Research Article A Method for Measurement of Absolute ...downloads.hindawi.com/journals/js/2015/503852.pdf · Research Article A Method for Measurement of Absolute Angular Position

Research ArticleA Method for Measurement of Absolute Angular Position andApplication in a Novel Electromagnetic Encoder System

Zijian Zhang1 Yangyang Dong2 Fenglei Ni3 Minghe Jin3 and Hong Liu3

1School of Software Engineering East China Normal University and School of Mechatronics EngineeringHarbin Institute of Technology Heilongjiang 150001 China2School of Mechanical Engineering Shanghai University of Engineering Science and Shenzhen Graduate SchoolHarbin Institute of Technology Heilongjiang 150001 China3State Key Laboratory of Robotics and System Harbin Institute of Technology (HIT) JQR Building High Technology Park of HITYikuang Street Nangang District Harbin Heilongjiang 150001 China

Correspondence should be addressed to Yangyang Dong dongyang1314126com

Received 14 July 2014 Accepted 4 April 2015

Academic Editor Aldo Minardo

Copyright copy 2015 Zijian Zhang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

For the encoders especially the sine-cosine magnetic ones a new method to measure absolute angular position is proposed inthe paper In the method the code disc of the encoder has only two circle tracks and each one was divided into 119873 and (119873 minus 1)equal code cells The cell angles changing from 0∘ to 360∘ between any two neighboring code cells are defined to represent anyposition on the code disc The position value of the same point can be represented by different cell angle values of different tracksand the absolute angular position of the point can be obtained by the difference value between the cell angle value of the outer trackand the inner one To validate the correctness of the method theoretically the derivation process of the method was provided Anelectromagnetic encoder system was designed and the experimental platform was established to test the methodThe experimentalresults indicate that the electromagnetic encoder can measure the absolute angular position Besides it shows that the method iseasy to be realized in algorithm and can reduce computational complexity and decrease dimension of the encoder

1 Introduction

Absolute angular position measurement is very importantin industrial applications and robotic systems [1] There aremany different methods for the absolute position sensorsespecially the encoders

For the optical encoders based on the arrangement formsof the photodetectors the methods can be divided into twotypes In the first type of the encoding method the pho-todetectors are arranged along the radial direction Onemethod of this type is the natural binary encoding methodwhich makes it easy for the encoders to obtain the absoluterotating angle However the encoders are prone to readingerrors especially the cross errors since more bits maychange between adjacent scale sectors [2]The gray encodingmethod another effective way to measure absolute angularposition is widely used in the optical encoders [3] Thismethod can eliminate the cross errors introduced by the

natural binary encoding method A shortcoming of thismethod is that it cannot read the angle value directly as thesignals should be translated into the natural binary code Asthe higher resolution of the encoder the more tracks arerequired in these twomethodsTherefore the encoders usingthis type of method are relatively large and complex [4] Inthe other type of the encodingmethod all the photodetectorsare arranged along the circumferential direction The matrixencoding is one method of this type According to it thecode disc of the encoder is divided into different fan-shapedareas Besides many reading heads lying in the same trackbut different areas are used to represent different bits ofthe position information The dimension of the encodercan be largely reduced However measuring errors causedby installation errors of the encoders will be larger thanthe other two methods mentioned above [5] The methodof pseudorandom encoding is used in [6] to measure theabsolute angular position In this method the encoder is

Hindawi Publishing CorporationJournal of SensorsVolume 2015 Article ID 503852 10 pageshttpdxdoiorg1011552015503852

2 Journal of Sensors

composed of the synchronization code track and the indexcode track which helps to decrease the size of the encoder [7]However the numbers of the reading heads and the slits willincrease exponentially with improving measuring precisionof the encoder Therefore the method is rarely used in thehigh-bit encoders The vernier-type encoders developed in[4 8] have two or three tracks on the code disc tomeasure theabsolute rotating position which are simple in constructionand compact with other ones However the method formeasuring absolute position angle should know the numbersof the slits that the encoder has rotated around which ishardly to be obtained in the real application [9] The M-codecoding method proposed in [10] can avoid gross errors and isuseful for minimizing the physical size of encoders Howeverto achieve higher resolution absolute encoders using M-code coding method need to incorporate a small slit pitchwhich prevents the encoderrsquos photodetector from obtainingthe fixed signal amplitude [11] With respect to the quasi-absolute encoding method the code disc of the encoder iscomposed of a cyclic code track and an index code track Allthe effective coding bits of the position are listed on the indexcode track Therefore there should be many photodetectorsalong the circumferential direction The method is useful forminimizing the dimension of the encoder but it needs a boot-strap process to obtain the first position code [12] Thereforein some extent it does not belong to the absolute positionmeasurement method Using the second-type method thedimension of the encoders can be largely reduced comparedwith the other two methods of the first type However thereare many photodetectors needed in these methods which isanother limitation to increase the measuring resolution anddecrease the dimensions of the encoders Comparingwith theoptical encoders the number of the absolute angular positionmeasurement methods of the magnetic encoders is smallFor the magnetic encoders with hollow shaft the encodertypically supports two output channels (channels A and B)which are dephased by 90 degrees with each other As theZ phase generated once per circle is used to produce thereference point or the zero point in the magnetic encodersmost types of the magnetic encoders are the relative positionsensors Although themultipole ones were studied in [13] themultipoles ones are difficult to bemagnetized [14] and tomeetthe requirements of small-size high-resolution and absoluteangle detection [15] For the type of the magnetic encoderswith shaft [16 17] only one circle of sine and cosine signalswill be generated and therefore the absolute angular positioncan be easily obtained by the inverse trigonometric functionsHowever this type of sensor cannot be fixed on the deviceswith hollow shaft such as the robot arms

Based on the analysis of the methods used in the opticalencoders developing an absolute angular position measure-mentmethodwhich can decrease dimensions of the encodersand the numbers of the reading heads is necessary Moreoverif the method can be used not only in optical encoders butalso in the magnetic or the electromagnetic encoders it ismuch better Based on this a novel absolute angular positionmeasurement method which can be widely used in differenttypes of encoders is proposed in the paper Without loss ofgenerality in the paper the attention will only focus on the

electromagnetic angular encoders in the validation sectionThe structure of the paper is organized as follows for a properanalysis of the method physical modeling and mathematicalvalidation process are required Therefore the paper startswith derivation process of the method in Section 2 Theanalytical results have been experimentally verified usinga novel electromagnetic sine-cosine encoder system andthe implementation details of the validation systems includ-ing the sensor system and the experimental platform arediscussed in Section 3 whereas the experimental data arepresented in Section 4 Final comments and conclusions arestated in Section 5

2 Derivation Process of the Method

According to the measurement method applied in the opticalencoders the method proposed in the paper to measureabsolute rotating angle needs the code disc of the encoder tohave two tracks Unlike other methods mentioned above thetwo tracks of the encoder can be divided into119873 and (119873 minus 1)equal code cells respectively Angle values of any point on thedisc are represented by the cell angle values which are definedin the paper instead of the unique binary codes used in otherencoders In each code cell the code angles are defined tochange from 0 to 360 degrees and any point within the samecode cell can be uniquely represented by this angle valueTherefore any point can be represented by two different anglevalues The absolute angular position can be easily obtainedfrom the difference values between code angle values of thesame point on different tracks If any point is representedby the same angle value of different tracks it is the absolutezero point Physical modeling and mathematical validationprocess are shown as follows

21 Physical Modeling of the Method As shown in Figure 1there are two circles that is 119874

1and 119874

2 rotating around

the same axis and they have the same absolute zero point119874 However there are 119873 relative zero points which are11986011 11986012 119860

1119873listed uniformly on the circle 119874

1 while

the circle 1198742is divided equally into (119873 minus 1) parts by the

relative zero points 11986021 11986022 119860

2(119873minus1) 1205791 and 1205792are the

angle values of 1198751and 119875

2relative to the nearest relative zero

points along the rotating direction In the model the cellangle values changing range between any two neighboringrelative zero points is defined from 0 to 360 degrees

Therefore the positions of any points 1198751and 119875

2on these

two circles relative to the absolute zero point 119874 can beexpressed as follows

1205791198751=1205791119873

+1198731 times2120587119873

+1205790

1205791 1205791198751 1205790 isin [0 2120587) 119873 = 1 2 3 119899 1198731 = 0 1 2 3 (119873 minus 1)(1)

1205791198752=

1205792(119873 minus 1)

+1198732 times2120587

(119873 minus 1)+ 1205791015840

0

1205792 1205791198752 1205791015840

0

isin [0 2120587) 119873 = 1 2 3 119899 1198732 = 0 1 2 (119873 minus 2) (2)

where1198731and119873

2are the numbers of the relative zero points

between 119874 and 1198751and 119874 and 119875

2 respectively 119875

1and 119875

2are

Journal of Sensors 3

(b) (c)(a)

O1

Axis

O

P1P2

O2

O

P1

1205791

O1

A11

A12

1205790

middot middot middot middot middot middot

O

O2

P2

1205792

A21

A22

A1(N1minus1)

