Research Article Development of Liquid Capacity …downloads.hindawi.com/journals/mpe/2016/4260397.pdfResearch Article Development of Liquid Capacity Measuring Algorithm Based on Data

Research ArticleDevelopment of Liquid Capacity Measuring Algorithm Based onData Integration from Multiple Sensors

Kiwoong Park Si-Kyoung Lee and Hyeon Cheol Kim

School of Electrical Engineering University of Ulsan Ulsan 44610 Republic of Korea

Correspondence should be addressed to Hyeon Cheol Kim hckim08ulsanackr

Received 15 March 2016 Revised 7 June 2016 Accepted 19 June 2016

Academic Editor Jean-Pierre Corriou

Copyright copy 2016 Kiwoong Park 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 research proposes an algorithm using a process of integrating data from multiple sensors to measure the liquid capacity inreal time regardless of the position of the liquid tankThe algorithm for measuring the capacity was created with a complementaryfilter using a Kalman filter in order to revise the level sensor data and IMU sensor data The measuring precision of the proposedalgorithm was assessed through repetitive experiments by varying the liquid capacity and the rotation angle of the liquid tankThe measurements of the capacity within the liquid tank were precise even when the liquid tank was rotated Using the proposedalgorithm one can obtain highly precise measurements and it is affordable since an existing level sensor is used

1 Introduction

Continuous or discontinuous measurements are used toassess the capacity of liquid contained in vehicles tankertrucks and industrial tanks Continuous measurements aredivided into the following types buoyancy weight mea-surement pressure capacitance ultrasound and radiationDiscontinuous measurements are divided into the follow-ing types conductivity heat transfer capacitance opticalultrasound and microwave [1ndash3] Some sensors used formeasuring liquid level are summarized in Table 1

The pressure level sensor a double-continuousmeasuringtype forms the core of the measurement market owing to itsaffordability simplicity of its gauge and its ability to workat high temperature and high pressure The ultrasound- andradar-type sensors are more efficient than others in measur-ing the methods but these sensors are not economical [4 5]

Most demand for current level sensors comes from watertreatment plants (sewage industrial water wastewater andblack water) a national key industry and the sector thatmea-sures the flow rate for vehicles and vessels The level sensoris especially adequate for measuring the flow rate of vesselsand the fuel of vehicles

The existing method for measuring the vehicle fuelinvolves measuring poles and floating box [6] This is struc-turally limited because it is immeasurable when the fuel tank

of the vehicle is below 10 In addition measuring errorsoccur because the fuel gets heeled over when the vehicle turnsor accelerates as shown in Figure 1

This paper proposes a new fuel-measuring algorithmusing a pressure level sensor that can be applied for fuel con-trol andmeasurements of vehiclesThe system can accuratelymeasure the fuel capacity below 10 by improving mechani-cal limits of the existing method in measuring the capacityFurthermore the measurement errors are minimized bycontinuously measuring and revising the slopping or heelingof fuel when the vehicle turns or accelerates This revisedalgorithm guarantees precision for laser- and ultrasound-type sensors In addition this algorithm is highly economicalbecause an existing pressure level sensor [7] has been used

2 Sensor Design

Theproposed sensor should have a characteristic of ameasur-ing algorithm of density and capacity for fuel measurementand a revised method in accordance with the fuel tankrsquoscondition acceleration and acceleration of gravity

Figure 2 demonstrates the system which this research hasproposed A pressure level sensor is used tomeasure capacitydensity and tilting angle and an inertial measurement unit(IMU) is used to revise the measuring value in accordance

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 4260397 12 pageshttpdxdoiorg10115520164260397

2 Mathematical Problems in Engineering

Table 1 Comparison of sensors used for measuring liquid level

Sensor type Scope of measurement [m] Precision Temperature limits [∘C] Pressure limits[kgfcm2]

Pressure type [9] 01ndash100 plusmn005 [FS] 70 mdashFloat type [10] Customizable 635mm 105 mdashDisplacement type [11] 03ndash3 plusmn2 [] 370 300Capacitance type [12] 01ndash4 plusmn01 [] 200 102Weight level type [13] 305 10 [mm] 49 mdash

Radiation type [14] 7 05 [] 50 No limitation(noncontact)

Ultrasonic type [15] 8ndash11 plusmn05 [] 70 3Radar type [16] 30 plusmn10 By the strength of windows

Level sensor

10

(a)

Measurement value

Measurement error

Levelsensor

120579

(b)

Measurement error

Level sensor

Measurementvalue

(c)

Figure 1 Disadvantages of current fuel-measuring methods for vehicles (a) Structural limits result in inability to measure the remaining10 of fuel (b) Fuel-measuring error on slopes (c) Fuel-measuring error when accelerating

with the rotation acceleration and acceleration of gravity [8]The unique approach of this paper is that the relative valuein accordance with the location of the level sensor and therotation angle data of the IMU are both used to measure theposition of a fuel tank

The reasons are as follows The IMU outputs a precisemeasurable value for the rotation angle and accelerationmea-surements but its disadvantage is that such values are accu-mulated with time by an integration constant In the case ofa level sensor it canmeasure the tilting angle and accelerationin accordance with the measuring value difference locatedstructurally but the measurements may not be accurate

The proposed algorithm of this paper guarantees pre-cision and credibility because it uses the Kalman filter torealize a complementary filter with precision despite the error

accumulation of the IMU and the imprecision and non-accumulation of errors of the level sensor

3 Sensor Measurement Algorithm for PreciseMeasurements of Sensors

31 Measuring Algorithm of Capacity As demonstrated inFigure 3 the capacity can be computed in accordance withthe container volume and the input capacity of the liquid

The mass can be computed using the pressure applied tothe bottom of a container with liquid First the calculationequation of the capacity is as follows

Pressure applied to the bottom of the container

prop liquid level times liquid density(1)

Mathematical Problems in Engineering 3

(i) High accuracy (i) Low accuracy (i) Capacity measurement(ii) Density measurement

Sensor integration algorithm using 3-axis IMU and level sensor

) h ( ) ( )(ii) Error propagation X(ii) Error propagation O

Figure 2 Algorithm applied in this assignment

(liquid density)(height)

20cm

P (kgm3)

A (m2) (area)

h (cm)

20 cm

Figure 3 Measuring method for capacity

The equation to compute the force applied to the bottomin accordance with the liquid is as follows [17]

119865 = 120588 times 119860ℎ times 119892 (119892 Gravitational acceleration) (2)

Therefore the pressure 119875 applied to the unit area of thebottom side can be defined as in the equation below [2]

119875 = 120588 times 119892 times ℎ

(If there is no internal pressure in the tank)

119875 minus 1198750= 120588 times 119892 times ℎ

(If there is an internal pressure in the tank)

(3)

In this control system the following method has beenused to measure the capacity

The sensor attached toward the 119909-axis and 119910-axis fromthe center of the target container is located on the oppositeside of the container The obtained measurement value isused to calculate the average value of each axis and the

Sensor 2

Sensor 1

Sensor 3

Sensor 4 Sensor 5Sensor 6

Sensor 7

Sensing unit

Liquid tank

Figure 4 Sensor location to measure the sensor

capacity is computed by comparing the accumulated datafrom experiments

A =Sensor1 out + Sensor2 out

2

B =Sensor3 out + Sensor4 out

2

C =Sensor

5 out + Sensor6 out

2

(4)

(A + B + C)

3

= The average value of the output voltage 997888rarr

The measuring comparison by the input data

(5)

Here the output value of each sensor does not accumulatebut the precision is lowThis sensor output value changeswithturning and accelerating and is used in the revised algorithmto measure the capacity

32 Measuring Tilting Angle with Level Sensors In Figure 4the sensor installed on the opposite side of each axis receives


Sensor 4

Sensor 7 Liquid tank

P1

P2

h1

h2

Figure 5 Liquid tank and sensor mimetic diagram to measuredensity

different pressures in accordancewith the turning and ideallythe sum of the sensors on the opposite side must be thesame as when the vehicle is not rotating The tilting angle iscomputed based on this definition

Sensor1 out + Sensor

2 outADC120579ref

= 119909-Axis Tilting Value


4 outADC120579ref


0 = 119911-Axis Tilting Value

(6)

In the equation above Sensor 1ndashSensor 6 are digitalvalues obtained by an ADC board used for implementationand ADC120579ref is the ADC value (constant) per tilting angleobtained from experiments Sensors 5 and 6 are only used asa means to increase the accuracy because they duplicate 119910-axis of rotationTherefore Sensors 5 and 6 are not used in thedata analysis

33MeasuringAlgorithmofDensity Figure 5 is the schematicdiagram of the attached sensor in the container in order tocompute the densityThe density is computedwith the signalsoutputted from the sensor attached to the container

Sensitivity of the sensor times Pressure = Output voltage 997888rarr

Output VoltageSensitivity of the sensor

= Pressure (pa)

120588 times 119892 times (ℎ2minus ℎ1) = 1198752minus 1198751997888rarr

120588 =1198751minus 1198752

119892 (ℎ2minus ℎ1)

