Gravity Turn Maneuver - Revision history

FelixMueller at 08:29, 27 July 2016

2016-07-27T08:29:42Z

2016-06-27T22:10:13Z

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	[[Category:MIOCP]]		[[Category:MIOCP]]
	[[Category:ODE model]]		[[Category:ODE model]]
			[[Category:Aeronautics]]
			[[Category:Path-constrained arcs]]

2016-03-03T17:39:45Z

References

← Older revision		Revision as of 17:39, 3 March 2016
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	== References ==		== References ==
			<biblist />
	~~CULLER, Glen J.; FRIED, Burton D. Universal gravity turn trajectories. Journal of Applied Physics, 1957, 28. Jg., Nr. 6, S. 672-676.~~

	<!--List of all categories this page is part of. List characterization of solution behavior, model properties, ore presence of implementation details (e.g., AMPL for AMPL model) here -->		<!--List of all categories this page is part of. List characterization of solution behavior, model properties, ore presence of implementation details (e.g., AMPL for AMPL model) here -->
	[[Category:MIOCP]]		[[Category:MIOCP]]
	[[Category:ODE model]]		[[Category:ODE model]]

2016-03-03T17:39:13Z

Physical description and model derivation

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	== Physical description and model derivation ==		== Physical description and model derivation ==

	For the purposes of this model, we start with the following ODE system proposed by Culler et. al.:		For the purposes of this model, we start with the following ODE system proposed by Culler et. al. in <bib id="Culler1957" />:

	<dd>		<dd>

2016-02-17T13:30:38Z

Mathematical formulation

← Older revision		Revision as of 13:30, 17 February 2016
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	<math>		<math>
	\begin{array}{llcll}		\begin{array}{llcll}
	\displaystyle \min_{T, m, v, q, h, d, u} & m_0 - m(T) \\[1.5ex]		\displaystyle \min_{T, m, v, \beta, h, \theta, u} & m_0 - m(T) \\[1.5ex]
	\mbox{s.t.} & \dot{m} & = & -\frac{F_{max}}{I_{sp} \cdot g_0} \cdot u ,\\		\mbox{s.t.} & \dot{m} & = & -\frac{F_{max}}{I_{sp} \cdot g_0} \cdot u ,\\
	& \dot{v} & = & (F - \frac{1}{2} \rho_0 \cdot e^{-\frac{h}{H}} A c_d v^2) \cdot \frac{1}{m} - g_0 \cdot \left(\frac{r_0}{r_0 + h}\right)^2 \cos \beta ,\\		& \dot{v} & = & (F - \frac{1}{2} \rho_0 \cdot e^{-\frac{h}{H}} A c_d v^2) \cdot \frac{1}{m} - g_0 \cdot \left(\frac{r_0}{r_0 + h}\right)^2 \cos \beta ,\\

2016-02-17T13:29:10Z

Mathematical formulation

← Older revision		Revision as of 13:29, 17 February 2016
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	\displaystyle \min_{T, m, v, q, h, d, u} & m_0 - m(T) \\[1.5ex]		\displaystyle \min_{T, m, v, q, h, d, u} & m_0 - m(T) \\[1.5ex]
	\mbox{s.t.} & \dot{m} & = & -\frac{F_{max}}{I_{sp} \cdot g_0} \cdot u ,\\		\mbox{s.t.} & \dot{m} & = & -\frac{F_{max}}{I_{sp} \cdot g_0} \cdot u ,\\
	& \dot{v} & = & (F - \frac{1}{2} \rho_0 \cdot e^{-\frac{h}{H}} A c_d v^2) \cdot \frac{1}{m} - g_0 \cdot \left(\frac{r_0}{r_0 + h}\right)^2 \cos \beta~~(t)~~ ,\\		& \dot{v} & = & (F - \frac{1}{2} \rho_0 \cdot e^{-\frac{h}{H}} A c_d v^2) \cdot \frac{1}{m} - g_0 \cdot \left(\frac{r_0}{r_0 + h}\right)^2 \cos \beta ,\\
	& \dot{\beta} & = & g_0 \cdot \left(\frac{r_0}{r_0 + h}\right)^2 \cdot \frac{\sin{\beta}}{v} - \frac{v \cdot \sin \beta}{r_0 + h} ,\\		& \dot{\beta} & = & g_0 \cdot \left(\frac{r_0}{r_0 + h}\right)^2 \cdot \frac{\sin{\beta}}{v} - \frac{v \cdot \sin \beta}{r_0 + h} ,\\
	& \dot{h} & = & v \cdot \cos \beta ,\\		& \dot{h} & = & v \cdot \cos \beta ,\\

