Algorithmic MEMS

Karl FriedrichBohringer, University of California at Berkeley, Berkeley, CA 94720

Bruce Randall Donald, Dartmouth Col lege, Hanover, NH 03755

Abstract

As improvements in fabrication technology for MEMS

micro electromechanical systems increase the avail-

ability and diversity of these micromachines, engineers

are de ning a growing numb er of tasks to which they

can b e put. The idea of carrying out tasks using large

co ordinated systems of MEMS units motivates the de-

velopment of automated, algorithmic metho ds for de-

signing and controlling these groups of devices. We

rep ort here on progress towards algorithmic MEMS,

taking on the challenge of design, control, and pro-

gramming of massively-parallel arrays of microactua-

tors.

We rep ort on these developments in this fo cused sur-

vey pap er, based on the research results originally re-

Figure 1: Sensorless parts orienting using force vec-

p orted in our 1994 pap er [24] and develop ed further

tor elds: The part reaches unique orientation after two

in [19,20,26,21,17,18,24,12,14,15, 9]. We describ e

subsequent squeezes. There exist such orienting strate-

how arrays of MEMS devices can move and p osition

gies for al l polygonal parts. See www.cs.dartmouth.edu/

tiny parts, suchasintegrated circuit chips, in exible

brd/demo/MicroManipulation for an animated simula-

and predictable ways by oscillatory or ciliary action.

tion.

The theory of programmable force elds can mo del this

action, leading to algorithms for various typ es of mi-

for their design and control. This pap er surveys

cromanipulation that require no sensing of where the

our progress towards algorithmic MEMS, by taking

part is. Exp eriments supp ort the theory. We also re-

on the challenge of design and control of massively-

p ort on how the theory and results can b e generalized

parallel micro actuator arrays. Our goal is to im-

to the macroscopic scale [11,10].

plement task-level, sensorless manipulation strategies

with arrays of microfabricated actuators. The theory

1 Intro duction

of programmable force elds [19] arguably represents

the rst systematic attack on massively-parallel, dis-

The recent, rapid innovations and improvements in

microfabrication technology have resulted in an ever-

growing complexityofavailable MEMS devices, man-

The theory of programmable force elds for micro ma-

nipulation tasks was intro duced in [24 ]. dating the use of automated, algorithmic metho ds

In P. Agarwal, L. Kavraki, M. Mason, editors, Robotics: The Algorithmic Perspective, A. K. Peters, Ltd. 1998.

K. F. Bohringer and B. R. Donald

When a part is placed on the array, the programmed

vector eld induces a force and momentuponit. Over

time, the part may come to rest in a dynamic equilib-

rium state. By chaining together sequences of vector y

(a) elds, the equilibrium states of a part in the eld ma

b e cascaded to obtain a desired nal state|for exam-

ple, this state may represent a unique orientation or

p ose of the part see Figures 1 and 2a-b. The re-

sulting strategies work from any initial con guration

p ose of the part, require no sensing, and enjoy e-

t planning algorithms. A system with such a be-

(b) cien

havior exhibits the feeding property [2]:

A system has the feeding property over a set

of parts P and a set of initial con gurations

I if, given any part P 2 P , there is some

output con guration q such that the system

(c)

can move P to q from any lo cation in I .

Figure 2: Sensorless sorting using forcevector elds: parts

of di erent sizes are rst centered and subsequently sepa-

In Section 5 we present several force vector elds that

rated depending on their size.

p ossess the feeding prop erty. Some of these elds are

more p owerful part feeders, as they allow the choice of

tributed manipulation based on geometric and physical

a sp eci c output con guration q. Our work on pro-

reasoning. Applications such as parts-feeding can be

grammable force elds is related to nonprehensile ma-

formulated in terms of the force elds required. Hence,

nipulation [32,61,35]: in b oth cases, parts are manip-

programmable force elds act as an abstraction bar-

ulated without form or force closure.

rier between applications requiring array micromanip-

Even though our elds are typically not smo oth, it

ulation and their implementation with MEMS devices.

is p ossible to de ne a potential for certain elds as a

Such abstraction barriers p ermit hierarchical design,

unique path integral, and to show that when an ar-

and allow application designs with greater indep en-

ray R p otential exists, it always lifts to a smo oth

dence from underlying device technology.

2 1

con guration space R S p otential for the ma-

We are interested in the algorithmic content of

nipulated part [19]. Vector elds with p otential have

MEMS control strategies. This pap er surveys our work

b een shown to b e theoretically well-suited for manipu-

on showing how to quantify it, by analyzing the com-

lation strategies, by classifying a sub-family of p oten-

plexity of 1 computing a manipulation plan, 2 the

tial elds in whichevery part has stable equilibria [19].

generated plans i.e., force eld sequences and 3 the

Hence, such elds have b een prop osed for manipula-

individual elds. We survey our results on obtaining

tion tasks in which we desire to cascade equilibria in

upp er and lower complexity b ounds on plans and elds,

order to uniquely p ose or orient a part for details see

and in nding trade-o s b etween these di erent kinds

[24,19,20]. The theory of programmable force elds

of complexity. We review how to solve the problems

p ermits calculation of stable equilibria for macroscopic

of planning and control of micro actuator arrays for

parts in p otential elds see Figure 15. In our ex-

a wide range of tasks, and present a tutorial on algo-

While this question has b een well-studied for a p oint

rithms that automatically generate manipulation plans

mass in a eld, the issue is more subtle when lifted to a

for translating, orienting, centering, and sorting small

body with nite area, due to the moment covector. See

parts [17,19]. [19 ] for details.

In P. Agarwal, L. Kavraki, M. Mason, editors, Robotics: The Algorithmic Perspective, A. K. Peters, Ltd. 1998.

Algorithmic MEMS

p eriments, we employed vector elds with p otential for 2 Research Growth

parts-orientation and -p osing tasks, and the theory was

When we b egan working in MEMS in 1991, it was

used to predict the equilibrium p oses of sp eci c parts

not immediately clear what fundamental algorithmic or

Figure 14. The p oses predicted by the equilibrium

computational problems arose in this new area. Even

analysis were observed in our exp eriments Figures 13-

after obtaining our rst results on the theory of pro-

left.

grammable force vector elds in 1993 [22], the received

Perhaps surprisingly, the theory also predicts the ex-

view in the community was that the chief computa-

istence of pathological elds which do not induce well-

tional issues in MEMS arose in 1 design, and 2

b ehaved equilibria. In particular, there exist p erfectly

simulation. Indeed, it is these problems that motivated

plausible vector elds which induce no stable equilib-

our initial e orts see, e.g., [5,6, 8, 7]: in 1990, Ralph

rium in very simple parts. Although these elds are

Merkle of Xerox PARC and Kris Pister of Berkeley

very simple, they result in limit cycles and quite com-

urged us to apply our results on the design and simu-

plex b ehavior. We implemented such elds on an or-

lation [33] of snap fasteners to MEMS [51].

ganic micro cilia array see Section 3.2 [59, 26, 21].

At that time, work on force elds for manipulation

Vector elds without p otential were employed to cast

had b een limited to the arti cial p otential elds rst

parts into limit cycles, e.g. \in nite" rotation using a

develop ed by Khatib, Ko ditschek, and Bro oks. While

skew-symmetric squeeze eld. The predicted b ehav-

p otential elds have b een widely used in rob ot con-

ior Figure 16 for such \unstable" vector elds was

trol [42, 43, 56, 53], micro-actuator arrays present us

also observed Figure 13-right. This shows that rather

with the abilitytoexplicitly program the applied force

complex|but p otentially useful|b ehavior can b e gen-

at every point in a vector eld. Whereas previous work

erated using very simple elds.

had develop ed control strategies with arti cial p oten-

tial elds, our elds are non-arti cial i.e., physical.

We survey recent exp eriments in implementing this

Arti cial p otential elds require a tight feedback lo op,

theory using microfabricated actuator arrays [19, 26,

in which, at each clo ck tick, the rob ot senses its state

21]. In these exp eriments, strategies were programmed

and lo oks up a control i.e., a vector using a state-

in a ne-grained SIMD fashion by sp ecifying planar

indexed navigation function i.e., a vector eld. In

force vector elds. These programmable elds were

contrast, physical p otential elds employ no sensing,

implemented bymoving the individual actuators in a

and the motion of the manipulated ob ject evolves op en-

cyclic, gait-like fashion. Motion in non-principal e.g.

lo op for example, like a particle in a gravity eld.

diagonal directions was e ected by a pairwise coupling

This alone makes our application of p otential eld the-

of the actuators to implement virtual actuators and

ory to micro-devices a di erent, and algorithmically

virtual gaits, analogous to the virtual legs employed

challenging enterprise. Such elds can be comp osed

by Raib ert's hopping and running rob ots [52]. The

using addition, sequential comp osition, or \parallel"

tasks of parts-translation, -rotation, -orientation, and

comp osition by sup erp osition of controls.

-centering were demonstrated using small IC dice.

