closed-form algorithms in hybrid positioning: myths and misconceptions
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Closed-form Algorithms in Hybrid Positioning: Myths and Misconceptions. Niilo Sirola Department of Mathematics Tampere University of Technology, Finland (currently with Taipale Telematics, Finland) [email protected]. Mobile positioning. Given a measurement model - PowerPoint PPT PresentationTRANSCRIPT
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Workshop for Positioning, Navigation and Communication 2010 22.04.23
Closed-form Algorithms in Hybrid Positioning: Myths and Misconceptions
Niilo SirolaDepartment of Mathematics
Tampere University of Technology, Finland
(currently with Taipale Telematics, Finland)[email protected]
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Workshop for Positioning, Navigation and Communication 2010 22.04.23
Mobile positioning
Given a measurement model y = h(x) + vand the measurements y
Find the position x that fits the measurements ”best”
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Workshop for Positioning, Navigation and Communication 2010 22.04.23
Iterative Least Squares
The Gauss-Newton or Taylor series or Iterative/Ordinary/Nonlinear least squares
Usual objections to Gauss-Newton1) initial guess: ”Selection of such a starting
point is not simple in practice”2) convergence is not assured3) computational load: ”as LS computation is
required in each iteration”
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Workshop for Positioning, Navigation and Communication 2010 22.04.23
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”
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Workshop for Positioning, Navigation and Communication 2010 22.04.23
Examples of closed-form methods
others are algebraic and more rigorous
sometimes come with a proof
sometimes can be implemeted by the reader
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Workshop for Positioning, Navigation and Communication 2010 22.04.23
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
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Workshop for Positioning, Navigation and Communication 2010 22.04.23
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…”
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Workshop for Positioning, Navigation and Communication 2010 22.04.23
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
-5000 0 5000
-5000
0
5000
x / m
y / m
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Workshop for Positioning, Navigation and Communication 2010 22.04.23
Tested algorithms
Candidate algorithms:1) ignore measurements – use the center of stations2) simple intersection3) range-Bancroft4) Gauss-Newton (with and without regularisation)5) Cheung (2006)6) Beck (2008)
All implemented in Matlab with similar level of optimization
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Workshop for Positioning, Navigation and Communication 2010 22.04.23
Results: RMS position error
100
101
102
103
104
100
102
104
measurement noise / m
rmse / m
meansimplebancroftgauss-newtongauss-newton-regbeckcheung
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Workshop for Positioning, Navigation and Communication 2010 22.04.23
Results: normalized error
100
101
102
103
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0.5
1
1.5
2
measurement noise / m
norm
aliz
ed e
rror
meansimplebancroftgauss-newtongauss-newton-regbeckcheung
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Workshop for Positioning, Navigation and Communication 2010 22.04.23
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
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Workshop for Positioning, Navigation and Communication 2010 22.04.23
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!
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Workshop for Positioning, Navigation and Communication 2010 22.04.23
3) Computational complexity
Matlab on a 1.4GHz Celeron laptop
station mean: instantsimple intersection: 0.3 ms/fixBancroft: 0.5 ms/fix Cheung: 0.8 ms/fix Beck: 1.2 ms/fix
Gauss-Newton: 1.2 – 1.5 ms/fix
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Workshop for Positioning, Navigation and Communication 2010 22.04.23
Bonus: flexibility
Closed-form methods:only ranges: okonly range differences: okmixed ranges and range differences: some choicesrange differences + a plane: at least one methodmixed ranges, range differences, planes, etc: … huh?
Gauss-Newton:combination of any (differentiable) measurements: OK
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Workshop for Positioning, Navigation and Communication 2010 22.04.23
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?