Cross-Coupling Errors of Micromachined Gyroscopes - University of ...

JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 13, NO. 2, APRIL 2004

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Cross-Coupling Errors of Micromachined Gyroscopes Eva Kanso, Andrew J. Szeri, and Albert P. Pisano, Member, ASME

Abstract—A model is developed for the response of a micromachined, rotary gyroscope subject to a general input motion. The governing equations are formulated for weakly nonlinear oscillations of the rotor, suspended above the moving substrate via elastic beams. The method of multiple scales is used to separate the slow and fast responses. This approach allows quick computation of the long-term behavior of the rotor without the need to integrate over fast oscillations. The power of the model to evince cross-coupling errors is demonstrated through examples. [1184] Index Terms—Cross-coupling, gyroscope, micromachined, multiple scales, parametric excitation.

I. INTRODUCTION

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HE MECHANICAL components of a micromachined gyroscope consist of a vibratory proof mass supported above a substrate via elastic beams. The proof mass is typically driven into linear or rotary oscillation, called drive motion, near its resonant frequency in order that a desirable amplitude is reached with minimum power input. The application of an external rotation to the substrate induces a second oscillation of the proof mass, either through Coriolis forces or gyroscopic couplings. This response, also called the sense motion, is measured by appropriate on-chip electronics. The measurement is used to estimate the angular motion of the substrate. Hence, understanding the dynamic coupling between the response of the proof mass and the free motion of the substrate is crucial to interpret properly the sensed response. Mathematical models that describe the dynamics of the proof mass have been developed for various designs of both linearly and rotary driven microgyroscopes, e.g., [1], [2]. Most of the early models were linear and focused on understanding the response given an input rotation designed to excite only the sense direction. That is, they ignored the remaining dynamic coupling due to a general substrate motion. However, these cross-couplings may produce an undesirable sense signal, called quadrature error, even in the absence of manufacturing imperfections [3].1 To this end, the elastic couplings that occur in the suspension beams of rotary gyroscopes have been studied extensively in [4]–[6]. There, a weakly nonlinear, torsional spring model is proposed to emulate the resistance of the suspension to a general rotation of the proof mass, or rotor. Here, we extend the suspension spring model to include the resistance of the beams to a rectilinear motion of the rotor perpendicular to the substrate, Manuscript received October 21, 2003. Subject Editor G. Stemme. The authors are with the Department of Mechanical Engineering, University of California, Berkeley, CA 94720-1774 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/JMEMS.2004.825293

1One finds a classification of the errors produced by manufacturing imperfections in [2].

also called the bounce motion. Subsequently, we formulate the equations governing weakly nonlinear oscillations of the vibratory rotor suspended to a substrate undergoing a general motion. These equations evince the effects of various cross-couplings on the sense response. In particular, they reflect the possibility of destabilization of the sense response through parametric excitation not only due to its coupling with the drive oscillation, noted in [5], but also due to its coupling with the bounce motion. Numerical integration of the equations governing the rotor motion is challenging due to the existence of two time scales which differ typically by more than 4–6 orders of magnitude. The first time scale is defined by the natural frequency of the , and the second by the input anproof mass . In this work, the gular velocity of the substrate method of multiple scales is used to separate the slow and fast responses. This approach is attractive because it allows quick computation of the long-term behavior of the rotor without the need to integrate over the fast oscillations. We remark that, in [2], the authors have averaged over the fast oscillations to obtain the equations governing the slow behavior predicted by the linear model. The organization of this paper is as follows: a model for the rotor and the suspension is described in Section II. The equations that govern the motion of the rotor are derived in Section III. A relative scaling of these equations is suggested in Section IV and a separation of time scales in the resulting equations is performed in Section V. In Section VI, the rotor response is examined numerically for several input motions of the substrate. Some remarks regarding the effects of parametric excitation on the rotor response are made in Section VII. In Section VIII, we summarize the findings of this work. II. DEVICE DESCRIPTION The mechanical components of the micromachined, dual-axis rate, rotary gyroscope consist of a polysilicon rotor suspended above the substrate via four elastic beams as shown in Fig. 1. The rotor is electromagnetically driven into rotational oscillation about an axis perpendicular to the substrate [see Fig. 2(a)]. The amplitude and frequency of the drive oscillation are controlled to remain constant despite any motion of the substrate. The gyroscope operates based on the principle that the angular momentum vector of the rotor, effectively a rigid body, tends to maintain a constant orientation, hence causing the rotor to undergo an out-of-plane tilt (or sense) response when an input rotation is applied to the substrate [see Fig. 2(b)]. The elastic suspension beams apply restoring forces and moments to the rotor without preventing it from rotating relative to the substrate. However, the mass center of the rotor is constrained to undergo only an out-of-plane (bounce) motion. Indeed, the in-plane motion of is neglected because the elastic

