A Low-Cost Photodiode Sun Sensor for CubeSat and Planetary ... · 2 InternationalJournalofAerospaceEngineering Figure1:A1UCubeSatwithsunsensor. Figure2:Microroverwithsunsensor. gridsensorarrayssuchasCCDsandphotodiodearrays,and

Hindawi Publishing CorporationInternational Journal of Aerospace EngineeringVolume 2013 Article ID 549080 9 pageshttpdxdoiorg1011552013549080

Research ArticleA Low-Cost Photodiode Sun Sensor for CubeSat andPlanetary Microrover

Mark A Post Junquan Li and Regina Lee

Department of Earth and Space Science and Engineering York University 4700 Keele St Toronto ON Canada M3J 1P3

Correspondence should be addressed to Junquan Li junquanlyorkuca

Received 21 July 2013 Accepted 4 November 2013

Academic Editor Hui Hu

Copyright copy 2013 Mark A Post 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

This paper presents the development of low-cost methodologies to determine the attitude of a small CubeSat-class satellite and amicrorover relative to the sunrsquos direction The use of commercial hardware and simple embedded designs has become an effectivepath for university programs to put experimental payloads in space for minimal cost and the development of sensors for attitudeand heading determination is often a critical part The development of two compact and efficient but simple coarse sun sensormethodologies is presented in this research A direct measurement of the solar angle uses a photodiode array sensor and slit maskAnother estimation of the solar angle uses current measurements from orthogonal arrays of solar cells The two methodologies aretested and compared on ground hardware Testing results show that coarse sun sensing is efficient even with minimal processingand complexity of design for satellite attitude determination systems and rover navigation systems

1 Introduction

One of the key problems in the development of attitude deter-mination and control systems (ADCS) for small satellites isthe use of attitude sensors small enough and efficient enoughto fit withinmass and power budgets One of the simplest andmost common sensors for attitude determination is the sunsensor [1] whichmeasures the angle of incident light from thesun with respect to an inertial body frame Both single-axisand dual-axis sensors are available though a dual-axis sensorcan be constructed from two compact single-axis sensorsresulting in lower component costs and processing require-ments [2 3] In other sun sensor applications both CCD andCMOS technology have been used to achieve fine pointingaccuracy [4 5] and softwaremethods can be used to increasethe accuracy of a sensor [6] Although several embedded sunsensors for nanosatellite ADCS hardware are now availableit is desirable in many nanosatellite development programsto develop the sun sensors in-house using commercial off-the-shelf (COTS) hardware [7] both for reasons of cost andto increase the potential for further research within their ownprogram Sun sensors are also used on planetary rovers whenother sensors such as magnetometers or GPS receivers arenot sufficient Measuring solar angles with a sun sensor is

a good way of estimating absolute orientation [8ndash10] Typicalrequirements include an accuracy on the order of 1 degreeand a field of view of 30 degrees or 60 degrees [11] Wide-field-of-view sun sensors [12] suitable for use on CubeSat andmicrorover platforms are still an open area of research and inmany cases simpler systems are desirable Low-cost sensorsfor research use are usually constructed by graduate studentsand researchers and must be efficient compact in size androbust enough to survive the space environment In thispaper simple low-cost and wide-field-of-view sun sensormethodologies are presented The sensors described here areunder development for the YUSEND (York University SpaceEngineering Nanosatellite Demonstration) mission at YorkUniversity [13 14] which focuses on CubeSat technologydevelopment such as for the 1U CubeSat shown in Figure 1as well as a microrover under development for the NorthernLight Mars Lander Mission shown in Figure 2 [15 16]

In this paper we outline the development of two coarsesun sensor methodologies that are compact and efficientenough for a CubeSat-class nanosatellites and microroversand can provide reliable solar angle information for embed-ded attitude determination and localizationThere are severalbasic methodologies that are in use for sun sensors includingthe use of Position-Sensitive Photodiodes (PSD) linear and

2 International Journal of Aerospace Engineering

Figure 1 A 1U CubeSat with sun sensor

Figure 2 Microrover with sun sensor

grid sensor arrays such as CCDs and photodiode arrays andthe measurement of sunlight on solar panels used for powerIn this work we make use of the latter two

First direct measurement of the solar angle is performedusing a photodiode array sensor placed below a slit designin the top that allows light to fall on the sensor elementSecond the solar angle is inferred using separate currentmeasurements from the array of solar cells used for powergeneration on the exterior of the vehicle In both casesthe solar angle is calculated by a microcontroller from thegeometry of the sun sensor with respect to the vehicle bodyFor this study it is assumed that at least a 90∘ field of viewis necessary for each sun sensor so that complete coverageof the exterior is possible with a sun sensor on each faceof a CubeSat To test and validate the sensors the sensorhardware is rotated on the spot by a rotary gimbal withangularmeasurement capability and simulated sunlight froma Schott 1150 Illuminator light source is provided at a fixedposition as shown in Figure 3 The nanosatellite model withsun sensor under test on an optical bench and the lightsource used is shown in Figure 4 Using embedded fixed-point processing estimations of the current body orientationare processed and made available by serial interface to an on-board computer which performs the localization and controldetermination operations [17]

Sun simulator

Rotary gimbal

Sun sensor

Solar cells

Direction ofsunlight

z

x y

CubeSat120579

Figure 3 Testing diagram for sun sensor hardware

Figure 4 Testing setup on optical bench with light source

2 Sensor Designs

21 Array Sensor Photodiode sun sensors have been usedoften for spacecraft and satellite attitude determination sys-tems [18 19] An established and simple method for directlysensing solar angles is to use a simple slit or an otheraperture design through which light is allowed to fall atone location on a spatially varying sensor dependent onthe incident angle of light This design uses a 256-elementlinear photodiode array which is both flexible in operationby allowing the distribution and magnitude of light to bedetermined and efficient by utilizing only a single line ofsensing elements rather than a matrix such as a Charge-Coupled Device (CCD) array While some systems use aPosition-SensitiveDiode (PSD) as a sensor an array increaseslinearity of measurement and provides more flexibility inprocessing of light distribution data The TAOS TSL1402Rintegrated linear photodiode array is used in this designReading of data and calculation of solar angles are performedby an Atmel ATMega168PA AVR microcontroller A blockdiagram of the connections used between the photodiodearray and microcontroller is shown in Figure 5

For single-axis sensing a simple linear slit in a mask overthe sensor is used as shown in Figure 6The slit is positionedvertically centered on the sensor element such that light fallson to the sensor array at angles up to 45∘ in every directionfor a 90∘ total field of view For a linear photodiode array with

