ir2014_20141020

University of Hamburg Department of Informatics

2.3 Fundamentals - Sensor characteristics 64-424 Intelligent Robotics

Outline2. Fundamentals

IntroductionSensor data acquisitionSensor characteristicsLiterature

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2.3 Fundamentals - Sensor characteristics 64-424 Intelligent Robotics

Sensor characteristics

An input signal might need to be converted multiple times untilthe sensor emits an electrical output signal

Example: Pressure on a fibre optic sensor1. Elongation occurs2. Refraction index changes3. Optical transmission properties change4. Photon flux is measured5. Electrical signal is output

We will consider the sensor to be a "black box" and will only lookat the relation between the input and output signal

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2.3.1 Fundamentals - Sensor characteristics - Transfer function 64-424 Intelligent Robotics

Transfer function

I The transfer function of a sensor represents the relationbetween stimulus and output quantity

I Each sensor has an ideal or theoretical relation between inputand output signal

DefinitionThe ideal relation between input and output signal of a sensor ischaracterized by the transfer function S = f (s)

I The output signal S represents the true value of the inputsignal s

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Transfer function (cont.)

I The ideal relation is true in case of an ideal design, materialand manufacture process

I Usually,I manufacturing accuracy,I material defects,I environmental influences,I wear and tear,I etc.

a�ect the ideal relation between stimulus and output signalI The actual relation is called: real transfer function

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I In most cases, the relation between stimulus and output signalof a sensor is one-dimensional and linear

Linear transfer function

S = a + b · s

I a is the output signal at an input signal of s = 0I b is the slopeI b is often called sensitivity

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Other possible transfer functions areI Logarithmic transfer function:

S = a + k · ln s

I Exponential transfer function:

S = a · eks

I Polynomial transfer function:

S = a0 + a1 · sk

(or any other polynomial equation of higher order)52


2.3.2 Fundamentals - Sensor characteristics - Regression 64-424 Intelligent Robotics

Approximation vs. Interpolation

A measurement series should be approximated using the simplestpossible function p(x)I Approximation:

The function p(x) shows a very good representation of thevalue pairs (xk , yk) (e.g. minimum mean square error)

p(xk) = yk does not need to be validI Interpolation:

The function p(x) shows an exact representation of the valuepairs

p(xk) = yk ; k = 1, 2, . . . , n must be valid

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Approximation of a transfer function

General problem: Measurement of a relation between twoquantities x and yI The easy case: Linear relation of x and y (e.g. voltage and

current on a resistor)

y = f (x) = a · x + b

I Coe�cients are calculated through linear regressionI In order to reduce the statistical error an adequate number of

measurements should be acquired

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Approximation of a transfer function (cont.)

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I Another quantity specifying the relation between x and y is theempirical correlation coe�cient rxy :

rxy =

Pni=1(xi � x)(yi � y)pPn

i=1(xi � x)2 Pni=1(yi � y)2

I The value range of the correlation coe�cient is from �1 to 1I The closer rxy is to either �1 or 1, the stronger the

corresponding linear dependency

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If the distribution of value pairs closely resembles a parabola, thetransfer function should be approximated through quadraticregressionI Quadratic relation of x and y

y = f (x) = ax2 + bx + c

I The vertical distance vi of the i-th value pair of this parabolaamounts to

vi = yi � f (xi) = yi � ax2i � bxi � c

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S(a; b; c) =nX

i=1(yi � ax2

i � bxi � c)2 �! Minimum

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I Some non-linear transfer functions are linear in a limitedinterval

I Therefore non-linear transfer functions can be approximated bymultiple linear functions

I The di�erence between the true and the linearly approximatedoutput signal should remain within a defined range

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I Many non-linear transfer functions can be reduced to a linearform through transformation

Example: exponential function

y = f (x) = a · ebx

Using the logarithm function we get:

ln y = ln(a · ebx ) = ln a + ln(ebx ) = ln a + bx

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I After linearization measured value pairs must be transformedbefore doing linear regression

I Being computationally simple this transformation is popular butit does not lead to the exact parameters a, b, . . .

