Project State Document: Inverse Pendulum on a Cart Simulation

1. Original Goals:

The initial goal was to derive the equations of motion for an inverse pendulum on a cart using Lagrangian mechanics. This involved:

• Defining the system and its degrees of freedom.
• Determining the kinetic and potential energy of the system.
• Formulating the Lagrangian.
• Applying Lagrange's equations to derive the equations of motion.

2. Lagrangian Formulation:

Given the generalized coordinates x (cart position) and θ (pendulum angle from the upward vertical), and system parameters M (cart mass), m (pendulum mass), and l (pendulum length), the Lagrangian was formulated as:

$L = \frac{1}{2} (M + m) \dot{x}^2 + \frac{1}{2} m l^2 \dot{\theta}^2 + m l \dot{x} \dot{\theta} \cos(\theta) - m g l \cos(\theta)$

Applying Lagrange's equations yielded the following equations of motion (with an external force F on the cart):

• Cart Equation: $(M + m) \ddot{x} + m l \ddot{\theta} \cos(\theta) - m l \dot{\theta}^2 \sin(\theta) = F$
• Pendulum Equation: $l \ddot{\theta} + \ddot{x} \cos(\theta) - g \sin(\theta) = 0$

3. State-Space Formulation:

The Lagrangian equations of motion were then converted into a state-space representation for simulation purposes. The state variables were defined as:

• $x_1 = x$ (cart position)
• $x_2 = \dot{x}$ (cart velocity)
• $x_3 = \theta$ (pendulum angle)
• $x_4 = \dot{\theta}$ (pendulum angular velocity)

This resulted in a set of non-linear first-order differential equations that describe the system's dynamics:

$\begin{equation} \frac{d}{dt} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix} = \begin{bmatrix} x_2 \\ \frac{F - m g \sin(x_3) \cos(x_3) + m l x_4^2 \sin(x_3)}{M + m \sin^2(x_3)} \\ x_4 \\ \frac{(M + m) g \sin(x_3) - F \cos(x_3) - ml x_4^2 \sin(x_3) \cos(x_3)}{l (M + m \sin^2(x_3))} \end{bmatrix} \end{equation}$

4. Current Goals Informing the Code:

The current goals driving the code development are to create an interactive web-based simulation of the inverse pendulum on a cart, allowing the user to influence the system through:

• Setting System Parameters: Input boxes allow users to define the cart mass, pendulum mass, and pendulum length.
• Applying a Base Force: A range slider allows the user to apply a constant horizontal force to the cart.
• Interactive Dragging: Clicking and dragging on the cart applies a force proportional to the distance between the mouse and the cart, providing a more intuitive control mechanism.

How can we analytically find stable perfectly damped PID parameters for both the cart and the pendulum?