## Saturday, 26 May 2012

### Minimum time value iteration solution of the double integrator

## Sunday, 13 May 2012

### Python simulation of the simple pendulum

Following on from the previous post, I've created some simulations of the simple pendulum without damping or control input. This work was designed more to be a test of my understanding of how to use the scipy integrator module, and to make sure the matplotlib plotting code worked correctly.

I'm pretty pleased with the results, and you can see a plot of the homoclinic orbits of pendulum starting in different initial states, and some videos of what that means, if you don't naturally think in phase space.

## Saturday, 12 May 2012

### Python simulation of the Van der Pol Oscillator

One of the things I'm doing at the moment is watching an excellent series of MIT OCW lectures by Assoc. Prof. Russ Tedrake on underactuated robotics. I've learned a lot watching the videos, which pick up at a point where some parts of my doctoral studies finished.

Just to pick something small that is at least tangentially related to the related to the course, I've attached a python snippet of code for generating the Van der Pol Oscillator. This system is significant in that it exhibits limit cycles, which are an important tool for reasoning about walking robots

import numpy as np import matplotlib.pyplot as plt from scipy.integrate import odeint mu = 0.2 def van_der_pol_oscillator_deriv(x, t): nx0 = x[1] nx1 = -mu * (x[0] ** 2.0 - 1.0) * x[1] - x[0] res = np.array([nx0, nx1]) return res ts = np.linspace(0.0, 50.0, 500) xs = odeint(van_der_pol_oscillator_deriv, [0.2, 0.2], ts) plt.plot(xs[:,0], xs[:,1]) xs = odeint(van_der_pol_oscillator_deriv, [-3.0, -3.0], ts) plt.plot(xs[:,0], xs[:,1]) xs = odeint(van_der_pol_oscillator_deriv, [4.0, 4.0], ts) plt.plot(xs[:,0], xs[:,1]) plt.gca().set_aspect('equal') plt.savefig('vanderpol_oscillator.png') plt.show()Which when simulated looks something like this: