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)
Workshop for Positioning, Navigation and Communication 2010

2. Workshop for Positioning, Navigation and Communication 2010
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
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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
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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

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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

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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
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10. Results: RMS position error
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11. Results: normalized error
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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
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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!
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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
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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
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16. Workshop for Positioning, Navigation and Communication 2010
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