Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Nanometer Scanning with Micrometer Sensing: Beating Quantization Constraints in Lissajous Trajectory Tracking

Matheus Lohse, Rafael S. Castro, Aurelio T. Salton, Minyue Fu

TL;DR

Addresses the problem of tracking Lissajous trajectories under output quantization by deriving continuous-time results for asymptotic tracking of periodic references using the Internal Model Principle ($H(s)$) and a positive real transfer function. Introduces artificial quantization in the reference path so that $\tilde{e}=q(r)-q(y)$ vanishes when $e=r-y=0$, enabling $e(t)\to 0$ in steady state. Provides a structured reference signal and a rank condition that guarantees unique recovery of $r(t)$ from $q(r(t))$ and demonstrates that, under PR, $e(t)\to 0$; applies the framework to 2D Lissajous tracking with sinusoidal components and frame-rate dependent frequencies. Shows in simulations that nanoscale tracking accuracy is achievable with a micrometer quantization step, and discusses how increasing $N$ (and frame rate) can reduce the scan resolution toward the nanometer scale, including a step-change demonstration for large scans.

Abstract

This paper addresses the task of tracking Lissajous trajectories in the presence of quantized positioning sensors. To do so, theoretical results on tracking of continuous time periodic signals in the presence of output quantization are provided. With these results in hand, the application to Lissajous tracking is explored. The method proposed relies on the internal model principle and dispenses perfect knowledge of the system equations. Numerical results show that an arbitrary small scanning resolution is achievable despite large sensor quantization intervals.

Nanometer Scanning with Micrometer Sensing: Beating Quantization Constraints in Lissajous Trajectory Tracking

TL;DR

Addresses the problem of tracking Lissajous trajectories under output quantization by deriving continuous-time results for asymptotic tracking of periodic references using the Internal Model Principle () and a positive real transfer function. Introduces artificial quantization in the reference path so that vanishes when , enabling in steady state. Provides a structured reference signal and a rank condition that guarantees unique recovery of from and demonstrates that, under PR, ; applies the framework to 2D Lissajous tracking with sinusoidal components and frame-rate dependent frequencies. Shows in simulations that nanoscale tracking accuracy is achievable with a micrometer quantization step, and discusses how increasing (and frame rate) can reduce the scan resolution toward the nanometer scale, including a step-change demonstration for large scans.

Abstract

This paper addresses the task of tracking Lissajous trajectories in the presence of quantized positioning sensors. To do so, theoretical results on tracking of continuous time periodic signals in the presence of output quantization are provided. With these results in hand, the application to Lissajous tracking is explored. The method proposed relies on the internal model principle and dispenses perfect knowledge of the system equations. Numerical results show that an arbitrary small scanning resolution is achievable despite large sensor quantization intervals.
Paper Structure (7 sections, 2 theorems, 28 equations, 7 figures)

This paper contains 7 sections, 2 theorems, 28 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Artificial quantization
  3. The Reference Signal
  4. Assymptotic tracking
  5. Design guidelines
  6. Application to Lissajous Tracking
  7. CONCLUSION

Key Result

Lemma 1

Consider $r(t)=r(t+T)$ in eq:ref2 and define $\mathcal{M}$ as the matrix containing $\phi(t)$ for all the $p$ time instants $t_1,\ t_2,\ \dots,\ t_p \in [0, T)$ when $r(t)$ crosses a quantization level: Assume $r(t)$ is such that $p\geq 2m+1$ by design and suppose that $\mathrm{rank}(\mathcal{M}) = (2m\times1)$, then $r(t)$ is unique in the sense that any other function achieves $q(r(t))=q(\bar{

Figures (7)

  • Figure 1: The challenging task of tracking a reference subject to output quantization: even if $e = r-y=0$ it is possible that $\tilde{e} = r- q(y)\neq 0$.
  • Figure 2: The addition of the artificial quantization in the reference path allows $\tilde{e} = r- q(y)\neq 0$ when $e = r-y=0$.
  • Figure 3: A typical pattern of the Lissajous trajectory given by the gray line along with the quantization regions delimited by the dashed lines.
  • Figure 4: Time evolution of the $\mathrm{x}$-axis in three plots. Top: displacement; Middle: errors; bottom: control effort.
  • Figure 5: Time evolution of the overall lissajous trajectory. The bottom right plot shows the euclidean error of the trajectory asymptotically approaching zero.
  • ...and 2 more figures

Theorems & Definitions (3)

  • Lemma 1: Reference
  • Theorem 1
  • Remark 1