After our rst pap er app eared in January 1994 [24],

The theory of programmable force elds and virtual it b ecame clear that there exist deep and challeng-

gaits gives a metho d for controlling a very large num- ing algorithmic problems in MEMS and programmable

ber of distributed actuators in a principled, geomet- vector elds, at the intersection of combinatorial algo-

ric, task-level fashion. Whereas many control theories rithms, geometry, dynamical systems, and distributed

for multiple indep endent actuators break down as the

A notable exception are the three-dimensional force

number of actuators b ecomes very large, our systems

elds used by Jo e and his collab orators at JPL [39 ], where

will only b ecome more robust as the actuators b ecome

AC magnetic elds are used to orient and assemble ferro-

magnetic parts { in 3D! denser and more numerous.

In P. Agarwal, L. Kavraki, M. Mason, editors, Robotics: The Algorithmic Perspective, A. K. Peters, Ltd. 1998.

K. F. Bohringer and B. R. Donald

systems. A urry of pap ers has emerged on new algo-

rithms, new analysis, and new arrayed devices for pro-

grammable vector elds. During the next few years,

the authors continued working with No el MacDon-

ald at the Cornell Nanofabrication Facility to develop

and test new arrays of MEMS microactuators for pro-

grammable vector elds [18, 18, 19, 20, 23]. At the

same time, weworked with Greg Kovacs' group at the

National Nanofabrication Facility at Stanford, to de-

velop a control system for MEMS organic ciliary arrays,

Figure 3: Prototype M-Chip. A large unidirectional ac-

and to p erform manipulation exp eriments with these

tuator array scanning electron microscopy. Each actu-

2 2

arrays to manipulate IC dice using array-induced force

ator is 180 240 m in size. Detail from a 1in array

elds [21,26]. In parallel, weworked with Ken Gold-

with more than 11,000 actuators. For more pictures on

b erg at Berkeley and Vivek Bhatt at Cornell to gener-

device design and fabrication see www.cs.dartmouth.edu/

alize the theory to macroscopic devices, by developing

brd/demo/MicroActuators.

algorithms for transversely vibrating plates in order to

implement programmable vector elds [11, 10]. Finally,

scopic wheels that can be driven and steered to ma-

Danny Halp erin, working with the authors, develop ed

nipulate large ob jects suchasboxes [46]. Such a sys-

new upp er and lower b ounds, output-sensitive algo-

tem can implement programmable vector elds. To-

rithms, and a precise computational-geometric anal-

gether with Howie Choset, they analyzed the resulting

ysis of the area bisectors arising in squeeze- eld algo-

dynamical system to obtain interesting results on con-

rithms [14,15,16].

trollability and programmable vector eld algorithms

based on conservative vs. non-conservative elds [47].

Other groups have also b een active in developing

Daniela Rus has develop ed three-dimensional recon g-

new devices, analysis, and algorithms. Ken Goldb erg

urable tilings of actuators that could deliver 3D pro-

worked with John Canny's group at Berkeley, to con-

grammable vector elds for manipulation and lo como-

tinue research on using vibrating plates for manipula-

tion [44]. Working with the Berkeley Sensors and Actu-

tion, showing that longitudinal vibrations can generate

ators Group BSAC, Karl Bohringer and Ken Gold-

a richvo cabulary of programmable vector elds [55]. In

b erg explored how MEMS devices employing electro-

addition, they develop ed sophisticated dynamic mo d-

static fringing elds can be used to implement pro-

els and dynamic simulators for b oth MEMS devices

grammable force elds for parts manipulation and self-

and macroscopic vibrating plates [54]. Lydia Kavraki

assembly [13, 25].

explored the power of continuous vector elds, and

demonstrated an elliptical p otential eld capable of

In short, there has b een an explosion of new and

p osing any part into one of two equilibrium states [40].

exotic arrayed devices for b oth MEMS manipulation

Peter Will and his colleagues at ISI have explored

and macroscopic manipulation. The theory of pro-

a number of di erent MEMS array designs, as well

grammable vector elds has b een applied and extended

as algorithms and analysis for programmable vector

to a variety of devices and systems. It is somewhat

elds [45, 28, 30, 29]. Andy Berlin, David Biegelsen,

remarkable that the same analysis to ols, elds, and al-

P. Cheung, and Warren Jackson at XeroxPARChave

gorithms apply to such a wide range of systems.

develop ed a novel MEMS microactuator array based

on controllable air jets, with integrated control and

3 Exp erimental Devices and Setup

sensing circuitry [4, 27]. Working at CMU, Bill Mess-

ner and Jonathan Luntz develop ed a small ro om whose Several groups have describ ed e orts to apply

o or is tiled with controllable, programmable, macro- MEMS micro electro mechanical system actuators to

In P. Agarwal, L. Kavraki, M. Mason, editors, Robotics: The Algorithmic Perspective, A. K. Peters, Ltd. 1998.

Algorithmic MEMS

Figure 4: Released asymmetric actuator for the M-Chip

scanning electron microscopy. Left: Dense grid 10 m

spacing with aluminum electrode underneath. Right: Grid

Figure 6: M-Chip prototype motion pixel consisting of

with 5 m high poles.

actuators oriented in four di erent directions.

3.1 Single-crystal silicon electrostatic actua-

tors

We now survey recent work on single-crystal silicon

electrostatic MEMS actuators, conducted in collab ora-

tion with No el MacDonald at the Cornell Nanofabrica-

tion facility [18,18,19,20,24,23].

The M-Chip M anipulation Chip, see Figure 3

Figure 5: Released M-Chip actuator consisting of single-

ingle-Crystal Silicon is fabricated using a Scream S

crystal silicon with 5 m high tips, suspended 5 m above

R eactiveEtching and Metallization pro cess develop ed

the silicon substrate.

in the Cornell Nanofabrication Facility [60, 58]. The

Scream pro cess is low-temp erature, and do es not in-

p ositioning, insp ection, and assembly tasks with small

terfere with traditional VLSI [57]. Hence it op ens

parts [50, 3, 37, 24, 45, for example]. However, the

the do or to building monolithic micro electromechani-

fabrication, control, and programming of micro-devices

cal systems with integrated microactuators and control

that can interact and actively change their environment

circuitry on the same wafer.

remains challenging. Problems arise from

The design is based on microfabricated torsional res-

onators [49,48]. Each unit device consists of a rectan-

1. the limited range of motion and force that can b e

gular grid etched out of single-crystal silicon susp ended

generated with microactuators,

by two ro ds that act as torsional springs Figure 4.

2. the lack of sucient sensor information with re-

The grid is ab out 200 m long and extends 120 m on

gard to manipulation tasks,

each side of the ro d. The ro ds are 150 m long. The

3. design limitations and geometric tolerances due to

current asymmetric design has 5 m high protruding

the fabrication pro cess, and

tips on one side of the grid that make contact with an

4. uncertain material prop erties and the lack of ade-

ob ject lying on top of the actuator Figure 5. The

quate mo dels for mechanisms at very small scales.

other side of the actuator consists of a denser grid

ab ovean aluminum electro de. If a voltage is applied

In the following subsections we describ e two di erent between silicon substrate and electro de, the dense grid

typ es of actuator arrays: single-crystal silicon SCS ab ove the electro de is pulled downward by the resulting

electrostatic actuators, and combined thermobimorph electrostatic force. Simultaneously the other side of the

and electrostatic organic micro cilia. device with the tips is de ected out of the plane by

In P. Agarwal, L. Kavraki, M. Mason, editors, Robotics: The Algorithmic Perspective, A. K. Peters, Ltd. 1998.

K. F. Bohringer and B. R. Donald

Electrostatic Plate Tip Wet Etch AccessVias High CTE Polyimide

TiW Resistor Low CTE Polyimide

Encapsulation/ Stiffening Layer

Base

(Curled out of plane when not heated)

Figure 8: Organic thermal and electrostatic microactua-

Figure 7: Portion of a polyimide cilia array SEM micro-

tor. Half of the upper polyimide and silicon nitride encapsu-

graph. Four orthogonal ly-oriented actuators are integrated

lation/sti ening layer are shown removed along the cilium's

into a motion pixel, which covers a surfacearea of approx-

axis of symmetry to show details.

imately 1:1 1:1 mm .

several m . Hence an ob ject can b e lifted and pushed

sideways by the actuator.

Because of its low inertia resonance in the high kHz

range the M-Chip can b e driven in a wide frequency

range from DC to several 100 kHz AC. The actuators

need not b e op erated at resonance: They can also b e

servoed to p erio dically \hit" an ob ject on top, hence

applying b oth lateral and vertical forces. Calculations,

simulations and exp eriments have shown that the force

generated with a torsional actuator is approximately

Figure 9: Polyimide cilia motion pixel SEM micrograph.

10 N , which corresp onds to a force-p er-area ratio of

Four actuators in a common center con guration make up

100 N =mm , large enough to levitate e.g. a piece of

a motion pixel. Each cilium is 430 m long and bends up

2 2

pap er 1 N =mm or a silicon wafer 10 N =mm .

to 120 m out of the plane.

