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Transactions on Mechatronics
IEEE/ASME TRANSACTIONS ON MECHATRONICS 1
Topology Optimization of Skeleton-reinforced Soft
Pneumatic Actuators for Desired Motions
Shitong Chen, Feifei Chen, Member, IEEE, Zizheng Cao, Yusheng Wang, Yunpeng Miao,
Guoying Gu, Member, IEEE, and Xiangyang Zhu, Member, IEEE
Abstract—Multimaterials with different modulus can endow
soft robots with embodied intelligence that deliver spatially-
varying deformation upon actuation. There is an increasing
need for design tools that can rigorously and efficiently gen-
erate material layouts for desired motions. Here, we present a
design paradigm for soft pneumatic multimaterial actuators by
attaching a stiffer material layer as skeleton to softer inflated
rubber, and develop a topology optimization based framework to
automatically generate the skeleton layout that leads the actuator
to achieve desired motions such as bending or twisting. Our
method is enabled by a dynamic level set function to describe and
track the topological change of the skeleton, large-deformation
analysis compatible with the varying skeleton layout, and a
gradient-based optimizer to govern the evolution of material
layout, with the geometric and material nonlinearities taken into
account. A forward geometric mapping and a backward design
velocity mapping are constructed to allow manipulating the level
sets on the planar space. We show that the design methodology
is capable of generating high-performance bending and twisting
actuators of cylindrical or customized cone shape. The simulation
and experiment results show that, the bending actuator achieves a
free bending angle 73°and blocking force 2.05 N, and the twisting
actuator achieves a large rotation angle of 143°.
Index Terms—Soft robots, soft pneumatic actuators, topology
optimization, multimaterial design.
I. INT ROD UC TI ON
In the past decade, the use of soft materials for building
robots has enabled a new generation of robots with un-
precedented flexibility, generally referred to as soft robots.
The inherent distributed compliance offered by freeform ge-
ometry and material diversity are imbuing soft bodies with
programmable mechanical properties to deliver desired com-
plex motion behaviors under physical stimuli. This embodied
mechanical intelligence has allowed for various innovative
applications, and in the meantime requires fundamentally new
design perspectives [1].
The motion behavior of a soft body is concurrently de-
termined by its geometry, material, and actuation, and these
factors are usually physically coupled. This is the case for
pneumatic actuators where the pressure loading is directly
related to the channel shape, known as design dependent
This work was supported by the National Natural Science Foundation of
China (Grants 51905340 and 91948302), and was sponsored by Shanghai
Sailing Program (19YF1422900). (Corresponding author: Feifei Chen.)
The authors are with State Key Laboratory of Mechanical System and Vi-
bration, Shanghai Jiao Tong University, and Robotics Institute, School of Me-
chanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China
(email: stonehot@sjtu.edu.cn; ffchen@sjtu.edu.cn; caozizheng@sjtu.edu.cn;
gloomysheng@sjtu.edu.cn; mythmyp@sjtu.edu.cn; guguoying@sjtu.edu.cn;
mexyzhu@sjtu.edu.cn).
loads. Spatially gradient materials are promising to create
complex motion behaviors that may be hard to achieve using
a single material, and usually leads to simple geometry and
compact bodies in applications such as actuators [2], end-
effectors [3] and jumping robots [4]. The complex interplay
among multiple materials, geometry profiles, and external
actuations in a highly nonlinear physical process makes the
design of soft actuators and robots with desired deformations
extremely difficult. Designers usually have to rely on intuitions
or experiences, and a mathematical approach to automate the
design process is in high demand.
Mechanical modeling of soft multimaterial systems has shed
light on the design optimization. Fiber-reinforced elastomers
for constructing pneumatic actuators provided excellent ex-
amples in which the elastomers are typically isotropic, while
the anisotropic fibers offer an avenue for exploring the defor-
mation mode of the actuator including extension, contraction,
bending, and twisting [5]–[7]. Connolly et al. developed an
analytical model to identify the fiber arrangements for tracking
a desired kinematic trajectory [8]. However, this mechanical
model can only describe regular layouts of fibers and thus is
not applicable to general multimaterial design of soft robots.
