# ELM-Assisted Particle Filter for INS/UWB-Integrated Quadrotor Positioning

Authors: Shuhui Bi; Fukun Li; Lei Wang; Yuan Xu; Jidong Feng
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

At present, with the wide application of the quadrotor, accurate positioning has become an increasingly important problem and needs to be considered. In this study, the inertial navigation system (INS) and ultra-wideband (UWB) technology are used together to collect the flight data of the quadrotor, and the particle filter (PF) algorithm will be employed to fuse the data information of the two sensors for reducing the error and obtaining the movement path of the quadrotor in the three dimensional (3D) space. Meanwhile, for making PF work more accurate, the extreme learning machine (ELM) is adopted to map the equation of state in the filtering process to make position information more reliable. In addition, ELM can also establish a new signal through mapping when UWB’s signal is interrupted, so that the whole system can work normally. To verify the effectiveness of the proposed method, a real experiment was carried out. The experimental results show that the ELM-assisted PF strategy has a good effect on INS/UWB-integrated navigation quadrotor positioning. When UWB signals are normal, compared with the single PF, the ELM-assisted PF is able to improve positioning accuracy by about 18.44%. When UWB’s signal is interrupted, compared with the least square support vector machine- (LS-SVM-) assisted PF, the ELM-assisted PF could improve positioning accuracy by about 1.15 m. On the whole, the proposed design algorithm not only improves the positioning accuracy but also can predict UWB’s signal when it is interrupted and thus make the filter work normally.


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## Body

## 1. Introduction

Quadrotor aircraft can adjust its attitude in the space by changing the speed. Because of the development of modern control technology, quadrotor aircraft has been more and more widely concerned. The stable flight of the quadrotor can be used in aerial photography, rescue and search, security, fire-fighting, pesticide spraying, construction, and other high-risk operating environments [1, 2]. Therefore, how to control the moving path and space position of the quadrotor, that is, how to achieve accurate positioning of the quadrotor, is a much more important subject that needs to be considered [3–9].The inertial navigation system measures the acceleration of the target in the reference frame and obtains velocity, heading angle, and position information by integrating it twice over time. The inertial navigation system is based on Newtonian mechanics, which belongs to the way of calculation navigation. It has good independence and continuity, does not depend on external information, nor does it radiate energy outward, and has low requirements for the working environment. However, due to the lack of reference, the solution position of the inertial navigation system affects each other, and the error will gradually increase with time accumulation; therefore, INS is not suitable for a long period, long distance navigation, and positioning. The UWB technology obtains the position information by collecting distance information between the reference node and the motion carrier, and the positions are independent from each other, so there is no time error accumulation problem. However, the signal transmission of the reference node is affected by the external environment, so there is the possibility of occlusion, deviation, or interruption. Therefore, it is difficult for a single navigation technology to meet the requirements of accurate positioning. For avoiding that, INS and UWB-integrated technology was proposed. Such multisensor fusion manner can give full play to their respective advantages, eliminate the error accumulation problem of INS, and ensure the continuity of signal data [10–14].Measurement data from the sensor are subjects to various kinds of noise, such as measurement environment, measuring technology, or hardware itself. Using the appropriate filtering algorithm can suppress the influence of bad noise, improve the signal quality, and make the positioning more accurate and reliable, so as to improve the performance of the whole navigation system. The particle filter algorithm is an effective sampling method, and it approximates the density function by calculating and comparing a random sample. In this method, the sample mean replaced the integral operation. The Monte Carlo method is its central idea, and probability is expressed by the particle set, which is widely used in practice [15, 16]. In the positioning process, in order to obtain more stable and superior effects, we can also use some learning strategies to match with filtering algorithms, such as the BP neural network, ELM, and LS-SVM.The extreme learning machine was proposed with the development of computer network technology. It was presented aiming to improve the backward propagation (BP) algorithm to improve low learning efficiency and simplify the setting of learning parameters. ELM is based on the feedforward neural network and improves the fuzzy neural network and thence has good adaptability in the supervised learning problem. In some current views, ELM has advantages in speed and generalization [17–20].In this study, the extreme learning machine will be adopted to assist the particle filter algorithm to calibrate the INS/UWB-integrated navigation system mounted on a quadrotor. In this process, the extreme learning machine plays two roles. ELM establishes the relationship between each state value in the filtering process, so as to make the whole model more stable. By establishing the mapping relationship between INS and UWB, even UWB’s signal is interrupted, and the necessary data needed for filtering can be provided to make the system work normally and improve the robustness of the whole system. In order to prove the availability of the proposed strategy, the experiment has been carried out in a real environment, and the experimental results prove the effectiveness of the proposed strategy.The main contributions of this study are as follows:(i)
A new PF algorithm based on the extreme learning machine is proposed to implement the fusion of INS and UWB’s data, and the mapping relationship of state values in the filtering process is built by training of the extreme learning machine, which improves the reliability of the positioning system.(ii)
The mapping between INS and UWB can be built by training of the extreme learning machine, and the signal compensation can be carried out when UWB’s signal is interrupted, so that the whole system can work normally.The remaining construction of this study is outlined as follows. The ELM-assisted PF strategy is explained in “INS/UWB-Integrated Navigation Quadrotor Positioning Scheme,” the PF filtering strategy for INS/UWB positioning and the ELM-assisted strategy are given in the “Particle Filtering Algorithm” and “The Scheme of ELM,” and the real test and error analysis are completed in “Test.” Finally, “Conclusion” describes the effectiveness of the proposed method for INS/UWB-integrated navigation quadrotor positioning.


