Hang Glider - Revision history

RobertLampel: /* Mathematical formulation */

2026-02-18T12:23:33Z

Mathematical formulation

← Older revision		Revision as of 12:23, 18 February 2026
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	\quad \dot{v}_x(t) & = & - \frac{L(t) \cdot w(t) + D(t) \cdot v_x(t)}{mv(t)}, \\		\quad \dot{v}_x(t) & = & - \frac{L(t) \cdot w(t) + D(t) \cdot v_x(t)}{mv(t)}, \\
	\quad \dot{v}_y(t) & = & \frac{L(t) \cdot v_x(t) - D(t) \cdot w}{mv(t)} - g, \\		\quad \dot{v}_y(t) & = & \frac{L(t) \cdot v_x(t) - D(t) \cdot w}{mv(t)} - g, \\
	\quad x(t) & \geq & 0 \ \quad & \forall t \in [0, T], \\		\quad x(t) & \geq & 0 \ \quad & \forall t \in [0, t_f], \\
	\quad v_x(t) & \geq & 0 \ \quad & \forall t \in [0, T], \\		\quad v_x(t) & \geq & 0 \ \quad & \forall t \in [0, t_f], \\
	\quad x(0) & = & (x_0, y_0, v_{x,0}, v_{y,0})^T, \\		\quad x(0) & = & (x_0, y_0, v_{x,0}, v_{y,0})^T, \\
	\quad y(t_f) & = & y_f \\		\quad y(t_f) & = & y_f \\
	\quad v_x(t_f) & = & v_{x,f} \\		\quad v_x(t_f) & = & v_{x,f} \\
	\quad v_y(t_f) & = & v_{y,f} \\		\quad v_y(t_f) & = & v_{y,f} \\
	\quad c_L(t) & \in & [0., 1.4] \ \quad & \forall t \in [0, T], \\		\quad c_L(t) & \in & [0., 1.4] \ \quad & \forall t \in [0, t_f], \\
	\quad t_f & \geq & 0		\quad t_f & \geq & 0
	\end{array}		\end{array}

2026-02-18T12:19:55Z

Mathematical formulation

← Older revision		Revision as of 12:19, 18 February 2026
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	\quad \dot{x}(t) & = & v_x(t),\\		\quad \dot{x}(t) & = & v_x(t),\\
	\quad \dot{y}(t) & = & v_y(t), \\		\quad \dot{y}(t) & = & v_y(t), \\
	\quad \dot{~~v_x~~}(t) & = & - \frac{L(t) \cdot w(t) + D(t) \cdot v_x(t)}{mv(t)}, \\		\quad \dot{v}_x(t) & = & - \frac{L(t) \cdot w(t) + D(t) \cdot v_x(t)}{mv(t)}, \\
	\quad \dot{~~v_y~~}(t) & = & \frac{L(t) \cdot v_x(t) - D(t) \cdot w}{mv(t)} - g, \\		\quad \dot{v}_y(t) & = & \frac{L(t) \cdot v_x(t) - D(t) \cdot w}{mv(t)} - g, \\
	\quad x(t) & \geq & 0 \ \quad & \forall t \in [0, T], \\		\quad x(t) & \geq & 0 \ \quad & \forall t \in [0, T], \\
	\quad v_x(t) & \geq & 0 \ \quad & \forall t \in [0, T], \\		\quad v_x(t) & \geq & 0 \ \quad & \forall t \in [0, T], \\

2025-11-28T10:42:16Z

Reference Solutions

← Older revision		Revision as of 10:42, 28 November 2025
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	Here is one local solution to the above control problem.		Here is one local solution to the above control problem.

