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MATER IALS ENG INEER ING

HRL Laboratories LLC, Malibu, CA 90265, USA.*Corresponding author. E-mail: cbchurchill@hrl.com

Churchill et al. Sci. Adv. 2016; 2 : e1500778 19 February 2016

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Dynamically variable negative stiffness structuresChristopher B. Churchill,* David W. Shahan, Sloan P. Smith,Andrew C. Keefe, Geoffrey P. McKnight

Dow

Variable stiffness structures that enable a wide range of efficient load-bearing and dexterous activity are ubiquitousin mammalian musculoskeletal systems but are rare in engineered systems because of their complexity, power,and cost. We present a new negative stiffness–based load-bearing structure with dynamically tunable stiffness.Negative stiffness, traditionally used to achieve novel response from passive structures, is a powerful tool toachieve dynamic stiffness changes when configured with an active component. Using relatively simple hardwareand low-power, low-frequency actuation, we show an assembly capable of fast (<10 ms) and useful (>100×) dynamicstiffness control. This approach mitigates limitations of conventional tunable stiffness structures that exhibit eithersmall (<30%) stiffness change, high friction, poor load/torque transmission at low stiffness, or high power activecontrol at the frequencies of interest. We experimentally demonstrate actively tunable vibration isolation and stiffnesstuning independent of supported loads, enhancing applications such as humanoid robotic limbs and lightweightadaptive vibration isolators.

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INTRODUCTION

We have developed an active variable stiffness vibration isolator capableof 100× stiffness changes and millisecond actuation times, independentof the static load. This performance surpasses existing mechanisms byat least 20× in either speed or useful stiffness change. Our designphilosophy was to combine active vibration control and nonlinearnegative stiffness (NS) into a benchtop demonstration system, althoughthe idea is applicable at scales from microns to meters. We startedwith a classical quasi-zero–stiffness (QZS) system consisting of positivestiffness (PS) and NS springs in parallel, which together add up to zerostiffness (1). We then added a solid-state actuator to control theamount of compression in the nonlinear mechanism, which providesNS. This let us control stiffness rapidly and continuously between thestiffness of the positive spring and nearly zero while still retainingexcellent isolation performance.

Our system builds on the capabilities of state-of-the-art variablestiffness systems in two ways: speed and amount of stiffness change.Mechanistic methods have inherently low performance at low stiff-ness, that is, zero force/torque at zero stiffness. This limits the practicalstiffness change to about 5× when supporting a load (2). Pneumaticair springs are easily tunable but trade off friction for volumetricefficiency, whereas smart material–based methods are fast but arelimited to about 30% stiffness change (3). Some NS systems havethe ability to change stiffness but only for initial assembly and zero-stiffness matching (4, 5). To our knowledge, dynamic stiffnessswitching has not been attempted at this scale and speed.

Dynamic stiffness control could be useful to several engineeringdisciplines that must adapt to large variations in loading rate andfrequency, as might be found in automotive (1, 3), aerospace, robotic,and assistive robotic systems. These platforms may use active isolatorswhen vibrations are large, mixed with shocks, or vary in their frequencyspectrum. Commercial active systems control dynamics through acounteracting force from an actuator, for example, hydraulics andvoice coil (1, 2). Cost, parasitic weight, and narrow-band effecton vibration and shock have limited their widespread adoption.

Semiactive vibration mitigation manipulates the stiffness, mass, ordamping of one or more of the system components. It is typicallymore efficient than active methods, which add energy to the systemto directly react to a disturbance. The most common semiactivemethod is to tune damper properties through changes in fluid viscosity(that is, magnetorheological fluid) or damper valve systems (6–8).Tuning the mass or moment of inertia has been demonstrated inthe case of dynamic vibration absorbers (9), but it is impractical forthe isolation of larger structures and vehicles because a traditionalisolator does not use a dynamic mass. Tuning of mechanical stiffnessis typically performed either through the insertion of additional springelements into the load path (for example, a jounce bumper on anautomotive suspension) or through geometric changes to springnetworks (2, 10). These methods work well when the supported massis small, for example, with robust tuned mass damper networks (11),but may be unsuitable for load-bearing applications when the positionof the isolated mass must be maintained independent of the stiffnesscharacteristics.