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with respect to the variables describing that motion. This nonlinearity is due to the change in the geometric configuration of the springs as the rotor undergoes a general motion—hence, it is called geometric nonlinearity. Finally, the effect of structural and air damping are modeled using torsional and linear dampers. III. EQUATIONS OF MOTION An exact description of the kinematics of the rotor is developed and followed by a derivation of the governing equations for weakly nonlinear oscillations of a rotor suspended to a moving substrate. A general motion of the rigid rotor can be decomposed into a pure rotation and a translational of the mass center . The angular velocity vector associated with the rotational motion can be expressed as (1) Fig. 1. Schematic plot that shows the mechanical components of the micromachined gyroscope: (a) top view of the device and (b) side view along the MN-section.

is the angular velocity vector of the rotor relative to where is that of the substrate. The position of the substrate and the mass center can be written as (2) where is the position of the mass center of the rotor relative is that of point . to a point of the substrate, and It is convenient to define two distinct orthonormal bases: and , , 2, 3, such that moves with the substrate and with the rotor; see Fig. 2. The rotation of the rotor relative to the substrate can be described using the 1-2-3 Euler angles, denoted by , and , respectively. In this set of generalized coordinates, the relative rotation is constructed via three successive simple rotations—a rotation about the -axis, followed by about the -axis, where , then about the -axis, where . Consequently, one can write

Fig. 2. The micro-gyroscope: (a) Reference configuration and (b) General motion. Note that the two bases f g, corotational with the substrate, and f g, corotational with the rotor, are chosen to coincide in the reference configuration.

E

e

(3) beams are much stiffer in extension than in bending. Torsional and linear springs are used to emulate the effect of the elastic suspension beams on the rotor tilt and bounce motions. Namely, the nonlinear torsional spring model developed in [4] based on the Cosserat rod theory is used to resist the tilt motion. In addition, four pre-stretched, identical, linear springs are assumed to connect to the end points of the beams, see Figs. 1 and 2. This configuration is desirable because, as moves away from , the linear springs produce a net moment that resists tilting of the rotor in addition to a net force that resists the rectilinear motion of . Hence, this configuration properly imitates the stiffening effect that occurs in the elastic beams. It is emphasized that the four linear springs are also elastically linear, that is, the spring force is proportional to extension. Yet, the net force and moment they produce in resisting the rotor motion are nonlinear

Clearly, the angular velocity strate can be expressed as

of the rotor relative to the sub(4)

where denotes the usual derivative with respect to time . are used Equations (3) and (4), and the definitions of and relative to the -basis to obtain the components of


(5)

KANSO et al.: CROSS-COUPLING ERRORS OF MICROMACHINED GYROSCOPES

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The angular velocity vector of the substrate can -basis using (3): be written relative to the

and and displacement , we use the model for the damping and the elastic suspension presented in Section II to obtain the and : following expressions for

(11)

(6) Let the position of the mass center relative to the origin be expressed as . Recall that and differentiate (2) twice with respect to time to derive the following expression for the acceleration vector of the mass center

Expressions for the right-hand sides of (9) are obtained using (1), (5) and (6). For small rotations, these expressions can be simplified considerably by, first replacing the trigonometric functions of , and by their first-order polynomial expansions, then discarding all nonlinear terms in , and and their time derivatives.4 Consequently, (9) reduce to

(7) In (7), denotes a time derivative corotational with the basis, that is, given an arbitrary vector , one has . The last term of (7) involves , the acceleration of point of the substrate. It is worth noting that the kinematics derived thus far is exact in the framework of the rigid rotor model and is independent of the model of the elastic suspension. and diThe elastic suspension is much stiffer in the rections. Hence, one can make the approximation that the mass . In center is subject to the constraints this case, the acceleration of (7) reduces to

