The mVSA-UT: a Miniaturized Differential Mechanism for aContinuous Rotational Variable Stiffness Actuator

Matteo Fumagalli, Eamon Barrett, Stefano Stramigioli and Raffaella Carloni

Abstract— In this paper, we present the mechanical design ofthe mVSA-UT, a miniaturized variable stiffness actuator. Theapparent output stiffness of this innovative actuation system canbe changed independently of the output position by varying thetransmission ratio between the internal mechanical springs andthe actuator output. The output stiffness can be tuned from zeroto almost infinite by moving a pivot point along a lever arm.The mVSA-UT is actuated by means of two motors, connectedin a differential configuration, which both work together tochange the output stiffness and the output position. The outputshaft can perform unbounded and continuous rotation. Thedesign ensures high output torque capability, light weight andcompact size to realize a multiple purpose actuation unit for agreat variety of robotic and biomechatronic applications.

I. INTRODUCTION

Variable stiffness actuators (VSAs) realize a new classof actuation systems, characterized by the property that theapparent output stiffness can be changed independently ofthe output position. Such actuators are particularly significantwhen implemented on robots that have to interact safelywith humans and have to feature properties such as energyefficiency, robustness and high dynamics. In particular, theseactuators find their application in different fields of roboticsand biomechatronics. In prosthetics or rehabilitation robots,for example, the introduction of a VSA allows the device toadapt to the task and to increase not only the efficiency ofthe actuation, but also the comfort of the patient [1]–[4].

Like series elastic actuators, VSAs have the intrinsiccapability to store and release energy during nominal tasks.Because of their ability to change their stiffness, however,they can adapt their mechanical properties; for instance toadjust their natural frequency or to perform precise posi-tioning tasks. Nevertheless, the presence of internal elasticelements limits their efficiency of transferring energy fromthe internal motors to the output. This motivates the researcheffort on their mechanical design and control.

In the literature, mechanical compliance has been imple-mented in different ways in VSAs. In the ‘Jack Spring’TM

actuator [2], the apparent output stiffness is varied bychanging the number of active coils of the internal spring.Other actuators, e.g., the MACCEPA 2.0 [5], the VSA-II [6]the VS-Joint [7], the ANLES [8] and the VSA-CubeBot[9] change the apparent output stiffness by varying thepretension of the internal nonlinear springs. Other actuators,

This work has been funded by the European Commission’s SeventhFramework Programme as part of the project VIACTORS under grant no.231554.{m.fumagalli, e.barrett, s.stramigioli, r.carloni}@utwente.nl, Faculty of

Electrical Engineering, Mathematics and Computer Science, University ofTwente, 7500 AE Enschede, The Netherlands.

Fig. 1. The mVSA-UT.

including the vsaUT [10], the vsaUT-II [11], the AwAS [12],the AwAS-II [13] and the HDAU [14], change the apparentoutput stiffness by changing the transmission ratio betweenthe internal linear springs and the actuator output.

In this paper, we present the novel design of the mVSA-UT, which realizes a compact rotational variable stiffnessactuator. As other VSAs, this system consist of a number ofinternal springs and a number of internal actuated degreesof freedom, which determine how the elastic elements areperceived at the actuator output. The mechanical structureof the mVSA-UT is such that the apparent output stiffnesscan be varied by changing the transmission ratio betweenthe internal elastic elements and the actuator output, namelyby implementing a lever arm of variable effective length.The length can be changed by moving the pivot point alongthe lever arm by means of a planetary gear system, whichrealizes a linear motion along the lever. By satisfying thiskinematic requirement, the actuator’s output stiffness can bechanged without changing the potential energy stored in theinternal elastic elements.

The extremely compact design of the mVSA-UT can beachieved by implementing the two internal degrees of free-dom, i.e., the two internal motors, in a differential configura-tion. This implies that, by combining two small motors, it ispossible to have high torque/speed capability on the actuatoroutput and an independent control of the apparent outputstiffness, which can be varied from almost zero to almostinfinite. An additional feature of the proposed mechanismconsists in the possibility of performing continuous rotationof the output shaft [15], which guarantees a wide range ofapplications of the system. Fig. 1 shows a picture of themVSA-UT prototype.