International Journal of Aerospace Engineering 3

ADC0

ADC1

ATMega168PAFiltering

CLK

SI1

SI0

33 V

TSL1402R

Arrays

AO1

AO0+minus

+minus

Figure 5 Diagram of photodiode array sensor

Sensor element

h

L

d

c

b

a

120579

120601

Figure 6 Sun angle sensing by photodiode array

length 119871 a light vector falls at an angle 120579 from the normalalong the element axis and at an angle 120601 from the normalalong the perpendicular to the element axis The photodiodevoltage output of the sensor falls at the point of contact withthe light vector at a distance 119889 from the center of the arrayIf the slit is positioned at a distance ℎ from the array thisdistance is simply

119889 = ℎ tan (120579) (1)

To ensure that |120579| le 45∘ the sensor distance should be

119867 = 119889 = 1198712 and to ensure |120601| le 45∘ the slit length should be

119887 = 2ℎ = 119871 It is preferable for higher precisionmeasurementsto use as small a slit width 119886 as possible without significantdiffractive effects but the material thickness around the slitis constrained by 119888 lt 119886 tan(120579) which limits how small theslit width 119886 can be as the beam of light will be ldquopinchedrdquo offat high angles In this design 119871 = 16mm and the dimensionsused were 119867 = 8mm and 119887 = 18mm In testing metal foilwith width 119888 = 05mm provided acceptable performanceusing a slit width of 119886 = 08mm

Linear arrays typically provide only one axis of attitudeestimation but because a pattern on the array can bemeasured it is also possible to measure elevation across mostangles by using an N-slit [4] as shown in Figure 7 By adding

h

L

c

120579120579

120575

120601120601

d1 d2

Figure 7 Sun angle sensing by photodiode array and N-slit

additional slits positioned at an angle 120575 to the central slit apattern with multiple illumination maxima will be projectedon to the sensor array and the transverse angle 120601 can bedetermined by the separations 119889

1and 119889

2of the central slit

and one of the side maximaAs the field of view of the sensor is limited more than

one sun sensor is needed to cover wide angles Two singleslit sensors or one Z-slit sensor can cover two solid anglesof 90∘ with a pyramidal volume of view and individual sidesof a CubeSat can be covered this way depending on missionpointing requirements For microrover use a full 180∘ of view(horizon to horizon) can be covered by four photodiodesattached to the frustum of square pyramid [19 20] In thecurrent study though a single N-slit sensor is being studiedfor partial sky coverage on the microrover due to spacelimitations

To estimate the orientation of a microrover from solarangles the angle of the body with respect to the groundand the angle of the sun with respect to the ground mustbe considered The former can be obtained with inertialmeasurements from an accelerometer at rest measuring thegravity vector and the latter can be obtained from solarephemeris data and current time To simplify the analysis weassume that the accelerometer is aligned with the sun sensorso that the angle 120579 is about the forward-pointing 119909-axis andthe angle 120601 is about the side-pointing 119910-axis The rover bodyangles about the 119909- and 119910-axes with respect to the groundframe 120579

119892and 120601

119892can be calculated from the gravity vector

(119892119909 119892119910 119892119911) using common aerospace relations as follows

120579119892= atan(

119892119910

119892119911

)

120601119892= atan(

minus119892119909

radic1198922119910

+ 1198922119911

)

(2)

To obtain the sensed horizontal azimuth120572119904of a vertically-

centered sun sensor measurement the angles 120579 and 120601 mustbe adjusted for rover body angle and projected on the hori-zontal plane using the quadrant-aware arctangent function asfollows

120572119904= atan2 (sin (120579 + 120579

119892) sin (120601 + 120601

119892)) (3)


1205793

1205792

1205791

Solar arrays

Figure 8 Sun angle sensing by solar cell output

The solar ephemeris data must be computed separatelyfrom an estimate of the current position which can be donewith a variety of available software The roverrsquos heading withrespect to true north 120572rover can then be calculated using thesolar azimuth 120572

119890and sensed azimuth 120572

119904by [9]

120572119890gt 120572119904997904rArr 120572rover = 120572

119890minus 120572119904

120572119890le 120572119904997904rArr 120572rover = 120572

119904minus 120572119890

(4)

22 Solar Current Sensing Due to the constraints on spaceand power available in a nanosatellite it is preferred to makeuse of sensing methodologies that focus on the processingof other available data rather than discrete sensors Onemethod of doing this is to sample the currents generated bythe nanosatellitersquos solar arrays information that is commonlyavailable on small satellites for peak-power tracking orbattery charge monitoring This has the advantage that arange of angles spanning a set of independently measurednoncoplanar solar panels can be measured without an exter-nal sensor but is in general less accurate than discrete sensorsdue to the nonlinearities involved In this study we considera cubic body with a fixed solar panel on each orthogonal faceas shown in Figure 8 For a CubeSat geometry no more thanthree solar panels are exposed to sunlight at a time in orbitif a point source and negligible reflection from the earth areassumed with angles 120579

1 1205792 and 120579

3 For simplicity a similar

body is assumed for a microrover case though no solar panelis present on the lower face

The linearity of this measurement varies depending onthe solar cells used Also nonlinearities are introduced byvariations in the load presented to the solar arrays For ananosatellite with a linear or pulse regulated battery chargesystem this generally arises from changes in battery chargerate as the battery state changes and can be compensated forby including solar cell voltage or an estimation of the chargesystem state in the solar calculation to determine total powerand current output

The current sensing circuit is constructed from a bankof differential amplifiers that are read by using the Analog-to-Digital Converter (ADC) channels available on the

ADC2

ADC3

ATMega168PA

Filtering

OPA2340

1 k

1 k

1 k

1 k

100 k

100 k

100 k

100 kI+

+

+

Iminus

minus

minus01 Ω

01 Ω

times3

Figure 9 Diagram of solar current sensors

ATMega168PA microcontroller that also reads the lineararray A current sense resistor of 01Ω creates a voltagedifference from current flowing from the solar panels whichis amplified by anOPA2340 rail-to-rail op-amp in differentialconfiguration with a gain of 100The output gain with respectto the solar panel current is then 10VA It is assumed thatno more than 500mA will be sourced from the nanosatellitesolar panel so an ADC reference of 5V can be used inmeasurement As custom-constructed solar panels oftenvary slightly in output it is still necessary to calibrate theADC measurements performed by the microcontroller Thecurrent sense amplifier circuit used for sensing is shown inFigure 9

The amount of current a solar panel produces depends onthe panel area 119860

0and changes in angle change the effective

area 119860119890of the solar panel intercepting solar energy For

sunlight with an even power density (Wm2) and constantloading or by using compensation calculations the effectivesolar current 119868

119890relative to the maximum solar current 119868

0

intercepted by the solar array varies with the incident angle120579 and can be expressed in general terms as a ratio

119868119890

1198680

=119860119890

1198600

= cos (120579) (5)

Current sensing results are much more noisy and lesslinear than the results from the photodiode array In par-ticular the ADC offsets and gains must be calibrated foreach panel separately to ensure that measurements can becompared Figure 16 shows the output from the solar panelsafter applying a moving average to remove noise By applying(5) and determining the angular quadrant around the vehiclethat the sun is in it is possible to extract an estimate ofrelative solar angle to each panel Combining several panelsallows determination of a solar angle with respect to the bodyat any angle observed by solar panels While this methodmay be considered generally less reliable than direct solarmeasurement it does allow solar angle measurement withoutdedicated sensors at a wider range of angles than a singleexternal sensor would be able to measure Equations (2) (3)and (4) can be applied for a rover so long as two orthogonalaxes of angular measurement can be obtained from the solarpanel geometry


0 50 100 150 200 2500

50

100

150

200

250

Light centroid position (sensor element)

Ligh

t mag

nitu

de (8

-bit

valu

e)

Figure 10 Example of linear array output

minus60

minus40

minus20

0

20

40

60Angular determination of linear array sun sensor

Ang

le (d

eg)

0 50 100 150 200 250Light centroid position (sensor element)

Figure 11 Sun angle 120579 results from linear array for 60∘ to minus60∘

3 Testing Results

31 Array Sensor Results Typical illuminationmeasurementsfrom the photodiode array sun sensor are shown in Figure 10The voltage output must be inverted so that illuminationbecomes positive Then a centroiding algorithm [4] is usedto determine the center of illumination across the sensor(shown with a vertical line) The centroid positions 119889 arethen used in (1) for each angle 119899 sampled to determine thecorresponding solar angle by

120579119899= atan(

119889

ℎ) (6)

The estimated centroided solar angle 120579 is plotted inFigure 11 with the actual solar angle used on the gimbal testapparatus shown as a straight line for comparison In generalvery good agreement is achieved between the estimated and

0

50

100

150

200

250

300

Ligh

t mag

nitu

de (8

-bit

valu

e)


Figure 12 Example of N-slit output at low 120579 angles

0

50

100

150

200

250

300Li

ght m

agni

tude

(8-b

it va

lue)


Figure 13 Example of N-slit output at high 120579 angles

actual solar angles It is worth noting that this angular data isnot filtered or smoothed but still achieves consistent tracking

If anN-slit is used to perform estimation of the transverseangle 120601 multiple illumination centroids are present onthe sensor as shown in Figure 12 and must be partitionedseparatelyHowever as the angle120601 increases from the verticalit is common for one of the illuminated regions to moveoff the sensor for moderately high angles of 120579 and for theother two illuminated regions to approach and ultimatelymerge together as seen in Figure 13 This poses a problemfor centroid partitioning as angle increases To mitigate thisproblem the outer boundaries of the illuminated areas aretracked and the centroids are constrained to be between theseboundaries and the midpoint between the boundaries Hav-ing a precise geometry for the N-slit is essential for accuracyin this case and some calibration must be performed for thealignment of the sensor andN-slit particularly for the angle 120575that the side slits make with the center slit Using the centroidfrom the center slit located at position 119889

2and the centroid of


0 5 10 15 20 25 30 35 40 450

5

10

15

20

25

30

35

40

45

Calc

ulat

ed an

gle (

deg)

Reference angle (deg)

Angular determination by array and N-slit

Figure 14 Transverse sun angle 120601 results from N-slit for 0∘ to 45∘

one of the side slits at position1198891 the angle120601 can be estimated

by using [4]

120601119899= atan(

1198892minus 1198891

ℎ tan (120575)) (7)

The estimated transverse solar angle 120601 obtained from N-slit measurements by using (7) is shown in Figure 14 Rea-sonable agreement is obtained but the error is higher than insimple linear slit measurements due to the greater complexityof constructing a precise N-slit Accurate measurements ofangles above approximately 35∘ could not be obtained dueto difficulties accurately partitioning and centroiding theilluminated areas on the sensor at high angles of 120601 The useof a wider N-slit or thinner material and slit width extendsthe measurable angles of this sensor

Using both the angles 120579 and 120601 and precalculated solarephemeris data a test of calculating the heading of themicrorover was conducted The sun sensor was mounted onthe microrover with the orientation aligned with the bodyframe as stated above and the microrover rotated from 165∘to 135∘ away from north The estimated heading angle for the30∘ sweep is plotted in Figure 15 with the externallymeasuredangle superimposed for reference To improve the accuracyof this measurement a 2-point moving average was used inprocessing

32 Solar Current Results Current measurement using highgains and ADC sensing generates much more noise thandigital sensing using a linear array as described aboveAlthough a constant current draw and capacitive decouplingof the amplifiers and microcontroller pins was used in thisstudy applying awindowed average to the data assuming slowchanges in angle was necessary to achieve consistent resultsFigure 16 shows the solar panel current output 119868

119899minus1 119868119899 and

119868119899+1

for three solar panels enumerated as 119899 minus 1 (facing the+119910 axis) 119899 (facing the +119909 axis) and 119899 + 1 (facing the minus119910

0 5 10 15 20 25 30 35minus165

minus160

minus155

minus150

minus145

minus140

minus135

minus130

Hea

ding

angl

e (de

g)


Heading angle by array and N-slit

Figure 15 Estimated microrover heading across 30∘ of rotation

20 40 60 80 100 120 140 1600

10

20

30

40

50

60

70

80

90Current from solar arrays

Curr

ent m

agni

tude

(8-b

it va

lue)

Angle (deg)

+y solar panel+x solar panel

minusy solar panel+z solar panel

Figure 16 Solar panel current measurements for minus180∘ to 180∘

axis) in the direction of increasing angle about a 1U CubeSatSmooth reference curves for the actual solar angles presentedin gimbal testingwith respect to each panel cos(120579

119899minus1) cos(120579

119899)

and cos(120579119899+1

) are superimposed for referenceAfter filtering the current 119868

119899from each panel 119899 the

quadrant that actual solar angle lies in with respect to thesatellite body must be determined The most straightforwardmethod of doing this is to simply identify which panelsare exposed to the most sunlight by comparing the relativesolar panel currents and assigning the appropriate sinusoidquadrant function using mapping functions As the greatestchange in illumination is present at high solar angles to eachpanel it is possible to determine the quadrant of a sinefunction for the satellite body frame angle 120579

119887by using only


0

20

40

60

80

100

120

140

160

180Angular determination by solar current

Calc

ulat

ed an

gle (

deg)

20 40 60 80 100 120 140 160Reference angle (deg)

Figure 17 Angle from single solar panel current for 0∘ to 180∘

119868119899minus1

and 119868119899+1

to determine the mapping for only the current119868119899as

119868119899minus1

gt 119868119899+1

997904rArr 120579119887= asin(

119868119899

max (119868119899))

119868119899minus1

le 119868119899+1

997904rArr 120579119887= asin(minus

119868119899

max (119868119899)) +

120587

2

(8)

Using (8) the current from each solar panel is used toobtain an estimated body frame angle 120579