(12) The mass center is constrained to move along , hence, -direction using (8), the balance of linear momentum in the reads as (13)

(8) where

is the total force in this direction

The balance of angular momentum of the rotor expressed in takes the following form: (14)

(9) In (9), are the components of the total moment applied to the rotor, and are the principal components of the rotor moment of inertia tensor, with

, with or , and The relationship between of the linear springs shown in Fig. 2(b) is prethe stiffness sented in Table II. Consequently, (13) becomes

(10) , for a perfectly symmetric rotor.2 It is worth noting that the rotor is driven into a sinusoidal oscillation whose amplitude is typically 0.08 (rad) while the tilt angles and and the bounce motion are limited by the gap between the rotor and the substrate, which means that . For such small rotations3 ,

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2Note that f g are, by construction, principal directions of the moment of inertia tensor; see Fig. 2(b). 3For small rotations, the distinction between f g and f g vanishes.

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(15) Dimensionless counterparts to (12) and (15) are obtained by introducing the dimensionless variables5 (16) 4One could repeat the same procedure to obtain polynomial approximation of any order. 5The angles , and are dimensionless.


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where is the natural frequency of the rotor in (rad/s).6 The derivatives with respect to time and dimensionless time are related as follows

TABLE I THE DIMENSIONS AND MATERIAL PROPERTIES OF THE POLYSILICON GYROSCOPE SKETCHED IN FIG. 1 ARE TAKEN FROM [5]

(17) Now, replace by and substitute (16) and (17) back into (12) and (15) in order to obtain the dimensionless equations of motion

TABLE II THE STIFFNESSES OF THE TORSIONAL SPRING MODEL ARE TAKEN FROM [4] WHILE K IS ESTIMATED BASED ON THE EULER-BERNOULLI BEAM THEORY IS DEFINED AS AS IN [1]. THE SYMBOL l l =l WHERE l IS THE LENGTH OF THE PRESTRETCHED LINEAR SPRING IN THE REFERENCE CONFIGURATION, SEE FIG. 2(a), AND l IS THE UNSTRETCHED LENGTH. . THE STIFFNESSES K : HERE, IS TAKEN TO BE AND K FOLLOW FROM THE GEOMETRY OF THE LINEAR SPRING MODEL SHOWN IN FIG. 2

1

1

1=( 0 )

1 = 2 6(10)

(18) and

(19) where the notation

is used to denote a derivative

.

IV. PARAMETER SCALING A positive scaling parameter is chosen to reflect the large of the rotor, magnitude of the natural frequency of oscillation , in comparison with the input angular velocity . That is, we choose such of the substrate, that (20)

given and for a perfectly symmetric rotor, the natural frequency is . Hence, by definition, the dimensionless linear stiffness (23) . Further, one can readily conclude that the discales as mensionless stiffnesses

Further, we assume that

(24) (21)

are also of

. However, one has

Consequently, a dimensionless set of variables , , and that describes the substrate motion is defined as follows: (22)

(25) and (26)

Based on the estimates in Tables I and II, we introduce a scaled set of dimensionless stiffness parameters. Note that, 6In fact, for the purpose of nondimensionalizing time, ! can be an arbitrary frequency in (rad/s).

where the notation or .

in (25), (26) denotes one of the angles ,



Estimates for the quality factors and that are reported in [4]. Hence, we lie in the range of propose the following scaling:

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At

, the equations are

(27) Recall that the rotor motion is restricted to small rotations , and and small displacement . Hence, one may write (28) where are of . Finally, we substitute (28) and (22)–(27) into (18) and (19) in order to obtain the scaled, dimensionless equations

(35) The solution to the leading order equations (34) is given by

(36)

(29) and

and are, respecwhere the frequencies tively, the natural frequencies of the tilt and bounce modes. The and are funcamplitudes , and and the phases , and tions of the slow time . We assume that and substitute (36) into (35). Elimination of the resulting secular terms gives rise to the following set of evolution equations for the slowly varying amplitudes and phases:

