Algorithmic MEMS

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 survey 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 1 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- 1 growing complexityofavailable MEMS devices, man- The theory of programmable force elds for micro manipulation 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 equilibrium 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- 2 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 2 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- 2 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 3 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 otential 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.

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