IEEE Robotics & Automation Magazine - September 2023 - 108

(a)
(c)
(e)
(b)
(d)
(f)
(g)
2.5 cm
FIGURE 1. A visual representation of the various robotics systems the toolbox can model. These systems include rigid-soft hybrid robots,
where the soft links can be approximated as Cosserat rods. (a) X-RHex, a hybrid robot with compliant legs. (b) Tuft Softworm, a soft robot
that relies on shape memory alloy actuators and frictional forces for locomotion. (c) PoseiDRONE, a hybrid robot that uses tendon-based
soft appendages to swim underwater and move on the seafloor [1]. (d) STIFF-FLOP, a pneumatic soft manipulator designed for use in
surgical procedures [11]. (e) FinRay, a closed-chain soft gripper with rigid connectors [12]. (f) A two-finger tendon-driven hybrid gripper
consisting of modular segments [13]. (g) A bioinspired hybrid flagellate robot for underwater applications [14]. The SoRoSim toolbox can
model all these classes of robots by including contact, friction, and fluidic interactions as custom external forces similar to the examples
in the " Modeling Applications " section.
discuss the computational performance of the toolbox and draw
conclusions and future directions of SoRoSim.
GOVERNING EQUATIONS
The GVS approach was recently introduced by Renda et al. [16]
in statics and Boyer et al. [17] in dynamics. It is based on a variable-strain
parametrization of soft links represented by Cosserat
rods, 1D slender rods that can bend, twist, stretch, and shear.
Cosserat's model is the most general rod model, as it accounts
for the orientation as well as the position of beam elements and
allows for all six modes of deformation to be considered during
analysis [2]. Since the strain is parameterized in the GVS
approach, it is easy to disable any deformation modes. Thus,
other beam theories can be kinematically reproduced. For
instance, SoRoSim users can enable the rotational modes along
g4(1)
g34
η1(X1)
r
g1(X1)
ξ4
g23
ξ1(X1)
g2(L2)
FIGURE 2. The proposed kinematics for a floating hybrid soft-rigid
chain.
10 IEEE ROBOTICS & AUTOMATION MAGAZINE SEPTEMBER 2023
8
with shear along y and z to create a Timoshenko beam. The
GVS model is also geometrically exact and generalizes the geometric
theory of rigid robotics to hybrid systems of soft and
rigid links with multidimensional joints, externally applied
point and distributed forces, and distributed actuation forces
[18]. In this section, we give a summary of the model and an
overview of the efficient computational techniques implemented
in the SoRoSim toolbox.
Consider a floating hybrid kinematic chain composed of
interconnected rigid and soft bodies represented by Cosserat
rods. The configuration of a soft body i (respectively, a rigid
body) with respect to its predecessor in the chain is defined as
a curve:
gg ()
ii ii i
():, ()XL XSE 3
$ !!=cm
ii
60
@ 7
[respectively, a point g SE 3
i !
Rr
1
(1)
()], mapping the body frame at
Xi to the body frame of the previous body at the reference
configuration, as demonstrated in Figure 2.
To study the exponential representation of (),
g Xii we introl
t
duce
its partial derivative with respect to space, ()= iipgg ,
6
and with respect to time, ()= htr where ()X Rii
.
gg ,
i Xi
i
i
hi
iiX
p
defines the strain twist in the body frame, ()X R6
r
i !
!
is the
velocity twist relative to the predecessor in the body frame,
and ()$X is the isomorphism from R6
g XX
to se(3). The space derivii
= Xit^h where
exp () ,
ative of ()g Xii is a matrix differential equation that can be
integrated in space according to ()
Xt
case, ()Xiipt
is the Magnus expansion of the field p .t For the rigid link
is constant in Xi and equal to the body frame
IEEE Robotics & Automation Magazine - September 2023

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