A Novel Type of Compliant and

Underactuated Robotic Hand for

Dexterous Grasping

Raphael Deimel Oliver Brock

August 24, 2015

Abstract

The usefulness and versatility of a robotic end-effector depends on the diversity of grasps it canaccomplish and also on the complexity of the con-trol methods required to achieve them. We believethat soft hands are able to provide diverse and robustgrasping with low control complexity. They possessmany mechanical degrees of freedom and are able toimplement complex deformations. At the same time,due to the inherent compliance of soft materials, onlyvery few of these mechanical degrees have to be con-trolled explicitly. Soft hands therefore may combinethe best of both worlds. In this paper, we presentRBO Hand 2, a highly compliant, underactuated,robust, and dexterous anthropomorphic hand. Thehand is inexpensive to manufacture and the morphol-ogy can easily be adapted to specific applications. Toenable efficient hand design, we derive and evaluatecomputational models for the mechanical propertiesof the hand’s basic building blocks, called PneuFlexactuators. The versatility of RBO Hand 2 is evalu-ated by implementing the comprehensive Feix taxon-omy of human grasps. The manipulator’s capabilitiesand limits are demonstrated using the Kapandji testand grasping experiments with a variety of objects ofvarying weight. Furthermore, we demonstrate thatthe effective dimensionality of grasp postures exceedsthe dimensionality of the actuation signals, illustrat-ing that complex grasping behavior can be achievedwith relatively simple control.

Figure 1: The RBO Hand 2 is a compliant, underac-tuated robotic hand, capable of dexterous grasping.It is pneumatically actuated and made of silicone rub-ber, polyester fibers, and a polyamide scaffold.

1 Introduction

Dexterous grasping is a prerequisite for task-dependent manipulation. By the term dexterous, werefer to the postural variability of the hand: thehigher this variability, the more dexterous we con-sider a hand (for examples of grasping postures, referto the grasp taxonomies presented in Cutkosky [1989]and Feix et al. [2009]). Such variability enables ver-satile grasping and manipulation: Small objects canbe picked up with pincer grasps, large objects withenveloping power grasps. Depending on the task, acylindrical side-grasp can be used to pick up a glassfor drinking, whereas a disk grasp from above is ap-propriate to lift it off a cluttered table.

In robotic hands, dexterous grasping capabilitiesare traditionally realized through complex, multi-jointed structures and sophisticated actuation mech-anisms. Such hands are expensive and difficult to

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design. They also require complex sensing and con-trol. Recently, underactuated hands with passivelycompliant parts have become a popular alternativein robot hand design. These hands perform cer-tain grasps robustly, have simpler mechanics, and re-quire simpler control due to underactuation. How-ever, there is one commonly assumed drawback ofcompliant hands: underactuation and passive com-pliance seem to render dexterous grasping difficult oreven impossible. The experiments performed withour novel hand indicate the opposite.

We present a novel type of compliant and underac-tuated hand based on soft robotic technology. Thishand, called RBO Hand 2, is capable of dexterousgrasping, it is easy to build, robust to unanticipatedimpact, inherently safe, low-cost, and easy to con-trol. These advantages are achieved by building al-most the entire hand out of soft, compliant materialsor structures, rather than of rigid parts. We believethat the combination of dexterous grasping capabilityand ease of manufacturing make the presented handwell-suited for enabling novel advances in graspingand manipulation.

Our design, shown in Fig. 1, purposefully maxi-mizes the hand’s passive compliance, while ensur-ing sufficient structural support to lift objects. Webelieve that this design choice is critical for robustgrasping: First, passive compliance facilitates obtain-ing force closure in power grasps [Dollar and Howe,2010, Deimel and Brock, 2013]. Second, passive com-pliance facilitates the use of contact with the envi-ronment to aid attaining a grasp. This strategy—the exploitation of environmental contact to reduceuncertainty—has been shown to increase grasp per-formance in humans as well as robots [Deimel et al.,2013, Kazemi et al., 2014]. For these reasons, passivecompliance is a key ingredient for robust grasping. Inthis paper, we want to show that in addition to thetwo aforementioned advantages, passively complianthands can also perform dexterous grasping. Indeed,our results indicate that passive compliance even fa-cilitates dexterous grasping.

An opposable thumb is important to achieve dex-terity in human and robotic hands. We evaluate thethumb dexterity of the RBO Hand 2 using the Ka-pandji test [Kapandji, 1986]. This test is commonly

used to evaluate thumb dexterity in human hands af-ter surgery. In addition, we show that the hand iscapable of enacting 31 out of 33 grasp postures ofthe human hand from the comprehensive Feix taxon-omy [Feix et al., 2009]. We evaluate the space of handposture exhibited by humans and the RBO Hand 2which we find to be similar. We also show thatfour actuation degrees of freedom suffice to achievea postural space with more than these for dimen-sions. This implies that the variability in graspingposture is only partially generated by the hand’s ac-tuation. The remaining variability is the result ofinteractions between hand and object. These interac-tions, we claim, are greatly simplified and enriched bythe extensive use of passive compliance in the hand’sdesign. These results indicate that dexterous grasp-ing is easier to achieve with passively compliant thanwith traditional, stiff-linked hands.

This paper extends our previous work [Deimel andBrock, 2014] in several important ways. In Sec-tion 6.3, we extend the analysis of the hand’s dex-terity by comparing its postural diversity to that ofthe human hand, given a set of diverse grasps. Thisis important, as it provides support for the state-ment that compliance enhances dexterity and ensuresthat the RBO Hand 2 has as diverse postures asthe human counterpart. We also present and vali-date a novel model for the actuator’s behavior in Ap-pendix A. This enables the validation of novel actua-tor designs prior to manufacturing. Furthermore, wesignificantly extended the description of the hand de-sign and production process in Section 4. Finally, wepublish the experimental data sets, videos, and highresolution images of the human and robotic experi-ments related to the Feix taxonomy in MultimediaExtensions 1 to 6.

2 Related Work

Many highly capable robotic hands exist. A his-torical overview, surveying robotic hands from overfive decades, provides an excellent overview [Con-trozzi et al., 2014]. An analysis of robot hand designswith respect to grasping capabilities was recently pre-sented by Grebenstein [2012]. As the notion of com-

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pliance is central to our hand design, we will limit ourdiscussion to hand designs that deliberately includethis concept.

We distinguish between two types of hands, ac-tively and passively compliant. The former can beachieved by using active control on fully actuatedor even hyper-actuated systems, where every degreeof freedom can be controlled. Examples are the im-pressive Awiwi hand [Grebenstein, 2012], the Shad-owRobot Shadow Dexterous Hand, and the SimLabAllegro Hand [Bae et al., 2012]. These hands achievedexterity and compliance through fast and accuratecontrol, which comes at the price of mechanical andcomputational complexity. As a result, these handstend to be mechanically complex and expensive tomanufacture. Mechanical complexity can also in-crease the probability of hardware failure.

The alternative is to make hands passively compli-ant by including elastic or flexible materials. Build-ing a passively compliant joint is much cheaper thanbuilding an actively controlled one in terms of costs,spatial volume, and mechanical complexity. Passivecompliance limits impact forces, a crucial propertyfor an end-effector designed to establish contact withthe world. More degrees of freedom can better adaptto the shape of an object greatly enhances grasp suc-cess and grasp quality. At the same time, the handcan be underactuated, effectively offloading controlto the physical embodiment of the hand.

A pioneering work in grasping with passive com-pliance was the soft gripper by Hirose and Umetani[1978]. Recently, a whole range of grippers andhands were built using passive compliance: the FRH-4 hand [Gaiser et al., 2008], the SDM hand and itssuccessor [Dollar and Howe, 2008, Ma et al., 2013,Odhner et al., 2014], the starfish gripper [Ilievskiet al., 2011], the THE Second Hand and the Pisa-IIT Soft Hand [Catalano et al., 2014], the ISR-SoftHand [Tavakoli and Almeida, 2014], the Pos-itive Pressure Gripper [Amend et al., 2012], theRBO Hand [Deimel and Brock, 2013], and theVelo Gripper [Ciocarlie et al., 2013]. A differentsource of inspiration was taken by Giannaccini et al.[2014], who built a compliant gripper inspired by theoctopus arm.

The practical realization of underactuated hands

is matched by theoretical approaches to analyze andevaluate their dexterity [Prattichizzo et al., 2012,Gabiccini et al., 2013]. The most promising mod-els rely on the hypothesis of a low-dimensional rep-resentation of grasp postures, called synergies [San-tello et al., 1998]. Odhner et al. [2014] simplify theunderactuated hand mechanics into compliance el-lipsoids at possible locations of contact points. How-ever, these approaches require accurate knowledge ofgrasp posture, contact point locations and contactforces. Given current sensor technologies, this infor-mation is difficult to obtain. Interestingly, humansare able to grasp under comparable conditions withstrongly impaired perception, e.g. with blurred sightand wearing a glove [Deimel et al., 2013]. This sug-gests that there is an alternative, perceptually lessdemanding representation of compliant behavior.