Each actuator can generate motion in one sp eci c

direction if it is activated; otherwise it acts as a pas-

We now survey recent work on using MEMS or-

sive frictional contact. Figure 3 shows a small sec-

ganic cilliary arrays [59] for massively parallel microa-

tion of a unidirectional actuator array, which consists

nipulation, conducted in collab oration with John Suh

of more than 11,000 individual actuators. The com-

and Greg Kovacs at the Stanford University National

bination and selective activation of several actuators

Nanofabrication facility [21, 26].

with di erent motion bias allows us to generate vari-

At Stanford, surface micromachining techniques

ous motions in discrete directions, spanning the plane

were used to create organic micro cilia arrays, con-

Figure 6. In initial manipulation exp eriments, small

sisting of p olyimide as the primary structural material

pieces of glass size: several mm , mass: ab out 1 mg

and aluminum as a sacri cial layer. The fabrication

were lifted within the motion range of the actuators

pro cess develop ed at the Center for Integrated Sys-

several m and pushed sideways by several 100 m .

tems, Stanford University was designed to b e compat-

The fabrication pro cess and mechanism analysis are

ible with CMOS or BiCMOS circuits which could be

describ ed in more detail in [24, 23,17].

pre-fabricated on a silicon substrate [59].

3.2 Combined thermobimorph and electro-

Actuator cilia. Each cilium consists of twolayers of

static p olyimide cilia arrays

p olyimide with di erent thermal expansion co ecients.

In P. Agarwal, L. Kavraki, M. Mason, editors, Robotics: The Algorithmic Perspective, A. K. Peters, Ltd. 1998.

Algorithmic MEMS

The cilium also contains a Titanium-Tungsten Ti:W motion

motion

resistive heater lo op for thermal actuation, Aluminum

electro des for electrostatic low-p ower hold-down, and

a silicon nitride encapsulation/sti ening layer Fig-

ure 8. For a detailed description of the fabrica-

tion pro cess see [59]. Vertical and horizontal displace-

ments of the cilia tips are a function of the thermal

mismatch in the actuator layers. For ro om temp era-

ture, these values can be calculated as 120 m

and 20 m [59]. Insp ection under the scanning

h 1 2

news neWs

electron microscop e SEM has veri ed these calcula-

up (off) down (on)

tions.

The lifting capacity of an actuator can b e estimated

the force required to de ect the actuator's tip to

as North

the substrate. The actuator load capacity has b een

East

calculated as F =76m , which gives a force-p er-area

l 2

2 West

ratio of 4 76 N =1:1 mm 250 N =mm .

South

phase 1 phase 2

Chip layout. The cilia array is comp osed of cells

Figure 10: Two-phase gait. The W actuator is repeatedly

motion pixels, each1:1 1:1 mm which contain four

switched on and o , while the other actuators always remain

orthogonally-oriented actuators Figure 9. On the

on, resulting in a news neWs sequence.

current cilia chip, the motion pixels are arranged in

an 8 8 array which o ccupies approximately 0:77 cm

using the MEMSA MEMS A rray language whichwe

of a 1 cm die. The four actuators of each pixel are

develop ed at Stanford. The control software includ-

indep endently activated by four thermal and four elec-

ing the MEMSA interpreter was written in PASCAL.

trostatic control lines. Four 8 8chips are diced and

Thermal and electrostatic control line signals are sent

packaged together to make a quad-shap ed 16 16 cilia

via the PC parallel p ort to D-typ e ip- ops which ac-

array device, with a total of 1024 cilia. The device it-

tivate p ower transistors. Currently up to 4 cilia arrays

self is attached to a hybrid package which is placed on

can b e controlled simultaneously by using a multiplexer

a heat sink and thermo-electric co oler Peltier e ect

with 2 address bits.

mo dule. The total input power to the chip can ex-

ceed 4 W , and without active co oling the package can

4 Low-level Control: Actuator Gaits

b ecome very hot. To observe the exp eriments, a long

working distance microscop e is connected to a CCD

We now survey some basic concepts of actuator gaits

camera, and a video cassette recorder is used to mon-

using MEMS arrays for massively parallel microanipu-

itor and record b oth the movements of an individual

lation, develop ed in collab oration with Jonathan Suh

cilium and the ob jects conveyed by the array.

and Greg Kovacs at the Stanford University National

Nanofabrication facility [21, 26].

Controller. The manipulation results describ ed b e-

To induce motion on a part that is placed on the ar-

low were accomplished with the cilia array device in-

ray, the cilia are actuated in a cyclic, gait-like fashion.

terfaced to an IBM 486 p ersonal computer. The PC

In each cycle, the part is moved in a certain direction

provides sp eed control via the drive pulse frequency

and directional control interactively via keyb oard or

MEMSA named after MENSA is the language for

mouse, or by actuator programs that can b e sp eci ed smart manipulation surfaces.

In P. Agarwal, L. Kavraki, M. Mason, editors, Robotics: The Algorithmic Perspective, A. K. Peters, Ltd. 1998.

K. F. Bohringer and B. R. Donald

motion motion

1 2 12 news neWs news neWS

up (off) down (on) up (off) down (on)

motion motion

34 34

nEWs nEws NEWS NEws

Figure 11: Four-phase gait consisting of the four-pattern Figure 12: Diagonal virtual gait consisting of the four-

sequence news neWs nEWs nEws. pattern sequence news neWS NEWS NEws. The N and E cilia,

and the W and S cilia, are coupled to form virtual cilia in

North-East and South-West directions.

by the motion of the actuators that are in contact with

it. The sp eed of the moving part dep ends on the hor-

motion in the East direction would b e news neWs see

izontal displacement of the actuators p er cycle as well

Figure 10.

as the frequency of cycle rep etition. It also dep ends on

the surface prop erties and weight of the moving part.

The four-phase gait consists of four di erent actu-

ation phases news neWs nEWs nEws such that motion

is induced during upward as well as downward strokes Task: translation of parts in principal direc-

of the cilia Figure 11, see also [3]. Note that the tions. The simplest gait is the two-phase gait, in

forces exerted on the moving part dep end on the state which all actuators of the same orientation rep eatedly

of the motion pixel: e.g. in the transition from phase 1 stroke the part while the remaining actuators are held

to phase 2 the cilium W moves up while the opp osing down. Assuming that the orthogonal cilia within a mo-

cilium e remains down. We denote the lateral force tion pixel are oriented at the principal compass p oints,

exerted on the part in this con guration f . Analo- let us use capital letters NEWS to denote the North,

W;e

gously, during transitions 2{3, 3{4, and 4{1 we observe East, West, and South actuators in the up p osition,

lateral forces f , f , and f , resp ectively. f and and lower-case letters news to denote the actuators in

E;W w;E e;w W;e

f are in p ositive x-direction, while f and f are the lowered p osition. Then the two-phase gait to e ect

e;w E;W w;E

In P. Agarwal, L. Kavraki, M. Mason, editors, Robotics: The Algorithmic Perspective, A. K. Peters, Ltd. 1998.

Algorithmic MEMS

negative. Furthermore, from this analysis it follows mo dels the device-part interaction well. Therefore,

that jf jjf j jf jjf j0. Hence we ex- when describing and predicting the motion of parts in

W;e e;w E;W w;E

p ect a relatively large motion step x during transi- force vector elds, wehave based our theory on a rst-

W;e

tion 1{2, and a smaller step x during transition 4{ order system see Section 5.

e;w

1, while during the other transitions the part remains

Silicon chips were moved with a motion resolution

at its current lo cation. This b ehavior has b een ob-

of a few m and sp eeds up to 200 m =sec . Four-phase

served in our exp eriments, where x was measured

W;e

gaits proved more e ective than two-phase gaits, be-

between 3 m and 10 m dep ending on input power,

cause during the downward motion in the two-phase

frequency, surface prop erties and weight of the part.

gait, the part tends to slip backwards. The four-phase

x was usually ab out twice as large as x .

W;e e;w

gait avoids this e ect, b ecause other cilia hold the part

in place during the transition 3{4. In the subsequent

Task: translation of parts in arbitrary direc-

downward motion in the transition 4{1, the part is also

tions. Motion in non-principal e.g. diagonal direc-

moved forward Figure 11.

tions is e ected by a pairwise coupling of two cilia

The diagonal gait also has the lowest power con-

of each pixel, implementing virtual cilia analogous to

sumption not considering electrostatic hold-down,

Raib ert's concept of virtual legs for hopping and run-

due to the fact that its duty cycle for cilia hold-down

ning rob ots [52]. Hence, several cilia can be co ordi-

is lowest 50 , compared to 75 for the principal

nated to emulate a virtual cilium, which generates a

four-phase gait, and 87.5 for the two-phase gait.

force corresp onding to the vector sum of its comp o-

As exp ected, diagonal virtual gaits induced the

nents. The diagonal gait to e ect motion in the North-

largest and fastest motion b ecause all four cilia of each

East direction would b e news neWS NEWS NEws where

pixel were activated, whereas in principal gaits only

the virtual cilia are NE and WS. Consequently, we ob-

two cilia are actively used, while the others havetobe

tain a virtual gait that moves the part in a diagonal

held down continuously Figure 11.

direction. Note that in a section view through the ar-

ray lo oking in the North-West direction, this gait lo oks

virtually identical to the four-phase gait depicted in

5 High-level Control: Vector Fields

Figure 11.

