FlatEarth

FlatEarth

Discipline:	Aereospace
Accession:	2014

Short description:

Launch of a satellite into circular orbit from a flat Earth where we assume a uniform gravitational field g.

Applicable solvers:

twpbvpc,twpbvplc,colsys,colnew,bvp_m2

Plots of the solution <- click to generate the plots of the solution and the textual output

Mathematical description:

We need to find the control variable $\alpha$ . The problem aims to minimize:

$J=t_f$

subject to:

$x=v_x$

$y=v_y$

$v_x=f(t)cos(\alpha)$

$v_y=f(t)sin(\alpha)-g$

with initial conditions:

$t_0=0 \quad x_0=0 \quad y_0=0$

$v_{x0}=0 \quad v_{y0}=0$

and final conditions:

$y_f=h \quad v_x(t_f)=v_c \quad v_y(t_f)=v_c$

with $h$ and $v_c$ height and speed of the circular orbit.

We define this problem with:

$z_1=x, z_2=y, z_3=v_x, z_4=v_y, z_5=\lambda_2, z_6=\lambda:4$

and obtaining the matrix:

$\begin{pmatrix} z_1 \\ z_2 \\ z_3 \\ z_4 \\ z_5 \\ z_6 \\ z_7 \end{pmatrix}' = \begin{pmatrix} z_3\frac{v_c}{h} \\ z_4\frac{v_c}{h} \\ acc\frac{1}{\lvert v_c \rvert \sqrt{1+z_6^2}} \\ acc\frac{1}{\lvert v_c \rvert \sqrt{1+z_6^2}} - \frac{g}{v_c} \\ 0 \\ -z_5\frac{v_c}{h}\\ 0 \end{pmatrix}$

with $acc=f(t)/m$ . We consider another equation $t_f=z_7$ .

The boundary conditions are obtained from:

$\begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix} \begin{pmatrix} z_1(0) \\ z_2(0) \\ z_3(0) \\ z_4(0) \\ z_5(0) \\ z_6(0) \\ z_7(0) \end{pmatrix} + \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \end{pmatrix} \begin{pmatrix} z_1(1) \\ z_2(1) \\ z_3(1) \\ z_4(1) \\ z_5(1) \\ z_6(1) \\ z_7(1) \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0 \\ 0 \\ h \\ v_c \\ 0 \end{pmatrix}$

Download:

Fortran code: FlatEarth.f
matlab code:
R code: FlatEarth.R

References
James M Longuski, José J. Guzmán, John E. Prussing Optimal Control with Aereospace Applications, Springer, Space Technology Library, v. 32, 2014 ISBN: 978-1-4614-8944-3 (Print) 978-1-4614-8945-0 (Online)

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FlatEarth

Short description:

Applicable solvers:

Mathematical description:

References
James M Longuski, José J. Guzmán, John E. Prussing Optimal Control with Aereospace Applications, Springer, Space Technology Library, v. 32, 2014 ISBN: 978-1-4614-8944-3 (Print) 978-1-4614-8945-0 (Online)

Test Set for BVP Solvers

BVP Solvers Login

Recent Posts

Recent Comments

FlatEarth

Short description:

Applicable solvers:

Mathematical description:

References James M Longuski, José J. Guzmán, John E. Prussing Optimal Control with Aereospace Applications, Springer, Space Technology Library, v. 32, 2014 ISBN: 978-1-4614-8944-3 (Print) 978-1-4614-8945-0 (Online)

References
James M Longuski, José J. Guzmán, John E. Prussing Optimal Control with Aereospace Applications, Springer, Space Technology Library, v. 32, 2014 ISBN: 978-1-4614-8944-3 (Print) 978-1-4614-8945-0 (Online)