The inclusion of compliance into the design ofrobotic hands has led to significant improvements inperforming power grasps. Very little work has ex-amined the effect of compliance and underactuationon the dexterity of a robotic hand. Closing this gapwill be the focus of this paper. Tavakoli et al. [2014]recently characterized the influence of reducing thenumber of actuated degrees of freedom on the num-ber of possible grasps for the ISR-SoftHand. Thisanthropomorphic hand relies on extensive complianceusing elastomeric joints and deformable finger pads.They found that about four to six actuated degreesof freedom are enough to enact a broad set of hu-man grasps and that an opposable thumb is crucialto achieve this, which corroborates our own findings.

3 PneuFlex Actuators

The RBO Hand 2 uses a highly compliant, pneumaticcontinuum actuator design, called PneuFlex, whichwas first presented in Deimel and Brock [2013]. Pneu-Flex actuators can be manufactured within a day anduse materials that are cheap and non-toxic. Fig. 2 il-lustrates the working principle. When inflating thecontained chamber with air the pressure forces thehull to elongate along the actuator. The bottomside contains an inelastic fabric to prohibit elonga-tion. This causes a difference in length between the

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(a) An inflated PneuFlex ac-tuator in front of deflatedones.

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

Figure 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 ofthe finger, into which an inelastic fabric is embedded. The helically wound threads stabilize the actuatorshape and relieve the rubber of nonfunctional strains, i.e. the inflation leads to bending rather than to radialexpansion.

top and bottom side and the actuator bends. Ra-dial fibers stabilize the actuator’s shape and greatlyincrease the attainable curvature.

Manufacturing Actuators A distinguishing fea-ture of the PneuFlex actuator is the integrated designpipeline, which enables rapid prototyping of actua-tors with widely varying properties without changingthe production process. The steps of the process areillustrated in Fig. 3.

To create an actuator from scratch, first a set ofplanes is defined along a line or curve, on whichthe local cross section of the actuator is defined.The principal shape parameters for each cross sec-tion (height, width, hull thickness) are determinedfrom the desired actuation ratio and stiffness at eachpoint along the actuator, e.g. by using the model pro-vided in Appendix A.

In the second step, the set of cross sections is trans-lated into a 3D model of a two-part mold for castingthe rubber body of the actuator. The model is pro-duced on a 3D printer. Because we need to separatethe mold from the cast, the bottom side of the actu-ator is not included.

In the next step, the top part is cast using theprinted mold and addition-cure silicone.

After unmolding, a silicone tube is inserted at aconvenient position into the top part and bonded toit. The tube enables us to easily connect the actuatorto the pneumatic control.

Afterwards the air chamber is closed by placing thetop part on a thin (1− 2 mm) sheet of freshly castsilicone that embeds a bendable but inextensible PETfabric.

When the bottom layer has cured, a sewing threadis wound around the actuator in form of a doublehelix. To fixate the thread in place, a thin layer ofaddition-cure silicone is applied to the top and bot-tom side. This step finishes the actuator.

The presented process enables us to freely changewidth and height of the actuator and the thickness ofthe rubber hull. We can also create straight as wellas curved actuators, as long as the bottom layer staysin a plane.

The actuator design space can be explored by vary-ing shape and size of the actuator, and by varying thethickness of the silicone hull at the top, side and bot-tom. Additionally, available silicone types let us varythe shear modulus by an order of magnitude. All ofthese parameters affect the bending behavior, stiff-ness, and limits of the actuator. If needed, a bellows-shaped hull extends the design space towards larger

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curvatures and lower stiffnesses. Curved actuatorscan realize three-dimensional motion. For example,the two palm actuators (actuators 6 and 7 in Fig. 4)are used to achieve thumb opposition. Differential in-flation of the two actuators provides additional dex-terity.

Actuators can also be packaged together, similarlyto muscle fibers, allowing for redundant actuation orvariation of the actuation strength. Joining actuatorsalso enables the implementation of multiple deforma-tion modes or the deliberate mixing of deformationmodes [Bishop-Moser et al., 2012].

Besides their flexible production process, PneuFlexactuators 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.

Modeling Actuators The PneuFlex actuatorshares many properties with other recently publishedcontinuum actuators, most notably the fast PneuNetactuators [Mosadegh et al., 2014] and the actuatorby Galloway et al. [2013]. In contrast to these actua-tor designs, the PneuFlex design and production pro-cess is optimized to provide freely adaptable cross sec-tions which determine actuation ratio and stiffness,the ability to include multiple separate air chambers,and to provide access to its internal space for theintegration of sensors and wiring.

We provide a detailed analysis of basic actuator be-havior in Appendix A and also propose design rulesfor successful actuator design in Section A.11. Ad-ditional insights can be drawn from related researchon continuum actuators. For example, Bishop-Moseret al. [2012] characterize all basic motions attainableby changing inclinations of the reinforcement helices.Others proposed approximate numeric models basedon twisted, one dimensional beams [Renda et al.,2012, Giorelli et al., 2012].

4 Hand Design

In this section, we describe the components of our softanthropomorphic hand (RBO Hand 2, see Fig. 1).The entire hand weighs 178 g and can carry a pay-load of up to 0.5 kg. Higher payload can easily be

Figure 4: The seven actuators of the soft anthropo-morphic hand: four fingers (1–4), thumb (5), and thepalm, consisting of two actuators (6, 7)

achieved, if necessary, as we will explain in Section 7.

Morphology The design space of possible handsis very large. For this hand, we chose an anthropo-morphic design in shape and size for three reasons.First, we know the human hand form enables dexter-ous grasping in humans. By starting with a human-hand-like morphology, we start with a proven handdesign. Second, many objects have been built formanipulation by a human hand and match the an-thropomorphic form factor. Third, we can use well-established grasp taxonomies and compare our de-signs with humans and many other robotic hands.

Control Pneumatic control of the PneuFlex actu-ators is based on a simple linear forward model forcomputing valve opening times. The model takes intoaccount the regulated supply pressure to achieve a de-sired channel pressure which corresponds to a desiredbending radius or contact force. Alternatives to thisdigital control are cylinder-based continuous controlsystems [Marchese et al., 2014]. Renda et al. [2012]demonstrate a computationally simple forward model

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Figure 3: Production steps for making a PneuFlex actuator with a custom stiffness and actuation ratioprofile.

Figure 5: Illustration of the finger geometry and itsprincipal parameters: κ (local curvature), M (mo-ment around the passive layer at the interface), d(top side rubber thickness) and c (circumference)

for an artificial octopus arm. This model can also beused for PneuFlex actuators.

For the experiments, control was implemented withindustry-grade air valves and a separate air supply.For a mobile system, the control system can easily beoptimized for size and weight, because the requiredair flows are an order of magnitude smaller than pro-vided by industrial grade valves. For the same reason,pressurized air can be effectively sourced by smallcompressors and small tanks. In systems where elec-tric energy is scarce, high-pressure tanks can providestorage of air with high energy density [Wehner et al.,2014].

Fingers The five fingers of our hand are singlePneuFlex actuators (see Fig. 4). The index, middle,ring, and little finger are 90 mm long and of identicalshape, the thumb actuator is 70 mm long. All fingersget narrower and flatter towards the finger tip. Byusing actuators as fingers, we can exploit the excel-lent compliance and robustness of the actuators andgreatly simplify the design.

The mechanical behavior of the finger can be de-

scribed by the local curvature κ and torsional stiff-ness M around the bottom side for short segmentsof the actuator and is determined by the geometry ofthe actuator cross section (see Fig. 5). Appendix Acontains an analysis of this simplified model of theactuator, which provides surprisingly simple rules todesign the ratio of curvature κ to pressure p at eachsegment along the actuator by varying the hull thick-ness d:

∆κ

∆p≈ 1

G · d,

where G is the shear modulus of the rubber and con-stant within an actuator.

Translational forces between actuator segments aremainly transmitted by the inelastic fabric of the bot-tom layer and therefore don’t need to be considered.The model in Appendix A also yields a rule of thumbfor the torsional stiffness of an actuator segment.For an approximately squared cross section, stiffnessscales with circumference c:

∆M

∆κ≈ G · d · c3.

As an example, for the fingers of the hand we choseto increase stiffness linearly from tip to base and keepthe actuation ratio constant. Such a profile has alsobeen used in the Soft Gripper [Hirose and Umetani,1978]. Using x as the distance from the tip along theactuator, the cross section at this point is defined by:

d(x) = const,

c(x) ∝ x13 .

