response [Figure 11(c)(3)]. The error is observed
to decrease slowly, allowing the tip to approach the desired
position and orientation.
DISCUSSION AND CONCLUSION
The computational performance of the toolbox depends on
problem parameters, such as the number of links, the DoF of
the system, strain order used, number of Gauss quadrature
points on soft links, number of actuators,
values of actuator strengths, external
forces, material, and geometric properties.
Here, we report the simulation
speed of the current version of the
SoRoSim toolbox. Table 2 summarizes
parameters that characterize the computational
performance of the toolbox. For
each analysis discussed in the " Toolbox
Validation " and " Modeling Applications "
sections, the table shows information,
including the total number of
links (N), total DoF (),Ndof maximum
strain order [( )],Op
order of the Zannah
collocation approximation [Z(O)], and
total number of points at which static or
dynamic (4) equations are evaluated
().Neva
For static analyses (highlighted
in gray), the seventh column of the table indicates the total
static equilibrium simulations performed ().Ns For dynamic
simulations, the column shows the real-time duration (0 to
)
"
FAST (2.5× FASTER THAN
REAL TIME) COMPUTATION
OF ACTUATOR
at 1.3 GHz and 1.5 GHz and with 16 GB of random-access memory.
The MATLAB version used for these analyses is R2021a.
The comparison of tmax and ts indicates that the toolbox performs
in real time or faster for most of the examples presented
in this article. This is the result of the geometrically exact
formulation, which requires the fewest DoF to estimate the
state of a robot. Fast (2.5× faster than real time) computation
of actuator strengths could be used for model-based inverse
dynamic control of real-world soft manipulators.
We will implement an implicit
time integrator to speed up the simulation
in a future version of the toolbox. Further
code optimization and implementation of
parallel computing can also increase the
toolbox performance.
It is important to emphasize that,
STRENGTHS COULD BE
USED FOR MODEL-BASED
INVERSE DYNAMIC CONTROL
OF REAL-WORLD
SOFT MANIPULATORS.
„
tmax of the dynamics problem.
MATLAB solvers used to solve the static or dynamic equations
are shown in the eighth column. The final column shows
the total simulation runtime ().ts We used a PC with the following
specifications for these analyses: Intel Core i7-1065G7 CPU
TABLE 2. A summary of the toolbox performance.
EXAMPLE
" Test 2: FEM Study of a Fixed-Fixed L-Shaped Beam " section
" Test
3: Dynamics of a Flexible Flying Rod " section
" Test 4: Energy Balance in SoRoSim Dynamics " section
" Test 4: Energy Balance in SoRoSim Dynamics " section*
" Case 1: Static Equilibrium " section
" Case 2: Contact Dynamics " section
" Underwater Robotics " section
" Design Optimization " section
" Case 1: Tip Pose Trajectory Tracking " section
" Case 2: Tip Pose Regulation Under Gravity " section
Static simulations are highlighted in gray.
*With undamped (D = 0) dynamics.
†With a time step of 0.01 s.
" Test 1: Fixed-Free Beam With a Follower Tip Force " section 1 5
N NDOF p(O) Z(O) NEVA
4
2 30
1 15
1 12
1 12
11 39
6 21
6 20
1 12
1 6
1 6
2
2
3
3
1
1
1
1
4
4
4
4
2
4
4
2
4
4
4
16
20
13
15
15
NS (NUMBER)/
TMAX (S)
130
1
7
5
5
143 5
96
30
15
9
5
2,887
15
15
16
10
SOLVER
fsolve
fsolve
ode45
ode15s
ode113
fsolve
ode15s
ode113
fsolve,
patternsearch
ode1†
ode1†
TS (S)
5.5
0.5
1
1.5
6
6
27
5
93
5
4
unlike FEM packages, the toolbox is suitable
only for systems whose soft links
can be modeled as Cosserat rods. Despite
this
limitation,
the geometrically exact
approach used in the toolbox makes it a
fast (due to a smaller number of DoFs) and
accurate tool for systems involving large
deformations, which is the case for the
majority of soft robotics applications. The
toolbox allows the modeling and analysis
of systems with open-, branched-, and closed-chain and interconnected
structures. Additionally, the toolbox provides the
user with plenty of ways to use the output data with existing
MATLAB packages and user-written MATLAB codes.
In summary, this article presented SoRoSim, an intuitive
MATLAB toolbox that uses the GVS approach to provide a
unified framework for the modeling, analysis, and control of
soft, rigid, and hybrid robots. It provides users with a level of
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