A1(N1+1)

A2(N2minus2)

A2(N2minus1)

A2(N2+1)

120579P1120579P2

A1N1

A1N1 A2N2

1205799984000

Figure 1 Two circles rotating around the same axis

two different points on these two circles 1205791198751

and 1205791198752

arethe absolute rotating angle values of the points 119875

1and 119875

2

1205791and 1205792are angle values of these points 119875

1and 119875

2 relative

to the nearest relative zero points 11986011198731

and 11986021198732

along therotating direction respectively

Based on the model if 1198751and 119875

2have the same rotating

angle values relative to the absolute zero point119874 and 1205790= 1205791015840

0

the absolute rotating angle position of any point on these twocircles can be calculated as follows

1205791198751

= 1205791198752

=

1205791 minus 1205792 if (1205791 minus 1205792) ge 0

1205791 minus 1205792 + 2120587 if (1205791 minus 1205792) lt 0

1205791 1205792 1205791198751 1205791198752 isin [0 2120587)

(3)

22Mathematical Validation of theMethod According to thephysical model above if 119875

1and 119875


angle values relative to the absolute zero point119874 1205791198751

is equalto 1205791198752 Therefore (1) is equal to (2) Consider

1205791198751=1205791119873

+1198731 times2120587119873

+1205790 = 1205791198752

=1205792

(119873 minus 1)+1198732 times

2120587(119873 minus 1)

+ 1205791015840

0

(4)

Then1205791 minus 1205792

=

1205791119873

+1198731 times2120587119873

minus 2120587 (1198731 minus 1198732) + (119873 minus 1) (12057910158400

minus 1205790)

1205792119873 minus 1

+21205871198731119873 minus 1

+2120587119873 (1198732 minus 1198731)

119873 minus 1+ 119873(120579

1015840

0

minus 1205790)

(5)

Taking (1) and (2) into (5)

1205791198751= 1205791 minus 1205792 + 2120587 (1198731 minus1198732) + (119873minus 1) (1205790 minus 120579

1015840

0

)

1205791198752= 1205791 minus 1205792 + 2120587 (1198731 minus1198732) +119873 (1205790 minus 120579

1015840

0

)

(6)

As known from the physical model established in the firstpart there are 119873 and (119873 minus 1) relative zero points on these

two different circles and1198731(1198732) is the number of the relative

zero points between any point 1198751(1198752) and the absolute zero

point119874Therefore there are two different numerical relation-ships between119873

1and119873

2as1198731= 1198732and119873

1= (1198732+ 1)

(1) When1198731= 1198732 (6) can be simplified as

1205791198751= 1205791 minus 1205792 + (119873minus 1) (1205790 minus 120579

1015840

0

)

1205791198752= 1205791 minus 1205792 +119873(1205790 minus 120579

1015840

0

)

(7)

(2) When1198731= (1198732+ 1) (6) can be simplified as

1205791198751= 1205791 minus 1205792 + 2120587+ (119873minus 1) (1205790 minus 120579

1015840

0

)

1205791198752= 1205791 minus 1205792 + 2120587+119873(1205790 minus 120579

1015840

0

)

(8)

If 1205790= 1205791015840

0

then

1205791198751= 1205791198752=

1205791 minus 1205792 when 1198731 = 1198732

1205791 minus 1205792 + 2120587 when 1198731 = 1198732 + 1(9)

where

1205791 1205792 1205791198751 1205791198752 isin [0 2120587)

119873 = 1 2 3 119899

1198731 = 0 1 2 (119873minus 1)

1198732 = 0 1 2 (119873minus 2)

(10)

The form of (9) is much similar to (3) However theestablishing conditions of the equations are different fromeach other Therefore in the next step we should find rela-tionships among119873

11198732and 1205791 1205792

According to the physical model there are two circleswhich have119873 and (119873minus1) relative zero pointsThe numericalrelationships between 120579

1(1205792) and 120579

1198751(1205791198752) can be established

in the following two equations


1205791 =

1198731205791198751 1198731 = 0 120579

1198751 isin [0 2120587119873

)

1198731205791198751 minus 2120587 1198731 = 1 120579

1198751 isin [2120587119873

4120587119873

)

1198731205791198751 minus 2 (119873 minus 1) 120587 1198731 = (119873 minus 1) 120579

1198751 isin [2 (119873 minus 1) 120587

119873 2120587)

(11)

1205792 =

(119873 minus 1) 1205791198752 1198732 = 0 120579

1198752 isin [0 2120587119873 minus 1

)

(119873 minus 1) 1205791198752 minus 2120587 1198732 = 1 120579

1198752 isin [2120587

119873 minus 1

4120587119873 minus 1

)

(119873 minus 1) 1205791198752 minus 2 (119873 minus 2) 120587 1198732 = (119873 minus 2) 120579

1198752 isin [2 (119873 minus 2) 120587

119873 minus 1 2120587)

(12)

From (11) minus (12) we get

(1205791 minus 1205792) =

1205791198751 gt 0 1198731 = 1198732 = 0 120579

1198751 isin [0 2120587119873

)

(1205791198751 minus 2120587) lt 0 1198731 = 1198732 + 1 = 1 120579

1198751 isin [2120587119873

2120587

119873 minus 1)

1205791198751 gt 0 1198731 = 1198732 = 1 120579

1198751 isin [2120587

119873 minus 14120587119873

)

(1205791198751 minus 2120587) lt 0 1198731 = 1198732 + 1 = 2 120579

1198751 isin [4120587

119873 minus 14120587119873

)

(1205791198751 minus 2120587) lt 0 1198731 = 1198732 + 1 = (119873 minus 1) 120579

1198751 isin [2 (119873 minus 1) 120587

119873 2120587)

(13)

Therefore the conclusion about the common conditionwhich is similar to (13) is generated Consider

1198731 = 1198732 997904rArr 1205791 gt 1205792

1198731 = 1198732 + 1 997904rArr 1205791 lt 1205792(14)

From (14) the numerical relationship between1198731and119873

2

is the sufficient condition to the relationship between 1205791and

1205792 However the relationship between 120579

1and 1205792cannot derive

relationship of1198731and119873

2 Therefore it should be done in the

following stepsAs 1205790

= 1205791015840

0

is assumed in the conclusion (5) can bechanged to the following form

119873(1205791 minus 1205792) = 2120587119873 (1198732 minus1198731) + 21205871198731 + 1205791 (15)

If 1205791ge 1205792 then

2120587119873 (1198732 minus1198731) + 21205871198731 + 1205791 ge 0 (16)

It is known from the physical model that the numericalrelationships between119873

1and119873

2are1198731= 1198732and119873

1= 1198732+

1

If1198731= 1198732 (16) changes to the following form

21205871198731 + 1205791 ge 0 (17)

If1198731= 1198732+ 1 then

2120587119873 (1198732 minus1198731) + 21205871198731 + 1205791 = 2120587 (1198731 minus119873)+ 1205791

∵ 1198731 = 0 1 2 3 (119873minus 1)

there4 2120587 (1198731 minus119873)+ 1205791 le 2120587 ((119873minus 1) minus119873) + 1205791

= minus 2120587+ 1205791

∵ 1205791 lt 2120587

there4 2120587119873 (1198732 minus1198731) + 21205871198731 + 1205791 lt 0

(18)

Therefore

1205791 ge 1205792 997904rArr 1198731 = 1198732

1205791 lt 1205792 997904rArr 1198731 = 1198732 + 1(19)


A B C D E F A B C D E F

The sensor andthe encoder

Working principleof the sensor

Output signalsof the sensor

Figure 2 Working principle of the sensor

Therefore from (14) and (19) the following conditions areequivalent

1205791 ge 1205792 lArrrArr 1198731 = 1198732

1205791 lt 1205792 lArrrArr 1198731 = 1198732 + 1(20)

Summarizing all the mathematical validation process ofthe method above (3) has been proved and the method hasbeen validated

3 Validation System of the Method

The validation system will be expatiated in two aspectsworking principle of the sensor and the validation systemincluding the encoder system and the experimental platform

31 Working Principle of the Encoder System Working prin-ciple of the encoder system is the law of electromagneticinduction whichmeans themutual inductance voltage can begenerated under the effect of the changingmagnetic field [18]As the detailed working principle has been presented in [19]a short description will be given Configuration of the highlyintegrated sensor and the code disc of the encoder system canbe seen in the left part of Figure 2

Five copper windings integrated into the sensor areshown in the middle part of the figure The larger one shownin the figure is injected by high frequency signals fromoutside and a time-varying magnetic field can be generatedTherefore the other four small helices located under thelarger winding will generate voltage signals The amplitudesof the voltage signals induced by the four small coils are equalto each other as they have the same dimension and all thedistances between central positions of the small coils and thelarger one are equal to each other Two of them with theopposite rotation directions are connected into a group tomagnify amplitudes of the signals Phase difference betweenvoltage signals generated by each group is 90 degrees as thereis a radius position difference between the correspondingcoils in each group As one of the copper sheets markedon the code disc moving under the four helices the highfrequency voltage signals will be generated As copper sheetsare with a certain dimension and are laid out with regulatedpositions relative to the sensor the regular signals which aresine and cosine signals can be generated if the maximum andminimum values of the signals are selected as shown in the

