small-rangehigh-precisionpositioningbasedontwo-point

Research ArticleSmall-Range High-Precision Positioning Based on Two-PointCoordination for Robot

Zhengping Li 1 Chaoliang Qin2 and Hao Shi3

1North China University of Technology Beijing 100144 China2Renesas Semiconductor Design (Beijing) Co Ltd Beijing 100085 China3)e Ministry of Public Security of Peoplersquos Republic of China )e First Research Institute Beijing 100048 China

Correspondence should be addressed to Zhengping Li lizpncuteducn

Received 20 April 2021 Revised 15 May 2021 Accepted 24 May 2021 Published 16 June 2021

Academic Editor Sang-Bing Tsai

Copyright copy 2021 Zhengping Li et al is 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

is paper proposed a two-point coordinated positioning algorithm Based on the assumption that the distance between twopoints was constant a fusion algorithm was introduced into the positioning process to enhance the positioning accuracy esimulation results showed that the proposed algorithm could reduce the RMS error to about 50 of the improved sinc in-terpolation-based positioning algorithm when the sampling frequency was 500MHz and the interpolation number was 19

1 Introduction

Accurate positioning is very important for robotic applica-tions [1] ere are several kinds of indoor positioning sys-tems e non-radio-based technologies mainly employcameras the location accuracy is low when there are obstaclesbefore the camera and the image processing algorithm needsa high-speed processor that makes the cost and the powerconsumption very high e commonly used approaches areradio-based technologies such as Wi-Fi and ultra-wideband(UWB) Wi-Fi-based indoor positioning system primarilyemploys the strength of the Wi-Firsquos access points (APs) andthe accuracy is about 2m which is too low to be used bymanyindoor applications [2ndash4] UWB-based positioning systemhas high accuracy due to the large bandwidth but because ofthe shadow fading and the random interferences the posi-tioning results are not stable Moreover time-based rangingtechnology was used in UWB positioning system and severalhandshaking processes are needed to get the range betweentwo nodes which makes the positioning frequency low and itcannot meet the requirements of some applications Manyresearch studies have been done on UWB-based localization[5ndash8] to enhance positioning stability Some research studiesemployed filters to enhance the positioning accuracy [8]some research studies were focused on the ranging error

elimination [5] and others attempted to fuse the data such asmoving state and IMU data with the UWB positing data toeliminate the unstability [6 7] and the problem is not solvedyet is paper proposed a two-point positioning data-fusingalgorithm for the applications with more than one targetnodes on an object [9]

is paper designed an improved positioning systemthat employed the proposed improved sinc interpolationalgorithm to enhance the positioning frequency of thesystem for TDOA values en the system estimated theposition of the target node using Chanrsquos algorithm and usedtwo-point coordination method to optimize the positioningresults As the result we can reduce the sampling frequencyto the maximum extent on the premise of ensuring accuracy

e rest of the paper is organized as follows Section 2describes a survey of the related research studies Section 3describes the related principle of the proposed algorithmSection 4 explains simulation and analysis of the algorithmand Section 5 draws conclusion

2 Related Works

With the rapid increase of data and multimedia services thedemand for positioning and navigation is increasing es-pecially in the complex indoor environment such as the

HindawiMobile Information SystemsVolume 2021 Article ID 8732006 6 pageshttpsdoiorg10115520218732006

airport hall exhibition hall warehouse supermarket li-brary and underground parking lot

TDOA (time difference of arrival) positioning is a kindof wireless positioning By measuring the time of arrival ofthe signal to the reference node the distance of the targetnode can be determined [10] Using the distance between thetarget node and the various reference nodes the location ofthe target node can be determined However the absolutetime is difficult to measure by comparing the time differencebetween the signals to the reference nodes we can make thehyperbola with focus on the reference node and the long axisof the distance difference e intersection of the hyperbolicis the location of the target node [11]

Based on TDOA Li and Wang put forward a new al-gorithm that can greatly improve the positioning accuracyeir system employs matched filter to calculate the TDOAvalue and does not need precise synchronization betweenthe transmitter and receivers that makes the TDOA valuemore accurate [12] In this paper we propose the two-pointcoordination algorithm to improve data processing etwo-point coordination algorithm uses two-point infor-mation to calculate the position of the target node while inthe Small Range High Precision Positioning Algorithm Basedon Improved Sinc only one-point information is used tocalculate the position of the target node so when we use two-point coordination algorithm to process positioning resultsit can improve the positioning accuracy

