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ErrorEstimationfortheLinearizedAuto-LocalizationAlgorithm•IntroductionLocalPositioningSystemsTypicalauto-localizationproblemknownpositionslocatethebeaconsOnlydistanceanonlinearleast-squareoptimizationalgorithmatrialanderrormethodExtendedKalmanFilterandanH-infinitefilterexternalpositioninginformationdead-reckoningadistancematrixaninterpolationschemeLinearizedAuto-Localization(LAL)algorithmneitherrequiresatrialanderrorapproximationnoranyexternalpositioninginformationallbeaconsarelocatedinaplaneparalleltotheplanecontainingthemobilenodetrajectoryByusingtheseconditions,allnonlineartermsofthetrilaterationequationscanbegroupedinadditionalvariablesinordertoobtainlinearauto-localizationequations.•estimatethesolution’sstandarddeviationoftheLALalgorithmbasedonafirstorderTaylorapproximation.dependsontheassumptionthatthesolutionhasaGaussiandistributionaparameterτmeasurethereliabilityoftheestimatedstandarddeviationimprovetheLALalgorithmLAL算法3DLPSs信标节点间距离（未知）信标节点与虚拟节点间距离（测出）LAL算法（依靠三边测量法）5维X向量的解只包含节点间距离由X可以表示出确定需要测量5段距离，测量6次，需提供6个虚拟节点。6个虚拟节点，可以由一个移动节点（例如无人机）来提供。在一个很大的区域内有多种移动路径。一旦确定了足够信标节点的位置，就变成了一个2D的定位问题。LAL算法的局限性：依靠MDS（多维标度分析），MDS需要到所有信标节点成对的距离组成的距离矩阵，运动的路径不能保证到所有信标节点的距离都能被测量的到。一些距离数据需要用其他方法估计。LaMsM(MDS的改进)，只需连接局部信标子集，形成局部图，然后再合并为包含所有信标的空间图。精度实际距离误差假设成0均值方差的的正态分布通过LAL可以估计的精确性，的误差分布不是正态的，直接的估计这个误差十分复杂，利用DSA微分灵敏度分析•DSAevaluatetheerrorofagivenfunctionoriginatedbytheuncertaintypresentonitsvariablesUsesafirstorderTaylorexpansiontherealdistributionofYisclosetoaGaussiandistribution,thehighertermsoftheTaylorseriescanbeneglected.applytheDSAtoobtainthevarianceoftheleastsquaressolutionofEquation(1)ErrorPerturbationinLeastSquaresSolutionsuseanapproximationproposedbyStewartbasedontheassumptionthattheerrorcanbemodelledasaGaussiandistributionallelementsofAandBcanbedependent.ObservedmatricesCovarianceMatrixoftheInter-BeaconDistancesthefinalvarianceerroroftheLALmethodthesimplestcaseallthemeasureddistanceshavethesamevariancegeometricdilutionofprecision(GDOP)inGPS衡量定位精度的很重要的一个系数，它代表GPS测距误差造成的接收机与空间卫星间的距离矢量放大因子。实际表征参与定位解的从接收机至空间卫星的单位矢量所勾勒的形体体积与GDOP成反比，故又称为几何精度因子。GDOP的数值越大，所代表的单位矢量形体体积越小，即接收机至空间卫星的角度十分相似导致的结果，此时的GDOP会导致定位精度变差DistanceDilutionofPrecision(DDOP)dependsontherelativepositionReliabilityoftheVarianceEstimationerrorsofapproximatetoazeromeanGaussiandistributionaparametermeasurethevalidityoftheestimationisnecessarywefoundthatevaluatingonlytheerrorperturbationontheleastsquaressolutionisenoughtoverifythereliabilityoftheestimatedvariance.Thisisbecauseamorerestrictedapproximationisusedforthepseudo-inverselinearizationthanfortheothernon-linearequations.complementaryorthogonalprojectionontotheorthogonalspaceofATheparameterτrepresentstheratiobetweentheerrorperturbationF,associatedbythenoisymeasurements,andtherealvalueofmatrixA+.Asthevalueincreases,theperturbationbecameincreasinglyimportantandcannotbedisregarded.whereErepresentstheexpectedvalueEvaluationoftheDSAMethodTherangingdatawasgeneratedwithanadditivewhiteGaussiannoisewithzeromeanandastandarddeviationof0.01m.inputerrorsoutputerrors1000timesofflineestimatesarethosesolutionsobtainedwiththeactualvaluesofthedistancemeasurementsonlineestimatesaresolutionsobtainedwithnoisydistancemeasurementsandtheestimatedpositionofthenodesobtainedbytheauto-localizationmethod.ReliabilityoftheVarianceEstimationinputnoiselowerWeightedLinearizedAuto-LocalizationAlgorithm(WLAL)EvaluationonanUltrasonicLPS0.23mm100Inter-BeaconDistancesEstimation0.23mmanimprovementof32.8%wasobtainedusingtheWLALalgorithmBeacons’PositionEstimationanimprovementof22%wasobtainedonthestandarddeviationonaxisXandYusingtheWLALalgorithm