Table 1 Look-up table of 119875

119904119894119899119875 0 01736 03420 sdot sdot sdot sdot sdot sdot minus01736 1cos119875 1 09848 09396 sdot sdot sdot sdot sdot sdot 09848 0119875 (∘) 0 10 20 sdot sdot sdot sdot sdot sdot 350 360

right part of Figure 2 To be easily described the generatingprocess of the signals can be divided into five steps from Ato E Taking the two red secondary coils for example whenthe copper sheet moves to positions A B and F it has littleinfluence or the same effect as the proceeding one on themagnetic fields of the secondary coilsTherefore the voltagesinduced by the coils are equal but opposite in directionand consequently the output voltage value is equal to zeroHowever as the copper sheet moves to points C and E theoutput sinusoidal signal obtains themaximumabsolute valueAs copper sheet moving to point E the output sinusoidalsignal is equal to zero owing to the fact that the sheet has thesame impact on themagnetic field of the two secondary coils

32 Validation System All the components of the validationsystem can be seen from Figure 3The system is mainly com-posed of five parts the encoder system the driven system thecalibration device the three-dimensional platform and theassistant components Configuration of the validation systemcan be seen in the lower left corner of Figure 3

The encoder system is mainly composed of two sensorsthe signal processing electrical circuit board and the codedisc Working principle of the encoder system has beenillustrated in the section aboveThe other parts including thesignal processing electrical circuit board and the code discwill be described in detailThe part of the system labeledA inthe figure is the electrical circuit board used to process signalsand communicate with the CPU (central processing unit)Each sensor will produce two groups of difference signalswhich are sin119875

1+ sin119875

1minus and cos119875

1+ cos119875

1minus separately

Before converting the analog signals to digital ones throughthe ADC (analog to digital converter) the signals have beenfiltered and changed to single signals from difference onesBesides the signal processing and the absolute position anglecalculation process are undertaken by the DSP (digital signalprocessor) Flow chart of the algorithm can be seen in theDSP part of the figure To easily calculate 119875

1(1198752) a look-

up table (Table 1) is established based on the relationships of


①

1 2 3

Analog signals process circuits

Sensor 1

Sensor 2

Algorithm flow chart

Pouter Pouter

A sin Pouter Acos PinnerA sin PinnerAcos Pouter

Looking upTable 1 Table 1

Looking up

Obtain the absolute position angle 120579

VDD

A sin P1

Acos P2

A sin P2

Acos P1

Analog todigital

converter

SPI

Digital signal processor (DSP)Serial peripheral interface (SPI)

Analog to digital converter (ADC)

Digital signal processor

Power system

+25V

+33V

+125V +33V +18V

+50V

SPISOMISPISIMO

SPISTESPICLK

The encoder

Magneticring

Magneticinductor

Positionof

sensor 1

Positionof

sensor 2

X Y

Z

Pouter minus Pinner

075

mm

33

mm

146 times 2466∘

085mm147 times 2449

∘33mmR265

R31

validation systemConfiguration of the

③

④

② ⑤

VDVV DPower converter (+33V +25V +125V +18V)

(A2)cos P1+

(A2)cos P1minus

(A2)cos P2+

(A2)cos P2minus

(A2)sin P1+

(A2)sin P1minus

(A2)sin P2+

(A2)sin P2minus

Figure 3 Validation system

sin119875(1198751 1198752) cos119875(119875

1 1198752) and 119875(119875

1 1198752) At last the absolute

position angle 119875 should be delivered to the CPU through theSPI (serial peripheral interface) modeThe look-up Table 1 of119875 can refer to Table 1The changing range of 119875 is divided into36 parts equally

The code disc another part of the encoder system islabeled B in Figure 3 Substrate material of the encoder iscopper-clad laminate Two circles of copper sheets are listedon it and the numbers of copper sheets in each track are 177and 176 to satisfy the requirement of the method Besidesthe rotating radiuses of these two sensorsrsquo centers are similarto those of the two circles of copper sheets respectivelyThe detailed parameters of the encoder can be seen in B inFigure 3

PartC of the system is the driven system As shown in thefigure the three main parts of the driven system includingthe controller board the driver board and the motor arenumbered sequentially from 1 to 3 in the figure The drivensystem can realize the purpose of position control speedcontrol and current control The code disc of the encodersystem is fixed on the driven system through the other

connection components to have the relative rotation with theelectrical circuit board

In fact part D a relative magnetic sensor used to cal-ibrate the absolute electromagnetic position sensor systemhas been fixed into the driven system for the purpose ofreducing dimensions of the system The resolution of themagnetic sensor is 14 bits and measuring precision is 00001degrees which can satisfy requirements of calibrating theelectromagnetic sensor system

To regulate the relative positions between the code discand the sensors the electrical circuit board has been fixed ona three-dimensional platform labeled E in the figure It canmove along 119909-axis 119910-axis and 119911-axis to ensure the coppersheets and the sensors have the same rotating axis Theresolution of the three-dimensional platform is 001mm

4 Experimental Results and Analysis

Based on the system established above the experiment hasbeen done All the original signals of these two sensorsmonitored by oscillograph can be seen in Figure 4 Thereare many cycles of signals shown in the upper window of


Period

Period

sin 1cos 1sin 2cos 2

Figure 4 Testing results monitored by oscillograph

the figure while the below one is the amplified window of allthe signals CH1 CH2 CH3 and CH4 represent sin1 cos1sin2 and cos2 respectively

There are 90 degrees of phase differences between thesignals generated by the same sensor such as sin1 and sin2Therefore according to the character of the inverse trigono-metric function the angle values of any point can be cal-culated and consequently the angular value curves can beobtained

Figure 5 gives us an illustration to the generating processof the absolute angular position The left part of the figureis the whole generating process of position while the rightpart is the amplification window to be easily observed InFigure 5(a) all the sine and cosine signals are illustratedand the amplitudes of the signal values are transformed tovary from minus1 to 1 through signal processing part of thesystem including hardware and software Figure 5(b) givesus an illustration to all the angle curves after the rotatorrotates 360 degrees The blue one has 177 circles and the redone has 176 circles Absolute angular position can be easilyobtained as shown in Figure 5(c) The blue one representsthe magnetic sensorrsquos angular position curve while the redone illustrates angle values measured by the electromagneticsensor system respectively To calibrate the electromagneticencoder amagnetic sensor has been fixed in the experimentalset-up In the experiment angle values measured by themagnetic sensor are defined to be the ideal values Thereforecomparing with results measured by the magnetic sensorthe errors of the electromagnetic encoder can be obtained asshown in Figure 5(d)

From Figure 5(d) it is easy to obtain that the measuringerrors of the electromagnetic sensor are less than 1∘ which isrelatively larger than other sensors Many different reasonscan lead to this From (3) the calculation equation of theabsolute angular position value in the electromagnetic sensorcan be expressed as follows

120579119875=

120579outer minus 120579inner if 120579outer ge 120579inner

120579outer minus 120579inner + 2120587 if 120579outer lt 120579inner

120579outer 120579inner 120579119875 isin [0 2120587)

(21)

According to the inverse trigonometric function 120579outerand 120579inner can be calculated by sine and cosine signals usingarctangent function which can be seen in the followingequation

120579change = 120579outer minus 120579inner

= arctan( sin 1cos 1

)minus arctan( sin 2cos 2

)

(22)

Therefore the absolute angular position value can beeasily obtained using the four groups of the original signals

There is no error caused by the method if 120579outer and 120579innerare the theoretic values However in fact the values are notequal to their ideal values which will cause errors to themeasuring precision of the sensor system In this systemthere are mainly two reasons that can lead to this

The first one is the approximation of the arctan functionAs is known the arctan function is not continuous curvesin the interval [0 2120587] and the theoretic values of 120579outer and120579inner cannot be acquired Besides the signals are analog onesand they should be transformed to the digital ones in thesignal processing part which are the approximation valuestoo However all the reasons above cannot be avoided in theapplication and they play limited roles in causing errors ofthe sensor Therefore they can be ignored in the analysis ofthe error causing reasons

The second one is the changes of all the signals includingthe amplitudes and phases Amplitude changes can be easilyobserved from Figure 4 The influence of the reason on theabsolute angular position can be expressed in the followingequation

120579actual = arctan(1198601119878 sin (1205791 + Δ1205791119878)

1198601119862 cos (1205791 + Δ1205791119862))

minus arctan(1198602119878 sin (1205792 + Δ1205792119878)

1198602119862 cos (1205792 + Δ1205792119862))

(23)

In the equation above sin1205791 cos120579

1 sin1205792 and cos120579

2cor-

respond to sin1 cos1 sin2 and cos2 respectively


5300 5320 5340 5360 5380 5400

0

05

1Amplified window of the signals

Sample point

Valu

e (de

g)

minus1

minus05

(a-a)

(b-a)

