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Automated design of pneumatic soft grippers through design-dependent
multi-material topology optimization
Josh Pinskier1, Prabhat Kumar2, David Howard1, and Matthijs Langelaar3
Abstract— In recent years, soft robotic grasping has rapidly
spread through the academic robotics community and pushed
into industrial applications. At the same time, multimaterial
3D printing has become widely available, enabling monolithic
manufacture of devices containing rigid and elastic section. We
propose a novel design technique which leverages both of these
technologies and is able to automatically design bespoke soft
robotic grippers for fruit-picking and similar applications. We
demonstrate the novel topology optimisation formulation which
generates multi-material soft gippers and is able to solve both
the internal and external pressure boundaries, and investigate
methods to produce air-tight designs. Compared to existing
methods, it vastly expands the searchable design space whilst
increasing simulation accuracy.
I. INTRODUCTION
Soft robotic grasping has emerged as an safe and effective
means for grasping fragile, flexible and fluctuating objects.
Their inherent defomrability enables them to conform to fit
the objects’ shape and distribute gripping force, hence gently
grasping even soft objects.
These soft grippers are often inspired by human hands,
which are seen as the gold standard in soft and dexterous
grasping. However, there is an increasing trend towards
non-anthropomoprhic designs, which enable diverse grasping
strategies and require controllable fewer degrees of freedom
(DOFs) [1]. Several mechanisms have been investigated for
their actuation including pneumatic [2], tendon-driven [3],
and granular (vacuum) jamming [4], [5].
Despite the diversity of grasping and actuation paraigms
available in the literature, most existing grippers are designed
by hand. They draw on human experience and biomimicry to
navigate the complexity of designing deformable devices to
generate high-quality designs [6]. The resultuing designs are
normally generic, emphasizing universal approaches rather
than bespoke designs [4]. However, real world applications
frequently require designs which are tailored to the specifics
of their task. Clearly a fruit picking robot requires a different
end effector to an assembly line robot or a human assis-
tance robot, and an apple-picking end-effector has different
requirements to a strawberry picker. Despite this obvious
1Robotics and Autonomous Systems Group, CSIRO Data61,
Brisbnae, Australia josh.pinskier@csiro.au,
david.howard@csiro.au
2Department of Mechanical and Aerospace Engineering, Indian
Institute of Technology Hyderabad, 502285 Telangana, India
pkumar@mae.iith.ac.in
3Department of Precision and Microsystems Engineering, TU Delft,
Delft, Netherlands m.langelaar@tudelft.nl
The authors thank Prof. Krister Svanberg for providing MATLAB codes
of the MMA optimizer.
Fig. 1. 3 Material optimised soft-gripper under 50kPa pressure (5x
deformation scale). Pressure is applied to the two faces on the left, causing
the jaws to close on the right. Material stiffnesses are: Red - 100MPa, Green
- 10MPa, Blue - 1MPa
need to produce bespoke soft end effectors, exiting auto-
mated design tools are limited and underexpolred. Methods
including simuluated and in-materio evolution have recently
proven successful in designing granular jamming grippers
[7], [8].
In contrast, Topology optimization (TO) is a general
purpose design tool, suitable to numerous actuation tech-
niques and physical domains [9]. It distributes material
inside a meshed (or pixelized/voxelized) space to identify
the topology with the best performance, and has designed
both pneumatic and tendon-driven soft grippers. [10], [6],
[11]. However, the methods presented in these works require
significant assumptions about the design space and actuation,
limiting both the accuracy of the simulation and the range
of realisable designs.
A. Topology Optimisation of Soft Grippers
The current state-of-the-art in topology optimised soft
grippers broadly falls into two categories:
1) Externally actuated grippers [12], [10]. These use an
exogenous displacement to drive their grasping be-
haviour, that is an externally routed cable or moving
surface.
2) Pressure actuated soft fingers without design depen-
dency [13], [14], [15]. These specify a pressure distri-
bution on a fixed surface, which must be prespecified
does not form part of the optimisation problem formu-
lation.
