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The International Journal of

Robotics Research

2016, Vol. 35(1–3) 161–185

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DOI: 10.1177/0278364915592961

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A novel type of compliant and

underactuated robotic hand for dexterous

grasping

Raphael Deimel and Oliver Brock

Abstract

The usefulness and versatility of a robotic end-effector depends on the diversity of grasps it can accomplish and also on

the complexity of the control methods required to achieve them. We believe that soft hands are able to provide diverse and

robust grasping with low control complexity. They possess many mechanical degrees of freedom and are able to implement

complex deformations. At the same time, due to the inherent compliance of soft materials, only very few of these mechani-

cal degrees have to be controlled explicitly. Soft hands therefore may combine the best of both worlds. In this paper, we

present RBO Hand 2, a highly compliant, underactuated, robust, and dexterous anthropomorphic hand. The hand is inex-

pensive to manufacture and the morphology can easily be adapted to specific applications. To enable efficient hand design,

we derive and evaluate computational models for the mechanical properties of the hand’s basic building blocks, called

PneuFlex actuators. The versatility of RBO Hand 2 is evaluated by implementing the comprehensive Feix taxonomy of

human grasps. The manipulator’s capabilities and limits are demonstrated using the Kapandji test and grasping experi-

ments with a variety of objects of varying weight. Furthermore, we demonstrate that the effective dimensionality of grasp

postures exceeds the dimensionality of the actuation signals, illustrating that complex grasping behavior can be achieved

with relatively simple control.

Keywords

Grasping, robotic hand, soft hand, human grasping, hand synergy, underactuation, pneumatics, actuator model,

Pneuflex, Feix taxonomy

1. Introduction

Dexterous grasping is a prerequisite for task-dependent

manipulation. By the term dexterous, we refer to the postural

variability of the hand: the higher this variability, the more

dexterous we consider a hand (for examples of grasping pos-

tures, refer to the grasp taxonomies presented in Cutkosky

(1989) and Feix et al. (2009)). Such variability enables versa-

tile grasping and manipulation: small objects can be picked

up with pincer grasps, large objects with enveloping power

grasps. Depending on the task, a cylindrical side grasp can

be used to pick up a glass for drinking, whereas a disk grasp

from above is appropriate to lift it off a cluttered table.

In robotic hands, dexterous grasping capabilities are tra-

ditionally realized through complex, multi-jointed struc-

tures and sophisticated actuation mechanisms. Such hands

are expensive and difficult to design. They also require

complex sensing and control. Recently, underactuated

hands with passively compliant parts have become a popu-

lar alternative in robot hand design. These hands perform

certain grasps robustly, have simpler mechanics, and require

simpler control due to underactuation. However, there is

one commonly assumed drawback of compliant hands:

underactuation and passive compliance seem to render dex-

terous grasping difficult or even impossible. The experi-

ments performed with our novel hand indicate the opposite.

We present a novel type of compliant and underactuated

hand based on soft robotic technology. This hand, called

RBO Hand 2, is capable of dexterous grasping, it is easy to

build, robust to unanticipated impact, inherently safe, low

cost, and easy to control. These advantages are achieved by

building almost the entire hand out of soft, compliant mate-

rials or structures, rather than of rigid parts. We believe that

the combination of dexterous grasping capability and ease

Berlin University of Technology, Berlin, Germany

Corresponding author:

Raphael Deimel, Berlin University of Technology, Marchstraße 23, MAR

5-1, Berlin 10587, Germany.

Email: raphael.deimel@tu-berlin.de

of manufacturing make the presented hand well-suited for

enabling novel advances in grasping and manipulation.

Our design, shown in Figure 1, purposefully maximizes

the hand’s passive compliance, while ensuring sufficient

structural support to lift objects. We believe that this design

choice is critical for robust grasping: First, passive compli-

ance facilitates obtaining force closure in power grasps

(Deimel and Brock, 2013; Dollar and Howe, 2010).

Second, passive compliance facilitates the use of contact

with the environment to aid attaining a grasp. This strategy,

the exploitation of environmental contact to reduce uncer-

tainty, has been shown to increase grasp performance in

humans as well as robots (Deimel et al., 2013; Kazemi

et al., 2014). For these reasons, passive compliance is a key

ingredient for robust grasping. In this paper, we want to

show that in addition to the two aforementioned advan-

tages, passively compliant hands can also perform dexter-

ous grasping. Indeed, our results indicate that passive

compliance even facilitates dexterous grasping.

An opposable thumb is important to achieve dexterity in

human and robotic hands. We evaluate the thumb dexterity

of the RBO Hand 2 using the Kapandji test (Kapandji,

1986). This test is commonly used to evaluate thumb

dexterity in human hands after surgery. In addition, we

show that the hand is capable of enacting 31 out of 33

grasp postures of the human hand from the comprehensive

Feix taxonomy (Feix et al., 2009). We evaluate the space of

hand posture exhibited by humans and the RBO Hand 2

which we find to be similar. We also show that four actua-

tion degrees of freedom suffice to achieve a postural space

with more than these four dimensions. This implies that the

variability in grasping posture is only partially generated by

the hand’s actuation. The remaining variability is the result

of interactions between hand and object. These interactions,

we claim, are greatly simplified and enriched by the exten-

sive use of passive compliance in the hand’s design. These

results indicate that dexterous grasping is easier to achieve

with passively compliant than with traditional, stiff-linked

hands.

This paper extends our previous work (Deimel and

Brock, 2014) in several important ways. In Section 6.3, we

extend the analysis of the hand’s dexterity by comparing its

postural diversity to that of the human hand, given a set of

diverse grasps. This is important, as it provides support for

the statement that compliance enhances dexterity and

ensures that the RBO Hand 2 has as diverse postures as the

Fig. 1. The RBO Hand 2 is a compliant, underactuated robotic hand, capable of dexterous grasping. It is pneumatically actuated and

made of silicone rubber, polyester fibers, and a polyamide scaffold.

162 The International Journal of Robotics Research 35(1–3)

human counterpart. We also present and validate a novel

model for the actuator’s behavior in Appendix B. This

enables the validation of novel actuator designs prior to

manufacturing. Furthermore, we significantly extended the

description of the hand design and production process in

Section 4. Finally, we publish the experimental data sets,

videos, and high resolution images of the human and

robotic experiments related to the Feix taxonomy in

Extensions 1–6.

2. Related work

Many highly capable robotic hands exist. A historical over-

view, surveying robotic hands from over five decades, pro-

vides an excellent overview (Controzzi et al., 2014). An

analysis of robot hand designs with respect to grasping cap-

abilities was recently presented by Grebenstein (2012). As

the notion of compliance is central to our hand design, we

will limit our discussion to hand designs that deliberately

include this concept.

We distinguish between two types of hands, actively and

passively compliant. The former can be achieved by using

active control on fully actuated or even hyper-actuated sys-

tems, where every degree of freedom can be controlled.

Examples are the impressive Awiwi hand (Grebenstein,

2012), the ShadowRobot Shadow Dexterous Hand, and the

SimLab Allegro Hand (Bae et al., 2012). These hands

achieve dexterity and compliance through fast and accurate

control, which comes at the price of mechanical and com-

putational complexity. As a result, these hands tend to be

mechanically complex and expensive to manufacture.

Mechanical complexity can also increase the probability of

hardware failure.

The alternative is to make hands passively compliant by

including elastic or flexible materials. Building a passively

compliant joint is much cheaper than building an actively

controlled one in terms of costs, spatial volume, and

mechanical complexity. Passive compliance limits impact

forces, a crucial property for an end-effector designed to

establish contact with the world. More degrees of freedom

can better adapt to the shape of an object greatly enhances

grasp success and grasp quality. At the same time, the hand

can be underactuated, effectively offloading control to the

physical embodiment of the hand.

A pioneering work in grasping with passive compliance

was the soft gripper by Hirose and Umetani (1978).

Recently, a whole range of grippers and hands were built

using passive compliance: the FRH-4 hand (Gaiser et al.,

2008), the SDM hand and its successor (Dollar and Howe,

2008; Ma et al., 2013; Odhner et al., 2014), the starfish

gripper (Ilievski et al., 2011), the THE Second Hand and

the Pisa-IIT Soft Hand (Catalano et al., 2014), the ISR-

SoftHand (Tavakoli and Almeida, 2014), the Positive

Pressure Gripper (Amend et al., 2012), the RBO Hand

(Deimel and Brock, 2013), and the Velo Gripper (Ciocarlie

et al., 2013). A different source of inspiration was taken by

Giannaccini et al. (2014), who built a compliant gripper

inspired by the octopus arm.

