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 14: / Line 14: @@
 <math>
   \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} \\
 \quad \dot{x}(t) & = & \dot{x}(t),\\
@@ Line 24: / Line 24: @@
 \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>