We b elieve that vector elds can be used as an ab-

Motion in arbitrary directions can be induced by

straction barrier between applications requiring array

alternating gaits that interleave principal or virtual

micro-manipulation and MEMS devices implementing

gaits of di erent directions. For example, a transla-

the requisite mechanical forces. That is, applications

tion at 25 from the x-axis requires motion in the y -

such as parts-feeding can b e formulated in terms of the

direction and x-direction at a ratio of tan 25 1:2.

vector elds required. This then serves as a sp eci -

Our control software determines the exact alternation

cation which the underlying MEMS device technology

analogously to the Bresenham line scan algorithm [36],

While exp eriments with our MEMS arrays indicate

which rasterizes lines at arbitrary angles, resulting in

they may be mo delled as rst order heavily-damp ed or

di erent elds that are interlaced in time.

essentially quasi-static systems, it is easy to imagine fu-

ture MEMS arrays that are highly dynamic and which

Exp eriments and results. In collab oration with

would require a detailed analysis of dynamic equilibria. In-

Jonathan Suh and Greg Kovacs at the Stanford Uni- deed, our vibrating plates exhibit more dynamics, although

empirically the quasi-static analysis has b een adequate so

versity National Nanofabrication facility [21, 26], a

far [11 , 10 ]. Some exp erimental evidence for quasi-staticity

large numb er of translation exp eriments have b een p er-

is discussed in [21 , 26], and the e ect of dynamics on the

formed in which two-phase and four-phase gaits were

quasi-static mo del is discussed in [20 ]. Research on dynamic

used to implement principal and virtual gaits. These

analysis of programmable vector elds is an active research

exp eriments show that a rst-order dynamical system area; see, e.g., [54 , 55, 28 , 30 , 29 , 47].

In P. Agarwal, L. Kavraki, M. Mason, editors, Robotics: The Algorithmic Perspective, A. K. Peters, Ltd. 1998.

K. F. Bohringer and B. R. Donald

Figure 14: Simulation of the alignment task with a squeeze

eld as shown in Figure 13-left.

pressed in vector eld terms, as \small" vector elds.

Restrictions in directional selectivity and magnitude

control can also be manifested as restrictions on the

vector eld abstraction resulting in discretization in

orientation or mo dulus. This means that MEMS de-

signers can p otentially ignore certain details of the ap- hing the re-

l plication pro cess, and instead fo cus on matc

quired vector eld sp eci cation. Then, once the capa-

bilities of MEMS actuator arrays were published as vec-

tor elds and tolerances, an application designer could

lo ok in a catalog to cho ose a device technology based on

the eld sp eci cation it promises to implement. This

would free application engineers from needing to know

much ab out pro cess engineering, in the same way that

ware and algorithm designers often abstract away

l soft

from details of the hardware. Such an abstraction bar-

rier could p ermit hierarchical design, and allow appli-

cation designs with greater indep endence from the un-

derlying device technology. At the same time, abstrac-

tion barriers could allow MEMS array technologies to

b e designed simultaneously with the abstract vector

eld control. This development pattern could b e sim-

concurrent design of VLSI pro cessors with

l ilar to the

their compilers, as is common in computer architecture.

Figure 13: Manipulation of silicon chips in programmable

vector elds induced by a micro cilia array microscope

We now survey some basic concepts and theory in

video images. Left: The chip is aligned with the verti-

high-level algorithmic control, and describ e exp eri-

cal squeeze line l marked by a dark line for clari cation.

ments to test the validity of the theory. The exp erimen-

Right: Rotating a square-shaped chip counterclockwise in a

tal results surveyed b elow in Sec. 5 were conducted in

skewed squeeze eld.

collab oration with Jonathan Suh and Greg Kovacs at

the Stanford University National Nanofabrication facil-

must deliver. Conversely, the capabilities of MEMS

ity [21,26]. In this survey,we emphasize the geometric

array technologies for actuation can b e formulated in

and algorithmic asp ects of the problem, as well connec-

terms of the vector elds they can implement. For ex-

tions and synergy with exp erimental work on design,

ample, limitations in force magnitude are naturally ex-

fabrication, and programming of MEMS arrays.

In P. Agarwal, L. Kavraki, M. Mason, editors, Robotics: The Algorithmic Perspective, A. K. Peters, Ltd. 1998.

Algorithmic MEMS

tion 5.1 has potential, of the form U z = f ds,

squeeze line

where is an arbitrary path to z from a xed refer-

encepoint. If d denotes the perpendicular distanceof

z from the squeeze line, then U d =jd j.

z z

Claim 4 [19] Let P bea connectedpolygonal part with

nite contact area and n vertices. Then in any squeeze

eld, P has E = O kn orientation equilibria, where

k is the maximum number of edges that a bisector of P

can cross. If P is convex, then the number of equilibria

is O n.

Figure 15: Equilibrium condition: To balance force and

Equilibria can be calculated numerically using the

moment acting on P in a unit squeeze eld, the two areas

metho d in Figure 15: Given an arbitrary part at a

P and P must beequal i.e., l must bisect the part, and

1 2

xed orientation, we translate it until its left and right

the line connecting the centers of area c and c must be

1 2

sections have equal size. If the resp ective centers of

perpendicular to the squeeze line l .

gravity lie on a line p erp endicular to the squeeze line,

then the part is in equilibrium. For p olygonal parts

5.1 Squeeze elds

there exist analytical metho ds to compute the equi-

libria exactly see [19] for a detailed algorithm and a

In [24], the authors prop osed a family of control strate-

derivation of the O kn b ound.

gies called squeeze elds and a planning algorithm for

Claim 5 [19] Let P be a polygon whose interior is

parts-orientation see Figures 1 and 15.

connected. There exists a sensorless alignment strategy

consisting of a sequence of squeeze elds that uniquely

De nition 1 [19] Given a straight line l , a squeeze

orients P up to symmetries.

eld f isatwo-dimensional force vector eld in which,

at each p oint, a unit force p oints p erp endicularly to-

Pro of: Claim 4 states that a squeeze eld brings a

wards l on l the force is zero.

p olygon P into one of E = O kn orientation equi-

libria. We de ne the squeeze function as this map-

We refer to the line l as the squeeze line , b ecause l

ping from original orientation to equilibrium orienta-

lies in the center of the squeeze eld.

tion. Hence, from Claim 4 it follows that the image of

the squeeze function is a set of E discrete values. Given

Assuming quasi-static motion, an ob ject will trans-

such a squeeze function, Goldb erg has presented an al-

late and rotate to an equilibrium con guration, as char-

gorithm for sensorless manipulation of p olygons [38]

acterized in Figure 15. To predict the equilibria, we

that constructs an orienting strategy with O E steps

assume a uniform force distribution over the surface of

in O E time. The output of this algorithm is a se-

P , which is a reasonable assumption for a at part that

quence of squeeze directions. When the corresp onding

is in contact with a large numb er of elastic actuators.

squeeze elds are applied to the part P , the set of p os-

sible orientations is successively reduced until a unique

De nition 2 A part P is in translational equilibrium

orientation up to symmetry is reached. For details

if the forces acting on P are balanced. P is in orienta-

see [19] and [38]. 2

tional equilibrium if the moments acting on P are bal-

anced. Total equilibrium is simultaneous translational

Task: orienting and aligning parts

and orientational equilibrium.

If a part is placed in a squeeze eld, it will trans-

Claim 3 [19] Every squeeze eld f see De ni- late and rotate until a stable equilibrium is reached

In P. Agarwal, L. Kavraki, M. Mason, editors, Robotics: The Algorithmic Perspective, A. K. Peters, Ltd. 1998.

K. F. Bohringer and B. R. Donald

Force equilibrium is only p ossible if the center of the

disk coincides with the squeeze line. In this p osition

the disk exp eriences a non-zero momentif 6=0.

Claim 9 A diagonal ly skewed eld = 1 induces

no stable equilibrium on a square-shapedpart.

For a pro of see [19].

Task: rotating parts

Figure 16: Unstable square-shaped part in a skewed

According to Claims 8 and 9, certain parts will rotate

squeeze eld = 1. The square with center on the

inde nitely in skewed squeeze elds Figure 16. Note

squeeze line wil l rotate inde nitely. Moreover, it has no

that even though our cilia device has more degrees of

stable equilibrium in this eld.

freedom, two areas of constant force are sucient to

implement a skewed eld, resulting in a very simple

Claim 4. Parts may exhibit several equilibria, hence

task-level rotation strategy. In particular, the rota-

after one squeeze the part orientation maybeambigu-

tion algorithm resulting from the application of skew-

ous. This ambiguity can b e removed with the strate-

symmetric squeeze elds is considerably simpler than

gies of Claim 5: by executing a sequence of squeezes

rotation algorithms prop osed in the MEMS literature

at particular angles, the part is uniquely oriented see

for example, the vortices suggested byFujita [37]orby

Figure 1.