## 2. INS/UWB-Integrated Navigation Quadrotor Positioning Scheme

The INS/UWB-integrated navigation quadrotor positioning scheme will be discussed for the following two situations.(i)
When UWB’s signal is available, INS and UWB measure the positionPoI and PoI of the target quadrotor, respectively. PF makes more accurate estimate δPo^ of the position error δPo by fusing the information of PoI and PoU to correct PoI and output the optimized target position. δPo is used as the observation vector in PF. In the filtering process, ELM establishes the mapping of the state deviation Δxt,t∈1,∞ generated at each moment of PF and correct Δx by the more accurate estimate Δx^ generated by ELM to reduce the state deviation. Meanwhile, the state vector x^t|t−1,t∈1,+∞ output from PF and δPo are taken as input and targets of ELM, and ELM keeps training to establish the mapping relationship between them. The whole system works normally, and ELM plays two roles: (1) to constantly correct Δxt along with PF, and ELM outputs deviation information; (2) through training, the mapping relationship between the Δxt output by PF and δPo is constantly established. At this time, ELM works offline and does not output observation information. The positioning strategy is shown in Figure 1.(ii)
When UWB’s signal is interrupted, UWB cannot providePoU and PF cannot work properly due to the lack of the observation vector δPo. At this point, ELM outputs the predicted observation vector δPo^ through the mapping established in the previous stage, and δPo^ is used as the observation vector of PF, so PF can work normally when UWB’s signal is interrupted. At this time, ELM works in the prediction stage and outputs observation information. The positioning strategy is shown in Figure 2.Figure 1
ELM-assisted PF data fusion strategy when UWB measurements are available.Figure 2
ELM-assisted PF data fusion strategy when UWB’s signal is interrupted.


