Simultaneous Force and Stiffness Control of a Pneumatic ...research.vuse.vanderbilt.edu/cim/pubs/journal/3... · Simultaneous Force and Stiffness Control of a Pneumatic Actuator This

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Xiangrong Shen

Michael Goldfarb

Department of Mechanical Engineering,Vanderbilt University,Nashville, TN 37235

Simultaneous Force and StiffnessControl of a Pneumatic ActuatorThis paper proposes a new approach to the design of a robot actuator with physicallyvariable stiffness. The proposed approach leverages the dynamic characteristics inherentin a pneumatic actuator, which behaves in essence as a series elastic actuator. By replac-ing the four-way servovalve used to control a typical pneumatic actuator with a pair ofthree-way valves, the stiffness of the series elastic component can be modulated indepen-dently of the actuator output force. Based on this notion, the authors propose a controlapproach for the simultaneous control of actuator output force and stiffness. Since theachievable output force and stiffness are coupled and configuration-dependent, the au-thors also present a control law that provides either maximum or minimum actuatoroutput stiffness for a given displacement and desired force output. The general controland maximum/minimum stiffness approaches are experimentally demonstrated and shownto provide high fidelity control of force and stiffness, and additionally shown to provide afactor of 6 dynamic range in stiffness. �DOI: 10.1115/1.2745850�

IntroductionThe modulation of actuator output stiffness can serve several

urposes in robotic applications, many of these motivated by bio-ogical motor control strategies. For example, research suggestshat humans achieve stable and effective interaction with a wideariety of environments by leveraging their ability to modulateoint impedance independently of joint torque �e.g. �1–4��, and aignificant body of research exists that highlights the role of vari-ble compliance in enhancing the energetic efficiency of mamma-ian locomotion �e.g. �5–7��. As proposed by Salisbury �8� andeneralized by Hogan �9�, one means of modulating actuator out-ut stiffness is via feedback control. Such an approach can effec-ively modulate actuator �or manipulator� output stiffness, but asith all feedback control systems, provides the desired character-

stics in a limited frequency range, and can jeopardize systemtability, especially in cases of noncollocated load sensing. Addi-ionally, closed-loop strategies offer little with respect to energeticfficiency, since feedback control is generally energetically non-onservative. In order to provide variable compliance withouthese limitations, several researchers have developed robot actua-ors with physically variable stiffness, which incorporate somepen-loop mechanism to enable simultaneous control of actuatororce and stiffness. Specifically, in order to implement a biologi-ally inspired strategy of interaction during robotic manipulationithout the limitations imposed by feedback control, Laurin-ovitz et al. �10� proposed the design of a variable stiffness ac-

uator that is loosely based on the configuration of the humanusculoskeletal system. Their approach incorporates two �non-

ackdrivable� motors for a single joint �similar to the agonist/ntagonist musculoskeletal relationships in animals�, each actuat-ng a tendon through a nonlinear spring. As with a biological

otor control system, the joint torque is a function of the differ-nce of the motor efforts, while the joint stiffness is a function ofhe sum of the motor efforts, thus providing simultaneous controlf joint torque and stiffness. Koganezawa and Yamazaki �11�,oganezawa and Ban �12�, and English and Russell �13,14� pro-osed a variable stiffness actuator design of a similar structure. Inrder to leverage the energetic benefits of variable compliance forobotic legged locomotion, Hurst et al. �15� presented a differentesign for a variable stiffness actuator. Like the previously de-

Contributed by the Dynamic Systems, Measurement, and Control Division ofSME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CON-

ROL. Manuscript received January 9, 2006; final manuscript received January 5,
007. Review conducted by Huei Peng.
ournal of Dynamic Systems, Measurement, and ControlCopyright © 20

aded 21 Mar 2008 to 129.59.75.233. Redistribution subject to ASME

scribed approaches, their design incorporates two motors for each�kinematic� degree-of-freedom, but unlike the agonist/antagonistapproaches, the design of Hurst et al. is essentially a series elasticactuator, where the stiffness of the series elastic element is modu-lated by the second motor �i.e., the second motor adjusts a springpretension, which modulates its stiffness�. Tonietti et al. �16� pre-sented the design of a variable stiffness actuator that is structurallydifferent but conceptually similar to the design of Hurst et al.Finally, in order to enhance the intrinsic safety of human/robotinteraction, Bicchi et al. �17� and Tonietti and Bicchi �18� pro-posed the use of an agonist/antagonist pair of McKibben artificialmuscles to provide simultaneous control of position and �open-loop� stiffness in a similar manner to a biological motor controlsystem. In essence they control the position via the difference ofactuator pressures, while they control the stiffness with the sum.