3 The Positioning Process andRelated Algorithm

31 )e Positioning Process e positioning steps are asfollows

Step 1 L fixed target nodes received the FM wavesignals from the reference node of which L was apositive integer e modulation signal of the FM wavewas a sawtooth signal and therefore the FM wave wascalled a sawtooth FM wave A cycle of the sawtooth FMsignal was called a chirpStep 2 we conducted amplitude limitation on the re-ceived signals sampled on M continuous chirps atinterval T and achieved the sample function xi(n) inwhich i 1 2 M and n 0 1 N N was thesample point number of each chirpStep 3 by using improved sinc interpolation algorithmto reconstruct the sample function we could get thereconstruction function y(k) of which k 0 1 (Nminus 1) (M+ 1) +MStep 4 sampling the original sawtooth FM waves atinterval T(M+ 1) and achieving the sample functionu(k) where k 0 1 (Nminus 1) (M+ 1) +M en wetook u(k) and y(k) to perform the cross-correlationoperation and obtained the correlation peak location Ai

of which i 0 1 LStep 5 using correlation peak location gap we couldcalculate the signal arrival time differencet21 t31 ti1 tL1 between the 2th 3th Lth

reference node and the 1th reference node Amongthem t21 t31 ti1 tL1 were the TDOA values ofwhich ti1 (Ai minus A1)lowastT Twas the time interval of thesample points in y(k)Step 6 the TDOA values and the coordinates of thereference nodes were put into Chanrsquos algorithm tocalculate the position of the target nodeStep 7 two-point coordination was used to optimizethe positioning results

32 )e Related Algorithm In Step 3 an improved sincinterpolation algorithm was mentioned and a detailed de-scription of the improved sinc interpolation algorithm waspresented in the Small Range High Precision PositioningAlgorithm Based on Improved Sinc Interpolation In Step 6the Chanrsquos algorithm was mentioned and a detailed de-scription of the Chanrsquos algorithm was presented in thePrecision Wireless Positioning Scheme in Small Range Basedon First-Order Difference and Correlation Inspectionerefore we need not repeat the algorithm here

In Step 7 the two-point coordination algorithm wasmentioned and a detailed description of the improved sincinterpolation algorithm was presented in this section

It is assumed that the distance of two target nodes isknown as h the coordinates of the two target nodes (x1prime y1prime)and (x2prime y2prime)are estimated using the location algorithm

If the distance between two points is greater than acertain distance this set of data is considered to be a grosserror and should be removed Namely

hprime

x1prime minus x2prime( 11138572

+ y1prime minus y2prime( 11138572

1113969

gtωh (1)

where ω is an empirical value that is greater than 1If (x1prime y1prime) and (x2prime y2prime) can be retained according to

(x1prime y1prime) and (x2prime y2prime) we estimate the location of the targetnode two times For example according to the coordinatesof the target node 1 (x1prime y1prime) to estimate the coordinates ofthe target node 2 (x2Prime y2Prime) as the distance between two nodesh is known it is assumed that the target node 2 (x2Prime y2Prime) is inthe circle with center point (x1prime y1prime) and a radius of h andtarget node 2 is also in the straight line with the two points(x1prime y1prime) and (x2prime y2prime)

As we know there are two intersection points of astraight line and a circle we choose the point that is closer to(x2prime y2prime) as the target node 2 (x2Prime y2Prime) We can estimate the(x2Prime y2Prime) according to the following equations

x minus x1prime( 11138572

+ y minus y1prime( 11138572

1113969

h



y minus y1prime( 1113857

y2prime minus y1prime( 1113857

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(2)

ere are two solutions to the equations and we choosethe final solution that is closer to (x2prime y2prime) e same can beused to solve the second-time estimation coordinates(x1Prime y1Prime) of the target node 1

2 Mobile Information Systems

So far the two estimation coordinates for each targetnode are obtained en the fusion algorithm is used to fusethe data of the two groups e following is a method ofcalculating the weight

