How to Implement Inertial Navigation System Equations in MATLAB?

[Pieter-S]

Dear :),
could some one please guide me in the right direction to answer the following questions:

1) using MATLAB, implement the differential equations that describe the evolution with time of along-track , cross-track and vertical position. (Inertial Navigation System).

and

2) for a speed of 250 m/s at a height above the Earth of 10km, calculate the evolution of x,y,z positions over a period of 5 minutes for an initial error in y of 1000 m.

3) for the same conditions, calculate the evolution of x, y and z over the same period (5 minutes)for an initial error in x of 10 m.




Relevant equations:

surfing on websites , I managed to find the following relevant equations:

d(phi)/dt = Vn/(Rm+h),

and d(lamda)/dt=(1/cos*phi(T))*VE/(Rp+h)

Vn and Ve , is the speed of the object in the north and east direction, obviously the speed is given, but not in which direction, so again I need to assume then that the speed is in the mentioned direction.



The attempted solution:

I could calculate these by using World Geodetic System (WGS) and assume Geocentric latitude angle (ΦC) and
Geodetic latitude angle (ΦT). But I am not sure if that is going to be acceptable, I think the question force me to believe that the speed is constant and the path is perfect circular.


any help is more than welcome.

thanks,

Pieter Haagendijk:)

 

[D H]

What are the relevant equations here? This is homework, after all. You should have followed the template. You don't need to do a google search for the relevant equations. You need to read your class notes or your textbook.

 

[Pieter-S]

Well, I was ill during the lecture sessions, and I don't have contact with my study mates. I sent an email to lecturer though, but he said that I could google this out.

King regards,

Pieter

 

[D H]

Google "Clohessy-Wiltshire equations".

 

[Pieter-S]

Thanks you for your prompt response ,

Hill–Clohessy–Wiltshire (HCW) describes two targets chasing each other. mine case is s simple mass point.

 

[D H]

The exact same equations apply to this problem.

Your case is not a simple point mass. You have the true orbit, which is unknown (but which you can model) and the orbit per your IMU. Unless your IMU state is updated per some state measurement, the true and computed orbits will diverge per the CW equations.

 

[Pieter-S]

I am so sorry to say, but this sounds like an music to my ears:(

 

[D H]

What did you mean by that? I don't think you really meant "music to my ears." That is an English idiom meaning roughly "exactly what I wanted to hear."

 

[Pieter-S]

Hi,
Well I thought in the first place that I could solve this problem by following your recommendation, At least that is what I hoped.

by looking further and reading some of those navigation books, the problem is complicated than it actually looks.

so still I struggle to give a good go.

cheers,

Pieter

 

[D H]

What have you tried?

Part 1 of your question asks you to "implement [in MATLAB] the differential equations that describe the evolution with time of along-track , cross-track and vertical position." Parts 2 and 3 implicitly require you to use your MATLAB script. You aren't going to make any progress on these two part of the problem until you have solved part 1.Have you done anything with regard to part 1? I gave a huge hint in post #4 regarding the differential equations that you need to implement.

 

[Pieter-S]

Yes, I did tried to implement those equation in simple m file, but it is not complete due initial conditions as this is unknown:(,

I do appreciate your response and patience:).

cheers,

Pieter

 

[D H]

You do not need to know the details of the initial conditions. They are irrelevant. You are given an initial speed and altitude. Assume a spherical Earth.

Suppose two real vehicles are separated by a small relative position and small relative velocity. The evolution of the relative states of these vehicles can be expressed in terms of a linearized set of differential equations.

Now suppose the state difference between those vehicles at some point in time is exactly equal to your given state error. The evolution of your error state is exactly the same as the state evolution of the relative states of these two vehicles.

Finally, suppose only one of these vehicles is real. The other is a fictitious vehicle whose state is given by your (erroneous) inertial navigation system. It doesn't matter than the other vehicle isn't real. The evolution of your error state is exactly the same as the state evolution of the relative states of the real vehicle and a fictitious one that is moving along with your navigated state.

 

[Pieter-S]

Thanks ever so much sir,

I am going to work on it, I will let you know if there still some uncertainties.

have a good night:)

Regards,
Pieter

 

[tribaljunkie]

Pieter you must be in my class, I have the same assignment. In class he gave us the equations:

ax (x is subscript)= xdotdot - (2v/r0)*zdot - (v/r0)^2*x
ay = ydotdot
az = zdotdot + (2v/r0)*xdot - (v/r0)^2

but he was very vague about the symbol meanings. i don't know what ax refers to, as xdotdot and ax are both accelerations. also there is no definition of v, which I am assuming is a velocity, but I am not sure whether the 250m/s initial condition is v or xdot.

once i have some clarity on this i imagine its just a case of inputting them into MATLAB and integrating them over the time frame. i just don't know where to input the error, is it just as simple as entering the correct initial conditions, as well as the correct initial conditions+error and plotting the two lines to demonstrate the difference? if anyone else recognises these equations and can shed some light on the definition of each term it would be much appreciated. i just need to know the difference between v and xdot and also between ax and xdotdot. the lecturer is too busy to see me before the deadline apparently...

 

[tribaljunkie]

hmmm, think i figured it out from the derivation. ax,ay,az are the body accelerations and xdotdot, ydotdot and zdotdot are the Earth axis accelerations. still, if someone could confirm that would be appreciated.