Design of a Novel Variable Stiffness Gripper Using Permanent Magnets

Amirhossein H. Memar, Nicholas Mastronarde and Ehsan T. Esfahani

Abstract— This paper presents the design of a novel variablestiffness gripper with two parallel fingers (jaws). Complianceof the system is generated by using permanent magnets as thenonlinear springs. Based on the presented design, the positionand stiffness level of the fingers can be adjusted simultaneouslyby changing the air gap between the magnets. The modeling ofmagnetic repulsion force and stiffness are presented and verifiedexperimentally. An experiment is also conducted to demonstratethe functionality of the gripper to improve safety when a fragileobject was grasped and the gripper collided with an obstacle.

I. INTRODUCTION

The capability of reliable grasping and handling objects

is essential to industrial robots as well as service robots for

physical human-robot interaction (pHRI). Moreover, safety

becomes an important issue when a robot handle fragile or

sharp objects in uncertain and dynamic environments. Con-

siderable research effort has been directed towards increasing

the human safety in pHRI [1]. For instance, the safety

analysis of soft-tissue injuries caused by sharp tools grasped

by a robotic arm has been experimentally investigated [2].

To address such safety issues, compliant interaction have

been introduced which can be categorized into active and

passive approaches. In most cases, compliance of stiff robots

is achieved by using active impedance control. However,

active control methods are particularly limited in terms

of shock tolerance and energy storage due to restricted

mechanical and electrical bandwidth [3]. Therefore, passive

compliance has been received increasing attentions in the

recent years to ensure the safety of grasped objects during

inadvertent collisions [4]. Inclusion of elastic elements into

the design of robotic joints/actuators reduces its stiffness

and ensures an infinite disturbance bandwidth. Depending

on whether the stiffness of a compliant joint is constant or

adjustable, it is called a Series Elastic Actuator (SEA) [5]

or a Variable Stiffness Actuator (VSA), respectively. SEAs,

on the other hand, have some limitations in terms of trade-

off between position accuracy and energy absorbing capacity

because these features are highly dependent to the stiffness

value which is constant in the case of SEA. To overcome

this issue, VSAs are proposed at a cost of an extra actuator.

VSAs provide users with not only the intrinsic character-

istics of a SEA (e.g. impact energy storage), but also a range

of adjustable stiffness. A low stiff setup allows the system

to provide a fine force control with light load capacity,

AH. Memar and ET. Esfahani are with the Department of Mechanicaland Aerospace Engineering, University at Buffalo - State University of NewYork, Buffalo, NY, 14260 USA. Nicholas Mastronarde is with the Depart-ment of Electrical Engineering, University at Buffalo - State University ofNew York, Buffalo, NY, 14260 USA. ahajiagh@buffalo.edu,ehsanesf@buffalo.edu

however, increasing the level of stiffness leads to the faster

transfer of the actuator power to the load, and consequently

faster position control and heavier load capacity. Several

different mechanisms for changing the stiffness of VSAs

have been proposed in the literature which are categorized

and summarized in [6]. The actuators of these mechanism

can be configured either in antagonistic or independent setup.

The most common design approaches and the associated

factors influencing their performance are presented in [7].

Despite various VSA mechanisms which are introduced in

the literature, robot grippers with variable stiffness actuation

and rigid fingers are relatively little studied. By utilizing

the concept of VSA in the design of robotic grippers, the

performance of manipulation tasks can be improved from

different point of views such as safety, grasp robustness and

the potential of performing dynamic manipulation. Intrinsic

compliance of such grippers allows the controllers to esti-

mate contact forces without the need for force sensors via

measuring the deflection of elastic elements. Moreover, by

employing a gripper with variable stiffness fingers mounted

to a commercial arm, some aspects of a fully VSA-based arm

can be achieved to accomplished a group of tasks such as

assembly, polishing, and improving safety during handling

fragile or sharp objects in pHRI.

