IEEE Robotics & Automation Magazine - December 2020 - 36

interactions, and control signals. Deimel et al. [66] investigated the feasibility of codesigning the morphology of
soft hands and their control strategies for grasping and
found that the codesign always outperformed the counterpart optimization limited to only one design domain.
Spielberg et al. [67] proposed a " learning-in-the-loop
optimization " design method that allows for the cooptimization of the controller and material parameters, using
differentiable simulation techniques. These works represent initial attempts to create an end-to-end design paradigm for soft robots and should be further generalized to
more complicated scenarios and validated through physical experiments.
Design Space Representation
To incorporate the physical design variables into an optimization model, a first prerequisite is the mathematical representation of the design
space, which is spanned by
the aforementioned design
In the framework of
variables. The representation should, in general, be
topology optimization,
able to describe all candidates in the full design
to describe an arbitrary
space in a unified mathematical framework.
topological shape, two
In the framework of
topology optimization, to
classes of representation
describe an arbitrary
topological shape, two
methods have been
classes of representation
methods have been widewidely used.
ly used. The first category
is density-based methods, where the design
variables are represented by the continuous " artificial density " of 0-1 [68], [69]. Depending on the physical problem, the spatially varying density may describe the
existence or removal of a material or an actuation field,
and its distribution is typically discretized by finite elements and interpolated using shape functions. The other
representation approach uses an implicit description of
boundaries to parameterize the geometry, i.e., level-setbased methods that implicitly define the interfaces
among material phases or actuation fields by iso-contours
of a level-set function [70]-[72]. This implicit function
enables a crisp description of the free-form boundaries. In
comparison with explicit boundary descriptions, level-set
functions enable the much more convenient tracking of
topological changes.
When dealing with structural shape and topology optimization on free-form surfaces, the conformal mapping theory
originating from differential geometry on the Riemannian
manifold can be combined with topology optimization theories to recast the manifold embedded in the 3D space as a 2D
topology optimization problem in the Euclidean space.
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Ye et al. [73] provided a unified level-set-based computational framework for the generative design of free-form
structures by conformally mapping the manifolds onto a 2D
rectangle domain where the level-set function is defined,
which allows for the convenient use of conventional computational schemes for level set methods.
Optimization Implementation
To explore the vast design space spanned by the geometry,
material, and actuation fields, optimization tools that can
automatically search for the optimal design candidates are
essential. Powerful optimization algorithms are expected to
refine the existing designs and, more importantly, create
novel free-form designs that are otherwise hardly attainable
by human intuitions or experiences. As summarized in
Table 1, we organize the referred works in the " Optimization
Model " section in terms of the design variable, report the
employed optimization methods, and briefly comment on
their generality and applications.
Simulation and Analysis
An important prerequisite to the implementation of
optimization algorithms is the simulation and analysis
tool that enables designers to evaluate the performance
of the current robot design. This prerequisite has been
very challenging for soft robots, mainly due to the nonlinearity, multiphysical coupling, and complex interplay
between multiple bodies and the environments. In general, one can hardly derive analytical (or semianalytical)
solutions for the kinematics of a soft robot but must
resort to numerical computation. The analytical solutions listed in Table 1 are case specific or simplified by
assumptions such as linearity. Nonlinear finite element
analysis has dominated because it can accurately capture
complex mechanical behaviors. However, nonlinear
solvers tend to suffer convergence issues and are usually
limited to relatively small deformations. In addition, the
computational cost is very high, which hinders efficient
evaluations of designs.
Many attempts have been made to perform fast and robust
simulation. Instead of computing the continuous deformation
fields, Hiller and Lipson [77] applied nonlinear relaxation
where the structure was represented as a network of basic elements, including springs, beams, and masses. However, the
parameter identification of the elements is a great challenge,
and various actuation technologies can hardly be incorporated into the framework.
Based on the finite element method (FEM), Duriez
and colleagues [78] have developed a well-known physics-based simulation engine, SOFA, which simulates the
deformation of soft robots by progressively solving a quasi-static equilibrium function for each sample time. The
method was recently further improved to achieve realtime computation with a reduced model [74] and has
been verified on real soft robots [Figure 6(a)]. The computation efficiency requires further improvement, and
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