def test_flat_default_output(self, vehicle_flat):
# Construct a flat system with the default outputs
flatsys = fs.FlatSystem(vehicle_flat.forward,
vehicle_flat.reverse,
vehicle_flat.updfcn,
inputs=vehicle_flat.ninputs,
outputs=vehicle_flat.ninputs,
states=vehicle_flat.nstates)
# 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
poly = fs.PolyFamily(6)
traj1 = fs.point_to_point(vehicle_flat, Tf, x0, u0, xf, uf, basis=poly)
traj2 = fs.point_to_point(flatsys, Tf, x0, u0, xf, uf, basis=poly)
# Verify that the trajectory computation is correct
T = np.linspace(0, Tf, 10)
x1, u1 = traj1.eval(T)
x2, u2 = traj2.eval(T)
np.testing.assert_array_almost_equal(x1, x2)
np.testing.assert_array_almost_equal(u1, u2)
# Run a simulation and verify that the outputs are correct
resp1 = ct.input_output_response(vehicle_flat, T, u1, x0)
resp2 = ct.input_output_response(flatsys, T, u1, x0)
np.testing.assert_array_almost_equal(resp1.outputs[0:2], resp2.outputs)
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_double_integrator(self):
# Define a second order integrator
sys = ct.StateSpace([[-1, 1], [0, -2]], [[0], [1]], [[1, 0]], 0)
flatsys = fs.LinearFlatSystem(sys)
# Define the endpoints of a trajectory
x1 = [0, 0]; u1 = [0]; T1 = 1
x2 = [1, 0]; u2 = [0]; T2 = 2
x3 = [0, 1]; u3 = [0]; T3 = 3
x4 = [1, 1]; u4 = [1]; T4 = 4
# Define the basis set
poly = fs.PolyFamily(6)
# Plan trajectories for various combinations
for x0, u0, xf, uf, Tf in [
(x1, u1, x2, u2, T2), (x1, u1, x3, u3, T3), (x1, u1, x4, u4, T4)]:
traj = fs.point_to_point(flatsys, x0, u0, xf, uf, Tf, 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, 100)
xd, ud = traj.eval(T)
t, y, x = ct.forced_response(sys, T, ud, x0)
np.testing.assert_array_almost_equal(x, xd, decimal=3)
def time_steering_point_to_point(basis_name, basis_size):
if basis_name == 'poly':
basis = flat.PolyFamily(basis_size)
elif basis_name == 'bezier':
basis = flat.BezierFamily(basis_size)
# Find trajectory between initial and final conditions
traj = flat.point_to_point(vehicle, Tf, x0, u0, xf, uf, basis=basis)
# 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])
def test_response(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)
# Compute the response the regular way
T = np.linspace(0, Tf, 10)
x, u = traj.eval(T)
# Recompute using response()
response = traj.response(T, squeeze=False)
np.testing.assert_equal(T, response.time)
np.testing.assert_equal(u, response.inputs)
np.testing.assert_equal(x, response.states)
def time_steering_cost():
# Define cost and constraints
traj_cost = opt.quadratic_cost(vehicle, None, np.diag([0.1, 1]), u0=uf)
constraints = [opt.input_range_constraint(vehicle, [8, -0.1], [12, 0.1])]
traj = flat.point_to_point(vehicle,
timepts,
x0,
u0,
xf,
uf,
cost=traj_cost,
constraints=constraints,
basis=flat.PolyFamily(8))
# 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])
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)
inputs=('v', 'delta'),
outputs=('x', 'y', 'theta'),
states=('x', 'y', 'theta'))
# Define the endpoints of the trajectory
x0 = [0., -2., 0.]
u0 = [10., 0.]
xf = [40., 2., 0.]
uf = [10., 0.]
Tf = 4
# Define a set of basis functions to use for the trajectories
poly = fs.PolyFamily(6)
# Find a trajectory between the initial condition and the final condition
traj = fs.point_to_point(vehicle_flat, x0, u0, xf, uf, Tf, basis=poly)
# Create the desired trajectory between the initial and final condition
T = np.linspace(0, Tf, 500)
xd, ud = traj.eval(T)
# Simulation the open system dynamics with the full input
t, y, x = ct.input_output_response(vehicle_flat, T, ud, x0, return_x=True)
# Plot the open loop system dynamics
plt.figure()
plt.suptitle("Open loop trajectory for kinematic car lane change")
# Plot the trajectory in xy coordinates
plt.subplot(4, 1, 2)
plt.plot(x[0], x[1])
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)
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)
# velocity $v$ and steering wheel angle $\delta$ at the endpoints. # In[ ]: # Define the endpoints of the trajectory x0 = [0., 2., 0.] u0 = [15, 0.] xf = [75, -2., 0.] uf = [15, 0.] Tf = xf[0] / uf[0] # Define a set of basis functions to use for the trajectories poly = fs.PolyFamily(8) # Find a trajectory between the initial condition and the final condition traj1 = fs.point_to_point(vehicle_flat, Tf, x0, u0, xf, uf, basis=poly) plot_vehicle_lanechange(traj1) # ### Change of basis function # In[ ]: bezier = fs.BezierFamily(8) traj2 = fs.point_to_point(vehicle_flat, Tf, x0, u0, xf, uf, basis=bezier) plot_vehicle_lanechange(traj2) # ### Added cost function # In[ ]: timepts = np.linspace(0, Tf, 12)
def test_kinematic_car(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
vehicle_flat = fs.FlatSystem(vehicle_flat_forward,
vehicle_flat_reverse,
vehicle_update,
vehicle_output,
inputs=('v', 'delta'),
outputs=('x', 'y', 'theta'),
states=('x', 'y', 'theta'))
# Define the endpoints of the trajectory
x0 = [0., -2., 0.]
u0 = [10., 0.]
xf = [100., 2., 0.]
uf = [10., 0.]
Tf = 10
# Define a set of basis functions to use for the trajectories
poly = fs.PolyFamily(6)
# Find trajectory between initial and final conditions
traj = fs.point_to_point(vehicle_flat, x0, u0, xf, uf, Tf, 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)
inputs=('v', 'delta'),
outputs=('x', 'y'),
states=('x', 'y', 'theta'))
# Define the endpoints of the trajectory
x0 = [0., -2., 0.]
u0 = [10., 0.]
xf = [40., 2., 0.]
uf = [10., 0.]
Tf = 4
# Define a set of basis functions to use for the trajectories
poly = fs.PolyFamily(6)
# Find a trajectory between the initial condition and the final condition
traj = fs.point_to_point(vehicle_flat, Tf, x0, u0, xf, uf, basis=poly)
# Create the desired trajectory between the initial and final condition
T = np.linspace(0, Tf, 500)
xd, ud = traj.eval(T)
# Simulation the open system dynamics with the full input
t, y, x = ct.input_output_response(vehicle_flat, T, ud, x0, return_x=True)
# Plot the open loop system dynamics
plt.figure(1)
plt.suptitle("Open loop trajectory for kinematic car lane change")
plot_results(t, x, ud)
#
# Approach #2: add cost function to make lane change quicker