In both of these cases, the actuation source is presprecified
and does not form part of the optimisation problem. Whilst
convenient, these assumptions do not reflect best-practice
design methods which use complex pneumatic chambers
and internal cable routing. To capture these features, the
loading point (magnitude, direction and location) should be
free to move with each iteration of the topology optimisation
solver. This design-dependency problem increases the solver
complexity and requires auxiliary physics equations to solve
and additional constraints to enphorse physical limits. A
small number of topology optimised soft grippers have
investigated design-dependent pressure optimisaion, but their
coarse physics approximations result in unrealisable designs,
with disconnected pressurised regions [16], [17].
Whilst the above methods have been demonstrated only
single-material optimisation, improvements in 3D printing
technology, enable the monolithic manufacture of arbitrarily
complex multi-material soft robots. By using two or more
materials, it is possible to strike a trade-off between the
flexibility and strength of the material, and increase the
overall strength of the device without comprimising on its
workspace. For a detailed review of soft robotic topology
optimisation see [6], [18].
To the best of the authors’ knowledge, there is currently no
method for creating multi-material pneumatically activated
soft robots using topology optimization.
B. Pressure-Loaded Topology Optimsation
Pressure-loaded topology optimsation is a problem which
extends beyond soft robotics. It has applications in the design
of pneumatically and hydraulically loaded structures like
pressure vessels, dams, pumps and ships. In these problems,
the fluid-solid boundary and hence the loading must move
during the optimisation. In density-based topology optimi-
sation, mesh elements are allowed to occupy a continuum
between solid and void [9]. Hence, the problem is commonly
approached either by attempting to explicitly identify a fluid-
solid boundary, or using a mixed fluid-solid fomulation [19],
[20]. Whilst the mixed-method overcomes the challenges of
identifying a unique boundary, it introduces an additional
material phase (solid/fluid/void rather than solid/void) whose
volume must be specified prior to optimisation. The current
state of the art method treats the continous density material
as a poroous media, and uses the Darcy method to estimate
fluid penetration as a function of density [20], [11].
Whilst the Darcy method is state of the art for pneu-
matically actuated mechanisms, there is no guarantee that
the design will be airtight. In both the Darcy method
and other pnumatic compliant mechanism formulations, the
optimisation frequently results in undesirable holes in the
structure. As there is no constraint placed on flow-rate,
these reduce stiffness and hence increase calculated output
displacement. Whilst this can be addressed using a material
filtering scheme, which forces a solid layer between the high
and low pressure regions [21], such a scheme is heavily
dependent on the optimisers initial conditions, and prevents
the formation of beneficial internal cavities.
C. Contributions
In this work, we present a novel method to design 3D
multi-material pressure-actuated soft grippers using topology
optimization. The method builds on our previous work into
pressure-loaded topology optimization using Darcy’s law
[20], [11] and the extended solid-isotropic material with
penalisation (SIMP) material model for the multi-material
modeling [22]. An example of a soft gripper designed using
this method is shown in Figure 1, it uses three materials with
stiffnesses of 1M P a,10 MP a and 100 MP a. Using the
multimaterial Darcy formulation, the solver converges to a
soft gripper which clamps together using several compliant
hinges.
The main contributions of this work are:
1) The first presentation of a multi-material topology
optimsation formulation for pneumatic soft robots
2) The development and investigation two new formula-
tions to generate sealed pneumatic actuators, based on
pressure regions and an energy penalty, respectively.
3) The design of several new multimaterial pressure-
actuated soft grippers.
Whilst we focuss on the application of this methology to
soft robotic grasping, it is generalisable to other pneumatic
compliant mechanism and soft robots.
II. TOPOLOGY OPTIMISATION FORMULATION
In this work, we use the density based SIMP method for
topology optimisation. The goal of topology optimisation is
to find a discrete material layout where each region contains
a unique material or is left void. To simplify the problem,
SIMP allows the design variable ρto occupy a continuum
from 0 to 1, and a penalty papplied to drive the results
towards a binary solution. For a single material problem this
is done using the SIMP interpolation law:
Ei= (1 −¯ρp)Emin + ¯ρpE1)(1)
where Emin is a small, non-zero constant used to prevent
singularities in material voids and E1is the elastic modulus
of the material used.