Liu and Will [45]. Vortices require at least four areas

of the array to b e pushing in di erent directions. That

Exp eriments and results. The long, thin part de-

is, vortices can be implemented using four triangular

picted in Figures 13-left and 14 exhibits a unique sta-

or rectangular regions, up on each of which the vector

ble equilibrium mo dulo 180 eld symmetry. When

eld is constant. Skewed elds p erform the same task

placed in a squeeze eld, its longitudinal axis aligns

with only two regions of constant force.

with the squeeze line. This dynamic pro cess is pre-

dicted by simulation in Figure 14, and veri ed in ex-

Exp eriments and results. Figure 13-right shows

p eriment see Figure 13-left. This part alignment ex-

video frames of a 3 3 mm IC chip rotating on the

p eriment has also b een p erformed with similar results

squeeze line of a skewed eld. During the exp eriments

for several other small pieces of glass and silicon of a

of approximately 10 min , several full rotations of the

few mm length and several mg of mass.

part were p erformed.

5.2 Skewed squeeze elds

5.3 Radial elds

De nition 6 [19] A skewed eld f is a vector eld

De nition 10 A radial eld f isatwo-dimensional

given by f x; y =signx1;, where 0 6= 2 R .

force vector eld such that f z=z=jzj if z 6= 0, and

f 0=0.

Claim 7 [19] No skewed squeeze eld has a potential.

Claim 11 [19] Aradial eld has a potential, U z=

jzj.

In askewed squeeze eld, it is easy to nd a circular

path along which the work integral is non-zero e.g.,

Claim 12 [19] Given a polygonal part P in a radial

along a circle with center on the squeeze line.

eld f , there exists a unique pivot p oint v of P such

that P is in equilibrium if and only if v coincides with

Claim 8 [19] A skewed eld induces no stable equilib-

the center of the radial eld.

rium on a disk-shapedpart for al l 6=0.

In P. Agarwal, L. Kavraki, M. Mason, editors, Robotics: The Algorithmic Perspective, A. K. Peters, Ltd. 1998.

Algorithmic MEMS

Claim 13 [19] Let P be apolygonal part with n ver- There is a direct relationship between equilibria in

tices, and let k be the maximum number of edges that a squeeze elds and area bisectors. Recall Figure 15: a

bisector of P can cross. Thereare at most E = O kn part is in force equilibrium if and only if the squeeze

stable equilibria in a eld of the form R + S if S is a line bisects the part into two sections of equal area.

squeeze eld, and is suciently smal l and positive.

De nition 14 [14] Let P b e a p olygon in the plane,

Pro ofs of the previous claims, and a numerical algo-

p ossibly with holes, and having n vertices in total. We

rithm to compute the pivot p ointisgiven in [19]. Note

denote by V the set of vertices of P . For a directed line

that Claim 13 results in strategies for unique part p os-

in the plane, we denote by h resp. h the

l r

2 2 2

ing in O E =O k n steps.

op en half-plane b ounded by on the left- resp. right-

hand-side of . The line is an area bisector of P if

Task: centering parts

the area of P \ h is equal to the area of P \ h .

l r

Radial elds can be used to center a part. With

the current four-quadrant cilia device, wehave imple-

A line partitions V into three sets twoof which

mented an approximation of an ideal radial eld similar

may b e empty: V \ h , V \ , and V \ h . Wesay

l r

to the eld in Figure 2-b. Note that this approximate

that two area bisectors of P are combinatorial ly distinct

radial eld has a p otential.

if the partitioning of V as ab ove induced by the two

bisectors is di erent. We say that two area bisectors

Exp eriments and results. Small silicon and glass

of P are combinatorial ly equivalent if they induce the

parts were centered using our cilia device. In this ex-

same partitioning of V . We assume that the p olygon

p eriment, high p ositioning accuracy in the m range

P is connected, and non-degenerate in the sense that

was hard to achieve, b ecause the center of the radial

the complementofP has the same b oundary as P .

eld coincides with the the lo cation of the dice edges.

An obvious upp er b ound on the numb er of distinct

Manual packaging of the four cilia chips leaves small

area bisectors of a p olygon with n vertices is O n |

gaps and non-planarities at these junctions. The next

each combinatorial equivalence class of area bisectors

generation cilia device will overcome this problem, b e-

is determined by a pair of vertices of the p olygon. In

cause it will allow us to implement the radial eld with

Section 6.2 we describ e how a p olygon with n vertices

a single chip. Furthermore, b ecause of its full pixel-

can have n distinct area bisectors [14]. Note that

wise programmability, the new chip will allow us to

the p olygon in this construction is simple.

closely approximate ideal radial elds.

We review an output-sensitive algorithm [14] for

computing an explicit representation of all the area bi-

6 Polygon Bisectors & Force Equilibria

sectors of a given p olygon, by constructing the bisector

curve de ned in a plane dual to the plane containing

In the previous section weintro duced vector elds for

the p olygon: the curve is the union of p oints dual

high-level control of micro actuator arrays. In particu-

to area bisectors in the primal plane. The algorithm

lar, it was stated that in a squeeze eld, p olygonal parts

pro ceeds by constructing the zone of the curve in

have a usually very small nite numb er of orientation

an arrangement of lines [34]inthe dual plane, where

equilibria see Claim 4. Because of this result, squeeze

each line of the arrangement is the dual of a vertex

elds playakey role in manipulation strategies. In this

of the p olygon. A sketchofthe algorithm is given in

section, we survey recent research on the combinatorial,

Section 6.3.

geometric, and algorithmic prop erties of squeeze elds,

by analyzing the area bisectors of a part. The results Area bisectors were considered by D az and

in Sec. 6 were obtained in collab oration with Danny O'Rourke [31]. However, their fo cus is on the contin-

Halp erin [14,15,16]. uous version of the ham-sandwich cut problem, and of

In P. Agarwal, L. Kavraki, M. Mason, editors, Robotics: The Algorithmic Perspective, A. K. Peters, Ltd. 1998.

K. F. Bohringer and B. R. Donald

a problem they intro duce of orthogonal four-sections;

see [31] for more details. The problem discussed here

can b e viewed as a continuous version of the well-known

k -set problem [34].

In the remainder of this section, we analyze the num-

b er of area bisectors, and hence b ound the number of

force equilibria. The algorithm for computing area bi-

sectors can b e used as a preliminary step in designing 4

alignment plans see Section 6.4.

3 0

6.1 Prop erties of area bisectors

In this section we review several prop erties of area bi-

Figure 17: A simple polygon with n vertices that has n

sectors of p olygons. Some of the pro ofs have b een omit-

combinatorial ly distinct bisectors.

ted here; they can b e found in [16].

Lemma 15 [14] Let P be a non-degenerate polygon

very close to c on a small circle whose center is c as

with n vertices. 1 There exist O n combinatorial ly

well, along two convex p olygonal chains.

distinct ways in which a line can partition P . 2 Let

We xaninteger m that we will determine later; for

A and B be the intersections of an area bisector with

the p olygon in the gure m = 3. The distance b etween

the boundary of the convex hul l of P . As the slopeof

the vertices v and v is the same for i =1;:::;m,

i i+1

varies from 1 to +1, A and B progress monoton-

0 0

and v and it is the same as the distance b etween v

i+1 i

ical ly counterclockwise on the boundary of the convex

for i =1;:::;m. The area of all triangles v u v for

i i i+1

hul l of P . 3 For every slope x there exists a unique

i = 1;:::;m is the same and is equal to the area of

bisector of P with slope x.

0 0 0

all triangles v u v for i = 1;:::;m. There are 2m

i i i+1

Lemma 16 [14] Let P be a polygon with n vertices.

vertices w near c and they are equally spaced on a

Let s be a point in R and let be a line that inter-

small circle centered at c. As can be easily veri ed,

sects r edges of P . The area bisectors of P that are

, there is a bisector for every pair of vertices v and v

combinatorial ly equivalent to and pass through s are

passing through these p oints that passes also through

determined by the roots of a polynomial equation of de-

the center p oint c. We next claim that as we rotate

gree r .

the bisector from v to v it will move o the center c

i i+1

and sweep m vertices w . The reason is that the angle

It can b e shown [16] that the bisectors of a p olygon P

0 0 0

6 6

u v v is larger than the angle u v v . Hence,

can b e describ ed by a piece-wise algebraic curve, where

i i i+1

as the bisector rotates, it will pro ceed `faster' on the

each piece is describ ed by a p olynomial whose degree

b ottom part of our p olygon than on the top part and

dep ends on the number of edges of P intersected by

therefore will sweep half of the vertices w on its way.

the corresp onding bisectors.

Finally m is chosen such that roughly n = 6m +

6.2 Lower b ound

8. The number of distinct area bisectors is evidently

2 2

m = n .

As argued ab ove, a p olygon with n vertices can haveat

most O n combinatorially distinct area bisectors. In

6.3 Output-sensitive algorithm

this section we present an example of a simple p olygon

with n vertices where the b ound n is attained [14].