## 3. Particle Filtering Algorithm

On the basis of INS/UWB integrated navigation, we use the particle filter algorithm to fuse sensor information. The state equation of particle filter is as follows:(1)ϕt|t−1δVt|t−1nδPt|t−1n∇t|t−1bεt|t−1b︸Xt|t−1=I3×303×303×303×3−I3×3CbnΔTSftnΔTI3×303×3I3×3CbnΔT03×303×3I3×3ΔTI3×303×303×303×303×303×3I3×303×303×303×303×303×3I3×3︸Atϕt−1δVt−1nδPt−1n∇t−1bεt−1b︸Xt−1+wt−1,(2)Sftn=0fUtn−fNtn−fUtn0fEtnfNtn−fEtn0,(3)Cbn=cosγ0−sinγ010sinγ0cosγ1000cosθsinθ0−sinθcosθcosψ−sinψ0sinψcosψ0001.In (1), t is the time index, ϕt, δVtn, and δPtn denote the errors of the INS’s attitude, velocity, and position, ∇tb,εtb represent the accelerometers bias and gyroscope drift employ. ΔT is the sample time, and ωt∼N0,Qt represent the noise of the system. In (2), (fUtn, fEtn, fNtn) is the acceleration up, east, and north of the coordinate system. In (3), Cbn is the rotation matrix, and θ, γ, ψ represent the rotation angle around x, y, and z axis.The observation equation of the particle filter is as follows.(4)Yt=Yt1Yt2Yt3Yt4Yt5Yt6T,δdti2︸Yti=dtIi2−dtUi2=2xtI−xtiδxti+2ytI−ytiδyti+2ztI−ztiδzti−δxti2+δyti2+δzti2︸gXt|t−1i+νdti︸νti.In the equation above,Yt is the observation vector of PF, δdti2 is the square of the difference between the distances collected by INS and UWB, δxti,δyti,δzti is the filter’s estimation of the INS position errors, xtI,ytI,ztI is the measurement position of INS, dtI,dtU is the distance obtained by INS and UWB, respectively, xti,yti,zti is the position of the ith RN, νti∼N0,Ri is the measurement noise, t is the time index, and superscript ii=1∼6 stands for the label of RN.


## 4. The Scheme of ELM

In the navigation positioning process, we use the PF algorithm to fuse the positioning information of the two sensors and reduce the bad noise and error. ELM is used to assist the PF algorithm. ELM has two functions here. One is to establish the mapping relationship between various state values in the PF filtering process to optimize the accuracy. The other is when UWB’s measurement being interrupted; the mapping relationship between the state value and the observation value is established to compensate the signal, so that the filter can work normally and enhance the stability of the system.


### 4.1. Scheme 1: Correction of State Deviation in the Filtering Process

In the particle filter state update process, the state valuexstate describes the state information of the motion quadrotor, while the state update of the filter xupdate represents the predicted value of the filter, and the particle filter outputs the filtering results by comparing the weight relation. The state value xstate and state update xupdate are two different descriptions of the target state, and there are deviations between them, given by(5)Δxt=xupdate,t−xstate,t,where t is the time index. Since the filtering effect will always be affected by the state value, so before the prediction stage, the state deviation Δx should be optimized. According to equation (1) and the real motion model, it can be known that Δx is affected by the previous moment at all moments, that is to say, the present Δxt is affected by Δx at all previous moments. The (5) is expressed as(6)Δxt=fΔxt−1,Δxt−2……Δx1,where f represents the relational function between them. It should be emphasized that f is unknown, so we use ELM to build this function. In the training model, the quantity of Δx keeps increasing with the change of time, and the training speed of ELM will decline rapidly. We usually set a training scale n to limit the number to the right of (6), for ensuring that the number of states affected by the previous state is the same. So, (6) can be written as(7)Δxt=fΔxt−1,Δxt−2……Δxt−n.Whent>=n+2, the training model is shown in Figure 3.Figure 3
ELM training process.In the filtering process, when time indext>=n+2, ELM collects training data Δxt−1−n,……,Δxt−2, Δxt−n,……,Δxt−1 that are generated by PF as input and output to establish the mapping relationship among them. Then, Δxt−n,……,Δxt−1 as the input of ELM and predict Δxt−n+1,……,Δxt through mapping established in the previous stage. The output state deviation array contains Δxt needed by PF state update at the next moment. Δxt is corrected by ELM. After entering PF, xupdate is calculated by (5) to obtain more accurate estimation information. The process is then repeated until filtering is complete. Thus, the state value of PF is constantly updated by ELM, and the current state value is jointly determined by the previous multiple state values, so the influence of noise is reduced and the positioning accuracy is improved.