(30) V. MULTIPLE SCALES ANALYSIS In this section, (29) and (30) are decomposed into components that are evolving at two distinct time scales: a rapid one , and a slow defined by the natural frequency of the rotor . one defined by the input angular velocity of the substrate The drive oscillation is regarded as given. The variables , and are written as polynomial expansions in (31) and functions of two time scales: a slow time and a fast time (32) It is important to note that the angular velocity and acceleraof the substrate and the linear acceleration are function tions only of the slow time . The time derivatives relative to can be expressed as

(33) where we use subscript notation to denote a derivative . Relations (31) and (33) are substituted into (29) and (30); and terms of the same order in are collected. At , one obtains

(34)

(37) In order to account for cases where the drive frequency is close but not equal to the resonant frequency , one may set , where arbitrary frequency of . The same procedure is then repeated to find the evolution equations of the slow behavior, which will have corresponding terms involving .


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Fig. 3. The rotor response to a constant angular velocity of the substrate: = 5, = 0, and = 0. The drive oscillation is set to = 5 sin( ), where = 10 . The linear acceleration a is zero. he bounce motion is z ( ) = 0 throughout. The cross-coupling induces a small error in the tilt , even in the absence of symmetry imperfections. The solid line tracing the envelope of corresponds to the numerical solution of the leading order equations (37) governing the slow behavior. Note that, at the leading order, the slow behavior does not capture the error in .

VI. NUMERICAL INVESTIGATION The behavior of the rotor is investigated for parameter values consistent with the scaling proposed in Section IV. The scaling . The drive frequency parameter is chosen to be is chosen equal to the resonance frequency, . The damping coefficients, unless otherwise specified, are set to (38) which correspond to quality factors of 1000. Based on Tables I and II and (23)–(26), one finds

(39) . In Fig. 3 In Figs. 3–8, the drive oscillation is set to is a plot of the rotor response when the substrate is moving with , and no accelconstant angular velocity direction, that is, . The initial conditions eration in the are taken to be identically zero. Note that the cross-coupling induces a small error in the tilt even in the absence of symmetry imperfections. Also, note that the leading-order equations governing the slow behavior predict properly the response but do

Fig. 4. The rotor response to a constant angular velocity of the substrate: = 5, = 0, and = 10. The drive oscillation is set to = 5 sin( ), where = 10 . The linear acceleration a is zero. The bounce motion is z ( ) = 0 throughout. The rotation has a subtle effect on the cross-coupling error in

the tilt .

not capture the much smaller error in . One needs to compute the next higher-order corrections in order to see such small errors. Fig. 4 shows the effect of an additional rotation on the response depicted in Fig. 3, while Fig. 5 illustrates the effect of a sinusoidal linear oscillation . For the chosen parameter values and initial conditions, these disturbances have subtle effects on the rotor response. Indeed, the effects are not captured at the leading-order equations of the slow behavior. Fig. 6 shows that the error in the tilt depicted in Fig. 3 becomes appreciable upon a small disturbance in the initial tilt . One also sees in Fig. 6 that the leading-order slow equations are able to describe adequately this error. Such erroneous tilt oscillations may occur even in the absence of an input rotation to the substrate, see Fig. 7 for nonzero initial condiand and Fig. 8 for , and tions . It is worth noting that the cross-coupling of the tilt with the bounce may cause erroneous tilt oscillations, even when the and drive motion is turned off, as shown in Fig. 9 for , and nonzero initial conditions and . This behavior, that is, the persistent response due to initial conditions, is a well-known property of parametrically excited systems such as the one in (29), (30) where the drive oscillaexplicitly excites the tilt and bounce responses, while tion these responses implicitly excite one another through the nonlinear elastic coupling. It is also well known that parametric excitation can induce unstable responses for certain parameter



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Fig. 6. The rotor response to a constant angular velocity of the substrate: = 5, = = 0. The drive oscillation is set to = 5 sin( ), where = 10 , and the linear acceleration to a = 0. An initial tilt = 0:01 is given. Note that, in comparison to Fig. 3 where = 0, a nonzero initial tilt induces a larger cross-coupling error in that is captured by the leading-order equations governing the slow behavior.

