little math problem October 2018

[ijabbott]

Both rovers start digging vertically downwards until they reach the centre of the planet.

[EugeneNine]

(10-20-2018 08:05 PM)Thomas Okken Wrote:  I'm sure this isn't optimal, but just to get started:

The greatest distance at which the rovers can see each other is D. The radius of the planet is R. The rovers can travel at a maximum speed V.

Assumptions: the planet is a perfect sphere, no water, no hills. Visibility is the same everywhere.

One rover stays put, the other starts searching. It searches in an expanding spiral, where each circuit is 2 * D from the last. I'm ignoring the details of the shape of that spiral near its ends.

The rover covers an area of 2 * D * V per unit of time. The area of the planet is 4 * pi * R^2, so it takes a time (4 * pi * R^2) / (2 * D * V) to search the entire planet, and the expected search time is half that.

what if both rovers started in a spiral? I'd think they would meet sooner depending on the speed and increasing radius of the spiral.

Though how would the rovers know they are making a spiral and not just going in a circle or random directions.

Seems like for any given planet we want to explore via rovers we should figure out the min number of gps satellites to deploy.

[Zaphod]

Quote:Though how would the rovers know they are making a spiral and not just going in a circle or random directions.

Extremely sensitive lateral acceleration sensors would tell you if you were spirialing in or out, or simply circling.

[Albert Chan]

Instead of finding each other, program one to to hide, the other to chase.

The chaser will visit an area, and place a flag to highest terrain.
Then he visit another area within eye-sight distance of first flag, and plant a second ...

The hider will also visit area with highest terrain.
If he see a flag, immediately turn around, and run in opposite direction

Eventually, the hider will be cornered ...

[Dave Shaffer]

(10-22-2018 06:33 PM)Zaphod Wrote:  

Quote:Though how would the rovers know they are making a spiral and not just going in a circle or random directions.

Extremely sensitive lateral acceleration sensors would tell you if you were spirialing in or out, or simply circling.

These could (should!) be part of an inertial navigation system (INS). Rovers thus equipped could readily navigate around a moon-sized "planet" and need NO help from other systems (GPS, stars, equator/pole, magnetic poles, etc.). If the planet is hard and smooth, there will be no hills on which to plant flags, and there will be no tracks left in the dust.

So, they should implement Thomas's spiral approach - the best so far for the case of no other initial knowledge about the planet. (Ian Abbott's idea of leaving "bread crumbs" behind is perhaps the next best approach, but you may need a goodly supply. An INS, and perhaps a string with a weight on it (defining "down"), would keep the rovers close enough to a great circle route.)

[Csaba Tizedes]

I think better, if "first" rover do not moves.
In this case the "second" need to search the "first" only that locations where the "second" was not earlier.
But how they decide which one is the "first" and which is the "second"?

[Csaba Tizedes]

(10-23-2018 06:02 PM)Csaba Tizedes Wrote:  I think better, if "first" rover do not moves.
In this case the "second" need to search the "first" only that locations where the "second" was not earlier.
But how they decide which one is the "first" and which is the "second"?

And the "second"'s route is a Peano curve - on a sphere.

[EugeneNine]

thats why I was thinking, have them both move in a spiral.

[Albert Chan]

(10-20-2018 06:48 PM)pier4r Wrote:  Communications between the rovers is impossible (unless they are in line of sight) ...

Does a personal rescue beacon considered "communication" ?

[Maximilian Hohmann]

Hello!

As I have a very pragmatic approach to life (and science) my first question would be: What sense does it make to land two rovers when they need to find each other and work together? One should be sufficient in the first place... And if you land two that would only make sense if they explore different parts of the planet!

But if it really needs to be done then going towards a pole would be my strategy. A simple (solid state) gyrocompass is all that's needed for that, no clear view of the sky/stars and no magnetic field are required. Depending on the size of the planet and on it's morphology that could take very long or be impossible (like on our planet untless the rovers can swim).

