KSP Caveman Celestial Navigation

[Moesly_Armlis]

KSP Caveman Celestial Navigation

 

KSP Caveman have an abhorrent fear of technological devices.

Never Trust A Computer You Cannot Bash!

 

So KSP caveman get by with what we have.  Using instinct along with trial and error.

Navigating interplanetary beyond Kerbin's backyard is sketchy.  No maneuver nodes makes an intercept risky. 

We have had KSP Caveman reach Duna but most are hesitant and stick with visiting Mun or Minmus.

The goal is to expand the KSP Caveman's set of tools for navigating beyond Kerbin. 

 

It should be possible to use the equation of an ellipse to create formula to assist in navigation.

The result should allow KSP caveman to travel interplanetary with confidence with the use of simple tools to measure on screen information and apply formulas to predict spacecraft and planet locations.

 

For those who are not familiar with the KSP Caveman challenge consider this as a method to utilize alien technology using primitive tools.  The navigation screen exposes a fantastic machine.  Considering that it is capable of tracking all large celestial bodies and spacecraft s in real time.  The first level navigation system indicates periapsis and  apoapsis accurately and if utilized properly can also determine orbit inclination.  We know the planets are on rails but if you ignore that and just imagine that the navigation screen is what our beloved Kerbals would interact with and see and they make do with the resources at hand. The goal is how to utilize the navigation screen to its fullest.  With just math theory and string this simple looking navigation system could allow KSP Caveman to complete a grand tour.

 

 

The Equation Of The Ellipse 

 

Definition of an ellipse.

 

 

 

From the focus take a parallel line from the minor axis and where the line intersects with the ellipse represents the focal length.

 

To perform calculations based on measurements from within the game we are going to need to use the equation of the ellipse.

 

x²/a² + y²/b² = 1

a = 1/2 major axis

b= 1/2 minor axis

Deriving the equation for the distance from the centre to focus.

c = sqrt (a²-b²)

 

The focal length is another piece of the puzzle.

L= 2b²/a

 

 

The values can be obtained by using the in game information available on the navigation map.

The value for 1/2 the major axis is from the centre to vertex.   

The apoapsis and periapsis inform us of the total length of the ellipse but it is the value above sea level.  Add in two times the planet radius and the total length of the major axis is solved.

 

2a = Periapsis + Apoapsis + 2 Planet Radius

To solve for  a take half of the major axis total length.

 

Solving for c is next and we use the value of a and subtract the periapsis of the orbiting spacecraft and the radius of the planet.

 One of the focus points of the ellipse corresponds to the planet centre.

The value of c refers to the centre of the ellipse to the focus.

c = a - (Periapsis + Planet Radius)

 

Solving for b is a little less complicated.

Rearranging the equation for the distance from centre to focus.

b = sqrt (a²-c²)

 

 

Example:

 

From the cheat menu set the orbit to 5 000 000 and 0.5 for the eccentricity.

 

 

Identifying the variables.

The periapsis equals 1 900 000 metres.

The apoapsis equals 6 900 000 metres.

The planet radius equals 600 000 metres.

All values are displayed in the navigation screen.

The value for a,b,c  are calculated.

 

2a = Periapsis + Apoapsis + 2 Planet Radius

2a = 1 900 000 m+ 6 900 000 m + 2 x 600 000 m

2a = 10 000 000 m

a = 5 000 000 m

c = a - (Periapsis + Planet Radius)

c = 5 000 000 m  -  (1 900 000 m + 600 000 m)

c = 2 500 000 m

b = sqrt (a²-c²)

b = sqrt (5 000 000²  m - 2 500 000² m)

b  = sqrt(18.75 Tm)

b = 4 330 127 m

 

Once b is solved it can be verified. 

 

The formula for focal length

L= 2b²/a

 

Measuring the space craft altitude and add the planet radius once over the focus point or centre of planet.

 

 

 

Altitude = 3 094 587 m

Planet Radius = 600 000 m

 

L= 2b²/a

The measurement needs to be doubled to represent the focal length.

2 (altitude + planet radius ) = 2b² / a

2 (3 094 587 m + 600 000 m) = 2(4 330 127)² m / 5 000 000 m

3 694 587 m compared to 3 749 999 m with some error.

Waiting a little longer reveals the actual position to measure the focal length.

 

 

The values a,b,c can now represent the ellipse in a equation form that will be manipulated mathematically. 

The formula that defines the ellipse: 

x²/a² + y²/b² = 1

The values for x, y represent the position of the space craft along the perimeter of the ellipse.

 

The Orbital period can be easily calculated once the spacecraft is at the opposite side and doubling the travel time.

This is not a very smart solution especially when travelling interplanetary. 

We need to find out the orbital period based on the instantaneous velocity of the space craft at a certain point (x,y) in the orbit.