I Only through minimization of the actual objective function canthe parameters be determined exactly

S(a; b; . . . ) =nX

i=1(yi � f (xi))

2

I In most cases a high computational e�ort is unavoidable inorder to determine the exact parameters numerically

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2.3.3 Fundamentals - Sensor characteristics - Multi-dimensional transfer functions 64-424 Intelligent Robotics

Multi-dimensional transfer functions

I The transfer function may depend on more than one stimulusExample: Infrared heat radiation sensor

U = G(T 4b � T 4

s ) (Stefan � Boltzmann � Law)

I G – constantI Tb – absolute temperature of the measured objectI Ts – absolute temperature of the sensor surfaceI U – output voltage

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2.3.3 Fundamentals - Sensor characteristics - Multi-dimensional transfer functions 64-424 Intelligent Robotics

Multi-dimensional transfer functions (cont.)

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2.3.4 Fundamentals - Sensor characteristics - Real transfer function 64-424 Intelligent Robotics

Real transfer function

I Compared to the ideal sensor model, real sensors are alwaysinaccurate

I Therefore, the transfer function of a real physical sensor iscalled: real transfer function

I Problem: Unlike the ideal transfer function the real transferfunction is usually neither linear nor monotonous

I Reasons: Di�erences in material and manufacturing process,design flaws, tolerances in production, . . .

I Nevertheless: Each sensor should work within the specifiedprecision

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Real transfer function (cont.)

I Allowed deviation from the ideal transfer function: ±�

I Deviation between ideal and real transfer function: ±�

� �

Example: Stimulus xI Ideal transfer function: y = fideal(x)I Real transfer function: y 0 = freal(x)

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Real transfer function (cont.)

Attention:If the ideal transfer function is used to map from the result y 0 tothe stimulus, the results are x 0 and � = x � x 0

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2.3.5 Fundamentals - Sensor characteristics - Full scale input/output 64-424 Intelligent Robotics

Span / Full scale input

DefinitionThe dynamic range of a stimulus which is converted by a sensor iscalled span or full scale input (FSI)

I The full scale input is specified as a relation between maximumand minimum input values

I It quantifies the lowest and highest possible value for a stimulusI An input signal outside the span may cause an unacceptably

high inaccuracy at best and damage the sensor at worst

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2.3.5 Fundamentals - Sensor characteristics - Full scale input/output 64-424 Intelligent Robotics

Full scale output

DefinitionThe full scale output (FSO) of a sensor is the interval of theoutput signal for the smallest and largest stimulus value within thespecified span

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2.3.6 Fundamentals - Sensor characteristics - Precision 64-424 Intelligent Robotics

Precision

I An important characteristic of a sensor is its precision or ratherits imprecision

I The precision describes the maximum deviation betweentheoretically ideal values and the ones output by the sensor

I Every measurement is a�ected by systematic and random errors

) see section Fundamentals - Sensor data acquisition

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Calibration errors

I Manufacturers calibrate new sensors after productionI The result is a systematic error: the calibration errorI The output of the sensor is shifted by a constant value for each

stimulusI This error is not necessarily evenly distributed across the span

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Calibration errors (cont.)

Example: Simple calibration procedureI A sensor has a linear transfer function, . . .I . . . but the slope of each manufactured sensor might be slightly

di�erent due to material fluctuations

I The manufacturer determines the slope through:I Application of two stimuli s1 and s2 to the sensorI Measurement of the corresponding output signals S1 and S2I Calculation of the slope based on the obtained value pairsI Problem: Due to measurement errors, the slope will deviate

from the real one if the pool of measured value pairs is chosentoo small

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Calibration errors (cont.)

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2.3.7 Fundamentals - Sensor characteristics - Hysteresis 64-424 Intelligent Robotics

Hysteresis error

DefinitionA hysteresis error is the deviation of the output signal for a certainstimulus value, depending on the direction that value is beingapproached from

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2.3.7 Fundamentals - Sensor characteristics - Hysteresis 64-424 Intelligent Robotics

Hysteresis error (cont.)