This paper presents an alternate configuration for an actuatorwith a physically variable output stiffness that offers a more com-pact package and requires less mechanical complexity than theaforementioned motor-spring designs. Unlike the work presentedin �17,18�, the proposed approach requires only a single actuatorrather than two. Further, as subsequently shown, the proposedapproach enables a greater dynamic range and bandwidth in thecontrol of stiffness and force, relative to that demonstrated in �18�.Like the work presented in �17,18�, the proposed approach lever-ages the open-loop behavior of a pneumatic actuator, which inher-ently provides a series elastic component via the compressible gasdynamics. A typical pneumatic actuator is controlled via a singlefour-way spool valve, and as such, the actuator output stiffness isnot controllable independently of the actuator force. By decou-pling the single four-way spool valve into a pair of three-wayvalves, however, the pressure in each cylinder chamber can beindependently controlled, and thus the actuator output stiffness ofthe cylinder actuator can be controlled independently of the outputforce. Like the agonist/antagonist systems previously described,the actuator force is a function of the difference between thechamber pressures, while the actuator output stiffness is a functionof their sum. Thus, the only additional hardware required is anextra valve �for each actuator�. As with the other �previously pro-posed� actuators, the stiffness is physical in nature �i.e., open-loop�, and therefore maintains its constitutive behavior throughoutthe frequency spectrum, with no potential for nonpassive behav-ior. It should be noted that Raibert in �19� suggests an actuatorconfiguration similar to the one proposed herein, but does notdescribe the simultaneous force and stiffness control of this con-
figuration. Rather, as utilized by Raibert, the actuator alternates
JULY 2007, Vol. 129 / 42507 by ASME

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etween a force source �when the valves are open� and a springwhen the valves are closed�. This paper presents an approach forhe simultaneous stiffness and force control of the pneumatic ac-uator, and presents experimental results that characterize theorce and stiffness tracking performance.

Modeling the Pneumatic ActuatorIn order to implement simultaneous force and stiffness control,two-input, two-output dynamic model of the actuator is briefly

escribed. For the proposed variable stiffness actuator, the twoodel outputs are actuator force and output stiffness. The two

nputs to the actuator are the respective valve commands. As in aypical pneumatic servo system, the commands are assumed to bealve areas, which are algebraically related to the valve spoolisplacement. Note that the servovalve spool dynamics are ne-lected, since these are typically significantly faster than the ac-uator dynamics. Finally, since mass flow rate through the valve isn algebraic function of the valve area, model formulation is sim-lified by assuming the model inputs are the respective mass flowates into �positive� or out of �negative� the respective sides of theneumatic cylinder. Specifically, modeling the flow through thealve as the flow of an ideal gas through a converging nozzle, theass flow is algebraically related to the valve area command by

he following relation:

m�Pu,Pd� = Av��Pu,Pd� �1�

here

��Pu,Pd� =��

RT� 2

� + 1��+1�/��−1�

CfPu

ifPd

Pu� Cr �choked�

� 2�

RT�� − 1��1 − �Pd

Pu��−1�/��Pd

Pu��1/��

CfPu

otherwise �unchoked�

�2�

nd Av is the valve area command, Pu and Pd are the upstream andownstream pressures, respectively, � is the ratio of specific heats,

is the universal gas constant, T is the gas temperature at therifice, Cf is the discharge coefficient of the valve �which ac-ounts for irreversible flow conditions�, and Cr is the pressureatio that divides the flow regimes into unchoked �subsonic� andhoked �sonic� flow through the orifice. Thus, by commanding thealve orifice areas, the servovalves are algebraically commandinghe mass flow rates into or out of each chamber of the cylinder,nd as such, the mass flow rates through the respective valves aressumed to be the two actuator inputs. Note that, since the twoalves are three-way valves, a positive valve command connectshe pressure supply to the cylinder so that mass flows into thehamber �defined as positive mass flow�, while a negative com-and connects the cylinder to exhaust, such that mass flows out of

he chamber �defined as negative mass flow�. The model is thuserived by describing the respective relationships between actua-or force and chamber mass flow, and between actuator outputtiffness and chamber mass flow. Based on the schematic of theneumatic actuator shown in Fig. 1, the force generated by thectuator is given by

