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August 25th, 2017, 03:29 AM   #1
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Numerical integration with 3 variables

I have 3 variables: v, b and h.
I know how to calculate v’, b’ and h’.
Starting the simulation from an initial condition, I need to calculate v, b and h after a given time using a numerical integrator (say RK4).

If I use the simple Euler method, I write;
v= v + dt * v’
b= b + dt * b’
h= h + dt * h’

but when I switch to RK4, I don’t know what to write. For example:
h’= v * cos(b)
and the RK4 function is of the form
y(t + dt)= rk4(dx, x, y(t)),
while the acceleration function is of the form
acc= f(x, y);
what should I write in place of x and y?

The simulation is explained here:
https://mintoc.de/index.php?title=Gravity_Turn_Maneuver
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August 25th, 2017, 04:04 AM   #2
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Quote:
Originally Posted by Cristiano View Post
I have 3 variables: v, b and h.
I know how to calculate v’, b’ and h’.
Starting the simulation from an initial condition, I need to calculate v, b and h after a given time using a numerical integrator (say RK4).

If I use the simple Euler method, I write;
v= v + dt * v’
b= b + dt * b’
h= h + dt * h’

but when I switch to RK4, I don’t know what to write. For example:
h’= v * cos(b)
and the RK4 function is of the form
y(t + dt)= rk4(dx, x, y(t)),
while the acceleration function is of the form
acc= f(x, y);
what should I write in place of x and y?

The simulation is explained here:
https://mintoc.de/index.php?title=Gravity_Turn_Maneuver
I'm not quite sure what differential equations you are trying to solve exactly. Can you write them explicitly?

This may be helpful.
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August 25th, 2017, 04:37 AM   #3
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Your post is useful when I have only one equation, but since I have a poor math background, when I need to solve a slightly complicate problem, I don't know what to do.

In this case, I need to solve the equation $\dot v = ...$ and $\dot b = ...$ showed here (please, don't ask me to rewrite those very long equations ).

For example, b'= rk4(dx, x, y), where dx and x are times and y is b, but I have that $\dot b = f(b, v, h)$. What should I write inside the function to calculate the acceleration for b'?
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August 25th, 2017, 08:37 AM   #4
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The resulting optimal control problem is given by



where $F_{max}$ is the maximum thrust of the vehicle's engine and $\varepsilon$ is a small number that is strictly greater than zero. The differential states of the problem are $m, v, \beta, h, \theta$ while $u$ is the control function.

Last edited by greg1313; August 25th, 2017 at 08:42 AM.
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August 25th, 2017, 09:08 AM   #5
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Is my link not working?
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September 3rd, 2017, 08:51 PM   #6
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Quote:
Originally Posted by Cristiano View Post
Is my link not working?
Sorry.. I didn't realise it was you who started the other thread I linked!

You definitely want to write the sim from scratch? Which language/software?

Also are you sure you need to use an RK algorithm? MATLAB has inbuilt functions for solving optimisation problems like the one in the link.. although to be honest with you, I'm not sure if these can be used for constraints that include differential equations.. interesting ..
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September 4th, 2017, 08:53 AM   #7
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Quote:
Originally Posted by Joppy View Post
You definitely want to write the sim from scratch? Which language/software?
In the meanwhile, I wrote my C++ simulation which uses several RK algorithms.

We have:
v’ = f(t, v)
g’ = f(g, v)
h’ = f(g, v)
x’ = f(g, v)

Let’s consider for a moment the RK2 algorithm. I do (in pseudo-code) [it seems that I don't know how to format the text to make it more readable]:

k1= stp * fv(t, v); k2= stp * fv(t + stp * .5, v + .5 * k1); v += k2;
k1= stp * fg(g, v); k2= stp * fg(g + .5 * k1, v); g += k2;
k1= stp * fh(g, v); k2= stp * fh(g, v); h += k2;
k1= stp * fx(g, v); k2= stp * fx(g, v); x += k2;

I use the updated v and the updated g as the input for the next functions; is that correct? Or should I update v and h after the calculation of g, h and x?

Quote:
Originally Posted by Joppy View Post
Also are you sure you need to use an RK algorithm? MATLAB has inbuilt functions for solving optimisation problems like the one in the link.. although to be honest with you, I'm not sure if these can be used for constraints that include differential equations.. interesting ..
I would use that simulation to understand how RK works and how to use it in any situation, without asking every time.
Since I always wrote a program for a new simulation, I cannot use MATLAB, Mathematica or whatever; I need to code in C++.
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