Set the actual horizontal coordinates of the target node 1x

x1prime x + v1

x1Prime x + v2(3)

where vi(i 1 2) is a random error and vi sim N(0 σ2i ) twoobservations are independent of each other

It is assumed that the final estimation results 1113954x of x are inlinear relationship with the first estimate x1prime and the secondestimate x1Prime and the 1113954x is the unbiased estimate of x

1113954x ω1x1primeω2x1Prime (4)

where Ω (ω1ω2) is the weight value of the estimatedvalue

Set the estimation error

1113957x x minus 1113954x (5)

Take the cost function 1113957x for the mean square error

J E 1113957x2

1113872 1113873 E x minus ω1 x + x1prime( 1113857 minus ω2 x + x1Prime( 11138571113858 11138592

1113966 1113967 (6)

As the 1113954x is the unbiased estimate of x

E(1113957x) E x minus ω1 x + x1prime( 1113857 minus ω2 x + x1Prime( 11138571113858 1113859 0 (7)

As E(v1) E(v2) 0 and E(x) E(1113954x)

ω2 1 minus ω1 (8)

en the cost function can be written as

J E 1113954x2

1113872 1113873 E ω21v

21 + 1 minus ω1( 1113857

2v22 + 2ω1 1 minus ω1( 1113857v1v21113960 1113961

(9)

As E(v21) σ21 and E(v22) σ22 v1 and v2are independentE(v1 v2) 0

en

J E 1113957x2

1113872 1113873 ω21σ

21 + 1 minus ω1( 1113857

2σ22 (10)

In order to obtain the minimum value of J and Ωderivatives

zJ

zΩ 0 (11)

e optimal weight value is

ωlowast1 σ22

σ22 + σ21

ωlowast2 σ21

σ22 + σ21

(12)

Optimal estimation is

1113954x σ22x1prime

σ22 + σ21+

σ21x1Prime

σ22 + σ21 (13)

In the same way the vertical coordinates can also solvede two-point coordination algorithm uses two-point

information to calculate the position of the target nodewhile in the Small Range High Precision Positioning Algo-rithm Based on Improved Sinc only one-point information isused to calculate the position of the target node so when weuse two-point coordination algorithm to process positioningresults it can improve the positioning accuracy

4 System Simulation and Analysis

In the simulation system the positioning area was deter-mined by the number of reference nodes and the more thenumber of reference nodes the larger the positioning areae reference nodes were stationary and they should bedistributed around the positioning area uniformly as muchas possible so that the system could get better positioningresults

In this simulation system it supposes that the posi-tioning range is 20m times 20m e coordinates of sevenreference nodes were (0 0) (0 20) (10 minus4) (20 0) (10 24)(20 20) and (minus4 10) e target node acted as a transmittere modulation signalrsquos frequency of the target node was1MHz e simulation supposes that the transmissionchannel was 6-path Rician channel that had 1 line-of-sight(LOS) path and 5 reflection paths e reflection paths werecaused by the multipath effect because of the signal re-flection diffraction and scattering e additional delay of 6paths were [0 311e9 711e9 1091e9 1731e9 and 2511e9](s) the additional attenuation were [0 minus1 minus9 minus10 minus15 andminus20] (db) and this was a common indoor channel In ad-dition the received signal is summed together of the LOSand reflection signals If the obstacles were on the LOS pathit should affect the TDOA value and cause TDOA errors

e positioning accuracy was measured with the root-mean-square error (RMSE) of positioning results which wasfrequently used at present (Figures 1ndash4) e positioningsystem simulation was done in different conditions throughMATLAB (Tables 1ndash3)

(1) e positioning accuracy and time with different sincinterpolation algorithms are shown in Figure 1In Figure 1 abscissa was three different algorithmsthe algorithm without any interpolation algorithmthe algorithm with nonimproved interpolation al-gorithm and the algorithm with improved inter-polation algorithm e ordinates were RMSE andtime Table 1 lists the details of each point in Figure 1e sampling frequency of the three algorithms inFigure 1 was 500MHz and the carrier frequency was100MHz From the simulation results it could be

