a robot gripper with variable stiffness actuation for

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS

A Robot Gripper with Variable StiffnessActuation for Enhancing Collision Safety

Amirhossein H. Memar, Student Member, IEEE, and Ehsan T. Esfahani, Member, IEEE

Abstract—This paper presents the design of a two-fingers variable stiffness gripper and demonstrates howthe adjustable compliance of the fingers can enhance ma-nipulation safety and robustness during collisions. Thecompliance between actuators and fingers are generatedby using repulsive magnets as preloaded nonlinear springssuch that by adjusting the air-gaps between the magnets,the position and stiffness of the fingers can be controlledsimultaneously. The capability of the proposed design incollision detection and reaction is demonstrated via twocollision scenarios. In the first experiment, a momentum-based collision detection algorithm in the context of com-pliant grasping is investigated. In the second experiment,an optimal reaction strategy is developed that exploitsthe adjustable stiffness characteristics of the gripper toimprove safety during collisions. The experimental resultsof the two scenarios indicate the effectiveness of the pro-posed gripper in enhancing collision safety.

Index Terms—Robotic gripper, variable stiffness actua-tion, magnetic spring, collision reaction.

I. INTRODUCTION

RECENTLY, there has been a growing interest in the de-sign and control of Variable Stiffness Actuators (VSAs)

as an alternative choice to their conventional stiff counterparts.Various VSA mechanisms have been proposed to mechanicallyadjust the joint stiffness by placing elastic elements likesprings between the actuator and load [1]. Although a variablestiffness actuation is generally realized at the cost of an extraactuator [2], the inherent compliance and energy storage capa-bility of VSAs offer two major advantages which rationalizethis cost. These advantages are as follow: (i) The elasticelements of VSAs act as a low-pass filter to external shocksthat can prevent serious damages to the robot or environmentby absorbing the collision energy and reducing impact forces.(ii) VSAs can enhance highly dynamic motions by exploitingtheir natural dynamics and energy storage capability withintheir elastic elements. In fact, the performance of tasks thatrequire releasing a large amount of energy over a short periodof time can be improved based on optimal stiffness controlmethods. Energy efficient hopping [3] and hitting [4] are twocommon examples of such dynamic tasks.

Manuscript received January 31, 2019; revised May 25, 2019 and July5, 2019; accepted August 9, 2019.

AH. Memar is with Facebook Reality Lab, Seattle, WA, USA. He waspreviously with the Department of Mechanical and Aerospace Eng. atUniversity at Buffalo, Buffalo, NY, USA (e-mail: [email protected]).

ET. Esfahani is with the Department of Mechanical and AerospaceEngineering, University at Buffalo, Buffalo, NY, 14260 USA (e-mail:[email protected]).

The development of a fully VSA-based robotic arm withmultiple degree-of-freedom (DoF) requires a highly complexdesign and an extra actuator at each joint (e.g., the DLRarm [5]). However, a VSA-based gripper in conjunction withconventional stiff robot arms can provide some characteristicsof a fully VSA-based robot arm at a lower cost and complexity.

A robot gripper with variable stiffness actuation can providethe capability of adjusting grasp stiffness based on the taskrequirements and expand the functionality of the conventionalsystems. The stiffness can be either increased to obtain ahigher positioning accuracy, or decreased when a compliantand safe interaction is required. For instance, the human safetyin physical human-robot interaction can be improved throughcompliant grasping, especially if handling potentially harmfulobjects, such as sharp tools and hot liquids [6]. Moreover, thegrasp stiffness can be adjusted with the aim to enhance thereliability of handling sensitive/fragile objects in uncertain anddynamic environments, where collisions are likely to occur.

To this end, various gripper mechanisms have been inves-tigated to realize variable stiffness actuation during objectgrasping. Kim et al. [7] used a rotary VSA [8] to actuate a1-DoF gripper. They demonstrated the benefits of using VSAsfor the grasping of fragile and heavy objects with compliantand stiff actuation. Zhang et al. [9] used a passive link witha progressive nonlinear stiffening to transfer force from thegripper actuator. They used the length of the passive link toestimate the external forces and to detect contacts with targetobjects. The progressive nonlinear stiffening of the passivelink also allowed the fingers to adapt their shapes to targetobjects as grasp force increases. However, the grasp stiffnessfor this design was not controllable and it was passivelychanging as a function of the link length. Wang et al. [10]proposed a compliant gripper to passively maintain a constantgrasp force and stiffness without using additional sensors andactive controllers. Since the constant grasp force was realizedthrough a statically balanced mechanism, a new design wasrequired for different desired force/stiffness characteristics.Li, et al. [11] used a motorized rotary flexure hinge shaftto change the effective moment of inertia of the fingers andaccordingly their stiffness. Sun et al. [12] developed a serialVSA based on an Archimedean spiral relocation mechanism.The relocation mechanism was used to change the positionof the pivot point of a lever and adjust the stiffness inde-pendently from the position of the rotary shaft. Instead ofusing VSA concept to mechanically adjust grasp stiffness,different alternative methods have been proposed to tunegripper material/structural stiffness such as using hydraulic or


e

42

3

e

e

1

21

123 4

M1M2

GraspedObject

( 1+ 2)/2 1

( 3+ 4)/2 ( 1+ 2)/2Δ Δ

2 1(a)