### 4.2. Scheme 2: Replenish Measurement Information When UWB Interrupted

Complicated or hostile indoor environment may lead to the interruption of UWB’s signal, and the observation vectorδPo of PF will not be available. Before signal interruption, UWB provides observation data, and ELM is used to establish the mapping between δPo and state value x^t|t−1,t∈1,+∞ output by PF. As in the previous section, this functional relationship is difficult to describe with exact mathematical equations. We use x^t|t−1 as the input training data of ELM and δPo as the training target, ELM through a certain amount of data training to establish the mapping relationship between them. In this way, in the absence of UWB’s signal, ELM can predict δPo^ through the x^t|t−1 and mapping relationship, so that the filter can work normally and ensure the stability of the system. It should be noted here that among the training data used, the x^t|t−1 output by the filter is obtained through the assistance of ELM, so it is more accurate than the simple PF.ELM provides the observation data-assisted PF process when UWB measurement is interrupted as shown in Figure4.Figure 4
ELM provides the observation data-assisted PF process.During the training stage, when UWB measurementPoU is available, PF provides x^t|t−1. The observation vector Yt=dtI−dtU is the training target and ELM establishes offline mapping through training. In the prediction stage, PoU of UWB is not available at this time, ELM predicts δPo^ by input x^t|t−1 and mapping relationship instead of Yt, and PF works normally by using δPo^ provided by ELM.


## 4.1. Scheme 1: Correction of State Deviation in the Filtering Process

In the particle filter state update process, the state valuexstate describes the state information of the motion quadrotor, while the state update of the filter xupdate represents the predicted value of the filter, and the particle filter outputs the filtering results by comparing the weight relation. The state value xstate and state update xupdate are two different descriptions of the target state, and there are deviations between them, given by(5)Δxt=xupdate,t−xstate,t,where t is the time index. Since the filtering effect will always be affected by the state value, so before the prediction stage, the state deviation Δx should be optimized. According to equation (1) and the real motion model, it can be known that Δx is affected by the previous moment at all moments, that is to say, the present Δxt is affected by Δx at all previous moments. The (5) is expressed as(6)Δxt=fΔxt−1,Δxt−2……Δx1,where f represents the relational function between them. It should be emphasized that f is unknown, so we use ELM to build this function. In the training model, the quantity of Δx keeps increasing with the change of time, and the training speed of ELM will decline rapidly. We usually set a training scale n to limit the number to the right of (6), for ensuring that the number of states affected by the previous state is the same. So, (6) can be written as(7)Δxt=fΔxt−1,Δxt−2……Δxt−n.Whent>=n+2, the training model is shown in Figure 3.Figure 3
ELM training process.In the filtering process, when time indext>=n+2, ELM collects training data Δxt−1−n,……,Δxt−2, Δxt−n,……,Δxt−1 that are generated by PF as input and output to establish the mapping relationship among them. Then, Δxt−n,……,Δxt−1 as the input of ELM and predict Δxt−n+1,……,Δxt through mapping established in the previous stage. The output state deviation array contains Δxt needed by PF state update at the next moment. Δxt is corrected by ELM. After entering PF, xupdate is calculated by (5) to obtain more accurate estimation information. The process is then repeated until filtering is complete. Thus, the state value of PF is constantly updated by ELM, and the current state value is jointly determined by the previous multiple state values, so the influence of noise is reduced and the positioning accuracy is improved.