Mobile Information Systems 3

seen that in the process of data processing thepositioning accuracy and positioning time of thedifferent degrees of improvement were comparedbetween Chauvenetrsquos criterion and coordinationalgorithm When we do not use any interpolationalgorithm the algorithm has a lower positioningaccuracy erefore the positioning accuracy can besignificantly improved when we use the coordinationalgorithm in the data processing However thepositioning time of the algorithm is very short so it isnot obvious that the positioning time is shortenedafter we use the coordination algorithm [13 14]When we use the improved interpolation algorithmthe algorithm has a lower positioning time ere-fore the positioning time can be significantly im-proved when we use the coordination algorithm inthe data processing However the positioning ac-curacy of the algorithm is very low so it is notobvious that the positioning accuracy is promotedafter we use the coordination algorithmenwe willanalyze the positioning accuracy in different situa-tions in detail

(2) e positioning accuracy in different sampling fre-quencies and interpolation points is shown inFigure 2In Figure 2 abscissa was carrier frequency and thevalues were 10MHz 20MHz 30MHz 40MHz50MHz 60MHz 70MHz 80MHz 90MHz and100MHz e ordinate was RMSE Table 2 lists thedetails of each point in Figure 2 e target node wasstationary and it could be at any place in the po-sitioning area From the simulation results it could

Without sinc Nonimproved sinc Improved sinc10ndash2

10ndash1

100

101

RMSE

(m)

Chauvenetrsquos criterion

Coordination algorithm

(a)

100

101

102

103

Tim

e (s)

Without sinc Nonimproved sinc Improved sinc



(b)

Figure 1 (a) RMSE and (b) time of different sinc interpolation algorithms

10 20 30 40 50 60 70 80 90 100Carrier frequency (MHz)

Sampling frequence 500MHz with Chauvenetrsquos criterionSampling frequence 500MHz with interpolation point 19 withChauvenetrsquos criterionSampling frequence 500MHz with two-point coordinationSampling frequence 500MHz with interpolation point 9 withtwo-point coordination

10ndash3

10ndash2

10ndash1

100

101

RMSE

(m)

Figure 2 RMSE of different algorithms in different samplingfrequencies and interpolation points

Table 2 RMSE of different algorithms in different sampling fre-quencies and interpolation points

CF(MHz)

RMSE (m)RMSEsquare

RMSEcircle

RMSEasterisk RMSE cross

10 38295 00125 36916 0011820 38724 00148 36835 0011330 38928 00107 36557 0010240 38995 00176 36531 0011250 38983 00184 36459 0010460 38196 00148 36184 0012170 38812 00119 36357 0008980 38886 00138 36198 0010990 38700 00174 36525 00124100 38979 00157 36554 00093

Table 1 RMSE and time of different sinc interpolation algorithms

RMSE square RMSE circleRMSE (m)

Without sinc 38979 34147Nonimproved sinc 34813 28058Improved sinc 00157 00142

Time (s)Without sinc 21740 19254Nonimproved sinc 151561 112563Improved sinc 1512365 1432465


be seen that the algorithm with improved sinc in-terpolation comparing with the algorithm withoutimproved sinc interpolation in positioning accuracyhad a very large enhancement e RMSE decreasedfrom about 3m to about 001m From the simulationresults it could be seen that the algorithm withimproved sinc interpolation with two-point coor-dination compared with the algorithm with im-proved sinc interpolation with Chauvenetrsquos criterionin positioning accuracy had certain enhancemente RMSE decreased from about 0015m to about0010m We could also see from the results that thecarrier frequency had little influence on the posi-tioning accuracy when the carrier frequency variesfrom 10MHz to 100MHz

(3) e positioning accuracy in different numbers ofinterpolation points is shown in Figure 3In Figure 3 abscissa was the numbers of interpo-lation points and the values were 0 9 and 19 eordinate was RMSE We set the carrier frequency to50MHz Table 3 lists the details of each point inFigure 3 From the simulation results it could beseen that when there were no interpolation pointswhether the Chauvenetrsquos criterion or two-pointcoordination is used the positioning accuracy isrelatively low e RMSE is over 30m When weinterpolated 9 points to the 500MHz1GHz sam-pling chips the positioning accuracy improved ob-viously e algorithm with two-point coordinationcompared with the algorithm with Chauvenetrsquoscriterion in positioning accuracy had certain en-hancement Since the positioning accuracy of thesampling frequency with 9 interpolation points to1GHz sampling chips was enough high there waslittle accuracy improvement when we interpolated 19points to the 1GHz sampling chips

