PyDy, short for Python Dynamics, is a tool kit written in the Python programming language that utilizes an array of scientific programs to enable the study of multibody dynamics. The goal is to have a modular framework and eventually a physics abstraction layer which utilizes a variety of backends 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. More recently we developed two packages, pydy.codegen and pydy.viz, for simulation and visualization of the models, respectively. This Python package contains these two packages and other tools for working with mathematical models generated from SymPy mechanics. The remaining tools currently used in the PyDy workflow are popular scientific Python packages such as NumPy, SciPy, IPython, Jupyter, ipywidgets, and matplotlib (i.e. the SciPy stack) which provide additional code for numerical analyses, simulation, and visualization.

PyDy has hard dependencies on the following software[1]:

• 2.7 <= Python < 3.0 or Python >= 3.5
• setuptools >= 20.7.0
• NumPy >= 1.11.0
• SciPy >= 0.17.1
• SymPy >= 0.7.6.1
• PyWin32 >= 219 (Windows Only)

PyDy has optional dependencies on these packages:

• 4.0.0 <= Jupyter Notebook < 5.0.0
• 4.0.0 <= ipywidgets < 5.0.0
• Theano >= 0.8.0
• Cython >= 0.23.4

The examples may require these dependencies:

• matplotlib >= 1.5.1
• version_information

It's best to install the SciPy Stack dependencies using the instructions provided on the SciPy website first. We recommend the conda package manager and the Anaconda distribution for easy cross platform installation.

Once the dependencies are installed, the latest stable version of the package can be downloaded from PyPi[2]:

$ wget https://pypi.python.org/packages/source/p/pydy/pydy-X.X.X.tar.gz

and extracted and installed[3]:

$ tar -zxvf pydy-X.X.X.tar.gz
$ cd pydy-X.X.X
$ python setup.py install

Or if you have the pip package manager installed you can simply type:

$ pip install pydy

Or if you have conda you can type:

$ conda install -c pydy pydy

Also, a simple way to install all of the optional dependencies is to install the pydy-examples metapackage using conda:

$ conda install -c pydy pydy-examples

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

Create a system to manage integration and specify numerical values for the constants and specified quantities. Here, we specify sinusoidal forcing:

from numpy import array, linspace, sin
from pydy.system import System

sys = System(kane,
             constants={mass: 1.0, stiffness: 1.0,
                        damping: 0.2, gravity: 9.8},
             specifieds={force: lambda x, t: sin(t)},
             initial_conditions={position: 0.1, speed: -1.0},
             times=linspace(0.0, 10.0, 1000))

Integrate the equations of motion to get the state trajectories:

y = sys.integrate()

Plot the results:

import matplotlib.pyplot as plt

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

The documentation is hosted at http://pydy.readthedocs.org but you can also build them from source using the following instructions.

To build the documentation you must install the dependencies:

• Sphinx
• numpydoc

To build the HTML docs, run Make from within the docs directory:

$ cd docs
$ make html

You can then view the documentation from your preferred web browser, for example:

$ firefox _build/html/index.html

This package provides code generation facilities. It generates functions that can numerically 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.

The models module provides some canned models of classic systems.

The System module provides a System class to manage simulation of a single system.

This package provides tools to create 3D animated visualizations of the systems. The visualizations utilize WebGL and run in a web browser. They can also be embedded into an IPython notebook for added interactivity.

The source code is managed with the Git version control system. To get the latest development version and access to the full repository, clone the repository from Github with:

$ git clone https://github.com/pydy/pydy.git

You should then install the dependencies for running the tests:

• nose: 1.3.7
• phantomjs: 1.9.0

It is typically advantageous to setup a virtual environment to isolate the development code from other versions on your system. There are two popular environment managers that work well with Python packages: virtualenv and conda.

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

$ mkvirtualenv pydy-dev
(pydy-dev)$ pip install numpy scipy cython nose theano sympy ipython "notebook<5.0" "ipywidgets<5.0" version_information
(pydy-dev)$ pip install matplotlib # make sure to do this after numpy
(pydy-dev)$ git clone git@github.com:pydy/pydy.git
(pydy-dev)$ cd pydy
(pydy-dev)$ python setup.py develop

Or with conda:

$ conda create -c pydy -n pydy-dev setuptools numpy scipy ipython "notebook<5.0" "ipywidgets<5.0" cython nose theano sympy matplotlib version_information
$ source activate pydy-dev
(pydy-dev)$ git clone git@github.com:pydy/pydy.git
(pydy-dev)$ cd pydy
(pydy-dev)$ conda develop .

The full Python test suite can be run with:

(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:

cd pydy/viz/static/js/tests && phantomjs run-jasmine.js SpecRunner.html && cd ../../../../../

Run the benchmark to test the n-link pendulum problem with the various backends:

$ python bin/benchmark_pydy_code_gen.py <max # of links> <# of time steps>

These are various related and similar Python packages:

• https://github.com/cdsousa/sympybotics
• https://pypi.python.org/pypi/Hamilton
• https://pypi.python.org/pypi/arboris
• https://pypi.python.org/pypi/PyODE
• https://pypi.python.org/pypi/odeViz
• https://pypi.python.org/pypi/ARS
• https://pypi.python.org/pypi/pymunk

If you make use of PyDy in your work or research, please cite us in your publications or on the web. This citation can be used:

Gilbert Gede, Dale L Peterson, Angadh S Nanjangud, Jason K Moore, and Mont Hubbard, "Constrained Multibody Dynamics With Python: From Symbolic Equation Generation to Publication", ASME 2013 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, 2013, 10.1115/DETC2013-13470.