The Fourth IEEE RAS/EMBS International Conferenceon Biomedical Robotics and BiomechatronicsRoma, Italy. June 24-27, 2012

978-1-4577-1200-5/12/$26.00 ©2012 IEEE 1943

II. REQUIREMENTSThe main goal of our work is to design a multipurpose,

compact and mechanically efficient VSA.Many of the VSAs present in the literature have a limited

range of output position due to their mechanical structure.In order to be multipurpose, the variable stiffness actuatorshould be capable of performing unbounded and continuousrotation. This feature increases the possibilities of applicationof VSAs on both robotic and biomechatronic fields.

Moreover, a compact design, i.e., lightweight and small,is an extremely important property of actuation systemsfor applications such as wearable devices, prostheses orexoskeletons, which require the mechanisms to be portable.For the purpose of having compactness, it is important touse small motors and to exploit their joint efforts, i.e., theirtorques, for both changing the output position and the outputstiffness. This requirement suggests to connect the internalactuated degrees of freedom in a differential configuration.

The third requirement follows the elaboration presentedin [10] and [16] by means of a port-based approach. Themechanism should realize a kinematic structure, such that theapparent output stiffness can be changed without injectingenergy into or extracting energy from the internal elastic el-ements. This property guarantees that all the energy suppliedby the internal actuated degrees of freedom can be used todo work on the output without being captured in the internalsprings. This characteristic is satisfied if the mechanicaldesign is based on a lever arm of variable effective length.As extensively analyzed in [11], a variable transmission ratiobetween the internal elastic elements and the actuator outputcan be realized by using a lever arm, if the position ofone of the three elements attached to the lever is varied,i.e., by moving the pivot point [11], [13], by changing theapplication point of the output force [10], [17] or by varyingthe attachment points of the internal springs [12]. It has beenshown in [11] that moving the pivot point along the lever armrealizes a more favorable design regarding the minimizationof mechanical work during stiffness changes.

In the next Section, we describe the mechanical design ofthe mVSA-UT, which fulfills the properties described above.

III. MECHANICAL DESIGNFig. 2 shows a sectioned CAD view of the mVSA-UT. Its

innovative mechanical design can be described in three mainlevels. In the first level, the variable stiffness mechanismis realized. It includes the output crankshaft (1), that isconnected to a lever (2), to which two linear springs (3)and a pivot pin (4) are also connected. The motion of thepivot pin is realized by a set of planetary gears (5) and (7).The second level is made of the differential mechanism thatactuates the output and pivot pin. It includes a second ringgear (10), planet gears (9) and the sun gear (8) which isconnected to the shaft of the planet carrier (6) of the firstlevel. In the third level, the internal motors (12) and (13)actuate the differential mechanism, by engaging the ring gear(10) and a second sun gear (11), that is also fixed to the shaftof the planet carrier (6).

9

1

4

5

7

13

3

2

11

12

6

8

10

Fig. 2. Sectioned CAD view of the mVSA-UT. The main parts of themechanism are: (1) the output, (2) the lever, (3) the springs, (4) the pivotpin, (5) the pivot gear, (6) the planet carrier, (7) the first ring gear, (8) thefirst sun gear, (9) three planet gears, (10) the second ring gear, (11) thesecond sun gear, the motors (12) and (13).

l

a

θ1

θ2

θ6

PpFsFp

Fo

Ps

Po

θ7

P1

P2

d70Peq

Fig. 3. Level 1 - The Variable Stiffness Mechanism: The output shaft rotatesaround the central axis of the mechanism and is connected to the lever armwith a crank. On the other side of the lever arm, two springs connect thelever to the frame, which can continuously rotate. The dotted spring isvirtual and represents the equivalent linear spring of the two springs.