(4) e positioning accuracy in different sampling fre-quencies is shown in Figure 4

In Figure 4 abscissa was the sampling frequency and thevalues were 250MHz 500MHz and 1000MHz (1GHz)eordinate was RMSE (m) We set the carrier frequency to50MHz Table 4 lists the details of each point in Figure 4From the simulation results it could be seen that the po-sitioning accuracy was not high in all three sampling fre-quencies when there were no interpolation points whetherthe Chauvenetrsquos criterion or two-point coordination is usede accuracy of the 250MHz sampling frequency with 919interpolation points was close to that of the 25GHz5GHzsampling frequency without interpolation points When thepositioning accuracy is relatively low the effect of using two-point coordination to improve the positioning accuracy isobviously compared with the Chauvenetrsquos criterion Whenthe accuracy is over about 30m using two-point coordi-nation can improve the positioning accuracy by about02mndash03m Due to the improved sinc algorithm the po-sitioning accuracy improves obviously e effect of usingtwo-point coordination to improve the positioning accuracy

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20The number of interpolation points

Sampling frequence 500MHz with Chauvenetrsquos criterionSampling frequence 500MHz with two-point coordinationSampling frequence 1GHz with Chauvenetrsquos criterionSampling frequence 1GHz with two-point coordination

10ndash3

10ndash2

10ndash1

100

101

RMSE

(m)

Figure 3 RMSE of different algorithms in different numbers ofinterpolation points

Table 3 RMSE of different algorithms in different numbers ofinterpolation points

RMSE (m)Interpolation numbers

0 9 19RMSE (square) 38983 16520 00184RMSE (circle) 36459 15649 00104RMSE (asterisk) 35430 00092 00072RMSE (cross) 34758 00084 00068

200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950 1000Sampling frequence (MHz)

10ndash3

10ndash2

10ndash1

100

101

RMSE

(m)

Interpolation point 0 with Chauvenetrsquos criterionInterpolation point 0 with two-point coordinationInterpolation point 9 with Chauvenetrsquos criterionInterpolation point 9 with two-point coordination



is not obvious comparing with the Chauvenetrsquos criterionWhen the accuracy is under 10m using two-point coor-dination can improve the positioning accuracy by about0001mndash0003m

e positioning accuracy and positioning time of thehave different degrees of improvement compared betweenChauvenetrsquos criterion which is used in the Small Range HighPrecision Positioning Algorithm Based on Improved Sinc andcoordination algorithm

5 Conclusion

is paper introduces the present situation and the futuredevelopment of the wireless location summarizes the relatedtechnologies and algorithms and proposes a coordinationlocalization algorithm e analysis and simulation resultsshow that if the coordination algorithm is used in the dataprocessing it can improve the positioning accuracy of thesystem e primary contribution was that a two-pointcoordination algorithm is proposed that could greatly in-crease positioning accuracy when the sampling frequencywas low

e problem is that when we use the improved sincinterpolation positioning algorithm the positioning accu-racy can meet the requirements but we have to wait for acertain amount of time to form a new chip to calculate theTDOA values even though we use the coordination algo-rithm to optimize the positioning time And when therewere more than one target nodes in the positioning area itwould take longer to estimate a position e next work willbe to continue to study the relationship between positioningaccuracy and positioning time

Data Availability

e data that support the findings of this study are availablefrom the corresponding author upon reasonable request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is paper was supported by the National Key Research andDevelopment Program (Program ID 2020YFC0811004)

References

[1] X Chen Z Liu J Wan and Z Li ldquoAggregated handoverauthentication for machine type communicationrdquo Journal ofOrganizational and End User Computing vol 31 no 3pp 83ndash96 2019

[2] Y Tao and L Zhao ldquoA novel system for WiFi radio mapautomatic adaptation and indoor positioningrdquo IEEE Trans-actions on Vehicular Technology vol 67 no 11 pp 1ndash10 2018