(b)

(c)

1- Magnet 2- Pulley 3- Linear Cart4- Linear Potentiometer 5- Linear Guide

M1 M2

(d)

1 3 5 2

4

150 mm

Fig. 1. (a) The proposed concept of the variable stiffness gripper. Passive motion of the gripper fingers: (b) Before grasp denoted by p1, and (c)after grasp denoted by ∆. The dotted red lines indicate the equilibrium points in the absence of the external forces. (d) Mechanical design andcomponents of the gripper.

pneumatic particle jamming [13], low-melting alloy [14], andshape memory polymer [15]. Despite interesting behaviors forsoft robotic applications, these methods are mostly suitable forperforming low-bandwidth and light manipulation tasks.

In our previous paper [16], we presented the initial design ofa two-finger gripper with variable stiffness actuation using per-manent magnets. The proposed actuation mechanism belongsto a category of VSAs known as antagonistic preloaded springswith antagonistic motors, which is similar to the bi-directionalagonistic-antagonistic motions of biological muscle-tendonunits [1]. It offers the capability of controlling the positionand stiffness of the gripper fingers simultaneously. In theproposed mechanism, the passive motions of the fingers aredecoupled such that (i) external forces exerted to each fingercan be estimated separately, and (ii) the entire grasping loop(fingers+object) can passively move based on their compliancelevel when there are external forces acting on the fingers or thegrasped object. These two features can significantly improvethe robustness of the grasp when external disturbances occur.The use of permanent magnets in repulsive configurationsprovides two main advantages over the conventional springs.First, they have a nonlinear force-displacement characteristicswhich is essential for antagonistic VSAs [2]. Second, theyprovide a non-contact interaction between moving componentsthat results in higher tolerances to misalignments and lessfriction hysteresis. Further, we demonstrated the application ofadjustable grasp stiffness to enhance a hammering task usingthe proposed gripper mechanism [17].

In this paper, we present the application of adjustable graspstiffness to improve grasp safety through collision detectionand reaction strategies. Our contribution can be summarized asfollow: First, the overall gripper design concept and modelingof the grasp stiffness are briefly reviewed in Section II. Thedesign modifications compared to [16] are discussed and afinite element model (FEM) to estimate magnet repulsionforces is presented and compared to experimental measure-ments. Then, collision detection and collision reaction usingthe adjustable grasp stiffness are investigated in Section IIIand IV, respectively, to demonstrate the effectiveness of theproposed gripper in enhancing collision safety. To achieve

a reliable and fast collision detection without the need forforce sensors, a momentum-based algorithm is adopted andformulated in the context of compliant grasping. For thispurpose, the modeling of magnet repulsion forces and motiondynamics of the grasped object are presented accordingly.Finally, by exploiting the adjustable grasp stiffness, an optimalstiffness problem is proposed and examined experimentally toimprove manipulation safety upon detection of collisions.

II. MODEL OF THE PROPOSED GRIPPER

A. Design

The design concept of the proposed variable stiffness grip-per is demonstrated in the schematic of Fig. 1a. It utilizespermanent magnets in repulsive configurations to serve as non-linear preloaded springs that can provides smooth transitionbetween different levels of stiffness.

Each finger and its associated magnets are mounted ona passive (non-actuated) slider whereas, the actuated mag-nets are mounted on motor driven sliders. In a quasi-staticscheme, the fingers tend to stay in their equilibrium points(midpoint between the actuated magnets) and their force-deflection characteristics are a function of the air-gaps betweenthe magnets. Thus, by adjusting the position of the actuatedmagnets, one can control the position and stiffness of thefingers simultaneously.

Considering a symmetric motion for the actuated magnets,two antagonistic motors (M1, M2) are sufficient for controllingthe positions of the 4 actuated magnets as shown in Fig. 1a.M1 controls the displacements of the outer magnets moving atthe same rate but opposite direction. M2 controls the positionof the inner actuated magnets in a similar manner. This designdecouples the passive deflection of the fingers from each otherwhich offers two main characteristics:

1) Before Grasp: Contact forces acting on each finger canbe sensed and estimated separately (Fig. 1b), which can help inthe grasping of objects with position uncertainties or irregularshapes by progressive stiffening of the fingers.

2) After Grasp: When an external force is exerted on thefingers or the grasped object, fingers can passively deflect


along their axis of motion (Fig. 1c) to protect the graspedobject and fingers from high impact forces.