5300 5320 5340 5360 5380 54000

50100150200250300350400 Angular value curves of the outer circle

Sample point

Valu

e (de

g)

Angular value curves of the outer circleAngular value curves of the inner circle

0 1000 2000 3000 4000 5000 6000 7000

0

05

1Signals of the sensors after processing

Sample point

(a)

Valu

e (de

g)

minus1

minus05

(b)

0 1000 2000 3000 4000 5000 6000 7000 80000


Valu

e (de

g)

0 1000 2000 3000 4000 5000 6000 7000 80000

200400 Angular value curves of the inner circle

177 circles

176 circles

Valu

e (de

g)

Sample point

(c)

0 1000 2000 3000 4000 5000 6000 7000 80000

50100150200250300350400 Angular value curves of the sensors

Sample point

Valu

e (de

g)

The magnetic sensorThe electromagnetic sensor

(c-a)

(d) (d-a)

5300 5310 5320 5330 5340 5350244245246247248249 Amplified window of angle curves

Sample point

Valu

e (de

g)


0 1000 2000 3000 4000 5000 6000 7000 8000

0

05 Errors of the electromagnetic sensor

Sample point

Valu

e (de

g)

Errors

minus05 5300 5310 5320 5330 5340 5350

0

05Amplified window of errors

Sample point

Valu

e (de

g)

Errors

minus05

sin 1cos 1

sin 2cos 2

sin 1cos 1

sin 2cos 2

Figure 5 Testing results


Consequently the errors of the encoder system can beshown in the following equation

Errors = 120579minus 120579actual

= arctan( sin 1205791cos 1205791


)

minus(

arctan(1198601119878 sin (1205791 + Δ1205791119878)

1198601119862 cos (1205791 + Δ1205791119862))

minusarctan(1198602119878 sin (1205792 + Δ1205792119878)

1198602119862 cos (1205792 + Δ1205792119862))

)

(24)

In the equation above the amplitudes of1198601119878and119860

11198621198602119878

and1198602119862

are not equal to each other in some cases The phasedifference of the signals such asΔ120579

1119878andΔ120579

1119862Δ1205792119878andΔ120579

2119862

will exist and the phase of the signals generated by the samesenor is not the same anymore Therefore the measurementerrors will be brought about Besides the angle errors will beamplified if there are errors in the arctan function

The changes of amplitudes and phases are caused bydifferent reasons such as the distance differences between theencoder and two sensors the sensors not perpendicular tothe encoder and the errors of the input voltage Therefore inthe near future the manufacturing and assembly precisionsof the sensor system and the experimental platform shouldbe improved All in all correctness of the method has beenvalidated by the experiment

5 Conclusion

To measure the absolute angular position a method includ-ing physical modeling and mathematical analysis has beenproposed in the paper Besides to validate the method anelectromagnetic encoder system and the testing platformhave been established Comparing the experimental resultsof the electromagnetic encoder with position informationobtained from a magnetic sensor the conclusion that themethod can be used to measure absolute angular position isobtained Some prominent characteristics of the method canbe listed as follows

(1) Using this method the structure of the encoder issimple and easy to be designed Besides the size ofthe encoder can largely be compacted For examplethe encoder system designed in the paper is based onan application in a robot arm The inner diameter isrequired to be 45mm Based on themethod the outerdiameter of the code disc is 70mm and the widthof the encoder system is just 59mm (sensor width09mm encoder disc width 15mm the electricalcircuit board 15mm the highest component 1mmand distance between the code disc and the sensors02mm) It is more compact than the other absolutesensors on the market such as the magnetic encoders

(2) Algorithm of the method is simple and easy to berealized The calculation algorithm can be decreasedas there is no need to change analog signals to digitalones

(3) It is friendly to customers As the exporting signals areanalog ones the suitable interpolation ratio can be setby the customers as they are willing to

(4) The method is suitable to be used in the encodersespecially when their output signals are sine-cosineanalog signals It is because the code angle valuewhich is defined as changing from 0∘ to 360∘ can beeasily obtained if full circles of sine and cosine signalscan be generated within a code cell

Although the purpose of validating the correctness of themethod has been achieved there are many limitations in thepaper and some further works should be done First mea-surement precision of the sensor system is about plusmn05 degreeswhich is low compared with other sensors such as the opticalsensors whose resolution can reach up to 34119890 minus 4 degrees(20 bits) It is mainly caused by manufacturing and assemblyerrors of the encoder system and the testing platform Thefurther work should be focused on improvingmanufacturingand assemblage precisions of all the systems On the otherhand the method is only applied to the electromagneticencoder system exporting sine-cosine signals In fact themethod has no demand for the forms of the signals whetherthey are the analog or the digital onesTherefore it may be analternative choice for the measurement of absolute angularposition used in other encoders such as the optical encodersFor example under conditions of nonincreasing dimensionsand complexity of the senor the measuring precision can beimproved if the encoder of the sensor is divided into severalequal sections This will be addressed in the near future

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The work was sponsored by the National Key Basic ResearchandDevelopment Program (973 Program) andNationalHighTechnology Research and Development Program of China(State 863 project) 2011AA7045041

References

[1] T Reininger F Welker and M von Zeppelin ldquoSensors inposition control applications for industrial automationrdquo Sensorsand Actuators A Physical vol 129 no 1-2 pp 270ndash274 2006

[2] E M Petriu ldquoReconsidering natural binary encoding forabsolute position measurement applicationrdquo IEEE Transactionson Instrumentation and Measurement vol 38 no 5 pp 1014ndash1016 1989

[3] K Jeong J Park and J S Yoon ldquoHigh-precision encoder usingmoire fringe and neural networkrdquo in Optomechatronic Systemsvol 4190 of Proceedings of SPIE pp 1ndash7 2001

[4] T Ueda F Kohsaka T Iino K Kazami and H NakayamaldquoOptical absolute encoder using spatial filterrdquo in Photomechan-ics and Speckle Metrology vol 0814 of Proceedings of SPIE pp217ndash221 San Diego Calif USA August 1987


[5] Z H F Cao ldquoMicro absolute matrix encoderrdquo Optics and FineMechanics vol 5 pp 65ndash70 1985

[6] E M Petriu ldquoAbsolute-type position transducers using a pseu-dorandom encodingrdquo IEEE Transactions on Instrumentationand Measurement vol IM-36 no 4 pp 950ndash955 1987

[7] EM Yeatman P J Kushner andD A Roberts ldquoUse of scanneddetection in optical position encodersrdquo IEEE Transactions onInstrumentation and Measurement vol 53 no 1 pp 37ndash442004

[8] F Kohsaka T Iino K Kazami H Nakayama and T UedaldquoMultiturn absolute encoder using spatial filterrdquo JSME Interna-tional Journal no 1 pp 94ndash99 1990

[9] S Wekhande and V Agarwal ldquoHigh-Resolution absoluteposition Vernier shaft encoder suitable for high-performancePMSM servo drivesrdquo IEEE Transactions on Instrumentation andMeasurement vol 55 no 1 pp 357ndash364 2006

[10] K Fujita T Nakayama and Y Matsuzoe ldquoRecent encodertechnologyrdquo Fuji Electric Review vol 46 pp 57ndash61 2000

[11] YMatsuzoe N Tsuji T Nakayama K Fujita and T YoshizawaldquoHigh-performance absolute rotary encoder using multitrackand M-coderdquo Optical Engineering vol 42 no 1 pp 124ndash1312003

[12] B W Edmister ldquoIndustrial applications of optical shaft encod-ersrdquo Proceedings of the Society of Photo-Optical InstrumentationEngineers vol 255 pp 99ndash105 1980

[13] Y Kikuchi F Nakamura H Wakiwaka H Yamada and YYamamoto ldquoConsideration of magnetization and detection onmagnetic rotary encoder using finite element methodrdquo IEEETransactions on Magnetics vol 33 no 2 pp 2159ndash2162 1997

[14] S-H Jeong S-H Rhyu B-I Kwon and B-T Kim ldquoDesign ofthe rotary magnetic position sensor with the sinusoidally mag-netized permanent magnetrdquo IEEE Transactions on Magneticsvol 43 no 4 pp 1837ndash1840 2007

[15] K Nakano T Takahashi and S Kawahito ldquoA CMOS smartrotary encoder using magnetic sensor arraysrdquo in Proceedings ofthe 2nd International Conference on Sensors (Sensors rsquo03) vol 1pp 206ndash209 IEEE October 2003

[16] S Lozanova and C Roumenin ldquoAngular position device with2D low-noise Hall microsensorrdquo Sensors and Actuators APhysical vol 162 no 2 pp 167ndash171 2010

[17] T Lan Y W Liu M H Jin S W Fan Z P Chen andH Liu ldquoStudy of ultra-miniature giant magneto resistancesensor system based on 3D static magnetic analysis techniquerdquoMeasurement vol 42 no 7 pp 1011ndash1016 2009

[18] G L Pollack and D R Stump Electromagnetism PearsonEducation Upper Saddle River NJ USA 2002