In the remainder of the Section, we describe these levelsof the mechanical structure and the operating principles.

A. Level 1 - The Variable Stiffness Mechanism

The first level of the mVSA-UT realizes the variablestiffness mechanism, similar to the one described in [11],and includes the parts (1) to (7) in Fig. 2. It is illustratedin Fig. 3. The operating principle of the mVSA-UT relieson a variable transmission between the internal springs (3)and the output crank (1), by connecting them via a lever (2)with variable length. The variable lever length is achieved bymoving the pivot pin (4) along a slot in the lever. A linearmotion of the pivot pin is accomplished with a planetary geartrain, where the diameter d7 of the ring gear (7) is twice aslarge as that of the pivot gear (5).

The ring gear forms part of a rotating frame, that defines

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the undeflected output position. The reference frame in 0 isfixed to it and its position is denoted by θ7. The pivot gearis actuated by the planet carrier (6), which position relativeto the ring gear is θ6 and influences the output stiffness.

The position l of the pivot pin along the diameter of thering gear is given by

l =d7

2cos θ6 (1)

As the pin slides in a slot in the lever arm, the motion ofthe lever arm is constrained to follow the relative motionsof the pivot pin and the output crank at Po. A deflectionθ1 of the output shaft produces a displacement of the pointPo = d7

2 [cos θ1 sin θ1]T and a rotation θ2 of the lever arm.

The distance a between the pivot position l and the outputposition Po and the angle θ2 are given by

a =

√(d7

2

)2

+ l2 − 2ld7

2cos θ1

θ2 = arcsin(d7

2asin θ1

)The two pretensioned springs act on the lever like a single

virtual linear spring, i.e., the dotted spring in Fig. 3, that isconnected to the reference frame at the equilibrium pointPeq = (P1 + P2) /2 =

[−d7

2 0]T

of the springs and to thelever arm at point Ps. Assuming a linear, zero length springwith elastic constant k, the spring force on the lever withrespect to 0 is given by

Fs = k (Peq − Ps)

where Ps = Po − d7

[cos θ2sin θ2

].

Let Rθ2 be the rotation matrix of the lever arm with respectto the reference frame 0, as defined by θ2. Then, it followsthat the spring forces along the lever arm at point Ps are

F2,Ps = RTθ2Fs :=[F x2,Ps

F y2,Ps

]If no friction is present between the pivot pin and the lever,

the force acting on the pivot pin is always perpendicular tothe lever arm. From a balance of the forces along the lever,and from the balance of the torques around the pivot pin, thereaction force on the lever in Po is given by

F2,Po =[ −F x2,Ps(

d7−aa

)F y2,Ps

]The perpendicular force on the output F y1,Po

can becalculated with the angle β = θ2 − θ1 between the leverand the crank. The resulting output torque acting around theoutput axis of rotation is given by

τ1 =d7

2F y1,Po

=d7

2[

cosβ sinβ]F2,Po

and therefore the output stiffness is

K =∂τ1∂θ1

(2)

ω10

d10

ω7

ω9

ω8

v10

v9

v8

d8

d9

Fig. 4. Level 2 - The Differential Mechanism: The second level of themVSA-UT is a set of planetary gears. This mechanism is used to obtain adifferential motion of the first ring gear (7) and the pivot gear (6).

B. Level 2 - The Differential Mechanism

The second level of the mVSA-UT differentially couplesthe two degrees of freedom of the variable stiffness mecha-nism, namely the neutral output position θ7 and the positionof the planet carrier θ6. By using a differential drive, bothinternal actuators can contribute to the output power. It is asecond set of planetary gears, made up by the parts (8) to(10) of Fig. 2 and illustrated in Fig. 4.