The most ubiquitous tunable stiffness structures are our own joints,which use antagonistic muscle contractions to vary joint stiffnesscontinuously. For example, your limbs will stiffen to lift a bowlingball but soften to paint with the tip of a brush. This functionalitywas initially identified in the field of humanoid robotics in softwareas impedance control (12), but because of stability problems (13),hardware solutions called variable stiffness or series elastic actuatorsarose. They consist of a pair of antagonistic nonlinear springs; thestiffness and load capacity of the joint are functions of the springcompression (2, 14, 15). The benefits of variable stiffness springs(or impedance control) include the ability to walk over uncertain terrain(16), safety while working with humans (17), energy efficiency (18, 19),and effective rehabilitation exoskeletons (20).

An alternative approach to manipulate stiffness is through NSstructures. NS structures exploit instability, such as buckling, snap-through, or phase change, to create a valuable mechanical non-linearity. Previously, they have been used in passive networks to createextreme properties in demonstration systems, such as extreme stiffnessand damping in composites (21, 22), enhanced broadband energy har-vesting (23, 24), QZS vibration isolation systems (1, 4, 25), enhancement

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of damping and energy management for earthquake engineering (5, 26),and enhanced MEMS (microelectromechanical systems) sensors (27).Despite considerable research interest, they have not found wide ac-ceptance, in part because they are difficult to make and their non-linear properties are highly sensitive to boundary conditions. We takeadvantage of that sensitivity, placing a solid-state actuator in the NSsystem to enable stiffness control of more than 100×, independentof the static load.

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RESULTS

At small deflections, our variable stiffness mechanism can be de-scribed with a simple single-degree-of-freedom (DOF) model.Following the work of Alabuzhev and Lakes (1, 21), Fig. 1B illus-trates a basic NS mechanism, also known as a “stiffness corrector.”A PS spring (k3) supports the payload, whereas a horizontal spring (k2)acts in the vertical direction through a snap-through mechanism.Held at its unstable point, this NS mechanism, which includes boththe k2 spring and the parallelogram linkage mechanism, cancels outthe PS to create a QZS or high-static–low-dynamic stiffness (HSLDS)system. Our novel addition to this work is to add a high-speed actuatorto the NS mechanism, which continuously adjusts the k2 spring pre-load, x0. This creates a highly variable stiffness system, one that canexhibit stiffness ranging from the k3 spring alone to as close to zero astolerance permits. At the small-deflection limit defined by eq. S5,the equation of motion for the vertical deflection (y) of the payloadmass (m) is

my:: þ cy

: þ k3 −2 k2L

x0

� �y ¼ F tð Þ ð1Þ

where c is an equivalent viscous damping coefficient, L is the length ofthe NS flexure beams, and F is an external force. Note that the equi-librium position y = 0 is constant for all stable x0.The transmissibilityof this single-DOF system in frequency space can still be modeled

Churchill et al. Sci. Adv. 2016; 2 : e1500778 19 February 2016

using a classical spring-mass-damper model, with the definition of avariable stiffness tuning ratio e = 1 − 2k2x0/(k3L)

Tr ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ cw

k3e

� �2

1− mw2

k3e

� �2þ cw

k3e

� �2

vuuuut ð2Þ

We built the hardware in Fig. 1A to demonstrate this variable stiff-ness system; static force displacement characteristics for the individualNS and PS components, as well as the assembled system, are shown inFig. 1D.Measured separately, the PS k3 spring has a stiffness of 100 N/mm.The NS component achieves any stiffness ranging from slightly positive10 N/mm to a negative −100 N/mm, equaling that of the PS compo-nent, up to a displacement of about ±0.25 mm.

