Experimental Validation of Complex Non-Minimum Phase Zeros in Flexure Mechanism Dynamics Leqing Cui, Dhanushkodi Mariappan, and Shorya Awtar Precision Systems Design Laboratory, Mechanical Engineering University of Michigan, Ann Arbor, MI 48109 BACKGROUND AND OBJECTIVE This research is motivated by the need to achieve large range, high precision, and high speed – all simultaneously – in multi-axis flexure mechanisms. Previous work has shown large range (10mm per axis) and nanometric precision (25nm) in an XY motion system based on a parallel-kinematic flexure mechanism [1]. This mechanism employs a systematic and symmetric layout of eight double parallelogram flexure modules (DPFM), resulting in a high degree of geometric decoupling between the X and Y motion axes. But experimental frequency response measurements of the X direction transfer function relevant for motion control exhibit complex non-minimum phase zeros (CNMP) at certain Y direction operating points. These CNMP zeros are highly detrimental to the closed-loop dynamic performance of the motion system including bandwidth and stability robustness [2]. To understand the conditions under which such CNMP zeros arise and determine if physical design decisions can eliminate these zeros, a representative XY flexure mechanism (Fig. 1) has been recently investigated [3]. This design is relatively simpler but has all the essential attributes of the original XY flexure mechanism [1], including multi-axis motion capability (X and Y) and at least two DPFMs in a nominally symmetric configuration that lead to closely spaced modes. The non- collocated transfer function from the force P to displacement X 1 was examined, as a function of cross-axis operating point Y 1o , via lumped- parameter non-linear dynamic modeling. A static Y direction force F 0 was assumed on the motion stage (stage 1) to establish this non-zero Y direction operating point (Y 1o ). It was shown that the geometric non-linearity associated with arc-length conservation in flexure beams along with the kinematic under-constraint in the DPFM result in a coupling between the Y motion of the secondary stages (2 and 3 in Fig.1) and the X motion of the motion stage for non-zero operating points (Y 1o ). Furthermore, the two closely spaced modal frequencies associated with the Y motions of the secondary stages of the two DPFM were varied by introducing an intentional parametric asymmetry (mass difference between stages 2 and 3, given by Δm 23 =m 2 /m 3 -1) to simulate practical manufacturing tolerances. This model showed that these closely spaced modes interact as the above mentioned coupling varies with the operating point and as the parametric asymmetry varies, giving rise to CNMP zeros under certain conditions. This is illustrated in Fig.2, which maps the existence of CNMP zeros against a range of operating points (Y 1o normalized with respect to beam length L) and parametric asymmetry (Δm 23 ). Black regions on this map indicate the presence of CNMP zeros. This map indicates that one can intentionally choose the physical design parameters (e.g. Δm 23 < 0) to eliminate the CNMP zeros, enabling better dynamic performance (i.e. higher speed, bandwidth, and stability robustness). This paper presents an experimental validation and confirmation of these modeling predictions. Y X Operating Point ① ② ⑤ ③ ④ (X 1 , Y 1 ) (X 2 , Y 2 ) (X 3 , Y 3 ) (X 4 ) (X 5 ) k 4 k 5 P F 0 Y 1o L Fig.1. Simple Representative 4 Degree-of-Freedom (SR4DoF) XY Flexure Mechanism -20 -15 -10 -5 0 5 10 15 20 Asymmetry Δm 23 =m 2 /m 3 -1 (%) -8 -6 -4 -2 0 2 4 6 8 Operating Point Y 1o (% of L) Case A Case B Experimental Points (MP) Experimental Points (CNMP) Simulation CNMP Region Simulation Surface Fig.2. Complex Non-Minimum Phase Zero Map