The sun gear (8) is fixed to the shaft of the planet carrier(6). The three planet gears (9) connect the sun (8) with thering gear (10). The shafts of the planets are connected to therotating frame (7). The velocity of the frame ω7 is given by

ω7 =v9

d8+d92

where v9 is the linear velocity of the shaft of the planet gears,d8 is the diamater of the sun gear and d9 the diameter of theplanets. Note that

v9 =v10 + v8

2

where v10 = d102 ω10 is the linear velocity of a point on

the ring gear with diameter d10 and rotating with angularvelocity ω10, and v8 = d8

2 ω8 is the linear velocity of a pointon the sun gear, rotating with angular velocity ω8.

The angular velocity ω7 of the frame can thus be expressedas a function of the angular velocity ω10 of the ring gear andof the angular velocity ω8 of the sun gear as

ω7 =d10

2(d8 + d9)ω10 +

d8

2(d8 + d9)ω8 (3)

The planet carrier (6) and the sun gear (8) are rigidlyconnected. However, ω6 refers to the velocity of the planetcarrier with respect to the the ring gear (7), i.e. the frame0 of Fig. 3, while ω7 and ω8 refer to an absolute frame ofreference. Using Eq. (3), ω6 can be expressed as

ω6 = ω8 − ω7

= − d102(d8+d9)

ω10 +(1 − d8

2(d8+d9)

)ω8

The motion of the planet carrier carrying the pivot gear,and the motion of the rotating frame of the VSA are thus

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ω10

d10

q13ω11

d13

.q12.

d11

d12

Fig. 5. Level 3 - The Actuation Stage: The two motors (12) and (13)perform the rotation of the two small gears, thus actuating q12 and q13.The two motors are fixed to the base, but allow the mVSA-UT to performcontinuous rotation.

linearly coupled, i.e.,[ω7

ω6

]=[α β−α 1 − β

] [ω10

ω8

]where {

α = d102(d8+d9)

β = d82(d8+d9)

C. Level 3 - The Actuation StageIn the third level, shown in Fig. 5, the internal actuators

are coupled to the ring gear (10) and the sun gear (8) of thedifferential mechanism by the parts (11) to (13) in Fig. 2.The first motor (12) engages the ring gear (10), while thesecond motor (13) engages a second sun gear (11), fixed tothe sun gear (8) via the shaft of the planet carrier (6).

The ring gear (10) and the sun gear (11) are driven by themotor pinions (12) and (13) with the transmission{

ω10 = d12d10q12

ω11 = −d13d11q13

(4)

where q12 and q13 are the angular velocities of the twomotors, d12 and d13 the diameters of the pinions, and d11

the diameter of the sun gear (11). The sun gear (11) doesnot have, in general, the same diameter as the sun gear (8).

Because ω11 = ω8, the transmission between the actuatorsand the differential mechanism expressed in Eq. (4) can bewritten in matrix form as[

ω10

ω8

]=

[d12d10

00 −d13

d11

] [q12q13

]leading to the overall transmission of the mVSA-UT[

ω7

ω6

]=

[d12d10α −d13

d11β

−d12d10α −d13

d11(1 − β)

] [q12q13

]= MT q (5)

With the transmission matrix MT the two inputs q12 andq13 can be chosen to achieve a desired position of theframe θ7 and pivot point θ6, thus yielding the desired outputequilibrium position and stiffness. Given the gear diametersused in the prototype, the internal transmission becomes

MT =[

0.197 −0.209−0.197 −0.556

]

(a) (b) (c)

(d) (e)

Fig. 6. Different spring configurations. 6(a), 6(b) and 6(c) show linearsprings with different attachement points on the lever and frame. 6(d) and6(e) show torsional springs

IV. STIFFNESS ANALYSIS

In this Section, the output stiffness characteristics of theproposed actuation system are analysed and design choicesregarding the spring configuration motivated.

The output stiffness in Eq. (2) depends on the way theinternal springs connect the lever to the frame of the variablestiffness mechanism shown in Fig. 3. It is determined bythe elongation of the springs for a output deflection θ1 andlinear pivot position x. Consequently the way the springs areattached should not only depend on constructional consider-ations, but also on the resulting output stiffness profiles.

