This project conducted a scientific investigation into the existence, consequences, and mitigation of complex non-minimum phase zeros in the dynamics of flexure mechanisms. Flexure mechanisms provide guided motion via elastic deformation of thin sections, as opposed to employing sliding or rolling joints seen in traditional linkage mechanisms. Because of their lack of friction, backlash, and assembly, flexure mechanisms are used extensively for motion guidance in several applications including precision motion stages, micro-electro mechanical systems (MEMS), and energy harvesting devices among others. In these applications, the flexure mechanism is generally integrated with sensors, actuators, drivers, and controls to provide desirable closed-loop dynamic performance such as high speed, command tracking, noise and disturbance rejection, and stability.

There has been an increasing need for simultaneously achieving large displacement range as well as superior dynamic performance in the application. However, these two desirable objectives strongly tradeoff against each other. Overcoming these tradeoffs requires a fundamental understanding of flexure mechanism dynamics over large displacement range. Such dynamics includes poles and zeros in the transfer function between the input and output of the system. The importance of poles in the dynamics of flexible systems (including flexure mechanisms) and their impact on closed-loop performance is historically well known. Furthermore, the mathematical and physical origins and meaning of these poles (i.e. natural modes) are also well understood. On the other hand, a comprehensive mathematical foundation and physical understanding of zeros was previously lacking. In particular, non-minimum phase (NMP) zeros are known to be highly detrimental to the closed-loop dynamic performance of the overall system and means to eliminate such zeros were highly desirable. A key outcome of this project has been the creation of new knowledge and understanding on the origins and behavior of NMP zeros in flexible systems including flexure mechanisms.

A   lumped-parameter modeling approach was developed to analytically model the dynamics of flexure mechanisms comprising the parallelogram or double parallelogram modules. This model captures the key relevant geometric nonlinearity in large-displacement flexure mechanics, which enabled a prediction of previously unexplained complex NMP zeros. The model establishes the existence of complex NMP zero sunder certain combinations of operating point and parametric asymmetry in the non-collocated transfer function of a representative flexure mechanism. This finding helps generate the design insight that, rather than an intuitively symmetric design, an intentional asymmetry in mass can avoid complex NMP zeros and make the system conducive to better dynamic performance. This overall investigation has relevance not just to flexure mechanisms but also to a broader range of flexible systems. This analytical modeling provides fundamental insights into the necessary or sufficient conditions to achieve certain desired zero dynamics e.g. eliminate NMP zeros, or achieve zero pole alternation.

This project has also led the development of an experimental setup comprising a flexure mechanism based motion system integrated with sensors, actuators, drivers and a real-time control system to validate the analytical modeling predictions and assumptions. This experimental setup will continue to serve as a valuable testbed for ongoing investigations and validation in flexible system dynamics and controls for several years after the project has ended. From a mechatronic system design stand-point, this experimental validation helps inform physical system design decisions in a deterministic manner. We have shown that independent of any other more complex design considerations (e.g. collocation of sensor and actuator, or addition of damping), complex NMP zeros can be eliminated very simply via an intentional use of mass asymmetry. This simple yet non-obvious physical system design tweak leads to better control system design and closed-loop dynamic performance.

This project creates a potentially broad impact in the dynamics and controls of flexible systems such as multi-axis flexure based nanopositioning systems, MEMS devices, kinetic energy harvesting systems, bridges, buildings, spacecraft structures, etc. An understanding of complex non-minimum phase (NMP) zeros is essential to moving this field forward. Furthermore, a relationship between NMP zeros, curve veering, and mode localization, as demonstrated via this research, has relevance to flexible systems well beyond the multi-axis flexure mechanisms considered here.

Student supervision and mentoring has been a major component of this project. Several graduate and undergraduate students were intellectually supported by this project. These students have gained significant professional skills in engineering research, mechanical design, dynamic system modeling and simulation, hardware design and fabrication, frequency domain system identification, modal testing, mechatronic system design and integration, technical documentation and professional communication.

Enabled by the results of this basic research, the PI has developed a strong partnership with the semiconductor manufacturing equipment industry with the goal of technology development. This research has directly led to a Partnerships for Innovation (PFI) - Research Partnership with an Industrial Partner to translate the new knowledge and innovations generated from this project towards creating next generation semiconductor wafer inspection and metrology systems with high throughput.

Last Modified: 12/30/2020
Modified by: Shorya Awtar