To measure the dynamic behavior of the system, we bolted it to anelectromagnetic shaker fitted with two single-DOF accelerometersmounted at the base (input y0) and the sprung mass (output ym).We tuned the apparatus to its highest stiffness state (−30 V to theactuator) and gradually increased the voltage to increase x0. Theresulting transmissibility data in Fig. 2A clearly show that the system’sresonance varies 10× from 160 to 16 Hz, which corresponds to a stiff-ness change of 100×. At our chosen excitation amplitude, the sensornoise floor lies at about −20 dB; at higher amplitudes, we expect thesystem to provide isolation up to 30 dB at frequencies below 400 Hz,the next structural resonance.

Assigning zero deflection, x0 = 0, at this voltage and given themeasured PS of the system with no preload, we estimated a dynamicmass of 0.106 kg. The hardware functions well with up to 5 kg ofsupported mass, but we chose this relatively small mass to bring thenatural frequency into the range of our test equipment. Figure 2 showsthe points surrounding each resonant peak (excluding points under−10 dB) fitted to the classical spring-mass-damper transmissibility(eq. 2) to estimate the system damping. These damping estimates,expressed as both the quality factor Q ¼ ffiffiffiffiffiffiffiffiffiffi

k3emp

=c and the viscous

x0 coarse adjustment

Piezo actuator

ym

Flexure-hinged negative stiffness (NS)snap-through mechanism

k3 positive stiffness (PS) plate spring

Lateral stiffener

Translation stage flexures

A B

DC

y (mm)

(N)F

–0.5 0 0.5–50

0

50

100

150

200y0 = 1.23 mm

–30 V

+150 V

150 V

–30 V

–30 V

–100 N/mm

10 N/mm

Fully assembled,

NS components alone

m

k2

x0

x

k3

ym

yb

Fb

Fm

y0

Lθ

Fig. 1. Design and performance of a dynamically variable stiffness structure. (A to D) Mechanical assembly (A and C), equivalent model (B), andmeasured force-displacement responses (D) for individual components. The vertical k3 (PS) spring is a simple steel plate. The horizontal k2 spring is

principally the compliance of the piezoelectric stack. The snap-through mechanism transforms the compression of the k2 spring into a verticalNS spring with a controllable stiffness between +10 and −100 N/mm. The assembled system stiffness is the sum of the NS and PS springs, resultingin a system with highly variable stiffness. Unlike other variable stiffness technologies, load capacity and stiffness are completely decoupled; in thisexample, the supported load is 130 N.

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damping equivalent c, are shown in Fig. 2B. The model above defines cas a constant, whereas Q is proportional to frequency with a slopem/c.As the fitted values show, this is a good assumption at most tuningvalues, with a constant value of c = 0.8 to 1 kg/s or dQ/dw = 0.9, validfor frequencies above 20 Hz. At the most extreme tuning, this assump-tion breaks down, and we measured increased damping from an un-known source, which could be interactions with the actuator powersupply or stiction from bolted joints.

Stiffness control largely divorces actuator bandwidth from excita-tion frequency, but most applications require stiffness to be changedat some minimum rate. For example, the system may need to reactto a sudden change in the excitation frequency/amplitude or protectitself from an anticipated shock load. Our piezoelectric-driven plat-

Churchill et al. Sci. Adv. 2016; 2 : e1500778 19 February 2016

form can change stiffness on demand at extremely high rates, yet stilllets us simulate slower, cheaper actuators. To stiffen, energy is re-leased from the system, so the only limit on transition speed is theinertia of the components. To soften, energy is added to the system,so actuator power becomes the limiting factor.

First, we show the system response at the most rapid switchingrate under the same broadband noise excitation as in Fig. 2. Duringsoftening in Fig. 3A, the system reaches 90% of its target position inonly 5 ms. Improved isolation is immediate, and acceleration amplitudedrops from 5 to <0.5 g. This is also reflected in the power spectrumin Fig. 3C, where the resonance peak shifts below 20 Hz and drops by10 dB owing to increased damping. The stiffening transformationin Fig. 3B is faster (<1 ms) because energy is being released from

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t (s)

(µm)∆ ∆

(g)am

1 2 3 4 50

40

80

120

160

–10

–5

0

5

10am

t (s)

(µm)∆

(g)am

3 4 5 6 70

40

80

120

160

–10

–5

0

5

10

∆

am

ω (Hz)

