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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}

 

 

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References
James M LonguskiJosé 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)