[19] Z Zhang F Ni Y Dong M Jin and H Liu ldquoA novel absoluteangular position sensor based on electromagnetismrdquo Sensorsand Actuators A Physical vol 194 pp 196ndash203 2013

International Journal of

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Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



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Journal of

Advances inOptoElectronics


Volume 2014

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



composed of the synchronization code track and the indexcode track which helps to decrease the size of the encoder [7]However the numbers of the reading heads and the slits willincrease exponentially with improving measuring precisionof the encoder Therefore the method is rarely used in thehigh-bit encoders The vernier-type encoders developed in[4 8] have two or three tracks on the code disc tomeasure theabsolute rotating position which are simple in constructionand compact with other ones However the method formeasuring absolute position angle should know the numbersof the slits that the encoder has rotated around which ishardly to be obtained in the real application [9] The M-codecoding method proposed in [10] can avoid gross errors and isuseful for minimizing the physical size of encoders Howeverto achieve higher resolution absolute encoders using M-code coding method need to incorporate a small slit pitchwhich prevents the encoderrsquos photodetector from obtainingthe fixed signal amplitude [11] With respect to the quasi-absolute encoding method the code disc of the encoder iscomposed of a cyclic code track and an index code track Allthe effective coding bits of the position are listed on the indexcode track Therefore there should be many photodetectorsalong the circumferential direction The method is useful forminimizing the dimension of the encoder but it needs a boot-strap process to obtain the first position code [12] Thereforein some extent it does not belong to the absolute positionmeasurement method Using the second-type method thedimension of the encoders can be largely reduced comparedwith the other two methods of the first type However thereare many photodetectors needed in these methods which isanother limitation to increase the measuring resolution anddecrease the dimensions of the encoders Comparingwith theoptical encoders the number of the absolute angular positionmeasurement methods of the magnetic encoders is smallFor the magnetic encoders with hollow shaft the encodertypically supports two output channels (channels A and B)which are dephased by 90 degrees with each other As theZ phase generated once per circle is used to produce thereference point or the zero point in the magnetic encodersmost types of the magnetic encoders are the relative positionsensors Although themultipole ones were studied in [13] themultipoles ones are difficult to bemagnetized [14] and tomeetthe requirements of small-size high-resolution and absoluteangle detection [15] For the type of the magnetic encoderswith shaft [16 17] only one circle of sine and cosine signalswill be generated and therefore the absolute angular positioncan be easily obtained by the inverse trigonometric functionsHowever this type of sensor cannot be fixed on the deviceswith hollow shaft such as the robot arms

Based on the analysis of the methods used in the opticalencoders developing an absolute angular position measure-mentmethodwhich can decrease dimensions of the encodersand the numbers of the reading heads is necessary Moreoverif the method can be used not only in optical encoders butalso in the magnetic or the electromagnetic encoders it ismuch better Based on this a novel absolute angular positionmeasurement method which can be widely used in differenttypes of encoders is proposed in the paper Without loss ofgenerality in the paper the attention will only focus on the

electromagnetic angular encoders in the validation sectionThe structure of the paper is organized as follows for a properanalysis of the method physical modeling and mathematicalvalidation process are required Therefore the paper startswith derivation process of the method in Section 2 Theanalytical results have been experimentally verified usinga novel electromagnetic sine-cosine encoder system andthe implementation details of the validation systems includ-ing the sensor system and the experimental platform arediscussed in Section 3 whereas the experimental data arepresented in Section 4 Final comments and conclusions arestated in Section 5

2 Derivation Process of the Method

According to the measurement method applied in the opticalencoders the method proposed in the paper to measureabsolute rotating angle needs the code disc of the encoder tohave two tracks Unlike other methods mentioned above thetwo tracks of the encoder can be divided into119873 and (119873 minus 1)equal code cells respectively Angle values of any point on thedisc are represented by the cell angle values which are definedin the paper instead of the unique binary codes used in otherencoders In each code cell the code angles are defined tochange from 0 to 360 degrees and any point within the samecode cell can be uniquely represented by this angle valueTherefore any point can be represented by two different anglevalues The absolute angular position can be easily obtainedfrom the difference values between code angle values of thesame point on different tracks If any point is representedby the same angle value of different tracks it is the absolutezero point Physical modeling and mathematical validationprocess are shown as follows

21 Physical Modeling of the Method As shown in Figure 1there are two circles that is 119874

1and 119874

2 rotating around

the same axis and they have the same absolute zero point119874 However there are 119873 relative zero points which are11986011 11986012 119860

1119873listed uniformly on the circle 119874

1 while

the circle 1198742is divided equally into (119873 minus 1) parts by the

relative zero points 11986021 11986022 119860

2(119873minus1) 1205791 and 1205792are the

angle values of 1198751and 119875

2relative to the nearest relative zero

points along the rotating direction In the model the cellangle values changing range between any two neighboringrelative zero points is defined from 0 to 360 degrees

Therefore the positions of any points 1198751and 119875

2on these

two circles relative to the absolute zero point 119874 can beexpressed as follows

1205791198751=1205791119873

+1198731 times2120587119873

+1205790

1205791 1205791198751 1205790 isin [0 2120587) 119873 = 1 2 3 119899 1198731 = 0 1 2 3 (119873 minus 1)(1)

1205791198752=

1205792(119873 minus 1)

+1198732 times2120587

(119873 minus 1)+ 1205791015840

0

1205792 1205791198752 1205791015840

0

isin [0 2120587) 119873 = 1 2 3 119899 1198732 = 0 1 2 (119873 minus 2) (2)

where1198731and119873

2are the numbers of the relative zero points

between 119874 and 1198751and 119874 and 119875

2 respectively 119875

1and 119875

2are


(b) (c)(a)

O1

Axis

O

P1P2

O2

O

P1

1205791

O1

A11

A12

1205790


O

O2

P2

1205792

A21

A22

A1(N1minus1)

A1(N1+1)

A2(N2minus2)

A2(N2minus1)

A2(N2+1)

120579P1120579P2

A1N1

A1N1 A2N2

1205799984000



and 1205791198752


1and 119875

2


1and 119875

2 relative


and 11986021198732





0


1205791198751

= 1205791198752

=

1205791 minus 1205792 if (1205791 minus 1205792) ge 0

1205791 minus 1205792 + 2120587 if (1205791 minus 1205792) lt 0

1205791 1205792 1205791198751 1205791198752 isin [0 2120587)

(3)


1and 119875




1205791198751=1205791119873

+1198731 times2120587119873

+1205790 = 1205791198752

=1205792

(119873 minus 1)+1198732 times

2120587(119873 minus 1)

+ 1205791015840

0

(4)

Then1205791 minus 1205792

=

1205791119873

+1198731 times2120587119873


minus 1205790)

1205792119873 minus 1

+21205871198731119873 minus 1

+2120587119873 (1198732 minus 1198731)

119873 minus 1+ 119873(120579

1015840

0

minus 1205790)

(5)



1015840

0

)


1015840

0

)

(6)





1and119873

2as1198731= 1198732and119873

1= (1198732+ 1)



1015840

0

)

1205791198752= 1205791 minus 1205792 +119873(1205790 minus 120579

1015840

0

)

(7)



1015840

0

)

1205791198752= 1205791 minus 1205792 + 2120587+119873(1205790 minus 120579

1015840

0

)

(8)

If 1205790= 1205791015840

0

then

1205791198751= 1205791198752=

1205791 minus 1205792 when 1198731 = 1198732

1205791 minus 1205792 + 2120587 when 1198731 = 1198732 + 1(9)

where

1205791 1205792 1205791198751 1205791198752 isin [0 2120587)

119873 = 1 2 3 119899

1198731 = 0 1 2 (119873minus 1)

1198732 = 0 1 2 (119873minus 2)

(10)


11198732and 1205791 1205792


1(1205792) and 120579




1205791 =

1198731205791198751 1198731 = 0 120579

1198751 isin [0 2120587119873

)

1198731205791198751 minus 2120587 1198731 = 1 120579

1198751 isin [2120587119873

4120587119873

)


1198751 isin [2 (119873 minus 1) 120587

119873 2120587)

(11)

1205792 =

(119873 minus 1) 1205791198752 1198732 = 0 120579

1198752 isin [0 2120587119873 minus 1

)

(119873 minus 1) 1205791198752 minus 2120587 1198732 = 1 120579

1198752 isin [2120587

119873 minus 1

4120587119873 minus 1

)


1198752 isin [2 (119873 minus 2) 120587

119873 minus 1 2120587)

(12)


(1205791 minus 1205792) =

1205791198751 gt 0 1198731 = 1198732 = 0 120579

1198751 isin [0 2120587119873

)

(1205791198751 minus 2120587) lt 0 1198731 = 1198732 + 1 = 1 120579

1198751 isin [2120587119873

2120587

119873 minus 1)

1205791198751 gt 0 1198731 = 1198732 = 1 120579

1198751 isin [2120587

119873 minus 14120587119873

)

(1205791198751 minus 2120587) lt 0 1198731 = 1198732 + 1 = 2 120579

1198751 isin [4120587

119873 minus 14120587119873

)

