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