[3] J W Jang and S-N Hong ldquoIndoor localization with WiFifingerprinting using convolutional neural networkrdquo in Pro-ceedings of the 10th International Conference on Ubiquitousand Future Networks pp 753ndash758 Prague Czech RepublicJuly 2018

[4] W Sun M Xue H Yu H Tang and A Lin ldquoAugmentationof fingerprints for indoorWiFi localization based on Gaussianprocess regressionrdquo IEEE Transactions on Vehicular Tech-nology vol 67 no 11 pp 10896ndash10905 2018

[5] M Stampa M Mueller D Hess and C Roehrig ldquoSemi-automatic calibration of UWB range measurements for anautonomous mobile robotrdquo in Proceedings of the 50th In-ternational Symposium on Robotics (ISR 2018) pp 300ndash305Munich Germany June 2018

[6] T-M Nguyen A H Zaini C Wang K Guo and L XieldquoRobust target-relative location with ultra-wideband rangingand communicationrdquo in Proceedings of the 2018 InternationalConference on Robotics and Automation (ICRA 2018)pp 2312ndash2319 Brisbane QLD Australia September 2018

[7] C Pierre R Chapuis R Aufrere J Laneurit and C DebainldquoRang-only based cooperative localization for mobile robotsrdquoin Proceedings of the 21st International Conference on Infor-mation Fusion (FUSION) pp 1933ndash1939 Cambridge UKJuly 2018

[8] Z Kasmi N Guerchali A Norrdine and J H SchillerldquoAlgorithms and position optimization for a decentralizedlocalization platform based on resource-constrained devicesrdquoJournal of IEEE Transactions on Mobile Computing vol 18no 8 pp 1731ndash1744 2018

[9] Y Huang W Sheng P Jin B Nie M Qiu and G Xu ldquoAnode-oriented discrete event scheduling algorithm based onfinite resource modelrdquo Journal of Organizational and EndUser Computing vol 31 no 3 pp 67ndash82 2019

[10] C-L Wei and C-T Ho ldquoExploring signaling roles of serviceprovidersrsquo reputation and competence in influencing per-ceptions of service quality and outsourcing intentionsrdquoJournal of Organizational and End User Computing vol 31no 1 pp 86ndash109 2019

[11] Y Yu Y Yao and X Cheng ldquoTDOA positioning technologyand practical applicationrdquo China Radio vol 11 pp 57-582014

[12] Z Li Z Wang Y Zhang and L Ma ldquoPrecision wirelesspositioning scheme in small range based on first-order dif-ference and correlation inspectionrdquo Journal of InformationTechnology Research vol 6 no 3 pp 1ndash15 2013

[13] M Zhou Y Wang Y Liu and Z Tian ldquoAn information-theoretic view of WLAN localization error bound in GPS-denied environmentrdquo IEEE Transactions on Vehicular Tech-nology vol 68 no 4 pp 4089ndash4093 2019

[14] M Zhou X Li Y Wang S Li Y Ding and W Nie ldquo6Gmulti-source information fusion based indoor positioning viaGaussian kernel density estimationrdquo IEEE Internet of )ingsJournal vol 10 no 99 p 1 2020

Table 4 RMSE of different algorithms in different samplingfrequencies

RMSE (m)SF (MHz)



airport hall exhibition hall warehouse supermarket li-brary and underground parking lot

TDOA (time difference of arrival) positioning is a kindof wireless positioning By measuring the time of arrival ofthe signal to the reference node the distance of the targetnode can be determined [10] Using the distance between thetarget node and the various reference nodes the location ofthe target node can be determined However the absolutetime is difficult to measure by comparing the time differencebetween the signals to the reference nodes we can make thehyperbola with focus on the reference node and the long axisof the distance difference e intersection of the hyperbolicis the location of the target node [11]

Based on TDOA Li and Wang put forward a new al-gorithm that can greatly improve the positioning accuracyeir system employs matched filter to calculate the TDOAvalue and does not need precise synchronization betweenthe transmitter and receivers that makes the TDOA valuemore accurate [12] In this paper we propose the two-pointcoordination algorithm to improve data processing etwo-point coordination algorithm uses two-point infor-mation to calculate the position of the target node while inthe Small Range High Precision Positioning Algorithm Basedon Improved Sinc only one-point information is used tocalculate the position of the target node so when we use two-point coordination algorithm to process positioning resultsit can improve the positioning accuracy