The resulting geometry of the actuator is illus-trated in Fig. 5. According to our model, the aspect

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ratio of the cross section does not strongly influencestiffness and actuation ratio. So to give the finger avisually more appealing shape, we set set the widthto:

width(x) ∝ x 18 .

Palm A key feature of the human hand is the op-posable thumb. We realize it in our hand by actu-ating the palm (see Fig. 4). The palmar actuatorcompound consists of two connected actuators. Itsbase shape is a circular section of 90◦ with 78 mmouter and 25 mm inner radius. The actuator curvesperpendicular to the passive layer. The stiffness aswell as the actuation ratio remain constant along thecurved actuator. They are also designed to be twiceas stiff as the fingers to account for the fact that thetwo actuators in the palm oppose four fingers. Fig. 8provides an impression of the possible thumb motionswhen the two palm actuators are inflated either to-gether or differentially.

In addition to enabling thumb opposition, the palmalso provides a compliant surface that, together withthe fingers, is used to enclose objects in various powergrasps. To augment this function, the fingers and thepalmar actuator are connected by a thin sheet of fiberreinforced silicone, covering the gap between palmactuators and fingers (shown in Fig. 1, but absent inin Fig. 4). This sheet transmits tensile forces betweenfingers and palm, and between adjacent fingers. Thisstabilizes the underlying scaffold during power graspsor for heavy loads, as shown in Fig. 11.

Thumb Like the other fingers, the thumb consistsof a single PneuFlex actuator. The thumb is shorterand twice as stiff, but also features a linear stiffnessprofile. A faithful imitation of how humans use theirthumb would require a negative curvature close tothe tip, as shown in Fig. 6, and would significantlyincrease complexity of manufacturing the thumb. Wetherefore deviated here from the human hand. In-stead of the inside of the thumb, we use the backside(dorsal side) as the primary contact surface for pincergrasps. This effectively changes the contact surfaceorientation by about 45–60◦, relative to the orienta-tion found in a human thumb, avoiding the need for

Figure 6: Difference in thumb configuration and fin-gertip use during a pincer grasp between a humanhand and the robotic hand

negative curvatures. As both sides of the PneuFlexactuator have similar surface characteristics (unlikehuman thumbs), this choice will not affect grasp qual-ity.

Scaffold The fingers and the palm are connectedto the wrist by individual, flexible struts as part ofa 3-D printed polyamide scaffold (2 mm thick, seeFig. 1). The intentionally flat cross section of thestruts enables deformation modes, such as archingthe palm and spreading the fingers. Space for the re-spective actuator is provisioned, but was not addedto the hand described here. The struts decouple dis-placement between fingers, further increasing passivecompliance of the hand. The flexibility of the strutslimits impact forces, while providing sufficient stiff-ness for heavy payloads without excessive deforma-tion (see Fig. 11).

The fingers and the palmar actuator compound arebonded to the supporting scaffold as shown in Fig. 1.The palm is supported by parts of the scaffold to

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increase its torsional stiffness during opposition withthe fingers.

Strength between thumb and fingers A flexi-ble, but inextensible band connects the base of the in-dex finger to that of the thumb (see Fig. 4). Similarlyto a muscle in human hands (adductor Pollicis), itenables increased contact forces between thumb andopposing finger, by reducing torques on the strutsat the wrist. The sheet connecting fingers and palmserves a similar role, especially for power grasps ofcylindrical objects with large diameter.

5 Grasp Dexterity

In this section, we evaluate the dexterous graspingcapabilities of the proposed hand. The most appro-priate evaluation would of course be in full-fledged,real-world grasping experiments. However, this re-quires the integration of hand and control with per-ception and grasp planning and would effectively bean evaluation of the integrated system. Here, we fo-cus on evaluating the capabilities offered by the hand.Furthermore, we have to resort to empirical methods.Accurate simulation of the complex, nonlinear defor-mations encountered in such a heterogeneous and softstructure is difficult to conduct and anyways requiresempirical experiments to validate the results.

5.1 Thumb Dexterity

Medical doctors employ the Kapandji test [Kapandji,1986] to assess thumb dexterity during rehabilitationafter injuries or surgery. This test was also used byGrebenstein for evaluating and improving the thumbdexterity of the Awiwi hand [Grebenstein, 2012]. Forthe Kapandji test, the human subject has to touch aset of easily identifiable locations on the fingers withthe tip of the thumb. These locations are shown inFig. 7. The total number of reachable locations servesas an indicator of overall thumb dexterity. A thumbis considered fully functional if it is able to reach alllocations.

To perform the Kapandji test on our hand, wemanually selected actuation pressures that would po-

Figure 7: The Kapandji test counts the number ofindicated locations that can be contacted with thethumb tip.

sition the thumb as desired. The six most importantpostures of the hand performing the test are shownin Fig. 8. The thumb tip could reach all but one lo-cation. Location 1 was not possible to reach becauseit would require a backwards bending of actuator 5(thumb). Still, the hand scores seven out of eightpoints, indicating a high thumb dexterity.

5.2 Grasp Postures

A common way of assessing the dexterous graspingcapabilities of hands is to demonstrate grasps for a setof objects. For example, the THE Second Hand wasevaluated with four objects and two grasp types [Gri-oli et al., 2012], the SDM hand on ten objects anda single grasp type [Dollar and Howe, 2008], theVelo Gripper on twelve objects and a single grasptype [Ciocarlie et al., 2013], and the Awiwi hand oneight objects and 16 grasp types [Grebenstein, 2012].We follow these examples in our evaluation.

We select grasp types and objects based on themost comprehensive grasp taxonomy to date, the Feixtaxonomy [Feix et al., 2009]. It covers the grasps mostcommonly observed in humans and therefore is a real-istic reference for assessing the dexterity necessary forcommon grasping tasks. The taxonomy encompasses33 grasp types, out of which the first 17 are identicalto the grasps in the Cutkosky taxonomy [Cutkosky,1989]. To demonstrate these 33 grasps, the originalpublication illustrates 17 different object shapes [Feix

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

5: Light Tool reference

(a) Failed Light Tool grasp: no force clo-sure

19: Distal Type reference

(b) Failed Distal Type grasp: functionallydefective

Figure 9: Grasping postures not successfully attainedby the robotic hand

0.0 0.2 0.4 0.6 0.80.0

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Figure 10: Scatter plots of the four actuation chan-nels for the actuation patterns of the 31 successfulgrasps. Darker color indicates overlapping dots.

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et al., 2009]. We therefore used 17 objects and 33grasp types to evaluate our hand.

We implemented the grasps from the Feix taxon-omy by defining appropriate actuation pressures andactuation sequences. When, due to collisions, simul-taneous actuation of all channels was not sufficientto reach the desired posture, we added an appro-priate pre-grasp posture. The commanded actuationpattern was then modified and tested iteratively toimprove the quality of the grasp in terms of graspstability and robustness against external forces, andto ensure the proper types and locations of contact.Grasp quality was judged by manually rotating andtranslating the hand, and by testing several repeti-tions of the actuation pattern.

To simplify the search for appropriate actuationpatterns, we combined the control of the seven actu-ators into four actuation channels. Channel A drivesactuators 1, 2, and 3 (small, ring, and middle fingers),channel B drives actuator 4 (index finger), channel Cdrives actuators 5 (thumb) and 7 (inner palm), andchannel D controls actuator 6 (outer palm). Thesechannels can be understood as the hand’s four grasp-ing synergies.

To perform a grasping experiment for a particu-lar grasp type, the experimenter triggers the actua-tion sequence to attain the pre-grasp posture, holdsthe object in the seemingly most appropriate locationrelative to the hand, and then triggers the actuationsequence for the grasping motion. The resulting pos-tures for each empirical actuation pattern are shownin Fig. 12; high resolution images are provided in Ex-tension 1. Out of 33 grasp types, the hand is able toperform 31 repeatably (three consecutive successfultrials). The two grasps that failed are the light toolgrasp and the distal type grasp.

The light tool grasp fails because the hand doesnot possess finger pulp that fills the cavity formed bythe maximally bent fingers, which causes the objectto slip. The distal type grasp fails because the result-ing grasp is nonfunctional with respect to proper useof the scissors, even though it is possible to put thesoft fingers through the scissors’ holes. Both graspfailures are shown in Fig. 9.

Fig. 10 shows a scatter plot of the actuation pat-terns for the 31 successfully achieved grasp types of

0.54 kg

(a) Cylinder off center

8N

(b) Tolerated disturbance

7N

(c) Tolerated disturbance

6N

(d) Tolerated disturbance

1.65 kg

(e) Support strength

Figure 11: Illustrations of grasping force capabilities:(a) finger strength and palm support strength, (b)–(d) tolerated disturbance forces in different directionsfor grasp 1, and (e) strength of the support providedby the scaffold

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the Feix taxonomy. The actuation patterns relate tofinal grasps, not pre-grasp postures. The plots indi-cate an even distribution of activation for all channelsand do not reveal obvious correlations that could beleveraged to further simplify actuation.

