Multi-Axis Resonant Filter Design using Frequency
Response Data applied to Industrial Scan Stage
Masahiro Mae, Wataru Ohnishi, Hiroshi Fujimoto

Koichi Sakata

The University of Tokyo
mmae@ieee.org, ohnishi@ieee.org, fujimoto@k.u-tokyo.ac.jp

Nikon Corporation
koichi.sakata@nikon.com

Abstract—Disturbance rejection of the high-precision scanstages is important in industrial lithography equipment. The aimof this paper is to develop an optimization method for designing
multi-axis resonant filters, that enhance the disturbance rejection
performance in scanning motion. The developed optimizationmethod explicitly defines resonant filters in structured representation and formulates the data-driven convex optimization problem.
The method enables the multi-axis resonant filter design with
iterative convex optimization using the frequency response dataof the six-degree-of-freedom experimental setup. Experimental
results on the industrial large-scale high-precision scan stage
demonstrate the performance improvement of the disturbancerejection in the scanning motion using the optimized resonant
filters.
Index Terms—Disturbance rejection, Loop shaping, Resonant
filter, MIMO system, Frequency response, Data-driven design,
Convex optimization

Fig. 1. Experimental setup of FPD lithography system [1].
F (s)Fky (jωkf , ρky )Xkr =1

ρky ,(kr ,2) (jωkf )2 + ρky ,(kr ,1) (jωkf ).
2(jωkf )2 + 2ζr,ky ,kr ωr,ky ,kr (jωkf ) + ωr,ky ,kr

κ
×K ×1

ρ(nr ,2) s + ρ(nr ,1) s2
s2 + 2ζr,nr ωr,nr s + ωr,nr

ws+2Fig. 1 shows the experimental setup of the industrial
FPD lithography system which is used for the productionof flat panel displays. In the setup, the MIMO large-scale
high-precision scan stage is implemented and it has 6-DOFs(x, y, θz , z, θx , θy ). The main scan stroke is the translation
along the x-axis.
The resonant filter has high-gained characteristics at thedesigned resonance frequency and effectively rejects the disturbance at the same frequency. In the decentralized resonant
filter for the MIMO controlled system, resonant filters in eachaxis are shown in Fig. 2 and are defined asnr,kyψ..
.

I. A PPROACH=1+Imρ(1,2) s2 + ρ(1,1) s2s2 + 2ζr,1 ωr,1 s + ωr,1+−1G(jωr ) G(jωr )F (jωr )0Re++Fig. 2. Block diagram of resonant filters ineach axis.

Fig. 3. Vector locus withresonant filter and robust
stability condition.

TABLE IE XPERIMENTAL ROOT MEAN SQUARE ERRORS OF 8 SCAN REGIONS IN
6-DOF S WITHOUT RESONANT FILTERS ( W / O ), WITH INITIAL RESONANTFILTERS ( INI ), AND WITH OPTIMIZED RESONANT FILTERS ( OPT ).
[count]w/oiniopt

ex498859ey1208865eθz
284134ez
116
111
107eθx616048eθy75108116R−1 E∥
F100 %
104 %87 %t ∥W

(1)

The tuning parameters, the resonance frequency, and the damping coefficient in each axis are ρky ,(kr ,:) , ωr,ky ,kr , and ζr,ky ,kr .
The initial resonant filters are designed as K times larger gainat the resonance frequency ωr and the vector locus recedesfrom (−1, j0) with a resonance circle as shown in Fig. 3.
The optimized resonant filters are designed with optimization
for the gain and phase of the resonant filters to improve theMIMO performance. The MIMO performance is evaluated by
the Frobenius norm of the normalized error frequency spectrum matrix. The original non-convex optimization problem isreformulated to the iterative convex optimization problem by
the Moore-Penrose inverse and sequential linearization.

II. R ESULTThe stage is moving along the main stroke x-axis withconstant velocity 0.5 m/s. From TABLE I, the experimental
results show that the MIMO performance with optimized
resonant filters outperforms that without resonant filters in
13 % and that with initial resonant filters in 17 %. Experimentson the industrial MIMO large-scale high-precision scan stage
demonstrate effective disturbance rejection performance withthe optimized resonant filters.
R EFERENCES[1] Nikon, “FPD Lithography Systems Lineup,” https://www.nikon.com/business/fpd/lineup/, 2024.