Stellar navigation | Navigation | Forum

Almost all of the navigation discussion so far has centered around inertial navigation, which is an important part of the story but by no means the whole thing.  Ideally we want to combine our inertial estimates of position and velocity (both translational and rotational) with as many other estimates as we can afford.  It feels like it's about time to start discussing other possible sources of input and navigation by the stars is a good candidate.  If anybody wants to discuss other options, please feel free to create threads for them.

How does stellar navigation measure up for precision when it comes to position and orientation?  Can we use an off-the-shelf solution for this or will we need to build our own?  Are there good, freely available databases of the sort we will need?

I also can't really think of how a star tracker could provide much in the way of position determination, but I didn't want to jump straight to the conclusion that it couldn't.

I remember the Earth /moon size and position idea.  I'm not sure how accurate it would be.  Somebody who understands more of the theory fo cameras should be able to figure out what sort of physical distance a single pixel represents at various distances from each body.  My intuition is that by the time we get far out, a single pixel may correspond to rather a large distance and hence this will only give us a fairly approximate reading – our image of the Earth will be the same number of pixels wide over a fairly long stretch of space.  But it can't hurt to throw it into the mix, a proper Kalman filter or the like is aware of how accurate the various measurements are so rough readings like this will not throw things off.  Also, while we are close to the Earth, it may actually be difficult to determine the boundaries of the Earth proper as opposed to its atmosphere.

Estimating total size and mass of a system incorporating these elements is an interesting problem and probably worth taking a crack at just to help refine our overall estimates of mass.  The MEMS vs mechanical debate for inertial navigation will have a huge influence on this, since mechanical inertial systems are orders of magnitude larger, heavier and more expensive than MEMS solutions.  I should think the cameras for the Earth and moon measurements could be fairly small but I'm not really in touch with camera technology.  Aside from having to shield the electronics from radiation, would anything else prevent an off-the-shelf camera working in space?

The idea is conceptually straightforward enough, but so far there has been little discussion of practical details.  My main concern is with accuracy.  You can measure the diameter of the visible Earth and moon only accurate to within one pixel.  When you're half way between the Earth and the moon, one pixel on Earth can be a long distance.  If this method ends up letting us estimate our position to within a ball of 50km radius or something like that then it isn't particularly useful (although mixing it in with more accurate measurements won't hurt if we take into account the error margin on the various measurements).

I feel like it shouldn't be too hard to figure out how accurate a measurement we can make with this approach, but I don't know enough about cameras to start working on it with confidence.  Using a pinhole camera model to do the maths would probably be sufficient enough to get an order of magnitude estimate, but it would need values for things like focal length and I wouldn't know where to start.  Also if we used a fisheye lense we'd need a good mathematical description of the distortion that they produce (although this probably wouldn't be too hard to find).

My gut says that the main utility of this would be as a back up if other navigation systems broke down.  I dunno.  If the inertial navigation system somehow died on the way to the moon I think the only option would be to abort the mission and wait for the free return trajectory to take us home.  Our other main source of position estimation will be round-time trip measurements for radio signals.  If those die I suppose that a combination of inertial measurements and this system might be accurate enough (although if the lost radio navigation because the entire comms system had died, once again a total abort would be about the only sensible option).

Figuring the rough accuracy of the radio solution should be a bit easier.  If the Earth/moon/sun diameter measurement system is on the same order of magnitude then we should definitely include it.

Okay, the actual procedure by which we make range estimates is this:

• Take a photo of Earth
• Count how many pixels wide the Earth (moon/whatever) appears (this may be hard with the Earth as the atmosphere presents a blury edge at closer ranges?).  Call this p_e.
• Knowing that the Earth's diameter is e_d km, and that the image is p_t pixels wide, deduce that the total field of view is 2x = (p_t / p_e)*e_d km wide.
• Knowing the angle a radians, deduce that the range is x / tan(a).

The source of innacuracy is that the number of pixels that the Earth's image is wide will be the same for several actual ranges: p_e will be the same for all ranges between l and l + delta-l (with delta-l dependent on l).  The big accuracy related question is: how big does delta-l get?

Here is a plot, using the values for a and p_t from above. The range l ranges from 192,000 km (mid way point) to 200,000 km.

As you can see, p_e stays constant for about 1000 km at a time, making this pretty useless as a ranging technique for most of the trip.

However, this is an extremely naive model. By using various kinds of lenses to zoom and stretch the image we could be able to improve on this a lot. This is well outside my current knowledge, though.

Also, this might work acceptably well at closer distances. Once we are within 1000km or so of the moon, this sort of measurement (using the moon, not Earth) may be accurate enough that mixing it in with our inertial estimates is actually a really useful technique. Which is good, because we will need accurate measurements at that point to time our capture burn.