To explore the vast design space of soft actuators, re-
searchers have investigated topology optimization of the
freeform material layout, driven by cables [9]–[11], electric
fields [12] or magnetic fields [13]. Among them, topology
optimization for soft pneumatic robots has been of great
interest due to their wide use. Ma et al. made an initial
attempt by modeling soft pneumatic materials with a fixed
frame and then optimizing the material densities of the frame
to generate desired motions [14], but the fixed frame does
not allow amorphous topological change. Chen et al. designed
a pneumatic bending actuator by optimizing internal cavities
with topological change [15], and found that the optimized
channel presents similar geometric features with Pneu-Nets
design [16]. Instead of relying on varying the cavity geometry
that involves cumbersome design-dependent loads, Zhang et
al. optimized the reinforced material placed on a soft pneu-
matic tube to produce desired bending motions [17].
We summarize the representative works on design of soft
pneumatic actuators and robots in Table I, with a special focus
on the design methodology, and it is found that the initial
attempts on topology optimization of soft pneumatic robots all
rely on linear optimization models [15], [17]. However, this
assumption of linearity is insufficient to capture the physical
complex behaviors of highly deformable soft materials. The
large deformation upon actuation typically induces significant
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Transactions on Mechatronics
IEEE/ASME TRANSACTIONS ON MECHATRONICS 2
TABLE I: Summary of current published works on design optimization of soft pneumatic robots.
Authors / Citation Design Space Mechanical Analysis Optimization Algorithm Application
Connolly et al. 2012 [8] Fiber angle Nonlinear N/A Bending, twisting actuators
Ma et al. 2017 [14] Frame material density Nonlinear Gradient-based Shape-matching
Chen et al. 2019 [15] Topology of channel Linear Gradient-based Bending actuator
Zhang et al. 2019 [17] Partial topology of reinforced material Linear Gradient-based Bending actuator
This work Full topology of skeleton layer Nonlinear Gradient-based Bending, twisting actuators
geometric nonlinearity, and the material hyperelasticity is
dominant for large strains, necessitating a nonlinear stress-
strain model. These nonlinearities are yet to be incorporated
into the optimization model to fully exploit the potential of
topology optimization for generative design of soft robots [1].
The paradigm shift in modeling and optimization translates
into addressing the challenge of nonlinear large deformation
analysis and sensitivity analysis in nonlinear regions through-
out the optimization process.
In this paper, resting on an actuator design paradigm by
attaching a stiffer material layer as skeleton to softer inflated
rubber, we develop a design methodology to automatically
generate the skeleton layout that leads the actuator to achieve
desired motions such as bending and twisting. The pneumatic
rubber embedded inside remains unchanged as the non-design
domain so that the pressure loading is design-independent. The
varying skeleton layout is described and tracked by a dynamic
level set function, and we construct a forward geometric
mapping from the planar regular design domain to the curved
shape of the physical skeleton layer and a backward design
velocity mapping, to allow manipulating the level sets on
the planar space. Combined with the nonlinear deformation
analysis of a design candidate, an adjoint sensitivity analysis
was conducted to infer how the desired motion quantitatively
depends on the skeleton profile, and thus the skeleton layout
can evolve toward the steepest descent direction to generate
the optimal design, without designer’s intervention.
Our optimized design is validated through both simulation
and experiments which agree well in both linear and nonlinear
regions. We demonstrate that the consideration of nonlinear-
ities delivers better design. Very interestingly, characteristic
structural features abstracted from the optimized design are
well explainable in terms of producing bending or twisting
motions, which provides new inspiration for designers. The
bending actuator prototype achieves a free bending angle
73°and blocking force 2.05 N, and the twisting actuator
achieves a large rotation angle of 143°. To demonstrate the
generality of our design method, we further extend the design
domain to cone-shaped actuators for desired bending or twist-
ing motions, and the simulation and experiment results well
validate the effectiveness.