The practical realization of underactuated hands is

matched by theoretical approaches to analyze and evaluate

their dexterity (Gabiccini et al., 2013; Prattichizzo et al.,

2012). The most promising models rely on the hypothesis

of a low-dimensional representation of grasp postures,

called synergies (Santello et al., 1998). Odhner et al. (2014)

simplify the underactuated hand mechanics into compliance

ellipsoids at possible locations of contact points. However,

these approaches require accurate knowledge of grasp pos-

ture, contact point locations and contact forces. Given cur-

rent sensor technologies, this information is difficult to

obtain. Interestingly, humans are able to grasp under com-

parable conditions with strongly impaired perception, e.g.

with blurred sight and wearing a glove (Deimel et al.,

2013). This suggests that there is an alternative, percep-

tually less demanding representation of compliant behavior.

The inclusion of compliance into the design of robotic

hands has led to significant improvements in performing

power grasps. Very little work has examined the effect of

compliance and underactuation on the dexterity of a robotic

hand. Closing this gap will be the focus of this paper.

Tavakoli et al. (2014) recently characterized the influence

of reducing the number of actuated degrees of freedom on

the number of possible grasps for the ISR-SoftHand. This

anthropomorphic hand relies on extensive compliance

using elastomeric joints and deformable finger pads. They

found that about four to six actuated degrees of freedom

are enough to enact a broad set of human grasps and that

an opposable thumb is crucial to achieve this, which corro-

borates our own findings.

3. PneuFlex actuators

The RBO Hand 2 uses a highly compliant, pneumatic con-

tinuum actuator design, called PneuFlex, which was first

presented in Deimel and Brock (2013). PneuFlex actuators

can be manufactured within a day and use materials that

are cheap and non-toxic. Figure 2 illustrates the working

principle. When inflating the contained chamber with air

the pressure forces the hull to elongate along the actuator.

The bottom side contains an inelastic fabric to prohibit

elongation. This causes a difference in length between the

top and bottom side and the actuator bends. Radial fibers

stabilize the actuator’s shape and greatly increase the attain-

able curvature.

3.1. Manufacturing actuators

A distinguishing feature of the PneuFlex actuator is the

integrated design pipeline, which enables rapid prototyping

of actuators with widely varying properties without chang-

ing the production process. The steps of the process are

illustrated in Figure 3.

To create an actuator from scratch, first a set of planes is

defined along a line or curve, on which the local cross

Deimel and Brock 163

section of the actuator is defined. The principal shape para-

meters for each cross-section (height, width, hull thickness)

are determined from the desired actuation ratio and stiffness

at each point along the actuator, e.g. by using the model

provided in Appendix B.

In the second step, the set of cross-sections is translated

into a 3D model of a two-part mold for casting the rubber

body of the actuator. The model is produced on a 3D

printer. Because we need to separate the mold from the

cast, the bottom side of the actuator is not included.

In the next step, the top part is cast using the printed

mold and addition-cure silicone.

After unmolding, a silicone tube is inserted at a conve-

nient position into the top part and bonded to it. The tube

enables us to easily connect the actuator to the pneumatic

control.

Afterwards the air chamber is closed by placing the top

part on a thin (1––2 mm) sheet of freshly cast silicone that

embeds a bendable but inextensible PET fabric.

When the bottom layer has cured, a sewing thread is

wound around the actuator in form of a double helix. To

fixate the thread in place, a thin layer of addition-cure sili-

cone is applied to the top and bottom side. This step

finishes the actuator.

The presented process enables us to freely change width

and height of the actuator and the thickness of the rubber

hull. We can also create straight as well as curved actuators,

as long as the bottom layer stays in a plane.

The actuator design space can be explored by varying

shape and size of the actuator, and by varying the thickness

of the silicone hull at the top, side and bottom. In addition,

available silicone types let us vary the shear modulus by an

order of magnitude. All of these parameters affect the bend-

ing behavior, stiffness, and limits of the actuator. If needed,

a bellows-shaped hull extends the design space towards

larger curvatures and lower stiffnesses. Curved actuators

can realize 3D motion. For example, the two palm actuators

(actuators 6 and 7 in Figure 4) are used to achieve thumb

opposition. Differential inflation of the two actuators pro-

vides additional dexterity.

(a) An inflated Pneu Flex ac-

tuatorin front of deflated

ones.

(b) Cut of a PneuFlex continuum actuator. (c) Functional parts

Fig. 2. Working principle and structure of a PneuFlex actuator: When inflated, the top of the finger, consisting of translucent silicone,

extends, thereby bending as its motion is constrained by the bottom of the finger, into which an inelastic fabric is embedded. The

helically wound threads stabilize the actuator shape and relieve the rubber of non-functional strains, i.e. the inflation leads to bending

rather than to radial expansion.

Fig. 3. Production steps for making a PneuFlex actuator with a custom stiffness and actuation ratio profile.

Fig. 4. The seven actuators of the soft anthropomorphic hand:

four fingers (1–4), thumb (5), and the palm, consisting of two

actuators (6, 7).

164 The International Journal of Robotics Research 35(1–3)

Actuators can also be packaged together, similarly to

muscle fibers, allowing for redundant actuation or variation

of the actuation strength. Joining actuators also enables the

implementation of multiple deformation modes or the

deliberate mixing of deformation modes (Bishop-Moser

et al., 2012).

In addition their flexible production process, PneuFlex

actuators are robust to impact and blunt collisions, are

inherently safe, and are not affected by dirt, dust, or liquids.

However, they can easily be cut or pierced.

3.2. Modeling actuators

The PneuFlex actuator shares many properties with other

recently published continuum actuators, most notably the

fast PneuNet actuators (Mosadegh et al., 2014) and the

actuator by Galloway et al. (2013). In contrast to these actua-

tor designs, the PneuFlex design and production process is

optimized to provide freely adaptable cross-sections which

determine actuation ratio and stiffness, the ability to include

multiple separate air chambers, and to provide access to its

internal space for the integration of sensors and wiring.

We provide a detailed analysis of basic actuator behavior

in Appendix B and also propose design rules for successful

actuator design in Section B.11. Additional insights can be

drawn from related research on continuum actuators. For

example, Bishop-Moser et al. (2012) characterize all basic

motions attainable by changing inclinations of the reinfor-

cement helices. Others proposed approximate numeric

models based on twisted, one-dimensional beams (Giorelli

et al., 2012; Renda et al., 2012).

4. Hand design

In this section, we describe the components of our soft

anthropomorphic hand (RBO Hand 2, see Figure 1). The

entire hand weighs 178 g and can carry a payload of up to

0.5 kg. Higher payload can easily be achieved, if necessary,

as we will explain in Section 7.

4.1. Morphology

The design space of possible hands is very large. For this

hand, we chose an anthropomorphic design in shape and

size for three reasons. First, we know the human hand form

enables dexterous grasping in humans. By starting with a

human-hand-like morphology, we start with a proven hand

design. Second, many objects have been built for manipula-

tion by a human hand and match the anthropomorphic form

factor. Third, we can use well-established grasp taxonomies

and compare our designs with humans and many other

robotic hands.

4.2. Control

Pneumatic control of the PneuFlex actuators is based on a

simple linear forward model for computing valve opening

times. The model takes into account the regulated supply

pressure to achieve a desired channel pressure which corre-

sponds to a desired bending radius or contact force.

Alternatives to this digital control are cylinder-based con-

tinuous control systems (Marchese et al., 2014). Renda

et al. (2012) demonstrate a computationally simple forward

model for an artificial octopus arm. This model can also be

used for PneuFlex actuators.

For the experiments, control was implemented with

industry-grade air valves and a separate air supply. For a

mobile system, the control system can easily be optimized

for size and weight, because the required air flows are an

order of magnitude smaller than provided by industrial

grade valves. For the same reason, pressurized air can be

effectively sourced by small compressors and small tanks.

In systems where electric energy is scarce, high-pressure

tanks can provide storage of air with high energy density

(Wehner et al., 2014).

4.3. Fingers

The five fingers of our hand are single PneuFlex actuators

(see Figure 4). The index, middle, ring, and little finger are

90 mm long and of identical shape, the thumb actuator is

70 mm long. All fingers get narrower and flatter towards

the finger tip. By using actuators as fingers, we can exploit

the excellent compliance and robustness of the actuators

and greatly simplify the design.

The mechanical behavior of the finger can be described

by the local curvature kand torsional stiffness Maround

the bottom side for short segments of the actuator and is

determined by the geometry of the actuator cross-section

(see Figure 5). Appendix B contains an analysis of this

simplified model of the actuator, which provides surpris-

ingly simple rules to design the ratio of curvature kto pres-

sure pat each segment along the actuator by varying the

hull thickness d:

’

Gd

where Gis the shear modulus of the rubber and constant

within an actuator.

