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digits wastothelogarithmic one.Othermathematicians (S.Goudsmit, W.Furry, H.Hurwitz, R.Pinkham, R.Raimi, Benford's lawgives expected patterns ofthedigits innu- andespecially byT.Hill) gavethetheoretical basis forBenmerical data.Itcanbeusedasa tooltodetect outliers ford's Law. forexample inasatestfortheauthenticity andreliability oftransaction level accounting data.BasedonBenford's lawtests forfirst twodigits, first three digits, last digits, II Mathematical approach lastdigit, last twodigits havebeenderived asanadditional analytical tool. Benford's lawisknownasa 'first Clearly theunderlying premise ofBenford's Law digit law', 'digit analisys' or'Benford-Newcomb phenom-isthatthesubject population ofquantities, expressed in enon'. Thesecond ordertestisananalysis ofthedigitthebase10andmoreorless arbitrary units, will befairly frequencies ofthedifferences between theordered (ranked) evenly distributed ona logarithmic scale. Thisisconvalues inadataset.Thedigit frequencies ofthese difer- firmed bythefactthattheexponents onthese constants encesapproximates thefrequencies ofBenford's Lawforarefairly uniformly distributed (atleast overseveral ormostdistributions oftheoriginal data.Fromsomeaudi-dersofmagnitude). Asaresult, theprobability ofthe tor's pointofviewitisveryimportant ifitispossible to leading digit beingdclearly approaches [4] useBenford 'slawtotrace siginificant changes indatain p dIB 10}loglo (d+ 1) loglo d someperiod oftimeordetect sometrends within them. log10 10- log10d

L. Banjanović-Mehmedović, I. Petrović, E. Ivanjko

Mobile robot localization is the problem of determining the pose (position and orientation) of a mobile robot under complex measurement uncertainties. The algorithm introduced here is based on the modified multiple model and exploits a soft gating of the measurements to reduce the computational requirements of the approach. The position part is based on an x and y histograms scan matching procedure, where x and y histograms are extracted directly from local occupancy grid maps using probability scalar transformation. The orientation part is based on the proposed obstacle vector transformation combined with polar histograms. Proposed algorithms are tested using a Pioneer 2DX mobile robot.

Global localization is the problem of determining the pose of a mobile robot under global uncertainty. The novel algorithm introduced here is based on

L. Banjanović-Mehmedović, I. Petrović, E. Ivanjko

In most applications, a mobile robot must be able to determine its position and orientation in the environment using only own sensors. Orientation estimation accuracy greatly influences the position estimation accuracy and is therefore crucial for a reliable mobile robot pose tracking. Our approach to orientation estimation is based on angle histograms matching. Angle histograms are obtained indirectly via Hough transformation combined with a non-iterative algorithm for determination of the end points and length of straight-line parts contained in obtained histograms. Sensors used for local occupancy grid generation are sonars. Test results with mobile robot Pioneer 2DX simulator show the capacity of this method.

L. Banjanović-Mehmedović, E. Ivanjko, Ivan Petrović

Global localization is the problem of determining the position of a mobile robot under global uncertainty. The algorithm introduced here is based on the multiple model (MM) and exploits a soft gating of the problem (SG) to reduce the computational requirements of the approach. This localization algorithm is based on combining histograms and Hough transform. Presented algorithm is tested using a Pioneer 2 DX mobile robot simulator

L. Banjanović-Mehmedović, I. Petrović, E. Ivanjko

In most applications, a mobile robot must be able to determine its position and orientation in the environment using only own sensors. The problem of pose tracking can be seen as a constituent part of the more general navigation problem. Our proposed approach is able to track the mobile robot pose without environment model. It is based on combining histograms and Hough transform (HHT). While histograms for position tracking (x and y histograms) are extracted directly from local occupancy grid maps, angle histogram is obtained indirectly via Hough transformation combined with a non-iterative algorithm for determination of end points and length of straight-line parts contained in obtained histograms. Histograms obtained at the actual mobile robot pose are compared to histograms saved at previous mobile robot poses to compute position displacement and orientation correction. Orientation estimation accuracy greatly influences the position estimation accuracy and is crucial for a reliable mobile robot pose tracking. Sensors used for local occupancy grid generation are sonars but other exteroceptive sensors like a laser range finder can also be used. Test results with mobile robot Pioneer 2DX simulator show the capacity of this method.

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