F = PaAa − PbAb − PatmAr �3�

here Pa and Pb are the absolute pressures inside each chamberf the actuator, Aa and Ab are the effective areas of each side ofhe piston, and Ar is the cross-sectional area of the piston rod. Inrder to recover the mass flow rate inputs, the force equation is
ifferentiated with respect to time
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F = PaAa − PbAb �4�

Assuming air is an ideal gas undergoing an isothermal process, therate of change of the pressure inside each chamber of the cylindercan be expressed as a function of mass flow rate as

P�a,b� =RT

V�a,b�m�a,b� −

P�a,b�

V�a,b�V�a,b� �5�

where P�a,b� is the absolute pressure inside each side of the cylin-der, m�a,b� is the mass flow rate command �where as previouslydescribed, a positive command indicates mass flowing into thechamber, negative indicates mass flowing out�, and V�a,b� is thevolume of each cylinder chamber. Thus, the dynamics from massflow input to the force output is given by

F =RTAa

Vama −

RTAb

Vbmb −

PaAa

VaVa +

PbAb

VbVb �6�

The volume in each chamber is a geometric function of pistondisplacement x, given by

V�a,b� = A�a,b��L

2± x� �7�

where L is the length of the actuator and x �L /2 �i.e., the cham-ber volume is never zero�. Thus, the dynamics from mass flowinput to force output can be written as a function of the measur-able states �i.e., the chamber pressures and piston displacement� as

F =RT

L

2+ x

ma −RT

L

2− x

mb −PaAax

L

2+ x

−PbAbx

L

2− x

�8�

The output stiffness of the actuator is defined by

K = −�F

�x�9�

where the actuator force is given by �3�. Substituting �3� into �9�yields

K = −��PaAa − PbAb − PatmAr�

�x= − Aa

�Pa

�x+ Ab

�Pb

�x�10�

The chamber pressures can be described as a function of displace-ment by assuming air is an ideal gas, such that

P�a,b� =m�a,b�RT

V�a,b��11�

Substituting �7� into �11� yields an expression for pressures as a

Fig. 1 Pneumatic actuator controlled by pair of three-wayvalves

function of piston displacement:

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P�a,b� =m�a,b�RT

A�a,b��L

2± x� �12�

ifferentiating with respect to x gives

�P�a,b�

�x= �

m�a,b�RT

A�a,b��L

2± x�2 �13�

hus, the stiffness can be written as a function of the respectiveasses of air in each cylinder chamber and the piston displace-ent as

K = RT� ma

�L

2+ x�2 +

mb

�L

2− x�2 �14�

n order to recover the mass flow rate command inputs, �14� isifferentiated with respect to time, which after simplificationased on substitution from Eq. �12� yields the dynamics fromass flow input to the stiffness output as a function of measured

tates,

K =RT

�L

2+ x�2ma +

RT

�L

2− x�2mb −

2PaAax

�L

2+ x�2 +

2PbAbx

�L

2− x�2

�15�

ote that the force dynamics �8� are influenced by the differencen valve commands �i.e., mass flow rates�, while the stiffness dy-amics �15� are influenced by the sum of the mass flow rates.