As shown in Fig. 1d, the gripper fingers and magnets aremounted on passive carts sliding on linear recirculating ballbearing sliders. Timing belts and pulleys are used as thetransmission mechanism to control the positions of the ac-tuated magnets with two identical gripper motors (DynamixelRX-24F). The symmetric displacement of the outer actuatedmagnets is achieved by connecting their sliding carts to bothsides of a timing belt. Similar approach is used for the inneractuated magnets.

To improve the feasible range of force-stiffness with com-parison to our previous design [16], fingers are mounted ontwo separate parallel rails and the resultant offset betweenthem is compensated through the oblique design of the fingers.Moreover, to increase the gripper stroke, the sliding cartholding the outer actuated magnets are placed on a third linearrail. The overall mechanical design is compact with preciseforce transmission and tolerance to misalignments.

The concept of using non-contact magnetic forces to controlthe force and stiffness of the grasp is simple and can providehigh reproducibility as well as opportunity for miniaturization.Indeed, in the industrial-level manufacturing, the complexityof the current design can be further reduced through cus-tomization of the mechanical and electrical components, andoptimized integration approaches.

B. Grasp Force and Stiffness Modulation

Let’s consider the right finger (finger-1) shown in Fig. 1b.Neglecting the effects of gravity and friction, the static equi-librium of the finger would be (u1+u2)/2 in the end-effectorcoordinate system (Oe) shown in Fig. 1b with red dashed-line.The passive deflection, p1, due to contact force fe1, can alsobe measured using the position feedback (x1) as (1),

p1 = x1 − (u1 + u2)/2 (1)

Since the repulsive magnets are equivalent to preloadedsprings, the static relation of (2) holds, where fe1 is the contactforce applied on the finger and fmi is the repulsive force ofith actuated magnet with the position ui.∑

f = fe1 + fm1 + fm2 = 0 (2)

The position of the magnet with respect to the finger definesthe direction of these forces. The contact force (fe1) andstiffness (K1) of the finger can be estimated by (3) and (4),

fe1(p1, u1, u2) = fm1 − fm2 (3)

K1(p1, u1, u2) =dfe1dp1

(4)

To compute the derivative in (4), an analytical expressionis required to express the magnet repulsion forces (fmi) as afunction of air-gaps that is pretested in section II-C.

C. Modeling of Magnet Repulsion Forces

Vokoun et al. [18] developed an analytical model to estimatethe repulsion force between two cylindrical magnets (with aradius Rm and height of Hm) as (5).

fm(d) =πµ0M

2R4m

4

[1

d2+

1

(d+ 2Hm)2− 2

(d+Hm)2

](5)

where d denotes the air-gap; µ0 is the permeability of vacuum(4π×10−7 T.m/A); and M is the magnetization of the mag-nets. It models the magnets as electric dipoles with uniformmagnetization, Equation (5) represent the interacting forcecomponents between the poles of two electric dipoles.

It has been previously observed that the dipole model tendsto overestimate the magnet repulsion forces particularly forsmall air-gaps [19]. Thus, to have a better estimation of magnetrepulsion forces, we conducted a Finite Element Analysis(FEA) using FEMM software [20].

Air

Y

X (symmetry axis)

NS N S

Fig. 2. The FEMM model for a pair of repulsive magnets. Heat mapindicates the numerical solution of magnetic flux density.

Figure2 demonstrates a sample FEMM model used in thisstudy, to numerically calculate the repulsion force between twopermanent magnets. An axial symmetry model about the centerof the cylindrical magnets is utilized and the magnetization isdirected along the symmetry axis (X). Relative permeabilityis set to 1.05 and 1.0 for the neodymium magnets and sur-rounding air, respectively. FEMM automatic mesh generatoris used to define mesh density. A solver precision of 10−8

is employed to ensure that the root mean square of fluxresiduals is less than this value. The interaction force betweenthe magnets are calculated numerically, for different magnetsizes within the gripper’s functional range of air-gap. Basedon the FEA results, cylindrical magnets with the diameterof 15 mm, thickness of 6 mm and magnetization strength of6.05×105 A/m are shown to provide 2 to 10 N forces withinthe gripper’s functional range of air-gap and therefore are se-lected. To validate the accuracy of the analytical dipoles modeland the numerical FEA, repulsion forces are measured in anexperiment conducted with the selected neodymium magnets.For this purpose, two magnets in repulsive configuration areattached to two sliders mounted on a same rail. One of thesliders is fixed to a force sensor, while the other one isactuated. The air-gap between the magnets is measured usinga linear sliding potentiometer sensor. The experimental resultsare shown in Fig. 3a.


