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349 lines (290 loc) · 14.4 KB
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"""flatsys_test.py - test flat system module
RMM, 29 Jun 2019
This test suite checks to make sure that the basic functions supporting
differential flat systetms are functioning. It doesn't do exhaustive
testing of operations on flat systems. Separate unit tests should be
created for that purpose.
"""
from distutils.version import StrictVersion
import numpy as np
import pytest
import scipy as sp
import control as ct
import control.flatsys as fs
import control.optimal as opt
class TestFlatSys:
"""Test differential flat systems"""
@pytest.mark.parametrize(
"xf, uf, Tf",
[([1, 0], [0], 2),
([0, 1], [0], 3),
([1, 1], [1], 4)])
def test_double_integrator(self, xf, uf, Tf):
# Define a second order integrator
sys = ct.StateSpace([[-1, 1], [0, -2]], [[0], [1]], [[1, 0]], 0)
flatsys = fs.LinearFlatSystem(sys)
# Define the basis set
poly = fs.PolyFamily(6)
x1, u1, = [0, 0], [0]
traj = fs.point_to_point(flatsys, Tf, x1, u1, xf, uf, basis=poly)
# Verify that the trajectory computation is correct
x, u = traj.eval([0, Tf])
np.testing.assert_array_almost_equal(x1, x[:, 0])
np.testing.assert_array_almost_equal(u1, u[:, 0])
np.testing.assert_array_almost_equal(xf, x[:, 1])
np.testing.assert_array_almost_equal(uf, u[:, 1])
# Simulate the system and make sure we stay close to desired traj
T = np.linspace(0, Tf, 100)
xd, ud = traj.eval(T)
t, y, x = ct.forced_response(sys, T, ud, x1, return_x=True)
np.testing.assert_array_almost_equal(x, xd, decimal=3)
@pytest.fixture
def vehicle_flat(self):
"""Differential flatness for a kinematic car"""
def vehicle_flat_forward(x, u, params={}):
b = params.get('wheelbase', 3.) # get parameter values
zflag = [np.zeros(3), np.zeros(3)] # list for flag arrays
zflag[0][0] = x[0] # flat outputs
zflag[1][0] = x[1]
zflag[0][1] = u[0] * np.cos(x[2]) # first derivatives
zflag[1][1] = u[0] * np.sin(x[2])
thdot = (u[0]/b) * np.tan(u[1]) # dtheta/dt
zflag[0][2] = -u[0] * thdot * np.sin(x[2]) # second derivatives
zflag[1][2] = u[0] * thdot * np.cos(x[2])
return zflag
def vehicle_flat_reverse(zflag, params={}):
b = params.get('wheelbase', 3.) # get parameter values
x = np.zeros(3); u = np.zeros(2) # vectors to store x, u
x[0] = zflag[0][0] # x position
x[1] = zflag[1][0] # y position
x[2] = np.arctan2(zflag[1][1], zflag[0][1]) # angle
u[0] = zflag[0][1] * np.cos(x[2]) + zflag[1][1] * np.sin(x[2])
thdot_v = zflag[1][2] * np.cos(x[2]) - zflag[0][2] * np.sin(x[2])
u[1] = np.arctan2(thdot_v, u[0]**2 / b)
return x, u
def vehicle_update(t, x, u, params):
b = params.get('wheelbase', 3.) # get parameter values
dx = np.array([
np.cos(x[2]) * u[0],
np.sin(x[2]) * u[0],
(u[0]/b) * np.tan(u[1])
])
return dx
def vehicle_output(t, x, u, params): return x
# Create differentially flat input/output system
return fs.FlatSystem(
vehicle_flat_forward, vehicle_flat_reverse, vehicle_update,
vehicle_output, inputs=('v', 'delta'), outputs=('x', 'y', 'theta'),
states=('x', 'y', 'theta'))
@pytest.mark.parametrize("poly", [
fs.PolyFamily(6), fs.PolyFamily(8), fs.BezierFamily(6)])
def test_kinematic_car(self, vehicle_flat, poly):
# Define the endpoints of the trajectory
x0 = [0., -2., 0.]; u0 = [10., 0.]
xf = [100., 2., 0.]; uf = [10., 0.]
