Cart Pendulum - Revision history

RobertLampel: /* Mathematical formulation */

2026-02-03T10:10:17Z

Mathematical formulation

← Older revision		Revision as of 10:10, 3 February 2026
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	<math>		<math>
	\begin{array}{lll}		\begin{array}{lll}
	\displaystyle \min_{u} && \int_{0}^{~~t_f~~} \alpha \cdot x(t)^2 + \beta \cdot (\theta(t) - \pi)^2 + \gamma \cdot u(t)^2 dt \\		\displaystyle \min_{u} && \int_{0}^{t_\mathrm{f}} \alpha \cdot x(t)^2 + \beta \cdot (\theta(t) - \pi)^2 + \gamma \cdot u(t)^2 dt \\
	\text{subject to} \\		\text{subject to} \\
	\quad \dot{x}(t) & = & \dot{x}(t),\\		\quad \dot{x}(t) & = & \dot{x}(t),\\
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	\quad \dot{x}(0) &=& 0, \\		\quad \dot{x}(0) &=& 0, \\
	\quad \dot{\theta}(0) &=& 0, \\		\quad \dot{\theta}(0) &=& 0, \\
	\quad x(t) &\in& [-2,2] \ &\quad \forall t \in [0,~~t_f~~], \\		\quad x(t) &\in& [-2,2] \ &\quad \forall t \in [0,t_\mathrm{f}], \\
	\quad u(t) &\in& [-30,30] \ &\quad \forall t \in [0,~~t_f~~], \\		\quad u(t) &\in& [-30,30] \ &\quad \forall t \in [0,t_\mathrm{f}], \\
	\end{array}		\end{array}
	</math>		</math>

RobertLampel: /* Mathematical formulation */

2026-02-03T09:15:43Z

Mathematical formulation

← Older revision		Revision as of 09:15, 3 February 2026
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	\quad \dot{\theta}(t) & = & \dot{\theta}(t), \\		\quad \dot{\theta}(t) & = & \dot{\theta}(t), \\
	\quad \ddot{x}(t) & = & \frac{u + m \cdot g \cdot \sin(\theta) \cdot \cos(\theta) + m \cdot \dot{\theta}^2 \cdot \sin(\theta)}{M + m \cdot (1 - \cos(\theta)^2)}, \\		\quad \ddot{x}(t) & = & \frac{u + m \cdot g \cdot \sin(\theta) \cdot \cos(\theta) + m \cdot \dot{\theta}^2 \cdot \sin(\theta)}{M + m \cdot (1 - \cos(\theta)^2)}, \\
	\quad \ddot{\theta}(t) & = & -g \cdot \sin(\theta) - \frac{u + m \cdot g \cdot \sin(\theta) \cdot \cos(\theta) + m \cdot \~~ddot~~{\theta}^2 \cdot \sin(\theta)}{M + m \cdot (1 - \cos(\theta)^2)} \cdot \cos(\theta), \\		\quad \ddot{\theta}(t) & = & -g \cdot \sin(\theta) - \frac{u + m \cdot g \cdot \sin(\theta) \cdot \cos(\theta) + m \cdot \dot{\theta}^2 \cdot \sin(\theta)}{M + m \cdot (1 - \cos(\theta)^2)} \cdot \cos(\theta), \\
	\quad x(0) &=& 0, \\		\quad x(0) &=& 0, \\
	\quad \theta(0) &=& 0, \\		\quad \theta(0) &=& 0, \\

RobertLampel: /* Miscellaneous and Further Reading */

2026-02-03T08:56:43Z

Miscellaneous and Further Reading

← Older revision		Revision as of 08:56, 3 February 2026
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	Image:Cart_Pendulum.png\| States and discretized control for a local optimum.		Image:Cart_Pendulum.png\| States and discretized control for a local optimum.
	</gallery>		</gallery>

	~~== Miscellaneous and Further Reading ==~~
	~~This formulation and a detailed description can be found in [[#openmdao\|[1]]].~~

	== References ==		== References ==

RobertLampel at 08:56, 3 February 2026

2026-02-03T08:56:19Z

← Older revision		Revision as of 08:56, 3 February 2026
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	The '''Cart Pendulum problem''' concerns a pendulum hinged to a mobile cart. The control objective is to transition the pendulum from a downward position to a stabilized, inverted state above the cart. In this formulation, the objective function is defined by a composite of least-squares terms that penalize the required horizontal motion, the distance of the pendulum's angle from the upward position, and the required control.		The '''Cart Pendulum problem''' concerns a pendulum hinged to a mobile cart. The control objective is to transition the pendulum from a downward position to a stabilized, inverted state above the cart. In this formulation, the objective function is defined by a composite of least-squares terms that penalize the required horizontal motion, the distance of the pendulum's angle from the upward position, and the required control.

	The implementation here is ~~taken~~ from [[#openmdao \| [1]]].		The implementation here is adapted from [[#openmdao \| [1]]] and [[#OCPjl \| [2]]].
	Its dynamics are given by a four-dimensional [[:Category:ODE model\|ODE model]].		Its dynamics are given by a four-dimensional [[:Category:ODE model\|ODE model]].