119887over a 180∘ arc

shown in Figure 17 It is evident that there are a discontinuityand higher inaccuracy near the angle 120579

119887= 90∘ which

corresponds to 120579119899

= 0 This is due to the sudden jump inassignment but also to the inaccuracy of determining anglesclose to the vertical Tomitigate this a revisedmapping givenin (9) can be used that takes advantage of the other solarpanelsrsquo contributions at high angles to increase the accuracyof measurement The estimated body frame angle 120579

119887for this

case is shown in Figure 18 The revised mapping is as follows

119868119899minus1

gt 119868119899gt 119868119899+1

997904rArr 120579119887= asin(

119868119899

max (119868119899))

119868119899gt 119868119899minus1

gt 119868119899+1

997904rArr 120579119887= asin(minus

119868119899minus1

max (119868119899minus1

)) +

120587

4

119868119899gt 119868119899+1

gt 119868119899minus1

997904rArr 120579119887= asin(

119868119899+1

max (119868119899+1

)) +

120587

4

119868119899+1

gt 119868119899gt 119868119899minus1

997904rArr 120579119887= asin(minus

119868119899

max (119868119899)) +

120587

2

(9)

It should be noted that this estimation is not as reliableif the distribution of solar panels over the body is not sym-metrically illuminated such as in the case of the microroverHence (8) is more appropriate for microrover use where lessuseful information is obtained at 120579

119887= 90∘ and (9) is more

appropriate for nanosatellite use

0

20

40

60

80

100

120

140

160


Calc

ulat

ed an

gle (

deg)


Figure 18 Angle from all solar panel currents for 0∘ to 180∘

minus10

minus8

minus6

minus4

minus2

0

2

4

6

8

10Angular error of linear array sun sensor

Erro

r (de

g)


Figure 19 Error in linear array 120579 estimation

4 Comparison of Results and Discussion

To effectively compare the twomethodologies described hereit is important to include the error of measurement withrespect to the known angles used during testing Figure 19shows the estimation error for the linear array angle mea-surement of 120579 from Figure 11 using (6) Figure 20 shows theestimation error of the transverse angle 120601 for the array whileusing an N-slit from Figure 14 using (7) and Figure 21 showsthe error in Figure 15 using an N-slit for heading estimationFinally Figure 22 shows the estimation error for the solarpanel current angle measurement from Figure 18 using (9)The linear array shows a maximum error of approximatelyplusmn5∘ overall with less consistency in the N-slit measurementwhile the solar current sensing shows a maximum error ofapproximately plusmn7∘ These are comparable results but thelinear array data is obtained by centroiding and otherwise


Transverse angular error in array with N-slit

0 5 10 15 20 25 30 35Angle (deg)

minus10

minus8

minus6

minus4

minus2

0

2

4

6

8

10

Erro

r (de

g)

Figure 20 Error in linear array with N-slit 120601 estimation

minus15

minus1

minus05

0

05

1

15

2

Erro

r (de

g)

Heading error using array with N-slit

0 5 10 15 20 25 30 35Angle (deg)

Figure 21 Error in heading angle estimation

Angular error in solar panel power

20 40 60 80 100 120 140 160Angle (deg)

minus10

minus8

minus6

minus4

minus2

0

2

4

6

8

10

Erro

r (de

g)

Figure 22 Error in solar current angle estimation

unfiltered while the solar current data requires significantfiltering to remove measurement noise Hence the use ofa discrete digital sensor is still expected to provide betterreliability and overall accuracy though with appropriatedata processing solar current measurement can also provideuseable and complimentary coarse angle measurements Thetracking accuracy and noise present in microrover headingestimation is comparable to the sensor laboratory tests butslightly lower as amoving averagewas used and indicates thatuseable heading information can be extracted using a singleN-slit sensor

5 Conclusions

We have implemented and compared two useful methodsfor coarse solar angle sensing Using only simple hardwareand embedded software implementation very coarse attitudeestimation results can be achieved using either photodiodearray or solar panel current measurement methodologies fornanosatellite attitude tracking or microrover navigation Thephotodiode array provides good overall accuracy to errorswithinplusmn5∘ without additional filtering and thus requiresmin-imal processing but can be improved beyond this measureif additional filtering is implemented Dual-axis sensing ispossible for a linear array using an N-slit configuration butprecise construction of the slit is essential and transverseangular measurements are more limited Solar panel currentmeasurements without the use of a discrete sensor canprovide angular approximations over the entire exterior of thevehicle to plusmn7∘ but require significant filtering and averagingof measurements and thus tend to be less accurate and moreprocessing-intensive

Sun sensor designs such as these are useable in universityand research hardware development programs due to theirsimplicity robustness and cost-effectiveness As testing ofboth sun sensor configurations was done in parallel bothsensors could also be used in parallel on a CubeSat ormicrorover to achieve higher accuracy under uncertain con-ditions Future work will include refinements to the design ofboth sun sensor methodologies and further improvements tolocalization and navigation

Conflict of Interests

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

References

[1] M Buonocore M Grassi and G Rufino ldquoAps-based miniaturesun sensor for earth observation nanosatellitesrdquoActa Astronau-tica vol 56 no 1-2 pp 139ndash145 2005

[2] A Ali and F Tanveer ldquoLow-cost design and development of 2-axis digital sun sensorrdquo Journal of Space Technology vol 1 no 12011

[3] P Ortega G Lopez-Rodrıguez J Ricart et al ldquoA miniaturizedtwo axis sun sensor for attitude control of nano-satellitesrdquo IEEESensors Journal vol 10 no 10 pp 1623ndash1632 2010


[4] M-SWei F Xing B Li andZ You ldquoInvestigation of digital sunsensor technology with an N-shaped slit maskrdquo Sensors vol 11no 10 pp 9764ndash9777 2011

[5] P Appel ldquoAttitude estimation from magnetometer and earth-albedo-corrected coarse sun sensormeasurementsrdquoActa Astro-nautica vol 56 no 1-2 pp 115ndash126 2005

[6] J P Enright and G Godard ldquoAdvanced sun-sensor processingand design for super-resolution performancerdquo in Proceedingsof the IEEE Aerospace Conference Big Sky Mont USA March2006

[7] S E Allgeier M Mahin and N G Fitz-Coy ldquoDesign andanalysis of a coarse sun sensor for pico-satellitesrdquo in Proceedingsof the AIAA Infotech at Aerospace Conference and Exhibit andAIAA Unmanned Unlimited Conference Seattle Wash Unitedstate April 2009

[8] R Volpe ldquoMars rover navigation results using sun sensor head-ing determinationrdquo in Proceedings of the IEEERSJ InternationalConference on Intelligent Robots and Systems (IROSrsquo99) vol 1pp 460ndash467 October 1999