In this paper, the design of a two finger gripper with

adjustable stiffness is discussed. This gripper takes advan-

tage of using magnetic repulsion force as the nonlinear

elastic element. The proposed concept corresponds to the

spring preload group of VSAs with antagonistic springs

and antagonistic motors. In this mechanism, simple linear

motion units are used to adjust the displacement of the

magnets by position control of two electric servo motors

such that equilibrium position and stiffness of the fingers

can be controlled simultaneously. The effectiveness of using

permanent magnets as the elastic elements in a VSA has been

shown in [8] by developing a magnetic torsional coupler. The

authors of this article characterized the developed coupler by

its compactness, reliability and variable compliance.

Although, most of the proposed VSAs in the literature

can be used as the actuator of a robotic gripper (e.g. [9]),

the particular design of the gripper presented in this paper

provides users with more functionalities such as low friction,

lightness and compactness. Moreover, this mechanism can

provide the controllers with contact force measurement for

each finger individually. Another characteristic of this design

is the compliance motion of the gripper when an object

is grasped and an external disturbance acts on one of the

fingers. This property can improve the robustness of grasping

when inadvertent collisions occur due to the arm movements.

2017 IEEE International Conference on Robotics and Automation (ICRA)Singapore, May 29 - June 3, 2017

978-1-5090-4633-1/17/$31.00 ©2017 IEEE 2818

II. DESIGN

A. Concept

The main objectives for the design of the gripper was

simultaneous stiffness and position control of the parallel

jaws using two identical actuators connected to compliant

elements. In addition, due to the need for attaching the

gripper to the end-effector of a lightweight robotic arm, there

was a limitation in terms of size and weight of the gripper.

One of the key features required for generating stiffness

variation with preload springs in an antagonistic mechanism

is the non-linearity of springs. In fact, this feature offers

the capability of smooth transition between different levels

of stiffness. In this project, permanent magnets in repulsive

configuration are employed to fulfill the requirement of non-

linear springs. In fact, magnets can provide the system with

not only compactness and lightness [8] but also non-contact

force interactions between the actuators and the load. Such

features make magnets a good candidate for replacement of

mechanical springs in manipulation applications by offering

a number of advantages such as precise force transmission

and tolerance to misalignments.

A schematic of the conceptual design for the gripper

mechanism is illustrated in Fig. 1. Based on the proposed

concept, the position of the fingers which are mounted on

passive magnets can be controlled via translational motion

of the actuated magnets. Basically, in the absence of external

forces and neglecting the effect of gravity, each finger tends

to stay in the equilibrium position which is at the middle of

the two actuated magnets. In this configuration both of the

magnetic springs and actuators are placed in an antagonistic

setup. As a result, simultaneous control of the position and

stiffness of the fingers can be achieved by position control

of the actuated magnets. To achieve a symmetric motion

of fingers, the displacements of the outer magnets must be

equal and in opposite direction, while having a same motion

synchronization for the inner magnets. Therefore, two motors

can satisfy the position control of the four actuated magnets

which are illustrated in Fig. 1. Through this design concept,

the position and stiffness setup of the fingers can vary by

changing the positions and air gaps between the passive and

actuated magnets.

Similar to the most of the VSAs presented in the literate,

the current system is capable of measuring contact forces

with no need for force/torque sensors and the magnitude

of a force acting in the direction of the finger axis can

be measured based on the deviation from the equilibrium

position. However, the presented design concept offers two

main characteristics for a gripper with parallel jaws.

The first characteristic is the capability of measuring

contact forces acting on each finger individually. To this end,

two linear position sensors (e.g. sliding potentiometer) are

required to measure the displacements of passive magnets.

By having the positions of the magnets, the direction and

magnitude of the contact forces can be estimated for each

finger using magnetic repulsion force models. This property

can improve the functionality of the gripper especially in the

N N N N NNS S S S S S

Actuated Magnets

Finger

End-Effector Flange

M1 M2

Object to be

grasped

Passive Magnet

Fig. 1: The upper image shows the schematic of the design

concept. Permanent magnets in repulsive configuration are

used as an alternative to the non-linear springs as illustrated

in the lower image. Two motors (M1 and M2) are used to

control the positions of the actuated magnets.

tasks with uncertainties in the location of the object before

the grasping task (see Fig. 2a).

