MPOPT

MPOPT is an open-source, extensible, customizable and easy to use python package that includes a collection of modules to solve multi-stage non-linear optimal control problems(OCP) using pseudo-spectral collocation methods.

The package uses collocation methods to construct a Nonlinear programming problem (NLP) representation of OCP. The resulting NLP is then solved by algorithmic differentiation based CasADi nlpsolver ( NLP solver supports multiple solver plugins including IPOPT, SNOPT, sqpmethod, scpgen).

Main features of the package are :

• Customizable collocation approximation, compatible with Legendre-Gauss-Radau (LGR), Legendre-Gauss-Lobatto (LGL), Chebyshev-Gauss-Lobatto (CGL) roots.
• Intuitive definition of single/multi-phase OCP.
• Supports Differential-Algebraic Equations (DAEs).
• Customized adaptive grid refinement schemes (Extendable)
• Gaussian quadrature and differentiation matrices are evaluated using algorithmic differentiation, thus, supporting arbitrarily high number of collocation points limited only by the computational resources.
• Intuitive post-processing module to retrieve and visualize the solution
• Good test coverage of the overall package
• Active development

Quick start

• Install from the Python Package Index repository using the following terminal command, then copy paste the code from example below in a file (test.py) and run (python3 test.py) to confirm the installation.

pip install mpopt

• (OR) Build directly from source (Terminal). Finally, make run to solve the moon-lander example described below.

git clone https://github.com/mpopt/mpopt.git --branch master
cd mpopt
make build
make run
source env/bin/activate

A sample code to solve moon-lander OCP (2D) under 10 lines

OCP :

Find optimal path, i.e Height ( $x_0$ ), Velocity ( $x_1$ ) and Throttle ( $u$ ) to reach the surface: Height (0m), Velocity (0m/s) from: Height (10m) and velocity(-2m/s) with: minimum fuel (u).

$$\begin{aligned} &\min_{x, u} & \qquad & J = 0 + \int_{t_0}^{t_f}u\ dt\\ &\text{subject to} & & \dot{x_0} = x_1; \dot{x_1} = u - 1.5\\ & & & x_0(t_f) = 0; \ x_1(t_f) = 0\\ & & & x_0(t_0) = 10; \ x_1(t_0) = -2\\ & & & x_0 \geq 0; 0 \leq u \leq 3\\ & & & t_0 = 0.0; t_f = \text{free variable} \end{aligned}$$

# Moon lander OCP direct collocation/multi-segment collocation

# from context import mpopt # (Uncomment if running from source)
from mpopt import mp

# Define OCP
ocp = mp.OCP(n_states=2, n_controls=1)
ocp.dynamics[0] = lambda x, u, t: [x[1], u[0] - 1.5]
ocp.running_costs[0] = lambda x, u, t: u[0]
ocp.terminal_constraints[0] = lambda xf, tf, x0, t0: [xf[0], xf[1]]
ocp.x00[0] = [10.0, -2.0]
ocp.lbu[0], ocp.ubu[0] = 0, 3
ocp.lbx[0][0] = 0

# Create optimizer(mpo), solve and post process(post) the solution
mpo, post = mp.solve(ocp, n_segments=20, poly_orders=3, scheme="LGR", plot=True)
x, u, t, _ = post.get_data()
mp.plt.show()

• Experiment with different collocation schemes by changing "LGR" to "CGL" or "LGL" in the above script.
• Update the grid to recompute solution (Ex. n_segments=3, poly_orders=[3, 30, 3]).
• For a detailed demo of the mpopt features, refer the notebook getting_started.ipynb

Resources

• Detailed implementation aspects of MPOPT are part of the master thesis.
• Documentation at mpopt.readthedocs.io/

Features and Limitations

While MPOPT is able to solve any Optimal control problem formulation in the Bolza form, the present limitations of MPOPT are,

• Only continuous functions and derivatives are supported
• Dynamics and constraints are to be written in CasADi variables (Familiarity with casadi variables and expressions is expected)
• The adaptive grid though successful in generating robust solutions for simple problems, doesn't have a concrete proof on convergence.

Authors

• Devakumar THAMMISETTY
• Prof. Colin Jones (Co-author)

License

This project is licensed under the GNU LGPL v3 - see the LICENSE file for details

Acknowledgements

• Petr Listov

Cite

• D. Thammisetty, “Development of a multi-phase optimal control software for aerospace applications (mpopt),” Master’s thesis, Lausanne, EPFL, 2020.

BibTex entry:

@mastersthesis{thammisetty2020development,
      title={Development of a multi-phase optimal control software for aerospace applications (mpopt)},
      author={Thammisetty, Devakumar},
      year={2020},
      school={Master’s thesis, Lausanne, EPFL}}