Comparing the analytical dipole model with the measuredforces indicates a significant difference, particularly for smallair-gaps (d/Hm<2) as expected. This is mainly due to theassumption of electric dipoles that makes the analytical modelof (5) significantly different from the actual magnetic repulsionwhen the air-gap to magnet height ratio (d/Hm) is small.Compared to the dipole model, the numerical results of FEMare closer to the experimental data. The difference betweenthe FEM results and actual measurements is mainly due tothe effect of non-ideal magnetic field of actual magnets.

Although the FEM simulation provide insight for the mag-net selection in the design process, it cannot provide an analyt-ical model necessary for collision detection and reactions. Toaddress this issue, a data driven approach is used to model themagnet repulsion force [17]. A modified inverse power law,as expressed in (6), is proposed for this purpose.

fm(d) =P1

d3 + P2d+ P3(6)

The power degree is chosen based on the two extremecases of air-gap. From (5), one can find that the repulsionforce is proportional to an inverse power law of degree four(fm∝1/d4) for large air-gaps (d�Hm), whereas it tends tobe (fm∝1/d2) for small air gaps (d�Hm). Considering thefunctional range of air-gap of the gripper fingers to be betweenthese extreme cases, a power degree of 3 has been chosenfor model fitting. The Matlab nonlinear least-squares withtrust-region method is used to estimate the model parametersP1, P2, P3 in (6) as 5.5×10−6 N.m3, 6.85×10−5 m2, and2.22×10−7 m3, respectively.

Figure 3b demonstrates the fitted mathematical model. Thelarge coefficient of determination (R2=.99) indicates the suit-ability of a power degree of n = 3 for our functional rangeof air-gap. The fitted model is then used as the repulsionforce function (fmi) for analytical modelings and simulations.Furthermore, a set of experiments is conducted to assess thevalidity of the fitted model when used in the gripper setupand identify the stiffness characteristics of the fingers. For thispurpose, the external force applied on finger-1 (fe1) and theassociated passive deflection (p1) are measured for differentstiffness setups via altering the distance between the actuatedmagnets (u1−u2). Figure 4a shows the experimental setupused for this purpose. A force sensor is used to measure the

5 10 15 20 25Air-gap (mm)

0

2

4

6

8

)N( ecroF

(a)

Forc

e (N

)

(b)5 10 15 20 25

Air-gap (mm)

0

2

4

6

8

Fig. 3. (a) Comparison of the measured repulsion force of the analytical(black) and the Finite Element models (green) with the experimentalloading (blue) and unloading (red). (b) Fitted Model compared to theaveraged loading and unloading experiments. The fitted model has RMSError of 0.06N.

external forces and the finger’s deflections are acquired fromthe sliding potentiometer sensor.

Figure 4b compares the experimental results of each stiff-ness setup with the corresponding estimated forces obtainedfrom the fitted model of (6). The small differences betweenthe measured and estimated forces indicate the reliabilityof the proposed magnetic repulsion model. As expected, bydecreasing the distance between the actuated magnets (smalleru1 − u2), the force-deflection curve of the finger representsa stiffer behavior. The use of nonlinear preloaded magneticsprings results in a non-constant force-stiffness relation suchthat the stiffness gradually increases as the finger deflectionincreases. This progressive stiffening is the main characteristicof antagonistic VSAs that provides smooth transition betweendifferent levels of stiffness and also protect the fingers fromreaching their motion limits.

In addition to magnetic model verification, the experimentaldata in Fig. 3a are also used to measure the frictional loss inthe linear sliders and investigate the hysteresis effect of themechanism based on the area enclosed within the loading andunloading curves. The root mean square (RMS) of the differ-ence between the loading and unloading curves was 0.21 Nwhich is less than 2.5% of the maximum force magnitude.This small difference indicates the low friction and hysteresiseffect in the mechanism as a result of using recirculating ballbearing sliders and permanent magnets. Indeed, lower frictionsenhance the dynamic modeling of the system which is essentialfor the collision detection algorithm in section III.

III. COLLISION DETECTIONEnhanced safety and robustness to collisions are the key

features of a compliant grasping that can be easily achievedbased on the low stiffness setups of the proposed gripper.In fact, the magnetic springs of the fingers can absorb thecontact energy and reduce the impact force caused by thecollision of the fingers or the grasped object with the envi-ronment. This reduces the collision damages to the gripper,grasped object, the environment and humans. Furthermore,the inherent compliance of the gripper can provide controllerwith the possibility of detecting collisions without the needfor additional force sensors. However, detecting collisions andswitching to a safe reaction strategy without force sensorsrequire a reliable and fast collision detection algorithm thatis presented in section III-A.

(a) (b)

Forc

e (N

)1 2

3

4x 10-3

( − )

1 − 2increases

1- End-Effector 2- Potentiometer3- Gripper Finger 4- F/T Sensor

Fig. 4. (a) Experimental setup for identifying finger’s stiffness charac-teristics. (b) Finger deflection versus external force measured in theexperiment (black line) and estimated form the fitted model (red dots).