 

The KSP Caveman Celestial Toolkit

To understand the concepts the cheat menu is being used. Certainly makes things easier to demonstrate but this needs to work within caveman tech levels.

The eccentricity can also help to assist in solving problems and was missed in the first part of this guide.

 

Using the formula for linear eccentricity.

e =  sqrt ( 1- b²/a² )

 

The above example eccentricity was set to 0.5 and the value for the major axis set to 5 000 000 

Rearranging the formula to solve for b:

b =  sqrt (a² (1- e²))

b = 4 330 127

 

But how does this help a KSP Caveman.  The solution needs to be something that a KSP Caveman can access.

 

The idea came to me while watching a video about ancient navigation techniques.

The knot is a measurement of speed over water and the etymology of the word knot refers to how a knotted drag line is used and the number of knots deployed over a set time indicates the speed of the vessel.

Having a string with knots evenly spaced is a suitable tool for the caveman. The string needs twelve equally spaced knots.  This string will assist in creating right angles.

The Special Right Angle Triangle

With sides of: three units, four units, and five units.  The sum of the smaller sides squared will equal the square of the hypotenuse.

a² + b² = c²

Along with twelve knots other combinations can be utilized.

3:	4	:5
5:	12	:13
8:	15	:17
7:	24	:25
9:	40	:41

 

The knotted string can form this special right angle triangle and using zoom to fit the triangle to the screen information.

In this picture the cheat menu represents the string.  The special triangle's right angle is nestled into the planets centre.  Focusing on the planet and zooming reveals the exact centre and zooming out does not change the point.  You can observe this by placing the mouse pointer over the centre and use the MINUS key to zoom out.

The ends of the string extend to the ellipse.  The markers for the periapsis and apoapsis are used to align the ellipse in a common geometric frame..It is very important to align the ellipse so that distortion is eliminated.  Use the mouse or cursor buttons to change the camera until a top down view is obtained.

 

 

The string can also be used to create measurements that are ratios compared to other measurements.  The zoom feature is also used to size the string to the on screen information.  

Angles can also be calculated and inclination can also be solved.

 

One more use of the string is drawing an ellipse.  Attached to each focus of the ellipse the perimeter can be outlined.

 

 

 

 

The Hohmann transfer calculations allow KSP Caveman to get close to an intercept.  Correction burns are usually done near the intercept.  This is very inefficient.  A Correction burn should happen close to the insertion point to be more fuel effective.

Once the orbit parameters are used to define the ellipse the string is used to take various measurements between the two bodies in orbit.

A correction burn can be calculated and through further measurement verify the intercept.   

When the vessel is over the focal point this might not be the most efficient place to do a correction burn but it is one that is easily marked. 

To illustrate how all these tools can be used I am crafting a hands on video. 

 Stay tuned!

 

 

 

 

 

Orbital Period

 

These calculations based on the space craft orbit parameters can help fill in the missing information. 

The equation of the ellipse can now be integrated and derivatives of the equation will give us the information to calculate orbital period.

The instantaneous velocity measured at any point along the ellipse ...

 

More to follow

Edited June 20, 2018 by Moesly_Armlis

[LordFerret]

[Moesly_Armlis]

@LordFerretAre you submitting a new KSP caveman challenge? Get to orbit using only potential energy!

[Gargamel]

Ahha!  Love the user name...  Thank you Ford Prefect for getting that edit into the Guide. 

[LordFerret]

11 hours ago, MoeslyArmlis said:

a new KSP caveman challenge?

That would be neat, although I'm not sure the game's physics and parts behavior would permit it. I'm not a rocket scientist though, nor Unity expert, so maybe someone here better qualified would know the answer to that.

 

That's a spiffy write-up. But, wait a minute.....

On 6/17/2018 at 3:41 AM, MoeslyArmlis said:

From the cheat menu set the orbit to ...

Cheat Menu?!?!?? How did KSP caveman gain access to this?!?!?? Did he have a little help, maybe?

[Moesly_Armlis]

12 hours ago, LordFerret said:

Cheat Menu?!?!?? How did KSP caveman gain access to this?!?!?? Did he have a little help, maybe?

Certainly no caveman would use the cheat menu.  It was only used for introductory demonstration of the concepts.

Wait till you see my Very Knot-EEE Solution!

 

12 hours ago, LordFerret said:

That would be neat, although I'm not sure the game's physics and parts behavior would permit it. I'm not a rocket scientist though, nor Unity expert, so maybe someone here better qualified would know the answer to that.

 

 

Nothing was said about which planet to orbit.  Remember the definition of planet is any world that orbits a star.  Moons are separate biomes and I consider that a different world.

 

Edited June 19, 2018 by MoeslyArmlis