(7)

where 120588 is liquid density 119892 is gravitational acceleration ℎ isheight and 119875 is pressure

Gyro ADC

HDR ADCtoLEVEL

Attitude Kalman filter Level Kalman filter

DIFF_level

Level

DIFF_angle

gn

Rn

Figure 6Measuringmethods of the rotation angle of the liquid tankand acting forces

The equation above is a basic equation to compute liquiddensity and 119875

1and 119875

2from the above equation are Sensor 7

and Sensor 4 in Figure 5 respectively [18 19]

4 Multiple Sensors Algorithm to Improve thePrecision of the Result Value of the Sensor

Figure 6 is a block diagram of the algorithm to accuratelymeasure the capacity when the liquid tank has rotated bythe force applied to it An attitude Kalman filter computesaccurate values while accumulating almost no errors byreceiving the rotation angle data from the IMU and levelsensor Furthermore a level Kalman filter precisely measuresthe capacity by using the obtained rotation angle data andlevel incrementThe holistic drift reduction (HDR) filter usedis excellent in avoiding drift in the obtained data from theIMU The acceleration of gravity is distributed to the gravityvalue of each axis in accordance with the fuel tank conditions

41 Revision of IMU Sensor When the output of the IMUsensor has an angular speed of 119887 120596119887 with a revised drift andscale is computed using the HDR algorithm

120596119887

= 119887

minus 119887119892 (8)

Here theHDRalgorithm is used to compute the bias value119887119892

42 Attitude Kalman Filter

421 Initialize The state variables to be used in the Kalmanfilter are indicated as a rotationmatrix and as Eulerian anglesand the covariance matrix is indicated as the covariance ofEulerian angles

State variables are as follows

119909 = 119877119899

119887

(State variables for the rotation matrix)

119910 = Λ119899

(State variables for the Eulerian angles) (9)


Covariance matrix of Eulerian angles is as follows

119875 =

[[[

[

12059000

1205900120579

1205900120595

1205900120579

120590120579120579

120590120579120595

1205900120593

120590120579120595

120590120595120595

]]]

]

(10)

State variables and covariancematrix are initialized asfollows

119896=0

= 1198683times3

119896=0

= [0 0 0]119879

119875119896=0

= 108

1198683times3

(11)

422 Attitude Prediction

(i) SystemModelThe angular speed measured by sensors hasdifferentials of the rotation matrix and the following relation

119899

119887

= 119877119899

119887

Ω119887

(12)

Here Ω119887 is the skew-symmetric matrix of 120596119887 and is definedas follows

Ω119887

= 120596119887

times

[[[

[

0 minus120596119911

120596119910

120596119911

0 minus120596119909

minus120596119910

120596119909

0

]]]

]

(13)

In addition the differentials of Eulerian angles have angularspeed and the following relation

Λ119899

= 119862minus1

1

120596119887

(14)

Here 119862minus1

1

is a matrix that converts the angular speedmeasured by the gyro sensor into Eulerian angles

119862minus1

1

=[[

[

1 sin 0 tan 0 cos 0 tan 120579

0 cos 0 minus sin 0

0 sin 0 cos 120579 cos 0 cos 120579

]]

]

(15)

If 120579 asymp plusmn1205872 in the matrix above it is cos 120579 asymp 0 whichgenerates the denominator of matrix element to be close to 0at times However such a case refers to the fact that when thesensor stands vertically this does not occur if one considersthe location and direction when attaching the sensor

(ii) Prediction Consider

minus

119896

= 119896minus1

(1 + Ω119887

Δ119905)

= 119896minus1

119877119911(120596119911Δ119905) 119877119910(120596119910Δ119905) 119877119909(120596119909Δ119905)

minus

119896

= 119896minus1

+ 119862minus1

1

120596119887

Δ119905

(16)

The covariance of Eulerian angles is updated as follows

119875minus

119896

= 119875119896minus1

+ 119862minus1

1

119876119862minus119879

1

(Δ119905)2

(17)

Here 119876 = diag(12059020

1205902

120579

1205902

120593

) is computed as follows

120590119860

= [1205900

120590120579

120590120595]119879

= 10 + 120596119887

(18)

423 Attitude Updates by Gravity

(i) Attitude Measurement Model of Sensors of Gravity Thematrix that indicates the attitude of sensors utilizing 0 and is revised as follows

119877119899

119887

larr997888 119877119910() 119877119909(0) 119877119899

119887

(19)

Furthermore Eulerian angles are revised as follows

Λ119899

larr997888 Λ119899

+ 119862minus1

0

[[[

[

0

0

]]]

]

(20)

Here 119862minus10

is as follows

119862minus1

0

=

[[[[[[

[

cos120595cos 120579

sin120595

cos 1205790

sin120595 cos120595 0

cos120595tan 120579

sin120595

tan 120579

1

]]]]]]

]

(21)

The attitude measurement model function ℎ(sdot) of the sensorutilizes Eulerian angles without changes as follows

ℎ (Λ119899

) = Λ119899

(22)

The attitude measurement value 119885119896is computed as follows

utilizing 0 and

119885119896= Λ119899

+ 119862minus1

0

[[[

[

0

0

]]]

]

(23)

(ii) Attitude Update One has

119896= 119877119910(119870120579Δ120579) 119877

119909(1198700Δ0) minus

119896

119896=

minus

119896

+ 119870119896(119911119896minus ℎ (

minus

119896

))

(24)

Here 1198700is column 1 row 1 element of 119870

119896 and 119870

120579is column

2 row 2 elements of 119870119896

119875119896= 119875minus

119896

= 119870119896119875minus

119896

119870119896= 119875minus

119896

(119875minus

119896

+ 119862minus1

0

119877119862minus119879

0

)

minus1

(25)

Here 119877 = diag(12059020

1205902

120579

1205902

120593

) is computed as below because thebenefit must increase when acceleration is near 1 g and itshould decrease when the size of acceleration exceeds 1 g

1205900= 120590120579= 01 +

10038161003816100381610038161003816

1003817100381710038171003817100381711989111988710038171003817100381710038171003817minus 1

10038161003816100381610038161003816+ 100

1003817100381710038171003817100381712059611988710038171003817100381710038171003817

120590120595

= 108

(26)


424 Attitude Update by Level Sensors (DIFF) The fueltank of a vehicle has its direction of motion set to the 119909-axis (heading) so the attitude may be updated with therotation angle data obtained by level sensors First velocityand rotation angle DIFF obtained from the IMU are broughtin Then from the rotation matrix 119877

119899

119887

the yaw angle of thevehicle can be computed as follows

Ψ120592= tanminus1 (119886

21

11988611

) (27)

Here 119886119894119895is the matrix ldquo119868rdquo of the rotation matrix 119877

119899

119887

and theelement of column ldquo119895rdquo

The matrix indicating the attitude of the fuel tank fromthe angle and direction of vehicle Ψ

120592measured by level

sensors is revised as follows

119877119899

119887

larr997888 119877119911(Ψ) 119877

119899

119887

Ψ cong Ψlevel minus Ψ

(28)


Λ119899

= Λ119899

+ 119862minus1

0

[[[

[

0

0

Ψ

]]]

]

(29)

The attitude measurement model function ℎ(lowast) of a fuel tankstill utilizes the Eulerian angles as follows

ℎ (Λ119899

) = Λ119899

(30)


utilizing Ψ

119885119896= Λ119899

+ 119862minus1

0

[[[

[

0

0

Ψ

]]]

]

(31)

(i) Attitude UpdateOne has

119896= 119877119911(119870ΨΨ) minus

119896

119896

= minus

119896

+ 119870119896(119911119896minus ℎ (

minus

119896

))

(32)

Here 119870Ψis a 3 times 3 element of 119870

119896

119877119896= 119875minus

119896

minus 119870119896119875minus

119896

119870119896= 119875minus

119896

(119875minus

119896

+ 119862minus1

0

119877119862minus119879

0

)

minus1

(33)

Here 119877 = diag(12059020

1205902

120579

1205902

120593


1205900= 120590120579= 108

120590120595

=2120590level

120598 + 119881IMU 120598 = 10

minus6

120590level = 10

(34)

425 Attitude Measurement of Vehicles by Gravity Since theacceleration of gravity is always toward the center of the earth119892 = (0 0 minus981ms2) is measured if another force is notapplied to the acceleration sensor The sensor attitude canbe revised by comparing the acceleration measured by theacceleration sensor and the acceleration of gravity Howeverthis is possible if gravity is the only force applied to theacceleration sensor Since it is not possible to separate theacceleration of gravity from other forces and measure it thesimplest method is to assess if the condition of the appliedforces other than the acceleration of gravity is 119886 asymp 119892

119886 = [119886119909

119886119910

119886119911] the acceleration measured by the

acceleration sensor includes various types of acceleration inaccordance with the acceleration of gravity and the accelera-tion of the sensor As an equation this would be as follows

119886 = V + V times 120596 + 119877119879

119892

(∵ 119877119879

= 119877minus1

)

=[[

[

V119909

V119910

V119911

]]

]

