Cart Pendulum: Difference between revisions

Cart Pendulum
State dimension:	1
Differential states:	3
Discrete control functions:	2

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Latest revision as of 10:10, 3 February 2026

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 adapted from [1] and [2]. Its dynamics are given by a four-dimensional ODE model.

Mathematical formulation

$\begin{array}{lll} \min_{u} & \int_{0}^{t_{f}} α \cdot x (t)^{2} + β \cdot (θ (t) - π)^{2} + γ \cdot u (t)^{2} d t \\ subject to \\ \dot{x} (t) & = & \dot{x} (t), \\ \dot{θ} (t) & = & \dot{θ} (t), \\ \ddot{x} (t) & = & \frac{u + m \cdot g \cdot \sin (θ) \cdot \cos (θ) + m \cdot {\dot{θ}}^{2} \cdot \sin (θ)}{M + m \cdot (1 - \cos (θ)^{2})}, \\ \ddot{θ} (t) & = & - g \cdot \sin (θ) - \frac{u + m \cdot g \cdot \sin (θ) \cdot \cos (θ) + m \cdot {\dot{θ}}^{2} \cdot \sin (θ)}{M + m \cdot (1 - \cos (θ)^{2})} \cdot \cos (θ), \\ x (0) & = & 0, \\ θ (0) & = & 0, \\ \dot{x} (0) & = & 0, \\ \dot{θ} (0) & = & 0, \\ x (t) & \in & [- 2, 2] & \forall t \in [0, t_{f}], \\ u (t) & \in & [- 30, 30] & \forall t \in [0, t_{f}], \end{array}$

Parameters

These fixed values are used within the model:

Symbol	Value	Description
$α$	10	Objective coefficient for $x$
$β$	50	Objective coefficient for $θ$
$γ$	0.5	Objective coefficient for $u$
$t_{f}$	4	Horizon of the control problem
$M$	1	Weight of the cart
$m$	0.1	Weight of the pendulum
$g$	9.81	Gravitational acceleration

Reference Solutions

Here is one local solution to the above control problem.

Reference solution plots
States and discretized control for a local optimum.

References

[1] Multidisciplinary Optimal Control Library: https://openmdao.org/dymos/docs/latest/examples/cart_pole/cart_pole.html
[2] OptimalControlProblems.jl: https://github.com/control-toolbox/OptimalControlProblems.jl/blob/main/ext/Descriptions/moonlander.md

@@ Line 5: / Line 5: @@
 }}
-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 adapted from [[#openmdao | [1]]] and [[#OCPjl | [2]]].
 Its dynamics are given by a four-dimensional [[:Category:ODE model|ODE model]].
@@ Line 14: / Line 14: @@
 <math>
   \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_\mathrm{f}} \alpha \cdot x(t)^2 + \beta \cdot (\theta(t) - \pi)^2 + \gamma \cdot u(t)^2 dt \\
   \text{subject to} \\
 \quad \dot{x}(t) & = & \dot{x}(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{\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 \theta(0) &=& 0, \\
 \quad \dot{x}(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}
 </math>
 </p>
+== Parameters ==
+These fixed values are used within the model:
+{| border="1" align="center" cellpadding="5" cellspacing="0"
+|- bgcolor=#c7c7c7
+! Symbol !! Value !! Description
+|-
+| align=center | <math>\alpha</math> || align=right | 10  || Objective coefficient for <math>x</math>
+|-
+| align=center | <math>\beta</math> || align=right | 50 || Objective coefficient for <math>\theta</math>
+|-
+| align=center | <math>\gamma</math> || align=right | 0.5 || Objective coefficient for <math>u</math>
+|-
+| align=center | <math>t_\mathrm{f}</math> || align=right | 4 || Horizon of the control problem
+|-
+| align=center | <math>M</math> || align=right | 1 || Weight of the cart
+|-
+| align=center | <math>m</math> || align=right | 0.1 || Weight of the pendulum
+|-
+| align=center | <math>g</math> || align=right | 9.81 || Gravitational acceleration
+|}
 == Reference Solutions ==
@@ Line 35: / Line 58: @@
 <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>
-== Miscellaneous and Further Reading ==
-This formulation and a detailed description can be found in [[#openmdao|[1]]].
 == 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:Bang bang]]