(1205791198751 minus 2120587) lt 0 1198731 = 1198732 + 1 = (119873 minus 1) 120579

1198751 isin [2 (119873 minus 1) 120587

119873 2120587)

(13)


1198731 = 1198732 997904rArr 1205791 gt 1205792

1198731 = 1198732 + 1 997904rArr 1205791 lt 1205792(14)


2







= 1205791015840

0


119873(1205791 minus 1205792) = 2120587119873 (1198732 minus1198731) + 21205871198731 + 1205791 (15)

If 1205791ge 1205792 then

2120587119873 (1198732 minus1198731) + 21205871198731 + 1205791 ge 0 (16)


1and119873

2are1198731= 1198732and119873

1= 1198732+

1


21205871198731 + 1205791 ge 0 (17)

If1198731= 1198732+ 1 then

2120587119873 (1198732 minus1198731) + 21205871198731 + 1205791 = 2120587 (1198731 minus119873)+ 1205791

∵ 1198731 = 0 1 2 3 (119873minus 1)


= minus 2120587+ 1205791

∵ 1205791 lt 2120587

there4 2120587119873 (1198732 minus1198731) + 21205871198731 + 1205791 lt 0

(18)

Therefore

1205791 ge 1205792 997904rArr 1198731 = 1198732

1205791 lt 1205792 997904rArr 1198731 = 1198732 + 1(19)








1205791 ge 1205792 lArrrArr 1198731 = 1198732

1205791 lt 1205792 lArrrArr 1198731 = 1198732 + 1(20)











1+ sin119875


1+ cos119875

1minus separately


1(1198752) a look-



①

1 2 3


Sensor 1

Sensor 2


Pouter Pouter



Looking up


VDD

A sin P1

Acos P2

A sin P2

Acos P1

Analog todigital

converter

SPI




Power system

+25V

+33V

+125V +33V +18V

+50V

SPISOMISPISIMO

SPISTESPICLK

The encoder

Magneticring

Magneticinductor

Positionof

sensor 1

Positionof

sensor 2

X Y

Z

Pouter minus Pinner

075

mm

33

mm

146 times 2466∘

085mm147 times 2449

∘33mmR265

R31


③

④

② ⑤


(A2)cos P1+

(A2)cos P1minus

(A2)cos P2+

(A2)cos P2minus

(A2)sin P1+

(A2)sin P1minus

(A2)sin P2+

(A2)sin P2minus


sin119875(1198751 1198752) cos119875(119875

1 1198752) and 119875(119875











Period

Period







120579119875=



120579outer 120579inner 120579119875 isin [0 2120587)

(21)





)

(22)






1198601119862 cos (1205791 + Δ1205791119862))


1198602119862 cos (1205792 + Δ1205792119862))

(23)


1 sin1205792 and cos120579

2cor-



5300 5320 5340 5360 5380 5400

0

05


Sample point

Valu

e (de

g)

minus1

minus05

(a-a)

(b-a)

5300 5320 5340 5360 5380 54000


Sample point

Valu

e (de

g)


0 1000 2000 3000 4000 5000 6000 7000

0

05


Sample point

(a)

Valu

e (de

g)

minus1

minus05

(b)

0 1000 2000 3000 4000 5000 6000 7000 80000


Valu

e (de

g)

0 1000 2000 3000 4000 5000 6000 7000 80000


177 circles

176 circles

Valu

e (de

g)

Sample point

(c)

0 1000 2000 3000 4000 5000 6000 7000 80000


Sample point

Valu

e (de

g)


(c-a)

(d) (d-a)


Sample point

Valu

e (de

g)


0 1000 2000 3000 4000 5000 6000 7000 8000

0


Sample point

Valu

e (de

g)

Errors

minus05 5300 5310 5320 5330 5340 5350

0


Sample point

Valu

e (de

g)

Errors

minus05

sin 1cos 1

sin 2cos 2

sin 1cos 1

sin 2cos 2





= arctan( sin 1205791cos 1205791


)

minus(

arctan(1198601119878 sin (1205791 + Δ1205791119878)

1198601119862 cos (1205791 + Δ1205791119862))


1198602119862 cos (1205792 + Δ1205792119862))

)

(24)


11198621198602119878

and1198602119862


1119878andΔ120579

1119862Δ1205792119878andΔ120579

2119862



5 Conclusion









Acknowledgments


References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










(b) (c)(a)

O1

Axis

O

P1P2

O2

O

P1

1205791

O1

A11

A12

1205790


O

O2

P2

1205792

A21

A22

A1(N1minus1)

A1(N1+1)

A2(N2minus2)

A2(N2minus1)

A2(N2+1)

120579P1120579P2

A1N1

A1N1 A2N2

1205799984000



and 1205791198752


1and 119875

2


1and 119875

2 relative


and 11986021198732





0


1205791198751

= 1205791198752

=

1205791 minus 1205792 if (1205791 minus 1205792) ge 0

1205791 minus 1205792 + 2120587 if (1205791 minus 1205792) lt 0

1205791 1205792 1205791198751 1205791198752 isin [0 2120587)

(3)


1and 119875




1205791198751=1205791119873

+1198731 times2120587119873

+1205790 = 1205791198752

=1205792

(119873 minus 1)+1198732 times

2120587(119873 minus 1)

+ 1205791015840

0

(4)

Then1205791 minus 1205792

=

1205791119873

+1198731 times2120587119873


minus 1205790)

1205792119873 minus 1

+21205871198731119873 minus 1

+2120587119873 (1198732 minus 1198731)

119873 minus 1+ 119873(120579

1015840

0

minus 1205790)

(5)



1015840

0

)


1015840

0

)

(6)





1and119873

2as1198731= 1198732and119873

1= (1198732+ 1)



1015840

0

)

1205791198752= 1205791 minus 1205792 +119873(1205790 minus 120579

1015840

0

)

(7)



1015840

0

)

1205791198752= 1205791 minus 1205792 + 2120587+119873(1205790 minus 120579

1015840

0

)

(8)

If 1205790= 1205791015840

0

then

1205791198751= 1205791198752=

1205791 minus 1205792 when 1198731 = 1198732

1205791 minus 1205792 + 2120587 when 1198731 = 1198732 + 1(9)

where

1205791 1205792 1205791198751 1205791198752 isin [0 2120587)

119873 = 1 2 3 119899

1198731 = 0 1 2 (119873minus 1)

1198732 = 0 1 2 (119873minus 2)

(10)


11198732and 1205791 1205792


1(1205792) and 120579




1205791 =

1198731205791198751 1198731 = 0 120579

1198751 isin [0 2120587119873

)

1198731205791198751 minus 2120587 1198731 = 1 120579

1198751 isin [2120587119873

4120587119873

)


1198751 isin [2 (119873 minus 1) 120587

119873 2120587)

(11)

1205792 =

(119873 minus 1) 1205791198752 1198732 = 0 120579

1198752 isin [0 2120587119873 minus 1

)

(119873 minus 1) 1205791198752 minus 2120587 1198732 = 1 120579

1198752 isin [2120587

119873 minus 1

4120587119873 minus 1

)


1198752 isin [2 (119873 minus 2) 120587

119873 minus 1 2120587)

(12)


(1205791 minus 1205792) =

1205791198751 gt 0 1198731 = 1198732 = 0 120579

1198751 isin [0 2120587119873

)

(1205791198751 minus 2120587) lt 0 1198731 = 1198732 + 1 = 1 120579

1198751 isin [2120587119873

2120587

119873 minus 1)

1205791198751 gt 0 1198731 = 1198732 = 1 120579

1198751 isin [2120587

119873 minus 14120587119873

)

(1205791198751 minus 2120587) lt 0 1198731 = 1198732 + 1 = 2 120579

1198751 isin [4120587

119873 minus 14120587119873

)

(1205791198751 minus 2120587) lt 0 1198731 = 1198732 + 1 = (119873 minus 1) 120579

1198751 isin [2 (119873 minus 1) 120587

119873 2120587)

(13)


1198731 = 1198732 997904rArr 1205791 gt 1205792

1198731 = 1198732 + 1 997904rArr 1205791 lt 1205792(14)


2







= 1205791015840

0


119873(1205791 minus 1205792) = 2120587119873 (1198732 minus1198731) + 21205871198731 + 1205791 (15)

If 1205791ge 1205792 then

2120587119873 (1198732 minus1198731) + 21205871198731 + 1205791 ge 0 (16)


1and119873

2are1198731= 1198732and119873

1= 1198732+

1


21205871198731 + 1205791 ge 0 (17)

If1198731= 1198732+ 1 then

2120587119873 (1198732 minus1198731) + 21205871198731 + 1205791 = 2120587 (1198731 minus119873)+ 1205791

∵ 1198731 = 0 1 2 3 (119873minus 1)


= minus 2120587+ 1205791

∵ 1205791 lt 2120587

there4 2120587119873 (1198732 minus1198731) + 21205871198731 + 1205791 lt 0

(18)