+

[[[

[

0 V119911

minusV119910

minusV119911

0 V119909

V119910

minusV119909

0

]]]

]

[[[

[

119908119909

119908119910

119908119911

]]]

]

+ 119892119911

[[

[

minus sin 120579

cos 120579 sin120601

cos 120579 cos120601

]]

]

(35)

5 Capacity Measuring Algorithmthrough Revised Rotation Angle Data andMeasurement Value of Level Pressure

Capacity is computed as a digital value through theADC board When the acquired ADC and capacity are =Sensoradc(ℓadc) the equation to compute the capacity is asfollows

51 Level Kalman Filter

511 System Model The difference value of the level sensormeasured in the coordinate system of the fuel tank isconverted into a navigation system

level119899

= 119877119899

119887

Diff119887level (36)

The formula above ismade into an equation of state as follows

[

level119899

level119899] = [

1 119877119899

119887

0 1

][

level119899

level119899] (37)

State variables are

119896= [

level119899

level119899] (38)


Table 2 26 PC Series performance characteristics [7]

Min Type Max UnitsExcitation mdash 10 16 VdcResponse time mdash mdash 10 msInput resistance 55 k 75 k 115 k OhmOutput resistance 15 k 25 k 30 k OhmWeight 2 Gram

Figure 7 Level sensor (side view of 26 PCSPS pressure sensorsused)

512 Prediction In the formula below119866 has a bias value dueto noise and the level 119865 has [

0 0

119868 0

] The equation depends onthe general Kalman equation

minus

119896

= 119860119896minus1

+ 119866119906119896Δ119905

119875minus

119896

= 119860119875119896minus1

119860119879

+ 119866119876119866119879

Δ1199052

119860 = 119868 + 119865Δ119905

119876 = diag (1205902

119891119909

1205902

119891119910

1205902

119891119911

1205902

119892119909

1205902

119892119910

1205902

119892119911

)

(39)

6 Experiment

Table 2 shows the specifications of the pressure sensor appliedto the system

The display device of the algorithm built on this researchhas a level sensor (Figure 7) to measure absolute pressureand the tilting angle an IMU (Figure 8) for measuring therotation angle acceleration and so forth a programmableAMP (Figure 9) that can consistently program the increaserate of pressure between sensors by amplifying the valuesof pressure sensors in the desired offset and a test bed(Figure 10)

Capacity Measurement of Distilled Water Using Level SensorsIn order to measure capacity using level sensors the sensoroutput must have a linear output in accordance with thecapacity To assess it the research team sequentially inserted1 Lndash11 L of water (distilled water density asymp 1) into the test bedand assessed each sensor value In order to avoid continuity

Figure 8 IMU analog device ADIS16365

Figure 9 AMP analog device AD8555

Figure 10 Test bed internally developed

and obtain objective data the experiment was conductedonce a day over three days After the experiment the teamremoved all distilled water from the test bed and conductedthe experiment again

Figure 11 demonstrates a 1 L75ADC value and 025V75ADC value Figure 11 indicates the difference of values inthe first second and third measurements and each sensorvalue with a margin of error of plusmn1 The output graph ofSensors numbers 5 and 6 differs from the output graph ofSensors numbers 1 2 3 and 4 owing to the installed locationSensor number 7 reacts when it is over 11 L because it waslocated on the upper part of the test bed to measure density

The level sensors used in this research are linear overalldemonstrating that capacity and density computation is pos-sible However the margin of error increases with rotationFigure 12 shows the output value measured three times at a0∘ 15∘ 30∘ and 45

∘ rotation and it indicates that the valueincreased or decreased significantly in accordance with therotation angle


1-liter to 11-liter linear increase sensor data (average SD)

Sensor 2

Sensor 1

Sensor 3



Sensor 5Sensor 6Sensor 7

Sensor 1Sensor 2 Sensor 3

Sensor 4

0100200300400500600700800900

Sens

or d

ata (

AD

C va

lue)

SD standard deviation

0100200300400500600700800

Sens

or d

ata (

AD

C va

lue)

2 3 4 5 6 7 8 9 10 111Water capacity (liter)


Figure 11 Output value of sensor in accordance with water quantity (1ndash11 L)

Sensor 3Sensor 4

Sensor 5Sensor 6

11-liter tilt (Sensor 3 derection)

0100200300400500600700800900

Sens

or d

ata (

AD

C va

lue)

15 30 450Tilt angle (deg)

(a)


Sensor 3Sensor 4

Sensor 5Sensor 6

0100200300400500600700800900

Sens

or d

ata (

AD

C va

lue)


(b)

Figure 12 Output graph of sensor in accordance with rotation (a) Tilt toward Sensor 3 (b) tilt toward Sensor 4


Revision of rotation angle and acceleration of gravity

Sensor 1Sensor 2Sensor 3Sensor 4


0

200

400

600

800

1000

Sens

or d

ata (

AD

C va

lue)

10 15 20 25 30 35 405Tilt angle (deg)

Figure 13 Measurement result graph using algorithm that hasrevised rotation angle and acceleration of gravity

05

101520253035404550

Mea

sure

d an

gle o

f IM

U (d

eg)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 171The number of repetitions

5deg10deg15deg20deg25deg

30deg35deg40deg45deg

Figure 14 Result graph of measuring the rotation angle using IMUdata

When tilting toward Sensor 3 the value of Sensor 3 andSensor 5 on the Sensor 3-axis increases whereas the value ofSensor 4 and Sensor 6 on the opposite side decreases Whentilting toward Sensor 4 the value of Sensor 4 and Sensor 6 onthe Sensor 4-axis increases whereas the value of Sensor 0 andSensor 2 on the opposite side decreases Sensor 1 and Sensor2 which are irrelevant to the tilting direction demonstratedno change and so the team has not included it here

The experiment results indicate that measuring capacityand density only with a level sensor generates a large marginof error when there is tilting involved Therefore it isinadequate to measure the capacity and density of a liquidusing only level sensor

When revising the rotation angle and acceleration ofgravity with the algorithm this paper proposes the measure-ment results are obtained in Figure 13

The research team inserted 11 L of distilled water intothe test bed tilted it by 5∘ and measured the sensor valueFigure 13 demonstrates a stable result compared with thesensor graph before the application of the algorithm This



05

101520253035404550

Mea

sure

d an

gle o

f lev

el se

nsor

(deg

)


Figure 15 Result graph of measuring the rotation angle using levelsensor data



05

101520253035404550

Mea

sure

d an

gle o

f sug

geste

d A

L (d

eg)


Figure 16 Result graph of measuring the rotation angle with theproposed algorithm

is because the accurate estimation of the rotation angle andits revision value were designed with a Kalman filter Thusestimating the rotation angle is important in this algorithm

The comparative results of the proposed algorithm thatestimates the rotation angle are as shown in Figure 14

Figure 14 demonstrates the measurements of rotationangles using only the IMU the measurement is carried outonce every minute As indicated in Figure 14 the marginof error continuously accumulated demonstrating a largedifference in value after 17min from the beginning Further-more when measuring the rotation angle with only levelsensors (Figure 15) the margin of error does not accumulateand is drifting as with the IMU data but the margin of erroris significantly large

Figure 16 shows the rotation angle using the algorithmproposed by this paper The margin of error is within plusmn5which is relatively accurate compared with existing methods


3(Reset)

Linearity of suggested AL (sensor output)



2 3 4 5 6 7 8 9 10 11 121

Water capacity (liter)

0100200300400500600700800

Sens

or o

utpu

t (A

DC

valu

e)

Figure 17 Measurement graph of capacity and density using the proposed algorithm

Accuracy of suggested AL (liter)

LiterMeasure liter

02468

101214

Mea

sure

d w

ater

capa

city

(lite

r)

2 3 4 5 6 7 8 9 10 11 12 131Water capacity (liter)

(a)

Accuracy of suggested AL (density)

Density

Mea

sure

d de

nsity

(kg

m3 )

0

02

04

06

08

1

12


(b)

Figure 18 Comparative graph 1 to assess the precision of capacity and density measurements (a) Accuracy graph of suggested algorithm forcapacity (b) accuracy graph of suggested algorithm for density

The results of measuring capacity and density using thealgorithm this paper has proposed are as shown in Figure 17

Figure 17 shows the output values of the experimentwhere 1 L was inserted as a sensor value without tiltingHowever themeasurements were not conducted sequentiallybecause each sensor has a slightly different minimum pres-sure value that computes the sensor output and because thesensor locations differ as well For precise measurementsthe research team inserted 3 L of distilled water and whenall sensors were reacting sufficiently the team initializedthe value and measured again As demonstrated in Fig-ure 17 accurate results from the measuring capacity could beobtained because the sensor value was close to the primaryequation and each sensor had less difference in value

Figure 18 is the computation of density and capacitywhich is very precise because it has a margin of error of onlyplusmn005 L compared with the actual capacity The density wasalso very precise with a margin of error of plusmn002 kgm3

Sensor number 4 and Sensor number 7 are used tomeasure the density Sensor number 7 is placed on the upperpart so it reacts only when the solution is more than 10 litersTherefore the density can be measured from 10 liters But if

the sensor can be placed lower we can measure the densityby using less fluid

In addition Figure 19 demonstrates the results of themeasurements with the same method as above but tilting by5∘