Translational forces between actuator segments are

mainly transmitted by the inelastic fabric of the bottom

layer and therefore do not need to be considered. The model

Fig. 5. Illustration of the finger geometry and its principal

parameters: k(local curvature), M(moment around the passive

layer at the interface), d(top side rubber thickness) and c

(circumference).

Deimel and Brock 165

in Appendix B also yields a rule of thumb for the torsional

stiffness of an actuator segment. For an approximately

squared cross-section, stiffness scales with circumference c:

’Gdc4

As an example, for the fingers of the hand we chose to

increase stiffness linearly from tip to base and keep the

actuation ratio constant. Such a profile has also been used

in the Soft Gripper (Hirose and Umetani, 1978). Using xas

the distance from the tip along the actuator, the cross-

section at this point is defined by

d(x)=const

c(x)}x1

The resulting geometry of the actuator is illustrated in

Figure 5. According to our model, the aspect ratio of the

cross section does not strongly influence stiffness and

actuation ratio. So to give the finger a visually more appeal-

ing shape, we set the width to

width(x)}x1

4.4. Palm

A key feature of the human hand is the opposable thumb.

We realize it in our hand by actuating the palm (see

Figure 4). The palmar actuator compound consists of two

connected actuators. Its base shape is a circular section of

90°with 78 mm outer and 25 mm inner radius. The actua-

tor curves perpendicular to the passive layer. The stiffness

as well as the actuation ratio remain constant along the

curved actuator. They are also designed to be twice as stiff

as the fingers to account for the fact that the two actuators

in the palm oppose four fingers. Figure 8 provides an

impression of the possible thumb motions when the two

palm actuators are inflated either together or differentially.

In addition to enabling thumb opposition, the palm also

provides a compliant surface that, together with the fingers,

is used to enclose objects in various power grasps. To aug-

ment this function, the fingers and the palmar actuator are

connected by a thin sheet of fiber-reinforced silicone, cov-

ering the gap between palm actuators and fingers (shown

in Figure 1, but absent in Figure 4). This sheet transmits

tensile forces between fingers and palm, and between adja-

cent fingers. This stabilizes the underlying scaffold during

power grasps or for heavy loads, as shown in Figure 11.

4.5. Thumb

Like the other fingers, the thumb consists of a single

PneuFlex actuator. The thumb is shorter and twice as stiff,

but also features a linear stiffness profile. A faithful imita-

tion of how humans use their thumb would require a nega-

tive curvature close to the tip, as shown in Figure 6, and

would significantly increase complexity of manufacturing

the thumb. We therefore deviated here from the human

hand. Instead of the inside of the thumb, we use the back-

side (dorsal side) as the primary contact surface for pincer

grasps. This effectively changes the contact surface orienta-

tion by about 45–60°, relative to the orientation found in a

human thumb, avoiding the need for negative curvatures.

As both sides of the PneuFlex actuator have similar surface

characteristics (unlike human thumbs), this choice will not

affect grasp quality.

4.6. Scaffold

The fingers and the palm are connected to the wrist by indi-

vidual, flexible struts as part of a three-dimensional printed

polyamide scaffold (2 mm thick, see Figure 1). The inten-

tionally flat cross section of the struts enables deformation

modes, such as arching the palm and spreading the fingers.

Space for the respective actuator is provisioned, but was

not added to the hand described here. The struts decouple

displacement between fingers, further increasing passive

compliance of the hand. The flexibility of the struts limits

impact forces, while providing sufficient stiffness for heavy

payloads without excessive deformation (see Figure 11).

The fingers and the palmar actuator compound are

bonded to the supporting scaffold as shown in Figure 1.

The palm is supported by parts of the scaffold to increase

its torsional stiffness during opposition with the fingers.

4.7. Strength between thumb and fingers

A flexible, but inextensible band connects the base of the

index finger to that of the thumb (see Figure 4). Similarly

Fig. 6. Difference in thumb configuration and fingertip use

during a pincer grasp between a human hand and the robotic

hand.

166 The International Journal of Robotics Research 35(1–3)

to a muscle in human hands (adductor Pollicis), it enables

increased contact forces between thumb and opposing fin-

ger, by reducing torques on the struts at the wrist. The sheet

connecting fingers and palm serves a similar role, espe-

cially for power grasps of cylindrical objects with large

diameter.

5. Grasp dexterity

In this section, we evaluate the dexterous grasping capabil-

ities of the proposed hand. The most appropriate evaluation

would of course be in full-fledged, real-world grasping

experiments. However, this requires the integration of hand

and control with perception and grasp planning and would

effectively be an evaluation of the integrated system. Here,

we focus on evaluating the capabilities offered by the hand.

Furthermore, we have to resort to empirical methods.

Accurate simulation of the complex, nonlinear deforma-

tions encountered in such a heterogeneous and soft struc-

ture is difficult to conduct and anyways requires empirical

experiments to validate the results.

5.1. Thumb dexterity

Medical doctors employ the Kapandji test (Kapandji, 1986)

to assess thumb dexterity during rehabilitation after injuries

or surgery. This test was also used by Grebenstein for evalu-

ating and improving the thumb dexterity of the Awiwi hand

(Grebenstein, 2012). For the Kapandji test, the human sub-

ject has to touch a set of easily identifiable locations on the

fingers with the tip of the thumb. These locations are shown

in Figure 7. The total number of reachable locations serves

as an indicator of overall thumb dexterity. A thumb is con-

sidered fully functional if it is able to reach all locations.

To perform the Kapandji test on our hand, we manually

selected actuation pressures that would position the thumb

as desired. The postures of the hand performing the test are

shown in Figure 8. The thumb tip could reach all but one

location. Location 1 was not possible to reach because it

would require a backwards bending of actuator 5 (thumb).

Still, the hand scores seven out of eight points, indicating a

high thumb dexterity.

5.2. Grasp postures

A common way of assessing the dexterous grasping cap-

abilities of hands is to demonstrate grasps for a set of

objects. For example, the THE Second Hand was evaluated

with 4 objects and 2 grasp types (Grioli et al., 2012), the

SDM hand on 10 objects and a single grasp type (Dollar

and Howe, 2008), the Velo Gripper on 12 objects and a sin-

gle grasp type (Ciocarlie et al., 2013), and the Awiwi hand

on 8 objects and 16 grasp types (Grebenstein, 2012). We

follow these examples in our evaluation.

We select grasp types and objects based on the most

comprehensive grasp taxonomy to date, the Feix taxonomy

(Feix et al., 2009). It covers the grasps most commonly

observed in humans and therefore is a realistic reference

for assessing the dexterity necessary for common grasping

tasks. The taxonomy encompasses 33 grasp types, out of

which the first 17 are identical to the grasps in the

Cutkosky taxonomy (Cutkosky, 1989). To demonstrate

these 33 grasps, the original publication illustrates 17 dif-

ferent object shapes (Feix et al., 2009). We therefore used

17 objects and 33 grasp types to evaluate our hand.

We implemented the grasps from the Feix taxonomy by

defining appropriate actuation pressures and actuation

sequences. When, due to collisions, simultaneous actuation

of all channels was not sufficient to reach the desired pos-

ture, we added an appropriate pre-grasp posture. The com-

manded actuation pattern was then modified and tested

iteratively to improve the quality of the grasp in terms of

grasp stability and robustness against external forces, and

to ensure the proper types and locations of contact. Grasp

quality was judged by manually rotating and translating the

hand, and by testing several repetitions of the actuation

pattern.

To simplify the search for appropriate actuation patterns,

we combined the control of the seven actuators into four

actuation channels. Channel A drives actuators 1, 2, and 3

(small, ring, and middle fingers), channel B drives actuator

4 (index finger), channel C drives actuators 5 (thumb) and

7 (inner palm), and channel D controls actuator 6 (outer

palm). These channels can be understood as the hand’s four

grasping synergies.

To perform a grasping experiment for a particular grasp

type, the experimenter triggers the actuation sequence to

attain the pre-grasp posture, holds the object in the see-

mingly most appropriate location relative to the hand, and

then triggers the actuation sequence for the grasping

motion. The resulting postures for each empirical actuation

pattern are shown in Figure 12; high-resolution images are

provided in Extension 1. Out of 33 grasp types, the hand is

able to perform 31 repeatedly (three consecutive successful

trials). The two grasps that failed are the light tool grasp

and the distal type grasp.

Fig. 7. The Kapandji test counts the number of indicated

locations that can be contacted with the thumb tip.

Deimel and Brock 167

The light tool grasp fails because the hand does not pos-

sess finger pulp that fills the cavity formed by the maxi-

mally bent fingers, which causes the object to slip. The

distal type grasp fails because the resulting grasp is non-

functional with respect to proper use of the scissors, even

though it is possible to put the soft fingers through the scis-

sors’ holes. Both grasp failures are shown in Figure 9.

