[Robot sensor application] Basic concept

02 Mar 2019 in Robotics / Robotics & Control on Robotics

Table of Contents

1. Sensor
   1. General Classification
   2. Terminology
2. Laplace transform
3. Inverse Laplace transform
4. Reference

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Sensor

• Device to measure the physical quantity of concern
  • position, velocity, acceleration torque, temperature
• Transducer
  • Device to convert from one kind of energy to another kind one.
    • light energy à electrical energy

General Classification

• Active vs Passive
  • Active: emit energy to environment
    • More robust, less efficient
  • Passive: passively receive energy from environment.
    • Less intrusive, but depends on environment e.g. light for camera
• Analog vs Digital
• Contact vs Non-contact

by physical quantity

Terminology

• Range, Span
  • The range of a transducer defines the limits between which the input can vary. n The span is the maximum value of the input minus the minimum value
  • if a thermometer cover -40°~ +50°C, Range is -40°~ +50°C and span is 90°C.
• Error:
  • measured value - true value.
• Accuracy:
  • guaranteed minimum and maximum error including all possible effects.
  • accuracy ± 2
• Repeatability/Reproducibility
  • ability to give the same output for repeated applications of the same input value.
  • described by % of full range.

• Sensitivity
  • The sensitivity is the relationship indicating how much output is produced if unit input is applied.
  • dy/dx
• Linearity
  • dy/dx is const for all x
• Dynamic range
  • cover wide measurement range as well as detect small changes of measure. \(\rightarrow\) wide Dynamic range.
  • logarithmic response \(\rightarrow\) wide dynamic range camera
• Response Time
  • Time needed to result in change output after changes in input
• Resolution
  • minimum input change can be recognized from output
• Bandwidth
  • input frequency limit can be read 70% magnitude of dc input
• Hysteresis error
  • hystera means a psychological disorder.
  • hysteresis represents exaggerated or uncontrollable errors of a device

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Laplace transform

• Purpose:
  • Differential Eq. in time domain \(\rightarrow\) Algebraic Eq. in complex domain
• Definition :

\[\begin{aligned} \textrm{F}(s) = \int^\infty _{0-} f(t) e^{-st} dt \end{aligned}\]

여기서, s는 복소변수 그리고 f(t) = 0, for t < 0

• Linearity of Laplace operator
  • Superposition(homogeneity, additivity)

\[\begin{aligned} & L[a f(t)] = a L[f(t)] \\ & L[f(t) + g(t)] = L[f(t)] + L[g(t)] \end{aligned}\]

Why Laplace transformation?

Table of Laplace Tr.

Formula

최종값 및 초기값의 정리

Inverse Laplace transform

• Laplace tr. \(\rightarrow\) algebraic eq.
  • if solution of the algebraic eq. is F(s)
  • A(s) : denominator of F(s)
    • Poles: roots of polynomial A(s) = 0
  • There are 3 cases in the poles of F(s)
    • (1) Distinct real poles
    • (2) Multiple real poles
    • (3) Complex poles
  • cf. Root of equation B(s)=0 are called zeros.

\[\begin{aligned} F(s) = \frac{B(s)}{A(s)} \end{aligned}\]

• Partial fraction expansion:

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Reference

• Seoultech, Robot sensor application