The model as described by �1�–�15� can be expressed in controlanonical form by defining the input vector such that it consists ofhe two mass flow rates, u= �ma mb �T, and by defining the outputector such that it consists of the force and stiffness, respectively,= �F K �T. As such, the system dynamics can be expressed as

x = f + bu �16�

here

f = � f1

f2 �17�

nd

b = �b11 b12

b21 b22 �18�

here

f1 = −PaAax

L

2+ x

−PbAbx

L

2− x

�19�

f2 = −2PaAax

�L

2+ x�2 +

2PbAbx

�L

2− x�2 �20�

b11 =RT

L

2+ x

�21�

b12 = −RT

L− x

�22�

2

ournal of Dynamic Systems, Measurement, and Control


b21 =RT

�L

2+ x�2 �23�

b22 =RT

�L

2− x�2 �24�

3 Sliding Mode Controller DesignGiven the two-input, two-output model provided by �16� and

�24�, a standard multi-input-multi-output sliding mode approachcan be utilized for the simultaneous control of actuator force andstiffness. Note that sliding mode control was chosen to accommo-date the model uncertainty in the nonlinear dynamics �e.g., theactual pressure dynamics will vary somewhat from the isothermalcondition assumed in the model derivation�. The bound for theuncertainty associated with the above model can be expressed by

f i − f i � Fi i = 1, 2 �25�

b = �I + ��b �ij � Dij i = 1, 2 j = 1, 2 �26�

where f i is the estimated value of f i, b is the estimated matrix ofb, I is the 2�2 identity matrix, and Dij is a positive constant.

In order to develop an MIMO sliding mode controller for thesystem, first define a sliding surface vector s as

s = �s1

s2 = �F − Fd

K − Kd �27�

where Fd and Kd are the desired actuator force and stiffness, re-spectively. The corresponding sliding conditions for the two statesare

1

2

d

dtsi

2 � − �isi i = 1, 2 �28�

where �i is a strictly positive constant. In order to satisfy thesliding conditions in the presence of the model uncertainty, theMIMO sliding mode control law �as described in �20�� is given by

u = b−1�xd − f − � sgn�s�� 29�

where � sgn�s� is the robustness vector �k1 sgn�s1� k2 sgn�s2��T,and xd is the time derivative of the desired input vector, which isdefined as xd= �Fd Kd �T .k1 and k2 are chosen such that

�1 − Dii�ki + �j�i

Dijkj = Fi + �j=1

2

Dijxdi − f j + �i i = 1, 2

�30�The existence of a solution to these equations is guaranteed by theFrobenius-Perron theorem, as described in �20�.

Finally, the commands to the control valves are calculated by

Av,a = �ma/��Ps,Pa� if ma 0

ma/��Pa,Patm� otherwise�31�

Av,b = �mb/��Ps,Pb� if mb 0

mb/��Pb,Patm� otherwise�32�

where ��Pu , Pd� is defined by �2�.

4 Force Control With Maximum/Minimum OutputStiffness

Many cases exist for the simultaneous control of force and stiff-ness for which one might desire either a maximum or a minimumactuator output stiffness. For example, during a kinematically con-
strained manipulation task �such as opening a drawer�, a manipu-
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ator should ideally maintain a low output stiffness, which mini-izes the effective gain of the disturbance transfer function from

osition error normal to the kinematic constraint to the resultingorce of interaction. Alternatively, for a manipulation task such asarving with a knife, the manipulator should ideally maintain aigh output stiffness, which will minimize the effective gain fromnvironmental disturbance force to a resulting error in desiredrajectory. As described by Eqs. �16�–�24�, the actuator outputtiffness and force are coupled, and thus the achievable stiffness atny given time depends on the actuator force and the actuatortate. This coupling is expressed more explicitly by substitutingq. �12� into Eq. �14�, so that the stiffness can be expressed in

erms of pressure and displacement as follows:

K =PaAa

L

2+ x

+PbAb

L

2− x

�33�

or a non-negative desired force Fd, the set of Eqs. �3� and �33�an be solved explicitly for K and Pa, such that the expression fortiffness becomes

K =Fd + PbAb + PatmAr

L

2+ x

+PbAb

L

2− x

�34�

ormally Pb is a positive value between atmospheric and supplyressure, and thus the minimum stiffness can be written as a func-ion of desired force and displacement as

Kmin =Fd + PatmAa

L

2+ x

+PatmAb

L

2− x

if Fd 0 �35�

ince Aa=Ab+Ar. Further analysis of other cases gives the follow-ng expressions for the maximum and minimum stiffness:

Kmax =�PsAa

L

2+ x

+− Fd + PsAa − PatmAr

L

2− x

if Fd �Ps − Patm�Ar

Fd + PsAb + PatmAr

L

2+ x

+PsAb

L

2− x

otherwise

�36�

ig. 2 Experimental setup for the second set of experiments,n which force and stiffness control were performed in the pres-nce of motion control. The experimental setup for the first setf experiments was essentially identical, except the inertial

oad was clamped at x=0.

