PyDy, short for Python Dynamics, is a tool kit written in and accessed through the Python programming language that utilizes an array of scientific tools to study multibody dynamics. The goal is to have a modular framework and eventually a physics abstraction layer which utilizes a variety of backend that can provide the user with their desired workflow, including:
- Model specification
- Equation of motion generation
- Simulation
- Visualization
- Publication
We started by building the SymPy mechanics package which provides an API for building models and generating the symbolic equations of motion for complex multibody systems and have more recently developed two packages, pydy.codegen and pydy.viz, for simulation and visualization of the models. The remaining tools currently used in the PyDy workflow are popular scientific Python packages such as NumPy, SciPy, IPython, and matplotlib (i.e. the SciPy stack) which provide additional code for numerical analyses, simulation, and visualization.
The PyDy workflow has hard dependecies on these Python packages:
- Python >= 2.7
- setuptools
SciPy Stack
- SymPy >= 0.7.4.1
- NumPy >= 1.6.1
- SciPy >= 0.9.0
- matplotlib >= 0.99.0
- IPython >= 0.13.0
It's best to install the SciPy Stack dependencies using the instructions provided on the SciPy website.
Once the dependencies are installed, the package can be installed from PyPi using:
$ easy_install pydy
or:
$ pip install pydy
For system wide installs you will need root permissions (perhaps prepend
commands with sudo).
You can also grab the source and then install[1].
Using the zip download:
$ wget https://github.com/pydy/pydy/archive/master.zip $ unzip pydy-master.zip $ cd pydy-master $ python setup.py install
Using Git:
$ git clone https://github.com/pydy/pydy.git $ cd pydy $ python setup.py install
| [1] | Note that this is the latest development version. Specific releases can be found here: https://github.com/pydy/pydy/releases or by checking out a tag with Git. |
Tests require nose:
- nose: 1.3.0
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 [email protected]:pydy/pydy.git (pydy-dev)$ cd pydy (pydy-dev)$ python setup.py develop
Run the tests:
(pydy-dev)$ nosetests
For the Javascript tests the Jasmine and blanket.js libraries are used. Both of these libraries are included in pydy-viz with the source. To run the Javascript tests, go to the javascript library directory:
$ cd pydy/viz/static/js
Then run a simple HTTP Server with Python (the server is required due to some cross browser issues with blanket.js):
$ python -m SimpleHTTPServer
Now visit http://localhost:8000/SpecRunner.html in a webgl compliant browser.
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>
Simply import the modules and functions when in a Python interpreter:
>>> from sympy import symbols >>> from sympy.physics import mechanics >>> from pydy import codegen, viz
The documentation is hosted at http://pydy-viz.readthedocs.org but you can also build them from source using the following instructions:
Requires:
- Sphinx
- numpydoc
pip install sphinx numpydoc
To build the HTML docs:
$ sphinx-build -b html docs/src docs/build
View:
$ firefox docs/build/index.html
This package 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.
To enable different code generation backends, you can install the various optional dependencies:
- Cython: >=0.15.1
- Theano: >=0.6.0
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.codegen.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()
Visualization of multibody systems generated with PyDy.