3. Getting started#

3.1. Solve moon-lander OCP in under 10 lines#

OCP :

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

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

# 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

For issues related to getting started examples, refer issues

Next steps: :doc:`./examples`_, :doc:`./notebooks`_