Figure 10 shows a scatter plot of the actuation patterns

for the 31 successfully achieved grasp types of the Feix

taxonomy. The actuation patterns relate to final grasps, not

pre-grasp postures. The plots indicate an even distribution

of activation for all channels and do not reveal obvious cor-

relations that could be leveraged to further simplify

actuation.

The evaluation presented in this section demonstrates

the hand’s ability to assume a variety of grasp postures.

This ability is comparable with that of other hands pre-

sented in the literature. We therefore believe that dexterous

grasping and compliance can indeed be combined in a

highly capable, compliant, underactuated robotic hand.

5.3. Grasping forces

While grasp quality and grasp strength was not the

driving design criterion for the hand, it is important to ver-

ify that a compliant hand is capable of lifting objects of

reasonable weight. To give the reader an intuition on the

capabilities of the hand, we provide a few tests regarding

grasping forces.

The heaviest objects used in the Feix taxonomy grasps

were the rectangular plate in grasp 22 (156 g), the metal

disk in grasp 10 (181 g), the wooden ball in grasp 26

(183 g), and the circular plate in grasp 30 (240 g). Note

that in grasps 26 and 30, the shown posture offers the least

structural support of possible hand poses. Figure 11 shows

two additional heavy objects, a wooden cylinder (541 g)

and a lead ball (1.650 g). Figure 11 also shows three differ-

ent directional disturbance forces on a cylinder which is

power-grasped with grasp 1. If forces above 6–8 N are

applied, the cylinder will slide in the hand.

5.4. Grasping in realistic settings

To further illustrate the effectiveness of the proposed hand,

we performed experiments with complete grasping

sequences, shown in Figure 13 and in Extension 2. In these

experiments, a human operator selects the appropriate

grasp, triggers the pre-grasp posture of the hand, places the

hand in the appropriate location, and then executes the

grasp. These experiments demonstrate that the proposed

hand, given appropriate perception and grasp planning

skills, is able to perform real-world grasps.

6. Compliance benefits dexterous grasping

In the previous section, we showed that our underactuated

and compliant hand is capable of dexterous grasping. In

this section we will investigate whether its compliance and

underactuation are beneficial or detrimental to attaining dif-

ferent grasp postures. If beneficial, control will be simpler

than the resulting behavior, i.e. actuation space is smaller

than grasp posture space. The dimensionality of the posture

space that exceeds the dimensionality of the actuation space

can be explained by the compliant interactions between

hand and object.

Fig. 8. The RBO Hand 2 succeeds in the Kapandji test for all but one position (position 1, lower right, showing best effort).

168 The International Journal of Robotics Research 35(1–3)

Fig. 9. Grasping postures not successfully attained by the robotic hand.

0.0 0.2 0.4 0.6 0.8

0.0

0.2

0.4

0.6

0.8

0.0 0.2 0.4 0.6 0.8

0.0

0.2

0.4

0.6

0.8

0.0 0.2 0.4 0.6 0.8

0.0

0.2

0.4

0.6

0.8

0.0 0.2 0.4 0.6 0.8

0.0

0.2

0.4

0.6

0.8

0.0 0.2 0.4 0.6 0.8

0.0

0.2

0.4

0.6

0.8

0.0 0.2 0.4 0.6 0.8

0.0

0.2

0.4

0.6

0.8

Fig. 10. Scatter plots of the four actuation channels for the actuation patterns of the 31 successful grasps. Darker color indicates

overlapping dots.

Deimel and Brock 169

Fig. 11. Illustrations of grasping force capabilities: (a) finger strength and palm support strength, (b)–(d) tolerated disturbance forces

in different directions for grasp 1, and (e) strength of the support provided by the scaffold.

170 The International Journal of Robotics Research 35(1–3)

Fig. 12. Enacted grasps of the Feix taxonomy, using empirically determined actuation patterns: Grasps are numbered according to the

Feix taxonomy (Feix et al., 2009); the hand failed to replicate grasps 5 (Light Tool) and 19 (Distal Type, Scissors).

Deimel and Brock 171

6.1. Postural diversity of the Feix taxonomy

To assess the dimensionality of the attainable grasp posture

space, we first have to assert that the grasp set we use to

sample from that space is diverse enough, i.e. that the

employed grasps span the space of possible grasps. For

this, we recorded humans doing Feix taxonomy grasps

using the method published by Santello et al. (1998), and

compare the results with existing published data sets (the

data published in Santello et al. (1998), the UNIPI data

set,

, and UNIPI-ASU data set.

In the experiment, we asked five healthy human partici-

pants to enact every grasp of the Feix taxonomy five times

while wearing a Cyberglove II data glove, using exactly the

same objects as used for the experiment in the previous

section. Participants were allowed to use the other hand to

assist in assuming the grasp posture, but had to achieve a

successful grasp in the sensorized hand without additional

support. The resulting postures were sampled 50 times

within 500 ms and averaged over samples and episodes.

We then performed dimensionality reduction by applying

principal component analysis (PCA) for each subject indi-

vidually to exclude inter-subject variance in accordance

with the data analysis used in Santello et al. (1998). We

then compared the resulting residual unexplained variances

with data from literature.

The results are shown in Figure 14. For four out of five

participants, the unexplained variances were higher than

those of the three independently published data sets, sug-

gesting that the grasps span the space of possible grasp

postures more effectively than the considerably larger but

less structured set of objects that was used for the data sets

we compare with.

The discrepancy between the published data sets and

ours may be explained by the fact that the Santello and

UNIPI data sets were recorded on grasping imagined

Fig. 13. Performing grasps using the grasp postures 25, 1, 9, 28 and 18: a human places the hand and then triggers the actuation of

the appropriate grasp; top, pre-grasp posture; middle, executed grasp; bottom, lifting object to show success.

1 2 3 4 5 6

Principal components

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Residual unexplained variance

participant 1

participant 2

participant 3

participant 4

participant 5

Dataset in Santello

(1998)

Dataset UNIPI-ASU

(May 2011)

Dataset UNIPI

(Oct. 2011)

Fig. 14. Residual variances of PCA on joint measurements taken

with data gloves. Solid lines denote published data sets using the

method of Santello et al. (1998), dotted lines denote the data

acquired using the Feix taxonomy and the objects shown in

Figure 12.

172 The International Journal of Robotics Research 35(1–3)

objects, while our data and the UNIPI-ASU data set are

recorded while grasping real objects. The additional var-

iance observed may come from the interaction between

hand and object, whereas with imagined objects, hand pos-

ture may be more related to actuation pattern than the

actual grasp posture. This interpretation of existing data is

consistent with our hypothesis that hand control can be

simpler than effected posture.

6.2. Choice of representation

Because the RBO Hand 2 does not have discrete joints, it is

not possible to assess its postural diversity using the method

of Santello et al. (1998), even though the hand is anthropo-

morphic. An alternative representation of hand posture

compatible with soft hand mechanics has been used for the

Human Grasp Database

experiment. In this experiment,

Romero et al. (2010) represented hand posture in terms of

fingertip position and orientation relative to the back of the

hand.

Before using this representation, we first have to estab-

lish that for assessing postural diversity it is comparable to

using joint angles. Figure 15 shows a comparison of the

residual unexplained variances of a PCA for fingertip posi-

tion and joint angles (from the previous experiment), both

acquired on human hands using the Feix taxonomy (31

grasps, excluding grasps 19 and 23 in Figure 12). The

graph shows that fingertip positions may be a more com-

pact representation than joint angles, but most importantly

not worse. Therefore, we can use fingertip position data as

a conservative estimate of joint posture diversity. Omitting

fingertip orientation data (represented as quaternions) from

the PCA did not significantly decrease unexplained var-

iance. It was therefore excluded to simplify acquisition of

comparable data on the RBO Hand 2 using a motion cap-

ture system.

Concluding the human experiments and comparative

study of published data, the Feix taxonomy appears to be a

good proxy for diverse grasp postures. In addition, we

found that fingertip position is a viable alternative to joint

angles for assessing postural diversity of human hands and

therefore also of anthropomorphic robot hands.

6.3. Postural diversity of RBO Hand 2

The previous subsection gives us a justification to use fin-

gertip positions of grasps from the Feix taxonomy to assess

the RBO Hand 2 in terms of grasp posture diversity. To

acquire the fingertip positions we used a motion capture

system. We placed 3 mm retroreflective markers on the

backside of the fingertips of the four fingers. On the thumb

tip, the marker was placed on the side to not interfere with

the pincer grasp. The marker placement can be seen in

Figure 16 as in the video Extension 4. Note that additional

markers were attached to the hand during recording, but

only the five fingertip markers were used for the

experiment.