As indicated in Figure 19 the margin of error of mea-surement increases in accordance with the tilting and themaximummargin of error was only 05 L Since it can be pre-sumed that the larger the target capacity formeasurement thelesser the error ratio this can be said to be a precise algorithm

As demonstrated in Figure 20 compared with the resultswhen there was no tilting there has rarely been any differencein density and the result was plusmn005 kgm3

For Figure 21 the program was implemented with agraphical user interface (GUI) and its resulting values werecontinuously obtained and observed

7 Conclusion

This research proposed an algorithm using a level sensor andIMU sensor to improve the limitations of the existing meth-ods of measuring the liquid capacity For this the research


Capacity relative error with rotation angle

Relative error (average)

0

01

02

03

04

05

Capa

city

relat

ive e

rror

(lite

r)10 15 20 25 30 35 405

Rotation angle (deg)

Figure 19 Comparative graph 2 to assess the precision of measurements of capacity and density

Density measurements with respect to rotation angle

5deg0deg

10deg15deg20deg



Den

sity

(kg

m3 )

0

02

04

06

08

1

12


Figure 21 GUI to observe the measurement values in real time


team installed level sensors and IMU sensors in a liquid tankand conducted sensing in real time The proposed algorithmutilized the strengths of each sensor and supplemented theirweaknesses which were assessed through experiments

A measurement evaluation of the liquid level sensor wasconducted by increasing and decreasing the liquid in thetank The team increased the water level by units of 1 L andanalyzed the output characteristics of up to 12 L As a resultthe system demonstrated a highly linear precision that waswithin plusmn05 The offset voltage of the liquid level sensorwas 935mV with a sensitivity of 3635mVL The outputcharacteristic in inclined environments resulted in a value of1 FS In addition the sensitivity to density measurementswas 108528 [kgm3] which is consistent with the computedvalue of the sensor output The reproducibility of the sensorshowed minute hysteresis but since there was repeatabilitya high-precision pressure level sensor was implemented as itcan ignore this repeatability

The research team believes that with the proposedalgorithm precise liquid capacity measurements in real timewould be possible even during movement or tilting of theliquid tank

Competing Interests

The authors declare no conflict of interests

References

[1] K Sohn J Kim S Cho and J Shim ldquoA study on the tank liquid-levelmonitoring sensor systems for large scaled vesselsrdquo Journalof the Korean Society of Marine Engineering vol 33 no 2 pp330ndash335 2009

[2] N Min ldquoSensor electronicsrdquo Dongilbook pp 386ndash401 2007[3] Level sensor Wikipedia httpsenwikipediaorgwikiLevel

sensor[4] S M Chandani and N A F Jaeger ldquoOptical fiber-based liquid

level sensorrdquo Optical Engineering vol 46 no 11 Article ID114401 2007

[5] J A Morris and C R Pollock ldquoA digital fiber-optic liquid levelsensorrdquo Journal of Lightwave Technology vol 5 no 7 pp 920ndash925 1987

[6] G Nagy Fuel Level Sensor Design from a System Perspective1997

[7] 26PC Series Honeywell httpsensinghoneywellcom[8] B Park Improving of the performance for wireless water level

controller using the full-duplex communication module [MSthesis] Mokpo National University Muan Republic of Korea2005

[9] OTT PLS-Pressure Level Sensor OTT httpwwwottcom[10] C4651-continuous float level sensor Madison httpsmadison-

cocom[11] Modulevel Pneumatic displacer liquid level controlMagnetrol

httpusmagnetrolcom[12] Capacitance Level measurement Liquicap FMI51 Endress+

hauser httpwwwendresscom[13] YO-YO Series SEMRAD httpwwwsemradcomau[14] FIBERTRAC31 VEGA httpswwwvegacom

[15] Rosemount 3100 Series Level Transmitters EMERSON httpwww2emersonprocesscom

[16] Rosemount 5400 Radar Level Transmitter EMERSONhttpwww2emersonprocesscom

[17] AKulkarni RNKarekar andRCAiyer ldquoLiquid level sensorrdquoReview of Scientific Instruments vol 76 no 10 Article ID 105108pp 1ndash5 2005

[18] K A P Menon and R Hariharan ldquoA new liquid level sensor forprocess-control applicationsrdquo IEEE Transactions on Instrumen-tation and Measurement vol 28 no 2 pp 155ndash158 1979

[19] J Bayliss ldquoApplying sensors for level controlrdquo Control andInstrumentation vol 16 no 7 pp 45ndash47 1984

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Table 1 Comparison of sensors used for measuring liquid level

Sensor type Scope of measurement [m] Precision Temperature limits [∘C] Pressure limits[kgfcm2]

Pressure type [9] 01ndash100 plusmn005 [FS] 70 mdashFloat type [10] Customizable 635mm 105 mdashDisplacement type [11] 03ndash3 plusmn2 [] 370 300Capacitance type [12] 01ndash4 plusmn01 [] 200 102Weight level type [13] 305 10 [mm] 49 mdash

Radiation type [14] 7 05 [] 50 No limitation(noncontact)

Ultrasonic type [15] 8ndash11 plusmn05 [] 70 3Radar type [16] 30 plusmn10 By the strength of windows

Level sensor

10

(a)

Measurement value

Measurement error

Levelsensor

120579

(b)

Measurement error

Level sensor

Measurementvalue

(c)

Figure 1 Disadvantages of current fuel-measuring methods for vehicles (a) Structural limits result in inability to measure the remaining10 of fuel (b) Fuel-measuring error on slopes (c) Fuel-measuring error when accelerating

with the rotation acceleration and acceleration of gravity [8]The unique approach of this paper is that the relative valuein accordance with the location of the level sensor and therotation angle data of the IMU are both used to measure theposition of a fuel tank

The reasons are as follows The IMU outputs a precisemeasurable value for the rotation angle and accelerationmea-surements but its disadvantage is that such values are accu-mulated with time by an integration constant In the case ofa level sensor it canmeasure the tilting angle and accelerationin accordance with the measuring value difference locatedstructurally but the measurements may not be accurate

The proposed algorithm of this paper guarantees pre-cision and credibility because it uses the Kalman filter torealize a complementary filter with precision despite the error

accumulation of the IMU and the imprecision and non-accumulation of errors of the level sensor

3 Sensor Measurement Algorithm for PreciseMeasurements of Sensors

31 Measuring Algorithm of Capacity As demonstrated inFigure 3 the capacity can be computed in accordance withthe container volume and the input capacity of the liquid

The mass can be computed using the pressure applied tothe bottom of a container with liquid First the calculationequation of the capacity is as follows

Pressure applied to the bottom of the container

prop liquid level times liquid density(1)







20cm

P (kgm3)

A (m2) (area)

h (cm)

20 cm





119875 = 120588 times 119892 times ℎ




(3)



Sensor 2

Sensor 1

Sensor 3


Sensor 7

Sensing unit

Liquid tank




2


2

C =Sensor

5 out + Sensor6 out

2

(4)

(A + B + C)

3



(5)




Sensor 4


P1

P2

h1

h2




2 outADC120579ref



4 outADC120579ref



(6)





= Pressure (pa)


120588 =1198751minus 1198752

119892 (ℎ2minus ℎ1)

(7)


Gyro ADC

HDR ADCtoLEVEL


DIFF_level

Level

DIFF_angle

gn

Rn



1and 119875






120596119887

= 119887

minus 119887119892 (8)





119909 = 119877119899

119887


119910 = Λ119899




119875 =

[[[

[

12059000

1205900120579

1205900120595

1205900120579

120590120579120579

120590120579120595

1205900120593

120590120579120595

120590120595120595

]]]

]

(10)


119896=0

= 1198683times3

119896=0

= [0 0 0]119879

119875119896=0

= 108

1198683times3

(11)



119899

119887

= 119877119899

119887

Ω119887

(12)


Ω119887

= 120596119887

times

[[[

[

0 minus120596119911

120596119910

120596119911

0 minus120596119909

minus120596119910

120596119909

0

]]]

]

(13)


Λ119899

= 119862minus1

1

120596119887

(14)

Here 119862minus1

1


119862minus1

1

=[[

[


0 cos 0 minus sin 0

0 sin 0 cos 120579 cos 0 cos 120579

]]

]

(15)



minus

119896

= 119896minus1

(1 + Ω119887

Δ119905)

= 119896minus1

119877119911(120596119911Δ119905) 119877119910(120596119910Δ119905) 119877119909(120596119909Δ119905)

minus

119896

= 119896minus1

+ 119862minus1

1

120596119887

Δ119905

(16)


119875minus

119896

= 119875119896minus1

+ 119862minus1

1

119876119862minus119879

1

(Δ119905)2

(17)

Here 119876 = diag(12059020

1205902

120579

1205902

120593


120590119860

= [1205900

120590120579

120590120595]119879

= 10 + 120596119887

(18)



119877119899

119887

larr997888 119877119910() 119877119909(0) 119877119899

119887

(19)