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2.3.8 Fundamentals - Sensor characteristics - Saturation 64-424 Intelligent Robotics

Saturation

I Nearly every sensor has a limited operating rangeI Many sensors have a linear transfer function, . . .I . . . but starting from a certain stimulus value, the desired

output is no longer generatedI That e�ect is called saturation

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2.3.8 Fundamentals - Sensor characteristics - Saturation 64-424 Intelligent Robotics

Saturation (cont.)

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2.3.9 Fundamentals - Sensor characteristics - Repeatability 64-424 Intelligent Robotics

Repeatability

I A sensor may produce di�erent output values under the sameconditions

I This type of error is called repeatabilityI Repeatability is usually determined as: Maximum distance � of

two output signals for the same input signalI Repeatability is specified proportionately to the full scale input

�r =�

Span · 100%

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2.3.9 Fundamentals - Sensor characteristics - Repeatability 64-424 Intelligent Robotics

Repeatability (cont.)

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2.3.10 Fundamentals - Sensor characteristics - Dead band 64-424 Intelligent Robotics

Dead band

A sensor has a dead band, if it outputs the same signal (usually 0)in a coherent range of the input signal

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2.3.11 Fundamentals - Sensor characteristics - Resolution 64-424 Intelligent Robotics

Resolution

DefinitionThe resolution is the smallest possible change of the stimulus thatis still detected by the sensor

I Examples: Potentiometer (resistance), laser range finder(angle), . . .

I The resolution may vary over the entire spanI The resolution of digital output is defined by the number of

bits (e.g. Audio: 8bit/16bit/. . . )I If the sensor does not have distinct resolution steps, it is said to

have a continuous or infinitesimal resolution80


2.3.12 Fundamentals - Sensor characteristics - Dynamic characteristics 64-424 Intelligent Robotics

Dynamic characteristics

I Previously mentioned characteristics describe a sensor’sbehavior for static input signals

I Variation of the input signal invalidates some of the presentedcharacteristics

I Reason: The sensor does not always provide an immediateresponse to the stimulus

I Therefore, a sensor does not always immediately output asignal corresponding to the stimulus

I Such e�ects are called the dynamic characteristics of a sensorI The associated errors are called dynamic errors

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Damping

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Damping factor

For oscillating cases a damping factor fd can be determined:

Definition

fd =FA =

AB =

BC = etc .

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2.3.13 Fundamentals - Sensor characteristics - Environmental factors 64-424 Intelligent Robotics

Environmental factors

I Ambient temperature (minimum and maximum)I Ambient air humidity (minimum and maximum)I Short- and long-term stability (drift) ) Long term drift may be

increased through pre-agingI Static and dynamic changes of electromagnetic fields,

gravitational forces, vibrations, radiation etc.I Self-heating (e.g. due to flow of current)

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2.3.14 Fundamentals - Sensor characteristics - Further sensor characteristics 64-424 Intelligent Robotics

Further sensor characteristics

I Reliability, e.g. mean time between failure (MTBF)I Certain properties important for the field of application:

I DesignI WeightI Form factorI PriceI . . .

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2.4 Fundamentals - Literature 64-424 Intelligent Robotics

Literature list

[1] Jacob Fraden.Handbook of Modern Sensors: Physics, Designs, andApplications, chapter 1+2, pages 1–52.Springer New York, 4. edition, 2010.

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3 Rotation / Motion 64-424 Intelligent Robotics

Outline1. Introduction2. Fundamentals3. Rotation / Motion

Optical encoderIncremental encoderRotary (angle) encoderResolverEncoder applicationsTachometerGyroscopeLiterature

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3.1 Rotation / Motion - Optical encoder 64-424 Intelligent Robotics

Optical encoder

I Optical encoders are based on a mask with transparent andopaque areas

I A ray of light is cast onto the mask and is registered by aphoto-resistor located on the opposite side

I The mask pattern is usually manufactured as a strip or a diskI Using a strip, a measurement of translation is obtained (change

of distance)I Using a disk, a measurement of rotation is obtained (change of

angle)I Observation of measurement values with regard to time yields a

measurement of velocity (linear/angular velocity)