II. GE OM ET RY MOD EL
A. Level Sets
The skeleton layer rests on a curved shape and is allowed
to change in topology to deliver the optimal design. Level
sets have been widely used in engineering applications for
Fig. 1: Representation of a level set function to describe the
skeleton layout.
describing and tracking topological shapes [18], and have been
well embedded in topology optimization methods [19]. Here,
we construct level sets defined in a signed distance function
whose zero contour represents the skeleton layout (see Fig. 1),
φ(X, t)>0,∀X∈Ω
φ(X, t)=0,∀X∈Γ
φ(X, t)<0,∀X∈D/(Ω ∪Γ)
(1)
where Ddenotes the unfolded rectangular design domain,
Ωdenotes the solid area of the skeleton, Γdenotes the
boundaries, X∈Drepresents the coordinates of a point
in question, and tdenotes pseudo-time for evolution. The
evolution of skeleton layout translates into the variation of zero
contour of the level set function, governed by the Hamilton-
Jacobi equation,
∂φ(X, t)
∂t =−|∇φ(X, t)|Vn(2)
where Vnrefers to the normal velocity at point X.
B. Domain Transformation
The curved shape the skeleton layer rests on can split into a
rectangular as the design domain on which we define the level
set function, and thus a geometry mapping is constructed. The
forward mapping of a vector is constructed by ac=Jarwith
arand acvectors in 2D and 3D space, respectively, and J
is the mapping matrix. The backward mapping of the moving
velocity is
Vr
n=Gnc·nrVc
n(3)
with Vr
nand Vc
nmoving velocity in 2D and 3D space,
respectively, nrand ncthe normal directions in 2D and 3D
surfaces, respectively, and Gis the inverse mapping of J. For
a cylindrical skeleton layer, the mapping is isometric, leading
to Vr
n=Vc
n.
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Transactions on Mechatronics
IEEE/ASME TRANSACTIONS ON MECHATRONICS 3
C. Domain Connectivity
The connectivity issue of the skeleton geometry arises when
it splits into the rectangular domain. Additional operations are
needed to ensure the signed distance property and connectivity.
Here, we construct an augmented level set function by peri-
odically tessellating the original level set function to ensure
the signed distance property on the cutting edge, and impose
their interconnectivity by parameterization. Parameterization
of level sets in B-splines provides a simple and fast way to
weld and polish the edges by directly restricting the weight
coefficients in B-splines [20]. We constrain the coefficients
related to the cutting edges of the design domain to guarantee
the connectivity of the level set function. The reader may refer
to [20] for the technical details. In this work, to ensure the
continuity of the design velocity which points to the boundary
normal direction, C1boundary connectivity is imposed.
III. OPT IM AL DE SI GN O F SKE LE TON LAYOUT
In this section, we will formulate the skeleton layout design
as a topology optimization problem and develop a computa-
tional framework to automate the iterative optimization.
A. State Equation
When the actuator undergoes deformation, the induced
displacement field udefines the deformation gradient by
F=I+∇uwith the gradient operator with respect to
the reference domain and Ithe second-order identity tensor.
We employ a hyperelastic model to characterize the material
nonlinearity of silicone rubbers and linear elastic model to
characterize the much stiffer material for the skeleton layer.
Without loss of generality, the generalized neo-Hookean model
[21] is adopted with the free energy density Wexpressed by,
W=µ
2trace(F·FT)
J2/3−3+κ
2(J−1)2(4)
where µand κdenote the initial shear modulus and bulk
modulus, respectively, and J= det(F).
In the context of geometric nonlinearity, we employ the
Green-Lagrange strain and the second Piola-Kirchhoff stress
measures which are work-conjugate. The state equation can
be derived based on principle of energy conservation,
a(u,v, φ) = l(u,v),∀v∈U(5)
with the variational structural form and load form
a(u,v, φ) = ZΩs
0
s(u) : ¯
E(u,v)dΩ + ZΩr
0
s(u) : ¯
E(u,v)dΩ
(6)
l(u,v) = pZΓr
0
v·mr·F−1JdΓ (7)
where sis the stress, Ωr
0∈R3is the undeformed region of
soft rubber, Ωs
0∈R3is the undeformed skeleton layer, v
is the virtual displacement field, Udenotes the kinematically
admissible space, mris the normal vector to the surface, Γr
0is
the inner surface, and pis the applied pressure. The variational
Lagrange strain tensor is
¯
E(u,v) = 1
2FT· ∇v+∇vT·F.(8)
Fig. 2: A skeleton-reinforced soft actuator for bending motion.