It is convenient to study the algorithmic problem in a

0 0

Consider Figure 17. All the vertices v ;v ;u and u dual plane: a line y =2xx y in the primal plane is

i i

lie on a circle whose center is at c. The vertices w lie transformed into the p ointx; y in the dual plane. A j

In P. Agarwal, L. Kavraki, M. Mason, editors, Robotics: The Algorithmic Perspective, A. K. Peters, Ltd. 1998.

Algorithmic MEMS

p ointx; y in the primal plane is transformed into the

exit p oint

line y =2xx y in the dual. The dual of an ob ject o

will be denoted by o . If O is a set of ob jects in the

plane, O will denote the set of dual ob jects.

p =B

Let P b e a p olygon with n vertices as de ned in the

Intro duction, namely connected, non-degenerate and

a b

p ossibly with holes. In the dual plane every vertex v of

P is transformed into a line v which is the collection

of all p oints dual to lines in the primal plane that pass

Figure 18: Ray shooting to determine the face f contain-

through v .

ing p a, and then nding the maximal pieces of inside

For any given direction there is a unique area bi-

f and its exit points from f b.

sector. We denote the oriented bisector of P that

makes an angle with the p ositive x-axis by B ,

which contains the p oint p := B .

and b ecause of symmetry con ne ourselves to the

range [=2;=2 for . We denote the collection of

Our next step is to construct the face f that contains

p oints dual to area bisectors of P in that range by .

the p oint p in AV . We describ e this pro cedure as-

Note that any b esides =2 corresp onds to an x-

suming f is b ounded; the extension to unb ounded faces

co ordinate in the dual plane.

is straightforward. We prepare in advance a data struc-

ture R V that supp orts ecientray sho oting among

The curve is a piece-wise algebraic andx -monotone

the lines V . We sho ot a ray from p in the upward ver-

curve this is proved in [16]. We call the bisector

tical direction and identify the line ` 2 V supp orting

curve of P , as it gives a complete sp eci cation of all

the edge of f ab ove p. See Figure 18a. We next pro-

the area bisectors of the p olygon P . We denote by

ceed in clo ckwise direction along the b oundary of f .

the number of maximal connected algebraic pieces in

From the p oint p 2 ` we sho ot a rayin AV along

1 1

, where the function describing each piece is de ned

` , and identify the line ` supp orting the next edge on

1 2

by the xed set of edges that the corresp onding set of

the b oundary of f , and so on until we have returned

bisectors cross. In this section we describ e an output-

to ` and thus have identi ed the entire face f .

sensitive algorithm from [14] to compute . Since we

aim for output-sensitivity,we cannot a ord to compute

Nowwe determine the maximal connected pieces of

the entire arrangement AV whose complexitymay

f \ . For each edge on b oundary of f we compute

be n . We will discover the maximal pieces of

the intersection of its supp orting line v with . This

in their order along , using two primitive op erations:

computation is equivalent to nding the bisectors that

, and intersection of ray shooting among the lines V

pass through the vertex v , and intersecting the edges

an algebraic curve with a straight line.

in E . By Lemma 16, this reduces to solving a p oly-

nomial equation of degree r , where r is the number of

We cho ose an arbitrary direction 2 [=2;=2

edges in E . We denote the time required to nd

and lo ok for the area bisector of P in that direction.

these ro ots by r .

This requires O n log n time since the p olygon may

We order the resulting intersections along thex -axis. have holes. Next, we obtain the set of edges crossed by

Since the curve isx -monotone, this ordered list of in- B . We denote by E the set of edges crossed by

tersections provides a description of the curve inside B . The set E determines a function :=

0 0

f , and indicates what are the neighb oring faces that describing the bisector curve in a neighb orho o d of ,

crosses. We mark each of these additional faces by the as long as the set of edges crossed by the bisector do es

p oint where crosses out of f . We call each such p oint not change. In the dual plane the function describ es

an exit point. See Figure 18b. the curve as long as we do not leave the face of AV

In P. Agarwal, L. Kavraki, M. Mason, editors, Robotics: The Algorithmic Perspective, A. K. Peters, Ltd. 1998.

K. F. Bohringer and B. R. Donald

Since f has already b een constructed, we know for Much remains to be done. We have found that in

each exit p oint of the line that contains it. There- practice, no non-symmetric part has more than 4 to-

fore we can construct each new face using ray sho oting tal equilibria i.e., force and moment. Finding tight

queries and pro ceed as ab ove. Wekeep a data struc- b ounds on the numb er of total equilibria|that is, tak-

ture that describ es all the faces of AV that have ing momentinto account|will b e critical in the future.

already b een constructed so that we do not construct

the same face twice.

7 Conclusions

The algorithm stops when wehave identi ed all the

Table 1 summarizes elds and algorithms for manipu-

intersection p oints of with lines in V , and so we

lation tasks with programmable vector elds, and in-

have also identi ed the zone of in AV , namely all

cludes some additional recent results.

the face of AV crossed by .

Less dicult tasks such as translation can be

Further details on the algorithm can b e found in [16].

achieved with relatively simple elds and without any

We summarize the algorithmic result in the following

planning. More complex tasks such as centering or

theorem.

unique orienting require increasingly complex elds.

However, planning complexity is e.g. higher for se-

Theorem 17 [14] Let P be a non-degenerate polygon

quences of squeeze elds, and lower for the more com-

possibly with holes with n vertices, and such that any

plex combined radial + squeeze elds for a thorough

line crosses at most r edges of P . For any ">0 we

discussion of these elds see [19]. This illustrates a

can nd a complete speci cation of the area bisectors

2=3 2=3+"

tradeo between mechanical complexity the dexter-

of P in time O n + r , where and

ity and controllability of actuator arrray elements and

r are as de ned above, and is the functional

computational complexity the algorithmic dicultyof

inverse of Ackermann's function. If P is rectilinear,

2=3 2=3+"

synthesizing a strategy.

then the algorithms runs in time O n . The

2=3 2=3+"

spacerequired by the algorithm is O n .

Universal feeder-orienter UFO devices. Our

research leads to the question ab out the existence of a

6.4 Moment equilibria and alignment plans

\universal feeder-orienter" UFO device that uniquely

Recall from the b eginning of this section that bisectors

p oses a part without the need of a clo ck, sensors, or

corresp ond to force equilibria of P in a squeeze eld.

programming [19,20]. It was shown in [19] that every

For total equilibrium, the moment acting on the part

connected p olygonal part P with n vertices has a -

has to be taken into account as well. In particular,

nite numb er of stable orientation equilibria when P is

not all of the force equilibrium con gurations will be

placed into a squeeze eld S . Based on this prop erty

moment equilibria. For each maximal piece b of the

we were able to generate manipulation strategies for

bisector curve there exists only a nite number of

unique part alignment. We then showed that by using

moment equilibria we omit the pro of here:

a combined radial and squeeze eld R + S , the number

of equilibria can be reduced to O kn. Using elliptic

Claim 18 [14] Let P be a polygon whose interior is

force elds f x; y = x; y such that 6= and

connected. Let be a bisector of P that intersects r

; 6= 0, this b ound can be reduced to two [41, 40].

edges of P . There exist O r lines that are combi-

An \inertial" squeeze eld f x; y = signxx ; 0

natorial ly equivalent to such that P is in total equilib-

uniquely orients a part mo dulo eld symmetry [21].

rium when coincides with the center line of a squeeze

In a stable equilibrium, the part's ma jor principal axis

eld.

of inertia lines up with the squeeze line to minimize the

second moment of inertia.

It follows that a squeeze eld induces a nite number

Do es there exist a universal eld that, for every part of total equilibria on a p olygonal part P .

In P. Agarwal, L. Kavraki, M. Mason, editors, Robotics: The Algorithmic Perspective, A. K. Peters, Ltd. 1998.

Algorithmic MEMS

task elds complexity

elds planning plan steps

translate constant constant magnitude | 1

and direction

center radial constant magnitude, | 1

continuous directions

2 4 2

uniquely orient seq. of squeezes piecewise constant O k n O kn

mag. and dir.

inertial smo oth magnitude O 1 O 1

piecewise constant dir.

2 2

uniquely p ose seq. of radial+squeeze piecewise continuous O k n O kn

mag. and dir.

elliptic smo oth mag. and dir. O 1 O 1

UFO continuous mag. and dir. | 1

Table 1: Fields and algorithms for manipulation tasks with programmable vector elds.

P , has only one unique equilibrium up to part sym- Acknowledgements

metry? Such a eld could b e used to build a universal

parts feeder [1] that uniquely p ositions a part without

We are very grateful to our collab orators and friends

the need of a clo ck, sensors, or programming.

