Double Cart Inverted Pendulum Mechanics

[CandidClam]

Homework Statement

Consider an inverted pendulum system as described in https://en.wikipedia.org/wiki/Inverted_pendulum. The system consists of a cart with mass M to which a pole of length l and mass m is affixed via a pivot point. Forces applied to the cart move it along a track. At any point in time the system can be described by the angle of the pole theta (radians, zero refers to an upright pole), the angular velocity of the pole theta_dot, the position of the cart along the track x, and the cart's velocity x_dot.

The following C++ code updates the state of the system given a force input and a timestep size tau:

    float temp =  (force + m * l * theta_dot * theta_dot * sin(theta)) / (M+m);
    float thetaacc = (gravity * sin(theta) - cos(theta) * temp) / (l * ((4.0/3.0) - m * cos(theta) * cos(theta) / (M+m)));
    float xacc = temp - m * l * thetaacc * cos(theta) / (M+m);

    // Update the four state variables, using Euler's method.
    x += tau * x_dot;
    x_dot += tau * xacc;
    theta += tau * theta_dot;
    theta_dot += tau * thetaacc;

This code comes from a reasonably reliable source so I assume it is correct. I wish to modify this system by adding a second cart atop which the first cart rests. This second, lower cart rests upon the track and has no pole, forming a stack of carts. Forces may now be applied independently to each cart.

The Attempt at a Solution

This attempted solution first calculates the movement of the lower cart based the force applied to the lower cart (lowerCartForce) then applies the lower cart's acceleration to the temp variable. (lower_x and lower_x_dot refer respectively to the position and velocity of the lower cart)

    // Lower cart physics calculations. This assumes wheel rigidity -- eg lower cart is not
    // influenced by movements of the upper cart.
    float lower_xacc = lowerCartForce / (lowerCartMass + M + m);
    lower_x += tau * lower_x_dot;
    lower_x_dot += tau * lower_xacc;

    // This next line is the only one that differs from the last code block
    float temp = lower_xacc + (force + m * l * theta_dot * theta_dot * sin(theta)) / (M+m);
    float thetaacc = (gravity * sin(theta) - cos(theta) * temp) / (l * ((4.0/3.0) - m * cos(theta) * cos(theta) / (M+m)));
    float xacc = temp - m * l * thetaacc * cos(theta) / (M+m);

    // Update the four state variables, using Euler's method.
    x += tau * x_dot;
    x_dot += tau * xacc;
    theta += tau * theta_dot;
    theta_dot += tau * thetaacc;

My question to these forums: Is the physics of the above system accurate? Specifically have I correctly transferred the lower cart forces to the upper cart?

 

[voko]

You will get better advice here if you first describe how you think the original code works and then how you intend to modify it - paying attention to the physics, not C++.

 

[CandidClam]

The original code was not my own creation and I've tried to puzzle through it several times without much success. However, here's an attempt:

The second and third lines of the code are calculating the acceleration of the pole and cart. I'm still trying to figure out what the temp variable actually represents. The 3rd line computes cart acceleration so the units should be m/s^2. Since temp appears in the 3rd line followed by subtraction, I feel that it's units should also be m/s^2. If anyone is able to understand what exactly temp is computing, that would in itself be a great help.