Furthermore, this mechanism provides compliance in the

direction of fingers axis when an object is grasped and

an external contact force exerts at the outer side of the

fingers (Fig. 2b). In this situation, the compliance of the

gripper can protect the object from high impact forces. This

property can also improve the safety aspects of grasping

particularly in industrials robots, where, fast stop methods

such as electromagnetic brakes rely mostly on mechanisms

without the capability of energy storage. In these cases,

braking after fast motions may lead to jerky oscillations of

the end-effector (due to the inherent stiffness of joints and

gear trains) that increase the risk of grasp failure.

Fi

Fe

(b)

(a)

∆1

∆2 ∆2

Fig. 2: Two types of compliant motion of the gripper fingers.

(a) Passive movement of a single finger due to an external

force. (b) Simultaneous passive movements of the fingers

when an external force acts on the outer side of a finger.

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The main limitations of the proposed design are mainly

related to the range of passive deflection and its energy

efficiency. Due to the translational nature of the fingers,

there is a limitation in terms of passive deflection range,

especially when the grasped object is too small or large.

In such cases, actuated magnets must be positioned very

closed to each other that limits the feasible length of passive

motion and the bandwidth of the stiffness setup. Moreover,

in VSAs with antagonistic springs and antagonistic motors,

the stiffness varies by storing energy in the springs which is

not retrievable and leads to reduced energy efficiency.

B. Mechanical Design

Each component is chosen or designed to fulfill the design

requirements discussed in the previous section. Fig. 3 illus-

trates the overall assembly model of the gripper. Timing belts

are used to transfer the rotational motion of the actuators

to translations of the magnets. Each pair of the actuated

magnets (the outer and inner magnets) are required to move

simultaneously in opposite directions and the same rates.

Thus, both sides of each belt are used to transfer actuator

forces to the magnets. Each motor axis is passed through

two pulleys, an active pulley fixated on the motor axis to

drive the belt and an idler pulley to support the other belt.

Each pulley has a pitch diameter of 25.4 mm.

Two identical servo motors (Dynamixel RX-24F) are used

to drive the magnets. The positions of the magnets are

controlled via position control of the servo motors with a

resolution of 0.29◦. Linear recirculating ball bearings and

guideways are used as the rail and guiding elements for the

translational motion of the passive and actuated magnets.

These components offer smooth precision motion with low

friction and high rigidity that is essential for an accurate

grasping task. Due to the non-contact nature of magnetic

forces, the only frictional term acting on the motion of the

fingers corresponds to the sliders.

8

7

6

4

3

2

1

50mm

80mm

5

1 Gripper finger 5 Linear guideway

2 Magnet holders 6 Slider block

3 Servo motor 7 Potentiometer

4 Idler/fixated pulleys 8 End-effector adapter

Fig. 3: Assembly model of the gripper. Two servo motors

actuated four magnets to adjust the position and stiffness of

two passive magnets which are connected to the fingers.

S N N S

NS N S

(a)

(b)

sh h

sh h

10-2 10-1 100 101 102

Log(s/h)

100

105

1010

Log(

F/F

0)

Fig. 4: Schematic of interaction between cylindrical magnets

(a) versus electric dipoles (b) in a repulsive configuration.

Graph (c) shows the log-log of the force vs. (s/h) given

by Eq. (1) (black line) compared to two limit cases. First,

the limit case of Fm ∝ 1/s2 for small distances (gray dotted

line). Second, the limit case of Fm ∝ 1/s4 for large distances

(grey dashed line). The blue area demonstrates the functional

range of magnets interaction for the designed gripper.

Two similar sliding potentiometer sensors with analog

output are attached to the guideways to measure the positions

of the fingers, while, the angular rotations of the servo motors

are used to calculate the displacements of the actuated mag-

nets. A 10-bit analog-to-digital interface is employed to con-

vert the output of the sliding potentiometers to displacement

values with a resolution of 0.06 mm. Identical neodymium

magnets with cylindrical shape are used as the passive and

active magnets of the system. The size and flux density of

the magnets can be chosen in order to meet the stiffness-

force requirements for specific applications. 3D printing was

used to create the gripper body and the connection links

between mechanical components. The gripper was mounted

at the end-effector of a lightweight 6-axis robotic arm which

was controlled and dynamically identified previously [10] for

the experimental purposes.