## 4.2. Scheme 2: Replenish Measurement Information When UWB Interrupted

Complicated or hostile indoor environment may lead to the interruption of UWB’s signal, and the observation vectorδPo of PF will not be available. Before signal interruption, UWB provides observation data, and ELM is used to establish the mapping between δPo and state value x^t|t−1,t∈1,+∞ output by PF. As in the previous section, this functional relationship is difficult to describe with exact mathematical equations. We use x^t|t−1 as the input training data of ELM and δPo as the training target, ELM through a certain amount of data training to establish the mapping relationship between them. In this way, in the absence of UWB’s signal, ELM can predict δPo^ through the x^t|t−1 and mapping relationship, so that the filter can work normally and ensure the stability of the system. It should be noted here that among the training data used, the x^t|t−1 output by the filter is obtained through the assistance of ELM, so it is more accurate than the simple PF.ELM provides the observation data-assisted PF process when UWB measurement is interrupted as shown in Figure4.Figure 4
ELM provides the observation data-assisted PF process.During the training stage, when UWB measurementPoU is available, PF provides x^t|t−1. The observation vector Yt=dtI−dtU is the training target and ELM establishes offline mapping through training. In the prediction stage, PoU of UWB is not available at this time, ELM predicts δPo^ by input x^t|t−1 and mapping relationship instead of Yt, and PF works normally by using δPo^ provided by ELM.


## 5. Test

In order to verify the effectiveness of the proposed strategy, a real experiment has been designed. The availability of the proposed strategy is proved by results of the simulation and error analysis.


### 5.1. Experimental Settings

The experimental site was selected in an empty space, andx,y,z represents the actual directions east, north, and up. The quadrotor is equipped with the inertial measurement unit (IMU) and UWB blind node (UWB BN) to provide reference for distance information. The quadrotor used in the experiment is shown in Figure 5. In order to better position the quadrotor, six UWB reference nodes (UWB RNs) were placed at known coordinate points. Three were placed high and three were placed low (height 2.7 m and 0.1 m, horizontal distances between each UWB RN were 2.3 m and 4.6 m, as shown in yellow, i=1−6). The UWB RN form a 3D space in which the target quadrotor moves along a planned path. The upper computer collects INS and UWB’s data at an interval of ΔT=0.02S. The actual test scene is shown in Figure 6.Figure 5
The quadrotor used in the experiment.Figure 6
The test environment.


### 5.2. Localization Errors

In this part, the effectiveness of the proposed strategy is verified by simulation calculation:Scheme 1: ELM-assisted PF can effectively reduce positioning errorScheme 2: when UWB’s signal is interrupted, ELM compensates the signal through the mapping relationship to make the filter work normally


#### 5.2.1. Error Analysis of Scheme 1: Correction of State Deviation

For scheme 1, the position informationPoI of INS and PoU of UWB can be calculated, respectively, through the data collected by the upper computer, and the flight trajectory in 3D space can be fitted out. The ELM-assisted PF algorithm mentioned in scheme 1 is used to process the data to reduce the influence of bad noise and improve the positioning accuracy. Here, an individual PF algorithm is also used to prove the superiority of the proposed strategy by comparison. The trajectory comparison diagram is shown in Figure 7.Figure 7
The trajectory comparison.The error reduction of the ELM-assisted PF model with that of individual PF was compared. The root mean square error (RMSE) has been used in thex, y, and z directions at all time indexes, which was calculated as(8)RMSEPot=13xt−xtref2+yt−ytref2+zt−ztref21/2,where RMSEPot means the RMSE of the position at time index t, xt,yt,zt is the estimated position in x, y, and z directions, and xtref,ytref,ztref is the reference position in x, y, and z directions at the time index t.After filtering, RMSE of UWB, PF, and ELM-assisted PF were calculated, and the results are shown in Figure8 and Table 1.Figure 8
RMSE comparison of UWB, PF, and PF + ELM.Table 1
Average RMSES produced by PF and PF + ELM in the east, north, and up.
RMSEm
East
North
UP
Mean
Improving
UWB
0.0692
0.0996
0.0906
0.0864
—-
PF
0.0600
0.0909
0.0800
0.0770
10.88 %
PF + ELM
0.0515
0.0768
0.0603
0.0628
27.31 %In Table1, mean represents the average value of the errors in the three directions. It can be seen that comparing with UWB, the PF algorithm error is reduced by 10.88% and the PF + ELM algorithm error is reduced by 27.31%. Compared with the PF algorithm, PF + ELM algorithm error is reduced by 18.44%. It should be emphasized that UWB positioning and PF are quite exact.Otherwise, the cumulative distribution function (CDF) is also used to describe the achievement rate of each algorithm. The comparison of CDF is shown in Figure9. As shown in the figure, the PF + ELM strategy achieves the ideal state faster than PF alone, indicating that for the INS/UWB combined navigation model, scheme 1 has a better effect on reducing noise and optimizing accuracy.Figure 9
CDF comparison of UWB, PF, and PF + ELM.