Tf = 10
# Find trajectory between initial and final conditions
traj = fs.point_to_point(vehicle_flat, Tf, x0, u0, xf, uf, basis=poly)
# Verify that the trajectory computation is correct
x, u = traj.eval([0, Tf])
np.testing.assert_array_almost_equal(x0, x[:, 0])
np.testing.assert_array_almost_equal(u0, u[:, 0])
np.testing.assert_array_almost_equal(xf, x[:, 1])
np.testing.assert_array_almost_equal(uf, u[:, 1])
# Simulate the system and make sure we stay close to desired traj
T = np.linspace(0, Tf, 500)
xd, ud = traj.eval(T)
# For SciPy 1.0+, integrate equations and compare to desired
if StrictVersion(sp.__version__) >= "1.0":
t, y, x = ct.input_output_response(
vehicle_flat, T, ud, x0, return_x=True)
np.testing.assert_allclose(x, xd, atol=0.01, rtol=0.01)
def test_flat_cost_constr(self):
# Double integrator system
sys = ct.ss([[0, 1], [0, 0]], [[0], [1]], [[1, 0]], 0)
flat_sys = fs.LinearFlatSystem(sys)
# Define the endpoints of the trajectory
x0 = [1, 0]; u0 = [0]
xf = [0, 0]; uf = [0]
Tf = 10
T = np.linspace(0, Tf, 500)
# Find trajectory between initial and final conditions
traj = fs.point_to_point(
flat_sys, Tf, x0, u0, xf, uf, basis=fs.PolyFamily(8))
x, u = traj.eval(T)
np.testing.assert_array_almost_equal(x0, x[:, 0])
np.testing.assert_array_almost_equal(u0, u[:, 0])
np.testing.assert_array_almost_equal(xf, x[:, -1])
np.testing.assert_array_almost_equal(uf, u[:, -1])
# Solve with a cost function
timepts = np.linspace(0, Tf, 10)
cost_fcn = opt.quadratic_cost(
flat_sys, np.diag([0, 0]), 1, x0=xf, u0=uf)
traj_cost = fs.point_to_point(
flat_sys, timepts, x0, u0, xf, uf, cost=cost_fcn,
basis=fs.PolyFamily(8),
# initial_guess='lstsq',
# minimize_kwargs={'method': 'trust-constr'}
)
# Verify that the trajectory computation is correct
x_cost, u_cost = traj_cost.eval(T)
np.testing.assert_array_almost_equal(x0, x_cost[:, 0])
np.testing.assert_array_almost_equal(u0, u_cost[:, 0])
np.testing.assert_array_almost_equal(xf, x_cost[:, -1])
np.testing.assert_array_almost_equal(uf, u_cost[:, -1])
# Make sure that we got a different answer than before
assert np.any(np.abs(x - x_cost) > 0.1)
# Re-solve with constraint on the y deviation
lb, ub = [-2, -0.1], [2, 0]
lb, ub = [-2, np.min(x_cost[1])*0.95], [2, 1]
constraints = [opt.state_range_constraint(flat_sys, lb, ub)]
# Make sure that the previous solution violated at least one constraint
assert np.any(x_cost[0, :] < lb[0]) or np.any(x_cost[0, :] > ub[0]) \
or np.any(x_cost[1, :] < lb[1]) or np.any(x_cost[1, :] > ub[1])
traj_const = fs.point_to_point(
flat_sys, timepts, x0, u0, xf, uf, cost=cost_fcn,
constraints=constraints, basis=fs.PolyFamily(8),
)
# Verify that the trajectory computation is correct
x_const, u_const = traj_const.eval(T)
np.testing.assert_array_almost_equal(x0, x_const[:, 0])
np.testing.assert_array_almost_equal(u0, u_const[:, 0])
np.testing.assert_array_almost_equal(xf, x_const[:, -1])
