IEEE Robotics & Automation Magazine - September 2016 - 58

Pressure
Source

Strut

With the substitution of (2)-(5) into (1) and integrating
over the hollow cross section, we get

Tip
p

A

t0
r0

l

A

P =0
∆P

C
S i
Section
Neutral Plane Cross
A-A
(Stiff Fabric)
(a)

Strut

B

t

B

Cross Sectio
Section
B-B

r

ρ = 1/κ
P >0
∆P
(b)
Effective Strut
(Hoop Strain
Constraint)

C

t
r

∆P
∆P > 0
C

Cross Sectio
Section
o
C-C

ρ = 1/κ

(c)
σ axial

t

y ∆P
P

MN
2rr
2

N

ρ = 1/κ

(d)
Figure 3. A schematic of one finger. (a) The rest state, with no
loads and no pressure; (b) the bending equilibrium state, where
the internal pressure is balanced by stretching the elastomer;
(c) the simplified model of the bending equilibrium state: struts
are replaced by a hoop strain constraint and other parameters
stay the same; and (d) a free-body diagram of a short section of
the finger actuator at the bending equilibrium state with both
pressure and an externally induced moment MN.

As per the standard pressure-vessel theory, in which the
fluid pressure is much smaller than the vessel (elastomer)
stresses, we neglect the radial stress, setting v radial = 0 . As
mentioned in regard to (2), we assume for simplicity that the
restraint against bulging can be represented as a circumferential stress per unit length, Ts . Again, for simplicity, we think of
these fibers as having modulus (force per unit length) E f , so
Ts =E f e hoop.
58

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IEEE ROBOTICS & AUTOMATION MAGAZINE

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September 2016

(5)

=-

E f - o 2 E f + Et
E f - o 2 E f -2Eot + Et
MN +
Tp .
3
2
3rr t (E E f + E t)
3t (E E f + E 2 t)
(6)

With no radial restraint (E f =0) and o ~0.5, which is the
common approximation for elastomeric materials, the coefficient of 3 p vanishes and the pressure has no effect on the
curvature or bending moment. Thus, for the elastomeric
fingers of this general design, the induction of curvature is
entirely dependent on the restraint tension, Ts ; with no circumferential constraint, in this case from struts but possibly from interior foam or circumferential fibers, there is no
bending due to pressure. The simplest approximation for
this restraint is to assume that inextensible fibers, struts or
what have you (E f =3) , in which case the relation between
the pressure, curvature, and net finger moment is
l

2
2
= - 1 - o3 M N + 1 - o Tp .
Et
3
3 Er r t

(7)

This linear-elastic small-strain composite-beam model
shows, for example, how the curvature increases with pressure and decreases with increasing elastomer modulus and
elastomer thickness. Most importantly, the model shows
the necessity of radial (circumferential) constraint in the elastomeric fingers for the pressure to cause bending, at least
to the extent that this small-strain linear theory applies.
When it is applied to long and narrow balloons (o = 0.5),
the theory says that, to the first order, there is no elongation with pressurization. In reality, however, with large
expansions, the long and narrow balloons do slightly elongate. Similarly, even without radial constraint, the elastomeric fingers will bend with large enough pressurization
(violating the simple small strain theory here).
The maximum net finger moment can also be derived
from (7) by setting l = 0 :
M N max = rr 3 Tp.

(8)

When substituting 3 p = 270 lpa and r = 10 mm , we get
M N max = 0.8 $ N $ m . For an 8-cm-long finger, this yields a
theoretical upper limit for the fingertip force of 10 N.
Manufacture of the Soft Orthotic Glove
Our orthotic glove is constructed by using a new rotational-casting technique followed by an overmolding
process. The optical-fiber sensor is also fabricated from
an innovative method.
Rotational Casting
There exist several methods to produce soft actuators [22],
each with its own drawbacks. Replica molding (sometimes
Table of Contents for the Digital Edition of IEEE Robotics & Automation Magazine - September 2016