No el MacDonald, John Suh, Greg Kovacs, Ken Gold-

b erg, and Danny Halp erin for sharing with us the re-

In [19], we prop osed a combined radial and \grav-

search surveyed in this pap er.

itational" eld R + G which might have this prop-

erty. is a small p ositive constant, and G is de ned

The authors would also like to thank Steve Glan-

as Gx; y =0; 1. This device design is inspired by

der, Rob ert Darling, Christopher Storment, and all

the \universal gripp er" in [1]. Such a eld could be

the memb ers of the Stanford Transducers Lab; the

obtained from a MEMS array that implements a unit

memb ers of No el MacDonald's research team; sta

radial force eld. Instead of rectangular actuators in a

and users of the Cornell Nanofabrication Facility, the

regular grid, triangular actuators could b e laid out in

memb ers of the Cornell Rob otics and Vision Lab

a p olar-co ordinate grid. The array could then b e tilted

CSRVL, the memb ers of Donald Lab at Dartmouth,

slightly to obtain the gravity comp onent. Hence sucha

Lydia Kavraki for fruitful discussions, and Jean-Claude

device would b e relatively easy to build. Alternatively,

Latomb e for his hospitality during our stay at the Stan-

a resonating sp eaker, or a vibrating disk-shap ed plate

ford Rob otics Lab oratory.

that is xed at the center, might be used to create a

radial force eld. Extensive simulations show that for

Supp ort for our research was provided in part by

every part wehave tried, one unique total equilibrium

the NSF under grants no. IRI-8802390, IRI-9000532,

is always obtained. We are working toward a rigorous

IRI-9201699, IRI-9530785, IRI-9896020, and by a Pres-

pro of of this exp erimental observation.

idential Young Investigator award to Bruce Donald, in

part by NSF/ARPA SGER no. IRI-9403903, in part by

Algorithmic MEMS In this pap er, we have sur- an NSF Challenges in Computer and Information Sci-

veyed a family of ideas in Algorithmic MEMS, relat- ence and Engineering CISE grant no. CDA-9726389,

ing to programmable force elds. Naturally, in sucha in part by an NSF CISE Postdo ctoral Asso ciateship

young and dynamic eld, we exp ect a myriad of new to Karl Bohringer no. CDA-9705022, and in part by

geometric and algorithmic problems to emerge in the the AFOSR, the Mathematical Sciences Institute, In-

next few years. tel Corp oration, and AT&T Bell lab oratories.

In P. Agarwal, L. Kavraki, M. Mason, editors, Robotics: The Algorithmic Perspective, A. K. Peters, Ltd. 1998.

K. F. Bohringer and B. R. Donald

References [11] K.-F. Bohringer, V. Bhatt, and K. Y. Goldb erg. Sen-

sorless manipulation using transverse vibrations of a

plate. In Proc. IEEE Int. Conf. on Robotics and Au-

[1] T. L. Ab ell and M. Erdmann. A universal parts feeder,

tomation ICRA, pages 1989{1996, Nagoya, Japan,

1996. Personal communication / in preparation.

May 1995. .

[2] S. Akella, W. H. Huang, K. M. Lynch, and M. T.

[12] K.-F. Bohringer, R. G. Brown, B. R. Donald, J. S.

Mason. Planar manipulation on a conveyor by a one

Jennings, and D. Rus. Distributed rob otic manipu-

joint rob ot with and without sensing. In International

lation: Exp eriments in minimalism. In O. Khatib et

Symposium of Robotics Research ISRR, 1995.

al., editor, Exp erimental Rob otics IV,Lecture Notes

in Control and Information Sciences 223, pages 11{25.

[3] M. Ataka, A. Omo daka, and H. Fujita. A biomimetic

Springer Verlag, Berlin, 1997. .

micro motion system. In Transducers | Digest Int.

Conf. on Solid-State Sensors and Actuators, pages 38{

[13] K.-F. Bohringer, M. B. Cohn, K. Goldb erg, R. Howe,

41, Paci co, Yokohama, Japan, June 1993.

and A. Pisano. Electrostatic self-assembly aided by ul-

trasonic vibration. In AVS 44th National Symposium,

[4] D. Biegelsen, W. Jackson, A. Berlin, and P. Cheung.

San Jose, CA, Oct. 1997.

Air jet arrays for precision p ositional control of exible

[14] K.-F. Bohringer, B. R. Donald, and D. Halp erin. The

media. In Int. Conf. on Micromechatronics for Infor-

area bisectors of a p olygon and force equilibria in pro-

mation And Precision Equipment MIPE'97,Tokyo,

grammable vector elds. In 13th ACM Symposium on

Japan, July 1997.

Computational Geometry, Nice, France, June 1997.

[5] K.-F. Bohringer. Computational asp ects of the design

[15] K.-F. Bohringer, B. R. Donald, and D. Halp erin. On

of micro-mechanical hinged structures. In AAAI Fal l

the area bisectors of a p olygon. In Second CGC Work-

Symposium Series | Design from Physical Principles,

shop on Computational Geometry, Durham, NC, Oct.

Working Notes, Boston, MA, Oct. 1992. Abstract. .

1997. Extended Abstract.

[6] K.-F. Bohringer. Application of computational top ol-

[16] K.-F. Bohringer, B. R. Donald, and D. Halp erin. On

ogy to the design of micro electromechanical struc-

the area bisectors of a p olygon. Discrete and Compu-

tures. In Proceedings of the NSF Design and Manufac-

tational Geometry, 1998. To app ear.

turing Systems Grantees Conference, Charlotte, NC,

[17] K.-F. Bohringer, B. R. Donald, and N. C. MacDon-

Jan. 1993. National Science Foundation. .

ald. Single-crystal silicon actuator arrays for micro

manipulation tasks. In Proc. IEEE Workshop on Mi-

[7] K.-F. Bohringer. A computational approach to the

cro ElectroMechanical Systems MEMS, pages 7{12,

design of micromechanical hinged structures. In

San Diego, CA, Feb. 1996. .

Proceedings of the ACM/SIGGRAPH Symposium on

Solid Modeling and Applications, Montr eal, Qu eb ec,

[18] K.-F. Bohringer, B. R. Donald, and N. C. MacDon-

Canada, May 1993. ACM Press. Extended abstract. .

ald. What programmable vector elds can and can-

not do: Vector eld algorithms for MEMS arrays

[8] K.-F. Bohringer. Computational asp ects of the de-

and vibratory parts feeders. In Proc. IEEE Int. Conf.

sign of micro electromechanical structures. In ICRA

on Robotics and Automation ICRA, pages 822{829,

Workshop on Geometric Algorithms for Manufactur-

Minneap olis, MN, Apr. 1996. .

ing,Atlanta, Georgia, May 1993. IEEE. .

[19] K.-F. Bohringer, B. R. Donald, and N. C. MacDon-

[9] K.-F. Bohringer. Programmable Force Fields for Dis-

ald. Upp er and lower b ounds for programmable vector

tributed Manipulation, and Their Implementation Us-

elds with applications to MEMS and vibratory plate

ing Microfabricated Actuator Arrays. PhD thesis,

parts feeders. In J.-P. Laumond and M. Overmars,

Cornell University, Department of Computer Science,

editors, Algorithms for Robotic Motion and Manipu-

Ithaca, NY 14853, Aug. 1997.

lation, pages 255{276. A. K. Peters, Wellesley, MA

02181, 1997. .

[10] K.-F. Bohringer, V. Bhatt, B. R. Donald, and K. Y.

Goldb erg. Sensorless manipulation using transverse [20] K.-F. Bohringer, B. R. Donald, and N. C. MacDonald.

vibrations of a plate. Algorithmica, 1998. To app ear in Programmable vector elds for distributed manipula-

Sp ecial Issue on Algorithmic Foundations of Rob otics. tion, with applications to MEMS actuator arrays and

In P. Agarwal, L. Kavraki, M. Mason, editors, Robotics: The Algorithmic Perspective, A. K. Peters, Ltd. 1998.

Algorithmic MEMS

along paths comp osed of straight line segments. In vibratory parts feeders. Int. Journal of Robotics Re-

IEEE Control and Decision Conference, USA, 1997. search, 1998. To app ear.

Submitted for review.

[21] K.-F. Bohringer, B. R. Donald, N. C. MacDonald,

[30] M. Coutinho and P. Will. Using dynamic vector force

G. T. A. Kovacs, and J. W. Suh. Computational meth-

elds to manipulate parts on an intelligent motion sur-

o ds for design and control of MEMS micromanipulator

face. In IEEE International Symposium on Automa-

arrays. Computer Science and Engineering, pages 17{

tion and Task Planning, Los Angeles, CA, 1997.

29, January { March 1997.

[31] M. D az and J. O'Rourke. Ham-sandwich sectioning of

[22] K.-F. Bohringer, B. R. Donald, R. Mihailovich, and

p olygons. In Proc. 2nd Canadian Conference on Com-

N. C. MacDonald. A geometric theory of manipulation

putational Geometry, pages 98{101, Ottawa, 1990.

and control for microfabricated actuator arrays. Tech-

nical Rep ort 93{87, Cornell University, Mathematical

[32] B. R. Donald, J. Jennings, and D. Rus. Information in-

Sciences Institute, Ithaca, NY 14853, Nov. 1993. .

variants for distributed manipulation. In K. Goldb erg,

D. Halp erin, J.-C. Latomb e, and R. Wilson, editors,

[23] K.-F. Bohringer, B. R. Donald, R. Mihailovich, and

International Workshop on Algorithmic Foundations

N. C. MacDonald. Sensorless manipulation using mas-

of Robotics WAFR, pages 431{459, Wellesley, MA,

sively parallel microfabricated actuator arrays. In

1995. A. K. Peters.

Proc. IEEE Int. Conf. on Robotics and Automation

ICRA, pages 826{833, San Diego, CA, May 1994. .