A. Grasped Object DynamicsFigure 5 shows a schematic representing the planar grasping

with the VSA-based gripper attached to the end-effector of arobot arm with n revolute joints and generalized coordinatesof q∈Rn. Considering a suitable grasp force, a firm and stablegrasp is assumed so that neither slipping nor rotating along thesqueeze axis is allowed. Two frames are defined to express theobject dynamics including the base frame Ob − xbybzb andthe end-effector frame Oe−xeyeze. The absolute translationalvelocity of the grasped object with respect to the end-effectorframe is described by (7).

peo = RT pb

o = RT(Jpq + S(ωb

e)rbe,o + R[x1, 0, 0]T

)(7)

where R is the rotation matrix of frame Oe with respectto frame Ob; Jp is the (3×n) Jacobian matrix of the armrelating the end-effector linear velocity to the joint velocities;ωb

e denotes the absolute angular velocity of the gripper withrespect to the base frame; S(.) is the skew-symmetric matrixoperator, and x1 is the relative velocity of the object withrespect to the end-effector frame. Note that x1=x2 since a firmgrasp is assumed. Based on the angular velocity and rotationmatrix properties [21], the relation (8) holds.

RS(ωee)R

T = S(Rωee) (8)

where ωee is the absolute angular velocity of the gripper rep-

resented in the end-effector frame (ωee = RTωb

e). Therefore,(7) can be simplified further as (9),

peo = RTJpq + S(ωe

e)ree,o + [x1, 0, 0]T (9)

The translational Newton-Euler formulation can express theobject’s equation of motion in the end-effector frame as (10).

m(peo + S(ωe

e)peo −RTg) = f (10)

where peo is the absolute translational acceleration of the

grasped object in the end-effector frame; m is the combinedmass of the grasped object and fingers; g is the gravitationalacceleration vector, and f=[fx, fy, fz]T is the overall forcevector acting on the grasped object in the end-effector frame.

For detecting collisions, we are mainly interested in thecompliant motions of the object along the fingers’ axis (xe).Thus, the equation of motion will be used only in xe coordi-nate. Therefore, only the first components of f will be requiredwhich can be expanded as (11).

Fig. 5. Schematic of the grasping with the variable stiffness gripper.

fx =

4∑i=1

fmi + Fvx1 + Fssgn(x1) + fe (11)

where fmi is the repulsive force of ith actuated magnet andhas been derived and discussed in Section II-C; fe is theprojection of external forces in xe coordinate; Fv and Fs arethe viscous and Coulomb friction coefficients, respectively. Aviscus coefficient of Fv=0.45 Ns/m is estimated for the slidersof the gripper mechanism based on the decay rate of slider’sfree oscillations. For this purpose, one of the gripper fingersis released from an initial deviation and the oscillations arerecorded using its potentiometer sensor. The Coulomb frictionis found to be very small and therefore is neglected.

B. Momentum-Based Collision DetectionWith an identified dynamic model, a straightforward choice

for collision detection is the use of position measurements toestimate fe according to (10) and (11). However, such directestimation requires numerical computation of accelerationterms (q and x1) which is not trivial in the presence of mea-surement noise. Luca and Mattone [22] proposed the use of amomentum observer to detect collisions in robot arms withoutthe need for force sensors. A similar concept is adopted inthis study and modified according to the motion dynamics ofthe grasped object and fingers’ compliance. For the object, theso-called residual in xe coordinate is defined by (12). It’s thedifference between the directly computed momentum at eachinstance and the one obtained from integrating the derivativeof the model-based momentum (14) over time.

r(t) = KI

[p(t)−

∫ t

0

(p(u) + r(u))du

](12)

where KI is a positive gain; p(t) represents the linear mo-mentum along xe as expressed in (13), and r(0) = 0.

p(t) = m(xeTe pe

o

)(13)

Using the equation of motion for the grasped object (10), thederivative of the momentum can also be computed by (14).

p(t)=mxeTe

[RTg − S(ωe

e)peo

]+

4∑i=1

fmi+Fvx1+Fssgn(x1) (14)

where xee is the coordinate vector of xe axis in the Oe

frame that is simply [1, 0, 0]T . From (10), (13) and (14), thedynamics of r can be represented by (15).

r(t) = KI(fe − r(t)) (15)

which represents a first-order filter with the gain KI . There-fore, physical collisions can be detected when r(t) exceedsome threshold (||r(t)||>rth). This threshold is defined toprevent potential false alarms due to measurement noise and/oruncertainties in the model. According to (15), the sign ofthe residual r(t) provides the directional information of feand it goes back to zero when the contact force (fe) is over.With regard to the collision detection, the main limitation ofthe proposed two-finger model is that it can only detect theexternal forces with non-zero components along xe coordinate.