#### 5.2.2. Error Analysis of Scheme 2: Replenish Measurement Information

For scheme 2, after the position informationPoI of INS and PoU of UWB are fitted, four interrupt regions #1,#2,#3,#4 were set to simulate the interruption of UWB’s signal, as shown in Figure 10.Figure 10
Reference trajectory and outage areas#1,#2,#3,#4.To demonstrate the performance of the proposed strategy, LS-SVM was also used. In the outage area, ELM and LS-SVM perform signal compensation, respectively, as shown in Figure11. Meanwhile, RMSEs of four areas are calculated, respectively, according to (8), as shown in Figure 12 and Table 2.Figure 11
Comparison of ELM and LS-SVM prediction values in outage areas.Figure 12
The RMSEs estimated by ELM and LS-SVM in outage areas.Table 2
Average RMSEs produced by ELM and LS-SVM in outages#1, #2, #3,#4.
RMSEm
#
1
#
2
#
3
#
4
Mean
PF + LS-SVM
1.3742
1.5339
0.9910
1.1561
1.2638
PF + ELM
0.1176
0.1046
0.0767
0.1502
0.1123As shown in Figure11, the red line represents the outage areas of the reference trajectory, the green line represents the result of ELM-assisted PF prediction, and the blue line represents the result of LS-SVM prediction. a,b,c,d indicate the outage areas #1,#2,#3,#4. Obviously, the difference between PF + ELM and the reference value is smaller, proving that compared with LS-SVM, the proposed strategy is more effective. According to Table 2, the error of PF + ELM is reduced by 1.15 m compared with LS-SVM.


## 5.1. Experimental Settings

The experimental site was selected in an empty space, andx,y,z represents the actual directions east, north, and up. The quadrotor is equipped with the inertial measurement unit (IMU) and UWB blind node (UWB BN) to provide reference for distance information. The quadrotor used in the experiment is shown in Figure 5. In order to better position the quadrotor, six UWB reference nodes (UWB RNs) were placed at known coordinate points. Three were placed high and three were placed low (height 2.7 m and 0.1 m, horizontal distances between each UWB RN were 2.3 m and 4.6 m, as shown in yellow, i=1−6). The UWB RN form a 3D space in which the target quadrotor moves along a planned path. The upper computer collects INS and UWB’s data at an interval of ΔT=0.02S. The actual test scene is shown in Figure 6.Figure 5
The quadrotor used in the experiment.Figure 6
The test environment.


## 5.2. Localization Errors

In this part, the effectiveness of the proposed strategy is verified by simulation calculation:Scheme 1: ELM-assisted PF can effectively reduce positioning errorScheme 2: when UWB’s signal is interrupted, ELM compensates the signal through the mapping relationship to make the filter work normally


