Equation

Car On the Hill Integration

$$\left\{ \begin{array}{lll}\dot{x}_{1} & = & x_{2}\\\dot{x}_{2} & = & -9.81\cdot\sin(\frac{1.1\cdot\sin(1.2\cdot x_{1})-1.2\cdot\sin(1.1\cdot x_{1})}{2})-0.7\cdot x_{2}+2\end{array}\right.$$

Initial set: $(x_{1})^{2}+(x_{2})^{2}\leq 1$

Search space: $[-2, 13]\times[-3,5]$

Figure

Code

from pyinvariant import *

# Define the search space
space = IntervalVector([[-2, 13],[-3, 5]])

# Create the grpah structure
smartSubPaving = SmartSubPaving(space)

# Create the Domain
dom_outer = Domain(smartSubPaving, FULL_WALL)
dom_outer.set_border_path_in(False)
dom_outer.set_border_path_out(False)
f_sep = Function("x[2]", "(x[0])^2+(x[1])^2-(1)^2")
s_outer = SepFwdBwd(f_sep, LEQ) # possible options : LT, LEQ, EQ, GEQ, GT
dom_outer.set_sep_input(s_outer);

dom_inner = Domain(smartSubPaving, FULL_DOOR)
dom_inner.set_border_path_in(True)
dom_inner.set_border_path_out(True)
s_inner = SepFwdBwd(f_sep, GEQ) # possible options : LT, LEQ, EQ, GEQ, GT
dom_inner.set_sep_output(s_inner);

# Create the Dynamics
f = Function("x[2]", "(x[1],-9.81*sin((1.1*sin(1.2*x[0])-1.2*sin(1.1*x[0]))/2.0)-0.7*x[1]+2.0)")
dyn = DynamicsFunction(f, FWD)

# Create the two Maze associated with the Domain and the dynamics
maze_inner = Maze(dom_inner, dyn)
maze_outer = Maze(dom_outer, dyn)

# Contract the system
for i in range(15):
	print(i)
	smartSubPaving.bisect()
	maze_outer.contract()
	maze_inner.contract()

# Visualization
visu = VibesMaze("Car on the Hill Intergation", maze_outer, maze_inner)
visu.setProperties(0,0,512,512)
visu.show()
visu.drawCircle(0, 0, 1, "black[red]");