Constrained double integrator¶

This example is intended to present PyTrajectory’s capabilities on handling system constraints. To do so, consider the double integrator which models the dynamics of a simple mass in an one-dimensional space, where a force effects the acceleration. The state space representation is given by the following dynamical system.

$\begin{eqnarray*} \dot{x_1} = x_2 \\ \dot{x_2} = u_1 \end{eqnarray*}$

A possibly wanted trajectory is the translation from $x_1(t_0 = 0) = 0$ to $x_1(T) = 1$ within $T = 2[s]$ . At the beginning and end the mass should stay at rest, that is $x_2(0) = x_2(2) = 0$ .

Now, suppose we want the velocity to be bounded by $x_{2,min} = 0.0 \leq x_2 \leq 0.65 = x_{2,max}$ . To achieve this PyTrajectory needs a dictionary containing the index of the constrained variable in $x = [x_1, x_2]$ and a tuple with the corresponding constraints. So, normally this would look like

>>> con = {1 : [0.0, 0.65]}

But, due to how the approach for handling system constraints is implemented, this would throw an exception because the lower bound of the constraints $x_{2,min}$ is equal to $x_2(0)$ and has to be smaller. So instead we use the dictionary

>>> con = {1 : [-0.1, 0.65]}

Source Code¶'''
This example of the double integrator demonstrates how to pass constraints to PyTrajectory.
'''
# imports
from pytrajectory import ControlSystem
import numpy as np

# define the vectorfield
def f(x,u):
    x1, x2 = x
    u1, = u
    
    ff = [x2,
          u1]
    
    return ff

# system state boundary values for a = 0.0 [s] and b = 2.0 [s]
xa = [0.0, 0.0]
xb = [1.0, 0.0]

# constraints dictionary
con = {1 : [-0.1, 0.65]}

# create the trajectory object
S = ControlSystem(f, a=0.0, b=2.0, xa=xa, xb=xb, constraints=con, use_chains=False)

# start
x, u = S.solve()