Therefore

1205791 ge 1205792 997904rArr 1198731 = 1198732

1205791 lt 1205792 997904rArr 1198731 = 1198732 + 1(19)








1205791 ge 1205792 lArrrArr 1198731 = 1198732

1205791 lt 1205792 lArrrArr 1198731 = 1198732 + 1(20)











1+ sin119875


1+ cos119875

1minus separately


1(1198752) a look-



①

1 2 3


Sensor 1

Sensor 2


Pouter Pouter



Looking up


VDD

A sin P1

Acos P2

A sin P2

Acos P1

Analog todigital

converter

SPI




Power system

+25V

+33V

+125V +33V +18V

+50V

SPISOMISPISIMO

SPISTESPICLK

The encoder

Magneticring

Magneticinductor

Positionof

sensor 1

Positionof

sensor 2

X Y

Z

Pouter minus Pinner

075

mm

33

mm

146 times 2466∘

085mm147 times 2449

∘33mmR265

R31


③

④

② ⑤


(A2)cos P1+

(A2)cos P1minus

(A2)cos P2+

(A2)cos P2minus

(A2)sin P1+

(A2)sin P1minus

(A2)sin P2+

(A2)sin P2minus


sin119875(1198751 1198752) cos119875(119875

1 1198752) and 119875(119875











Period

Period







120579119875=



120579outer 120579inner 120579119875 isin [0 2120587)

(21)





)

(22)






1198601119862 cos (1205791 + Δ1205791119862))


1198602119862 cos (1205792 + Δ1205792119862))

(23)


1 sin1205792 and cos120579

2cor-



5300 5320 5340 5360 5380 5400

0

05


Sample point

Valu

e (de

g)

minus1

minus05

(a-a)

(b-a)

5300 5320 5340 5360 5380 54000


Sample point

Valu

e (de

g)


0 1000 2000 3000 4000 5000 6000 7000

0

05


Sample point

(a)

Valu

e (de

g)

minus1

minus05

(b)

0 1000 2000 3000 4000 5000 6000 7000 80000


Valu

e (de

g)

0 1000 2000 3000 4000 5000 6000 7000 80000


177 circles

176 circles

Valu

e (de

g)

Sample point

(c)

0 1000 2000 3000 4000 5000 6000 7000 80000


Sample point

Valu

e (de

g)


(c-a)

(d) (d-a)


Sample point

Valu

e (de

g)


0 1000 2000 3000 4000 5000 6000 7000 8000

0


Sample point

Valu

e (de

g)

Errors

minus05 5300 5310 5320 5330 5340 5350

0


Sample point

Valu

e (de

g)

Errors

minus05

sin 1cos 1

sin 2cos 2

sin 1cos 1

sin 2cos 2





= arctan( sin 1205791cos 1205791


)

minus(

arctan(1198601119878 sin (1205791 + Δ1205791119878)

1198601119862 cos (1205791 + Δ1205791119862))


1198602119862 cos (1205792 + Δ1205792119862))

)

(24)


11198621198602119878

and1198602119862


1119878andΔ120579

1119862Δ1205792119878andΔ120579

2119862



5 Conclusion









Acknowledgments


References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










1205791 =

1198731205791198751 1198731 = 0 120579

1198751 isin [0 2120587119873

)

1198731205791198751 minus 2120587 1198731 = 1 120579

1198751 isin [2120587119873

4120587119873

)


1198751 isin [2 (119873 minus 1) 120587

119873 2120587)

(11)

1205792 =

(119873 minus 1) 1205791198752 1198732 = 0 120579

1198752 isin [0 2120587119873 minus 1

)

(119873 minus 1) 1205791198752 minus 2120587 1198732 = 1 120579

1198752 isin [2120587

119873 minus 1

4120587119873 minus 1

)


1198752 isin [2 (119873 minus 2) 120587

119873 minus 1 2120587)

(12)


(1205791 minus 1205792) =

1205791198751 gt 0 1198731 = 1198732 = 0 120579

1198751 isin [0 2120587119873

)

(1205791198751 minus 2120587) lt 0 1198731 = 1198732 + 1 = 1 120579

1198751 isin [2120587119873

2120587

119873 minus 1)

1205791198751 gt 0 1198731 = 1198732 = 1 120579

1198751 isin [2120587

119873 minus 14120587119873

)

(1205791198751 minus 2120587) lt 0 1198731 = 1198732 + 1 = 2 120579

1198751 isin [4120587

119873 minus 14120587119873

)

(1205791198751 minus 2120587) lt 0 1198731 = 1198732 + 1 = (119873 minus 1) 120579

1198751 isin [2 (119873 minus 1) 120587

119873 2120587)

(13)


1198731 = 1198732 997904rArr 1205791 gt 1205792

1198731 = 1198732 + 1 997904rArr 1205791 lt 1205792(14)


2







= 1205791015840

0


119873(1205791 minus 1205792) = 2120587119873 (1198732 minus1198731) + 21205871198731 + 1205791 (15)

If 1205791ge 1205792 then

2120587119873 (1198732 minus1198731) + 21205871198731 + 1205791 ge 0 (16)


1and119873

2are1198731= 1198732and119873

1= 1198732+

1


21205871198731 + 1205791 ge 0 (17)

If1198731= 1198732+ 1 then

2120587119873 (1198732 minus1198731) + 21205871198731 + 1205791 = 2120587 (1198731 minus119873)+ 1205791

∵ 1198731 = 0 1 2 3 (119873minus 1)


= minus 2120587+ 1205791

∵ 1205791 lt 2120587

there4 2120587119873 (1198732 minus1198731) + 21205871198731 + 1205791 lt 0

(18)

Therefore

1205791 ge 1205792 997904rArr 1198731 = 1198732

1205791 lt 1205792 997904rArr 1198731 = 1198732 + 1(19)








1205791 ge 1205792 lArrrArr 1198731 = 1198732

1205791 lt 1205792 lArrrArr 1198731 = 1198732 + 1(20)











1+ sin119875


1+ cos119875

1minus separately


1(1198752) a look-



①

1 2 3


Sensor 1

Sensor 2


Pouter Pouter



Looking up


VDD

A sin P1

Acos P2

A sin P2

Acos P1

Analog todigital

converter

SPI




Power system

+25V

+33V

+125V +33V +18V

+50V

SPISOMISPISIMO

SPISTESPICLK

The encoder

Magneticring

Magneticinductor

Positionof

sensor 1

Positionof

sensor 2

X Y

Z

Pouter minus Pinner

075

mm

33

mm

146 times 2466∘

085mm147 times 2449

∘33mmR265

R31


③

④

② ⑤


(A2)cos P1+

(A2)cos P1minus

(A2)cos P2+

(A2)cos P2minus

(A2)sin P1+

(A2)sin P1minus

(A2)sin P2+

(A2)sin P2minus


sin119875(1198751 1198752) cos119875(119875

1 1198752) and 119875(119875











Period

Period







120579119875=



120579outer 120579inner 120579119875 isin [0 2120587)

(21)





)

(22)






1198601119862 cos (1205791 + Δ1205791119862))


1198602119862 cos (1205792 + Δ1205792119862))

(23)


1 sin1205792 and cos120579

2cor-



5300 5320 5340 5360 5380 5400

0

05


Sample point

Valu

e (de

g)

minus1

minus05

(a-a)

(b-a)

5300 5320 5340 5360 5380 54000


Sample point

Valu

e (de

g)


0 1000 2000 3000 4000 5000 6000 7000

0

05


Sample point

(a)

Valu

e (de

g)

minus1

minus05

(b)

0 1000 2000 3000 4000 5000 6000 7000 80000


Valu

e (de

g)

0 1000 2000 3000 4000 5000 6000 7000 80000


177 circles

176 circles

Valu

e (de

g)

Sample point

(c)

0 1000 2000 3000 4000 5000 6000 7000 80000


Sample point

Valu

e (de

g)


(c-a)

(d) (d-a)


Sample point

Valu

e (de

g)


0 1000 2000 3000 4000 5000 6000 7000 8000

0


Sample point

Valu

e (de

g)

Errors

minus05 5300 5310 5320 5330 5340 5350

0


Sample point

Valu

e (de

g)

Errors

minus05

sin 1cos 1

sin 2cos 2

sin 1cos 1

sin 2cos 2





= arctan( sin 1205791cos 1205791


)

minus(

arctan(1198601119878 sin (1205791 + Δ1205791119878)

1198601119862 cos (1205791 + Δ1205791119862))


1198602119862 cos (1205792 + Δ1205792119862))

)

(24)


11198621198602119878

and1198602119862


1119878andΔ120579

1119862Δ1205792119878andΔ120579

2119862



5 Conclusion









Acknowledgments


References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
















1205791 ge 1205792 lArrrArr 1198731 = 1198732

1205791 lt 1205792 lArrrArr 1198731 = 1198732 + 1(20)











