Talk:Global Positioning System/Archive 8: Difference between revisions

Revision as of 00:05, 6 March 2015

Revision as of 00:05, 26 March 2015

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:The anisotropic propagation means the surfaces of constant light-time (or constant phase) are not spheres, and we should not describe them as such. I don't know why you think I'm suggesting that the least squares method can't solve the navigation equations; on the contrary I have repeatedly insisted the opposite-- even when you yourself did not believe it. Repeatedly calling other editors stupid or unkowledgeable is not acceptable on wikipedia; if you continue like this you can expect increased resistance and eventually administrative action. Stop. [[User:Siafu|siafu]] ([[User talk:Siafu|talk]]) 21:30, 1 February 2015 (UTC)

::The equations do not operate in 3D space, since they have 4 independent variables (coordinates). The time bias is the fourth dimension necessary in the solution process.−[[User:Woodstone|Woodstone]] ([[User talk:Woodstone|talk]]) 10:30, 2 February 2015 (UTC)

== Least squares solved iteratively or directly ==

I am confused by the current suggested opposition between least squares and closed form. In all cases with more than 4 satellites, a least squares set of equations is solved. This can be done either by an iterative or a direct method. Newton-Raphson is a commmon iterative method. Bancroft's method is direct (also called closed form), but still solves the Least Squares equations.−[[User:Woodstone|Woodstone]] ([[User talk:Woodstone|talk]]) 13:50, 3 February 2015 (UTC)

:It seems the true distinction is between linearized and non-linear algorithms. Bancroft's pseudo-inverse is only a stepping stone to solve a quadratic equation; also his matrix of coefficients (denoted A in the body of the article) is not the usual [[Jacobian matrix]] (denoted H in his abstract). Background: the ranging equation is non-linear in the unknown position coordinates (it's a square root of squared coordinate differences). The usual least squares procedure involves linearizing the ranging equation about an initial position solution, then iterating until the approximation converges (which depends on the accuracy of the initial approximation). The closed-form solutions don't require an initial solution and make use of the non-linear (quadratic) terms involved in the ranging equation. If you're out in interplanetary orbit you'd be better off using a closed-form algorithm, at least to initialize your Kalman filter or batch estimation. [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 15:02, 3 February 2015 (UTC)

:It occurs to me that even using a supposedly "closed-form" solution would require some amount of iteration in the coordinate frame rotation step. [[User:Siafu|siafu]] ([[User talk:Siafu|talk]]) 17:32, 3 February 2015 (UTC)

== Clock bias ==

A clock "bias" is an ambiguous word. Satnav literature uses clock "advance" as the definition of the bias, as opposed against the current description of this article. [[User:Kkddkkdd|Kkddkkdd]] ([[User talk:Kkddkkdd|talk]]) 07:34, 1 February 2015 (UTC)

:The bias can be negative as well as positive-- it's the result of accumulated stochastic error-- so "advance" is not an accurate descriptor. Most of the literature actually uses the term "bias", as is done in the article. [[User:Siafu|siafu]] ([[User talk:Siafu|talk]]) 14:03, 1 February 2015 (UTC)

::Indeed, it's a signed bias: advance or delay. You could also be called simply "error", although "bias" hints better at the fact that it's not a zero-mean error. [[User:Fgnievinski|Fgnievinski]] ([[User talk:Fgnievinski|talk]]) 19:01, 1 February 2015 (UTC)

:::Or maybe "offset"? But if most of the sources use "bias" I would go with that. [[User:Kendall-K1|Kendall-K1]] ([[User talk:Kendall-K1|talk]]) 19:27, 1 February 2015 (UTC)

::::I've seen offset used, but it's basically synonymous with "bias", so I would not be surprised to see it used interchangeably with bias even in the same source. I would not object to doing that here, either, if the prose starts getting ungainly. [[User:Siafu|siafu]] ([[User talk:Siafu|talk]]) 21:33, 1 February 2015 (UTC)

::: That's right. It's a signed bias. Satnav literature uses "advance" as the definition of its sign as follows (note the minus sign contrary to the current description of this article): [[User:Kkddkkdd|Kkddkkdd]] ([[User talk:Kkddkkdd|talk]]) 12:03, 14 February 2015 (UTC)

:::: "the true reception time is <math>\, t_\text{ri} = \tilde{t}_\text{ri} - b</math>"