The evaluation presented in this section demon-strates the hand’s ability to assume a variety of grasppostures. This ability is comparable with that ofother hands presented in the literature. We thereforebelieve that dexterous grasping and compliance canindeed be combined in a highly capable, compliant,underactuated robotic hand.

5.3 Grasping Forces

While grasp quality and grasp strength was not thedriving design criterion for the hand, it is importantto verify that a compliant hand is capable of liftingobjects of reasonable weight. To give the reader anintuition on the capabilities of the hand, we providea few tests regarding grasping forces.

The heaviest objects used in the Feix taxonomygrasps were the rectangular plate in grasp 22 (156 g),the metal disc in grasp 10 (181 g), the wooden ballin grasp 26 (183 g), and the circular plate in grasp 30(240 g). Note that in grasps 26 and 30, the shownposture offers the least structural support of possi-ble hand poses. Fig. 11 shows two additional heavyobjects, a wooden cylinder (541 g) and a lead ball(1.650 g). Fig. 11 also shows three different di-rectional disturbance forces on a cylinder which ispower-grasped with grasp 1. If forces above 6-8 Nare applied, the cylinder will slide in the hand.

5.4 Grasping in Realistic Settings

To further illustrate the effectiveness of the pro-posed hand, we performed experiments with com-plete grasping sequences, shown in Fig. 13 and inExtension 2. In these experiments, a human operatorselects the appropriate grasp, triggers the pre-graspposture of the hand, places the hand in the appropri-ate location, and then executes the grasp. These ex-periments demonstrate that the proposed hand, givenappropriate perception and grasp planning skills, isable to perform real-world grasps.

6 Compliance Benefits Dexter-ous Grasping

In the previous section, we showed that our under-actuated and compliant hand is capable of dexterousgrasping. In this section we will investigate whetherits compliance and underactuation are beneficial ordetrimental to attaining different grasp postures. Ifbeneficial, control will be simpler than the result-ing behavior, i.e. actuation space smaller is smallerthan grasp posture space. The dimensionality of theposture space that exceeds the dimensionality of theactuation space can be explained by the compliantinteractions between hand and object.

6.1 Postural Diversity of the Feix Tax-onomy

To assess the dimensionality of the attainable graspposture space, we first have to assert that the graspset that we use to sample from that space is diverseenough, i.e. that the employed grasps span the spaceof possible grasps. For this, we recorded humans do-ing Feix taxonomy grasps using the method publishedby Santello et al. [1998], and compare the resultsto existing published data sets (the data publishedin Santello et al. [1998], the UNIPI data set1, andUNIPI-ASU data set2.

In the experiment, we asked five healthy humanparticipants to enact every grasp of the Feix taxon-omy five times while wearing a Cyberglove II dataglove, using exactly the same objects as used forthe experiment in the previous section. Participantswere allowed to use the other hand to assist in as-suming the grasp posture, but had to achieve a suc-cessful grasp in the sensorized hand without addi-tional support. The resulting postures were sam-pled 50 times within 500 ms and averaged over sam-ples and episodes. We then performed dimensionalityreduction by applying principal component analysis(PCA) for each subject individually to exclude inter-subject variance in accordance with the data analysisused in Santello et al. [1998]. We then compared the

1Data set available at http://handcorpus.org2Data set available at http://handcorpus.org

11

1: Large Diam-

eter

2: Small Diam-

eter

3: Medium

Wrap

4: Adducted

Thumb

5: Light Tool

(failed)

6: Prismatic 4

Finger

7: Prismatic 3

Finger

8: Prismatic 2

Finger

9: Palmar

Pinch

10: Power Disk

11: Power

Sphere

12: Precision

Disk

13: Precision

Sphere

14: Tripod 15: Fixed Hook

16: Lateral 17: Index F.

Ext.

18: Extension 19: Distal

Type (failed)

20: Writing

Tripod

21: Tripod

Variation

22: Parallel

Ext.

23: Adduction

Grip

24: Tip Pinch 25: Lateral

Tripod

26: Sphere 4

Finger

27: Quadpod 28: Sphere 3

Finger

29: Stick 30: Palmar

31: Ring 32: Ventral 33: Inferior

Pincer

Figure 12: Enacted grasps of the Feix taxonomy, using empirically determined actuation patterns: Graspsare numbered according to the Feix taxonomy [Feix et al., 2009]; the hand failed to replicate grasps 5 (LightTool) and 19 (Distal Type, Scissors)

12

Figure 13: Performing grasps using the grasp postures 25, 1, 9, 28 and 18: a human places the hand and thentriggers the actuation of the appropriate grasp; top: pre-grasp posture, middle: executed grasp, bottom:lifting object to show success.

resulting residual unexplained variances with datafrom literature.

The results are shown in Fig. 14. For four outof five participants, the unexplained variances werehigher than those of the three independently pub-lished data sets, suggesting that the grasps span thespace of possible grasp postures more effectively thanthe considerably larger but less structured set of ob-jects that was used for the data sets we compare with.

The discrepancy between the published data setsand ours may be explained by the fact that the San-tello and UNIPI data sets were recorded on graspingimagined objects, while our data and the UNIPI-ASUdata set are recorded while grasping real objects.The additional variance observed may come from theinteraction between hand and object, whereas withimagined objects, hand posture may be more relatedto actuation pattern than the actual grasp posture.This interpretation of existing data is consistent withour hypothesis that hand control can be simpler thaneffected posture.

6.2 Choice of Representation

Because the RBO Hand 2 does not have discretejoints, it is not possible to assess its postural diver-sity using the method of Santello et al. [1998], eventhough the hand is anthropomorphic. An alternativerepresentation of hand posture compatible with softhand mechanics has been used for the Human GraspDatabase3 experiment. In this experiment, Romeroet al. [2010] represented hand posture in terms of fin-gertip position and orientation relative to the back ofthe hand.

Before using this representation, we first have to es-tablish that for assessing postural diversity it is com-parable to using joint angles. Fig. 15 shows a compar-ison of the residual unexplained variances of a Prin-cipal Component Analysis for fingertip position andjoint angles (from the previous experiment), both ac-quired on human hands using the Feix taxonomy (31grasps, excluding grasps 19 and 23 in Fig 12). Thegraph shows that fingertip positions may be a morecompact representation than joint angles, but most

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

13

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

Res

idual

unex

pla

ined

vari

ance

participant 1

participant 2

participant 3

participant 4

participant 5

Dataset in Santello(1998)

Dataset UNIPI-ASU(May 2011)

Dataset UNIPI(Oct. 2011)

Figure 14: Residual variances of PCA on joint mea-surements taken with data gloves. Solid lines denotepublished data sets using the method of Santello et al.[1998], dotted lines denote the data acquired usingthe Feix taxonomy and the objects shown in Fig. 12.

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

Res

idual

unex

pla

ined

vari

ance

participant 1

participant 2

participant 3

participant 4

participant 5

Human Grasp Database(fingertip positions)

Figure 15: Residual variances of PCA on grasp pos-tures enacted by humans using data from differentacquisition methods, joint angles and fingertip posi-tions. Solid line denotes data from the Human GraspDatabase [Romero et al., 2010], dotted lines denotethe data acquired using the Feix taxonomy and theobjects shown in Fig. 12

importantly not worse. Therefore we can use finger-tip position data as a conservative estimate of jointposture posture diversity. Omitting fingertip orien-tation data (represented as quaternions) from thePrincipal Component Analysis did not significantlydecrease unexplained variance. It was therefore ex-cluded to simplify acquisition of comparable data onthe RBO Hand 2 using a motion capture system.

Concluding the human experiments and compara-tive study of published data, the Feix taxonomy ap-pears to be a good proxy for diverse grasp postures.Additionally, we found that fingertip position is a vi-able alternative to joint angles for assessing posturaldiversity of human hands and therefore also of an-thropomorphic robot hands.

14

Figure 16: Screen shot from video Extension 4, show-ing the marker placement on the fingertips for MotionCapture.

6.3 Postural Diversity of RBO Hand 2

The previous subsection gives us a justification to usefingertip positions of grasps from the Feix taxonomyto assess the RBO Hand 2 in terms of grasp posturediversity. To acquire the fingertip positions we useda motion capture system. We placed 3 mm retrore-flective markers on the backside of the fingertips ofthe four fingers. On the thumb tip, the marker wasplaced on the side to not interfere with the pincergrasp. The marker placement can be seen in Fig. 16as in the video Extension 4. Note that additionalmarkers were attached to the hand during recording,but only the five fingertip markers were used for theexperiment.