[9] A Trebi-Ollennu T Huntsberger Y Cheng E T BaumgartnerB Kennedy and P Schenker ldquoDesign and analysis of a sunsensor for planetary rover absolute heading detectionrdquo IEEETransactions on Robotics and Automation vol 17 no 6 pp 939ndash947 2001

[10] P Furgale J Enright and T Barfoot ldquoSun sensor navigation forplanetary rovers theory and field testingrdquo IEEE Transactions onAerospace and Electronic Systems vol 47 no 3 pp 1631ndash16472011

[11] I Maqsood and T Akram ldquoDevelopment of a low cost sunsensor using quadphotodioderdquo in Proceedings of the IEEEIONPosition Location and Navigation Symposium (PLANS rsquo10) pp639ndash644 Indian Wells Calif USA May 2010

[12] J D Francisco J Quero J Garca C L Tarrida P R Ortegaand S Bermejo ldquoAccurate andwide-field-of-viewMEMS-basedsun sensor for industrial applicationsrdquo IEEE Transactions onIndustrial Electronics vol 59 no 12 pp 4871ndash4880 2012

[13] J Li M A Post and R Lee ldquoDesign of attitude control systemsfor CubeSat-class nanosatelliterdquo Journal of Control Science andEngineering vol 2013 Article ID 657182 15 pages 2013

[14] R Lee H Chesser M Cannata M Post and K Kumar ldquoMod-ular attitude control system design for CubeSat applicationrdquo inProceedings of the 16th Bi-Annual Astronautics Conference of theCanadian Aeronautics and Space Institute (CASI ASTRO rsquo12)Quebec Canada April 2012

[15] N Navarathinam R Lee K Borschiov and B Quine ldquoNorth-ern light drill for Mars performance evaluationrdquo Acta Astro-nautica vol 68 no 7-8 pp 1234ndash1241 2011

[16] M A Post M A L R and B M Quine ldquoBeaver micro-roverdevelopment for the Northern light mars landerrdquo in Proceedingsof the 16th Bi-Annual Astronautics Conference of the CanadianAeronautics and Space Institute (CASI ASTRO rsquo12) QuebecCanada April 2012

[17] S Chouraqui M Benyettou and M A Si ldquoSensor vectorsmodeling for small satellite attitude determinationrdquo Journal ofApplied Sciences vol 5 no 10 pp 1739ndash1743 2005

[18] M A Post J Li and R Lee ldquoNanosatellite sun sensor attitudedetermination using low cost hardwarerdquo in Proceedings of the23th AASAIAA Space Flight Mechanics Meeting (AASAIAArsquo13) Kauai Hawaii USA February 2013

[19] J C Springmann and J W Cutler ldquoOptimization of direc-tional sensor orientation with application to photodiodes for

spacecraft attitude determinationrdquo in Proceedings of the 23thAASAIAA Space Flight Mechanics Meeting (AASAIAA rsquo13)Kauai Hawaii USA February 2013

[20] T Tambo M Shibata Y Mizuno and T Yamauchi ldquoSearchmethod of sun using fixed five photodiode sensorrdquo IEEJ Trans-actions on Sensors andMicromachines vol 129 no 2 pp 53ndash592009

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014


Shock and Vibration


Mechanical Engineering

Advances in


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of


Distributed Sensor Networks


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Advances inOptoElectronics


Volume 2014

RoboticsJournal of


Chemical EngineeringInternational Journal of


Control Scienceand Engineering

Journal of


Antennas andPropagation




Navigation and Observation



Figure 1 A 1U CubeSat with sun sensor

Figure 2 Microrover with sun sensor

grid sensor arrays such as CCDs and photodiode arrays andthe measurement of sunlight on solar panels used for powerIn this work we make use of the latter two

First direct measurement of the solar angle is performedusing a photodiode array sensor placed below a slit designin the top that allows light to fall on the sensor elementSecond the solar angle is inferred using separate currentmeasurements from the array of solar cells used for powergeneration on the exterior of the vehicle In both casesthe solar angle is calculated by a microcontroller from thegeometry of the sun sensor with respect to the vehicle bodyFor this study it is assumed that at least a 90∘ field of viewis necessary for each sun sensor so that complete coverageof the exterior is possible with a sun sensor on each faceof a CubeSat To test and validate the sensors the sensorhardware is rotated on the spot by a rotary gimbal withangularmeasurement capability and simulated sunlight froma Schott 1150 Illuminator light source is provided at a fixedposition as shown in Figure 3 The nanosatellite model withsun sensor under test on an optical bench and the lightsource used is shown in Figure 4 Using embedded fixed-point processing estimations of the current body orientationare processed and made available by serial interface to an on-board computer which performs the localization and controldetermination operations [17]

Sun simulator

Rotary gimbal

Sun sensor

Solar cells

Direction ofsunlight

z

x y

CubeSat120579

Figure 3 Testing diagram for sun sensor hardware

Figure 4 Testing setup on optical bench with light source

2 Sensor Designs

21 Array Sensor Photodiode sun sensors have been usedoften for spacecraft and satellite attitude determination sys-tems [18 19] An established and simple method for directlysensing solar angles is to use a simple slit or an otheraperture design through which light is allowed to fall atone location on a spatially varying sensor dependent onthe incident angle of light This design uses a 256-elementlinear photodiode array which is both flexible in operationby allowing the distribution and magnitude of light to bedetermined and efficient by utilizing only a single line ofsensing elements rather than a matrix such as a Charge-Coupled Device (CCD) array While some systems use aPosition-SensitiveDiode (PSD) as a sensor an array increaseslinearity of measurement and provides more flexibility inprocessing of light distribution data The TAOS TSL1402Rintegrated linear photodiode array is used in this designReading of data and calculation of solar angles are performedby an Atmel ATMega168PA AVR microcontroller A blockdiagram of the connections used between the photodiodearray and microcontroller is shown in Figure 5

For single-axis sensing a simple linear slit in a mask overthe sensor is used as shown in Figure 6The slit is positionedvertically centered on the sensor element such that light fallson to the sensor array at angles up to 45∘ in every directionfor a 90∘ total field of view For a linear photodiode array with


ADC0

ADC1


CLK

SI1

SI0

33 V

TSL1402R

Arrays

AO1

AO0+minus

+minus


Sensor element

h

L

d

c

b

a

120579

120601



119889 = ℎ tan (120579) (1)





h

L

c

120579120579

120575

120601120601

d1 d2



1and 119889





119892and 120601



120579119892= atan(

119892119910

119892119911

)

120601119892= atan(

minus119892119909

radic1198922119910

+ 1198922119911

)

(2)



120572119904= atan2 (sin (120579 + 120579

119892) sin (120601 + 120601

119892)) (3)


1205793

1205792

1205791

Solar arrays




119904by [9]

120572119890gt 120572119904997904rArr 120572rover = 120572

119890minus 120572119904

120572119890le 120572119904997904rArr 120572rover = 120572

119904minus 120572119890

(4)