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Kmin =�Fd + PatmAa

L

2+ x

+PatmAb

L

2− x

if Fd 0

PatmAa

L

2+ x

+− Fd + PatmAb

L

2− x

otherwise

�37�

As such, the cases of minimum and maximum actuator stiffnesscould be implemented by utilizing either Eq. �36� or �37�, respec-tively, to determine the desired stiffness command to the previ-ously described MIMO controller for a given desired force. Suchan approach, however, operates outside of the control loop, andthus does not account for disturbances or tracking error �i.e., doesnot take into consideration the actual force, or more accurately,the actual cylinder pressures�.

In order to implement a more robust approach to obtaining amaximum/minimum actuator stiffness �i.e., one that is sensitive todisturbances and tracking error�, the authors developed a controlapproach that operates inside the control loop on the control com-mands �i.e., on the mass flow rates�, rather than the aforemen-tioned command generator described by Eqs. �36� and �37�.Rather than specify a given stiffness at any given time, the pro-posed approach leverages the presence of saturation in stiffness�i.e., the presence of Eqs. �36� and �37�� and the fact that stiffnessis a strictly positive quantity. Given these, the stiffness can bemade and maintained at its maximum �in general fluctuating�value by maximizing its time rate of change �i.e., driving it asquickly as possible to its maximum value�. Similarly, the stiffnesscan be made and maintained at its minimum �in general fluctuat-ing� value by minimizing its time rate of change �i.e., driving it as

Fig. 3 Simultaneous force and stiffness tracking, each of
4.0 Hz sinusoid, with piston rod constrained at x=0


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uickly as possible to its minimum value�. As described by Eq.15�, the time rate of change of stiffness is linearly influenced byhe �input� mass flow rates, and as such the maximum or mini-

um time derivative of stiffness can be obtained via the linearptimization of

J = b21ma + b22mb �38�

here b21 and b22 are defined by �23� and �24�. For stable forceracking, the mass flow rates should satisfy the sliding condition28� for i=1,

1

2

d

dts1

2 � − �1s1 �39�

his condition can be satisfied by the following equation betweenhe inputs:

b11ma + b12mb = − f1 + Fd − k1 sgn�s1� �40�

here b11 and b12 are given by �21� and �22�, f1 is given by �19�,˙

d is the time derivative of the desired actuator force, and k1 is theobustness gain satisfying �39�. Moreover, the inputs ma and mbre bounded by an upper limit for which the corresponding valves fully open to the air supply and a lower limit for which theorresponding valve is fully open to the exhaust. These constraintsan be expressed as

ma,min � ma � ma,max �41�

mb,min � mb � mb,max �42�

here m�a,b��min,max� are functions of the state as given by �1�,valuated at the maximum valve area. Specifically, m�a,b��min,max�re given by

m�a,b�max = Av,max��Ps,P�a,b�� 43�

nd

m�a,b�min = − Av,max��P�a,b�,Patm� �44�

here Ps is the supply pressure and Patm is the exhaust pressure.ote that the negative sign for m�a,b� min pertains to the directionf mass flow, as previously discussed. Thus, the whole probleman be defined as maximizing or minimizing the objective func-ion J as defined by �38� under the constraints �40�–�42�.

The optimization problem defined above can be solved with ainear programming approach. The control inputs for maximizing