III. MODELING

A. Magnetic Repulsion Force

Analytical modeling of the attractive and repulsive forces

between two magnets is not a trivial task. It depends on

a number of parameters such as flux density, geometry,

orientation and air gap between the magnets. However, with

the assumption of uniform magnetization, a mathematical

model is proposed in [11]. This model estimates the magnetic

repulsion between two aligned cylindrical magnets with a

radius of R and height of h as described in (1),

Fm(s) =πμ0M

2R4

4

[1

s2+

1

(s+ 2h)2− 2

(s+ h)2

](1)

where, Fm is the repulsive force, s is the air gap between

the magnets, μ0 is the permeability of vacuum (4π × 10−7

T.m/A), and M is the magnetization of the magnets. For the

case of employed neodymium magnets, the magnetization is

approximately equal to 6.05×105 (A/m) as reported by the

manufacturer’s datasheet.

Equation (1) is composed of the force interaction com-

ponents between two electric dipoles (Fig. 4b), where the

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first term inside the bracket corresponds to the force be-

tween the nearest poles (N-N), the second term denotes the

force between distant poles (S-S) and the third term is the

attraction force between the opposite poles (N-S). Based on

this relation, the repulsion force for large air gaps (s� h) is

proportional to an inverse power law of the form Fm ∝ 1/s4

, while it becomes Fm ∝ 1/s2 for small air gaps (s � h).

This is illustrated in Fig. 4c by the help of a log-log graph.

Since Eq. (1) is based on the assumption of uniform mag-

netization and ideal magnets, an experiment was conducted

to verify the accuracy of this model for the specific case of

our gripper. A force sensor was used to measure the actual re-

pulsive forces between two cylindrical neodymium magnets

in different distances. In this setup, the same linear unit and

guiding elements used in the gripper were used to generate

translational motion between two magnets. This gives the

possibility to investigate the effect of force hysteresis caused

by the friction in a loading and unloading motion.

0 0.005 0.01 0.015

Air Gap (m)

0

5

10

15

20

For

ce (

N)

Loading CurveUnloading CurveMagnetic Depoles Model

Fig. 5: The dotted blue and dashed red curve represent the

measured repulsion force of a loading and unloading motion,

respectively. The black curve demonstrates the corresponding

analytical force given by Eq. (1).

Fig. 5 illustrates the experimental measurements of mag-

netic repulsion versus air gap between magnets in a loading

and unloading motion. The frictional losses in the mechanism

can be measured by calculating the area enclosed within

the loading and unloading curves as a hysteresis effect.

However, small differences between loading and unloading

curves reported in Fig. 5 demonstrates a low frictional loss

in the gripper mechanism. The root mean square (RMS) of

the difference between the two curves is 0.34 N which is less

than 1.5% of the maximum force. The small deviations in the

force measurements are mainly due to the minor fluctuations

of the friction related to recirculating balls of the sliders [12].

Comparison between the experimental measurements and

the curve generated by Eq. (1) reveals a significant difference

between the actual and analytical model especially for small

air gap to magnet height ratios ( sh < 1.5). Disassociation of

the experimental and analytical model of Eq. (1) for small

distances between the magnets was also observed in previous

research (e.g. [13]) and this suggests the use of curve fitting

methods to model repulsion force for stiffness modeling

and taking into account the effect of non-ideal magnets. A

modified inverse power law expressed in Eq. (2) is used for

fitting the experimental measurements. In this model, a, band c are the model parameters, p is the power degree, and

s is the air gap between the magnets.

Fm(s) =a

sp + bs+ c(2)

With reference to Fig. 4c, it is illustrated that for the

case of our gripper, log10(s/h) is in the range −1 to 1.

Therefore, a value about 3 for p is expected to show a good

fit. To fit a model to the experimental data and find the

model parameters, the Matlab nonlinear least-squares solver

“lsqnonlin” with trust-region method is employed. Eq. (2)

with p = 3 demonstrated a good fit and the result of the

fitted curve is shown in Fig. 6.