B. Optimization Model
The skeleton layout is optimized in order to produce as
closely as possible the desired displacements. The topology
optimization model is formulated by
min
ΩJobj =1
2X
i∈A
ui−ud
i
2
subject to a(u,v)−l(u,v)=0,∀v∈U,
sc−f|D|= 0
(9)
where Jobj is the design objective, udis the desired displace-
ment field, Adenotes the set of points of interest, sc=RΩs
0dΩ
defines the skeleton volume, and the volume constraint plays
regularization roles when performing optimization, with fand
|D|the given volume fraction for the skeleton and the total
volume of the design domain.
C. Sensitivity Analysis
To investigate how the desired motion quantitatively de-
pends on the skeleton layout, we carry out sensitivity analysis
with the adjoint method. The Lagrangian is formulated by
L=Jobj +a(u,w, φ)−l(u,w) + λ(sc−f|D|)(10)
where w∈Udenotes the adjoint displacement field, and λ
is the Lagrange multiplier for penalizing the constraint. The
derivative of Lagrangian with respect to pseudo time is
˙
Jobj =X
i∈Aui−ud
i˙
ui(11)
˙a(u,w, φ) = ZΩs
0˙
s(u) : ¯
E(u,w) + s(u) : ˙
¯
E(u,w)dΩ
+ZΓs
0
s(u) : ¯
E(u,w)VndΩ
(12)
˙
l(u,w) = pZΓr
0w·(mr·˙
F−1)J+w·(mr·F−1)˙
JdΓ (13)
˙sc=ZΓs
0
VndΓ (14)
where Γs
0is the boundary of the skeleton. Without loss of
generality, ˙
wis set to be zero. Since ˙
uis unknown, all terms
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Transactions on Mechatronics
IEEE/ASME TRANSACTIONS ON MECHATRONICS 4
Fig. 3: Flowchart of designing the skeleton layout by topology optimization.
containing ˙
uare collected and their sum is set to be zero
by selecting a unique w, which yields the so-called adjoint
equation. After mathematical derivations, the adjoint equation
numerically translates into a linear finite element equation
K(u)w=P(15)
with Kthe tangent stiffness matrix of the actuator which is
function of the displacement field, and Pis an assembled
vector consisting of the magnitude of ud−uat degrees
of freedom of interest and zero otherwise. In this work, a
bending actuator is designed and the maximal bending motion
is pursued at the endpoint, i.e. |ud|=∞, in which case P
is equivalently a unit concentrated load by noting that (15) is
a linear equation. The adjoint displacement wis solved such
that the terms containing ˙
uare eliminated. Then the derivative
of the Lagrangian ˙
Lis rewritten by
˙
L=ZΓs
0s(u) : ¯
E(u,w) + λVndΓ (16)
To this end, for ensuring the steepest descent direction, the
normal velocities on the skeleton boundaries can be selected,
Vn(X) = −s(u) : ¯
E(u,w) + λ,∀X∈Γs
0.(17)
It can be observed from (15)-(17) that, the sensitivity and
design velocity are function of the induced displacement field
which is load-dependent, necessitating the consideration of the
nonlinear physical deformation process.
D. Design Workflow
In the design workflow, the skeleton layout is iteratively
renewed by optimization procedure, as shown in Fig. 3. The
optimizer starts from an initial skeleton profile, followed by
nonlinear finite element analysis to evaluate the deformation
behavior of the actuator. The adjoint displacement is solved
by (15). Thereafter, the sensitivity analysis is conducted to
generate the design velocity field based on which the skeleton
layout is updated by solving the Hamilton-Jacobian equation.
The movement distance satisfies the Courant-Friedrichs-Lewy
condition for numerical stability. The above process is repeated
until the design candidate fulfills the predefined convergence
criterion.
The volume constraint is set to be 0.5 during the optimiza-
tion and is imposed by updating the Lagrange multiplier. We
adopt the following updating strategy,
λk+1 =λk+ξk(ςk−f|D|), ξk+1 =γξk, k = 1,2, .., Ns(18)
with ξkthe penalization factor at the kth step, γ≥1an
expansion parameter and Ns= 200 the prescribed iteration
limit. The optimization algorithm is terminated when the
relative difference of the objective function values between
two successive iterations becomes lower than 10−3.
