What is $\alpha \sin(\theta) + \beta \frac{d \theta}{d t}$ in the inverted pendulum problem?

Question

I am preparing for an exam in neural networks. As an example for self-organizing maps they showed the inverted pendulum problem where you want to keep the pole vertical:

Now the part which I don't understand:

$$f(\theta) = \alpha \sin(\theta) + \beta \frac{\mathrm{d} \theta}{\mathrm{d} t}$$ Let $x= \theta$, $y=\frac{\mathrm{d} \theta}{\mathrm{d} t}$, $z=f$.

Solution with SOM:

• three-dimensional surface in $(x,y,z)$
• adapt two-dimensional SOM to surface
• Method of control
  • For a given $(x,y)$ find neuron $k$ for wich $w_k = [w_{k1}, w_{k2}, w_{k2}, w_{k3}]$
  • $f(\theta)$ is then $w_{k3}$

I guess we use the SOM to learn the function $f$. However, I would like to understand where $f$ comes from / what it means in this model.

Answer 1

The function $f$ comes from the equation of motion for the inverted pendulum problem (inverted pendulum alone, not including the motion of the wheeled platform). If you consider your figure but ignore the side-to-side motion of the cart, then the equilibrium of moments about the hinge is:

$\sum M = m g l \sin \theta - b\frac{d \theta}{dt} $

Where $m$ is the mass of the black point mass at the tip of the pendulum and $b$ is the linear damping coefficient (due to friction at the hinge).

The sum of moments is equal to the inertial reaction (equivalent of $F = ma$):

$\sum M = m l^2 \frac{d^2 \theta}{dt^2}$

Equating these two leads to the equation of motion for the inverted pendulum:

$\frac{d^2 \theta}{dt^2} = \frac{g}{l} \sin \theta - \frac{b}{m l^2} \frac{d \theta}{dt}$

Substituting $\alpha = \frac{g}{l}$ and $\beta = -\frac{b}{m l^2}$, we then recover the form of the function $f$ in your question:

$\frac{d^2 \theta}{dt^2} = \alpha \sin \theta + \beta \frac{d \theta}{dt}$

(It should again be stressed that this analysis does not include the side-to-side motion of the cart.)