If there is an atmosohere the rovers could release balloons on thin copper wires serving as antennas to get into contact with each other. That would be my solution as an areospace engineer. Again a gyrocompass on each rover together with timing the sunrise/sunset individually would give them an idea about their relative locations. This of course will only work if the length of the wire antennas (a few hundred metres at best) will allow radio contact. Using medium or long waves several thousand kilometers can be covered.

Regards
Max

[pier4r]

(10-23-2018 07:44 PM)Maximilian Hohmann Wrote:  But if it really needs to be done then going towards a pole would be my strategy. A simple (solid state) gyrocompass is all that's needed for that, no clear view of the sky/stars and no magnetic field are required. Depending on the size of the planet and on it's morphology that could take very long or be impossible (like on our planet untless the rovers can swim).

how does the gytocompass knows where is the pole? Shouldn't it be calibrated first?

[Maximilian Hohmann]

Hello!

(10-24-2018 03:26 PM)pier4r Wrote:  how does the gytocompass knows where is the pole? Shouldn't it be calibrated first?

Look no further than here: https://en.wikipedia.org/wiki/Gyrocompass

Regards
Max

NB: On places like Amazon you can buy rugged commercial solid state gyrocompasses which would suit your rovers well - maybe the heavy enclosure could be swapped against a lightweight one: https://www.amazon.com/Maretron-Solid-St...B0141OIADY

[Thomas Okken]

(10-24-2018 03:43 PM)Maximilian Hohmann Wrote:  NB: On places like Amazon you can buy rugged commercial solid state gyrocompasses which would suit your rovers well - maybe the heavy enclosure could be swapped against a lightweight one: https://www.amazon.com/Maretron-Solid-St...B0141OIADY

That device is not a gyrocompass. The terminology in that listing is misleading; the manufacturer's web site is better: https://www.maretron.com/products/ssc300.php
The use of the term "rate gyro" in conjunction with the device being a compass makes it sound like a gyrocompass, but a rate gyro is actually just a turn rate measuring device. Note that they also mention calibrating the device for deviation; deviation is the term for the difference between true heading and magnetic heading.

[pier4r]

Well it doesn't matter much. Even if they don't go to the pole but they go to the same point of the planet's surface, assuming that are reliable, I guess we found an additional way to solve it without too much help from outside the planet.

I thought that a gyrocompass was a gyroscope. The thing is cool. (Also a gyroscope is cool)

[sasa]

(10-23-2018 07:44 PM)Maximilian Hohmann Wrote:  What sense does it make to land two rovers when they need to find each other and work together?

None. This is quite hypothetical "search and destroy" task...

First of all, the size of the planet/sphere with flat and hard surface with no obstacle is quite important, otherwise that would take forever. Secondly, designated point - navigated with magnetic/gyro compass never mind, it is the same approach...

If the diameter is about 10000 km with assumed constant speed of 10km/h and 0% defects, it would take at most about 10000*3.14/10/24/2 ~ 66 days.

However, constant speed of 10km/h for theoretically maximum 15700km (+ for extra tasks later) require quite a bit of energy to generate. If the imaginary planet is near the sun, theoretically can be powered on solar energy and battery... However, then many factors are further involved, included: size and efficiency of solar panels, then total mass, speed of rotation and gravity, atmosphere... To avoid all these details, it would assume nulear power generator, which is not quite small, or huge amount of fuel tanks to carry...

Curiosity, for instance, is about 3x2 meters and carry small radioisotope thermoelectric generator which generate only about 100W.. Its maximum speed is 200m per day...

Many factors are important in order to design the actual specific mission in details, but for a brain teaser, proper initial approach is more than enough.

[EdS2]

Gyrocompass: great answer! If the body is spinning and has enough gravity, each rover can unambiguously get to a "North" pole. No need for magnetic fields, clear skies, or a constellation of satellites.