[33] B. R. Donald and D. K. Pai. The motion of compli-

antly connected rigid b o dies in contact. Int. Journal

[24] K.-F. Bohringer, B. R. Donald, R. Mihailovich, and

of Robotics Research, July 1991.

N. C. MacDonald. A theory of manipulation and

control for microfabricated actuator arrays. In Proc.

[34] H. Edelsbrunner. Algorithms in Combinatorial Geom-

IEEE Workshop on Micro ElectroMechanical Systems

etry,volume 10 of EATCS Monographs on Theoretical

MEMS, pages 102{107, Oiso, Japan, Jan. 1994. .

Computer Science. Springer Verlag, Heidelb erg, Ger-

many, 1987.

[25] K.-F. Bohringer, K. Goldb erg, M. B. Cohn, R. Howe,

and A. Pisano. Parallel microassembly with elec-

[35] M. A. Erdmann. An exploration of nonprehensile two-

trostatic force elds. In Proc. IEEE Int. Conf. on

palm manipulation: Planning and execution. Tech-

Robotics and Automation ICRA, Leuven, Belgium,

nical rep ort, Carnegie Mellon University, Pittsburgh,

May 1998.

PA, 1996.

[26] K.-F. Bohringer, J. W. Suh, B. R. Donald, and

[36] J. D. Foley,A.Van Dam, Feiner, and Hughes. Com-

G. T. A. Kovacs. Vector elds for task-level distributed

puter Graphics: Principles and Practice. Addison

manipulation: Exp eriments with organic micro actua-

Wesley, 2nd edition, 1996.

tor arrays. In Proc. IEEE Int. Conf. on Robotics and

Automation ICRA, pages 1779{1786, Albuquerque,

[37] H. Fujita. Group work of microactuators. In Inter-

New Mexico, Apr. 1997.

national Advanced Robot Program Workshop on Mi-

cromachine Technologies and Systems, pages 24{31,

[27] P. Cheung, A. Berlin, D. Biegelsen, and W. Jack-

Tokyo, Japan, Oct. 1993.

son. Batch fabrication of pneumatic valve arrays by

combining mems with printed circuit b oard technol-

[38] K. Y. Goldb erg. Orienting p olygonal parts with-

ogy. In Proc. Symposium on Micro-Mechanical Sys-

out sensing. Algorithmica, 102/3/4:201{225, Au-

tems, ASME International Mechanical Engineering

gust/Septemb er/Octob er 1993.

Congress and Exhibition, pages 16{21, Dallas, TX,

[39] B. Jo e. Manipulation and identi cation of ob jects by

Nov. 1997.

magnetic forces. In Proc. International Symposium on

[28] M. Coutinho and P. Will. The intelligent motion

Magnetic Suspension Technology, NASA Conference

surface: a hardware/software to ol for the assembly

Publication 3152 part 2, pages 617{639, Langley,VA,

of meso-scale devices. In Proc. IEEE Int. Conf. on

Aug. 1991.

Robotics and Automation ICRA, Albuquerque, New

[40] L. Kavraki. Part orientation with programmable vec-

Mexico, Apr. 1997.

tor elds: Two stable equilibria for most parts. In

[29] M. Coutinho and P. Will. Op en lo op dynamic con- Proc. IEEE Int. Conf. on Robotics and Automation

trol of parts moving on an intelligent motion surface ICRA, Albuquerque, New Mexico, Apr. 1997.

In P. Agarwal, L. Kavraki, M. Mason, editors, Robotics: The Algorithmic Perspective, A. K. Peters, Ltd. 1998.

K. F. Bohringer and B. R. Donald

[41] L. E. Kavraki. On the numb er of equilibrium place- [52] M. H. Raib ert, J. K. Ho dgins, R. R. Playter, and R. P.

ments of mass distributions in elliptic p otential elds. Ringrose. Animation of legged maneuvers: jumps,

Technical Rep ort STAN-CS-TR-95-1559, Department somersaults, and gait transitions. Journal of the

of Computer Science, Stanford University, Stanford, Robotics Society of Japan, 113:333{341, 1993.

CA 94305, 1995.

[53] J. Reif and H. Wang. So cial p otential elds: A dis-

tributed b ehavioral control for autono omous rob ots.

[42] O. Khatib. Real time obstacle avoidance for manip-

In K. Goldb erg, D. Halp erin, J.-C. Latombe, and

ulators and mobile rob ots. Int. Journal of Robotics

R. Wilson, editors, International Workshop on Algo-

Research, 51:90{99, Spring 1986.

rithmic Foundations of Robotics WAFR, pages 431{

[43] D. E. Ko ditschek and E. Rimon. Rob ot navigation

459. A. K. Peters, Wellesley, MA, 1995.

functions on manifolds with b oundary. Advances in

[54] D. Reznik, S. Brown, and J. F. Canny. Dynamic sim-

Applied Mathematics, 1988.

ulation as a design to ol for a microactuator array. In

[44] K. Kotay, D. Rus, M. Vona, and C. McGray. The self-

Proc. IEEE Int. Conf. on Robotics and Automation

recon gurable rob otic molecule. In Proc. IEEE Int.

ICRA, Albuquerque, NM, Apr. 1997. .

Conf. on Robotics and Automation ICRA, Belgium,

[55] D. Reznik, J. F. Canny, and K. Y. Goldb erg. Analy-

May 1998.

sis of part motion on a longitudinally vibrating plate.

[45] W. Liu and P. Will. Parts manipulation on an intelli-

In IEEE/RSJ Int. Workshop on Intel ligent Robots &

gent motion surface. In IEEE/RSJ Int. Workshop on

Systems IROS, Grenoble, France, Sept. 1997.

Intel ligent Robots & Systems IROS, Pittsburgh, PA,

[56] E. Rimon and D. Ko ditschek. Exact rob ot navigation

1995.

using arti cial p otential functions. IEEE Transactions

[46] J. E. Luntz and W. Messner. A distributed control

on Robotics and Automation, 85, Octob er 1992.

system for exible materials handling. IEEE Control

[57] K. A. Shaw and N. C. MacDonald. Integrating

Systems, 171, Feb. 1997.

SCREAM micromechanical devices with integrated

[47] J. E. Luntz, W. Messner, and H. Choset. Parcel ma-

circuits. In Proc. IEEE Workshop on Micro Electro

nipulation and dynamics with a distributed actuator

Mechanical Systems MEMS, San Diego, CA, Feb.

array: The virtual vehicle. In Proc. IEEE Int. Conf.

1996.

on Robotics and Automation ICRA, pages 1541{

[58] K. A. Shaw, Z. L. Zhang, and N. C. MacDonald.

1546, Albuquerque, New Mexico, Apr. 1997.

SCREAM I: A single mask, single-crystal silicon pro-

[48] R. E. Mihailovich and N. C. MacDonald. Dissipation

cess for micro electromechanical structures. In Trans-

measurements of vacuum-op erated single-crystal sili-

ducers | Digest Int. Conf. on Solid-State Sensors and

con resonators. Sensors and Actuators, 1996.

Actuators,Paci co, Yokohama, Japan, June 1993.

[49] R. E. Mihailovich, Z. L. Zhang, K. A. Shaw, and N. C.

[59] J. W. Suh, S. F. Glander, R. B. Darling, C. W. Stor-

MacDonald. Single-crystal silicon torsional resonators.

ment, and G. T. A. Kovacs. Combined organic thermal

In Proc. IEEE Workshop on Micro Electro Mechani-

and electrostatic omnidirectional ciliary microactuator

cal Systems MEMS, pages 155{160, Fort Lauderdale,

array for ob ject p ositioning and insp ection. In Proc.

FL, Feb. 1993.

Solid State Sensor and Actuator Workshop, Hilton

Head, NC, June 1996.

[50] K. S. J. Pister, R. Fearing, and R. Howe. A planar

air levitated electrostatic actuator system. In Proc.

[60] Z. L. Zhang and N. C. MacDonald. An RIE pro cess for

IEEE Workshop on Micro ElectroMechanical Systems

submicron, silicon electromechanical structures. Jour-

MEMS, pages 67{71, Napa Valley, California, Feb.

nal of Micromechanics and Microengineering, 21:31{

1990.

38, Mar. 1992.

[51] R. Prasad, K.-F. Bohringer, and N. C. MacDonald.

[61] N. B. Zumel and M. A. Erdmann. Nonprehensile two

Design, fabrication, and characterization of SCS latch-

palm manipulation with non-equilibrium transitions

ing snap fasteners for micro assembly. In Proceed-

between stable states. In Proc. IEEE Int. Conf. on

ings of the ASME International Mechanical Engineer-

Robotics and Automation ICRA, Minneap olis, MN,

ing Congress and Exposition IMECE, San Francisco,

Apr. 1996.

California, Nov. 1995. .

In P. Agarwal, L. Kavraki, M. Mason, editors, Robotics: The Algorithmic Perspective, A. K. Peters, Ltd. 1998.