(Time) optimal path following for differentially flat systems
The software is developed in the Python programming language and uses CasADi as a framework for symbolic computations. Furthermore CasADi provides an interface to IPOPT, a software package for large-scale nonlinear optimization. For installation instructions regarding these software packages, the user is referred to the CasADi homepage. CasADi now offers binaries, which simplify the installation procedure considerably.
Run
>> sudo python setup.py install
to install our software.
The class FlatSystem defines a flat system. A flat system S with m flat outputs using derivatives up to order k is defined as
S = FlatSystem(m, k)
Its symbolic flat output variables are stored in the instance attribute
y. For example, the j-th derivative of flat output i of system S
is S.y[i, j]. The instance method set_state allows to express the state
x of the system S as a function of the flat outputs:
x = S.set_state(expr[, name=None])
Optionally, the state can be given a name. The computation of time
derivatives is facilitated with the instance method dt:
xdot = S.dt(x)
As an example, we model an overhead crane.
from casadi import *
from pathfollowing import *
S = FlatSystem(2, 4)
G = 9.81
y1, dy1, ddy1 = S.y[0, 0], S.y[0, 1], S.y[0, 2]
y2, dy2, ddy2 = S.y[1, 0], S.y[1, 1], S.y[1, 2]
u1 = S.set_state(y1 + y2 * ddy1 / (G - ddy2), 'u1')
u2 = S.set_state(y2 * sqrt(1 + (ddy1 / (G - ddy2)) ** 2), 'u2')
theta = S.set_state(arctan(-ddy1 / (G - ddy2)), 'theta')
du1 = S.set_state(S.dt(u1), 'du1')
du2 = S.set_state(S.dt(u2), 'du2')
For the flat system S an instance of a path following problem is created via
P = PathFollowing(S)
The symbolic path coordinate and its time derivatives are stored in the
instance attribute s. The reference path is set by the instance method
set_path(expr[, r2r=False])
where expr is a list in which each component of the flat output is expressed as
a function of s[0]. When r2r is set True, the path is reparameterized such
that a rest-to-rest transition is performed.
Inequality constraints are set using the instance method
set_constraint(expr, lb, ub)
where lb and ub are the lower and upper bounds. Furthermore, various
solver options are set through
set_options(’option’: value)
Aside from all supported options of IPOPT in CasADi, the following options are available
-
N: The number of discretization steps (default: 199) -
Ts: The sampling frequency, used when transforming the solution back to time (default: 0.01 s). -
reg: A regularization factor added to the goal function to avoid singular arcs in the solution (default: 1e-20)
Finally, the instance method
solve()
solves the problem. The solution is stored in the instance attribute
sol. The instance method
plot()
plots all states as defined in the flat system and inequality constraints.
Continuing the example from previous section, we define a path following problem for the overhead crane tracking a circular trajectory with its load. The velocities of the trolley and hoisting mechanisms are constrained to [-5,5] and [-2.5,2.5] respectively.
P = PathFollowing(S)
path = [
0.25 * sin(2 * pi * P.s[0]),
0.25 * cos(2 * pi * P.s[0]) + 0.5
]
P.set_path(path)
P.set_constraint(du1, -5, 5)
P.set_constraint(du2, -2.5, 2.5)
P.set_options({'reg': 1e-10})
P.solve()
P.plot()
Check out the examples directory!