IV. NUM ER IC AL IM PL EM EN TATIO N
In this section, we will introduce how the large deformation
analysis is addressed for the design candidates which may
take amorphous shapes during the optimization process, and
investigate how the geometric and material nonlinearities tailor
the optimization results.
A. Large Deformation Analysis
For the nonlinear finite element analysis compatible with
varying skeleton layouts, we customized an automatic numer-
ical framework by connecting Matlab and Abaqus. We adopt
shell elements (S3) to discretize the thin-wall skeleton layer
and C3D4H elements (4-node linear tetrahedron and hybrid)
for soft rubber. The numerical results show that shell elements
have good interaction properties with the inner rubber part,
leading to better convergence.
When calling Abaqus to perform analysis, a key issue is
to coordinate varying level set descriptions and the finite
element mesh in Abaqus. People usually keep the geometry
unchanged and vary the material properties by interpolation,
giving rise to artificial intermediate materials, as in [15],
[17]. These artificial materials increase the node scale and the
computation cost, and more importantly, the weak materials
which represent voids are prone to suffering mesh distortions
and then cause severe convergence issues. In this work, instead
of relying on use of weak materials, we deactivate the shell
elements corresponding to voids by evaluating the nodal level
set function to obtain distinct skeleton layout, leading to
greatly improved computational convergence and efficiency.
Specifically, an element is deactivated when the mean of its
nodal level-set function values is negative, expressed by
X
Ni∈Ωe
φ(Ni)<0,for element e(19)
when Ωedenotes the domain of element e. The generated
shell mesh well reproduces the skeleton layout defined by zero
contour of the level set function, and the boundary quality can
be tuned by the resolution of the shell mesh.
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Fig. 4: Optimization results: history of (a) design objective and (b) volume fraction, and (c) three design candidates.
The geometry dimensions for analysis and experiments are
specified here. The bending actuator is of length L=82.5
mm, outer perimeter 2πR =50 mm. The skeleton layer is
of uniform thickness 1 mm. The inner soft rubber tube is of
uniform thickness 3.18 mm. The material properties for the
inner soft rubber are: shear modulus µ=184 kPa and bulk
modulus κ=18.35 MPa. Its Young’s modulus at small strains
is 0.55 MPa. The material properties for the skeleton are:
Young’s modulus 19.46 MPa, and Poisson’s ratio 0.495. The
reader may refer to Sec. V. A for details of material selection.
B. Gradually Loading Scheme
Considering the intermediate design candidates during op-
timization may easily suffer convergence issues, the applied
pressure is gradually increased with the optimization iterations.
This operation is similar to [22] in which the linear opti-
mization result is taken as the starting point for the nonlinear
optimization. At the first stage, the applied pressure is 2.5
kPa, the induced strain is small, and thus the actuator exhibits
a linear response. The optimization conducted at this low
pressure is referred to as the linear region, as shown in Fig.
4. When the optimization converges in the linear region, we
increase the pressure to a high level, i.e. 60 kPa, and the
optimization continues and leads to significant improvement
up to 11.0% in terms of the concerned bending displacement.
At high pressures, the actuator undergoes large deformation
and exhibits considerable geometric and material nonlinearities
which are well incorporated into the optimizer to further refine
the skeleton layout during nonlinear regions.
C. Optimization Results
The evolving process of the skeleton layout is shown in
Fig. 4. The optimization starts with an initial design which
contains many small circles to allow for sufficient geometric
flexibility for potential topological evolutions. The holes in the
initial design merge into complex topological shapes that are
hard to perceive by intuition. Three key states are shown in
Fig. 4, including their bending configurations and the induced
stress. Several key structural features are observed from the
final optimized design, which are also well explainable. First,
the circumferential bars well constrain the radical inflation.
Second, as expected, more materials are distributed on the
compressive part in order to induce bending, functioning as the
limiting layer as in many pneumatic actuators [16]. Third, very
interestingly, it is observed that the skeleton connection on the
compressive side results in parallelogram shape which may
function as a mechanism to transfer circumferential expansions
into compression and further enhance bending motions, as
suggested by [23]. This finding sheds light into the innovative
design of the limiting layer.