The motion capture system has a spatial precision of

0.05 mm, as measured using 205 frames recorded for a sta-

tionary hand (at 50 Hz). For each grasp, two trials are

recorded, fingertip positions are extracted and averaged

over the trials before applying PCA. The data set from the

Human Grasp Database is subjected to the same procedure.

The results are shown in Figure 17. The postural dimen-

sionality exhibited by the RBO Hand 2 matches the human

hand data well. This strongly supports our claim that our

robotic hand is as dexterous as a human hand with respect

to attainable grasp posture, taking the Feix taxonomy as a

reference.

A second observation can be made by considering the

dimensionality of control required to implement the grasp

postures, which is indicated by the dotted line and gray

1 2 3 4 5 6

Principal components

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Residual unexplained variance

participant 1

participant 2

participant 3

participant 4

participant 5

Human Grasp Database

(fingertip positions)

Fig. 15. Residual variances of PCA on grasp postures enacted

by humans using data from different acquisition methods, joint

angles and fingertip positions. Solid line denotes data from the

Human Grasp Database (Romero et al., 2010), dotted lines

denote the data acquired using the Feix taxonomy and the objects

shown in Figure 12.

Fig. 16. Screen shot from video Extension 4, showing the

marker placement on the fingertips for Motion Capture.

Deimel and Brock 173

area in Figure 17. The space of grasp postures is of higher

dimensionality than the actuation space, which is of dimen-

sion four. Where does this increase come from? It must be

introduced by the diverse shapes of the grasped objects.

The interactions between the hand and the object differenti-

ate different postures. This differentiation is facilitated by

the hand’s ability to adapt to objects compliantly. The dif-

ferences between imagined and real grasps in human

experiments, discussed in Section 6.1, corroborate this

hypothesis. Tavakoli et al. (2014) also obtain the minimum

of four to six actuated degrees of freedom to implement

dexterous grasping for their anthropomorphic, compliant

hand.

The diversity and consistency of evidence we presented

here strongly suggests that compliant hands benefit dexter-

ous grasping.

The presented evaluation, based on published data sets,

measurements of human grasps, and of grasps with the

RBO Hand 2, paints a consistent picture: the robotic hand

performs similar to a human hand, both in terms of grasp

posture diversity, and in terms of covering sets of grasps,

again, using the Feix taxonomy as a reference.

This diversity in grasping posture is achieved by the

RBO Hand 2 with only four actuated degrees of freedom.

This is possible because the interaction of the compliant

degrees of freedom with the diverse objects introduces

additional variance in the posture.

7. Limitations

Adopting a novel technology in a new application, like

continuum actuators in the design of soft dexterous hands,

opens up new possibilities but also leads to new limitations

and challenges that need to be considered carefully.

7.1. Grasp forces and payload

Continuum actuators, when constructed with reinforced

rubber and actuated hydraulically, are in principle capable

of exerting extremely large forces. For example, an actuator

made out of car tire rubber with steel-fiber reinforcements

and hydraulic actuation would probably be able to exert

grasping forces exceeding 100 N. It would also be straight-

forward to make a much stronger hand with the current

production process by choosing stiffer rubbers and thicker

hulls. In the current hand design, we chose to use very soft

actuators, to investigate the effect of compliance, to

increase safety, and to make manufacturing convenient. We

chose pneumatic actuation over other fluidic options as it is

much simpler and cleaner to operate in a lab environment.

7.2. Grasp stiffness

While grasp forces, as discussed above, refer to the magni-

tude of forces exerted on the object, grasp stiffness refers to

the hand’s ability to maintain a grasp posture in the pres-

ence of external forces. Naturally, a soft hand built for max-

imum compliance does not create an extraordinarily stiff

grasp. Low grasp stiffness has the advantage of reducing

the peak forces encountered at the contact points between

object and hand. It also reduces the probability of slip when

objects collide or upon jerky wrist motion. At the same

time, it also places a limit on the forces the hand can exert

on the environment through the grasped object, for exam-

ple. This negatively affects tasks where those forces must

be high. We therefore need to balance and these two com-

peting properties.

In case grasp stiffness proves to be the limiting factor for

certain tasks, there are several methods available to selec-

tively increase actuator stiffness (Wall et al., 2015). But

they increase design complexity and production costs, and

therefore should be avoided if possible. Another simple

method to increase grasp stiffness is to select power grasps

instead of precision grasps.

In the context of grasp stiffness, we can understand the

exploration of soft hand designs as searching for a lower

bound on grasp stiffness that still is able to provide suffi-

cient grasp force. Future research will investigate how grasp

stiffness can be increased while maintaining compliance

where necessary.

7.3. Pneumatics

Our hand relies on additional external pneumatic compo-

nents for control. These components are cheap and readily

available in industry-grade quality. However, they are over-

sized for the low pressures, small volumetric flow, and size

constraints of robotic hands. Miniaturizing and integrating

electrically actuated valves directly into the hand, possibly

even into the actuator, would greatly simplify integration

into predominantly electromechanic robots.

1 2 3 4 5 6

Principal components

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Residual unexplained variance

Human Grasp Database

(fingertip positions)

RBO Hand 2

(fingertip positions)

RBO Hand 2

(actuation signal)

Fig. 17. Dimensionality of fingertip positions for a human hand,

the RBO Hand 2, and its respective control signal. The gray

region indicates where the dimensionality of fingertip posture

exceeds control dimensionality.

174 The International Journal of Robotics Research 35(1–3)

Long-term autonomy arguably is easier to achieve with

pneumatic systems. In contrast to electrical power systems,

where no good solution for long-term untethered operation

exists, the technology exists to make small, quickly refill-

able air tanks or even to directly convert chemical energy

(Wehner et al., 2014).

Mobility can easily be obtained by using compact, small

compressors, as the average rate of airflow for operating

the RBO Hand 2 is very low and peaks can be serviced by

small air tanks. The use of electrical energy storage also

often simplifies the integration into existing system.

7.4. Precision and repeatability of grasps

While actuation patterns can be reproduced with high preci-

sion, the interaction with the object or features of the envi-

ronment during a grasp can introduce substantial variations

in the final grasp posture. While this is often understood to

be a disadvantage, we view it as an important feature of the

design, leading to robust grasping performance. Therefore,

we believe one must carefully differentiate between preci-

sion and repeatability versus grasp robustness. The hand

presented in this paper deliberately trades the former for the

latter.

7.5. Sensing

The compliance of the materials makes local, dense sen-

sing for proprioception and contact forces important but

also very difficult. Also, sensor technologies compatible

with soft actuators are currently not available commercially.

While it is easy to integrate air pressure sensors, it would

be very desirable to integrate strain and touch sensors too.

This is a topic of active research as current electronic sen-

sor technology is predominantly designed for rigid struc-

tures, but working solutions for stretchable electronics start

to appear, making an integration with PneuFlex actuators

feasible in the near future (Culha et al., 2014; Gerratt et al.,

2014; Rahimi et al., 2014).

7.6. Modeling hand mechanics

Modeling the whole hand poses certain challenges: an

accurate mechanical state is difficult to obtain due to nonli-

nearities arising from large deformation and anisotropic

structure of its components. In addition, the actuators inten-

tionally provide a large number of deformation modes,

which increases hand complexity even further and makes

sensing the hand’s complete mechanical state very difficult.

Because of these features, we are currently not able to

provide a quantitative analysis of grasp quality based on

mechanical models as it is state of the art for hands with

rigid links. We therefore qualitatively assess grasp quality

in Section 5.3 and also provide two videos in Multimedia

Extensions 5 and 6 that illustrate the attainable quality of

grasps.

To start closing the gap in modeling soft continuum

actuators, we present a model for computing key mechani-

cal properties of PneuFlex actuators in Appendix B. While

that model is not able to estimate all parameters necessary

to create a faithful simulation, it can be used to straightfor-

wardly customize deformation modes and stiffness profiles

of the actuators before building them. Appendix B also

evaluates the influence of several nonlinear phenomena on

actuator behavior and suggests several, easy to follow

design guidelines to avoid common failure modes.

Yet another consequence of the complex and highly vari-

able deformations that happen during grasping is that most

existing grasp planners cannot be applied, as they rely on

complete and accurate geometric and kinematic models of

hand and environment. While it might be possible to per-

form simulations of the hand’s deformation using finite ele-

ment methods, these are computationally too complex to

employ them in search-based grasp planning.