A. Multimaterial Modeling
To model multiple materials for the gripper mechanisms,
we apply the extended SIMP interpolation technique [9].
In this formulation, one design variable is assigned to each
material. For example, in the two-material case, the scheme
with the modified SIMP formulation can be written as:
Ei= (1 −¯ρp
i1)Emin + ¯ρp
i1((1 −¯ρp
i2)E1+ ¯ρp
i2E2),(2)
where E1and E2are moduli of material 1 and material 2,
respectively. ¯ρidenotes the physical variable corresponding
to design variable ρi, and the SIMP parameter pis set to
3. {¯ρi1= 1,¯ρi2= 1}gives the second material, whereas
{¯ρi1= 1,¯ρi2= 0}provides the first material. Thus, ¯ρi1is
called the topology variable, i.e., it decides the topology
of the evolving design, whereas ¯ρi2decides the candidate
material. Similarly the three-material case can be described
by:
Ei= (1 −¯ρp
i1)Emin + ¯ρp
i1[((1 −¯ρp
i2)E1+
¯ρp
i2((1 −¯ρp
i3)E2+ ¯ρp
i3E3))],(3)
where E3is the modulus of material 3. Using three materials,
{¯ρi1= 1,¯ρi2= 0,¯ρi3= 0},{¯ρi1= 1,¯ρi2= 1,¯ρi3= 0}
and {¯ρi1= 1,¯ρi2= 1,¯ρi3= 1}give respectively material 1,
material 2 and material 3.
To remove non-physical checkerboard patterns and inter-
mediate (i.e non-binary) densities from the final design, we
use density and hyperbolic projection filters as in [9], [23].
B. Pressure load modeling
This section describes the pressure load modeling scheme
using the Darcy law with the conceptualized drainage term.
The method developed here for pneumatic soft robotic
optimisations builds on our previous work into the Darcy
method, a detailed description of which can be found in [20],
[11]. It conceptualises the continiuous design variable ¯ρas a
porous medium, and uses Darcy’s law to calculate pressure
losses. In it, the flux q(volumetric fliud flow rate across a
unit area) is defined by the flow coefficient K(¯ρi1)and the
pressure difference ∇pas:
q=−κ
µ∇p=−K(¯ρi1)∇p(4)
As the topology of the multimaterial structure (whether there
is a material or void) is determined by ¯ρi1, the flux solely
depends on ¯ρi1, regardless of the number of materials. Hence,
the flow coefficient of element iis calculated as
K(¯ρi) = Kv1−(1 −Ks
Kv
)H(¯ρi1, βκ, ηκ),(5)
where
H(¯ρi1, βκ, ηκ) = tanh (βκηκ) + tanh (βκ(¯ρi1−ηκ))
tanh (βκηκ) + tanh (βκ(1 −ηκ)) ,
(6)
Ksand Kvare flow coefficients of solid and void phases.
respectively, and ηκand βκshape the distribution of K( ¯ρi).
Finally, a drainage term, Qdrain, is added. It helps achieve
the natural pressure field variation by draining pressure from
internal cavities:
Qdrain =−DsH( ¯ρi1, βd, ηd)(¯ρe)(p−pstm)(7)
where Ds is drainage coefficient and patm is the atmospheric
pressure. The net flow of the system is given by the equilib-
rium equation:
∇ · q−Qdrain = 0.(8)
. Which is solved using the finite element method to find the
equilibrium pressure distribution and transform the pressure
distribution p, to a global force Fto solve the mechanical
equilibruim equation:
Ku =F=−Tp (9)
where uand Kare the global displacement vector and
stiffness matrix, and Ttranforms elemental pressures to
nodal forces. By using two physical equation to solve for
the equilibrium pressure and displacement, the formulation
determines the pressure boundary at each iteration.
C. Problem formulation
The final optimisation problem is formulated using:
min
ρ
−suout
(SE)1/n
such that: Ap =0
Ku =F=−Tp
nel
X
i=1
vi¯ρi1≤(vf1+vf2+vf3)
nel
X
i=1
vi
nel
X
i=1
vi¯ρi2≤vf2
nel
X
i=1
vi
nel
X
i=1
vi¯ρi3≤vf3
nel
X
i=1
vi
0≤¯ρ≤1
Data: vf1, vf2, vf3, E1, E2, E3, p
,(10)
where uout and SE indicate output displacement and strain
energy, respectively. sis the consistent scaling parameter. A
is the global flow matrix, which is found by assembling (8).