The rotor response to a constant angular velocity of the substrate: = = = 0. The drive oscillation is set to = 5 sin( ), where = 10 , and the linear acceleration to a = 50cos(20 ). The linear acceleration of the substrate, coupled to the rotational dynamics of the rotor, induces a bounce oscillation z (t). Fig. 5. 5,

values which would cause temporary malfunctioning or permanent damage to the gyroscope. Therefore, a stability analysis of the system (29), (30) would be of great interest for both designing and operating the micro-gyroscope. One approach would be to generate stability diagrams, either numerically or using a perturbation method to obtain approximate analytic expressions for the transition curves, that identify the regions of the parameter space7 for which the solutions of the system (29), (30) are stable. Another method would be to study equilibria in the phase space of the slow equations, their stability type and bifurcations upon changes in parameter values. These ideas are beyond the scope of the present work.

7Note

that the parameter space is higher-dimensional. To make the problem more tractable, one could regard some parameters as fixed, hence, reducing the dimensionality of the space.

Fig. 7. Erroneous tilt oscillations in the absence of an input rotation to the substrate ( = 0) and linear acceleration (a = 0). The drive oscillation is set to = 5 sin( ), where = 10 . Initial tilts = and = are used. The bounce motion is z ( ) = 0 throughout.


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Fig. 8. Erroneous tilt and bounce oscillations in the absence of an input rotation to the substrate ( = 0) and linear acceleration (a = 0). The drive oscillation is set to = 5 sin( ), where = 10 . Initial tilts = and = and initial linear offset z = are used.

VII. REMARKS ON THE EFFECTS OF PARAMETRIC EXCITATION ON THE ROTOR RESPONSE In order to investigate the consequences of the parametric excitation due to the drive and bounce oscillations on the stability of the tilt motion, we eliminate the tilt-tilt nonlinear coupling and examine the response of the tilt when the sub, and strate is moving at constant angular velocity: . In this case, (29) becomes

Fig. 9. Erroneous tilt oscillations in the absence of an input rotation to the substrate, that is, = 0. The drive oscillation is set to zero, = 0, and the = are linear acceleration to a = 50cos(20 ). Initial tilts = and used.

substrate in a similar way to controlling the drive oscillation. Hence, (40) reads as

(41) where

(42) (40) Further, in (40), we consider both and as given. The assumption that the bounce motion is prescribed is valid because, theoretically, one can adjust by varying the electro-magnetic field between the rotor and

Equation (41) is a quasiperiodically driven Mathieu equation with cubic nonlinearity and external forcing. This equation has been extensively studied [7]–[10] for the case of no substrate . In [7], it is shown that in the nonmotion, that is, when linear Mathieu equation, one obtains finite-amplitude motion,



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which may be quasiperiodic or chaotic, due to the nonlinear resonance in contrast to the linear Mathieu equation where instability implies unboundedness. The boundedness of solutions in the nonlinear equations may be exploited in designing the micro-gyroscope to ensure that, even when the sense signal is irrelevant, the rotor will not enter a dangerous zone of unbounded tilt resonance that eventually destroys the device. throughout), When the bounce motion is neglected ( (41) agrees with the model presented in [5] for the tilt motion. In that work, the authors report on the possibility of destabilization of the tilt response due to parametric excitation by the drive oscillation, especially because the tilt natural frequency is tuned to match the drive natural frequency. Note that the frequency matching is done in order to maximize the sensitivity of the device to angular rotation of the substrate by making such rotation excite a resonant tilt motion. Equation (41) suggests that one could simultaneously control the bounce motion to ensure safe operation without compromising the sensitivity of the device. In [5], the validity of the torsional spring model was experimentally checked for the gyroscope shown in Fig. 1 and the parameter values in Table I. The experiment is designed to change the electro-magnetic moment driving the oscillation and measure the corresponding amplitude and frequency of . Consequently, the experimental results are compared to a weakly nonlinear model of the form