8. Conclusions

We have presented a compliant, underactuated, and dexter-

ous anthropomorphic robotic hand based on soft robotics

technology. The hand is able to achieve 31 of 33 grasp pos-

tures from a state-of-the-art human grasp taxonomy. To

evaluate the dexterity of the opposable thumb, we per-

formed the Kapandji test, in which the hand achieves seven

out of eight possible points. We illustrated the hand’s excel-

lent payload to weight ratio, as it is able to lift objects of

nearly three times its own weight. We also presented real-

world grasping experiments to demonstrate the hand’s cap-

abilities in a realistic setting.

We believe that compliance is crucial to enable robust

grasping in robotic hands. We provided support for this

statement by showing that the dimensionality of the achiev-

able postural space is significantly larger than the dimen-

sionality of the hand’s actuation space. We explain this

observation with the hand’s ability to mechanically comply

to the shape of the grasped object: The final grasping pos-

ture is the result of the hand’s actuation together with com-

pliant interactions between the hand and the object. We

found that several grasping experiments with humans are

consistent with this interpretation. We therefore conclude

that compliance in robotic hands, when used correctly, can

facilitate not only robustness in power grasps but also

dexterity.

In addition to enhancing dexterity, the use of soft robotic

technology renders the hand robust to impact and blunt col-

lisions and makes it inherently safe and suitable for working

environments containing dirt, dust, or liquids. The effort,

complexity, and cost of building the hand are significantly

lower than for existing hand technologies. The hand pre-

sented here can be built in 2 days, using materials worth

less than US$100. Both actuator and hand structure are eas-

ily adaptable to specific application domains. We therefore

believe that this novel way of building robotic hands

Deimel and Brock 175

significantly lowers the barrier to entry in the field of grasp-

ing and manipulation research.

Funding

This work was supported by the Alexander von Humboldt founda-

tion through an Alexander von Humboldt professorship (funded

by the German Federal Ministry of Education and Research), the

SOMA project (European Commission, grant number H2020-ICT-

645599) and the German Research Foundation (DFG, award num-

ber BR 2248/3-1).

Notes

1. Data set available at http://handcorpus.org

2. Data set available at http://handcorpus.org

3. Data set available at http://grasp.xief.net/

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Appendix A: Index to Multimedia Extensions

Archives of IJRR multimedia extensions published prior to

2014 can be found at http://www.ijrr.org, after 2014 all

videos are available on the IJRR YouTube channel at http://

www.youtube.com/user/ijrrmultimedia

Appendix B: Derivation of an approximate

PneuFlex model

To facilitate the design of PneuFlex actuators, we present

a computational model for its deformation. Overall, the

deformation of such an actuator is complex. Nevertheless,

surprisingly simple design rules can be derived for

designing the most important actuator properties, namely,

actuation ratio and rotational stiffness around the actuated

axis. This appendix proposes a suitable formalization, an

analysis of simplifications taken, and a basic experimen-

tal evaluation of the resulting equations.

B.1 Formalization of the PneuFlex actuator

geometry

Figure 18 shows the parameterization of a small segment

of the actuator. To simplify the model, we ignore the side

walls and assume a rectangular cross-section. Let x,z,dbe

the length, width, and thickness of a segment of the actua-

tor, respectively.

Width and height of the actuator are assumed to not

change due to the helical thread around the actuator. This

assumption is facilitated by selecting an approximately cir-

cular cross-section (e.g. a square) for the shape of actuator

cross-section, as the radial fibers will always deform the

shape into a circle to balance the radial pressure.

Due to its embedded fabric, the bottom layer also has a

fixed length x. The only possible deformation left is to

Table of Multimedia Extensions

Extension Media type Description

1 Images High resolution images of grasps from the Feix taxonomy in Figure 12

2 Data Fingertip positions of RBO Hand 2 for the Feix taxonomy

3 Data Joint angles of five human participants executing grasps from the Feix taxonomy recorded with a

Cyberglove II

4 Video Video of how grasp postures for grasps from the Feix taxonomy were obtained

5 Video Video of example grasps from a tabletop under human control

6 Video Video of manipulating two heavy objects

Fig. 18. Simple model of an actuator segment.

Deimel and Brock 177

stretch the silicone layer while bending the bottom layer,

which is illustrated in Figure 19

The interfaces of the segment are rotated to each other

by the angle uand the bottom layer curves with the radius

r=x

u. As we also have configurations with u= 0 (when

the actuator is deflated), we will rather express equations in

terms of the curvature kof the bottom layer:

r=u

x)u=kx

We can also express the curvature on the top side of the

actuator:

ktop =

k+h

Both kand k

top

are defined on the same angle u.

Therefore, we can compute a relationship between angle u

and stretch l

u=utop

kx=

k+hl1x

l11

ð1Þ

l1=1+khð2Þ

u=x

hl11ðÞ ð3Þ

Volumes. To aid readability of the analysis, we define sev-

eral constant volumes that do not change under deforma-

tion. These volumes can be computed from the basic

actuator geometry:

The volume of the effectively incompressible silicone in

the active layer of the actuator:

Vsil =zxd

The volume of the deflated air chamber:

Vch =zx(hd)

The total volume of the deflated actuator:

Vact =zxh=h

dVsil

=Vch +Vsil

The total volume of air contained in the deflated actuator

plus any volumes connected to it, such as supply tubes:

Vair =Vsupply +Vch =Vsupply +Vact Vsil

The symbol Vwill denote the actual volume of the air

chamber, which is dependent on the deformation. Therefore

it is a function of actuator curvature. The total actual vol-

ume of the actuator is V+V

sil

Volume change. For computing the energy stored by the

compressed gas (air) within the actuator, we need to com-

pute the actual volume with respect to actuator curvatures.

We can do this by first calculating the total volume of a

flexed actuator segment:

V+Vsil =zu

2ppx



px



=Vact 1+h



As u

x=k, we can express actuator curvature in terms of air

chamber volume and initial geometry as

V+Vsil =Vact 1+h

2k



V+Vsil

Vact

1



hVVch

Vact



With Equation (1) we can also compute the relationship

between l

and the actuator volume:

l1=1+2VVch

Vact

 ð4Þ

This leads to the first insight: actuator curvature and

stretch of the silicone rubber are linearly proportional to

the gas volume:

Fig. 19. Parameterization of an actuator segment, illustrated at different curvatures. Here x,V

sil

and hstay constant while l,rand u

change.

178 The International Journal of Robotics Research 35(1–3)

∂k=

Vact h∂V

∂l1=

Vact

∂V

Strain tensor invariants. The energy stored in the rubber

during deformation is modeled using strain tensor invar-

iants (Gent, 2012). The strain tensor invariants are related

to the orthogonal stretches l

J1=l2

1+l2

2+l2

33

J2=l2

1l2

2+l2

1l2

3+l2

2l2

33

J3=l1l2l31

To compute them, we first define actuator-specific rela-

tions between stretches l

and l

in three principal

directions, which are aligned to x,hand z, respectively.

Silicones are effectively incompressible, therefore we

can assume a constant volume:

l1l2l3=1

The actuator’s radial size does not change either because

of the reinforcement helices. We therefore set the circumfer-

ential stretch l

= 1, and obtain the relationship

l2=l1

This deformation is also called pure shear. Using these

relations, both J

and J

reduce to

J2=J1=l2

1+l2

12ð5Þ

and because of the incompressibility assumption:

J3=0

Simplifications and limitations. To keep the model sim-

ple, many potentially important effects were not included.

We assume a uniform strain energy density within the rub-

ber hull, which is acceptable for moderately thin rubber

hulls (i.e. d\0.5h). This assumption is modeled and dis-

cussed in Section B.6.

We use a neo-Hookean material model. This ignores

higher-order deformation effects. The error is less than

3.4% though, as discussed in Section B.7.

We ignore material stiffening. The consequences are dis-

cussed in Section B.8. The resulting error is typically less

than 7.7%.

Finally, we also ignore the side walls, i.e. hull parts of

the actuator which are stretched only at fractions of l

. The

ramifications are discussed in Section B.9.

The model can also be invalidated by compressive forces

applied externally. They remove fiber tension and therefore

usually lead to buckling. This limits the usefulness of the

model for simulation and planning. The main purpose is to

provide simple equations for designing actuator behavior

though. Here the limitations are acceptable.

B.2 Statically stable actuator configurations

Using the minimum potential energy principle we can

derive statically stable configurations for the actuator First,

we need to define all relevant forms of work in the system:

gas compression;

elastic rubber deformation;

work added by an external load.

The total potential energy of the actuator is

W=Wair +Wsil +Wload ð6Þ

Equilibrium is reached, when the gradient of work is

zero, which is in our case very simple as we will describe

the actuator state with the single variable l

Dl1

The remainder of this section derives each work compo-

nent from the definitions of the previous section and yields

the equation for statically stable configurations.

Gas compression work. For computing the gas com-

pression work, we need to differentiate between two differ-

ent control regimes.