If you have any question about installation, usage, etc, feel free send a message to our public mailing list or visit our Gitter chatroom.

If you think there’s a bug or you would like to request a feature, please open an issue on Github.

• Added a new example for anthropomorphic arm. [PR #406]
• Fixed errors in the differential drive example. [PR #405]
• Added a new example for a scara arm. [PR #402]
• Fixed errors due to backwards incompatible changes with various dependencies. [PR #397]
• ODEFunctionGenerator now works with no constants symbols. [PR #391]

• Bumped minimum Jupyter notebook to 4.0 and restricted to < 5.0. [PR #381]
• Removed several deprecated functions. [PR #375]
• Bumped minimum required hard dependencies to Ubuntu 16.04 LTS package versions. [PR #372]
• Implemented ThreeJS Tube Geometry. [PR #368]
• Improved circle rendering. [PR #357]
• kwargs can be passed from System.generate_ode_function to the matrix generator. [PR #356]
• Lagrangian simple pendulum example added. [PR #351]
• Derivatives can now be used as specifies in System. [PR #340]
• The initial conditions can now be adjusted in the notebook GUI. [PR #333]
• The width of the viz canvas is now properly bounded in the notebook. [PR #332]
• Planes now render both sides in the visualization GUI. [PR #330]
• Adds in more type checks for System.times. [PR #322]
• Added an OctaveMatrixGenerator for basic Octave/Matlab printing. [PR #323]
• Simplified the right hand side evaluation code in the ODEFunctionGenerator. Note that this change comes with some performance hits. [PR #301]

• Removed the general deprecation warning from System. [PR #262]
• Don't assume user enters input in server shutdown. [PR #264]
• Use vectorized operations to compute transformations. [PR #266]
• Speedup theano generators. [PR #267]
• Correct time is displayed on the animation slider. [PR #272]
• Test optional dependencies only if installed. [PR #276]
• Require benchmark to run in Travis. [PR #277]
• Fix dependency minimum versions in setup.py [PR #279]
• Make CSE optional in CMatrixGenerator. [PR #284]
• Fix codegen line break. [PR #292]
• Don't assume Scene always has a System. [PR #295]
• Python 3.5 support and testing against Python 3.5 on Travis. [PR #305]
• Set minimum dependency versions to match Ubuntu Trusty 14.04 LTS. [PR #306]
• Replace sympy.phyics.mechanics deprecated methods. [PR #309]
• Updated installation details to work with IPython/Jupyter 4.0. [PR #311]
• Avoid the IPython widget deprecation warning if possible. [PR #311]
• Updated the mass-spring-damper example to IPy4 and added version_information. [PR #312]
• The Cython backend now compiles on Windows. [PR #313]
• CI testing is now run on appveyor with Windows VMs. [PR #315]
• Added a verbose option to the Cython compilation. [PR #315]
• Fixed the RHS autogeneration. [PR #318]
• Improved the camera code through inheritance [PR #319]

• Introduced conda builds and binstar support. [PR #219]
• Dropped support for IPython < 3.0. [PR #237]
• Added support Python 3.3 and 3.4. [PR #229]
• Bumped up the minimum dependencies for NumPy, SciPy, and Cython [PR #233].
• Removed the partial implementation of the Mesh shape. [PR #172]
• Overhauled the code generation package to make the generators more easily extensible and to improve simulation speed. [PR #113]
• The visualizer has been overhauled as part of Tarun Gaba's 2014 GSoC internship [PR #82]. Here are some of the changes:
  • The JavaScript is now handled by AJAX and requires a simple server.
  • The JavaScript has been overhauled and now uses prototype.js for object oriented design.
  • The visualizer can now be loaded in an IPython notebook via IPython's widgets using Scene.display_ipython().
  • A slider was added to manually control the frame playback.
  • The visualization shapes' attributes can be manipulated via the GUI.
  • The scene json file can be edited and downloaded from the GUI.
  • pydy.viz generates two JSONs now (instead of one in earlier versions). The JSON generated from earlier versions will not work in the new version.
  • Shapes can now have a material attribute.
  • Model constants can be modified and the simulations can be rerun all via the GUI.
  • Switched from socket based server to python's core SimpleHTTPServer.
  • The server has a proper shutdown response [PR #241]
• Added a new experimental System class and module to more seamlessly manage integrating the equations of motion. [PR #81]

• Switched to a conda based Travis testing setup. [PR #231]
• When using older SymPy development versions with non-PEP440 compliant version identifiers, setuptools < 8 is required. [PR #166]
• Development version numbers are now PEP 440 compliant. [PR #141]
• Introduced pull request checklists and CONTRIBUTING file. [PR #146]
• Introduced light code linting into Travis. [PR #148]

• Unbundled unnecessary files from tar ball.

• Merged pydy_viz, pydy_code_gen, and pydy_examples into the source tree.
• Added a method to output "static" visualizations from a Scene object.
• Dropped the matplotlib dependency and now only three.js colors are valid.
• Added joint torques to the n_pendulum model.
• Added basic examples for codegen and viz.
• Graceful fail if theano or cython are not present.
• Shapes can now use sympy symbols for geometric dimensions.