(dB)T r

0 100 200 300 400–40

0

40

ω (Hz)

(dB)T r

0 100 200 300 400–40

0

40

A

C D

B

Fig. 3. Rapid stiffness switching. (A to D) Rapid softening (A and C) and stiffening (B and D) of the adaptive stiffness platform under white noise baseexcitation. The top row shows time histories of the mass acceleration (dots) and piezoelectric actuator motion (solid blue line). The bottom row shows the

frequency response, for a short 0.5-s window before and after the stiffness switch. The window center for each spectrogram is indicated by the verticallines on the top row. am, acceleration amplitude.

ωn (Hz)

(kg/cs) Q

Q

c

0 50 100 150 2000

0.5

1

1.5

2

2.5

0

50

100

150

200

250

BA

0 20 40 60 80 100 120 140 160 180 200–40

–20

0

20

40

60

Tr(dB)

f (Hz)

Increasing piezo voltage

–30 V

150 V

SDOF model (variable c)

Data

Fig. 2. Dynamic performance. (A) Transmission data and estimated linear model with best-fit damping and natural frequency for each stiffness tuningvoltage. The best-fit values are plotted in (B), in terms of both damping coefficient and quality factor. SDOF, single-DOF; f, frequency.

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the system into the new 160-Hz natural frequency, which rings imme-diately after the stiffness change.

When changing stiffness in a tonal excitation environment, it isvaluable to “hop” over troublesome excitation frequency bands withoutresonating. This sets a lower limit on actuator power. Whereas creatinga single rule to cover all possible environments depends on the tonespectrum and system damping, we have found that for the case oflow damping, excitation and resonance regions should overlap nomore than 10 oscillation periods.

To illustrate this effect, the base excitation was changed fromconstant-power white noise to a dual-sine excitation with tones at50 and 100 Hz. Next, the actuator was swept between 20 and 150 Hzat various rates of voltage change. Figure 4 shows time histories of the50- and 100-Hz spectrum lines, as measured with a moving flat top

Churchill et al. Sci. Adv. 2016; 2 : e1500778 19 February 2016

window, 0.5 s wide (512 samples). The system starts at a natural fre-quency of 20 Hz, so the 50- and 100-Hz tones are isolated with 8 and25 dB, respectively. In the time period of 4 to 6 s, the system is in itsstiff state, and the tones sit below isolator resonance so transmission isclose to 0 dB at 50 Hz and slightly higher at the closer 100-Hz tone. At6 s, the isolator is softened, and the system returns to its isolating state.When softened quickly, in 100 ms (black line), the system does notresonate measurably, but at slower rates, the tones are increasinglytransmitted, first at 100 Hz and then at 50 Hz. At the lowest rate,the isolator risks nonlinear behavior (stiffness amplification), as describedin eq. S5, due to the excessive displacement amplitude at resonance.

DISCUSSION

Advanced lightweight materials are increasingly finding their way intotransportation platforms to achieve low mass and high stiffness. Theyare expected to isolate multiple different vibration environments, buttheir increased structural efficiency tends to only increase transmissionof shock and vibration to the payload. To compensate, we seek newmechanisms of vibration control that are inherently energy efficientand scalable to a variety of loads and environmental frequencies. Inthe area of robotics, we also need high-quality variable stiffness jointsfor safety, power efficiency, and natural movement.

We have introduced a new concept of dynamic stiffness controlusingNS coupledwith an actuator to build a system that controls stiffnessthrough two orders ofmagnitude. Thismethodmanages high-frequencydynamic loads with simple, low-power, low-frequency inputs, whichcan inform new solutions to long-standing problems. We imaginerobot manipulators that can dynamically vary stiffness to successful-ly manipulate a scalpel and lift a patient from their bed with the samejoint or semipassive mechanical isolators that avoid multiple inputfrequencies as encountered in high-efficiency transportation platforms.

The scope of this paper demonstrates the basic concept of activestiffness control and experimentally validates a simple model forthe system’s dynamic behavior. Ongoing work focuses on multiaxisloading, actuator efficiency and optimization, and robust handlingof low-frequency load variations. Scaling this approach to meet theneeds of these varied applications will require individually tailoredNS components as well as actuators and control systems.