Time (sec)0 2 4 6 8 10 12

Line

ar V

eloc

ity

(m/s

)A

ngul

ar V

eloc

ity

(rad

/s)

-1

1

0

-0.2

0.2

0

(b)(a)

Fig. 6. (a) Intentional collisions with the gripper during arm motions toassess the collision detection algorithm. (b) Tracked linear and angularvelocity profiles of the end-effector in the collision detection experiment.

C. Experimental Results

Two experiments are conducted to evaluate the collision de-tection algorithm. In both cases, the arm follows a predefinedtrajectory while a cuboid object with a mass of 115 g wasgrasped by the gripper. The first experiment was collision-free whereas the second one involved two collisions with thegrasped object at different instances. The impact forces areapplied manually using a soft flexible ruler as shown in Fig. 6a.

Both experiments are carried out with a same referencetrajectory of the robotic arm. For the predefined trajectoryof the arm, we considered the most challenging case wherethe dynamic terms of (14) are excited. For this purpose,we utilizeed the reference trajectory suggested for dynamicparameter identification of this robotic arm [23]. The resultingangular and linear velocity of the gripper with respect to thebase are illustrated in Fig. 6b. The first 2 seconds of theseprofiles correspond to a Hermite interpolation between thejoints zero homing position and the starting point of theexciting trajectory.

The deviations of the object from the static equilibrium(before arm movement) as a result of end-effector motions

Fig. 7. Object displacements from the equilibrium point in the exper-imental trials with collisions and collision-free cases. Correspondingresiduals of each case are shown in the 2nd row. The collision instancesare highlighted in red. The dashed blue lines are the residual thresholds.

and collisions are shown in the first row of Fig. 7. Thecorresponding residuals are computed based on (12) with adetection gain of KI=40. The obtained results are representedin the second row of Fig. 7. A residual threshold of rth=0.15 Nis found to be large enough to avoid false alarms due tomeasurement noises and modeling uncertainty.

It should be noted that a heavier object may improve thedetection performance through increasing the signal to noiseratio (SNR) of the residual calculation. In fact, the frictionalterms in (14) are one of the main sources of uncertainties suchas stick-slip. Therefore, for a larger mass (m), the ratio of thefrictional to inertial terms will decrease and result in a higherSNR. The weight of the grasped object in this study is chosenbased on the resolution of the fingers’ position sensors. Infact, for heavy objects, the passive deflections of the fingersdecrease and a high sensor resolution is required to calculatethe object’s momentum.

IV. COLLISION REACTION

Upon the detection of an undesired physical collision, therobot controller must switch from the control law associatedwith the nominal task execution to a safe reaction controllaw [24]. The simplest reaction strategy is utilizing a faststop mechanism by either engaging mechanical brakes at eachjoint or commanding zero velocity at the controller level [25].However, such methods would not eliminate the arm contactwith the environment/human and the jerky arm motions causedby the fast stop may cause instability in the grasp.

To address this issue, different active reaction methods forrobot arms have been proposed to improve collision safetyparticularly in physical human-robot interaction. For instance,a series of “reflex” strategies have been proposed in theliterature in which the robot links are pushed away by applyingan amplified collision torque along the direction of the contactforces at the joint level (e.g., [26], [27]). Most of the proposedreflex reaction methods rely on the torque control of robotjoints as a function of the computed residual vector. Thesecond group of reaction strategies are based on admittancecontrol schemes which requires the availability of low-levelhigh-gain control loops. In this case, a robot velocity commandis generated in response to the estimated residual vector bymimicking the robot’s behavior as a mass-damper system. Asa result, the robot motion will be slowed down until reachinga rest position away from the contact area (e.g., [28], [29]).

Since the collision reaction problem is extensively studiedfor robot arms, in this study, we only focused on the reactionstrategy at the gripper level which is utilized upon recognitionof a collision with the gripper fingers to ensure a stablegrasp. Even in the presence of a collision detection algorithm,direct physical contacts and passive deflections of the fingersare unavoidable due to the robot inertia and joint elasticity.Thus, the adjustable stiffness of the gripper can be utilizedto enhance the safety and robustness of the grasp duringcollisions. This will lead to a nonlinear optimization problemwith the actuated magnets’ positions (or grasp stiffness) as thedesign parameters.


TABLE ILIST OF CONSTRAINT VALUES.

Constraint u1max u2min wm wf umax

Value 76 mm 0 mm 6 mm 22 mm 0.15 m/s

A. Problem Formulation and Numerical SimulationWe assumed the collision happens to one of the fingers when

an object is grasped by the gripper. Due to the collision, bothfingers will deflect from their equilibrium in the direction ofcontact force and along xe (as depicted in Fig. 1c).