1 1205792 and 120579





ADC2

ADC3

ATMega168PA

Filtering

OPA2340

1 k

1 k

1 k

1 k

100 k

100 k

100 k

100 kI+

+

+

Iminus

minus

minus01 Ω

01 Ω

times3








0


119868119890

1198680

=119860119890

1198600

= cos (120579) (5)



0 50 100 150 200 2500

50

100

150

200

250


Ligh

t mag

nitu

de (8

-bit

valu

e)


minus60

minus40

minus20

0

20

40


Ang

le (d

eg)



3 Testing Results


120579119899= atan(

119889

ℎ) (6)


0

50

100

150

200

250

300

Ligh

t mag

nitu

de (8

-bit

valu

e)



0

50

100

150

200

250

300Li

ght m

agni

tude

(8-b

it va

lue)







0 5 10 15 20 25 30 35 40 450

5

10

15

20

25

30

35

40

45

Calc

ulat

ed an

gle (

deg)





by using [4]

120601119899= atan(

1198892minus 1198891

ℎ tan (120575)) (7)




119899minus1 119868119899 and

119868119899+1


0 5 10 15 20 25 30 35minus165

minus160

minus155

minus150

minus145

minus140

minus135

minus130

Hea

ding

angl

e (de

g)




20 40 60 80 100 120 140 1600

10

20

30

40

50

60

70

80


Curr

ent m

agni

tude

(8-b

it va

lue)

Angle (deg)





119899minus1) cos(120579

119899)

and cos(120579119899+1




119887by using only


0

20

40

60

80

100

120

140

160


Calc

ulat

ed an

gle (

deg)



119868119899minus1

and 119868119899+1


119868119899minus1

gt 119868119899+1

997904rArr 120579119887= asin(

119868119899

max (119868119899))

119868119899minus1

le 119868119899+1

997904rArr 120579119887= asin(minus

119868119899

max (119868119899)) +

120587

2

(8)


119887over a 180∘ arc


119887= 90∘ which



119887for this


119868119899minus1

gt 119868119899gt 119868119899+1

997904rArr 120579119887= asin(

119868119899

max (119868119899))

119868119899gt 119868119899minus1

gt 119868119899+1

997904rArr 120579119887= asin(minus

119868119899minus1

max (119868119899minus1

)) +

120587

4

119868119899gt 119868119899+1

gt 119868119899minus1

997904rArr 120579119887= asin(

119868119899+1

max (119868119899+1

)) +

120587

4

119868119899+1

gt 119868119899gt 119868119899minus1

997904rArr 120579119887= asin(minus

119868119899

max (119868119899)) +

120587

2

(9)


119887= 90∘ and (9) is more


0

20

40

60

80

100

120

140

160


Calc

ulat

ed an

gle (

deg)



minus10

minus8

minus6

minus4

minus2

0

2

4

6

8


Erro

r (de

g)







0 5 10 15 20 25 30 35Angle (deg)

minus10

minus8

minus6

minus4

minus2

0

2

4

6

8

10

Erro

r (de

g)


minus15

minus1

minus05

0

05

1

15

2

Erro

r (de

g)


0 5 10 15 20 25 30 35Angle (deg)



20 40 60 80 100 120 140 160Angle (deg)

minus10

minus8

minus6

minus4

minus2

0

2

4

6

8

10

Erro

r (de

g)



5 Conclusions





References
























VLSI Design



RotatingMachinery





Shock and Vibration



Advances in







Journal of





SensorsJournal of







Volume 2014

RoboticsJournal of





Journal of









ADC0

ADC1


CLK

SI1

SI0

33 V

TSL1402R

Arrays

AO1

AO0+minus

+minus


Sensor element

h

L

d

c

b

a

120579

120601



119889 = ℎ tan (120579) (1)





h

L

c

120579120579

120575

120601120601

d1 d2



1and 119889





119892and 120601



120579119892= atan(

119892119910

119892119911

)

120601119892= atan(

minus119892119909

radic1198922119910

+ 1198922119911

)

(2)



120572119904= atan2 (sin (120579 + 120579

119892) sin (120601 + 120601

119892)) (3)


1205793

1205792

1205791

Solar arrays




119904by [9]

120572119890gt 120572119904997904rArr 120572rover = 120572

119890minus 120572119904

120572119890le 120572119904997904rArr 120572rover = 120572

119904minus 120572119890

(4)


1 1205792 and 120579





ADC2

ADC3

ATMega168PA

Filtering

OPA2340

1 k

1 k

1 k

1 k

100 k

100 k

100 k

100 kI+

+

+

Iminus

minus

minus01 Ω

01 Ω

times3








0


119868119890

1198680

=119860119890

1198600

= cos (120579) (5)



0 50 100 150 200 2500

50

100

150

200

250


Ligh

t mag

nitu

de (8

-bit

valu

e)


minus60

minus40

minus20

0

20

40


Ang

le (d

eg)



3 Testing Results


120579119899= atan(

119889

ℎ) (6)


0

50

100

150

200

250

300

Ligh

t mag

nitu

de (8

-bit

valu

e)



0

50

100

150

200

250

300Li

ght m

agni

tude

(8-b

it va

lue)







0 5 10 15 20 25 30 35 40 450

5

10

15

20

25

30

35

40

45

Calc

ulat

ed an

gle (

deg)





by using [4]

120601119899= atan(

1198892minus 1198891

ℎ tan (120575)) (7)




119899minus1 119868119899 and

119868119899+1


0 5 10 15 20 25 30 35minus165

minus160

minus155

minus150

minus145

minus140

minus135

minus130

Hea

ding

angl

e (de

g)




20 40 60 80 100 120 140 1600

10

20

30

40

50

60

70

80


Curr

ent m

agni

tude

(8-b

it va

lue)

Angle (deg)





119899minus1) cos(120579

119899)

and cos(120579119899+1




119887by using only


0

20

40

60

80

100

120

140

160


Calc

ulat

ed an

gle (

deg)



119868119899minus1

and 119868119899+1


119868119899minus1

gt 119868119899+1

997904rArr 120579119887= asin(

119868119899

max (119868119899))

119868119899minus1

le 119868119899+1

997904rArr 120579119887= asin(minus

119868119899

max (119868119899)) +

120587

2

(8)


119887over a 180∘ arc


119887= 90∘ which



119887for this


119868119899minus1

gt 119868119899gt 119868119899+1

997904rArr 120579119887= asin(

119868119899

max (119868119899))

119868119899gt 119868119899minus1

gt 119868119899+1

997904rArr 120579119887= asin(minus

119868119899minus1

max (119868119899minus1

)) +

120587

4

119868119899gt 119868119899+1

gt 119868119899minus1

997904rArr 120579119887= asin(

119868119899+1

max (119868119899+1

)) +

120587

4

119868119899+1

gt 119868119899gt 119868119899minus1

997904rArr 120579119887= asin(minus

119868119899

max (119868119899)) +

120587

2

(9)