MATERIALS AND METHODS

Following the variable stiffness hardware pictured in Fig. 1A, the PS(k3) spring lies at the base and consists of a laser-cut steel planarspring. A snap-through mechanism lies directly above the k3 spring.It consists of a parallel bar linkage system with notch flexure hinges atthe ends to provide a frictionless pinned connection. The bar length is24.1 mm; the flexure notches are 0.15 mm thick. The left-hand side ofthe mechanism is grounded, whereas the other side is connected to apiezoelectric actuator, model APC PSt 150/10/20 VS15, held in ahorizontal parallel flexure translation stage. The right-hand side ofthe actuator is grounded through a fine-thread adjustment screw,which adjusts the initial system preload x0. This preload ensures thatthe actuator is in constant contact with the translation stage, as well asoperating in the best displacement range for maximum stiffnesschange. As the actuator expands, it increases the horizontal compres-sion force acting on the snap-through mechanism, thus increasing the

t (s)

(dB)T r50

0 2 4 6 8 10–30

–20

–10

0

10

20

30

t (s)

(µm)∆

0 2 4 6 8 100

50

100

150

t (s)

(dB)T r100

0 2 4 6 8 10–30

–20

–10

0

10

20

30

B

A

C

3000 ms

3000 ms

3000 ms

500 ms

500 ms

100 ms

100 ms

100 ms500 ms

Fig. 4. Isolator transmissibility during a continuous stiffening thensoftening sweep through simultaneous tonal excitations of 50 and

100Hz. (A) 50-Hz line transfer function history during sweep times of100, 500, and 3000 ms. (B) 100-Hz line history. (C) Piezo position (x0) history.

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NS measured in the vertical direction. Lateral stiffeners are designed toincrease the first global bending mode frequency of the entire device,expanding the bandwidth of single-DOF operation. Most of thecontribution to the horizontal spring k2 is from the stiffness of thepiezoelectric stack itself, which was measured to have an open-circuitstiffness of 15 N/mm. Together, the piezo and snap-through mechanismform the NS component. The center of the snap-through mechanismand the k3 spring are connected in parallel by a vertical-threaded rod.Nuts along the rod adjust the k3 preload, y0.

Before full dynamic experiments, the individual quasistatic force-displacement measurements in Fig. 1D were measured in a uniaxialhydraulic load frame (MTS model 858). First, the k3 spring wasmeasured alone with other components removed. It had a PS of100 N/mm. Next, the k3 spring was removed, and the NS componentwas tested at various levels of preload x0. With the actuator removedentirely, the flexures alone were measured to add 43 N/mm stiffness.Next, the piezo stack was set to its minimum −30 V and manuallyloaded with a set screw to about x0 = 50 mm. Force-displacementcurves in the bottom of Fig. 1D were taken adjusting the screw, onlychanging x0 through the actuator from about 50 to 240 mm. The NScomponent showed stiffness ranging from slightly positive to −100N/mm, enough to perfectly cancel the stiffness of the k3 spring. Finally,the k3 spring was coupled with a small vertical preload y0 to dem-onstrate load-holding ability. The top of Fig. 1D shows the static re-sponse of the fully assembled system.

For dynamic experiments, the validation system was bolted to anelectromagnetic shaker (Unholtz-Dikie; rated at 900 N, 85 g, and50 mm displacement) and fitted with two single-axis accelerometersmounted at the base (input) and the sprung mass (output).

For the data in Fig. 2, the system was excited with an open-looprandom input spanning 0 to 400Hz. Deflections were kept sufficientlylow (<0.5 mm) so as not to create any amplitude-dependent properties(W ≅ 1 from eq. S5). At each stiffness (or actuator voltage), the trans-missibility of the system was reported as the ratio of accelerations be-tween the base and supported mass, from 0 to 200Hz.