### 5.2.1. Error Analysis of Scheme 1: Correction of State Deviation

For scheme 1, the position informationPoI of INS and PoU of UWB can be calculated, respectively, through the data collected by the upper computer, and the flight trajectory in 3D space can be fitted out. The ELM-assisted PF algorithm mentioned in scheme 1 is used to process the data to reduce the influence of bad noise and improve the positioning accuracy. Here, an individual PF algorithm is also used to prove the superiority of the proposed strategy by comparison. The trajectory comparison diagram is shown in Figure 7.Figure 7
The trajectory comparison.The error reduction of the ELM-assisted PF model with that of individual PF was compared. The root mean square error (RMSE) has been used in thex, y, and z directions at all time indexes, which was calculated as(8)RMSEPot=13xt−xtref2+yt−ytref2+zt−ztref21/2,where RMSEPot means the RMSE of the position at time index t, xt,yt,zt is the estimated position in x, y, and z directions, and xtref,ytref,ztref is the reference position in x, y, and z directions at the time index t.After filtering, RMSE of UWB, PF, and ELM-assisted PF were calculated, and the results are shown in Figure8 and Table 1.Figure 8
RMSE comparison of UWB, PF, and PF + ELM.Table 1
Average RMSES produced by PF and PF + ELM in the east, north, and up.
RMSEm
East
North
UP
Mean
Improving
UWB
0.0692
0.0996
0.0906
0.0864
—-
PF
0.0600
0.0909
0.0800
0.0770
10.88 %
PF + ELM
0.0515
0.0768
0.0603
0.0628
27.31 %In Table1, mean represents the average value of the errors in the three directions. It can be seen that comparing with UWB, the PF algorithm error is reduced by 10.88% and the PF + ELM algorithm error is reduced by 27.31%. Compared with the PF algorithm, PF + ELM algorithm error is reduced by 18.44%. It should be emphasized that UWB positioning and PF are quite exact.Otherwise, the cumulative distribution function (CDF) is also used to describe the achievement rate of each algorithm. The comparison of CDF is shown in Figure9. As shown in the figure, the PF + ELM strategy achieves the ideal state faster than PF alone, indicating that for the INS/UWB combined navigation model, scheme 1 has a better effect on reducing noise and optimizing accuracy.Figure 9
CDF comparison of UWB, PF, and PF + ELM.


### 5.2.2. Error Analysis of Scheme 2: Replenish Measurement Information

For scheme 2, after the position informationPoI of INS and PoU of UWB are fitted, four interrupt regions #1,#2,#3,#4 were set to simulate the interruption of UWB’s signal, as shown in Figure 10.Figure 10
Reference trajectory and outage areas#1,#2,#3,#4.To demonstrate the performance of the proposed strategy, LS-SVM was also used. In the outage area, ELM and LS-SVM perform signal compensation, respectively, as shown in Figure11. Meanwhile, RMSEs of four areas are calculated, respectively, according to (8), as shown in Figure 12 and Table 2.Figure 11
Comparison of ELM and LS-SVM prediction values in outage areas.Figure 12
The RMSEs estimated by ELM and LS-SVM in outage areas.Table 2
Average RMSEs produced by ELM and LS-SVM in outages#1, #2, #3,#4.
RMSEm
#
1
#
2
#
3
#
4
Mean
PF + LS-SVM
1.3742
1.5339
0.9910
1.1561
1.2638
PF + ELM
0.1176
0.1046
0.0767
0.1502
0.1123As shown in Figure11, the red line represents the outage areas of the reference trajectory, the green line represents the result of ELM-assisted PF prediction, and the blue line represents the result of LS-SVM prediction. a,b,c,d indicate the outage areas #1,#2,#3,#4. Obviously, the difference between PF + ELM and the reference value is smaller, proving that compared with LS-SVM, the proposed strategy is more effective. According to Table 2, the error of PF + ELM is reduced by 1.15 m compared with LS-SVM.