The optimization is gradient-based and thus naturally de-
pends on the initial guess. Therefore, we investigate how the
initial design affects the final generated skeleton layout. Three
other different initial profiles are tested as shown in Fig. 5.
It is found that all initial designs result in the circumferential
bars, but the skeleton layout on the compressive side depends
on whether holes are initially placed on the boundary. Besides,
the optimization results present the size effect which means
the final delivered geometric complexities depend on the initial
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Transactions on Mechatronics
IEEE/ASME TRANSACTIONS ON MECHATRONICS 6
Fig. 5: Three different initial designs (the first row) and their
corresponding optimization results (the second row).
hole density, which may inherently be related to the non-
convexity of the displacement-oriented optimization problem.
Based on these phenomena, we finally select the initial design
as shown by state I in Fig. 4 which outperforms the three
candidates in Fig. 5.
We further investigate how the optimized skeleton depends
on the volume constraint. The volume fraction is increased
from 0.3 to 0.7 with an interval of 0.1, and the associated
optimized designs and their performance are compared as
shown in Fig. 6. As the volume fraction increases from 0.3 to
0.5, the design produces enhanced bending benefiting from
stronger constraining effect imposed by the hard material
distributed on the compressive side. Thereafter, an evident
decrease in bending motion is observed with the increasing
volume fraction. On the one hand, the accumulation of mate-
rial around the stripes suppresses the tensile deformation. On
the other hand, more materials distributed on the compressive
side increase the whole stiffness of the actuator. Thus, the
volume fraction is set 0.5 for the final design.
V. EXPE RI ME NT S AN D APP LI CATIONS
A. Material Selection and Actuator Fabrication
It can be inferred from dimensionless analysis that the
performance of the actuator partially depends on the material
modulus ratio of the skeleton to the soft rubber. A skeleton
layer too soft cannot exert a constraining effect, while a
skeleton layer too rigid would increase the overall actuator
stiffness. Thus, there exists an optimal material ratio which
can be roughly estimated by finite element analysis based
on a selected skeleton layout, falling between 25 and 40.
In consideration of the available materials off-the-shelf, we
choose Dragon Skin 20 (Smooth-On, USA) as the soft rubber
material whose viscosity, ductility and strength are satisfactory
and HeiCast 8400 (Shore 90A, WeNext Technology, China)
as the skeleton material. The ratio of their Young’s modulus
is around 35. The cylindrical shape soft layer is fabricated by
injection molding. The material mixing and injection processes
Fig. 6: The optimization solution depends on the volume
fraction. The optimized designs and deformed shapes at 60
kPa are plotted as insets.
are performed simultaneously in a vacuum environment. The
skeleton layer is fabricated by compound mold techniques.
The fabricated parts as shown in Fig. 7(a) are assembled to
build a soft bending actuator prototype.
B. Experiment Setup
The characterization of the actuator involves the control of
pressure and the measurements of displacement and force. The
whole experiment setup is shown in Fig. 7(b). The soft actuator
is fixed to a rigid basement and is connected to an analog
pneumatic close-loop controller (SMC, Japan). The motion of
the actuator is tracked by a motion capture system (OptiTrack,
USA). We use two phosphor markers [see Fig. 7(b)] adhering
to the fingertip to capture the displacements. An S-beam load
cell (ARIZON China) with customized fixture matching the
shape of fingertip is deployed to record the contact force
generated by the actuator upon pressurization.
C. Bending Actuator Characterization
Fig. 7(c) shows the free travel trajectory. The actuator
prototype is pressurized at a speed of 4.8 kPa/s to simu-
late a quasi-static process. We repeat five loading circles in
the pressure range of 0-120 kPa. The free motion exhibits
excellent repeatability, as indicated by the narrow error bar
in Fig. 7(c). The bending angle increases with the applied
pressure, and reaches 73° under 120 kPa. A nonlinear finite
element simulation is conducted which matches well with
the trajectories in the range of 0-90 kPa with the maximal
deviation less than 4.0%.