2016-02-17T13:28:08Z

Mathematical formulation

← Older revision		Revision as of 13:28, 17 February 2016
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	\displaystyle \min_{T, m, v, q, h, d, u} & m_0 - m(T) \\[1.5ex]		\displaystyle \min_{T, m, v, q, h, d, u} & m_0 - m(T) \\[1.5ex]
	\mbox{s.t.} & \dot{m} & = & -\frac{F_{max}}{I_{sp} \cdot g_0} \cdot u ,\\		\mbox{s.t.} & \dot{m} & = & -\frac{F_{max}}{I_{sp} \cdot g_0} \cdot u ,\\
	~~& \dot{v} & = & \frac{F - \frac{1}{2} \rho_0 \cdot e^{-\frac{h}{H}} A c_d v^2}{m} - g_0 \cdot \left(\frac{r_0}{r_0 + h}\right)^2 \cos \beta(t) ,\\~~
	& \dot{v} & = & (F - \frac{1}{2} \rho_0 \cdot e^{-\frac{h}{H}} A c_d v^2) \cdot \frac{1}{m} - g_0 \cdot \left(\frac{r_0}{r_0 + h}\right)^2 \cos \beta(t) ,\\		& \dot{v} & = & (F - \frac{1}{2} \rho_0 \cdot e^{-\frac{h}{H}} A c_d v^2) \cdot \frac{1}{m} - g_0 \cdot \left(\frac{r_0}{r_0 + h}\right)^2 \cos \beta(t) ,\\
	& \dot{\beta} & = & g_0 \cdot \left(\frac{r_0}{r_0 + h}\right)^2 \cdot \frac{\sin{\beta}}{v} - \frac{v \cdot \sin \beta}{r_0 + h} ,\\		& \dot{\beta} & = & g_0 \cdot \left(\frac{r_0}{r_0 + h}\right)^2 \cdot \frac{\sin{\beta}}{v} - \frac{v \cdot \sin \beta}{r_0 + h} ,\\

2016-02-17T13:27:33Z

Mathematical formulation

← Older revision		Revision as of 13:27, 17 February 2016
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	\mbox{s.t.} & \dot{m} & = & -\frac{F_{max}}{I_{sp} \cdot g_0} \cdot u ,\\		\mbox{s.t.} & \dot{m} & = & -\frac{F_{max}}{I_{sp} \cdot g_0} \cdot u ,\\
	& \dot{v} & = & \frac{F - \frac{1}{2} \rho_0 \cdot e^{-\frac{h}{H}} A c_d v^2}{m} - g_0 \cdot \left(\frac{r_0}{r_0 + h}\right)^2 \cos \beta(t) ,\\		& \dot{v} & = & \frac{F - \frac{1}{2} \rho_0 \cdot e^{-\frac{h}{H}} A c_d v^2}{m} - g_0 \cdot \left(\frac{r_0}{r_0 + h}\right)^2 \cos \beta(t) ,\\
			& \dot{v} & = & (F - \frac{1}{2} \rho_0 \cdot e^{-\frac{h}{H}} A c_d v^2) \cdot \frac{1}{m} - g_0 \cdot \left(\frac{r_0}{r_0 + h}\right)^2 \cos \beta(t) ,\\
	& \dot{\beta} & = & g_0 \cdot \left(\frac{r_0}{r_0 + h}\right)^2 \cdot \frac{\sin{\beta}}{v} - \frac{v \cdot \sin \beta}{r_0 + h} ,\\		& \dot{\beta} & = & g_0 \cdot \left(\frac{r_0}{r_0 + h}\right)^2 \cdot \frac{\sin{\beta}}{v} - \frac{v \cdot \sin \beta}{r_0 + h} ,\\
	& \dot{h} & = & v \cdot \cos \beta ,\\		& \dot{h} & = & v \cdot \cos \beta ,\\

2016-02-17T13:26:01Z

Mathematical formulation

← Older revision		Revision as of 13:26, 17 February 2016
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	</dd>		</dd>

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

	== Parameters ==		== Parameters ==

2016-02-17T13:25:44Z

Mathematical formulation

← Older revision		Revision as of 13:25, 17 February 2016
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	</dd>		</dd>

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

	== Parameters ==		== Parameters ==