Table 1 Model and controller param

arameter Description

sSupply pressure

aPiston area side a

bPiston area side b

rRod area

v maxMaximum valve area

fDischarge coefficient

rPressure ratio

Ratio of specific heatsUniversal gas constant

Gas temperature at the orifice

1Force error robustness gain

2Stiffness error robustness gain

pOuter loop proportional gain

dOuter loop derivative gain

tiffness are given by



ma =�ma,max

if− f1 + Fd − k1 sgn�s1� − b11ma,max

b12� mb,max

− f1 + Fd − k1 sgn�s1� − b12mb,max

b11

otherwise

�45�and

mb =�− f1 + Fd − k1 sgn�s1� − b11ma,max

b12

if− f1 + Fd − k1 sgn�s1� − b11ma,max

b12� mb,max

mb,max

otherwise

�46�The control inputs for minimizing stiffness are given by

ma =�ma,min

if− f1 + Fd − k1 sgn�s1� − b11ma,min

b12 mb,min

− f1 + Fd − k1 sgn�s1� − b12mb,min

b11

otherwise

�47�

and

mb =�− f1 + Fd − k1 sgn�s1� − b11ma,min

b12

if− f1 + Fd − k1 sgn�s1� − b11ma,min

b12 mb,min

mb,min

otherwise

�48�

The corresponding commands to the control valves are calculatedby �31� and �32�.

5 Experimental ResultsThe force/stiffness controller was implemented on an experi-

rs for experimental implementation

Value Unit

653 kPa572 mm2

540 mm2

32 mm2

12.57 mm2

0.80.51.4

0.287 kJ/kg K293 K

6�103 kg m/s3

1.5�105 kg/s3

1000 N/m185 N s/m

ete

mental setup to demonstrate and validate the approach. The ex-

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erimental setup, which is shown schematically in Fig. 1 and in ahotograph in Fig. 2, consists of a 2.7 cm �1 1

16 in.� inner diam-ter, 10 cm �4 in.� stroke pneumatic cylinder �Numatics 1062D04-4A� controlled by a pair of proportional servovalves �FESTOPYE-5-M5-010-B�, each configured as a three-way valve as

hown in Fig. 1, and each supplied with air at an absolute pressuref 653 kPa �95 psia�. The setup is instrumented with pressureensors �FESTO SDE-16-10V/20mA�, which measure the pres-ure in each cylinder chamber. Two sets of experiments were per-ormed. In the first set, the piston rod was clamped to the tableuch that the piston displacement was constrained to remain at x

ig. 4 Force tracking of 0.25 Hz sinusoid and simultaneoustiffness tracking of 4.0 Hz sinusoid with piston rod con-trained at x=0. The pressure trajectories „in chambers a and, respectively… that generate the force and stiffness trackingre shown below the force and stiffness plots.

0, and the desired force and stiffness were commanded directly

30 / Vol. 129, JULY 2007


to the MIMO controller. In the second set of experiments �thesetup for which is depicted in Fig. 2�, rather than command theforce directly, the force command was derived indirectly from amotion control outer loop, while the stiffness was commandeddirectly. Although this approach is arguably a less direct assess-ment of force control relative to clamping the piston and provid-ing a desired force trajectory, it demonstrates force control in thepresence of motion, and thus involves all terms in the force dy-namics described in �8�. In these experiments, the output of thepneumatic cylinder was connected to a linear slide, upon which a10 kg mass was mounted, while the displacement of the slide �and

Fig. 5 Force tracking of 4.0 Hz sinusoid and simultaneousstiffness tracking of 0.25 Hz sinusoid with piston rod con-strained at x=0. The pressure trajectories „in chambers a andb, respectively… that generate the force and stiffness trackingare shown below the force and stiffness plots.

actuator� was measured with a linear potentiometer �Midori LP-



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00F�, as depicted in Fig. 2. The model and controller parameterssed for the force/stiffness controller are listed in Table 1. The PDontrol gains used in the position control loop in the motion con-rol set of experiments are also listed in the table. Note that the

odel parameters �i.e., the first ten entries in the table� were eithereasured, calculated, or found via standard references, whereas

he controller gains �i.e., the last four entries� were obtained byxperimental tuning to balance tracking performance and sensoroise �i.e., higher gains provide better performance and betterobustness to model uncertainty, but introduce more sensor noise

Fig. 6 Force and stiffness tracking with force commshow actual and desired position tracking of a 0.5tracking of the force command resulting from the pmultaneous stiffness tracking for commanded sinusstiffness error.

nto the output vector�. Finally, it should be noted that actuator



output stiffness, which cannot be measured directly, was com-puted based on �33� and the combination of measured chamberpressures and piston displacement.