np.testing.assert_array_almost_equal(uf, u_const[:, -1])
# Make sure that the solution respects the bounds (with some slop)
for i in range(x_const.shape[0]):
assert np.all(x_const[i] >= lb[i] * 1.02)
assert np.all(x_const[i] <= ub[i] * 1.02)
# Solve the same problem with a nonlinear constraint type
nl_constraints = [
(sp.optimize.NonlinearConstraint, lambda x, u: x, lb, ub)]
traj_nlconst = fs.point_to_point(
flat_sys, timepts, x0, u0, xf, uf, cost=cost_fcn,
constraints=nl_constraints, basis=fs.PolyFamily(8),
)
x_nlconst, u_nlconst = traj_nlconst.eval(T)
np.testing.assert_almost_equal(x_const, x_nlconst)
np.testing.assert_almost_equal(u_const, u_nlconst)
def test_bezier_basis(self):
bezier = fs.BezierFamily(4)
time = np.linspace(0, 1, 100)
# Sum of the Bezier curves should be one
np.testing.assert_almost_equal(
1, sum([bezier(i, time) for i in range(4)]))
# Sum of derivatives should be zero
for k in range(1, 5):
np.testing.assert_almost_equal(
0, sum([bezier.eval_deriv(i, k, time) for i in range(4)]))
# Compare derivatives to formulas
np.testing.assert_almost_equal(
bezier.eval_deriv(1, 0, time), 3 * time - 6 * time**2 + 3 * time**3)
np.testing.assert_almost_equal(
bezier.eval_deriv(1, 1, time), 3 - 12 * time + 9 * time**2)
np.testing.assert_almost_equal(
bezier.eval_deriv(1, 2, time), -12 + 18 * time)
# Make sure that the second derivative integrates to the first
time = np.linspace(0, 1, 1000)
dt = np.diff(time)
for N in range(5):
bezier = fs.BezierFamily(N)
for i in range(N):
for j in range(1, N+1):
np.testing.assert_allclose(
np.diff(bezier.eval_deriv(i, j-1, time)) / dt,
bezier.eval_deriv(i, j, time)[0:-1],
atol=0.01, rtol=0.01)
# Exception check
with pytest.raises(ValueError, match="index too high"):
bezier.eval_deriv(4, 0, time)
def test_point_to_point_errors(self):
"""Test error and warning conditions in point_to_point()"""
# Double integrator system
sys = ct.ss([[0, 1], [0, 0]], [[0], [1]], [[1, 0]], 0)
flat_sys = fs.LinearFlatSystem(sys)
# Define the endpoints of the trajectory
x0 = [1, 0]; u0 = [0]
xf = [0, 0]; uf = [0]
Tf = 10
T = np.linspace(0, Tf, 500)
# Cost function
timepts = np.linspace(0, Tf, 10)
cost_fcn = opt.quadratic_cost(
flat_sys, np.diag([1, 1]), 1, x0=xf, u0=uf)
# Solving without basis specified should be OK
traj = fs.point_to_point(flat_sys, timepts, x0, u0, xf, uf)
x, u = traj.eval(timepts)
np.testing.assert_array_almost_equal(x0, x[:, 0])
np.testing.assert_array_almost_equal(u0, u[:, 0])
np.testing.assert_array_almost_equal(xf, x[:, -1])
np.testing.assert_array_almost_equal(uf, u[:, -1])
# Adding a cost function generates a warning
with pytest.warns(UserWarning, match="optimization not possible"):
traj = fs.point_to_point(
flat_sys, timepts, x0, u0, xf, uf, cost=cost_fcn)
# Make sure we still solved the problem
x, u = traj.eval(timepts)
np.testing.assert_array_almost_equal(x0, x[:, 0])
np.testing.assert_array_almost_equal(u0, u[:, 0])
np.testing.assert_array_almost_equal(xf, x[:, -1])