Λ119899

larr997888 Λ119899

+ 119862minus1

0

[[[

[

0

0

]]]

]

(20)

Here 119862minus10

is as follows

119862minus1

0

=

[[[[[[

[

cos120595cos 120579

sin120595

cos 1205790

sin120595 cos120595 0

cos120595tan 120579

sin120595

tan 120579

1

]]]]]]

]

(21)


ℎ (Λ119899

) = Λ119899

(22)


utilizing 0 and

119885119896= Λ119899

+ 119862minus1

0

[[[

[

0

0

]]]

]

(23)


119896= 119877119910(119870120579Δ120579) 119877

119909(1198700Δ0) minus

119896

119896=

minus

119896

+ 119870119896(119911119896minus ℎ (

minus

119896

))

(24)


119896 and 119870

120579is column


119875119896= 119875minus

119896

= 119870119896119875minus

119896

119870119896= 119875minus

119896

(119875minus

119896

+ 119862minus1

0

119877119862minus119879

0

)

minus1

(25)

Here 119877 = diag(12059020

1205902

120579

1205902

120593


1205900= 120590120579= 01 +

10038161003816100381610038161003816

1003817100381710038171003817100381711989111988710038171003817100381710038171003817minus 1

10038161003816100381610038161003816+ 100

1003817100381710038171003817100381712059611988710038171003817100381710038171003817

120590120595

= 108

(26)



119899

119887


Ψ120592= tanminus1 (119886

21

11988611

) (27)


119899

119887





119877119899

119887

larr997888 119877119911(Ψ) 119877

119899

119887


(28)


Λ119899

= Λ119899

+ 119862minus1

0

[[[

[

0

0

Ψ

]]]

]

(29)


ℎ (Λ119899

) = Λ119899

(30)


utilizing Ψ

119885119896= Λ119899

+ 119862minus1

0

[[[

[

0

0

Ψ

]]]

]

(31)


119896= 119877119911(119870ΨΨ) minus

119896

119896

= minus

119896

+ 119870119896(119911119896minus ℎ (

minus

119896

))

(32)


119896

119877119896= 119875minus

119896

minus 119870119896119875minus

119896

119870119896= 119875minus

119896

(119875minus

119896

+ 119862minus1

0

119877119862minus119879

0

)

minus1

(33)

Here 119877 = diag(12059020

1205902

120579

1205902

120593


1205900= 120590120579= 108

120590120595

=2120590level

120598 + 119881IMU 120598 = 10

minus6

120590level = 10

(34)


119886 = [119886119909

119886119910



119886 = V + V times 120596 + 119877119879

119892

(∵ 119877119879

= 119877minus1

)

=[[

[

V119909

V119910

V119911

]]

]

+

[[[

[

0 V119911

minusV119910

minusV119911

0 V119909

V119910

minusV119909

0

]]]

]

[[[

[

119908119909

119908119910

119908119911

]]]

]

+ 119892119911

[[

[

minus sin 120579

cos 120579 sin120601

cos 120579 cos120601

]]

]

(35)





level119899

= 119877119899

119887



[

level119899

level119899] = [

1 119877119899

119887

0 1

][

level119899

level119899] (37)

State variables are

119896= [

level119899

level119899] (38)






0 0

119868 0


minus

119896

= 119860119896minus1

+ 119866119906119896Δ119905

119875minus

119896

= 119860119875119896minus1

119860119879

+ 119866119876119866119879

Δ1199052

119860 = 119868 + 119865Δ119905

119876 = diag (1205902

119891119909

1205902

119891119910

1205902

119891119911

1205902

119892119909

1205902

119892119910

1205902

119892119911

)

(39)

6 Experiment













Sensor 2

Sensor 1

Sensor 3





Sensor 4

0100200300400500600700800900

Sens

or d

ata (

AD

C va

lue)


0100200300400500600700800

Sens

or d

ata (

AD

C va

lue)




Sensor 3Sensor 4

Sensor 5Sensor 6


0100200300400500600700800900

Sens

or d

ata (

AD

C va

lue)


(a)


Sensor 3Sensor 4

Sensor 5Sensor 6

0100200300400500600700800900

Sens

or d

ata (

AD

C va

lue)


(b)






0

200

400

600

800

1000

Sens

or d

ata (

AD

C va

lue)

10 15 20 25 30 35 405Tilt angle (deg)


05

101520253035404550

Mea

sure

d an

gle o

f IM

U (d

eg)











05

101520253035404550

Mea

sure

d an

gle o

f lev

el se

nsor

(deg

)





05

101520253035404550

Mea

sure

d an

gle o

f sug

geste

d A

L (d

eg)








3(Reset)




2 3 4 5 6 7 8 9 10 11 121


0100200300400500600700800

Sens

or o

utpu

t (A

DC

valu

e)



LiterMeasure liter

02468

101214

Mea

sure

d w

ater

capa

city

(lite

r)


(a)


Density

Mea

sure

d de

nsity

(kg

m3 )

0

02

04

06

08

1

12


(b)











7 Conclusion





0

01

02

03

04

05

Capa

city

relat

ive e

rror

(lite

r)10 15 20 25 30 35 405




5deg0deg

10deg15deg20deg



Den

sity

(kg

m3 )

0

02

04

06

08

1

12







Competing Interests


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra















20cm

P (kgm3)

A (m2) (area)

h (cm)

20 cm





119875 = 120588 times 119892 times ℎ




(3)



Sensor 2

Sensor 1

Sensor 3


Sensor 7

Sensing unit

Liquid tank




2


2

C =Sensor

5 out + Sensor6 out

2

(4)

(A + B + C)

3



(5)




Sensor 4


P1

P2

h1

h2




2 outADC120579ref



4 outADC120579ref



(6)





= Pressure (pa)


120588 =1198751minus 1198752

119892 (ℎ2minus ℎ1)

(7)


Gyro ADC

HDR ADCtoLEVEL


DIFF_level

Level

DIFF_angle

gn

Rn



1and 119875






120596119887

= 119887

minus 119887119892 (8)





119909 = 119877119899

119887


119910 = Λ119899




119875 =

[[[

[

12059000

1205900120579

1205900120595

1205900120579

120590120579120579

120590120579120595

1205900120593

120590120579120595

120590120595120595

]]]

]

(10)


119896=0

= 1198683times3

119896=0

= [0 0 0]119879

119875119896=0

= 108

1198683times3

(11)



119899

119887

= 119877119899

119887

Ω119887

(12)


Ω119887

= 120596119887

times

[[[

[

0 minus120596119911

120596119910

120596119911

0 minus120596119909

minus120596119910

120596119909

0

]]]

]

(13)


Λ119899

= 119862minus1

1

120596119887

(14)

Here 119862minus1

1


119862minus1

1

=[[

[


0 cos 0 minus sin 0

0 sin 0 cos 120579 cos 0 cos 120579

]]

]

(15)



minus

119896

= 119896minus1

(1 + Ω119887

Δ119905)

= 119896minus1

119877119911(120596119911Δ119905) 119877119910(120596119910Δ119905) 119877119909(120596119909Δ119905)

minus

119896

= 119896minus1

+ 119862minus1

1

120596119887

Δ119905

(16)


119875minus

119896

= 119875119896minus1

+ 119862minus1

1

119876119862minus119879

1

(Δ119905)2

(17)

Here 119876 = diag(12059020

1205902

120579

1205902

120593


120590119860

= [1205900

120590120579

120590120595]119879

= 10 + 120596119887

(18)



119877119899

119887

larr997888 119877119910() 119877119909(0) 119877119899

119887

(19)


Λ119899

larr997888 Λ119899

+ 119862minus1

0

[[[

[

0

0

]]]

]

(20)

Here 119862minus10

is as follows

119862minus1

0

=

[[[[[[

[

cos120595cos 120579

sin120595

cos 1205790

sin120595 cos120595 0

cos120595tan 120579

sin120595

tan 120579

1

]]]]]]

]

(21)


ℎ (Λ119899

) = Λ119899

(22)


utilizing 0 and

119885119896= Λ119899

+ 119862minus1

0

[[[

[

0

0

]]]

]

(23)


119896= 119877119910(119870120579Δ120579) 119877

119909(1198700Δ0) minus

119896

119896=

minus

119896

+ 119870119896(119911119896minus ℎ (

minus

119896

))

(24)


119896 and 119870

120579is column


119875119896= 119875minus

119896

= 119870119896119875minus

119896

119870119896= 119875minus

119896

(119875minus

119896

+ 119862minus1

0

119877119862minus119879

0

)

minus1

(25)

Here 119877 = diag(12059020

1205902

120579

1205902

120593


1205900= 120590120579= 01 +

10038161003816100381610038161003816

1003817100381710038171003817100381711989111988710038171003817100381710038171003817minus 1

10038161003816100381610038161003816+ 100

1003817100381710038171003817100381712059611988710038171003817100381710038171003817

120590120595

= 108

(26)



119899

119887


Ψ120592= tanminus1 (119886

21

11988611

) (27)