The motion capture system has a spatial precisionof 0.05 mm, as measured using 205 frames recordedfor a stationary hand (at 50 Hz). For each grasp, twotrials are recorded, fingertip positions are extractedand averaged over the trials before applying PrincipalComponent Analysis. The data set from the HumanGrasp Database is subjected to the same procedure.The results are shown in Fig. 17. The postural di-mensionality exhibited by the RBO Hand 2 matchesthe human hand data well. This strongly supportsour claim that our robotic hand is as dexterous as ahuman hand with respect to attainable grasp posture,taking the Feix taxonomy as a reference.

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

Res

idu

al

un

exp

lain

edvari

an

ce

Human Grasp Database(fingertip positions)

RBO Hand 2(fingertip positions)

RBO Hand 2(actuation signal)

Figure 17: Dimensionality of fingertip positions fora human hand, the RBO Hand 2, and its respectivecontrol signal. The gray region indicates where thedimensionality of fingertip posture exceeds control di-mensionality.

A second observation can be made by consideringthe dimensionality of control required to implementthe grasp postures, which is indicated by the dottedline and gray area in Fig. 17. The space of grasp pos-tures is of higher dimensionality than the actuationspace, which is of dimension four. Where does thisincrease come from? It must be introduced by thediverse shapes of the grasped objects. The interac-tions between the hand and the object differentiatedifferent postures. This differentiation is facilitatedby the hand’s ability to adapt to objects compliantly.The differences between imagined and real grasps inhuman experiments, discussed in Section 6.1, corrob-orate this hypothesis. Tavakoli et al. [2014] also ob-tain the minimum of four to six actuated degrees offreedom to implement dexterous grasping for theiranthropomorphic, compliant hand.

The diversity and consistency of evidence we pre-sented here strongly suggests that compliant handsbenefit dexterous grasping.

The presented evaluation, based on published data

15

sets, measurements of human grasps, and of graspswith 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 termsof covering sets of grasps, again, using the Feix tax-onomy as a reference.

This diversity in grasping posture is achieved bythe RBO Hand 2 with only four actuated degreesof freedom. This is possible because the interactionof the compliant degrees of freedom with the diverseobjects introduces additional variance in the posture.

7 Limitations

Adopting a novel technology in a new application,like continuum actuators in the design of soft dexter-ous hands, opens up new possibilities but also leadsto new limitations and challenges that need to beconsidered carefully.

Grasp Forces and Payload Continuum actua-tors, when constructed with reinforced rubber andactuated hydraulically, are in principle capable of ex-erting extremely large forces. For example, an ac-tuator made out of car tire rubber with steel-fiberreinforcements and hydraulic actuation would proba-bly be able to exert grasping forces exceeding 100 N.It would also be straightforward to make a muchstronger hand with the current production processby choosing stiffer rubbers and thicker hulls. In thecurrent hand design, we chose to use very soft actu-ators, to investigate the effect of compliance, to in-crease safety, and to make manufacturing convenient.We chose pneumatic actuation over other fluidic op-tions as it is much simpler and cleaner to operate ina lab environment.

Grasp Stiffness While grasp forces, as discussedabove, refer to the magnitude of forces exerted onthe object, grasp stiffness refers to the hand’s abil-ity to maintain a grasp posture in the presence ofexternal forces. Naturally, a soft hand built for max-imum compliance does not create an extraordinarilystiff grasp. Low grasp stiffness has the advantage ofreducing the peak forces encountered at the contact

points between object and hand. It also reduces theprobability of slip when objects collide or upon jerkywrist motion. At the same time, it also places a limiton the forces the hand can exert on the environmentthrough the grasped object, for example. This nega-tively affects tasks where those forces must be high.We therefore need to balance and these two compet-ing properties.

In case grasp stiffness proves to be the limitingfactor for certain tasks, there are several methodsavailable to selectively increase actuator stiffness [V.et al., 2015]. But they increase design complexity andproduction costs, and therefore should be avoided ifpossible. Another simple method to increase graspstiffness is to select power grasps instead of precisiongrasps.

In the context of grasp stiffness, we can understandthe exploration of soft hand designs as searching fora lower bound on grasp stiffness that still is able toprovide sufficient grasp force. Future research willinvestigate how grasp stiffness can be increased whilemaintaining compliance where necessary.

Pneumatics Our hand relies on additional exter-nal pneumatic components for control. These com-ponents are cheap and readily available in industry-grade quality. However, they are are over-sized forthe low pressures, small volumetric flow, and size con-straints of robotic hands. Miniaturizing and integrat-ing electrically actuated valves directly into the hand,possibly even into the actuator, would greatly sim-plify integration into predominantly electromechanicrobots.

Long-term autonomy arguably is easier to achievewith pneumatic systems. In contrast to electricalpower systems, where no good solution for long-termuntethered operation exists, the technology exists tomake small, quickly refillable air tanks or even to di-rectly convert chemical energy [Wehner et al., 2014].

Mobility can easily be obtained by using compact,small compressors, as the average rate of airflow foroperating the RBO Hand 2 is very low and peaks canbe serviced by small air tanks. The use of electricalenergy storage also often simplifies the integrationinto existing system.

16

Precision and Repeatability of grasps Whileactuation patterns can be reproduced with high pre-cision, the interaction with the object or features ofthe environment during a grasp can introduce sub-stantial variations in the final grasp posture. Whilethis is often understood to be a disadvantage, we viewit as an important feature of the design, leading torobust grasping performance. Therefore, we believeone must carefully differentiate between precision andrepeatability versus grasp robustness. The hand pre-sented in this paper deliberately trades the former forthe latter.

Sensing The compliance of the materials makeslocal, dense sensing for proprioception and contactforces important but also very difficult. Also, sensortechnologies compatible with soft actuators are cur-rently not available commercially. While it is easy tointegrate air pressure sensors, it would be very desir-able to integrate strain and touch sensors too. This isa topic of active research as current electronic sensortechnology is predominantly designed for rigid struc-tures, but working solutions for stretchable electron-ics start to appear, making an integration with Pneu-Flex actuators feasible in the near future [Rahimiet al., 2014, Culha et al., 2014, Gerratt et al., 2014].

Modeling Hand Mechanics Modeling the wholehand poses certain challenges: an accurate mechan-ical state is difficult to obtain due to nonlinearitiesarising from large deformation and anisotropic struc-ture of its components. Additionally, the actuatorsintentionally provide a large number of deformationmodes, which increases hand complexity even furtherand makes sensing the hand’s complete mechanicalstate very difficult.

Because of these features, we are currently not ableto provide a quantitative analysis of grasp qualitybased on mechanical models as it is state of the artfor hands with rigid links. We therefore qualitativelyassess grasp quality in Section 5.3 and also providetwo videos in Multimedia Extension 5 and 6 that il-lustrate the attainable quality of grasps.

To start closing the gap in modeling soft contin-uum actuators, we present a model for computing

key mechanical properties of PneuFlex actuators inAppendix A. While that model is not able to es-timate all parameters necessary to create a faithfulsimulation, it can be used to straightforwardly cus-tomize deformation modes and stiffness profiles of theactuators before building them. The Appendix alsoevaluates the influence of several nonlinear phenom-ena on actuator behavior and suggests several, easyto follow design guidelines to avoid common failuremodes.

Yet another consequence of the complex and highlyvariable deformations that happen during graspingis that most existing grasp planners cannot be ap-plied, as they rely on complete and accurate geomet-ric and kinematic models of hand and environment.While it might be possible to perform simulationsof the hand’s deformation using finite element meth-ods, these are computationally too complex to em-ploy them in search-based grasp planning.

8 Conclusions

We presented a compliant, underactuated, and dex-terous anthropomorphic robotic hand based on softrobotics technology. The hand is able to achieve 31of 33 grasp postures from a state-of-the-art humangrasp taxonomy. To evaluate the dexterity of the op-posable thumb, we performed the Kapandji test, inwhich the hand achieves seven out of eight possiblepoints. We illustrated the hand’s excellent payload toweight ratio, as it is able to lift objects of nearly threetimes its own weight. We also presented real-worldgrasping experiments to demonstrate the hand’s ca-pabilities in a realistic setting.

We believe that compliance is crucial to enable ro-bust grasping in robotic hands. We provided supportfor this statement by showing that the dimensional-ity of the achievable postural space is significantlylarger than the dimensionality of the hand’s actu-ation space. We explain this observation with thehand’s ability to mechanically comply to the shapeof the grasped object: The final grasping posture isthe result of the hand’s actuation together with com-pliant interactions between the hand and the object.We found that several grasping experiments with hu-

17

mans are consistent with this interpretation. Wetherefore conclude that compliance in robotic hands,when used correctly, can facilitate not only robust-ness in power grasps but also dexterity.