3 The Positioning Process andRelated Algorithm

31 )e Positioning Process e positioning steps are asfollows

Step 1 L fixed target nodes received the FM wavesignals from the reference node of which L was apositive integer e modulation signal of the FM wavewas a sawtooth signal and therefore the FM wave wascalled a sawtooth FM wave A cycle of the sawtooth FMsignal was called a chirpStep 2 we conducted amplitude limitation on the re-ceived signals sampled on M continuous chirps atinterval T and achieved the sample function xi(n) inwhich i 1 2 M and n 0 1 N N was thesample point number of each chirpStep 3 by using improved sinc interpolation algorithmto reconstruct the sample function we could get thereconstruction function y(k) of which k 0 1 (Nminus 1) (M+ 1) +MStep 4 sampling the original sawtooth FM waves atinterval T(M+ 1) and achieving the sample functionu(k) where k 0 1 (Nminus 1) (M+ 1) +M en wetook u(k) and y(k) to perform the cross-correlationoperation and obtained the correlation peak location Ai

of which i 0 1 LStep 5 using correlation peak location gap we couldcalculate the signal arrival time differencet21 t31 ti1 tL1 between the 2th 3th Lth

reference node and the 1th reference node Amongthem t21 t31 ti1 tL1 were the TDOA values ofwhich ti1 (Ai minus A1)lowastT Twas the time interval of thesample points in y(k)Step 6 the TDOA values and the coordinates of thereference nodes were put into Chanrsquos algorithm tocalculate the position of the target nodeStep 7 two-point coordination was used to optimizethe positioning results

32 )e Related Algorithm In Step 3 an improved sincinterpolation algorithm was mentioned and a detailed de-scription of the improved sinc interpolation algorithm waspresented in the Small Range High Precision PositioningAlgorithm Based on Improved Sinc Interpolation In Step 6the Chanrsquos algorithm was mentioned and a detailed de-scription of the Chanrsquos algorithm was presented in thePrecision Wireless Positioning Scheme in Small Range Basedon First-Order Difference and Correlation Inspectionerefore we need not repeat the algorithm here

In Step 7 the two-point coordination algorithm wasmentioned and a detailed description of the improved sincinterpolation algorithm was presented in this section

It is assumed that the distance of two target nodes isknown as h the coordinates of the two target nodes (x1prime y1prime)and (x2prime y2prime)are estimated using the location algorithm

If the distance between two points is greater than acertain distance this set of data is considered to be a grosserror and should be removed Namely

hprime


+ y1prime minus y2prime( 11138572

1113969

gtωh (1)

where ω is an empirical value that is greater than 1If (x1prime y1prime) and (x2prime y2prime) can be retained according to

(x1prime y1prime) and (x2prime y2prime) we estimate the location of the targetnode two times For example according to the coordinatesof the target node 1 (x1prime y1prime) to estimate the coordinates ofthe target node 2 (x2Prime y2Prime) as the distance between two nodesh is known it is assumed that the target node 2 (x2Prime y2Prime) is inthe circle with center point (x1prime y1prime) and a radius of h andtarget node 2 is also in the straight line with the two points(x1prime y1prime) and (x2prime y2prime)

As we know there are two intersection points of astraight line and a circle we choose the point that is closer to(x2prime y2prime) as the target node 2 (x2Prime y2Prime) We can estimate the(x2Prime y2Prime) according to the following equations


+ y minus y1prime( 11138572

1113969

h



y minus y1prime( 1113857

y2prime minus y1prime( 1113857

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(2)

ere are two solutions to the equations and we choosethe final solution that is closer to (x2prime y2prime) e same can beused to solve the second-time estimation coordinates(x1Prime y1Prime) of the target node 1




x1prime x + v1

x1Prime x + v2(3)






1113957x x minus 1113954x (5)


J E 1113957x2


1113966 1113967 (6)



As E(v1) E(v2) 0 and E(x) E(1113954x)

ω2 1 minus ω1 (8)