119887= 90∘ and (9) is more


0

20

40

60

80

100

120

140

160


Calc

ulat

ed an

gle (

deg)



minus10

minus8

minus6

minus4

minus2

0

2

4

6

8


Erro

r (de

g)







0 5 10 15 20 25 30 35Angle (deg)

minus10

minus8

minus6

minus4

minus2

0

2

4

6

8

10

Erro

r (de

g)


minus15

minus1

minus05

0

05

1

15

2

Erro

r (de

g)


0 5 10 15 20 25 30 35Angle (deg)



20 40 60 80 100 120 140 160Angle (deg)

minus10

minus8

minus6

minus4

minus2

0

2

4

6

8

10

Erro

r (de

g)



5 Conclusions





References
























VLSI Design



RotatingMachinery





Shock and Vibration



Advances in







Journal of





SensorsJournal of







Volume 2014

RoboticsJournal of





Journal of









1205793

1205792

1205791

Solar arrays




119904by [9]

120572119890gt 120572119904997904rArr 120572rover = 120572

119890minus 120572119904

120572119890le 120572119904997904rArr 120572rover = 120572

119904minus 120572119890

(4)


1 1205792 and 120579





ADC2

ADC3

ATMega168PA

Filtering

OPA2340

1 k

1 k

1 k

1 k

100 k

100 k

100 k

100 kI+

+

+

Iminus

minus

minus01 Ω

01 Ω

times3








0


119868119890

1198680

=119860119890

1198600

= cos (120579) (5)



0 50 100 150 200 2500

50

100

150

200

250


Ligh

t mag

nitu

de (8

-bit

valu

e)


minus60

minus40

minus20

0

20

40


Ang

le (d

eg)



3 Testing Results


120579119899= atan(

119889

ℎ) (6)


0

50

100

150

200

250

300

Ligh

t mag

nitu

de (8

-bit

valu

e)



0

50

100

150

200

250

300Li

ght m

agni

tude

(8-b

it va

lue)







0 5 10 15 20 25 30 35 40 450

5

10

15

20

25

30

35

40

45

Calc

ulat

ed an

gle (

deg)





by using [4]

120601119899= atan(

1198892minus 1198891

ℎ tan (120575)) (7)




119899minus1 119868119899 and

119868119899+1


0 5 10 15 20 25 30 35minus165

minus160

minus155

minus150

minus145

minus140

minus135

minus130

Hea

ding

angl

e (de

g)




20 40 60 80 100 120 140 1600

10

20

30

40

50

60

70

80


Curr

ent m

agni

tude

(8-b

it va

lue)

Angle (deg)





119899minus1) cos(120579

119899)

and cos(120579119899+1




119887by using only


0

20

40

60

80

100

120

140

160


Calc

ulat

ed an

gle (

deg)



119868119899minus1

and 119868119899+1


119868119899minus1

gt 119868119899+1

997904rArr 120579119887= asin(

119868119899

max (119868119899))

119868119899minus1

le 119868119899+1

997904rArr 120579119887= asin(minus

119868119899

max (119868119899)) +

120587

2

(8)


119887over a 180∘ arc


119887= 90∘ which



119887for this


119868119899minus1

gt 119868119899gt 119868119899+1

997904rArr 120579119887= asin(

119868119899

max (119868119899))

119868119899gt 119868119899minus1

gt 119868119899+1

997904rArr 120579119887= asin(minus

119868119899minus1

max (119868119899minus1

)) +

120587

4

119868119899gt 119868119899+1

gt 119868119899minus1

997904rArr 120579119887= asin(

119868119899+1

max (119868119899+1

)) +

120587

4

119868119899+1

gt 119868119899gt 119868119899minus1

997904rArr 120579119887= asin(minus

119868119899

max (119868119899)) +

120587

2

(9)


119887= 90∘ and (9) is more


0

20

40

60

80

100

120

140

160


Calc

ulat

ed an

gle (

deg)



minus10

minus8

minus6

minus4

minus2

0

2

4

6

8


Erro

r (de

g)







0 5 10 15 20 25 30 35Angle (deg)

minus10

minus8

minus6

minus4

minus2

0

2

4

6

8

10

Erro

r (de

g)


minus15

minus1

minus05

0

05

1

15

2

Erro

r (de

g)


0 5 10 15 20 25 30 35Angle (deg)



20 40 60 80 100 120 140 160Angle (deg)

minus10

minus8

minus6

minus4

minus2

0

2

4

6

8

10

Erro

r (de

g)



5 Conclusions





References
























VLSI Design



RotatingMachinery





Shock and Vibration



Advances in







Journal of





SensorsJournal of







Volume 2014

RoboticsJournal of





Journal of









0 50 100 150 200 2500

50

100

150

200

250


Ligh

t mag

nitu

de (8

-bit

valu

e)


minus60

minus40

minus20

0

20

40


Ang

le (d

eg)



3 Testing Results


120579119899= atan(

119889

ℎ) (6)


0

50

100

150

200

250

300

Ligh

t mag

nitu

de (8

-bit

valu

e)



0

50

100

150

200

250

300Li

ght m

agni

tude

(8-b

it va

lue)







0 5 10 15 20 25 30 35 40 450

5

10

15

20

25

30

35

40

45

Calc

ulat

ed an

gle (

deg)





by using [4]

120601119899= atan(

1198892minus 1198891

ℎ tan (120575)) (7)




119899minus1 119868119899 and

119868119899+1


0 5 10 15 20 25 30 35minus165

minus160

minus155

minus150

minus145

minus140

minus135

minus130

Hea

ding

angl

e (de

g)




20 40 60 80 100 120 140 1600

10

20

30

40

50

60

70

80


Curr

ent m

agni

tude

(8-b

it va

lue)

Angle (deg)





119899minus1) cos(120579

119899)

and cos(120579119899+1




119887by using only


0

20

40

60

80

100

120

140

160


Calc

ulat

ed an

gle (

deg)



119868119899minus1

and 119868119899+1


119868119899minus1

gt 119868119899+1

997904rArr 120579119887= asin(

119868119899

max (119868119899))

119868119899minus1

le 119868119899+1

997904rArr 120579119887= asin(minus

119868119899

max (119868119899)) +

120587

2

(8)


119887over a 180∘ arc


119887= 90∘ which



119887for this


119868119899minus1

gt 119868119899gt 119868119899+1

997904rArr 120579119887= asin(

119868119899

max (119868119899))