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3.2 Rotation / Motion - Incremental encoder 64-424 Intelligent Robotics

Incremental encoder

I The mask of an incremental encoder consists of equidistant,transparent and opaque areas equal in size

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Incremental encoder (Function principle)

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Incremental encoder (Structure)

I In most cases, an incremental encoder uses infraredlight-emitting diodes

I Receivers operate in the spectral range of 820nm to 960nmI Patterned disks are usually made out of laminated plasticI Advantages: The disks are light-weight, have a low inertia

moment and are resistant to shock and vibrationI Disadvantage: The temperature window for reliable operation

is quite narrowI Disks intended for high temperature use cases are

manufactured from perforated metal

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3.2.1 Rotation / Motion - Incremental encoder - Schmitt trigger 64-424 Intelligent Robotics

Schmitt trigger

I A Schmitt trigger is often used for:I Debouncing of switchesI Shaping/conditioning of signals

I It converts an analog input signal into a square-wave signalI The output voltage UO flips to UO(high) when reaching an input

voltage of UI = UONI If the input voltage decreases to UI = UOFF , the output

voltage flips back to UO(low)

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Schmitt trigger (Voltage curve)

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Schmitt trigger (Characteristic curve)

I The transfer characteristic shown in the figure is called voltagehysteresis or switching hysteresis

I It determines/defines the levels of input voltage on which theoutput voltage changes to the maximum/minimum value

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Schmitt trigger (Further applications)

I Transmission of digital signals over long distance channels(signal re-shaping)I Long distance transmission leads to unclean signal edges

(low-pass characteristic)I Schmitt trigger regenerates signal edges and signal levels

I Clock generator/oscillator:I Combination of Schmitt trigger, resistor and capacitor (RC

element)

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3.2.2 Rotation / Motion - Incremental encoder - Detection of rotation direction 64-424 Intelligent Robotics

Detection of rotation direction

I Using two light-emitting diodes and photo-resistors, thedirection of a rotation can be detected

I If the disk is being rotated clockwise (CW), signal A leadsI If the disk is being rotated counter-clockwise (CCW), signal B

leads

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3.3 Rotation / Motion - Rotary (angle) encoder 64-424 Intelligent Robotics

Rotary (angle) encoder

I Contrary to an incremental encoder, a rotary (angle) encoderprovides absolute angles as its output signal

I Rotary encoders use disks with a binary encoded patternI Several light-emitting diodes and photo-resistors are used to

scan the diskI One unique binary code word is allocated to each resolution

stepI Resolution directly a�ects the measurement accuracy of the

rotary encoder

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Patterned disks for rotary encoders

5 bit binary code = 32 resolution steps

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Patterned disks for rotary encoders (cont.)

10 bit binary code = 1024 resolution steps

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Rotary (angle) encoder (Function principle)

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Rotary (angle) encoder (Output signal)

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Rotary (angle) encoder (Applications)

Rotary encoders are used within systems that require absoluteprecision and cannot a�ord re-calibration proceduresI Robotic manipulatorsI Positioning systems

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3.4 Rotation / Motion - Resolver 64-424 Intelligent Robotics

Resolver

I A resolver is another option to measure absolute anglesI The design configuration of a resolver corresponds to that of a

double-stranded rotating field machine

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Resolver (cont.)

I The rotor winding of a resolver is usually attached to the motorshaft

I The rotor winding is powered with the alternating voltageUR1,R2 using brushes, abrasive rings, etc.

I The exciter field induces a voltage into the stator windings US1and US2

I This voltage shows a phase shift 'S as opposed to theexcitation voltage

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Resolver (cont.)

I Resolvers in servo drives use high supply frequencies rangingfrom 5 kHz to 20 kHz

I The rotation angle is determined based on the resolver signalsusing basic trigonometric relations

I The angular position ✏ is calculated from the amplitudes ↵1and ↵2:

Angular position

✏ = arctan ↵1↵2

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Resolver (cont.)

I Using micro-controllers the output signals of the resolver areusually sampled directly

I Resolvers are inexpensive transducers, that are used whenmoderate requirements regarding motion dynamics and angularaccuracy apply

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