J E 1113954x2

1113872 1113873 E ω21v

21 + 1 minus ω1( 1113857

2v22 + 2ω1 1 minus ω1( 1113857v1v21113960 1113961

(9)


en

J E 1113957x2

1113872 1113873 ω21σ

21 + 1 minus ω1( 1113857

2σ22 (10)


zJ

zΩ 0 (11)


ωlowast1 σ22

σ22 + σ21

ωlowast2 σ21

σ22 + σ21

(12)



σ22 + σ21+

σ21x1Prime

σ22 + σ21 (13)












10ndash1

100

101

RMSE

(m)



(a)

100

101

102

103

Tim

e (s)




(b)




10ndash3

10ndash2

10ndash1

100

101

RMSE

(m)



CF(MHz)

RMSE (m)RMSEsquare

RMSEcircle


10 38295 00125 36916 0011820 38724 00148 36835 0011330 38928 00107 36557 0010240 38995 00176 36531 0011250 38983 00184 36459 0010460 38196 00148 36184 0012170 38812 00119 36357 0008980 38886 00138 36198 0010990 38700 00174 36525 00124100 38979 00157 36554 00093












10ndash3

10ndash2

10ndash1

100

101

RMSE

(m)






10ndash3

10ndash2

10ndash1

100

101

RMSE

(m)






5 Conclusion



Data Availability




Acknowledgments


References
















RMSE (m)SF (MHz)





x1prime x + v1

x1Prime x + v2(3)






1113957x x minus 1113954x (5)


J E 1113957x2


1113966 1113967 (6)



As E(v1) E(v2) 0 and E(x) E(1113954x)

ω2 1 minus ω1 (8)


J E 1113954x2

1113872 1113873 E ω21v

21 + 1 minus ω1( 1113857

2v22 + 2ω1 1 minus ω1( 1113857v1v21113960 1113961

(9)


en

J E 1113957x2

1113872 1113873 ω21σ

21 + 1 minus ω1( 1113857

2σ22 (10)


zJ

zΩ 0 (11)


ωlowast1 σ22

σ22 + σ21

ωlowast2 σ21

σ22 + σ21

(12)



σ22 + σ21+

σ21x1Prime

σ22 + σ21 (13)












10ndash1

100

101

RMSE

(m)



(a)

100

101

102

103

Tim

e (s)




(b)




10ndash3

10ndash2

10ndash1

100

101

RMSE

(m)



CF(MHz)

RMSE (m)RMSEsquare

RMSEcircle


10 38295 00125 36916 0011820 38724 00148 36835 0011330 38928 00107 36557 0010240 38995 00176 36531 0011250 38983 00184 36459 0010460 38196 00148 36184 0012170 38812 00119 36357 0008980 38886 00138 36198 0010990 38700 00174 36525 00124100 38979 00157 36554 00093












10ndash3

10ndash2

10ndash1

100

101

RMSE

(m)






10ndash3

10ndash2

10ndash1

100

101

RMSE

(m)






5 Conclusion



Data Availability




Acknowledgments


References
















RMSE (m)SF (MHz)






10ndash1

100

101

RMSE

(m)



(a)

100

101

102

103

Tim

e (s)




(b)




10ndash3

10ndash2

10ndash1

100

101

RMSE

(m)



CF(MHz)

RMSE (m)RMSEsquare

RMSEcircle


10 38295 00125 36916 0011820 38724 00148 36835 0011330 38928 00107 36557 0010240 38995 00176 36531 0011250 38983 00184 36459 0010460 38196 00148 36184 0012170 38812 00119 36357 0008980 38886 00138 36198 0010990 38700 00174 36525 00124100 38979 00157 36554 00093












10ndash3

10ndash2

10ndash1

100

101

RMSE

(m)






10ndash3

10ndash2

10ndash1

100

101

RMSE

(m)






5 Conclusion



Data Availability




Acknowledgments


References
















RMSE (m)SF (MHz)









10ndash3

10ndash2

10ndash1

100

101

RMSE

(m)






10ndash3

10ndash2

10ndash1

100

101

RMSE

(m)






5 Conclusion



Data Availability




Acknowledgments


References
















RMSE (m)SF (MHz)





5 Conclusion



Data Availability




Acknowledgments


References
















RMSE (m)SF (MHz)



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