The output torques and stiffnesses have been analysed fordifferent spring configurations, shown in Fig. 6, using bothlinear and torsional springs. Configuration 6(a) represents thedesign that was chosen in the mVSA-UT, where two linearsprings are attached at right angles to the end of the lever.In configuration 6(b) the springs are also attached to the endof the lever, but their equilibrium point lies in the center ofthe ring gear. In configuration 6(c) the springs are connectedat right angles to the center of the lever. Configurations 6(d)and 6(e) use torsional springs between the lever and outputand between the lever and frame, respectively.

Fig. 7 shows the output stiffness in the equilibrium posi-tion, i.e., θ1 = 0, as a function of the pivot point positionx of the lever, where x = l + d7/2 with respect to Eq. (1).Even though the output stiffness also depends on the outputposition, it is far more sensitive to the pivot position. Thestiffness calculations for configurations 6(a), 6(b) and 6(c)follow Section III-A, where the torque on the output shaftis obtained by an equilibrium of forces and moments of thelever. The lever is constrained to follow the output crank andthe pivot pin, causing an elongation of the springs and thusa spring force. The stiffness is then obtained by taking thepartial derivative of the output torque τ1 with respect to theoutput deflection θ1, as given by Eq. (2). In the case of thetorsional springs in 6(d) and 6(e), the torque on the outputis calculated with an equilibrium of forces and moments ofthe lever that experiences a torque (θ2 − θ1) · c and θ2 · c,

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00

x

K(x

)|θ1=

0

Con�g. e)

d 7d /27

Con�g. a)Con�g. c)

Con�g. b)

Con�g. d)

Fig. 7. Output stiffness K as function of the linear pivot position x fordifferent spring configurations. K scales linearly with the stiffness of theinternal springs.

(a) (b)

Fig. 8. Plots of the output torque (a) and the output stiffness (b) as functionof θ1 and θ6 for the mVSA-UT. Note that the plots are bounded due to thesingularity at θ6 = 0◦.

respectively, where c is the torsional stiffness of the spring.All profiles show an asymptote for x = d7, i.e. theoreti-

cally infinite stiffness, because the pivot pin then coincideswith the connection between the lever and the output crank.

However, only configurations 6(a) and 6(e) show a mono-tonic stiffness that has no local minimum between 0 andd. The other curves do show a minimum around d7

2 andare steeper for x > d7/2. Their shape after the minimumresembles that of the stiffness for configuration 6(a), if itwere horizontally compressed. The configurations 6(b), 6(c)and 6(d) only seem to utilize half the length of the lever.This means that for these configurations, half the lengthof the variable stiffness mechanism cannot be used to varythe stiffness significantly, but also that the stiffness is moresensitive to θ6 in the other half. Assuming a given precisionof θ6, then the stiffness can be more finely tuned in a moremonotonic and continuous way when using configurations6(a) and 6(e). Note that configuration 6(e) does not reachzero stiffness, because a deflection of the output alwaysresults in a deflection of the lever. The other configurationsachieve zero stiffness by attaching the springs to the lever ina way that they are not elongated for a certain pivot position.

Configuration 6(a) thus shows the best output stiffnesscharacteristics. Furthermore it turns out that it is not lesscompact than the other configurations, because the spaceused to place the springs is not lost, but used for the motionof the lever and for constructing the frame of the ring gear.

In Fig. 8, the output torque and stiffness are shown as

functions of both the output deflection θ1 and of the planetcarrier angle θ6. Note the peak close to the singularity atθ6 = 0 and also that not the whole plane is reachable;especially the area of negative stiffness is inaccessible dueto mechanical stops constraining the lever and output.

To be able to realize configuration 6(a), linear springsare needed that can be stretched extremely far. Because thespace for the springs is limited, the stroke of the lever nearlyreaches the connection points of the springs and the frame.This results in the springs being elongated to several timestheir rest length. Rubber springs can meet these requirements,however, care has to be taken in the selection of the material.The rubber should be strong enough, resistant to the greaseused to lubricate the actuator and keep its elastic propertiesover a longer periode of time.

