Trajectory optimization for minimum range error in bearings-only source localization

Abstract

This paper compares two performance indices for computing optimal observer paths for the bearings-only source localization problem, for constant velocity sources. Previous work on this problem is based on maximizing the determinant of the Fisher information matrix (FIM) of the estimation problem. This paper considers minimizing the trace of a weighted sum of the Cramer-Rao lower bound (CRLB) of current source position error. Quasi-Newton optimization is used to compare optimal observer paths, given the goal of minimizing current position error. Significant differences in optimal paths are observed, arid the CRLB trace is found to yield smaller range error.'The prohleni of li)c,iiting R constait; velocsity 11011-in;trieiiveriiig sourcc rising only bearing rneasrirernmts is known to be tiependcnt on t h : observer's trajectory. Iri a typical c>st,imation problem, a freely rrioving observer rwords bearing inea.surein(;iii.~8 to t,he source through a passive sensor and mtimattts i~lie sourct:'~ unknown position. Alt,lioiigh i~ ~tuniber of authors liavt, studied the tr:ijwtorics that, niakc, this ~)rot)l(*ni ot)sc:rvtible [I-31, the problcni of finding the opt inial o1)scrvcr path has rccr.ivd little iItt>cmtiori. 'lypicdly it is suggested that t,Iics obstmw t>xtv.utes ti t:xtrcmic chiillgo in rlirt,ct,ion for tho hwrings-oiilv locitlizat iori algorit hrns to prforin well. h i i t mort: needs to kx~ siiitf o i i how t,o use observcr trajcctories to improve rstirnatiori &orit lini pcrfortnaim. t,hc> cjur:st,ioii of tiow to properly define optiiriitlity in the bmriiigs-only localization p r o b le~n. The opt,imality criterion iised here is based on minimizing the trace of Crarrit;r-R;to lower bound (CRLB).The CRLB dcfintd in 131 is i'. 1owt.r twund on tht: error covariarice of the estirnatioii 1)robltm. ;tnd gives a criterion of optiniality t lint is indtqji:nrierit~ of the estiniat,ioIi algorithm. Tlir CRLB t,ritc*ts approach makes intuitive s i x w sincc the, t r a w of t hr! c.ov;iriarice niiitrix is the nio;tsurf' for defining optiinality in Iialman filters [SI, antl the t1ir~)retiid co\.ariitncc. r1iittri.u for two well known Cartesiaii rst iniation algorithms. t l~ Sttrrisfielti and thc Maxiniiini 1,ikolihood rnetliocls [(j:. itre itlso t . h~ S U I I~ ii.5 tho C'RLB.A previous study on calculating optimitl paths is based on maximizing the dctermin;tiit of the Fisher inforrriation mat,rix (FIM) of t,he estirnation problem [7]. the inverse of the CRLB. That approach minimizes the area of a confidence ellipsoid around the estimates of the initial p u sitiori and initial velocity. Using area as a performance measure may favor solutions with highly eccentric confidence ellipsoids. This behavior may be troublesome in sourcr localizat,ion, since the largest ambiguity in the confidence ellipsoid often corresponds to the unknown range variablc, whilc the smaller axis often corresponds to the known bearing measurements.The performance criterion in this piaper is the tracci of the soiircv position CRLB. The tracr of the cov...

Stochastic Real-Time Optimal Control for Bearing-Only Trajectory Planning

“…To form a suitable localization performance index, the directional information in the matrix must be compressed into a scalar value upon which to optimize. Maximization of the determinant of the FIM, a determinant lower bound, or a determinant approximation [2,5,6,[14][15][16] effectively minimizes the volume of the uncertainty ellipsoid around the target estimate, but highly eccentric ellipsoid shapes can result [17]. Similar options (sometimes only semantically different) include the trace of the CRLB [18], the trace of the covariance matrix [3], maximization of the smallest FIM eigenvalue [19], minimization of the trace of the inverse of the FIM [20], or minimization of the differential entropy of the posterior target density (equivalent to maximizing the FIM determinant for the Gaussian case) [21].…”

“…subject to the dynamic constraints: (16) the path constraints: (17) and the boundary conditions: (18) with equality constraints imposed via a second constraint on the additive inverse. One advantage of incorporating final covariance as a boundary constraint in the optimal control problem is that any performance index can be used that best fits the situation.…”

Optimal Maneuvering of Seismic Sensors for Localization of Subsurface Targets

“…l / is the signal reflected from the j th target which can be estimated using (13). Once we remove the first target using (20), the FIM will be reduced to a 22 matrix.…”

“…Various measures of Fisher information are possible, giving different design criteria, but this paper concentrates on D-optimal design which maximizes the determinant of the Fisher information matrix. An example of using the method of optimal experiments for sensor placement can be found in [12], [13], which deals with the movement of sensors used in direction-of-arrival (DOA) estimation to localize a source. The D-optimal criterion is equivalent to minimizing the trace of the Cramer-Rao lower bound (CRLB), so the result is an optimal observer (sensor) path that localizes a moving source.…”

Trajectory Optimization for Target Localization Using Small Unmanned Aerial Vehicles

“…21 Oshman and Davidson 33 also use the determinant of the FIM to show enhanced target estimation. Other optimization functions include the standard deviation value for the range, 10 the trace of the CRLB, 16,17 and the determinant of the error covariance matrix. 13,28 Other optimization approaches considered by Liu 27 and by Passerieux and Van Cappel 35 involve classical optimal control techniques, however, these results do not easily scale to more complex target motions or higher order dynamics.…”