Representative results of the first set of experiments �desiredforce and stiffness commanded directly and the piston displace-ment clamped� are shown in Figs. 3–5. Figure 3 shows simulta-neous force and stiffness tracking, where the force command is a4.0 Hz sinusoid with 50 N amplitude, and the stiffness commandis also a 4.0 Hz sinusoid, varying between 7.0 and 9.0 N/mm.

ds generated by the position control loop. The plotssinusoidal motion and corresponding error, force

ition control loop and corresponding error, and si-al stiffness variation of 0.5 Hz, with corresponding

anHzosoid

Note that, despite unmatched initial conditions, the trajectories

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onverge fairly quickly to the desired trajectories. Figures 4 and 5emonstrate the ability of the system to independently track forcend stiffness. Specifically, Fig. 4 shows 0.25 Hz force trackingnd simultaneous 4.0 Hz stiffness tracking �with the same ampli-ude variations in Fig. 3�, while Fig. 5 shows the reverse. In ad-ition to the force and stiffness trajectories, the pressure variationsn each cylinder chamber are also shown in Figs. 4 and 5. As islear in the pressure plots, the force is a function of the pressureifference, while the stiffness is a function of the pressure sum.

Representative results of the second set of experiments �desired

Fig. 7 Force and stiffness tracking with force commshow actual and desired position tracking of a 2.0tracking of the force command resulting from the pmultaneous stiffness tracking for commanded sinusstiffness error.

32 / Vol. 129, JULY 2007


force commanded inside of a PD motion control loop, desiredstiffness commanded directly� are shown in Figs. 6 and 7. Figure6 shows the force and stiffness tracking when the force commandsare generated by a PD motion controller tracking a 20 mm ampli-tude, 0.5 Hz sinusoidal motion trajectory, and when the stiffnesscommands are generated directly as a 0.5 Hz sinusoidal variationin stiffness between 7.0 and 9.0 N/mm. Specifically, Fig. 6 showsthe desired and actual motion, along with the motion trackingerror; the desired and actual force, along with the force tracking

s generated by the position control loop. The plotssinusoidal motion and corresponding error, force

ition control loop and corresponding error, and si-al stiffness variation of 4.0 Hz, with corresponding

andHzosoid

error; and the desired and actual stiffness, along with the stiffness



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racking error. Note that the force command is closely representa-ive of the position tracking error. Both the nonsinusoidal shape ofhe position error �and force command� and the fact that the maxi-um errors occur at motion reversals indicate the clear presence

f Coulomb friction in the system, presumably in the piston seals.ote that the maximum force tracking error is approximately 5%,hile the maximum stiffness tracking error is approximately 2%

relative to the amplitude variation� with the major difference be-ng the significantly higher frequency content of the force trajec-ory, due to the aforementioned Coulomb friction.

Figure 7 shows the force and stiffness tracking when the force

ig. 8 Force tracking with commands generated by the posi-ion control loop, while maximizing the output stiffness of thectuator. The plots show actual and desired position trackingf a 0.5 Hz sinusoidal motion, force tracking corresponding tohe force command resulting from the position control loop,nd the corresponding stiffness of the actuator.

ommands are generated by the same PD motion controller track-



ing a 2.0 Hz sinusoidal motion trajectory, and when the stiffnesscommands are generated directly as a 4.0 Hz sinusoidal variationin stiffness �with the same amplitude variations represented in Fig.6�. Like Fig. 6, Fig. 7 shows the desired and actual motion, alongwith the motion tracking error; the desired and actual force, alongwith the force tracking error; and the desired and actual stiffness,along with the stiffness tracking error. Note that the position errorand corresponding force command are more sinusoidal in thiscase, since the dynamics become more inertially dominated �and

Fig. 9 Force tracking with commands generated by the posi-tion control loop, while minimizing the output stiffness of theactuator. The plots show actual and desired position trackingof a 0.5 Hz sinusoidal motion, force tracking corresponding tothe force command resulting from the position control loop,and the corresponding stiffness of the actuator.

hence proportionally less influenced by friction� due to the higher

JULY 2007, Vol. 129 / 433


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ccelerations. Note that both the force and stiffness maximumracking errors are approximately 10%, relative to the amplitudef variation.