## 5.2.1. Error Analysis of Scheme 1: Correction of State Deviation

For scheme 1, the position informationPoI of INS and PoU of UWB can be calculated, respectively, through the data collected by the upper computer, and the flight trajectory in 3D space can be fitted out. The ELM-assisted PF algorithm mentioned in scheme 1 is used to process the data to reduce the influence of bad noise and improve the positioning accuracy. Here, an individual PF algorithm is also used to prove the superiority of the proposed strategy by comparison. The trajectory comparison diagram is shown in Figure 7.Figure 7
The trajectory comparison.The error reduction of the ELM-assisted PF model with that of individual PF was compared. The root mean square error (RMSE) has been used in thex, y, and z directions at all time indexes, which was calculated as(8)RMSEPot=13xt−xtref2+yt−ytref2+zt−ztref21/2,where RMSEPot means the RMSE of the position at time index t, xt,yt,zt is the estimated position in x, y, and z directions, and xtref,ytref,ztref is the reference position in x, y, and z directions at the time index t.After filtering, RMSE of UWB, PF, and ELM-assisted PF were calculated, and the results are shown in Figure8 and Table 1.Figure 8
RMSE comparison of UWB, PF, and PF + ELM.Table 1
Average RMSES produced by PF and PF + ELM in the east, north, and up.
RMSEm
East
North
UP
Mean
Improving
UWB
0.0692
0.0996
0.0906
0.0864
—-
PF
0.0600
0.0909
0.0800
0.0770
10.88 %
PF + ELM
0.0515
0.0768
0.0603
0.0628
27.31 %In Table1, mean represents the average value of the errors in the three directions. It can be seen that comparing with UWB, the PF algorithm error is reduced by 10.88% and the PF + ELM algorithm error is reduced by 27.31%. Compared with the PF algorithm, PF + ELM algorithm error is reduced by 18.44%. It should be emphasized that UWB positioning and PF are quite exact.Otherwise, the cumulative distribution function (CDF) is also used to describe the achievement rate of each algorithm. The comparison of CDF is shown in Figure9. As shown in the figure, the PF + ELM strategy achieves the ideal state faster than PF alone, indicating that for the INS/UWB combined navigation model, scheme 1 has a better effect on reducing noise and optimizing accuracy.Figure 9
CDF comparison of UWB, PF, and PF + ELM.


## 5.2.2. Error Analysis of Scheme 2: Replenish Measurement Information

For scheme 2, after the position informationPoI of INS and PoU of UWB are fitted, four interrupt regions #1,#2,#3,#4 were set to simulate the interruption of UWB’s signal, as shown in Figure 10.Figure 10
Reference trajectory and outage areas#1,#2,#3,#4.To demonstrate the performance of the proposed strategy, LS-SVM was also used. In the outage area, ELM and LS-SVM perform signal compensation, respectively, as shown in Figure11. Meanwhile, RMSEs of four areas are calculated, respectively, according to (8), as shown in Figure 12 and Table 2.Figure 11
Comparison of ELM and LS-SVM prediction values in outage areas.Figure 12
The RMSEs estimated by ELM and LS-SVM in outage areas.Table 2
Average RMSEs produced by ELM and LS-SVM in outages#1, #2, #3,#4.
RMSEm
#
1
#
2
#
3
#
4
Mean
PF + LS-SVM
1.3742
1.5339
0.9910
1.1561
1.2638
PF + ELM
0.1176
0.1046
0.0767
0.1502
0.1123As shown in Figure11, the red line represents the outage areas of the reference trajectory, the green line represents the result of ELM-assisted PF prediction, and the blue line represents the result of LS-SVM prediction. a,b,c,d indicate the outage areas #1,#2,#3,#4. Obviously, the difference between PF + ELM and the reference value is smaller, proving that compared with LS-SVM, the proposed strategy is more effective. According to Table 2, the error of PF + ELM is reduced by 1.15 m compared with LS-SVM.


## 6. Conclusion

In this study, ELM-assisted PF for INS/UWB combined navigation quadrotor positioning strategy is studied. In this scheme, INS and UWB acquire the position information of the motion quadrotor, respectively, PF performs data fusion and filtering, and ELM is used to assist the filter. ELM establishes the mapping between the difference deviation of each state in the filtering process to correct state deviation. Meanwhile, during the normal operation of PF, ELM keeps training and establishing the relationship between the state vector output by PF and the observation vector. When UWB’s signal is interrupted, the observation vector fails; under this condition, ELM makes signal prediction and provides observation signal to make the filter work normally. The experimental results show that the proposed ELM-assisted PF strategy can effectively correct the state deviation, further reduce the position error on the basis of PF, and improve the positioning accuracy by 18.44% compared with single PF. When the signal is abnormal, PF can also work normally through the observation vector provided by ELM. Compared with LS-SVM, the positioning accuracy provided by ELM improved by 1.15 m.



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*Source: 9739345-2022-06-13.xml*