119899

119887





119877119899

119887

larr997888 119877119911(Ψ) 119877

119899

119887


(28)


Λ119899

= Λ119899

+ 119862minus1

0

[[[

[

0

0

Ψ

]]]

]

(29)


ℎ (Λ119899

) = Λ119899

(30)


utilizing Ψ

119885119896= Λ119899

+ 119862minus1

0

[[[

[

0

0

Ψ

]]]

]

(31)


119896= 119877119911(119870ΨΨ) minus

119896

119896

= minus

119896

+ 119870119896(119911119896minus ℎ (

minus

119896

))

(32)


119896

119877119896= 119875minus

119896

minus 119870119896119875minus

119896

119870119896= 119875minus

119896

(119875minus

119896

+ 119862minus1

0

119877119862minus119879

0

)

minus1

(33)

Here 119877 = diag(12059020

1205902

120579

1205902

120593


1205900= 120590120579= 108

120590120595

=2120590level

120598 + 119881IMU 120598 = 10

minus6

120590level = 10

(34)


119886 = [119886119909

119886119910



119886 = V + V times 120596 + 119877119879

119892

(∵ 119877119879

= 119877minus1

)

=[[

[

V119909

V119910

V119911

]]

]

+

[[[

[

0 V119911

minusV119910

minusV119911

0 V119909

V119910

minusV119909

0

]]]

]

[[[

[

119908119909

119908119910

119908119911

]]]

]

+ 119892119911

[[

[

minus sin 120579

cos 120579 sin120601

cos 120579 cos120601

]]

]

(35)





level119899

= 119877119899

119887



[

level119899

level119899] = [

1 119877119899

119887

0 1

][

level119899

level119899] (37)

State variables are

119896= [

level119899

level119899] (38)






0 0

119868 0


minus

119896

= 119860119896minus1

+ 119866119906119896Δ119905

119875minus

119896

= 119860119875119896minus1

119860119879

+ 119866119876119866119879

Δ1199052

119860 = 119868 + 119865Δ119905

119876 = diag (1205902

119891119909

1205902

119891119910

1205902

119891119911

1205902

119892119909

1205902

119892119910

1205902

119892119911

)

(39)

6 Experiment













Sensor 2

Sensor 1

Sensor 3





Sensor 4

0100200300400500600700800900

Sens

or d

ata (

AD

C va

lue)


0100200300400500600700800

Sens

or d

ata (

AD

C va

lue)




Sensor 3Sensor 4

Sensor 5Sensor 6


0100200300400500600700800900

Sens

or d

ata (

AD

C va

lue)


(a)


Sensor 3Sensor 4

Sensor 5Sensor 6

0100200300400500600700800900

Sens

or d

ata (

AD

C va

lue)


(b)






0

200

400

600

800

1000

Sens

or d

ata (

AD

C va

lue)

10 15 20 25 30 35 405Tilt angle (deg)


05

101520253035404550

Mea

sure

d an

gle o

f IM

U (d

eg)











05

101520253035404550

Mea

sure

d an

gle o

f lev

el se

nsor

(deg

)





05

101520253035404550

Mea

sure

d an

gle o

f sug

geste

d A

L (d

eg)








3(Reset)




2 3 4 5 6 7 8 9 10 11 121


0100200300400500600700800

Sens

or o

utpu

t (A

DC

valu

e)



LiterMeasure liter

02468

101214

Mea

sure

d w

ater

capa

city

(lite

r)


(a)


Density

Mea

sure

d de

nsity

(kg

m3 )

0

02

04

06

08

1

12


(b)











7 Conclusion





0

01

02

03

04

05

Capa

city

relat

ive e

rror

(lite

r)10 15 20 25 30 35 405




5deg0deg

10deg15deg20deg



Den

sity

(kg

m3 )

0

02

04

06

08

1

12







Competing Interests


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Sensor 4


P1

P2

h1

h2




2 outADC120579ref



4 outADC120579ref



(6)





= Pressure (pa)


120588 =1198751minus 1198752

119892 (ℎ2minus ℎ1)

(7)


Gyro ADC

HDR ADCtoLEVEL


DIFF_level

Level

DIFF_angle

gn

Rn



1and 119875






120596119887

= 119887

minus 119887119892 (8)





119909 = 119877119899

119887


119910 = Λ119899




119875 =

[[[

[

12059000

1205900120579

1205900120595

1205900120579

120590120579120579

120590120579120595

1205900120593

120590120579120595

120590120595120595

]]]

]

(10)


119896=0

= 1198683times3

119896=0

= [0 0 0]119879

119875119896=0

= 108

1198683times3

(11)



119899

119887

= 119877119899

119887

Ω119887

(12)


Ω119887

= 120596119887

times

[[[

[

0 minus120596119911

120596119910

120596119911

0 minus120596119909

minus120596119910

120596119909

0

]]]

]

(13)


Λ119899

= 119862minus1

1

120596119887

(14)

Here 119862minus1

1


119862minus1

1

=[[

[


0 cos 0 minus sin 0

0 sin 0 cos 120579 cos 0 cos 120579

]]

]

(15)



minus

119896

= 119896minus1

(1 + Ω119887

Δ119905)

= 119896minus1

119877119911(120596119911Δ119905) 119877119910(120596119910Δ119905) 119877119909(120596119909Δ119905)

minus

119896

= 119896minus1

+ 119862minus1

1

120596119887

Δ119905

(16)


119875minus

119896

= 119875119896minus1

+ 119862minus1

1

119876119862minus119879

1

(Δ119905)2

(17)

Here 119876 = diag(12059020

1205902

120579

1205902

120593


120590119860

= [1205900

120590120579

120590120595]119879

= 10 + 120596119887

(18)



119877119899

119887

larr997888 119877119910() 119877119909(0) 119877119899

119887

(19)


Λ119899

larr997888 Λ119899

+ 119862minus1

0

[[[

[

0

0

]]]

]

(20)

Here 119862minus10

is as follows

119862minus1

0

=

[[[[[[

[

cos120595cos 120579

sin120595

cos 1205790

sin120595 cos120595 0

cos120595tan 120579

sin120595

tan 120579

1

]]]]]]

]

(21)


ℎ (Λ119899

) = Λ119899

(22)


utilizing 0 and

119885119896= Λ119899

+ 119862minus1

0

[[[

[

0

0

]]]

]

(23)


119896= 119877119910(119870120579Δ120579) 119877

119909(1198700Δ0) minus

119896

119896=

minus

119896

+ 119870119896(119911119896minus ℎ (

minus

119896

))

(24)


119896 and 119870

120579is column


119875119896= 119875minus

119896

= 119870119896119875minus

119896

119870119896= 119875minus

119896

(119875minus

119896

+ 119862minus1

0

119877119862minus119879

0

)

minus1

(25)

Here 119877 = diag(12059020

1205902

120579

1205902

120593


1205900= 120590120579= 01 +

10038161003816100381610038161003816

1003817100381710038171003817100381711989111988710038171003817100381710038171003817minus 1

10038161003816100381610038161003816+ 100

1003817100381710038171003817100381712059611988710038171003817100381710038171003817

120590120595

= 108

(26)



119899

119887


Ψ120592= tanminus1 (119886

21

11988611

) (27)


119899

119887





119877119899

119887

larr997888 119877119911(Ψ) 119877

119899

119887


(28)


Λ119899

= Λ119899

+ 119862minus1

0

[[[

[

0

0

Ψ

]]]

]

(29)


ℎ (Λ119899

) = Λ119899

(30)


utilizing Ψ

119885119896= Λ119899

+ 119862minus1

0

[[[

[

0

0

Ψ

]]]

]

(31)


119896= 119877119911(119870ΨΨ) minus

119896

119896

= minus

119896

+ 119870119896(119911119896minus ℎ (

minus

119896

))

(32)


119896

119877119896= 119875minus

119896

minus 119870119896119875minus

119896

119870119896= 119875minus

119896

(119875minus

119896

+ 119862minus1

0

119877119862minus119879

0

)

minus1

(33)

Here 119877 = diag(12059020

1205902

120579

1205902

120593


1205900= 120590120579= 108

120590120595

=2120590level

120598 + 119881IMU 120598 = 10

minus6

120590level = 10

(34)


119886 = [119886119909

119886119910



119886 = V + V times 120596 + 119877119879

119892

(∵ 119877119879

= 119877minus1

)

=[[

[

V119909

V119910

V119911

]]

]

+

[[[

[

0 V119911

minusV119910

minusV119911

0 V119909

V119910

minusV119909

0

]]]

]

[[[

[

119908119909

119908119910

119908119911

]]]

]

+ 119892119911

[[

[

minus sin 120579

cos 120579 sin120601

cos 120579 cos120601

]]

]

(35)





level119899

= 119877119899

119887



[

level119899

level119899] = [

1 119877119899

119887

0 1

][

level119899

level119899] (37)

State variables are

119896= [

level119899

level119899] (38)