In addition to enhancing dexterity, the use of softrobotic technology renders the hand robust to impactand blunt collisions and makes it inherently safe andsuitable for working environments containing dirt,dust, or liquids. The effort, complexity, and cost ofbuilding the hand are significantly lower than for ex-isting hand technologies. The hand presented herecan be built in two days, using materials worth lessthan 100 US$. Both actuator and hand structureare easily adaptable to specific application domains.We therefore believe that this novel way of buildingrobotic hands significantly lowers the barrier to entryin the field of grasping and manipulation research.

9 Acknowledgments

We gratefully acknowledge financial support by theAlexander von Humboldt foundation through anAlexander von Humboldt professorship (funded bythe German Federal Ministry of Education and Re-search). We are equally grateful for the funding pro-vided by the SOMA project (European Commission,H2020-ICT-645599) and the German Research Foun-dation (DFG, award number BR 2248/3-1).

A Derivation of an Approxi-mate PneuFlex Model

To facilitate the design of PneuFlex actuators, wepresent a computational model for its deformation.Overall, the deformation of such an actuator is com-plex. Nevertheless, surprisingly simple design rulescan be derived for designing the most important ac-tuator properties, namely, actuation ratio and rota-tional stiffness around the actuated axis. This ap-pendix proposes a suitable formalization, an analy-ses of simplifications taken, and a basic experimentalevaluation of the resulting equations.

Figure 18: Simple model of an actuator segment

A.1 Formalization of the PneuFlexActuator Geometry

Fig. 18 shows the parameterization of a small seg-ment of the actuator. To simplify the model, we ig-nore the side walls and assume a rectangular crosssection. Let x, z, d be the length, width, and thick-ness of a segment of the actuator respectively.

Width and height of the actuator are assumed tonot change due to the helical thread around the ac-tuator. This assumption is facilitated by selecting anapproximately circular cross section (e.g. a square)for the shape of actuator cross section, as the radialfibers will always deform the shape into circle to bal-ance the radial pressure.

Due to its embedded fabric, the bottom layer alsohas a fixed length x. The only possible deformationleft is to stretch the silicone layer while bending thebottom layer, which is illustrated in Fig. 19

The interfaces of the segment are rotated to eachother by the angle ϕ and the bottom layer curves withthe radius r = x

ϕ . As we also have configurations with

ϕ = 0 (when the actuator is deflated), we will ratherexpress equations in terms of the curvature κ of thebottom layer:

κ =1

r=ϕ

x⇒ ϕ = κ ·x

We can also express the curvature on the top sideof the actuator:

κtop =1

1κ + h

18

Figure 19: Parameterization of an actuator segment, illustrated at different curvatures. x, Vsil and h stayconstant while λ, r and ϕ change.

Both κ and κtop are defined on the same angle ϕ.Therefore we can compute a relationship between an-gle ϕ and stretch λ1:

ϕ = ϕtop

κ ·x =1

1κ + h

·λ1x

κ =λ1 − 1

h(1)

λ1 = 1 + κ ·h (2)

ϕ =x

h· (λ1 − 1) (3)

Volumes To aid readability of the analysis, we de-fine several constant volume that do not change underdeformation. These volumes can be computed fromthe basic actuator geometry:

The volume of the effectively incompressible sili-cone 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 deflatedactuator plus any volumes connected to it, such assupply tubes:

Vair = Vsupply+V ch = Vsupply + Vact − Vsil

The Symbol V will denote the actual volume of theair chamber, which is dependent on the deformation.Therefore it is a function of actuator curvature. Thetotal actual volume of the actuator is V + Vsil.

Volume Change For computing the energy storedby the compressed gas (air) within the actuator, weneed to compute the actual volume with respect toactuator curvatures. We can do this by first calculat-ing the total volume of a flexed actuator segment:

V + Vsil = z · ϕ2π·

(π ·(x

ϕ+ h

)2

− π ·(x

ϕ

)2)

= Vact ·(

1 +h

2

ϕ

x

)As ϕ

x = κ, we can express actuator curvature interms of air chamber volume and initial geometry as:

V + Vsil = Vact ·(

1 +h

2·κ)

κ =2

h

(V + VsilVact

− 1

)=

2

h·(V − VchVact

)

19

With Equation 1 we can also compute the relation-ship between λ1 and the actuator volume:

λ1 = 1 + 2 ·(V − VchVact

)(4)

This leads to the first insight: actuator curvatureand stretch of the silicone rubber are linearly propor-tional to the gas volume:

∂κ =2

Vact ·h· ∂V

∂λ1 =2

Vact· ∂V

Strain Tensor Invariants The energy stored inthe rubber during deformation is modeled usingstrain tensor invariants [Gent, 2012]. The strain ten-sor invariants are related to the orthogonal stretchesλ1, λ2, λ3:

J1 = λ21 + λ22 + λ23 − 3

J2 = λ21λ22 + λ21λ

23 + λ22λ

23 − 3

J3 = λ1 ·λ2 ·λ3 − 1

To compute them, we first define actuator-specificrelations between stretches λ1, λ2, λ3 in three prin-cipal directions, which are aligned to x, h and z re-spectively.

Silicones are effectively incompressible, thereforewe can assume a constant volume:

λ1 ·λ2 ·λ3 = 1

The actuator’s radial size does not change eitherbecause of the reinforcement helices. We thereforeset the circumferential stretch λ3 = 1, and get therelationship:

λ2 = λ−11

This deformation is also called pure shear. Usingthese relations, both J1 and J2 reduce to:

J2 = J1 = λ21 + λ−21 − 2 (5)

And because of the incompressibility assumption:

J3 = 0

Simplifications and Limitations To keep themodel simple, many potentially important effectswere not included:

We assume a uniform strain energy density withinthe rubber hull, which is acceptable for moderatelythin rubber hulls (i.e. d < 0.5h). This assumption ismodeled and discussed in Section A.6.

We use a Neo-Hookean material model. This ig-nores higher order deformation effects. The error isless than 2% though, as discussed in Section A.7.

We ignore material stiffening. The consequencesare discussed in Section A.8. The resulting error istypical less than 7.7%.

Finally, we also ignore the side walls, i.e. hull partsof the actuator which are stretched only at fractionsof λ1. The ramifications are discussed in Section A.9.

The model can also be invalidated by compressiveforces applied externally. They remove fiber tensionand therefore usually lead to buckling. This limitsthe usefulness of the model for simulation and plan-ning. The main purpose is to provide simple equa-tions for designing actuator behavior though. Herethe limitations are acceptable.

A.2 Statically Stable Actuator Con-figurations

Using the Minimum Potential Energy Principle wecan derive statically stable configurations for the ac-tuator First, we need to define all relevant forms ofwork 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 (6)

Equilibrium is reached, when the gradient of workis zero, which is in our case very simple as we will

20

describe the actuator state with the single variableλ1:

∆W

∆λ1= 0

The remainder of this section derives each workcomponent from the definitions of the previous sec-tion and yields the equation for statically stable con-figurations.

Gas Compression Work For computing the gascompression work, we need to differentiate betweentwo different control regimes:

In the mathematically simpler case, pressure is heldconstant under deformation: p(V ) = p. This is eitherdone actively by using a control system or passivelyby using a big reservoir connected to the actuatorvolume, which attenuates the effect of volume changewithin the actuator on pressure.

In the second, more common case the total en-closed gas mass in the system is held constant e.g.when using pneumatic valves. Pressure changes ac-cording to the ideal gas equation: p(V ) = nRT · 1V .A closed gas volume increases the stiffness of the ac-tuator slightly. But for the sake of brevity, the lattercase will not be derived here.

In both cases, the work done by changing the vol-ume of a gas from V1 to V2 is:

Wair = WV1−

V2∫V1

p(V ) · dV

In the case of constant gas pressure, we get a simpleequation:

Wair = W0 − p · (V )

δWair

δλ1= −p · δV

δλ1

δWair

δλ1= −p · 1

2Vact (7)

Rubber Deformation Work The deformationwork of the silicone rubber Wsil is modeled as a Neo-Hookean solid model with coefficient C10 = G

2 :

Wsil =

∫Vsil

G

2· J1 · δV

G is the material’s shear modulus, and Vsil thevolume of the silicone. Choosing this simple modelover more complex ones is discussed in Section A.7.We will further assume a uniform deformation, andthus uniform strain energy density (i.e. uniform J1)throughout the hull, as justified in Section A.6. Wecan then calculate the total strain energy as:

Wsil =G

2· J1 ·Vsil

Using Equation 5, we can express the work gradientw.r.t. stretch λ1:

Wsil =G

2· (λ21 + λ−21 − 2) ·Vsil

δWsil

δλ1= G ·

(λ1 − λ−31

)·Vsil (8)

Load Work For a given actuator segment, externalload is applied on the interfaces to the two adjacentsegments. By attaching our frame of reference to oneinterface, load work can can be computed by onlyconsidering the motion and force of the other inter-face.