In the mathematically simpler case, pressure is held con-

stant under deformation: p(V)=p. This is either done

actively by using a control system or passively by using a

big reservoir connected to the actuator volume, which

attenuates the effect of volume change within the actuator

on pressure.

In the second, more common case the total enclosed gas

mass in the system is held constant e.g. when using pneu-

matic valves. Pressure changes according to the ideal gas

equation: p(V)=nRT 1

V. A closed gas volume increases

the stiffness of the actuator slightly. But for the sake of

brevity, the latter case will not be derived here.

In both cases, the work done by changing the volume of

a gas from V

to V

Wair =WV1ð

p(V)dV

In the case of constant gas pressure, we obtain a simple

equation:

Wair =W0p(V)

dWair

dl1

=pdV

dl1

dWair

dl1

=p1

2Vact

ð7Þ

Rubber deformation work. The deformation work of the

silicone rubber W

sil

is modeled as a neo-Hookean solid

model with coefficient C10 =G

Deimel and Brock 179

Wsil =ð

Vsil

2J1dV

Gis the material’s shear modulus, and V

sil

the volume of the

silicone. Choosing this simple model over more complex

onesisdiscussedinSectionB.7.Wewillfurtherassumea

uniform deformation, and thus uniform strain energy density

(i.e. uniform J

) throughout the hull, as justified in Section

B.6. We can then calculate the total strain energy as

Wsil =G

2J1Vsil

Using Equation (5), we can express the work gradient

with respect to stretch l

Wsil =G

2(l2

1+l2

12)Vsil

dWsil

dl1

=Gl1l3



Vsil

ð8Þ

Load work. For a given actuator segment, external load is

applied on the interfaces to the two adjacent segments. By

attaching our frame of reference to one interface, load work

can be computed by only considering the motion and force

of the other interface.

The load work can further be split up into the work done

by translatory forces and rotary moments. Translatory

forces are transmitted by the inelastic fibers of reinforce-

ment helix and passive layer. If we assume completely

inelastic fibers, those forces do not contribute any work.

Rotations, on the other hand, do contribute work, and we

can integrate the contribution along the bottom layer:

Wload =ZMuðÞdu

For the model, it is more convenient to integrate over x

instead of ualong the actuator segment. We can rewrite the

integral to

Wload =ZMuxðÞðÞ

∂uxðÞ

∂xdx

For short enough actuator segments (small x) we can

assume constant, averaged moment along the whole seg-

ment, i.e. M(x)=M. In addition we can substitute the deri-

vative of Equation (3) for ∂uxðÞ

∂x:

Wload =ZMl11

hdx

=Ml11

hx+W0

From this equation, we can compute the work gradient

with respect to stretch l

dWload

dl1

=Mx

hð9Þ

Minimum total potential energy. By computing the local

minimum of Equation (6) with respect to l

and substitut-

ing with Equations (7), (8) and (9), we obtain the equation

describing stable actuator states:

dWair

dl1

dWsil

dl1

dWload

dl1

0=pVact

2+Gl1l3



Vsil +Mload x

ð10Þ

B.3 Stiffness

The Stiffness of an actuator segment is expressed by the

change in moment M

load

with respect to curvature k. From

Equation (10) we can compute M

load

it explicitly:

Mload =h2z

2pl1l3



Ghdz

As we assume a constant pressure regime, the first term

vanishes when differentiating:

dMload

dl1

=Ghdz(1+3l4

We then substitute with the derivative of Equation 2

which is ∂l

=h∂k. In addition, we can set z=h, as the

width of the actuator is usually approximately its height.

We arrive at the equation describing the stiffness of the

actuator given its shape and deformation:

dMload

dk =h3Gd1+3l4



The last, nonlinear term predicts a strong stiffening

when a PneuFlex actuator is straight or even negatively

curved. This sudden stiffening is indeed observed with

actual actuators. But they also tend to buckle under such

loads.

Scaling law. When scaling an actuator, the ratios d

hand z

stay constant. The equation then becomes:

dMload

dk =h4Gd

h1+3l4





Stiffness therefore scales to the fourth power of actuator

size. This gives us a powerful lever to adjust the strength of

an actuator.

Non-squared cross-sections. Because the stable config-

uration of the helical thread from the applied uniform radial

pressure is a circle, the cross-section will always deform

into one given high enough air pressure.

It therefore makes sense to use the actuator circumfer-

ence cinstead of height and width to compute actuator

stiffness:

dMload

dk =c



GdðÞ1+3l4



180 The International Journal of Robotics Research 35(1–3)

For designing actuator stiffness, we can approximate the

nonlinear term with 1:

dMload

dk =c



GdðÞ ð11Þ

B.4 Actuation ratio

The actuation ratio dk

dpis the change of curvature given an

increase in pressure while assuming zero load. It can also

be computed from Equation (10):

p=l1l3



Gd

dl1

dp=1

1+3l4

ðÞ

Gd

ð12Þ

Using Equation (2) we can substitute l

and dl

. We obtain

the actuation ratio

dp=

1+31+hkðÞ

41

Gdð13Þ

So according to the simple model, the actuation ratio is

inversely proportional to the silicone’s shear modulus, and

its thickness. The actuation ratio also has a nonlinear com-

ponent with respect to the actual curvature. At small curva-

tures the actuation ratio is considerably lower than at higher

curvatures. The nonlinearity can be linearized though by a

non-circular cross-section, as discussed in Section B.5. For

actuator design we can conveniently drop the nonlinear

term and arrive at a simple design rule for computing the

inverse actuation ratio:

’

Gdð14Þ

Note that this equation is independent of both hand z

(and, therefore, also circumference c). The actuation ratio

is not dependent on the size of the actuator cross-section!

We can therefore set an actuation ratio profile using thick-

ness and shear modulus of the rubber hull, and then set the

stiffness profile with

dMload

dk =c



1

1+3l4

ð15Þ

B.5 Justifying simplification: Linear actuation

ratio

Equation (12) contains the nonlinear term l1l3



. The

model therefore predicts pressure to be nonlinearly related

to actuator curvature. Interestingly though, the actuation

ratio can be linear at moderate pressures, as was observed

by Deimel and Brock (2013). We believe that this is caused

by a non-circular cross-section.

When pressure increases, the helical thread tensions and

always deforms the actuator cross-section into a circle. But

until then, the actuator expands in three dimensions instead

of one, which increases the strain energy in the rubber hull

faster with respect to stretch l

. The effect is more pro-

nounced with less circular cross-sections, but can also be

elicited by a loosely wound helical thread. We can investi-

gate the resulting effects by introducing a correction factor

for the strain energy gradient:

Dl1

ðÞ=(1k0)l11ðÞ

l11ðÞ

2+k2

+k0ð16Þ

The factor k

’[1.5.0] defines the relative increase

of the gradient for a deflated actuator, while the factor

’0.025 determines at which elongation the effect is

halved. The latter is probably dependent on the wall’s thick-

ness and rubber stiffness. The equation was chosen because

unlike simpler models the correction factor has a limited

range, a finite integral, and no poles.

Then D(l

) is plugged into Equation (12):

p=D(l1)l1l3



2Gd

The impact of different k

on actuation ratio is illustrated

in Figure 20. The actuator can behave almost linearly at

higher pressures. An interesting application of this effect

may be to simplify actuator control.

B.6 Justifying simplification: Uniform strain

energies within rubber hull

In the model we assume a uniform strain energy density

throughout the hull. This needs to be checked though, as

the principal stretches are not uniform at all.

Formalization. For an infinitesimal volume of elasto-

mer within the hull, we define dto be the radial distance of

the volume from the outer boundary of the hull in an unde-

formed actuator, while d0denotes the actual radial distance

in the deformed actuator.

When the actuator bends, the volume moves radially

outwards as the wall thins. This motion influences the

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Normalized pressure p·h

2·G·d

1.00

1.05

1.10

1.15

1.20

1.25

λ1

model

k1=0.02

k1=0.03

k1=0.04

linearization

Fig. 20. A non-circular cross-section effectively linearizes the

actuation ratio at moderate curvatures.

Deimel and Brock 181

stretches encountered. As the radial displacement is limited

by the helical reinforcement fibers, we can assume the

outer boundary of the rubber hull to not move radially at

all.

