opty utilizes symbolic descriptions of ordinary differential equations expressed with SymPy to form the constraints needed to solve optimal control and parameter identification problems using the direct collocation method and non-linear programming. In general, if one can express the continuous first order ordinary differential equations of the system as symbolic expressions opty will automatically generate a function to efficiently evaluate the dynamical constraints and a function that evaluates the sparse Jacobian of the constraints, which have been optimized for speed and memory consumption. The translation of the dynamical system description to the NLP form, primarily the formation of the constraints and the Jacobian of the constraints, manually is a time consuming and error prone process. opty eliminates both of those issues.


  • Both implicit and explicit forms of the first order ordinary differential equations and differential algebraic equations are supported, i.e. there is no need to solve for the derivatives of the dependent variables.

  • Backward Euler or Midpoint integration methods.

  • Supports both trajectory optimization and parameter identification.

  • Easy specification of bounds on free variables.

  • Easily specify additional “instance” constraints.

  • Automatic parallel execution using openmp if installed.

  • Built with support of sympy.physics.mechanics and PyDy in mind.


The required dependencies are as follows:

  • python 2.7 or 3.5+

  • sympy >= 1.0.0

  • ipopt >= 3.11

  • numpy >= 1.8.1

  • scipy >= 0.14.1

  • cython >= 0.20.1

  • cyipopt >= 0.1.7

To run all of the examples the following additional dependencies are required:

  • matplotlib >= 1.3.1

  • pydy >= 0.3.0

  • pytables

  • pandas

  • yeadon

  • openmp

Currently only Linux and Mac are officially supported. Although, it should be possible to install this on Windows with an appropriate Cython compilation toolchain and IPOPT installed from binaries or custom compliation.

If you are installing on Linux or Mac, the easiest way to get started is to install Anaconda (or Miniconda) and use conda to install opty and any desired optional dependencies from the Conda Forge channel, e.g.:

$ conda config --add channels conda-forge
$ conda install opty matplotlib pytables pandas yeadon pydy

If you are using Windows or want a custom installation of any of the dependencies, e.g. Ipopt, you must first install Ipopt along with it’s headers. For example, on Debian based systems you can use the package manager:

$ sudo apt-get install coinor-libipopt1v5 coinor-libipopt-dev

or prebuilt binaries can be downloaded from

For customized installation (usually desired for performance) follow the instructions on the IPOPT documentation to compile the library. If you install to a location other than /usr/local on Unix systems you will likely have to set the LD_LIBRARY_PATH so that you can link to IPOPT when installing cyipopt.

Once Ipopt is installed and accessible, install conda then create an environment:

$ conda create -n opty-custom pip numpy scipy cython sympy
$ source activate opty-custom
(opty-custom)$ pip install
(opty-custom)$ pip install opty

If you want to develop opty, create a conda environment with all of the dependencies installed:

$ conda config --add channels conda-forge
$ conda create -n opty-dev python sympy numpy scipy cython ipopt cyipopt matplotlib pytables pydy pandas
$ source activate opty-dev

Next download the opty source files and install with:

(opty-dev)$ conda develop /path/to/opty


(opty-dev)$ cd /path/to/opty
(opty-dev)$ python develop


There are several examples available in the examples directory. For example, the optimal torque to swing up a pendulum with minimal energy can be run with:

$ python examples/


The work was partially funded by the State of Ohio Third Frontier Commission through the Wright Center for Sensor Systems Engineering (WCSSE), by the National Science Foundation under Grant No. 1344954, and by National Center of Simulation in Rehabilitation Research 2014 Visiting Scholarship at Stanford University.

Indices and tables