Figures 8 and 9 show the force tracking corresponding to a.5 Hz sinusoidal command in motion while maximizing stiffnessnd minimizing stiffness, respectively. Note that the maximumnd minimum stiffnesses are not constant, but rather are functionsf the cylinder configuration as expressed in �14�. Note also thefrequency doubling” effect in the stiffness variation, due to thequared terms in �14�. The variation in stiffness between the twoonditions �maximum relative to minimum� is approximately aactor of 6, and as such provides a significant dynamic range inchievable stiffness. The dynamic range of the stiffness could bencreased further by using a higher supply pressure �relative to the0 psig used in these experiments�.

ConclusionThis paper proposes the use of a pneumatic actuator as a vari-

ble stiffness actuator. These actuators are well suited to such aask, since they are inherently series elastic actuators, and throughhe use of two three-way servovalves rather than a single four-wayalve, the output stiffness can be controlled simultaneously withhe actuation force. The net result is a robot actuator capable of

odulated output stiffness with a minimal amount of associatedardware. This paper describes an approach to simultaneouslyontrol actuator force and stiffness, and presented experimentalesults that demonstrate the effectiveness of the proposed ap-roach. Among the performance attributes are a stiffness trackingandwidth equivalent to the force tracking bandwidth, and an ap-roximate factor of 6 dynamic range in stiffness �which could bencreased with an increased supply pressure�.

eferences�1� Hogan, N., 1985, “The Mechanics of Multi-Joint Posture and Movement Con-

trol,” Biol. Cybern., 52, pp. 315–331.�2� Hogan, N., 1984, “Adaptive Control of Mechanical Impedance by Coactiva-

tion of Antagonist Muscles,” IEEE Trans. Autom. Control, 29�8�, pp. 681–690.

�3� Hogan, N., 1980, “Tuning Muscle Stiffness can Simplify Control of NaturalMovement,” in Advances in Bioengineering, ASME Winter Annual Meeting,
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�4� Mussa-Ivaldi, F. A., Hogan, N., and Bizzi, E., 1985, “Neural, Mechanical, andGeometric Factors Subserving Arm Posture in Humans,” J. Neurosci., 5, pp.2732–2743.

�5� Alexander, R., 1990, “Three Uses for Springs in Legged Locomotion,” Int. J.Robot. Res., 9�2�, pp. 53–61.

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�7� Taylor, C. R., and Heglund, N. C., 1982, “Energetics and Mechanics of Ter-restrial Locomotion,” Annu. Rev. Physiol., 44, pp. 97–107.

�8� Salisbury, J. K., 1980, “Active Stiffness Control of a Manipulator in CartesianCoordinates,” in Proceedings of the IEEE Conference on Decision and Con-trol, pp. 383–388.

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�11� Koganezawa, K., and Yamazaki, M., 1999, “Mechanical Stiffness Control ofTendon-Driven Joints,” in Proceedings of the IEEE/RSJ International Confer-ence on Intelligent Robots and Systems, pp. 818–825.

�12� Koganezawa, K., and Ban, S., 2002, “Stiffness Control of AntagonisticallyDriven Redundant DOF Manipulator,” in Proceedings of the IEEE/RSJ Inter-national Conference on Intelligent Robots and Systems, pp. 2280–2285.

�13� English, C. E., and Russell, D., 1999, “Mechanics and Stiffness Limitations ofa Variable Stiffness Actuator for Use in Prosthetic Limbs,” Mech. Mach.Theory, 34�1�, pp. 7–25.

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�15� Hurst, J. W., Chestnutt, J. E., and Rizzi, A. A., 2004, “An Actuator withPhysically Variable Stiffness for Highly Dynamic Legged Locomotion,” inProceedings of the IEEE International Conference on Robotics and Automa-tion, pp. 4662–4667.

�16� Tonietti, G., Schiavi, R., and Bicchi, A., 2005, “Design and Control of aVariable Stiffness Actuator for Safe and Fast Physical Human/Robot Interac-tion,” in Proceedings of the IEEE International Conference on Robotics andAutomation, pp. 528–533.

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�18� Tonietti, G., and Bicchi, A., 2002, “Adaptive Simultaneous Position and Stiff-ness Control for a Soft Robot Arm,” in Proceedings of the IEEE/RSJ Interna-tional Conference on Intelligent Robots and Systems, pp. 1992–1997.

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