0 0

119868 0


minus

119896

= 119860119896minus1

+ 119866119906119896Δ119905

119875minus

119896

= 119860119875119896minus1

119860119879

+ 119866119876119866119879

Δ1199052

119860 = 119868 + 119865Δ119905

119876 = diag (1205902

119891119909

1205902

119891119910

1205902

119891119911

1205902

119892119909

1205902

119892119910

1205902

119892119911

)

(39)

6 Experiment













Sensor 2

Sensor 1

Sensor 3





Sensor 4

0100200300400500600700800900

Sens

or d

ata (

AD

C va

lue)


0100200300400500600700800

Sens

or d

ata (

AD

C va

lue)




Sensor 3Sensor 4

Sensor 5Sensor 6


0100200300400500600700800900

Sens

or d

ata (

AD

C va

lue)


(a)


Sensor 3Sensor 4

Sensor 5Sensor 6

0100200300400500600700800900

Sens

or d

ata (

AD

C va

lue)


(b)






0

200

400

600

800

1000

Sens

or d

ata (

AD

C va

lue)

10 15 20 25 30 35 405Tilt angle (deg)


05

101520253035404550

Mea

sure

d an

gle o

f IM

U (d

eg)











05

101520253035404550

Mea

sure

d an

gle o

f lev

el se

nsor

(deg

)





05

101520253035404550

Mea

sure

d an

gle o

f sug

geste

d A

L (d

eg)








3(Reset)




2 3 4 5 6 7 8 9 10 11 121


0100200300400500600700800

Sens

or o

utpu

t (A

DC

valu

e)



LiterMeasure liter

02468

101214

Mea

sure

d w

ater

capa

city

(lite

r)


(a)


Density

Mea

sure

d de

nsity

(kg

m3 )

0

02

04

06

08

1

12


(b)











7 Conclusion





0

01

02

03

04

05

Capa

city

relat

ive e

rror

(lite

r)10 15 20 25 30 35 405




5deg0deg

10deg15deg20deg



Den

sity

(kg

m3 )

0

02

04

06

08

1

12







Competing Interests


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119875 =

[[[

[

12059000

1205900120579

1205900120595

1205900120579

120590120579120579

120590120579120595

1205900120593

120590120579120595

120590120595120595

]]]

]

(10)


119896=0

= 1198683times3

119896=0

= [0 0 0]119879

119875119896=0

= 108

1198683times3

(11)



119899

119887

= 119877119899

119887

Ω119887

(12)


Ω119887

= 120596119887

times

[[[

[

0 minus120596119911

120596119910

120596119911

0 minus120596119909

minus120596119910

120596119909

0

]]]

]

(13)


Λ119899

= 119862minus1

1

120596119887

(14)

Here 119862minus1

1


119862minus1

1

=[[

[


0 cos 0 minus sin 0

0 sin 0 cos 120579 cos 0 cos 120579

]]

]

(15)



minus

119896

= 119896minus1

(1 + Ω119887

Δ119905)

= 119896minus1

119877119911(120596119911Δ119905) 119877119910(120596119910Δ119905) 119877119909(120596119909Δ119905)

minus

119896

= 119896minus1

+ 119862minus1

1

120596119887

Δ119905

(16)


119875minus

119896

= 119875119896minus1

+ 119862minus1

1

119876119862minus119879

1

(Δ119905)2

(17)

Here 119876 = diag(12059020

1205902

120579

1205902

120593


120590119860

= [1205900

120590120579

120590120595]119879

= 10 + 120596119887

(18)



119877119899

119887

larr997888 119877119910() 119877119909(0) 119877119899

119887

(19)


Λ119899

larr997888 Λ119899

+ 119862minus1

0

[[[

[

0

0

]]]

]

(20)

Here 119862minus10

is as follows

119862minus1

0

=

[[[[[[

[

cos120595cos 120579

sin120595

cos 1205790

sin120595 cos120595 0

cos120595tan 120579

sin120595

tan 120579

1

]]]]]]

]

(21)


ℎ (Λ119899

) = Λ119899

(22)


utilizing 0 and

119885119896= Λ119899

+ 119862minus1

0

[[[

[

0

0

]]]

]

(23)


119896= 119877119910(119870120579Δ120579) 119877

119909(1198700Δ0) minus

119896

119896=

minus

119896

+ 119870119896(119911119896minus ℎ (

minus

119896

))

(24)


119896 and 119870

120579is column


119875119896= 119875minus

119896

= 119870119896119875minus

119896

119870119896= 119875minus

119896

(119875minus

119896

+ 119862minus1

0

119877119862minus119879

0

)

minus1

(25)

Here 119877 = diag(12059020

1205902

120579

1205902

120593


1205900= 120590120579= 01 +

10038161003816100381610038161003816

1003817100381710038171003817100381711989111988710038171003817100381710038171003817minus 1

10038161003816100381610038161003816+ 100

1003817100381710038171003817100381712059611988710038171003817100381710038171003817

120590120595

= 108

(26)



119899

119887


Ψ120592= tanminus1 (119886

21

11988611

) (27)


119899

119887





119877119899

119887

larr997888 119877119911(Ψ) 119877

119899

119887


(28)


Λ119899

= Λ119899

+ 119862minus1

0

[[[

[

0

0

Ψ

]]]

]

(29)


ℎ (Λ119899

) = Λ119899

(30)


utilizing Ψ

119885119896= Λ119899

+ 119862minus1

0

[[[

[

0

0

Ψ

]]]

]

(31)


119896= 119877119911(119870ΨΨ) minus

119896

119896

= minus

119896

+ 119870119896(119911119896minus ℎ (

minus

119896

))

(32)


119896

119877119896= 119875minus

119896

minus 119870119896119875minus

119896

119870119896= 119875minus

119896

(119875minus

119896

+ 119862minus1

0

119877119862minus119879

0

)

minus1

(33)

Here 119877 = diag(12059020

1205902

120579

1205902

120593


1205900= 120590120579= 108

120590120595

=2120590level

120598 + 119881IMU 120598 = 10

minus6

120590level = 10

(34)


119886 = [119886119909

119886119910



119886 = V + V times 120596 + 119877119879

119892

(∵ 119877119879

= 119877minus1

)

=[[

[

V119909

V119910

V119911

]]

]

+

[[[

[

0 V119911

minusV119910

minusV119911

0 V119909

V119910

minusV119909

0

]]]

]

[[[

[

119908119909

119908119910

119908119911

]]]

]

+ 119892119911

[[

[

minus sin 120579

cos 120579 sin120601

cos 120579 cos120601

]]

]

(35)





level119899

= 119877119899

119887



[

level119899

level119899] = [

1 119877119899

119887

0 1

][

level119899

level119899] (37)

State variables are

119896= [

level119899

level119899] (38)






0 0

119868 0


minus

119896

= 119860119896minus1

+ 119866119906119896Δ119905

119875minus

119896

= 119860119875119896minus1

119860119879

+ 119866119876119866119879

Δ1199052

119860 = 119868 + 119865Δ119905

119876 = diag (1205902

119891119909

1205902

119891119910

1205902

119891119911

1205902

119892119909

1205902

119892119910

1205902

119892119911

)

(39)

6 Experiment













Sensor 2

Sensor 1

Sensor 3





Sensor 4

0100200300400500600700800900

Sens

or d

ata (

AD

C va

lue)


0100200300400500600700800

Sens

or d

ata (

AD

C va

lue)




Sensor 3Sensor 4

Sensor 5Sensor 6


0100200300400500600700800900

Sens

or d

ata (

AD

C va

lue)


(a)


Sensor 3Sensor 4

Sensor 5Sensor 6

0100200300400500600700800900

Sens

or d

ata (

AD

C va

lue)


(b)






0

200

400

600

800

1000

Sens

or d

ata (

AD

C va

lue)

10 15 20 25 30 35 405Tilt angle (deg)


05

101520253035404550

Mea

sure

d an

gle o

f IM

U (d

eg)











05

101520253035404550

Mea

sure

d an

gle o

f lev

el se

nsor

(deg

)





05

101520253035404550

Mea

sure

d an

gle o

f sug

geste

d A

L (d

eg)








3(Reset)




2 3 4 5 6 7 8 9 10 11 121


0100200300400500600700800

Sens

or o

utpu

t (A

DC

valu

e)



LiterMeasure liter

02468

101214

Mea

sure

d w

ater

capa

city

(lite

r)


(a)


Density

Mea

sure

d de

nsity

(kg

m3 )

0

02

04

06

08

1

12


(b)











7 Conclusion





0

01

02

03

04

05

Capa

city

relat

ive e

rror

(lite

r)10 15 20 25 30 35 405




5deg0deg

10deg15deg20deg



Den

sity

(kg

m3 )

0

02

04

06

08

1

12







Competing Interests


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119899

119887


Ψ120592= tanminus1 (119886

21

11988611

) (27)


119899

119887





119877119899

119887

larr997888 119877119911(Ψ) 119877

119899

119887


(28)


Λ119899

= Λ119899

+ 119862minus1

0

[[[

[

0

0

Ψ

]]]

]

(29)


ℎ (Λ119899

) = Λ119899

(30)


utilizing Ψ

119885119896= Λ119899

+ 119862minus1

0

[[[

[

0

0

Ψ

]]]

