Mathematics 1 Sirola: A Versatile Algorithm for Local Positioning in Closed Form 15.6.2015 A Versatile Algorithm for Local Positioning in Closed Form Niilo.

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Presentation transcript:

Mathematics 1 Sirola: A Versatile Algorithm for Local Positioning in Closed Form A Versatile Algorithm for Local Positioning in Closed Form Niilo Sirola Institute of Mathematics Tampere University of Technology FINLAND

Mathematics 2 Sirola: A Versatile Algorithm for Local Positioning in Closed Form Introduction Range and range difference measurements Systems of three quadratic surfaces Closed-form solution Several solutions (up to nine!)

Mathematics 3 Sirola: A Versatile Algorithm for Local Positioning in Closed Form Local Positioning Local stations -E.g. cellular base stations -Nonlinear behavior Distant stations -E.g. positioning satellites -Range can be linearized with negligible error

Mathematics 4 Sirola: A Versatile Algorithm for Local Positioning in Closed Form Geometric Model 1.Plane range to distant station range difference between distant stations altitude 2.Sphere range to local station 3.Paraboloid range difference between distant and local station 4.Hyperboloid range difference between local stations quadrics

Mathematics 5 Sirola: A Versatile Algorithm for Local Positioning in Closed Form Intersection Types 1/4 Intersection of three planes: linear system at most 1 distinct solution

Mathematics 6 Sirola: A Versatile Algorithm for Local Positioning in Closed Form Intersection Types 2/4 Intersection of two planes and a quadric: two planes intersect in a line 0-2 distinct solutions second-degree polynomial

Mathematics 7 Sirola: A Versatile Algorithm for Local Positioning in Closed Form Intersection Types 3/4 Intersection of a plane and two quadrics: up to four distinct solutions

Mathematics 8 Sirola: A Versatile Algorithm for Local Positioning in Closed Form Intersection Types 4/4 Intersection of three quadrics: roots of a 9th degree polynomial usually reduces to a simpler case

Mathematics 9 Sirola: A Versatile Algorithm for Local Positioning in Closed Form Geometric Reduction When intersection of two surfaces is planar, one can be replaced with the intersection plane

Mathematics 10 Sirola: A Versatile Algorithm for Local Positioning in Closed Form Algorithm 1.Acquire measurements 2.Transform to geometric form 3.Use local assumption to linearize distant measurements 4.Linearize further by intersection planes 5.Solve intersection(s)

Mathematics 11 Sirola: A Versatile Algorithm for Local Positioning in Closed Form Comparison Against Least Squares Pros: gives all solutions, LS just one very small computational cost LS may perform poorly for strongly nonlinear cases Cons: can use only three measurements, LS uses all no variance estimates

Mathematics 12 Sirola: A Versatile Algorithm for Local Positioning in Closed Form Testing vs. Least Squares Iteration Impact of LS starting point (4 measurements: overdetermined system) LS starting point Base station True position

Mathematics 13 Sirola: A Versatile Algorithm for Local Positioning in Closed Form Testing vs. Least Squares Iteration LS started from both direct solutions, the one with the smallest residual chosen

Mathematics 14 Sirola: A Versatile Algorithm for Local Positioning in Closed Form Conclusions position solution by intersecting three surfaces that are either planes or (branches of) quadrics of revolution can be used as a preliminary step to obtain a set of starting points for LS iteration multiple solutions future research: choosing the correct position

Mathematics 15 Sirola: A Versatile Algorithm for Local Positioning in Closed Form Equations plane: quadric of revolution: normal vector stationdirectrix vector

Mathematics 16 Sirola: A Versatile Algorithm for Local Positioning in Closed Form Quadric Surface Types -(point)sphere -(point)(ellipsoid) -(line)paraboloid hyperboloid branch cone branch hyperboloid branch