1 / 16

Closed-form Algorithms in Hybrid Positioning: Myths and Misconceptions

Closed-form Algorithms in Hybrid Positioning: Myths and Misconceptions. Niilo Sirola Department of Mathematics Tampere University of Technology, Finland (currently with Taipale Telematics, Finland) niilo.sirola@taipaletelematics.com. Mobile positioning. Given a measurement model

jody
Download Presentation

Closed-form Algorithms in Hybrid Positioning: Myths and Misconceptions

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Closed-form Algorithms in Hybrid Positioning: Myths and Misconceptions Niilo Sirola Department of Mathematics Tampere University of Technology, Finland (currently with Taipale Telematics, Finland) niilo.sirola@taipaletelematics.com Workshop for Positioning, Navigation and Communication 2010

  2. Mobile positioning • Given a measurement model • y = h(x) + v • and the measurements • y • Find the position x that fits the measurements ”best” Workshop for Positioning, Navigation and Communication 2010

  3. Iterative Least Squares • The Gauss-Newton or Taylor series or Iterative/Ordinary/Nonlinear least squares • Usual objections to Gauss-Newton • initial guess: ”Selection of such a starting point is not simple in practice” • convergence is not assured • computational load: ”as LS computation is required in each iteration” Workshop for Positioning, Navigation and Communication 2010

  4. Examples of closed-form methods • geometrically inspired methods – easy to explain and visualise ”replace each intersection with a line then solve the linear LS problem” Workshop for Positioning, Navigation and Communication 2010

  5. Examples of closed-form methods • others are algebraic and more rigorous • sometimes come with a proof • sometimes can be implemeted by the reader Workshop for Positioning, Navigation and Communication 2010

  6. Least-squares vs least-quartic solution • Least-squares solution: • Find x such that ‖y – h(x)‖2 is as small as possible • Least-quartic solution: • ‖y2 – h(x) 2‖2 • easier to solve analytically, but the solution is not least squares solution • -> is non-optimal in variance sense Workshop for Positioning, Navigation and Communication 2010

  7. Closed-form methods? • Some are not even in closed form…. • ”…first assume there is no relationship between x,y, and r1 … The final solution is obtained by imposing the relationship.. via another LS computation” • ”we can first use (14) to obtain an initial solution…” Workshop for Positioning, Navigation and Communication 2010

  8. Testing method • testing range-only methods • 12 by 12 kilometer simulated test field, six ranging beacons • independent and identically distributed Gaussian measurement noise • noise sigma sweeps from 1 m to 10 km • 1000 position fixes with random true position for each noise level Workshop for Positioning, Navigation and Communication 2010

  9. Tested algorithms • Candidate algorithms: • ignore measurements – use the center of stations • simple intersection • range-Bancroft • Gauss-Newton (with and without regularisation) • Cheung (2006) • Beck (2008) • All implemented in Matlab with similar level of optimization Workshop for Positioning, Navigation and Communication 2010

  10. Results: RMS position error Workshop for Positioning, Navigation and Communication 2010

  11. Results: normalized error Workshop for Positioning, Navigation and Communication 2010

  12. Back to the objections against GN… • 1) requires an initial guess • so do several ”closed-form” methods • in practical applications rough position usually known from the context: physical constraints, station positions, etc. • possible to use a closed-form solution as a starting point Workshop for Positioning, Navigation and Communication 2010

  13. 2) Convergence not guaranteed • Depends on the quality of the initial guess • Regularisation helps • Sanity checks recommended • - probably should use some with closed-form methods as well! Workshop for Positioning, Navigation and Communication 2010

  14. 3) Computational complexity • Matlab on a 1.4GHz Celeron laptop • station mean: instant • simple intersection: 0.3 ms/fix • Bancroft: 0.5 ms/fix • Cheung: 0.8 ms/fix • Beck: 1.2 ms/fix • Gauss-Newton: 1.2 – 1.5 ms/fix Workshop for Positioning, Navigation and Communication 2010

  15. Bonus: flexibility • Closed-form methods: • only ranges: ok • only range differences: ok • mixed ranges and range differences: some choices • range differences + a plane: at least one method • mixed ranges, range differences, planes, etc: … huh? • Gauss-Newton: • combination of any (differentiable) measurements: OK Workshop for Positioning, Navigation and Communication 2010

  16. Conclusions • Gauss-Newton was found to be competitive against several closed-form solutions • Additional bonus points: • Handles also correlated noise • Robust numerics • Gives an error estimates • Extends to time series -> Extended Kalman Filter • Questions? Workshop for Positioning, Navigation and Communication 2010

More Related