For Fig. 3, two separate power supplies were set to +150 and −30 V.A single-pull double-throw switch between the supplies and the ac-tuator allowed rapid switching between the two supplies, whereasdiodes protected each supply from reverse-current surge. Time res-olution of system stiffness was limited by the digital sample win-dow length because a good natural frequency measurement cannot beresolved in the relatively small 0.5-s window range. To compensate, weused a strain gauge directly attached to the actuator to directly measureend motion, providing a separate real-time readout of system state.

For Fig. 4, the actuator was attached to a voltage-controlled powersupply (ThorLabs MDT693; maximum current, 60 mA). Instead ofrandom noise, the shaker was excited with dual tones at 50 and100 Hz, still taking care to keep the absolute amplitude for yb below±0.5 mm. The stiffness was changed continuously by commandingthe power supply with a linear voltage ramp between 0 and 150 V atrates corresponding to 100−, 500−, and 3000−ms sweeps.

SUPPLEMENTARY MATERIALSSupplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/2/2/e1500778/DC1Modeling dynamic stiffness controlEq. S1. Nonlinear system equation of motion.

Churchill et al. Sci. Adv. 2016; 2 : e1500778 19 February 2016

Eq. S2. Nonlinear nondimensional equation of motion.Eq. S3. Stiffness change ratio.Eq. S4. Maximum practical stiffness change.Eq. S5. Nonlinear frequency amplification factor.Eq. S6. Linear nondimensional equation of motion.Eq. S7. Linear system transmissibility.Table S1. Model parameters and nondimensional substitutions.Reference (28)

REFERENCES AND NOTES1. P. Alabuzhev, A. Gritchin, Vibration Protecting and Measuring Systems with Quasi-Zero Stiff-

ness (CRC Press, Boca Raton, FL, 1989).2. E. I. Rivin, Handbook on Stiffness & Damping in Mechanical Design (ASME Press, New York,

2010).3. M. R. Jolly, J. D. Carlson, B. C. Muñoz, A model of the behaviour of magnetorheological

materials. Smart Mater. Struct. 5, 607–614 (1996).

4. D. L. Platus, Negative-stiffness-mechanism vibration isolation systems. SPIE Vib. ControlMicroelect. Opt. Metrol. 1619, 44 (1991).

5. D. T. R. Pasala, A. A. Sarlis, S. Nagarajaiah, A. M. Reinhorn, M. C. Constantinou, D. Taylor,Adaptive negative stiffness: A new structural modification approach for seismic protection.J. Struct. Eng. 139, 1112–1123 (2013).

6. D. Karnopp, Active and semi-active vibration isolation. J. Vib. Acoust. 117, 177–185(1995).

7. A. Preumont, Vibration Control of Active Structures (Springer, Berlin, 2011).8. G. Z. Yao, F. F. Yap, G. Chen, W. H. Li, S. H. Yeo, MR damper and its application for semi-active

control of vehicle suspension system. Mech. Mater. 12, 963–973 (2002).9. J. Q. Sun, M. R. Jolly, M. A. Norris, Passive, adaptive, and active tuned vibration absorbers—

A survey. J. Mech. Des. 117, 234–242 (1995).10. S. Nagarajaiah, E. Sonmez, Structures of semiactive variable stiffness single/multiple tuned

mass dampers. J. Struct. Eng. 133, 67–77 (2007).11. S. Nagarajaiah, N. Varadarajan, Short time Fourier transform algorthim for wind response

control of buildings with variable stiffness TMD. Eng. Struct. 27, 431–441 (2005).12. N. Hogan, Impedance control: An approach to manipulation: Part II—Implementation. J. Dyn.

Syst. Meas. Control 107, 8–16 (1985).13. N. Hogan, Adaptive control of mechanical impedance by coactivation of antagonist

muscles. IEEE Trans. Autom. Control 29, 681–690 (1984).14. R. Ham, T. G. Sugar, B. Vanderborght, K.W. Hollander, D. Lefeber, Compliant actuator designs.