We also characterize the load capabilities of the actuator
prototype by evaluating the blocking force. As shown in Fig.
7(d), the blocking force increases with the applied pressure,
peaking at 2.05 N under 120 kPa. It is noted that, although
the power requirement is not directly included in the design
objective of the optimization model (9), the skeleton still
attains considerable stiffness which leads to good load capa-
bilities. The inner soft rubber as the non-design domain owns
considerable stiffness to withstand the gravity, and thus the
skeleton layer must be stiffer to dominate the bending motion,
which leads to the remarkable load capability of the actuator.
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Fig. 7: Experiment results: (a) components; (b) the bending actuator prototype; (c) free motion and (d) blocking force tests.
VI. EX TE NS IO N AN D DIS CU SSION
A. Extension
To demonstrate the generality of the design method, we
further extend the cases studies to more types of motions
and more complex design domain. Bending and twisting
motions of a cone-shaped actuator are investigated by firstly
constructing geometry and velocity mappings from the regular
2D design domain to the 3D cone surface according to (3),
and by secondly modifying the design objective to capture
the desired motion. It is worth noting that, different from
cylindrical surface, the geometry mapping is not isometric and
the design velocity mapping is not uniform. The upper base
radius of the cone is 5.57 mm, and the bottom radius is 7.95
mm. The other parameters and the boundary conditions keep
unchanged with the case in Sec. IV.
Fig. 8 shows the optimized design of the cone-shaped actu-
ator toward bending and twisting motions, and the simulation
and experimental results. Specifically, for bending motion, the
design pattern is similar to the case in Sec. IV except that
the geometric features vary in size. For twisting motion, the
design objective is to maximize the tangential displacements of
points on the edge, and the optimized solution generates chiral
patterns which are also physically reasonable. The experiment
shows that the actuator twists by 143° under 60 kPa. The
smoothness of the skeleton pattern is again guaranteed by the
domain continuity operation. Here, it is emphasized that our
design method provides a rational and quantitative guidance
for design of soft bodied actuators and robots.
Fig. 8: Skeleton optimization for cone-shaped actuator’s (a)
bending or (b) twisting motion. From left to right: the skeleton
layout, simulation result and experimental result.
B. Discussion
The design space can be further explored. For example, the
thickness dimension has been designed to tune the spatially-
varying bending abilities toward desired deformations [24].
The skeleton layout plays an important role in modulating
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the global motion behavior, while its non-uniform thickness
is expected to exert major impact on local deformations. In
this work, we only carry out surface design because we focus
more on the global motion and the thickness variations may
cause convergence issues in finite element analysis for large
deformation. Nevertheless, we believe that, the incorporation
of thickness will further explore the design space to deliver
better design, and may become necessary to more complicated
design problems such as shape-matching in which many
local features are pursued. It is also noted that, although
only bending and twisting deformations are investigated, the
proposed method is readily applicable to more complicated
desired motions by directly modifying the design objective.
When dealing with general free-form surfaces, the key is
still to establish the geometric mapping and inverse design
velocity mapping, except that the mappings may not be
constructed explicitly but numerically. The conformal mapping
theory can be combined with topology optimization theories to
recast the manifold embedded in the 3D space as a 2D topol-
ogy optimization problem in the Euclidean space. Ye et al. pro-
vided 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 [25]. This operation allows for
the convenient use of conventional computational schemes for
level set methods by solving the modified Hamilton-Jacobi
equation.
VII. CON CL US IO N
We build an actuator design paradigm by attaching a stiffer
material layer as skeleton to softer inflated rubber, and develop
a topology optimization based design framework to automat-
ically generate the skeleton layout that leads the actuator to
achieve desired motions such as bending and twisting. To cap-
ture large deformation that typically occurs in soft robots, our
optimizer incorporates the geometric and material nonlineari-
ties, and the generated design contains reliable and explainable
structural features that provide new insights for designers. The
simulation and experiment results agree well in both linear
and nonlinear regions, and validate that our designed actuator
achieves remarkable bending motions with considerable force
capabilities. We further investigate the design of cone-shaped
actuators toward bending and twisting motions to demonstrate
the generality of the proposed method. In the future, we hope
to further generalize the design methodology to free-form
surfaces for creating soft-bodied robots of customizable shapes
capable of undergoing more complex motion behaviors.