]

(31)


119896= 119877119911(119870ΨΨ) minus

119896

119896

= minus

119896

+ 119870119896(119911119896minus ℎ (

minus

119896

))

(32)


119896

119877119896= 119875minus

119896

minus 119870119896119875minus

119896

119870119896= 119875minus

119896

(119875minus

119896

+ 119862minus1

0

119877119862minus119879

0

)

minus1

(33)

Here 119877 = diag(12059020

1205902

120579

1205902

120593


1205900= 120590120579= 108

120590120595

=2120590level

120598 + 119881IMU 120598 = 10

minus6

120590level = 10

(34)


119886 = [119886119909

119886119910



119886 = V + V times 120596 + 119877119879

119892

(∵ 119877119879

= 119877minus1

)

=[[

[

V119909

V119910

V119911

]]

]

+

[[[

[

0 V119911

minusV119910

minusV119911

0 V119909

V119910

minusV119909

0

]]]

]

[[[

[

119908119909

119908119910

119908119911

]]]

]

+ 119892119911

[[

[

minus sin 120579

cos 120579 sin120601

cos 120579 cos120601

]]

]

(35)





level119899

= 119877119899

119887



[

level119899

level119899] = [

1 119877119899

119887

0 1

][

level119899

level119899] (37)

State variables are

119896= [

level119899

level119899] (38)






0 0

119868 0


minus

119896

= 119860119896minus1

+ 119866119906119896Δ119905

119875minus

119896

= 119860119875119896minus1

119860119879

+ 119866119876119866119879

Δ1199052

119860 = 119868 + 119865Δ119905

119876 = diag (1205902

119891119909

1205902

119891119910

1205902

119891119911

1205902

119892119909

1205902

119892119910

1205902

119892119911

)

(39)

6 Experiment













Sensor 2

Sensor 1

Sensor 3





Sensor 4

0100200300400500600700800900

Sens

or d

ata (

AD

C va

lue)


0100200300400500600700800

Sens

or d

ata (

AD

C va

lue)




Sensor 3Sensor 4

Sensor 5Sensor 6


0100200300400500600700800900

Sens

or d

ata (

AD

C va

lue)


(a)


Sensor 3Sensor 4

Sensor 5Sensor 6

0100200300400500600700800900

Sens

or d

ata (

AD

C va

lue)


(b)






0

200

400

600

800

1000

Sens

or d

ata (

AD

C va

lue)

10 15 20 25 30 35 405Tilt angle (deg)


05

101520253035404550

Mea

sure

d an

gle o

f IM

U (d

eg)











05

101520253035404550

Mea

sure

d an

gle o

f lev

el se

nsor

(deg

)





05

101520253035404550

Mea

sure

d an

gle o

f sug

geste

d A

L (d

eg)








3(Reset)




2 3 4 5 6 7 8 9 10 11 121


0100200300400500600700800

Sens

or o

utpu

t (A

DC

valu

e)



LiterMeasure liter

02468

101214

Mea

sure

d w

ater

capa

city

(lite

r)


(a)


Density

Mea

sure

d de

nsity

(kg

m3 )

0

02

04

06

08

1

12


(b)











7 Conclusion





0

01

02

03

04

05

Capa

city

relat

ive e

rror

(lite

r)10 15 20 25 30 35 405




5deg0deg

10deg15deg20deg



Den

sity

(kg

m3 )

0

02

04

06

08

1

12







Competing Interests


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra














0 0

119868 0


minus

119896

= 119860119896minus1

+ 119866119906119896Δ119905

119875minus

119896

= 119860119875119896minus1

119860119879

+ 119866119876119866119879

Δ1199052

119860 = 119868 + 119865Δ119905

119876 = diag (1205902

119891119909

1205902

119891119910

1205902

119891119911

1205902

119892119909

1205902

119892119910

1205902

119892119911

)

(39)

6 Experiment













Sensor 2

Sensor 1

Sensor 3





Sensor 4

0100200300400500600700800900

Sens

or d

ata (

AD

C va

lue)


0100200300400500600700800

Sens

or d

ata (

AD

C va

lue)




Sensor 3Sensor 4

Sensor 5Sensor 6


0100200300400500600700800900

Sens

or d

ata (

AD

C va

lue)


(a)


Sensor 3Sensor 4

Sensor 5Sensor 6

0100200300400500600700800900

Sens

or d

ata (

AD

C va

lue)


(b)






0

200

400

600

800

1000

Sens

or d

ata (

AD

C va

lue)

10 15 20 25 30 35 405Tilt angle (deg)


05

101520253035404550

Mea

sure

d an

gle o

f IM

U (d

eg)











05

101520253035404550

Mea

sure

d an

gle o

f lev

el se

nsor

(deg

)





05

101520253035404550

Mea

sure

d an

gle o

f sug

geste

d A

L (d

eg)








3(Reset)




2 3 4 5 6 7 8 9 10 11 121


0100200300400500600700800

Sens

or o

utpu

t (A

DC

valu

e)



LiterMeasure liter

02468

101214

Mea

sure

d w

ater

capa

city

(lite

r)


(a)


Density

Mea

sure

d de

nsity

(kg

m3 )

0

02

04

06

08

1

12


(b)











7 Conclusion





0

01

02

03

04

05

Capa

city

relat

ive e

rror

(lite

r)10 15 20 25 30 35 405




5deg0deg

10deg15deg20deg



Den

sity

(kg

m3 )

0

02

04

06

08

1

12







Competing Interests


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Sensor 2

Sensor 1

Sensor 3





Sensor 4

0100200300400500600700800900

Sens

or d

ata (

AD

C va

lue)


0100200300400500600700800

Sens

or d

ata (

AD

C va

lue)




Sensor 3Sensor 4

Sensor 5Sensor 6


0100200300400500600700800900

Sens

or d

ata (

AD

C va

lue)


(a)


Sensor 3Sensor 4

Sensor 5Sensor 6

0100200300400500600700800900

Sens

or d

ata (

AD

C va

lue)


(b)






0

200

400

600

800

1000

Sens

or d

ata (

AD

C va

lue)

10 15 20 25 30 35 405Tilt angle (deg)


05

101520253035404550

Mea

sure

d an

gle o

f IM

U (d

eg)











05

101520253035404550

Mea

sure

d an

gle o

f lev

el se

nsor

(deg

)





05

101520253035404550

Mea

sure

d an

gle o

f sug

geste

d A

L (d

eg)








3(Reset)




2 3 4 5 6 7 8 9 10 11 121


0100200300400500600700800

Sens

or o

utpu

t (A

DC

valu

e)



LiterMeasure liter

02468

101214

Mea

sure

d w

ater

capa

city

(lite

r)


(a)


Density

Mea

sure

d de

nsity

(kg

m3 )

0

02

04

06

08

1

12


(b)











7 Conclusion





0

01

02

03

04

05

Capa

city

relat

ive e

rror

(lite

r)10 15 20 25 30 35 405




5deg0deg

10deg15deg20deg



Den

sity

(kg

m3 )

0

02

04

06

08

1

12







Competing Interests


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra













0

200

400

600

800

1000

Sens

or d

ata (

AD

C va

lue)

10 15 20 25 30 35 405Tilt angle (deg)


05

101520253035404550

Mea

sure

d an

gle o

f IM

U (d

eg)











05

101520253035404550

Mea

sure

d an

gle o

f lev

el se

nsor

(deg

)





05

101520253035404550

Mea

sure

d an

gle o

f sug

geste

d A

L (d

eg)








3(Reset)




2 3 4 5 6 7 8 9 10 11 121


0100200300400500600700800

Sens

or o

utpu

t (A

DC

valu

e)



LiterMeasure liter

02468

101214

Mea

sure

d w

ater

capa

city

(lite

r)


(a)


Density

Mea

sure

d de

nsity

(kg

m3 )

0

02

04

06

08

1

12


(b)











7 Conclusion





0

01

02

03

04

05

Capa

city

relat

ive e

rror

(lite

r)10 15 20 25 30 35 405




5deg0deg

10deg15deg20deg



Den

sity

(kg

m3 )

0

02

04

06

08

1

12







Competing Interests


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










3(Reset)




2 3 4 5 6 7 8 9 10 11 121


0100200300400500600700800

Sens

or o

utpu

t (A

DC

valu

e)



LiterMeasure liter

02468

101214

Mea

sure

d w

ater

capa

city

(lite

r)


(a)


Density

Mea

sure

d de

nsity

(kg

m3 )

0

02

04

06

08

1

12


(b)











7 Conclusion





0

01

02

03

04

05

Capa

city

relat

ive e

rror

(lite

r)10 15 20 25 30 35 405




5deg0deg

10deg15deg20deg



Den

sity

(kg

m3 )

0

02

04

06

08

1

12







Competing Interests


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












0

01

02

03

04

05

Capa

city

relat

ive e

rror

(lite

r)10 15 20 25 30 35 405




5deg0deg

10deg15deg20deg



Den

sity

(kg

m3 )

0

02

04

06

08

1

12







Competing Interests


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Competing Interests


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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