np.testing.assert_array_almost_equal(uf, u[:, -1])
# Try to optimize with insufficient degrees of freedom
with pytest.warns(UserWarning, match="optimization not possible"):
traj = fs.point_to_point(
flat_sys, timepts, x0, u0, xf, uf, cost=cost_fcn,
basis=fs.PolyFamily(6))
# Make sure we still solved the problem
x, u = traj.eval(timepts)
np.testing.assert_array_almost_equal(x0, x[:, 0])
np.testing.assert_array_almost_equal(u0, u[:, 0])
np.testing.assert_array_almost_equal(xf, x[:, -1])
np.testing.assert_array_almost_equal(uf, u[:, -1])
# Solve with the errors in the various input arguments
with pytest.raises(ValueError, match="Initial state: Wrong shape"):
traj = fs.point_to_point(flat_sys, timepts, np.zeros(3), u0, xf, uf)
with pytest.raises(ValueError, match="Initial input: Wrong shape"):
traj = fs.point_to_point(flat_sys, timepts, x0, np.zeros(3), xf, uf)
with pytest.raises(ValueError, match="Final state: Wrong shape"):
traj = fs.point_to_point(flat_sys, timepts, x0, u0, np.zeros(3), uf)
with pytest.raises(ValueError, match="Final input: Wrong shape"):
traj = fs.point_to_point(flat_sys, timepts, x0, u0, xf, np.zeros(3))
# Different ways of describing constraints
constraint = opt.input_range_constraint(flat_sys, -100, 100)
with pytest.warns(UserWarning, match="optimization not possible"):
traj = fs.point_to_point(
flat_sys, timepts, x0, u0, xf, uf, constraints=constraint,
basis=fs.PolyFamily(6))
x, u = traj.eval(timepts)
np.testing.assert_array_almost_equal(x0, x[:, 0])
np.testing.assert_array_almost_equal(u0, u[:, 0])
np.testing.assert_array_almost_equal(xf, x[:, -1])
np.testing.assert_array_almost_equal(uf, u[:, -1])
# Constraint that isn't a constraint
with pytest.raises(TypeError, match="must be a list"):
traj = fs.point_to_point(
flat_sys, timepts, x0, u0, xf, uf, constraints=np.eye(2),
basis=fs.PolyFamily(8))
# Unknown constraint type
with pytest.raises(TypeError, match="unknown constraint type"):
traj = fs.point_to_point(
flat_sys, timepts, x0, u0, xf, uf,
constraints=[(None, 0, 0, 0)], basis=fs.PolyFamily(8))
# Unsolvable optimization
constraint = [opt.input_range_constraint(flat_sys, -0.01, 0.01)]
with pytest.raises(RuntimeError, match="Unable to solve optimal"):
traj = fs.point_to_point(
flat_sys, timepts, x0, u0, xf, uf, constraints=constraint,
basis=fs.PolyFamily(8))
# Method arguments, parameters
traj_method = fs.point_to_point(
flat_sys, timepts, x0, u0, xf, uf, cost=cost_fcn,
basis=fs.PolyFamily(8), minimize_method='slsqp')
traj_kwarg = fs.point_to_point(
flat_sys, timepts, x0, u0, xf, uf, cost=cost_fcn,
basis=fs.PolyFamily(8), minimize_kwargs={'method': 'slsqp'})
np.testing.assert_allclose(
traj_method.eval(timepts)[0], traj_kwarg.eval(timepts)[0],
atol=1e-5)
# Unrecognized keywords
with pytest.raises(TypeError, match="unrecognized keyword"):
traj_method = fs.point_to_point(
flat_sys, timepts, x0, u0, xf, uf, solve_ivp_method=None)