[4] W. Davis, “Mechanical Analysis and Design of Vibratory Micromachined gyroscopes,” Ph.D. dissertation, Univ. California, Berkeley, 1999. [5] W. O. Davis, O. M. O’Reilly, and A. P. Pisano, “On the nonlinear dynamics of tether suspensions for MEMS,” ASME J. Vibr. Acoust., 2003, submitted for publication. [6] W. O. Davis, A. P. Pisano, and O. M. O’Reilly, Designing Tether Suspensions for MEMS in the Presence of Nonlinearities, 2003, submitted for publication. [7] R. Zounes and R. Rand, “Subharmonic resonance in the nonlinear Mathieu equation,” Int. J. Nonlinear Mechan., vol. 37, pp. 43–73, 2002. , “Global behavior of a nonlinear quasiperiodic Mathieu equation,” [8] Nonlinear Dynam., vol. 27, pp. 87–105, 2002. [9] L. Ng and R. Rand, “Bifurcations in a Mathieu equation with cubic nonlinearities,” Chaos, Solitons, Fractals, vol. 14, pp. 173–181, 2002. [10] R. Rand, K. Guennoun, and M. Belhaq, “2:2:1 Resonance in the quasiperiodic Mathieu equation,” Nonlinear Dynam., vol. 31, pp. 367–374, 2003.

Eva Kanso received the Ph.D. degree from the University of California, Berkeley in mechanical engineering in 2003. She is a Postdoctoral Scholar at the California Institute of Technology, Pasadena. Her research interests include mechanics and nonlinear dynamical systems.

(43) postulated for the drive oscillation. Note that, in (43), and is a sinusoidal drive moment. Indeed, the response obtained from (43) is in good agreement with the experimental results [5, Figs. 7 and 8]. We note that, if one makes use of the procedure of Section III, an equation governing the oscillation can be derived that is identical to (43) when the cross-coupling terms with the tilt and bounce are eliminated. VIII. CLOSURE The dynamic response of a micromachined rotary gyroscope to a general input motion was analyzed. A mathematical model was developed for weakly nonlinear oscillations of the rotor. The model takes into account the elastic couplings in the suspension beams which connect the rotor to the substrate. A multiple scales analysis was conducted to separate the slow response of the rotor due to a general input motion of the substrate from the fast oscillations due to the high drive frequency. Numerical simulations show that the responses obtained from the equations governing the slow behavior are in agreement with those of the full equations. REFERENCES [1] T. Juneau, “Micromachined Dual Input Rate Gyroscope,” Ph.D. dissertation, Univ. California, Berkeley, 1997. [2] A. Shkel, R. T. Howe, and R. Horowitz, “Modeling and simulation of micromachined gyroscopes in the presence of imperfections,” in Proc. International Conference on Modeling and Simulation of Microsystems, Puerto Rico, Apr. 1999. [3] M. Braxmaier, A. Gaiber, A. Schumacher, I. Simon, J. Frech, H. Sandmaier, and W. Lang, “Cross-coupling of the oscillation modes of vibratory gyroscopes,” in Proc. International Conference on Solid State Sensors, Actuators, and Microsystems, Boston, MA, June 2003.

Andrew J. Szeri received the Ph.D. degree in theoretical and applied mechanics from Cornell University, Ithaca, NY, in 1988. He is a Professor in the Department of Mechanical Engineering and in the Program in Applied Science and Technology at the University of California at Berkeley. In 1997, he joined the Berkeley Faculty after several years on the faculty of the University of California at Irvine. His research interests include nonlinear dynamics and fluid mechanics. He serves on the editorial boards of the Journal of Nonlinear Science and the Journal of the Acoustical Society of America.

Albert (“Al”) P. Pisano received the B.S., M.S., and Ph.D. (1981) degrees in mechanical engineering from Columbia University, New York, NY. He holds the FANUC Chair of Mechanical Systems in the Department of Mechanical Engineering at the University of California at Berkeley and is a recent inductee of the National Academy of Engineering. From 1997 to 1999, he served as Program Manager for the MEMS program at the Defense Advanced Research Projects Agency (DARPA), Arlington, VA. He is jointly appointed to the Department of Electrical Engineering and Computer Science and he serves as the Director of the Electronics Research Laboratory at the University of California at Berkeley. He is also one of the Directors of the Berkeley Sensor and Actuator Center (BSAC). Prior to joining the faculty at the University of California at Berkeley, he held research positions with Xerox Palo Alto Research Center, Singer Sewing Machines Corporate R&D Center and General Motors Research Labs. His research interests and activities at University of California at Berkeley include MEMS for micropower generation, MEMS for drug reconstitution and delivery, MEMS for RF components, MEMS sensors embedded in metal structures, MEMS inertial instruments, and MEMS actuators for disk drives. Dr. Pisano is a Member of the National Academy of Engineers and the American Society of Mechanical Engineers (ASME).