IEEE Robot. Autom. Mag. 16, 81–94 (2009).15. G. A. Pratt, M. M. Williamson, Series elastic actuators. Proc. Int. Conf. Intelligent Robots and

Systems 1, 399–406 (1995).16. J. H. Park, Impedance control for biped robot locomotion. IEEE Robot. Autom. Mag. 17,

870–882 (2001).17. M. Zinn, B. Roth, O. Khatib, J. K. Salisbury, A new actuation approach for human friendly

robot design. Int. J. Robot. Res. 23, 379–398 (2004).18. D. J. Braun, M. Howard, S. Vijayakumar, Exploiting variable stiffness in explosive movement

tasks, in Robotics: Science and Systems VII (MIT Press, Cambridge, MA, 2012).19. L. C. Visser, R. Carloni, S. Stramigioli, Energy-efficient variable stiffness actuators. IEEE Trans.

Robot. 27, 865–875 (2011).

20. J. A. Blaya, H. Herr, Adaptive control of a variable-impedance ankle-foot orthosis to assistdrop-foot gait. IEEE Trans. Neural Sys. Rehabil. Eng. 12, 24–31 (2004).

21. R. S. Lakes, T. Lee, A. Bersie, Y. C. Wang, Extreme damping in composite materials withnegative-stiffness inclusions. Nature 410, 565–567 (2001).

22. Y. C. Wang, R. S. Lakes, Extreme stiffness systems due to negative stiffness elements. Am. J. Phys.72, 40–50 (2004).

23. F. Cottone, H. Vocca, L. Gammaitoni, Nonlinear energy harvesting. Phys. Rev. Lett. 102, 080601(2009).

24. R. L. Harne, K. W. Wang, A review of the recent research on vibration energy harvesting viabistable systems. Smart Mater. Struct. 22, 023001 (2013).

25. R. A. Ibrahim, Recent advances in nonlinear passive vibration isolators. J. Sound Vib. 314,371–452 (2008).

26. A. A. Sarlis, D. Pasala, M. Constantinou, A. Reinhorn, S. Nagarajaiah,D. Taylor, Negativestiffness device for seismic protection of structures. J. Struct. Eng. 139, 1124–1133(2013).

27. R. Harne, K. Wang, Mass detection via bifurcation sensing with multistable microelectro-mechanical system, presented at the ASME 2013 Conference on Smart Materials, AdaptiveStructures and Intelligent Systems, Snowbird, UT, 16 to 18 September 2013.

28. P. J. Holmes, D. A. Rand, The bifurcations of Duffing’s equation: An application ofcatastrophe theory. J. Sound Vib. 44, 237–253 (1976).

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Acknowledgments: We appreciate the continued support of our LLC partners Boeing andGeneral Motors in helping to motivate this research. We acknowledge J. Ensberg forperforming structural analyses, which improved the performance of the demonstrationhardware. We also appreciate the helpful discussions with and technical assistance of ourcolleagues C. Henry, G. Herrera, J. Mikulsky, G. Chang, W. Carter, and L. Momoda. Funding:This work was performed at HRL Laboratories LLC, using internal funding. Author contribu-tions: G.P.M., C.B.C., and D.W.S. provided the research motivation and system models. C.B.C.,S.P.S., and A.C.K. designed, assembled, and tested the variable stiffness hardware. Competinginterests: The authors declare that they have no competing interests. Data and materialsavailability: All data needed to evaluate the conclusions in this paper are present in the paper

Churchill et al. Sci. Adv. 2016; 2 : e1500778 19 February 2016

and/or the Supplementary Materials. Additional data and raw data are available upon requestfrom C.B.C. (cbchurchill@hrl.com).

Submitted 11 June 2015Accepted 7 December 2015Published 19 February 201610.1126/sciadv.1500778

Citation: C. B. Churchill, D. W. Shahan, S. P. Smith, A. C. Keefe, G. P. McKnight, Dynamicallyvariable negative stiffness structures. Sci. Adv. 2, e1500778 (2016).

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Dynamically variable negative stiffness structuresChristopher B. Churchill, David W. Shahan, Sloan P. Smith, Andrew C. Keefe and Geoffrey P. McKnight

DOI: 10.1126/sciadv.1500778 (2), e1500778.2Sci Adv 

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