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Code generation for multibody dynamic systems.

Project description

PyDy Code Generation

This distribution provides code generation facilities for PyDy. For now, it generates functions that can evaluate the right hand side of the ordinary differential equations generated with sympy.physics.mechanics with three different backends: SymPy’s lambdify, Theano, and Cython.

Dependencies

Required

  • Python: 2.7 (Python 3+ may work)
  • setuptools
  • NumPy: >=1.6.1
  • SymPy: >=0.7.3

Optional

  • Cython: >=0.15.1
  • Theano: >=0.6.0
  • SciPy: >=0.9 (only for full examples)
  • matplotlib: >=0.99 (only for full examples)

There are a variety of methods to install these packages. Refer to the SciPy Stack installation instructions for details.

Installation

Once the dependencies are installed, the package can be installed from PyPi using:

$ easy_install pydy-code-gen

or:

$ pip install pydy-code-gen

You can also grab the source and then install[1].

Using the zip download:

$ wget https://github.com/PythonDynamics/pydy-code-gen/archive/master.zip
$ unzip pydy-code-gen-master.zip
$ cd pydy-code-gen-master
$ python setup.py install

Using Git:

$ git clone https://github.com/PythonDynamics/pydy-code-gen.git
$ cd pydy-code-gen
$ python setup.py install
[1]Note that this is the latest development version. Specific releases can be found here: https://github.com/PythonDynamics/pydy-code-gen/releases or by checking out a tag with Git.

Usage

This is an example of a simple 1 degree of freedom system: a mass, spring, damper system under the influence of gravity and a force:

/ / / / / / / / /
-----------------
  |    |     |   | g
  \   | |    |   V
k /   --- c  |
  |    |     | x, v
 --------    V
 |  m   | -----
 --------
    | F
    V

Derive the system:

from sympy import symbols
import sympy.physics.mechanics as me

mass, stiffness, damping, gravity = symbols('m, k, c, g')

position, speed = me.dynamicsymbols('x v')
positiond = me.dynamicsymbols('x', 1)
force = me.dynamicsymbols('F')

ceiling = me.ReferenceFrame('N')

origin = me.Point('origin')
origin.set_vel(ceiling, 0)

center = origin.locatenew('center', position * ceiling.x)
center.set_vel(ceiling, speed * ceiling.x)

block = me.Particle('block', center, mass)

kinematic_equations = [speed - positiond]

force_magnitude = mass * gravity - stiffness * position - damping * speed + force
forces = [(center, force_magnitude * ceiling.x)]

particles = [block]

kane = me.KanesMethod(ceiling, q_ind=[position], u_ind=[speed],
                     kd_eqs=kinematic_equations)
kane.kanes_equations(forces, particles)

Store the expressions and symbols in sequences for the code generation:

mass_matrix = kane.mass_matrix_full
forcing_vector = kane.forcing_full
constants = (mass, stiffness, damping, gravity)
coordinates = (position,)
speeds = (speed,)
specified = (force,)

Now generate the function needed for numerical evaluation of the ODEs. The generator can use various back ends: lambdify, theano, or cython:

from pydy_code_gen.code import generate_ode_function

evaluate_ode = generate_ode_function(mass_matrix, forcing_vector, constants,
                                     coordinates, speeds, specified,
                                     generator='lambdify')

Integrate the equations of motion under the influence of a specified sinusoidal force:

from numpy import array, linspace, sin
from scipy.integrate import odeint

x0 = array([0.1, -1.0])
args = {'constants': array([1.0, 1.0, 0.2, 9.8]),
        'specified': lambda x, t: sin(t)}
t = linspace(0.0, 10.0, 1000)

y = odeint(evaluate_ode, x0, t, args=(args,))

Plot the results:

import matplotlib.pyplot as plt

plt.plot(t, y)
plt.legend((str(position), str(speed)))
plt.show()

Development Environment

Development Dependencies

  • nose: 1.3.0

Installation

The following installation assumes you have virtualenvwrapper and all the dependencies needed to build the packages:

$ mkvirtualenv pydy-dev
(pydy-dev)$ pip install numpy scipy cython nose theano sympy
(pydy-dev)$ pip install matplotlib # make sure to do this after numpy
(pydy-dev)$ git clone git@github.com:PythonDynamics/pydy-code-gen.git
(pydy-dev)$ cd pydy-code-gen
(pydy-dev)$ python setup.py develop

Run the tests:

(pydy-dev)$ nosetests

Run the benchmark to test the n-link pendulum problem.:

(pydy-dev)$ python bin/benchmark_pydy_code_gen.py <max # of links> <# of time steps>

Release Notes

0.1.0

  • Initial version.

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