RobertLampel: /* References */

2026-02-03T08:55:39Z

References

← Older revision		Revision as of 08:55, 3 February 2026
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	== References ==		== References ==
	<span id="openmdao">[1]</span> Multidisciplinary Optimal Control Library: https://openmdao.org/dymos/docs/latest/examples/~~moon_landing~~/~~moon_landing~~.html		<span id="openmdao">[1]</span> Multidisciplinary Optimal Control Library: https://openmdao.org/dymos/docs/latest/examples/cart_pole/cart_pole.html <br>
			<span id="OCPjl">[2]</span> OptimalControlProblems.jl: https://github.com/control-toolbox/OptimalControlProblems.jl/blob/main/ext/Descriptions/moonlander.md
	[[Category:MIOCP]]		[[Category:MIOCP]]
	[[Category:Bang bang]]		[[Category:Bang bang]]

RobertLampel at 08:54, 3 February 2026

2026-02-03T08:54:07Z

← Older revision		Revision as of 08:54, 3 February 2026
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	}}		}}

	The '''Cart Pendulum problem''' concerns a pendulum hinged to a mobile cart. The control objective is to transition the pendulum from a downward position to a stabilized, inverted state above the cart. In this formulation, the objective function is defined by a composite of least-squares terms.		The '''Cart Pendulum problem''' concerns a pendulum hinged to a mobile cart. The control objective is to transition the pendulum from a downward position to a stabilized, inverted state above the cart. In this formulation, the objective function is defined by a composite of least-squares terms that penalize the required horizontal motion, the distance of the pendulum's angle from the upward position, and the required control.

	The implementation here is taken from [[#openmdao \| [1]]].		The implementation here is taken from [[#openmdao \| [1]]].

RobertLampel: /* Reference Solutions */

2026-02-03T08:51:39Z

Reference Solutions

← Older revision		Revision as of 08:51, 3 February 2026
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	<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">		<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">
	Image:~~Moon_Landing~~.png\| States and discretized control for a local optimum~~. The free end time <math>t_f</math> was modeled using the additional control <math>t</math>~~.		Image:Cart_Pendulum.png\| States and discretized control for a local optimum.
	</gallery>		</gallery>

RobertLampel: /* Parameters */

2026-02-03T08:47:39Z

Parameters

← Older revision		Revision as of 08:47, 3 February 2026
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	! Symbol !! Value !! Description		! Symbol !! Value !! Description
	\|-		\|-
	\| align=center \| <math>\alpha</math> \|\| align=right \| ~~-0.75~~ \|\| ~~Nonlinear~~ coefficient		\| align=center \| <math>\alpha</math> \|\| align=right \| 10 \|\| Objective coefficient for <math>x</math>
	\|-		\|-
	\| align=center \| <math>c</math> \|\| align=right \| 1 \|\| ~~Damping~~ coefficient		\| align=center \| <math>\beta</math> \|\| align=right \| 50 \|\| Objective coefficient for <math>\theta</math>
	\|-		\|-
	\| align=center \| <math>t_\~~mathrm{f}~~</math> \|\| align=right \| 8 \|\| ~~Horizon of the control problem~~		\| align=center \| <math>\gamma</math> \|\| align=right \| 0.5 \|\| Objective coefficient for <math>u</math>
	\|-		\|-
	\| align=center \| <math>~~\varepsilon_~~\mathrm{~~reg~~}</math> \|\| align=right \| ~~0.01~~ \|\| ~~Regularization~~ of ~~Fisher matrix~~		\| align=center \| <math>t_\mathrm{f}</math> \|\| align=right \| 4 \|\| Horizon of the control problem
	\|-		\|-
	\| align=center \| <math>~~\mathcal{U}~~</math> \|\| align=right \| [-1~~,1]~~ \|\| ~~Bounds~~ of ~~control function~~		\| align=center \| <math>M</math> \|\| align=right \| 1 \|\| Weight of the cart
	\|-		\|-
	\| align=center \| <math>~~\mathcal{W}~~</math> \|\| align=right \| [0,1] \|\| ~~Bounds~~ of ~~measurement function~~		\| align=center \| <math>m</math> \|\| align=right \| 0.1 \|\| Weight of the pendulum
	\|-		\|-
	\| align=center \| <math>~~M_1, M_2~~</math> \|\| align=right \| 2 \|\| ~~Maximum measurement time~~		\| align=center \| <math>g</math> \|\| align=right \| 9.81 \|\| Gravitational acceleration
	\|}		\|}

RobertLampel: /* Mathematical formulation */

2026-02-03T08:40:39Z

Mathematical formulation

← Older revision		Revision as of 08:40, 3 February 2026
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	<math>		<math>
	\begin{array}{lll}		\begin{array}{lll}
	\displaystyle \min_{u} && \int_{0}^{t_f} \alpha \cdot x(t)^2 + \beta \cdot (\theta - \pi)^2 + \gamma \cdot u(t)^2 dt \\		\displaystyle \min_{u} && \int_{0}^{t_f} \alpha \cdot x(t)^2 + \beta \cdot (\theta(t) - \pi)^2 + \gamma \cdot u(t)^2 dt \\
	\text{subject to} \\		\text{subject to} \\
	\quad \dot{x}(t) & = & \dot{x}(t),\\		\quad \dot{x}(t) & = & \dot{x}(t),\\

RobertLampel: /* Mathematical formulation */

2026-02-03T08:40:23Z

Mathematical formulation

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 </math>
 </p>
 == Reference Solutions ==