0 0.005 0.01 0.015

Air Gap (m)

0

10

20

For

ce (

N)

MeasurementsFitted Model

Fig. 6: Experimental data (green points) and the fitted model

(black line) of magnetic repulsion force based on Eq. (2) for

p = 3 and identified parameters as a = 4.5e-6, b = 8.7e-5and c = 1.5e-7 (R2 = 0.998 and RMSE = 0.29N )

B. Stiffness

Based on the antagonistic actuation design of the gripper,

translational motions of the fingers can be controlled through

a combination of actuation inputs, whereas the antagonistic

setup of the preload magnetic springs provides the system

with the possibility of changing the stiffness setup while

keeping constant the position (before grasp) or contact force

(after grasp) of the fingers. In fact, passive magnets tend

to stay in the static equilibrium position between actuated

magnets and altering the distance between them leads to

the change of force-deflection characteristics of the fingers.

Lets assume that for one of the fingers the positions of the

actuated magnets are defined as x1 and x2. In the case

of identical actuated magnets and neglecting the effect of

gravity and friction, the static equilibrium position of the

finger (xf ) equals to (x1 + x2)/2 (See Fig. 7a). However,

when the gripper grasps an object or an external force acts

on it, the equilibrium position of the finger alters (Fig. 7b).

x1 x2xf

wmwm wf

Fe

s1 d12 s2

(a) (b)

Fig. 7: The left and right images represent the static equilib-

rium positions of the finger in the absence and presence of

an external force, respectively.

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Considering the magnet repulsion as the main forcing factor,

the following static equation holds,∑F = Fe + Fm1 + Fm2 = 0 (3)

where, Fe is the external force and Fm1 and Fm2 denote

repulsive forces of the actuated magnets associated with

coordinates x1 and x2. By considering the magnetic re-

pulsion as a function of air gap between the magnets and

substituting (2) in (3), the magnitude of the external force

can be estimated as Fm(s1)− Fm(s2) shown in (4),

Fe(s1, s2) =1

s31 + bs1 + c− 1

s32 + bs2 + c(4)

where, sn denotes the distance from surfaces of the actuated

magnet n to the passive magnet illustrated in Fig. 7, while a,

b and c are the parameters of the fitted model. The stiffness

for a given state can be described using the chain rule,

K(s1, s2) =dFe

dxf=

dFm1

dxf− dFm2

dxf

=dFm1

ds1

ds1dxf

− dFm2

ds2

ds2dxf

= dFm(s1) + dFm(s2) (5)

where,

dFm(s) =−a(3s2 + b)

(s3 + bs+ c)2(6)

The distances between magnets (s1 and s2) can be calcu-

lated in terms of magnet positions (x1, x2 and xf ) as follows,

s1 = xf − x1− 0.5(wm + wf ) (7)

s2 = x2 − xf − 0.5(wm + wf ) (8)

where, wm and wf are the width of the holders of actuated

and passive magnets, respectively, as shown in Fig. 7. Conse-

quently, the above expressions of the force and stiffness can

be rewritten as a function of the positions of the actuated

and passive magnets. For the presented design, the magnet

positions can be easily measured from motor rotations and

sliding potentiometer sensors.

In order to assess the reliability of the fitted model for

repulsion force estimation and examine the stiffness charac-

teristics of the gripper, an experimental setup was assembled

which is shown in Fig. 8. In this setup, the gripper was

attached to the end-effector of a robotic arm and an external

force was applied to one of the fingers due to the motion of

the arm. The applied force to the finger was measured using a

force/torque sensor, while the passive deflection of the finger

was acquired from its position sensor simultaneously.

By changing the distance between the actuated magnets

(d12), exerted forces were measured for different stiffness se-

tups. Experimental results and their deviation from estimated

forces obtained by Eq. (4) are shown in Fig. 9. The estimated

forces demonstrate good agreements with the measured ones

by an average RMS error of 0.43 N which is less than 2%

of the maximum measured force.

Due to the preload spring setup and nonlinearity of the

magnetic repulsion force, the stiffness increasingly varies by

Force Sensor

End-effector

Flange

Gripper Finger

Position Sensor

Fig. 8: Setup for force-deflection measurements.

applying more deflection which results in a non constant

force-stiffness relation. This stiffening effect can protect the

gripper from reaching the mechanical limits, however, real-

time position control of the magnets is necessary for the tasks

in which constant stiffness is desired.