119868119899gt 119868119899minus1

gt 119868119899+1

997904rArr 120579119887= asin(minus

119868119899minus1

max (119868119899minus1

)) +

120587

4

119868119899gt 119868119899+1

gt 119868119899minus1

997904rArr 120579119887= asin(

119868119899+1

max (119868119899+1

)) +

120587

4

119868119899+1

gt 119868119899gt 119868119899minus1

997904rArr 120579119887= asin(minus

119868119899

max (119868119899)) +

120587

2

(9)


119887= 90∘ and (9) is more


0

20

40

60

80

100

120

140

160


Calc

ulat

ed an

gle (

deg)



minus10

minus8

minus6

minus4

minus2

0

2

4

6

8


Erro

r (de

g)







0 5 10 15 20 25 30 35Angle (deg)

minus10

minus8

minus6

minus4

minus2

0

2

4

6

8

10

Erro

r (de

g)


minus15

minus1

minus05

0

05

1

15

2

Erro

r (de

g)


0 5 10 15 20 25 30 35Angle (deg)



20 40 60 80 100 120 140 160Angle (deg)

minus10

minus8

minus6

minus4

minus2

0

2

4

6

8

10

Erro

r (de

g)



5 Conclusions





References
























VLSI Design



RotatingMachinery





Shock and Vibration



Advances in







Journal of





SensorsJournal of







Volume 2014

RoboticsJournal of





Journal of









0 5 10 15 20 25 30 35 40 450

5

10

15

20

25

30

35

40

45

Calc

ulat

ed an

gle (

deg)





by using [4]

120601119899= atan(

1198892minus 1198891

ℎ tan (120575)) (7)




119899minus1 119868119899 and

119868119899+1


0 5 10 15 20 25 30 35minus165

minus160

minus155

minus150

minus145

minus140

minus135

minus130

Hea

ding

angl

e (de

g)




20 40 60 80 100 120 140 1600

10

20

30

40

50

60

70

80


Curr

ent m

agni

tude

(8-b

it va

lue)

Angle (deg)





119899minus1) cos(120579

119899)

and cos(120579119899+1




119887by using only


0

20

40

60

80

100

120

140

160


Calc

ulat

ed an

gle (

deg)



119868119899minus1

and 119868119899+1


119868119899minus1

gt 119868119899+1

997904rArr 120579119887= asin(

119868119899

max (119868119899))

119868119899minus1

le 119868119899+1

997904rArr 120579119887= asin(minus

119868119899

max (119868119899)) +

120587

2

(8)


119887over a 180∘ arc


119887= 90∘ which



119887for this


119868119899minus1

gt 119868119899gt 119868119899+1

997904rArr 120579119887= asin(

119868119899

max (119868119899))

119868119899gt 119868119899minus1

gt 119868119899+1

997904rArr 120579119887= asin(minus

119868119899minus1

max (119868119899minus1

)) +

120587

4

119868119899gt 119868119899+1

gt 119868119899minus1

997904rArr 120579119887= asin(

119868119899+1

max (119868119899+1

)) +

120587

4

119868119899+1

gt 119868119899gt 119868119899minus1

997904rArr 120579119887= asin(minus

119868119899

max (119868119899)) +

120587

2

(9)


119887= 90∘ and (9) is more


0

20

40

60

80

100

120

140

160


Calc

ulat

ed an

gle (

deg)



minus10

minus8

minus6

minus4

minus2

0

2

4

6

8


Erro

r (de

g)







0 5 10 15 20 25 30 35Angle (deg)

minus10

minus8

minus6

minus4

minus2

0

2

4

6

8

10

Erro

r (de

g)


minus15

minus1

minus05

0

05

1

15

2

Erro

r (de

g)


0 5 10 15 20 25 30 35Angle (deg)



20 40 60 80 100 120 140 160Angle (deg)

minus10

minus8

minus6

minus4

minus2

0

2

4

6

8

10

Erro

r (de

g)



5 Conclusions





References
























VLSI Design



RotatingMachinery





Shock and Vibration



Advances in







Journal of





SensorsJournal of







Volume 2014

RoboticsJournal of





Journal of









0

20

40

60

80

100

120

140

160


Calc

ulat

ed an

gle (

deg)



119868119899minus1

and 119868119899+1


119868119899minus1

gt 119868119899+1

997904rArr 120579119887= asin(

119868119899

max (119868119899))

119868119899minus1

le 119868119899+1

997904rArr 120579119887= asin(minus

119868119899

max (119868119899)) +

120587

2

(8)


119887over a 180∘ arc


119887= 90∘ which



119887for this


119868119899minus1

gt 119868119899gt 119868119899+1

997904rArr 120579119887= asin(

119868119899

max (119868119899))

119868119899gt 119868119899minus1

gt 119868119899+1

997904rArr 120579119887= asin(minus

119868119899minus1

max (119868119899minus1

)) +

120587

4

119868119899gt 119868119899+1

gt 119868119899minus1

997904rArr 120579119887= asin(

119868119899+1

max (119868119899+1

)) +

120587

4

119868119899+1

gt 119868119899gt 119868119899minus1

997904rArr 120579119887= asin(minus

119868119899

max (119868119899)) +

120587

2

(9)


119887= 90∘ and (9) is more


0

20

40

60

80

100

120

140

160


Calc

ulat

ed an

gle (

deg)



minus10

minus8

minus6

minus4

minus2

0

2

4

6

8


Erro

r (de

g)







0 5 10 15 20 25 30 35Angle (deg)

minus10

minus8

minus6

minus4

minus2

0

2

4

6

8

10

Erro

r (de

g)


minus15

minus1

minus05

0

05

1

15

2

Erro

r (de

g)


0 5 10 15 20 25 30 35Angle (deg)



20 40 60 80 100 120 140 160Angle (deg)

minus10

minus8

minus6

minus4

minus2

0

2

4

6

8

10

Erro

r (de

g)



5 Conclusions





References
























VLSI Design



RotatingMachinery





Shock and Vibration



Advances in







Journal of





SensorsJournal of







Volume 2014

RoboticsJournal of





Journal of










0 5 10 15 20 25 30 35Angle (deg)

minus10

minus8

minus6

minus4

minus2

0

2

4

6

8

10

Erro

r (de

g)


minus15

minus1

minus05

0

05

1

15

2

Erro

r (de

g)


0 5 10 15 20 25 30 35Angle (deg)



20 40 60 80 100 120 140 160Angle (deg)

minus10

minus8

minus6

minus4

minus2

0

2

4

6

8

10

Erro

r (de

g)



5 Conclusions





References
























VLSI Design



RotatingMachinery





Shock and Vibration



Advances in







Journal of





SensorsJournal of







Volume 2014

RoboticsJournal of





Journal of




























VLSI Design



RotatingMachinery





Shock and Vibration



Advances in







Journal of





SensorsJournal of







Volume 2014

RoboticsJournal of





Journal of









VLSI Design



RotatingMachinery





Shock and Vibration



Advances in







Journal of





SensorsJournal of







Volume 2014

RoboticsJournal of





Journal of








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