1. 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 Sirola Institute of Mathematics Tampere University of Technology FINLAND

2. Mathematics 2 Sirola: A Versatile Algorithm for Local Positioning in Closed Form 15.6.2015 Introduction Range and range difference measurements Systems of three quadratic surfaces Closed-form solution Several solutions (up to nine!)

3. Mathematics 3 Sirola: A Versatile Algorithm for Local Positioning in Closed Form 15.6.2015 Local Positioning Local stations -E.g. cellular base stations -Nonlinear behavior Distant stations -E.g. positioning satellites -Range can be linearized with negligible error

4. Mathematics 4 Sirola: A Versatile Algorithm for Local Positioning in Closed Form 15.6.2015 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

5. Mathematics 5 Sirola: A Versatile Algorithm for Local Positioning in Closed Form 15.6.2015 Intersection Types 1/4 Intersection of three planes: linear system at most 1 distinct solution

6. Mathematics 6 Sirola: A Versatile Algorithm for Local Positioning in Closed Form 15.6.2015 Intersection Types 2/4 Intersection of two planes and a quadric: two planes intersect in a line 0-2 distinct solutions second-degree polynomial

7. Mathematics 7 Sirola: A Versatile Algorithm for Local Positioning in Closed Form 15.6.2015 Intersection Types 3/4 Intersection of a plane and two quadrics: up to four distinct solutions

8. Mathematics 8 Sirola: A Versatile Algorithm for Local Positioning in Closed Form 15.6.2015 Intersection Types 4/4 Intersection of three quadrics: roots of a 9th degree polynomial usually reduces to a simpler case

9. Mathematics 9 Sirola: A Versatile Algorithm for Local Positioning in Closed Form 15.6.2015 Geometric Reduction When intersection of two surfaces is planar, one can be replaced with the intersection plane

10. Mathematics 10 Sirola: A Versatile Algorithm for Local Positioning in Closed Form 15.6.2015 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)

11. Mathematics 11 Sirola: A Versatile Algorithm for Local Positioning in Closed Form 15.6.2015 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

12. Mathematics 12 Sirola: A Versatile Algorithm for Local Positioning in Closed Form 15.6.2015 Testing vs. Least Squares Iteration Impact of LS starting point (4 measurements: overdetermined system) LS starting point Base station True position

13. Mathematics 13 Sirola: A Versatile Algorithm for Local Positioning in Closed Form 15.6.2015 Testing vs. Least Squares Iteration LS started from both direct solutions, the one with the smallest residual chosen

14. Mathematics 14 Sirola: A Versatile Algorithm for Local Positioning in Closed Form 15.6.2015 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

15. Mathematics 15 Sirola: A Versatile Algorithm for Local Positioning in Closed Form 15.6.2015 Equations plane: quadric of revolution: normal vector stationdirectrix vector

16. Mathematics 16 Sirola: A Versatile Algorithm for Local Positioning in Closed Form 15.6.2015 Quadric Surface Types -(point)sphere -(point)(ellipsoid) -(line)paraboloid hyperboloid branch cone branch hyperboloid branch