Ongoing research on the control of mechanical systems is concerned with the design of nonlinear control laws using methods based on differential geometry and the principles of Lagrangian and Hamiltonian mechanics. Current research continues to widen the range of applications, and at the same time new applications such as controlling groups of autonomous vehicles have suggested new extensions of the theory. Because geometric methods facilitate the analysis of the ways in which control laws depend on physical length scales, they have proven to be especially useful in controlling very small-scale systems.

Controlled motions of rotating heavy chains provide a rich set of examples of ways in which active control can be used to modify bifurcations in nonlinear systems.

High-frequency oscillatory control designs based on a new theory of averaged Lagrangian systems have been used in experiments on the stabilization of very small pendulum mechanisms like the one depicted above.

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Since the theory shows exactly how such control designs change with respect to the characteristic length scales of the systems to which they are applied, recent work has been focused on applications to MEMS systems such as silicon diaphragm piston actuators.

Current research on controlling arrays (such as the one depicted above) is focused on developing techniques for handling massively parallel feedback loops in setting where increasingly severe communications constraints limit the amount of data that can be passed through each feedback channel. Work is also aimed at finding ways to most effectively control the nonlinear dynamics of each actuator and in particular to deal with the snap-through instability.

Recall that the snap-through instability for such a piston microactuator is a result of the nonlinear (inverse-square) force created by an applied voltage (electrostatic charge). By applying a constant voltage to deflect the diaphragm, one can achieve a deflection no larger than 1/3 of the gap between the undeformed diaphragm and the substrate. A voltage less than the critical voltage will deflect the diaphragm less than 1/3 of the gap, while a voltage greater than or equal to the critical voltage will cause the diaphragm to crash onto the substrate. In principle, one could effectively triple the stroke of such an electrostatic actuator by implementing real-time feedback between charge and position. Current research is aimed at applying recently developed methods of averaging for Lagrangian systems to design control laws which can in principle remove the snap-through instability. Details are available in the following references.

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Selected Publications

• S. Weibel & J.Baillieul, "Oscillatory Control of Bifurcations in Rotating Chains,'' Proc. of the 1997 American Control Conference, Albuquerque, NM, pp. 2313-2317.
• S. Weibel, T. Kaper, & J. Baillieul, ``Global Dynamics of a Rapidly Forced Cart and Pendulum,'' Nonlinear Dynamics, 13:131-170, July, 1997.
• S.P. Weibel & J. Baillieul, ``Scale Dependence in the Oscillatory Control of Micromechanisms,'' in the 1998 IEEE Conf. on Decision & Control, December 16-18, Tampa, FL. Published by the IEEE, Piscataway, NJ, pp. 3058-3063.
• J. Baillieul, ``A control design which respects characteristic length scales in smart systems and smart structures,'' 1999. Proceedings of SPIE 6-th Annual Int'l Symposium on Smart Structures and Smart Materials, March 1-4, Newport Beach, CA, Volume 3667, pp. 202-210.
• K. Nonaka & J. Baillieul, ``Open Loop Robust Oscillatory Stabilization of a Two-wire System inside the Snap-through Instability Region,'' The 2001 IEEE Conference on Decision and Control, Orlando, FL, December, 2001, pp.1334- 1341.
• T. Sugimoto, M. Horenstein, K. Nonaka & J. Baillieul. ``Bi-Directional Electrostatic Actuator Operated with Charge Control,'' Proceedings of the 2002 Joint Electrostatics Society of America/Institute of Electrostatics Japan Annual Conference, June 25-28, 2002, Northwestern University, Evanston, IL. (Also submitted to Journal of Electrostatics (Elsevier).)
• J. Baillieul, ``Control Methods for Very Small-scale Devices,'' in Proc. of the 2003 Soc. for Experimental Mechanics Annual Conf. and Exposition, Charlotte, NC, June-2-4, 2003, pp. 111-116.