The load work can further be split up into thework done by translatory forces and rotary moments.Translatory forces are transmitted by the inelasticfibers of reinforcement helix and passive layer. If weassume completely inelastic fibers, those forces do notcontribute any work. Rotations, on the other hand,do contribute work, and we can integrate the contri-bution along the bottom layer:

Wload =

∫M (ϕ) · dϕ

For the model, it is more convenient to integrateover x instead of ϕ along the actuator segment. Wecan rewrite the integral to:

21

Wload =

∫M (ϕ (x)) · ∂ϕ (x)

∂xdx

For short enough actuator segments (small x) wecan assume constant, averaged moment along thewhole segment, i.e. M(x) = M . Additionally we can

substitute the derivative of Equation 3 for ∂ϕ(x)∂x :

Wload =

∫M · λ1 − 1

h· dx

= M · λ1 − 1

h·x+W0

From this equation, we can compute the work gra-dient w.r.t. stretch λ1:

δWload

δλ1= M · x

h(9)

Minimum Total Potential Energy By comput-ing the local minimum of Equation 6 w.r.t. λ1 andsubstituting with Equations 7, 8 and 9, we obtainthe equation describing stable actuator states:

0 = δWair

δλ1+ δWsil

δλ1+ δWload

δλ1

0 = −p · Vact

2 +G ·(λ1 − λ−31

)Vsil +Mload · xh(10)

A.3 Stiffness

The Stiffness of an actuator segment is expressed bythe change in moment Mload w.r.t. curvature κ. FromEquation 10 we can compute Mload it explicitly:

Mload =h2 · z

2· p−

(λ1 − λ−31

)·G ·h · d · z

As we assume a constant pressure regime, the firstterm vanishes when differentiating:

δMload

δλ1= −G ·h · d · z · (1 + 3 ·λ−41 )

We then substitute with the derivative of Equa-tion 2 which is ∂λ1 = h · ∂κ. Additionally, we can

set z = h, as the width of the actuator is usuallyapproximately its height. We arrive at the equationdescribing the stiffness of the actuator given its shapeand deformation:

δMload

δκ= h3 ·G · d ·

(1 + 3λ−41

)The last, nonlinear term predicts a strong stiffening

when a PneuFlex actuator is straight or even nega-tively curved. This sudden stiffening is indeed ob-served with actual actuators. But they also tend tobuckle under such loads.

Scaling Law When scaling an actuator, the ratiosdh and z

h stay constant. The equation then becomes:

δMload

δκ= h4 ·

(G · d

h·(1 + 3λ−41

))Stiffness therefore scales to the fourth power of ac-

tuator size. This gives us a powerful lever to adjustthe strength of an actuator.

Non Squared Cross Sections Because the stableconfiguration of the helical thread from the applieduniform radial pressure is a circle, the cross sectionwill always deform into one given high enough airpressure.

It therefore makes sense to use the the actuator cir-cumference c instead of height and width to computeactuator stiffness:

δMload

δκ=( c

4

)3· (G · d) ·

(1 + 3λ−41

)For designing actuator stiffness, we can approxi-

mate the nonlinear term with 1:

δMload

δκ=( c

4

)3· (G · d) (11)

A.4 Actuation Ratio

The actuation ratio δκδp is the change of curvature

given an increase in pressure while assuming zeroload. It can also be computed from Equation 10:

22

p =(λ1 − λ−31

)·G · d

h(12)

δλ1δp

=1(

1 + 3λ−41

)·G · dh

Using Equation 2 we can substitute λ1 and δλ1.We get the actuation ratio:

δκ

δp=

1

1 + 3 (1 + hκ)−4 ·

1

G · d(13)

So according to the simple model, the actuationratio is inversely proportional to the silicone’s shearmodulus, and its thickness. The actuation ratio alsohas a nonlinear component with respect to the actualcurvature. At small curvatures the actuation ratio isconsiderably lower than at higher curvatures. Thenonlinearity can be linearized though by a non circu-lar cross section, as discussed in Section A.5. For ac-tuator design we can conveniently drop the nonlinearterm and arrive at a simple design rule for computingthe inverse actuation ratio:

δκ

δp≈ 1

G · d(14)

Note that this equation is independent of both hand z (and therefore also circumference c). The ac-tuation ratio is not dependent on the size of the actu-ator cross section! We can therefore set an actuationratio profile using thickness and shear modulus of therubber hull, and then set the stiffness profile with:

δMload

δκ=( c

4

)3· 1δκδp

·(1 + 3λ−41

)(15)

A.5 Justifying Simplification: LinearActuation Ratio

Equation 12 contains the nonlinear term(λ1 − λ−31

).

The model therefore predicts pressure to be non-linearly related to actuator curvature. Interestinglythough, the actuation ratio can be linear at moder-ate pressures, as it was observed in Deimel and Brock

[2013]. We believe that this can be caused by a noncircular cross section.

When pressure increases, the helical thread ten-sions and always deforms the actuator cross sectioninto a circle. But until then, the actuator expands inthree dimensions instead of one, which increases thestrain energy in the rubber hull faster with respect tostretch λ1. The effect is more pronounced with lesscircular cross sections, but can also be elicited by aloosely wound helical thread. We can investigate theresulting effects by introducing a correction factor forthe strain energy gradient:

D (λ1) = (1− k0) · (λ1 − 1)2

(λ1 − 1)2

+ k21+ k0 (16)

The factor k0 ≈ [1 . . . 5.0] defines the relative in-crease of the gradient for a deflated actuator, whilethe factor k1 ≈ 0.025 determines at which elonga-tion the effect is halved. The latter is probably de-pendent on the wall’s thickness and rubber stiffness.The equation was chosen because unlike simpler mod-els the correction factor has a limited range, a finiteintegral, and no poles.D(λ1) is then plugged into Equation 12:

p = D(λ1) ·(λ1 − λ−31

)· 2 ·G · d

h

The impact of different k0 on actuation ratio is il-lustrated in Fig. 20. The actuator can behave almostlinearly at higher pressures. An interesting applica-tion of this effect may be to simplify actuator control.

A.6 Justifying Simplification: Uni-form Strain Energies WithinRubber Hull

In the model we assume a uniform strain energy den-sity throughout the hull. This needs to be checkedthough, as the principal stretches are not uniform atall.

Formalization For an infinitesimal volume of elas-tomer within the hull, we define d to be the radial

23

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Normalized pressure p · h2·G·d

1.00

1.05

1.10

1.15

1.20

1.25

λ1

modelk1 = 0.02k1 = 0.03k1 = 0.04linearization

Figure 20: A non circular cross section effectivelylinearizes the actuation ratio at moderate curvatures.

distance of the volume from the outer boundary ofthe hull in an undeformed actuator, while d′ denotesthe actual radial distance in the deformed actuator.

When the actuator bends, the volume moves ra-dially outwards as the wall thins. This motion in-fluences the stretches encountered. As the radialdisplacement is limited by the helical reinforcementfibers, we can assume the outer boundary of the rub-ber hull to not move radially at all.

Due to the smaller radius the longitudinal stretchis:

λ1 = 1 + (κ ·h− κ · d′) (17)

The circumferential stretch can be calculated bythe radial displacement given a position d with re-spect to the undeformed position d:

δλ3 =π (h− d′)π (h− d)

− π (h− d′ − δd′)π (h− d)

= − δd′

h− d

⇒ λ3 =−d′

h− d+ C

Setting the boundary condition λ3 = 1 at the outer

boundary of the hull yields:

λ3 =h− d′

h− d

Finally, we can derive λ2 by using the incompress-ibility assumption λ1λ2λ3 = 1 and get:

λ2 =h− d

κ ·h2 + κ · d′2 − (2 ·κ ·h+ 1) d′ + h

To normalize our calculations, we can express theprincipal stretches in terms of the dimensionless ra-tios d′

h , dh , and κ·h:

λ1 = 1 + κ·h ·(

1− d′

h

)λ2 =

1− dh

κ·h+ κ·h ·(d′

h

)2 − (2 ·κ·h+ 1) d′

h + 1

λ3 =1− d′

h

1− dh

With λ2 we can express the relation between d′

and d as a linear differential equation using λ2 as thegradient of displacement:

δd′

δd= λ2

δd′

δd=

1− dh(

d′

h

)2κ·h− d′

h (2 ·κ·h+ 1) + 1 + κ·h

This differential equation determines the positiond′ of a packet of rubber, and therefore its deforma-tion.

Strain Energy Distribution To analyze thestrain energy, the differential equation was solved nu-merically. The boundary conditions were set on theouter boundary of the hull at d′ = 0 to λ2 = 1 andλ3 = 1

λ1.

