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Flat_Moon

 

Flat_Moon
Contributor:  J.R. Cash, Davy Hollevoet, F. Mazzia, A. N. Abdo
Discipline:  Aerospace
Accession:  2014

 

Short description:

This problem is about the optimal-time launching of a satellite into orbit from flat Moon without atmospheric drag.

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 consider 

    \[F/m\]

as a constant.

We need to find the control variable \alpha. in order to minimize:

    \[J=t_f\]

subject to:

    \[x=v_x\]

    \[y=v_y\]

    \[v_x=(F/m)cos(\alpha)\]

    \[v_y=(F/m)sin(\alpha)-g\]

with initial conditions:

    \[t_0=0 \quad x_0=0 \quad y_0=0\]

 

    \[v_{x}=0 \quad v_{y}=0\]

and final conditions

    \[\tau _1=y_f-h \quad \tau _2=v_x(t_f)-v_c=0 \quad \tau_ 3=v_y(t_f)=0\]

We define this problem with:

z_1=x, z_2=y, z_3=v_x, z_4=v_y, z_5=\alpha, z_6=\beta: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 \\ z_4 \\ Acos(z_5) \\ z_6 cos(z_5) \\ -z_6 cos(z_5) \\ (z_6)^2 cos(z_5) \\ 0 \end{pmatrix}

with A=F/m. We consider the 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 \\ 0 \\ 0 \\ 0 \end{pmatrix}

 

 

 

 

 

 

 

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References

James M LonguskiJosé J. Guzmán,  John E. Prussing Optimal Control with Aerospace Applications, Springer,