To find an optimal gripper control input at each instance(u1,2), once a collision has been detected, an optimizationproblem is formulated as (16).

minu1,2

J = α(fg − fgd) + (1− α)fe (16)

subject to,

max(x1, x2) +

wf+wm

2 ≤ u1 ≤ u1max

u2min ≤ u2 ≤ min(x1, x2) +wf+wm

2

|u1,2| ≤ umax

fg ≥ fgd

(17)

where J is a linearly weighted multi-objective cost functionaiming to minimize a combination of the contact force withthe external environment (fe) and the deviation of the graspforce (fg) from its desired value (fgd). Minimizing fe reducesthe contact forces and potential harms to the environment (e.g.,workspace or human operator), whereas maintaining the graspforce fg improves the grasp robustness and reduces the riskof damage to the grasped object due to the high compressionforces between fingers caused by the collision. Coefficientα assigns a weight (0 to 1) to each objective according toits importance. For instance, one may consider a large α forphysical human-robot interaction scenarios like handing-oversharp objects, or a small α when the grasped object is fragile.

Regarding the problem constraints formulated in (17), thefirst two constraints are imposed by the mechanical motionlimitations of the gripper mechanism. The geometrical param-eters wf and wm are the width of the fingers and actuatedmagnets, respectively, as shown in Fig. 1a. The third constraintis the maximum motor speed. Finally, the last constraintensures an active contact between the fingers and the graspedobject with a minimum force of fgd. The numerical valuesassociated with the constraints are chosen based on the actualgripper’s geometries and are listed in Table I.

To investigate the performance and limitations of the pre-sented collision reaction strategy, the defined optimizationproblem is solved numerically for two different α valuesand the results are compared to a constant stiffness case. Tothis end, a collision with an stiff obstacle is simulated inwhich, upon the collision, the optimal gripper inputs (u1,2)are computed as a function of the compliant displacements ofthe fingers (x1,2) caused by the robot inertia. The contact force(fe) and the grasp force (fg) are computed in a quasi-staticmanner by (18) and (19), respectively.

fe =

4∑i=1

fmi= fm(d2) + fm(d3)− fm(d1)− fm(d4) (18)

Gra

spin

g Fo

rce

(N)

Con

tact

For

ce (N

) u 1

(mm

) u 2

(mm

)

Fig. 8. Results of the simulated collision reaction for the case of constantstiffness (solid-black) and variable stiffness with an optimization factor ofα=0.05 and α=0.95. The vertical green line corresponds the maximumstroke of the outer actuated magnets (u1 = u1max).

fg =

{fm(d1)− fm(d2), if x1 > x2.

fm(d3)− fm(d4), otherwise.(19)

where fmi is the repulsion force of ith actuated magnet that ismodeled by (6) as a function of air-gap between associatedmagnets (di). The air-gap between each pair of repulsivemagnets can be easily measured based on the position feed-back from the actuated magnets and gripper fingers (e.g.,d1 = u1−x1−wf+wm

2 ). The simulation parameters for thecollision reaction are listed in Table II, where Lobj is the lengthof grasped object and ∆ = (x1 − x2)/2 is the simultaneousdeflection of the fingers (refer to Fig. 1c).

TABLE IILIST OF SIMULATION PARAMETERS IN COLLISION REACTION.

Parameter u1(t0) u2(t0) fgd Lobj ∆Value 72 mm 0 mm 2 N 80 mm [0-6] mm

The optimization problem (16) is solved numerically ateach time instance using Matlab nonlinear programming solverwith sequential quadratic programming algorithm. Figure 8illustrates the optimal gripper inputs (u1,2) and correspondingforces (fe and fg) with respect to ∆. The optimization resultscan be described before/after the point at which the outermagnets reach their maximum stroke (u1=u1max). These tworegions are separated by a vertical green line in Fig. 8.

Before reaching the maximum stroke of u1, the contactand grasp forces are remained less than the case of constant-stiffness for both optimal reactions cases with small andlarge α values. Interestingly, the optimized gripper actuationfor both cases are identical in this region regardless of theobjective weights. This suggests that, as long as u1 ≤ u1max,the optimization results for any given α 6=1 lies on the equality


Con

tact

For

ce (N

)B

uckl

ing

(mm

)

(a) (b)

1- Spaghetti Stick2- Obstacle3- Force/Torque Sensor

12

3

10

8

6

4

2

03

2

1

00 1 2 3 4 5 6

(mm)Δ

Sim. Exp.

Breaking Point

Fig. 9. (a) Experimental setup for collision reaction. (b) Collision reactionresults for constant and variable stiffness strategies. The first graphrepresents the contact force (fe) measured by an external force sensor.The sudden decrease in the contact force of constant-stiffness for∆ > 5.8 mm is due to the breaking of the grasped spaghetti stick. Thesecond graph represents decrease in the length of spaghetti stick dueto its buckling after collision.

condition of the last constraint (fg ≤ fgd), while the graspstiffness reduces as ∆ increases. However, a trade-off betweenthe contact and grasp force is evident after outer magnets reachtheir maximum stroke u1 > u1max. In this state, maintaining aconstant grasp force can only be obtained at the cost of bearinghigh contact forces with the environment. In this case, thecontact force may even exceed the case of constant-stiffness. Infact, to save grasps object from exceeding compression forcewith a high α, the stiffness of fingers increases as ∆ increases.