Due to the smaller radius the longitudinal stretch is

l1=1+khkd0

ðÞ ð17Þ

The circumferential stretch can be calculated by the

radial displacement given a position d0with respect to the

undeformed position d:

dl3=

phd0

ðÞ

phdðÞ

phd0dd0

ðÞ

phdðÞ

=dd0

hd

)l3=d0

hd+C

Setting the boundary condition l

= 1 at the outer

boundary of the hull yields

l3=hd0

hd

Finally, we can derive l

by using the incompressibility

assumption l

= 1 and get

l2=hd

kh2+kd022kh+1ðÞd0+h

To normalize our calculations, we can express the prin-

cipal stretches in terms of the dimensionless ratios d0

h,d

h,and

kh:

l1=1+kh1d0



l2=

1d

kh+khd0



22kh+1ðÞ

h+1

l3=

1d0

1d

With l

we can express the relation between d0and das

a linear differential equation using l

as the gradient of

displacement:

dd0

dd=l2

dd0

dd=

1d



2khd0

h2kh+1ðÞ+1+kh

This differential equation determines the position d0of a

packet of rubber, and therefore its deformation.

Strain energy distribution. To analyze the strain energy,

the differential equation was solved numerically. The

boundary conditions were set on the outer boundary of the

hull at d0=0tol

= 1 and l3=1

l1.

Figure 21 shows the strain energy density with respect

to the normalized depth d

hwithin the hull and at different

normalized curvatures kh. We can see that the strain

energy density stays surprisingly flat even for moderately

thick hulls and at very strong curvatures.

The reason for this surprising result can be understood

when looking at the principal stretches when the rubber

moves outwards radially (thinning the wall).

Circumferential stretch l

increases, but at the same time

radial stretch l

decreases as the packet gets more com-

pressed. Also l

decreases with the distance from the bot-

tom layer.

The analysis shows that as long as the hull thickness is

less than half the actuator height, we can assume a uniform

strain energy distribution for modeling. Staying below this

limit also avoids material fatigue of the rubber on the inside

of the air chamber.

B.7 Justifying simplification: Neo-Hookean

deformation model

An alternative to the simple neo-Hookean deformation

model used in our model is the generalized Mooney–Rivlin

model for incompressible hyperelastic materials. It states a

polynomial approximation of strain energy density, using

the strain tensor invariants:

dv =X

i,j=0

Cij Ji

1Jj

Silicone rubbers have rather small coefficients for C

and C

though. The coefficients published by Meier

et al. (2005) have the following relations:

C01’

50 C10

C20’

500 C10

C02’0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

radial position d

0.0

0.2

0.4

0.6

0.8

κ·h=0.5

κ·h=0.4

κ·h=0.3

κ·h=0.2

κ·h=0.1

κ·h=0.0

Fig. 21. Strain energy density distribution within the rubber hull

for different normalized curvatures.

182 The International Journal of Robotics Research 35(1–3)

Ignoring the coefficients C

and C

(i.e. setting

them to 0) yields the neo-Hookean model.

We can bound the error to strain energy when assuming

a maximum stretch of l

\3 (which relates to an actuator

bending at the radius of half its height), which given

Equation (5) bounds the strain tensor values to

J1\7:11

J2\7:11

The terms dropped from the Mooney–Rivlin model are

bounded to

C01 J2\0:02 C10 J1

C20 J2

1\0:014 C10 J1

which results in a total error of less than 3.4%.

B.8 Justifying simplification: Ignore material

stiffening

Elastomers exhibit a stiffening at large stretches. This is

modeled by augmenting the strain energy function with an

additional parameter. The Gent model (Gent, 2012) aug-

ments the strain energy function with a logarithmic term:

W=G

2J1lmax

ðÞ

ln 1J1l1

ðÞ

J1lmax

ðÞ



Vsil

2Jmln 1J1



Vsil

With the additional material parameter l

max

which is the

stretch where the material exhibits unlimited stiffness.

The rubber used for the PneuFlex actuator typically has

max

’10, which results in

Jm’100

When computing the strain energy gradient using the

Gent model, we obtain

dWGent

sil

dl1

=C1JmVsil 1

1l2

1+l2

12

2l12l3

1l2

1+l2

12

dWsil

dl1

For plausible values of l

\3andJm = 100, the

ignored stiffening factor can be bounded to

dWGent

sil

dl1

\1:077 dWsil

dl1

So ignoring the material stiffening introduces an error of

less than 7.7%.

Fig. 22. Example of four fingers with different actuation ratios, inflated to 46.3 kPa. The overlaid circle segments indicate the

constant curvature along the bottom layer.

0 20 40 60 80 100

Pressure [kPa]

100

120

140

160

180

Tip orientation [

]

2.5 mm

3.5 mm

4.5 mm

5.5 mm

3.5 (model)

4.5 (model)

5.5 (model)

Fig. 23. Tip orientation versus inflation pressure for different

rubber hull thicknesses. Dotted lines indicate model estimates

based on the 2.5 mm measurement.

Deimel and Brock 183

B.9 Justifying simplification: Ignoring the side

walls

For analyzing the impact of ignoring the side walls, we can

conceptually split the cross section of a real PneuFlex

actuator into many small parts, of which each behaves

according to the simple model.

As there is no air below the side walls (only more rub-

ber), there is no additional force applied by gas pressure.

The amount of deformation work dWsil

dl1does increase, but

the J

also drops off quadratically when approaching the

bottom layer. By using a constant ratio between the thick-

ness of the side wall and the top side of the actuator, the

error being made when computing the actuation ratio can

be made constant.

For the stiffness, the side walls play even less of a role,

as it scales with h

B.10 Experimental validation

To validate the design rules we developed, we conducted

two experiments. The first one validates the equation for

the actuation ratio, the second the equation for actuator

stiffness.

B.10.1 Actuators with varying d

The model predicts that the actuation ratio is only depen-

dent on hull thickness dand the shear modulus of the rub-

ber. To test this, we built four actuators with the same shape

but with a dof 2.5 mm, 3.5 mm, 4.5 mm and 5.5 mm,

respectively.

Figure 22 shows the actuators at a pressure of 46.3 kPa.

Despite the change in height and width, the curvatures of

the bottom layers are constant along the actuators, as indi-

cated by the circle segments. We can therefore validate the

model’s prediction that height and width do not influence

actuation ratio. Please note that the circle segments are

placed on top of the edges of the bottom layers, where the

fibers of the embedded fabric are most stressed.

Figure 23 shows the relationship between pressure and

fingertip orientation, which is an aggregated measure of

the curvature along the actuator. The curves for each

actuator show the nonlinear behavior predicted by

Equation (13), i.e. an increase in actuation ratio towards

higher curvatures.

The dotted lines in Figure 23 indicate the actuation ratio

when scaling the measurements of the thinnest actuator

(2.5 mm) to the thickness of the other actuators according

to our model. Model and measurement agree well for

3.5 mm and 4.5 mm. At 5.5 mm it is clearly visible that

the nonlinearity of the actuation ratio increases, making the

actuator stiffer than expected at low pressures.

B.10.2 Actuator with linear stiffness

To validate Equations (11) and (15), we can apply a force

at the tip of an actuator. The actuator has a constant actua-

tion ratio along its main axis. The contact force will create

a bending moment to segments of the actuator that

increases linearly with the segment’s distance from the con-

tact point. At small curvatures, i.e. an almost straight fin-

ger, the distance along the bottom layer will approximate

the Euclidean distance well. For a straight actuator, we can

therefore assume the moment along the actuator to be

increasing linearly. If the actuator has a linear stiffness pro-

file along the actuator and a constant actuation ratio too,

then the load moment and actuated moment will cancel out

and yield a constant curvature. The constant actuation ratio

is demonstrated in Figure 22.

Figure 24 shows an actuator at four different pressures.

The curvature stays constant during a large range of infla-

tion pressures, with the highest pressure corresponding to

about a 360°rotation if there was no contact. The actuator

therefore has a linear stiffness profile, which validates

Equation (11).

The upwards rotation of the whole finger relative to the

fixture is caused by the rubber cap which closes the air

chamber at the base of the finger. This also causes the small

curvature changes between different pressures. Figure 24(d)

shows a slightly stronger curvature close to the tip. This

may be caused by difference between contact point location

and the point where the stiffness profile reaches zero stiff-

ness, which is exactly at the tip.

Fig. 24. The finger is blocked by a fingertip contact while being inflated. An actuator with a linearly decreasing stiffness profile will

show a constant curvature along the actuator under such load.

184 The International Journal of Robotics Research 35(1–3)

B.11 Summary

The presented theoretic model yields two simple design

rules (Equations (14) and (15)) for designing actuation

ratios and stiffnesses along a PneuFlex actuator. We also

validated the scaling behavior of the model in two

experiments.

Based on the analysis, we can give the following recom-

mendations for choosing geometric parameters.

Active layer thickness should stay less than half the

actuator height: d

h\0:5.

To achieve a minimal bending radius of 1.5 times

actuator height, the rubber should have an elongation

at break of at least 500%.

Material stiffening can usually be ignored with silicone

rubber, the error is typically less than 7.7%.

Deimel and Brock 185