We use three linear volume constraints using the definitions
¯ρi1,¯ρi2and ¯ρi3described above. The first constraint controls
the total amount of the solid state, whereas the second and
third give the material amount of phase 2 and phase 3.
vf1k, vf2and vf3denote the volume fraction for material 1,
material 2 and material 3, respectively.
The cost function is selected to balance the dual require-
ments of maximising the deformation of the gripper, and
maintaining a design which is stiff enough to grasp and hold
objects. Here n= 8 is used to place a soft penalty on the
design’s stiffness. In this work vf1= 0.3,vf2= 0.2, and
vf3= 0.2unless otherwise stated, and the input pressure is
50 kPa and the materials are given stiffnesses E1= 1 MPa,
E2= 10 MPa, and E3= 100 MPa
III. SOF T GRIPPERS DESIGN
To demonstrate the method and motivate the need for
airtightness, this section investigates the design of pressure
actuated grippers using the multimaterial Darcy formulation.
The design space of the grippers is presented in Figure
2(a). Pressure is applied from the left face, with the output
direction shown on the right. To simplify the design space
and reduce computation time, 2 planes of symmetry are used,
reducing the size of the workspace.
The resulting design is illustrated in , showing the un-
deformed and defomed configuration. In it, a solid face is
formed on the left side, which absorbs the pressure. The
internal strains are then transferred to the output face via a
series of compliant hinges, one in the centre of the gripper,
and four on the outer edges. To give the design stiffness, thin
sections of the stiffest material E3are used in each of the
hinges, and joined by the softer materials E1and E2. Whilst
quite elegant, the design illustrates two issues with existing
pressure optimisation methods. The first is that the optimiser
frequently falls into a local minima in which the pressurised
fluid is not allowed to penetrate deeply into the structure,
preventing the formation of more complex, higher perform-
ing designs. The second is that without careful consideration
of the design space, the optimiser generates holes in the final
design which spuriously increases performance by reducing
stiffness in undesired locations. In this case, resealing the
device is fairly trivial, but in more complex designs, doing
so adversely affects performance. Hence, design methods are
needed which drive closed designs.
(a) (b)
(c) (d)
Fig. 2. 3 Material optimised soft-gripper with stiffnesses: Red - 100MPa,
Green - 10MPa, Blue - 1MPa,(a) Design domain (b) Undeformed Side-view
(c) Undeformed top-view (d) Deformed (5x deformation scale)
IV. AIRTIGHT DESIGN
To generate closed designs, we investigate and compare
three methods, and apply them to soft finger design. In
soft fingers, the pressure laod is often applied via a central
channel in the design space. Whilst this forces pressure
deeper into the design and enhances performance, it also
increases its succeptability to hole generation. Viewed from
the perspective of the optimsation problem, sealed chambers
increase the stiffness of the device and restrict output motion.
We propose two new methods for generating sealed designs:
1) A heuristic approach, which adds material to the final
design along the median pressure contour.
2) A penalty approach, which adds an energy term to
the cost function and drives the optimisation to reduce
pressure loss.
The first approach leverages the advantages of the Darcy
method, which calculates the internal pressure distribution
between the inlet and outlet points. Where a face is unsealed,
there will be a smooth gradient of pressure flowing from
the inlet to outlet. However, closed regions have a sharper
pressure boundary. Hence by adding material along the line
0.5∗(pin −pout)we close open regions without significantly
impacting regions which already have material.
The second approach is more rigorous, but remains suc-
ceptible to local minima. Using the equilibrium flow from
the Darcy equation, we are able to calculate the energy
transferred from inlet to outlet. In a closed system there
would be no flow, hence no energy transferred. However,
using the Darcy method, a small flow will always arise. We
use this energy value as a penalty term in the cost function,
such that we seek to minimise:
min
ρ
−suout
Et(SE)1/n (11)
where Etis the total energy loss.