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Transactions on Mechatronics
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Shitong Chen received the B.E. degree in aircraft
design engineering from Xi’an Jiaotong University,
Xi’an, China, in 2018. He is currently working
toward the Ph.D. degree in mechanical engineering
at Shanghai Jiao Tong University, Shanghai, China.
His research interests include computational de-
sign of soft robots.
Feifei Chen (Member, IEEE) received the B.E.
degree in mechanical engineering from University of
Science and Technology of China in 2013, and Ph.D.
degree in mechanical engineering from National
University of Singapore in 2018. He joined Shanghai
Jiao Tong University in 2018 and currently is an
Associate Professor (tenure-track) with the School
of Mechanical Engineering.
His research interests have been focused on design
thoery and methods for soft robots.
Zizheng Cao is currently working toward the B.E.
degree in mechanical engineering at Shanghai Jiao
Tong University, Shanghai, China.
His research interests include design, manufac-
ture, and simulation of soft robots.
Yusheng Wang is currently working toward the B.E.
degree in mechanical engineering at Shanghai Jiao
Tong University, Shanghai, China.
His research interests include computational me-
chanics, robotics and smart materials.
Yunpeng Miao received the B.E. degree in me-
chanical engineering from East China University of
Science and Technology, Shanghai, China, in 2019.
He is currently working toward the M.S. degree
in mechanical engineering at Shanghai Jiao Tong
University, Shanghai, China.
His research interests include geometric modeling
and optimization of soft robots.
Guoying Gu (Member, IEEE) received the B.E.
degree (with honors) in electronic science and tech-
nology, and the Ph.D. degree (with honors) in
mechatronic engineering from Shanghai Jiao Tong
University, Shanghai, China, in 2006 and 2012,
respectively. Since October 2012, Dr. Gu has worked
at Shanghai Jiao Tong University, where he is
currently appointed as a Professor of School of
Mechanical Engineering. He was a Humboldt Fel-
low with University of Oldenburg, Germany. He
was a Visiting Scholar at Massachusetts Institute
of Technology, National University of Singapore and Concordia University.
His research interests include soft robotics, bioinspired and wearable robots,
smart materials sensing, actuation and motion control. He is the author or
co-author of over 90 publications, which have appeared in Science Robotics,
Science Advances, IEEE/ASME Trans., Advanced Functional Materials, Soft
Robotics, etc., as book chapters and in conference proceedings.
Dr. Gu received the National Science Fund for Distinguished Young
Scholars in 2020. Now he serves as Associate Editor of IEEE Transactions on
Robotics and IEEE Robotics and Automation Letters. He has also served for
several journals as Editorial Board Member, Topic Editor, or Guest Editor, and
several international conferences/symposiums as Chair, Co-Chair, Associate
Editor or Program Committee Member.
Xiangyang Zhu (Member, IEEE) received the B.S.
degree in automatic control from Nanjing Institute
of Technology, Nanjing, China, in 1985, the M.Phil.
degree in instrumentation engineering and the Ph.D.
degree in control engineering, both from Southeast
University, Nanjing, China, in 1989 and 1992, re-
spectively. From 1993 to 1994, he was a postdoctoral
research fellow with Huazhong University of Sci-
ence and Technology, Wuhan, China. He joined the
Department of Mechanical Engineering, Southeast
University, as an associate professor in 1995. Since
June 2002, he has been with the School of Mechanical Engineering, Shanghai
Jiao Tong University, Shanghai, China, where he is currently a chair professor
and the director of the Robotics Institute. His research interests include
robotic manipulation planning, neuro-interfacing and neuro-prosthetics, and
soft robotics. He has published more than 200 papers in international journals
and conference proceedings.
Dr. Zhu has received a number of awards including the National Science
Fund for Distinguished Young Scholars from NSFC in 2005, and the Cheung
Kong Distinguished Professorship from the Ministry of Education in 2007.
He currently serves on the editorial board of the IEEE Transactions on
Cybernetics and Bio-design and Manufacturing.
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