B. Experimental Results

The performance of the proposed collision reaction methodis evaluated through a set of collision experiments that arethe same as the simulation cases. This included a constant-stiffness and variable-stiffness with α=0.05 and 0.95. Figure9a illustrates the experimental setup used for this purpose,which consists the same components as the setup in sectionII-C. A force sensor is used to measure the external forcesapplied to the finger. The grasp object was a spaghetti stickwhich was very fragile. It could buckle and even break at highlevel of grasping forces. The experimental parameters were thesame as simulation parameters.

Reduction in the object length, caused due to the bucklingof the spaghetti, is measured through the fingers’ positionsensors. This buckling can serve as an indicator of disturbancesin the grasp force. A straight trajectory was tracked by theend-effector for the collision reaction experiment such thata maximum end-effector velocity of 20 cm/s was reachedbefore the contact point. Then, the arm decelerated upon thecollision with -2.5 m/s2. These values are chosen according tothe speed limit of the gripper motors. The contact betweenthe gripper and the aluminum profile (attached to the forcesensor) leads to the passive deflection of the fingers (∆). Theresults are presented in Fig. 9b with respect to this passivedeflection similar to the simulation results. Also, snapshot of

the experiment for the three different conditions are shown inFig. 10. The location of the active magnets are highlighted andthe status of the grasped object is indicated in red.

Figure 9b compares the results of collision reaction exper-iment with 3 different stiffness settings. The large α valuerepresents a reaction toward minimizing the deviation ofgrasp force from its desired value (2 N), whereas the small αprioritizes the minimization of contact forces. Comparing theexperimental results with the simulation ones, similar trend inall three cases can be observed. With the constant stiffness,the collision test results in the breakage of spaghetti stickindicating an increased grasp force due to the collision andfingers’ compliance. Contrary to the simulation results, thedivergence between the reactions with α=0.05 and 0.95 areobserved for ∆ ≈ 4 mm that is mainly due to the actuatordynamics. The largest passive deflection ∆ measured forα=0.95 is around 5.6 mm. This is due to the fact that thelarge external contact force of this reaction has led to somedeflections in the structure of the gripper as well.

V. CONCLUSIONIn this paper, we presented the design and modeling of a

two-finger variable stiffness gripper with the aim of enhancingcollision safety. In the design of gripper, the compliancebetween gripper fingers and actuators were generated using re-pulsive magnetic springs. The position control of the actuatedmagnets provided the capability of adjusting the position andforce-stiffness characteristics of the fingers simultaneously.

To demonstrate the effectiveness of the design, the ad-justable stiffness of the fingers was exploited to enhancecollision safety in the context of robotic manipulation. For

Con

sta

nt

Sti

ffn

ess

Va

ria

ble

Sti

ffn

ess

a=.95

Initial contact Mid-collisioin Terminal collision

BreakSevere Buckle

Mild BuckleVa

ria

ble

Sti

ffn

ess

a=.05

Fig. 10. Snapshots of the three cases in collision reaction. The actuatedmagnets are highlighted and their movement direction is indicated by anarrow. Full video at https://youtu.be/tYahoQgSiRM


this purpose, first, a momentum-based collision detectionalgorithm was adopted to identify collision instances relyingon only compliant deflections of the fingers without the needfor force sensors. Then, a collision reaction strategy wasstudied in the form of an optimal stiffness problem andconsidering fingers’ stiffness as the design parameters. Theeffectiveness of adjusting fingers’ stiffness was demonstratedby defining a simple linearly weighted cost function as thereaction objective. However, one may consider a differentobjective function or consider the motion dynamics of thegrasped object to further improve results.

It should be mentioned that there are also some limita-tions associated with the proposed gripper. First, due to theantagonistic actuation, the feasible range of grasp stiffnessdepends on the minimum required grasp force as discussedin section II-B. For instance, low stiffness behaviors cannotbe achieved for large grasp forces. Second, the performanceof collision reaction highly depends on the dynamic featuresof the actuators and accuracy of the fingers’s position sensors.A proper reaction to fast collisions requires quicker magnetpositioning within short durations. Third, the compliance ofthe fingers and accordingly the collision detection and reactionproblems were limited to the a single DoF. Only collisionswith non-zero force components along the fingers’ axis couldbe detected with the proposed parallel gripper. To addressthis issue, our design concept can be extended to highernumbers of DoF with multiple fingers. As a future direction,one can further study the adjustable grasp stiffness in dynamicmanipulation tasks, such as hammering and re-grasping, in thecontext of optimal control framework.

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a robot gripper with variable stiffness actuation for

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