Fig. 21 shows the strain energy density w.r.t. thenormalized depth d

h within the hull and at differentnormalized curvatures κ·h. We can see that the strain

24

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

radial position dh

0.0

0.2

0.4

0.6

0.8

J1

κ·h = 0.5κ·h = 0.4κ·h = 0.3κ·h = 0.2κ·h = 0.1κ·h = 0.0

Figure 21: Strain energy density distribution withinthe rubber hull for different normalized curvatures.

energy density stays surprisingly flat even for moder-ately thick hulls and at very strong curvatures.

The reason for this surprising result can be under-stood when looking at the principal stretches whenthe rubber moves outwards radially (thinning thewall). Circumferential stretch λ3 increases, but at thesame time radial stretch λ2 decreases as the packetgets more compressed. Also λ1 decreases with thedistance from the bottom layer.

The analysis shows that as long as the hull thick-ness is less than half the actuator height, we can as-sume a uniform strain energy distribution for mod-eling. Staying below this limit also avoids materialfatigue of the rubber on the inside of the air chamber.

A.7 Justifying Simplification: Neo-Hookean Deformation Model

An alternative to the simple Neo-Hookean defor-mation model used in our model is the generalizedMooney-Rivlin model for incompressible hyperelas-tic materials. It states a polynomial approximationof strain energy density, using the strain tensor in-variants:

dW

dv=

n∑i,j=0

Cij · J i1 · Jj2

Silicone rubbers have rather small coefficients for

C01, C02 and C20 though. The coefficients publishedby Meier et al. [2005] have the following relations:

C01 ≈ 1

50·C10

C20 ≈ 1

500·C10

C02 ≈ 0

Ignoring the coefficients C01, C02 and C20 (i.e. set-ting them to 0) yields the Neo-Hookean model.

We can bound the error to strain energy whenassuming a maximum stretch of λ1 < 3 (which re-lates to an actuator bending at the radius of half itsheight), which given Equation 5 bounds the straintensor values to:

J1 < 7.11

J2 < 7.11

The terms dropped from the Mooney-Rivlin modelare bounded to:

C01 · J2 < 0.02 ·C10 · J1C20 · J2

1 < 0.014 ·C10 · J1

Which results in a total error of less than 2%.

A.8 Justifying Simplification: IgnoreMaterial Stiffening

Elastomers exhibit a stiffening at large stretches.This is modeled by augmenting the strain energyfunction with an additional parameter. The Gentmodel [Gent, 2012] augments the strain energy func-tion with a logarithmic term:

W =G

2· J1 (λmax) ln

(1− J1 (λ1)

J1 (λmax)

)·Vsil

=G

2· Jm ln

(1− J1

Jm

)·Vsil

With the additional material parameter λmaxwhich is the stretch where the material exhibits un-limited stiffness.

25

The rubber used for the PneuFlex actuator typi-cally has λmax ≈ 10, which results in:

Jm ≈ 100

When computing the strain energy gradient usingthe Gent model, we get:

δWGentsil

δλ1= C1 · Jm ·Vsil ·

1

1− λ21+λ

−21 −2

Jm

· 2λ1 − 2 ·λ−31

Jm

=1

1− λ21+λ

−21 −2

Jm

· δWsil

δλ1

For plausible values of λ1 < 3 and Jm = 100, theignored stiffening factor can be bounded to:

δWGentsil

δλ1< 1.077 · δWsil

δλ1

So ignoring the material stiffening introduces anerror of less than 7.7%.

A.9 Justifying Simplification: Ignor-ing the Side Walls

For analyzing the impact of ignoring the side walls,we can conceptually split the cross section of a realPneuflex actuator into many small parts, of whicheach behaves according to the simple model.

As there is no air below the side walls (only morerubber), there is no additional force applied by gaspressure.

The amount of deformation work δWsil

δλ1does in-

crease, but the J1 also drops off quadratically whenapproaching the bottom layer. By using a constantratio between the thickness of the side wall and thetop side of the actuator, the error being made whencomputing the actuation ratio can be made constant.

For the stiffness, the side walls play even less of arole, as it scales with h3.

A.10 Experimental Validation

To validate the design rules we developed, we con-ducted two experiments. The first one validates the

equation for the actuation ratio, the second the equa-tion for actuator stiffness.

A.10.1 Actuators with Varying d

The model predicts that the actuation ratio is onlydependent on hull thickness d and the shear modulusof the rubber. To test this, we built four actuatorswith the same shape but with a d of 2.5 mm, 3.5 mm,4.5 mm and 5.5 mm respectively.

Fig. 22 shows the actuators at a pressure of46.3 kPa. Despite the change in height and width,the curvatures of the bottom layers are constant alongthe actuators, as indicated by the circle segments.We can therefore validate the model’s prediction thatheight and width do not influence actuation ratio.Please note that the circle segments are placed on topof the edges of the bottom layers, where the fibers ofthe embedded fabric are most stressed.

Fig. 23 shows the relationship between pressureand fingertip orientation, which is an aggregatedmeasure of the curvature along the actuator. Thecurves for each actuator show the nonlinear behaviorpredicted by Eq. 13, i.e. an increase in actuation ratiotowards higher curvatures.

The dotted lines in Fig. 23 indicate the actuationratio when scaling the measurements of the thinnestactuator (2.5 mm) to the thickness of the other actu-ators according to our model. Model and measure-ment agree well for 3.5 mm and 4.5 mm. At 5.5 mmit is clearly visible that the nonlinearity of the actua-tion ratio increases, making the actuator stiffer thanexpected at low pressures.

A.10.2 Actuator with Linear Stiffness

To validate Equation 11 and 15, we can apply a forceat the tip of an actuator. The actuator has a constantactuation ratio along its main axis. The contact forcewill create a bending moment to segments of the ac-tuator that increases linearly with the segment’s dis-tance from the contact point. At small curvatures,i.e. an almost straight finger, the distance along thebottom layer will approximate the euclidean distancewell. For a straight actuator, we can therefore as-sume the moment along the actuator to be increasing

26

Figure 22: Example of four fingers with different actuation ratios, inflated to 46.3 kPa. The overlaid circlesegments indicate the constant curvature along the bottom layer.

0 20 40 60 80 100

Pressure [kPa]

0

20

40

60

80

100

120

140

160

180

Tip

ori

enta

tion

[◦]

2.5 mm

3.5 mm

4.5 mm

5.5 mm

3.5 (model)

4.5 (model)

5.5 (model)

Figure 23: Tip orientation versus inflation pressurefor different rubber hull thicknesses. Dotted lines in-dicate model estimates based on the 2.5 mm measure-ment

linearly. If the actuator has a linear stiffness profilealong the actuator and a constant actuation ratio too,then the load moment and actuated moment will can-cel out and yield a constant curvature. The constantactuation ratio is demonstrated in Fig. 22.

Fig. 24 shows an actuator at four different pres-sures. The curvature stays constant during a largerange of inflation pressures, with the highest pres-sure corresponding to about a 360◦ rotation if therewas no contact. The actuator therefore has a linearstiffness profile, which validates Equation 11.

The upwards rotation of the whole finger relativeto the fixture is caused by the rubber cap which closesthe air chamber at the base of the finger. This alsocauses the small curvature changes between differentpressures. Fig. 24 (d) shows a slightly stronger cur-vature close to the tip. This may be caused by dif-ference between contact point location and the pointwhere the stiffness profile reaches zero stiffness, whichis exactly at the tip.

A.11 Summary

The presented theoretic model yields two simple de-sign rules (Equations 14 and 15) for designing actua-tion ratios and stiffnesses along a PneuFlex actuator.We also validated the scaling behavior of the modelin two experiments.

Based on the analysis, we can give the followingrecommendations for choosing geometric parameters:

27

(a) 0 kPa (b) 17.3 kPa (c) 41.6 kPa (d) 80.1 kPa

Figure 24: The finger is blocked by a fingertip contact while being inflated. An actuator with a linearlydecreasing stiffness profile will show a constant curvature along the actuator under such load.

• active layer thickness should stay less than halfthe actuator height: d

h < 0.5

• To achieve a minimum bending radius of 1.5times actuator height, the rubber should havean elongation at break of at least 500%

• Material stiffening can usually be ignored withsilicone rubber, the error is typically less than7.7%

B Multimedia Extensions

Nr. Type Description1 Images High resolution images of grasps

from the Feix taxonomy inFig. 12

2 Data Fingertip positions ofRBO Hand 2 for the Feixtaxonomy

3 Data Joint angles of five human partic-ipants executing grasps from theFeix taxonomy recorded with aCyberglove II

4 Video Video of how grasp postures forgrasps from the Feix taxonomywere obtained

5 Video Video of example grasps from atabletop under human control

6 Video Video of manipulating two heavyobjects

28

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