	<gallery caption="Reference solution plots" widths="~~180px~~" heights="~~140px~~" perrow="1">		<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">
	Image:Hang_Glider.png\| States and discretized control for a local optimum. The initial guess for <math>t_f</math> was chosen as 100.		Image:Hang_Glider.png\| States and discretized control for a local optimum. The initial guess for <math>t_f</math> was chosen as 100.
	</gallery>		</gallery>

2025-11-25T10:28:28Z

References

← Older revision		Revision as of 10:28, 25 November 2025
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	== References ==		== References ==
	<span id="OCPjl">[1]</span> Caillau, J.-B., Cots, O., Gergaud, J., & Martinon, P. OptimalControlProblems.jl: a collection of optimal control problems with ODE's in Julia. https://github.com/control-toolbox/OptimalControlProblems.jl/blob/main/ext/Descriptions/~~robbins~~.md<br>		<span id="OCPjl">[1]</span> Caillau, J.-B., Cots, O., Gergaud, J., & Martinon, P. OptimalControlProblems.jl: a collection of optimal control problems with ODE's in Julia. https://github.com/control-toolbox/OptimalControlProblems.jl/blob/main/ext/Descriptions/glider.md<br>


	[[Category:MIOCP]]		[[Category:MIOCP]]
	[[Category:ODE model]]		[[Category:ODE model]]

2025-11-25T10:27:57Z

Reference Solutions

← Older revision		Revision as of 10:27, 25 November 2025
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	<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="1">		<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="1">
	Image:Hang_Glider.png\| States and discretized control for a local optimum.		Image:Hang_Glider.png\| States and discretized control for a local optimum. The initial guess for <math>t_f</math> was chosen as 100.
	</gallery>		</gallery>

2025-11-25T10:27:11Z

Reference Solutions

← Older revision		Revision as of 10:27, 25 November 2025
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	<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="1">		<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="1">
	Image:~~Robbins~~.png\| States and discretized control for a local optimum.		Image:Hang_Glider.png\| States and discretized control for a local optimum.
	</gallery>		</gallery>

2025-11-25T10:10:49Z

Mathematical formulation

← Older revision		Revision as of 10:10, 25 November 2025
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	<math>		<math>
	\begin{align}		\begin{align}
	r(t) &= \left( \frac{x(t)}{~~r_0~~} - 2.5 \right)^2, \\		r(t) &= \left( \frac{x(t)}{r_c} - 2.5 \right)^2, \\
	U_\text{updraft}(x(t)) &= u_c\, (1 - r(t)) \cdot \exp\left(-r(t)\right), \\		U_\text{updraft}(x(t)) &= u_c\, (1 - r(t)) \cdot \exp\left(-r(t)\right), \\
	w(t) &= v_y(t) - U_\text{updraft}(x(t)), \\		w(t) &= v_y(t) - U_\text{updraft}(x(t)), \\

2025-11-25T10:10:17Z

Mathematical formulation

2025-11-25T10:09:05Z

Mathematical formulation

2025-11-25T10:08:21Z

Mathematical formulation

← Older revision		Revision as of 10:08, 25 November 2025
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	r(t) &= \left( \frac{x(t)}{r_0} - 2.5 \right)^2, \\		r(t) &= \left( \frac{x(t)}{r_0} - 2.5 \right)^2, \\
	U_\text{updraft}(x(t)) &= u_c\, (1 - r(t)) \cdot \exp\left(-r(t)\right), \\		U_\text{updraft}(x(t)) &= u_c\, (1 - r(t)) \cdot \exp\left(-r(t)\right), \\
	w(t) &= v_y(t) - U_\text{updraft}(x), \\		w(t) &= v_y(t) - U_\text{updraft}(x(t)), \\
	v(t) &= \sqrt{v_x(t)^2 + w(t)^2}, \\		v(t) &= \sqrt{v_x(t)^2 + w(t)^2}, \\
	D(t) &= \frac{1}{2} \rho S (c_0 + c_1 c_L(t)^2) \cdot v(t)^2, \\		D(t) &= \frac{1}{2} \rho S (c_0 + c_1 c_L(t)^2) \cdot v(t)^2, \\