1+ sin119875


1+ cos119875

1minus separately


1(1198752) a look-



①

1 2 3


Sensor 1

Sensor 2


Pouter Pouter



Looking up


VDD

A sin P1

Acos P2

A sin P2

Acos P1

Analog todigital

converter

SPI




Power system

+25V

+33V

+125V +33V +18V

+50V

SPISOMISPISIMO

SPISTESPICLK

The encoder

Magneticring

Magneticinductor

Positionof

sensor 1

Positionof

sensor 2

X Y

Z

Pouter minus Pinner

075

mm

33

mm

146 times 2466∘

085mm147 times 2449

∘33mmR265

R31


③

④

② ⑤


(A2)cos P1+

(A2)cos P1minus

(A2)cos P2+

(A2)cos P2minus

(A2)sin P1+

(A2)sin P1minus

(A2)sin P2+

(A2)sin P2minus


sin119875(1198751 1198752) cos119875(119875

1 1198752) and 119875(119875











Period

Period







120579119875=



120579outer 120579inner 120579119875 isin [0 2120587)

(21)





)

(22)






1198601119862 cos (1205791 + Δ1205791119862))


1198602119862 cos (1205792 + Δ1205792119862))

(23)


1 sin1205792 and cos120579

2cor-



5300 5320 5340 5360 5380 5400

0

05


Sample point

Valu

e (de

g)

minus1

minus05

(a-a)

(b-a)

5300 5320 5340 5360 5380 54000


Sample point

Valu

e (de

g)


0 1000 2000 3000 4000 5000 6000 7000

0

05


Sample point

(a)

Valu

e (de

g)

minus1

minus05

(b)

0 1000 2000 3000 4000 5000 6000 7000 80000


Valu

e (de

g)

0 1000 2000 3000 4000 5000 6000 7000 80000


177 circles

176 circles

Valu

e (de

g)

Sample point

(c)

0 1000 2000 3000 4000 5000 6000 7000 80000


Sample point

Valu

e (de

g)


(c-a)

(d) (d-a)


Sample point

Valu

e (de

g)


0 1000 2000 3000 4000 5000 6000 7000 8000

0


Sample point

Valu

e (de

g)

Errors

minus05 5300 5310 5320 5330 5340 5350

0


Sample point

Valu

e (de

g)

Errors

minus05

sin 1cos 1

sin 2cos 2

sin 1cos 1

sin 2cos 2





= arctan( sin 1205791cos 1205791


)

minus(

arctan(1198601119878 sin (1205791 + Δ1205791119878)

1198601119862 cos (1205791 + Δ1205791119862))


1198602119862 cos (1205792 + Δ1205792119862))

)

(24)


11198621198602119878

and1198602119862


1119878andΔ120579

1119862Δ1205792119878andΔ120579

2119862



5 Conclusion









Acknowledgments


References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










①

1 2 3


Sensor 1

Sensor 2


Pouter Pouter



Looking up


VDD

A sin P1

Acos P2

A sin P2

Acos P1

Analog todigital

converter

SPI




Power system

+25V

+33V

+125V +33V +18V

+50V

SPISOMISPISIMO

SPISTESPICLK

The encoder

Magneticring

Magneticinductor

Positionof

sensor 1

Positionof

sensor 2

X Y

Z

Pouter minus Pinner

075

mm

33

mm

146 times 2466∘

085mm147 times 2449

∘33mmR265

R31


③

④

② ⑤


(A2)cos P1+

(A2)cos P1minus

(A2)cos P2+

(A2)cos P2minus

(A2)sin P1+

(A2)sin P1minus

(A2)sin P2+

(A2)sin P2minus


sin119875(1198751 1198752) cos119875(119875

1 1198752) and 119875(119875











Period

Period







120579119875=



120579outer 120579inner 120579119875 isin [0 2120587)

(21)





)

(22)






1198601119862 cos (1205791 + Δ1205791119862))


1198602119862 cos (1205792 + Δ1205792119862))

(23)


1 sin1205792 and cos120579

2cor-



5300 5320 5340 5360 5380 5400

0

05


Sample point

Valu

e (de

g)

minus1

minus05

(a-a)

(b-a)

5300 5320 5340 5360 5380 54000


Sample point

Valu

e (de

g)


0 1000 2000 3000 4000 5000 6000 7000

0

05


Sample point

(a)

Valu

e (de

g)

minus1

minus05

(b)

0 1000 2000 3000 4000 5000 6000 7000 80000


Valu

e (de

g)

0 1000 2000 3000 4000 5000 6000 7000 80000


177 circles

176 circles

Valu

e (de

g)

Sample point

(c)

0 1000 2000 3000 4000 5000 6000 7000 80000


Sample point

Valu

e (de

g)


(c-a)

(d) (d-a)


Sample point

Valu

e (de

g)


0 1000 2000 3000 4000 5000 6000 7000 8000

0


Sample point

Valu

e (de

g)

Errors

minus05 5300 5310 5320 5330 5340 5350

0


Sample point

Valu

e (de

g)

Errors

minus05

sin 1cos 1

sin 2cos 2

sin 1cos 1

sin 2cos 2





= arctan( sin 1205791cos 1205791


)

minus(

arctan(1198601119878 sin (1205791 + Δ1205791119878)

1198601119862 cos (1205791 + Δ1205791119862))


1198602119862 cos (1205792 + Δ1205792119862))

)

(24)


11198621198602119878

and1198602119862


1119878andΔ120579

1119862Δ1205792119878andΔ120579

2119862



5 Conclusion









Acknowledgments


References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Period

Period







120579119875=



120579outer 120579inner 120579119875 isin [0 2120587)

(21)





)

(22)






1198601119862 cos (1205791 + Δ1205791119862))


1198602119862 cos (1205792 + Δ1205792119862))

(23)


1 sin1205792 and cos120579

2cor-



5300 5320 5340 5360 5380 5400

0

05


Sample point

Valu

e (de

g)

minus1

minus05

(a-a)

(b-a)

5300 5320 5340 5360 5380 54000


Sample point

Valu

e (de

g)


0 1000 2000 3000 4000 5000 6000 7000

0

05


Sample point

(a)

Valu

e (de

g)

minus1

minus05

(b)

0 1000 2000 3000 4000 5000 6000 7000 80000


Valu

e (de

g)

0 1000 2000 3000 4000 5000 6000 7000 80000


177 circles

176 circles

Valu

e (de

g)

Sample point

(c)

0 1000 2000 3000 4000 5000 6000 7000 80000


Sample point

Valu

e (de

g)


(c-a)

(d) (d-a)


Sample point

Valu

e (de

g)


0 1000 2000 3000 4000 5000 6000 7000 8000

0


Sample point

Valu

e (de

g)

Errors

minus05 5300 5310 5320 5330 5340 5350

0


Sample point

Valu

e (de

g)

Errors

minus05

sin 1cos 1

sin 2cos 2

sin 1cos 1

sin 2cos 2





= arctan( sin 1205791cos 1205791


)

minus(

arctan(1198601119878 sin (1205791 + Δ1205791119878)

1198601119862 cos (1205791 + Δ1205791119862))


1198602119862 cos (1205792 + Δ1205792119862))

)

(24)


11198621198602119878

and1198602119862


1119878andΔ120579

1119862Δ1205792119878andΔ120579

2119862



5 Conclusion









Acknowledgments


References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










5300 5320 5340 5360 5380 5400

0

05


Sample point

Valu

e (de

g)

minus1

minus05

(a-a)

(b-a)

5300 5320 5340 5360 5380 54000


Sample point

Valu

e (de

g)


0 1000 2000 3000 4000 5000 6000 7000

0

05


Sample point

(a)

Valu

e (de

g)

minus1

minus05

(b)

0 1000 2000 3000 4000 5000 6000 7000 80000


Valu

e (de

g)

0 1000 2000 3000 4000 5000 6000 7000 80000


177 circles

176 circles

Valu

e (de

g)

Sample point

(c)

0 1000 2000 3000 4000 5000 6000 7000 80000


Sample point

Valu

e (de

g)


(c-a)

(d) (d-a)


Sample point

Valu

e (de

g)


0 1000 2000 3000 4000 5000 6000 7000 8000

0


Sample point

Valu

e (de

g)

Errors

minus05 5300 5310 5320 5330 5340 5350

0


Sample point

Valu

e (de

g)

Errors

minus05

sin 1cos 1

sin 2cos 2

sin 1cos 1

sin 2cos 2





= arctan( sin 1205791cos 1205791


)

minus(

arctan(1198601119878 sin (1205791 + Δ1205791119878)

1198601119862 cos (1205791 + Δ1205791119862))


1198602119862 cos (1205792 + Δ1205792119862))

)

(24)


11198621198602119878

and1198602119862


1119878andΔ120579

1119862Δ1205792119878andΔ120579

2119862



5 Conclusion









Acknowledgments


References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












= arctan( sin 1205791cos 1205791


)

minus(

arctan(1198601119878 sin (1205791 + Δ1205791119878)

1198601119862 cos (1205791 + Δ1205791119862))


1198602119862 cos (1205792 + Δ1205792119862))

)

(24)


11198621198602119878

and1198602119862


1119878andΔ120579

1119862Δ1205792119878andΔ120579

2119862



5 Conclusion









Acknowledgments


References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation



























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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