V. AIRTIGHT SOF T FINGERS
The design space of the soft fingers is presented in 3(a). It
is fixed around the edges on the left side and pressure enters
via a central cavity, a single symmetry face is used to reduce
the problem size. The aim is to maximise the bending on the
right side.
A. Heuristic Skin
An example of the design of the bending soft finger
is shown in Figure 3. Without any closure method, the
material is distributed roughly from stiffnest to softest, with
the stiffest material placed around the fixed side. Bending
is increased by placing holes at the top and sides of the
structure. However, a closed structure is easily regenerated
using the heuristic methods
(a) (b)
(c) (d)
(e) (f)
Fig. 3. 3 Material optimised soft-finger with stiffnesses: Red - 100MPa,
Green - 10MPa, Blue - 1MPa,(a) Design domain (b) Undeformed (c)
Deformed (5x deformation scale) (d) Optimised Pressure Distribution (Un-
deformed) (e) Implied pressure boundary (f) Complete design with sealed
chamber
B. With Skin
The surface can also be inserted as part of the optimisation
problem by creating a non-design domain on the boundary of
the optimisation region and assigning it to have stiffness E1.
This guarantees air cannot leak, but will produce suboptimal
solutions as the external boundary must bend and expand to
generate deformation. In contrast, pnuenets, a state of the art
designs have sinusodal profiles which localise bending. This
result is illustrated in Figure 4.
(a) (b)
(c) (d)
Fig. 4. 3 Material optimised soft-finger with casing - stiffnesses: Red -
100MPa, Green - 10MPa, Blue - 1MPa, (a) Design domain (b) Undeformed
Side-view (casing not shown) (c) Undeformed top-view (casing not shown)
(d) Deformed half model, showing casing (5x deformation scale)
C. Energy Penalty
Finally, the same design is presented using the energy
penalty method. Here, the optimiser has reduced the overall
amount of air leakage by using the low stiffness material E1
to close sections of the chamber which contribute least to
bending. The result is not a totally closed design, but one
where the open areas have been greatly reduced. This uses
the same design space as the heuristic skin The efficacy of
(a) (b)
Fig. 5. Energy Penalised soft-gripper(a) Undeformed (b) Deformed (5x
deformation scale)
this of this penalty can be increased by allowing more or
lower stiffness materials to be included as is illustrated in
6. When using Vf1= 0.2, there is insufficient material to
meaningfully close the design, but at Vf1= 0.4an almost
sealed chamber emerges with only a small opening around
the fixed side. Whilst the remaining opening is not ideal, it
is relatively simple to close in postprocessing
(a) (b)
Fig. 6. Energy Penalised soft-gripper with (a) Vf1= 0.2(b) Vf1= 0.4
D. Numerical Comparison
We compare the two proposed closure methods by cal-
culating their output displacement, strain energy, mechan-
ical work done, and energy loss across 9 different out-
put stiffnesses (springs placed at the output face) from
0.1N/mto1000N/m. The results are presented in Figure 7.
Unsurprisingly, the unconstrained (no skin) optimisation has
by far the highest bending, strain energy, work done and
energy loss. Ignoring the energy loss, the design performs ex-
tremenly well. In contrast, the closed design space performs
extremely poorly, and is a poor choice of closure method.
Of the two methods discussed in this work, the heuristic
gives the best performance, with a relatively large output
displacement and low strain energy and energy loss. Whilst
the energy penalty shows promise, it is impeded by the
gradient based solve, and minimum length scales of topology
optimisation, which precent the formation of thin skins.
VI. DISCUSSION AND CONCLUSION
Guaranteeing closure remains a problem in this method,
and something which has not been solved in existing re-
search. We discussed two new methods for generated closed
or near closed soft robots. Of the two, the heuristic approach
outperforms the optimisation method. However the latter
approach is worthy of further investigation. In addition,
this work presented a multi-material method for pneumatic
topology optimisation and presented several new soft gripper
designs. By tweaking the design space and choosing different
output points and stiffnesses, the method is able to generate
bespoke grippers for specific applications such as fruit har-
vesting or human interaction. We aim to integrate it into soft
robotic pick and place systems and experimentally verify its
performance in the future.
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