The stiffness bandwidth at different amount of external

loads describes the stiffness variation performance of a

variable stiffness mechanism. For the case of developed

gripper, however, the range of stiffness and force bandwidths

vary by the positions of the fingers (or equivalently the

length of the grasped object). This affects the stiffness-force

performance of the gripper particularly in the case of small

grasped objects, where the griper fingers are too close to

each other. Fig. 10a illustrates the stiffness-force diagram

for the feasible object lengths. This drawback is a result of

reduction in the range of feasible passive deflection due to the

translational motion of the antagonistic magnets and parallel

jaws of the gripper. The stiffness-force diagram of the gripper

for a specific length of object is also shown in Fig. 10b. This

figure visualizes the minimum and maximum bands of the

feasible stiffness at different forces.

IV. COLLISION EXPERIMENT

To verify the functionality of the gripper in terms of

compliant motion during collision, an experimental study is

conducted. For this purpose, the gripper was mounted to the

end-effector of a robotic arm. A predefined trajectory was

tracked by the arm and a fragile object (an egg) was grasped

0 0.005 0.01 0.015

Deflection (m)

0

5

10

15

20

25

For

ce (

N)

(1)(2)(3)(4)(5)(6)(7)(8)

MeasurementsFitted Model

Fig. 9: Measured and estimated force versus deflection.

Different curves correspond to various setups of stiffness

obtained from changing the distance between the actuated

magnets (d12). Starting from 32 mm, d12 is decreased with

the step sizes of (1:2) = 5mm, (2:3) = 5mm, (3:4) = 3mm,

(4:5) = 3mm, (5:6) = 3mm, (6:7) = 3mm and (7:8) = 1.5mm.

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0 5 10 15 20 25

Force (N)

0

0.5

1

1.5

2

2.5

Stif

fnes

s (N

/m)

104

(b)

Fig. 10: a) the stiffness-force relation for the feasible range

of object lengths. b) Cross section of for object length of

8cm. The lower and upper curves demonstrate the stiffness

bandwidth in the range of feasible forces.

by the gripper. After completion of the grasping task, the

gripper collided with a heavy object at a speed of 25 cm/s.

This experiment scenario examines the simultaneous passive

deflection of the fingers (Fig. 2b), in a collision during pick

and place task. A collision detection algorithm was used

to stop the robotic arm by triggering the electromagnetic

brakes at the robot joints. The absolute velocity of the

simultaneous motion of the fingers was considered as the

collision threshold. A 5-points backward finite difference

method was applied on a moving window of the sensor data

to estimate the velocity. Experimentally, a threshold of 32

mm/s was found to perform reliably for collision detection.

Fig. 11 shows the sequence of the collision in three

instants recorded before and after the collision, while Fig.

12 illustrates the passive motion of the fingers and estimated

external forces caused by the collision. The compliance of

the mechanism resulted in simultaneous displacements of

the two fingers while the grasped object remained safe and

stable. The small differences in the deflections after the

collision show a small amount of slippage of the object

between the fingers due to the impact forces. The oscillations

after the collision were mainly due to the compliance of the

gripper and also the fast-stop mechanism of the arm which

caused vibrations in the elastic harmonic drives of the joints.

(t = 1)

(t = 2) (t = 3)

Fig. 11: Collision experiment. Changes in the length of the

yellow lines demonstrate the compliant motion of the fingers.

~~ ~~

Fin

ger

Dis

plac

emen

t (m

m)

(a) (b)

Fig. 12: a) Passive motions of the fingers caused by the

collision b) Estimated impact force.

V. CONCLUSION AND FUTURE WORK

This paper present the design aspects of a novel variable

stiffness gripper with two fingers. Magnetic springs in a

repulsive configuration are used as the non-linear preload

springs with antagonistic actuators setup. Two servo motors

adjust the position and stiffness of the gripper simultaneously

by changing the position of the magnets and accordingly air

gaps between them. The proposed design can also estimate

the external force at each finger independently. A data

driven approach was used to model and verify the magnetic

repulsion force and stiffness. Finally, compliance of the

proposed gripper is experimentally tested in a collision task.

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