V. ROBUST DESIGN

Even though small dimensions were a design objective, theactuator needs to be mechanically robust to prevent failure.

A challenge in miniaturisation is the fact, that stresses incomponents, e.g. bending or torsion stresses of a shaft, donot scale linearly, but with the third power of the diameter.A balanced design was achieved by analyzing how big theexpected loads on certain components are, and laying outtheir dimensions accordingly. In this way it was possible todesign a robust system with small dimensions. The designwas guided and validated by strength calculations and finiteelement analyses for critical parts, especially in the first levelof the actuator. The results of the FEM analysis of the carrier,output and pivot are depicted in Fig. 9.

The material of the components was chosen accordingto the required strength and light weight, among otherconsiderations like low friction or machinability. In any caseit was made sure that the nominal stresses in the parts donot exceed the yield strengths of the material. Componentswith high stresses are made out of tool steel type 1.2510with a yield strength of 400 N/mm2. The output shaft, theinternal lever and all the gears are made of this steel, as wellas the shafts of the planet gears. Because of their complex,but planar, geometry the gears for the prototype were sparkeroded. The pivot pin is made from the end of a hardenedsteel drill. The carrier, which experiences smaller stresses, ismade of stainless steel type 1.4305 for easier machining. Thelower parts of the internal rotating frame and the housing thathave contact with the planet gears and the lower ring gearare made of bronze to minimize friction and to achieve goodmachinability. Bronze offers favorable friction properties,however, it is also rather heavy. The remaining parts of theframe and the housing are made of aluminium alloy 3.1645for low weight, high strength and good machinability.

All rotating parts are mounted with ball bearings, apartfrom the ring gear of the differential stage and the planets,which are mounted directly onto shafts. The bearings wereselected because their dimensions made a compact con-struction of the mechanism possible. Still their load ratingsexceed the nominal bearing loads, which ensures mechanicalrobustness, as well as low friction and precision.

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Fig. 9. Results of the FEM analysis of the carrier, output and pivot

VI. ACTUATION AND INSTRUMENTATION

The design of the mVSA-UT is modular, so that differentmotors can be used as the internal actuators. This results infreedom of design choices concerning the output speed andstrength, dimensions, motor types or control strategies for theactuator. The chosen motors must merely carry the correctpinion (13 teeth and modulus 0, 5 mm) and have the correctdistance between their axes to properly engage the lower sunand ring gear. They can be fixed to the base of the housing,e.g. with an adapter piece that is screwed to the housing.

The current prototype of the mVSA-UT is equipped withtwo micro-servo motors, the HET RC DS-001MG and DS-002MG, the only difference between those two being theinternal transmission. The current motors can provide thenominal output torque of 1 Nm of the actuator, the speed ofthe mVSA-UT can be increased by using faster motors.

A DC motor control board, based on a ATMEGA328micro-controller and a Polulu DC driver, perform joint levelposition control of the motors at 1kHz.

VII. CONCLUSIONS

In this paper, we presented the design of the mVSA-UT, which realizes a compact variable stiffness actuatorthat can perform continuous rotation at the output shaft.The mechanical structure of the mVSA-UT is such thatthe apparent output stiffness can be varied from zero toalmost infinite by changing the transmission ratio betweenthe internal elastic elements and the actuator output, namelyby implementing a lever arm of variable effective length bymeans of a set of planetary gears.

The working principle of the variable stiffness mechanismhas been realized in a compact design thanks to the imple-mentation of a differential configuration of the two internal

TABLE ISPECIFICATIONS OF THE MVSA-UT PROTOTYPE

Stall torque 1 NmNo-load speed 4/5 π rad/sStiffness change duration 0.54 secActuated range continuousPassive range ± π/4 rad (±45)Weight 100 gDimensions (excl. shaft) 30× 32× 49 mm3

motors. Table I reports the specification of the mVSA-UTand the behavior of the system is shown in the attached video.

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