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flight_trajectory_reconstruction.py
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flight_trajectory_reconstruction.py
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# Borrowed from tether-model project
import casadi as ca
import numpy as np
def plot_flight_sections2(ax, df, use_flag_turn=False):
# Different shade colors for different turns
if isinstance(ax, np.ndarray):
ax = ax.reshape(-1)
else:
ax = [ax]
for a in ax:
y0, y1 = a.get_ylim()
a.set_ylim([y0, y1])
if use_flag_turn:
a.fill_between(df.time, y0, y1, where=df['flag_turn'], facecolor='lightsteelblue', alpha=0.5) # lightgrey
else:
a.fill_between(df.time, y0, y1, where=df['pattern_section'] == 2, facecolor='lightgrey', alpha=0.5)
a.fill_between(df.time, y0, y1, where=df['pattern_section'] == 0, facecolor='lightsteelblue', alpha=0.5)
for i_ph in range(6):
mask = df['phase'] == i_ph
if mask.sum() > 0:
a.axvline(df.loc[mask.idxmax(), 'time'], linestyle='--', color='grey')
def setup_integrator_kinematic_model(tf, solver='idas'):
# ODE for kinematic model
r = ca.SX.sym('r', 4)
v = ca.SX.sym('v', 4)
x = ca.vertcat(r, v)
a = ca.SX.sym('a', 4)
dx = ca.vertcat(v, a)
# Create an integrator
ode = {'x': x, 'ode': dx, 'p': a}
intg = ca.integrator('intg', solver, ode, {'tf': tf})
return intg
def find_acceleration_matching_kite_trajectory(df, solver='idas'):
n_intervals = df.shape[0]-1
tf = .1
intg = setup_integrator_kinematic_model(tf, solver)
opti = ca.casadi.Opti()
# Decision variables for states
states = opti.variable(n_intervals+1, 8)
# Decision variables for control vector
controls = opti.variable(n_intervals, 4)
# Gap-closing shooting constraints
for k in range(n_intervals):
res = intg(x0=states[k, :], p=controls[k, :])
opti.subject_to(states[k+1, :].T == res["xf"])
# Initial guesses
mea_pos = df[['rx', 'ry', 'rz']].values
opti.set_initial(states[:, :3], mea_pos)
opti.set_initial(states[:, 3], df.kite_distance.values)
mea_speed = df[['vx', 'vy', 'vz']].values
opti.set_initial(states[:, 4:7], mea_speed)
opti.set_initial(states[:, 7], df.ground_tether_reelout_speed.values)
accelerations = df[['kite_1_ax', 'kite_1_ay', 'kite_1_az']].values
opti.set_initial(controls[:, :3], accelerations[:-1, :3])
# Norm of cartesian coordinates should be equal to the radial coordinate
opti.subject_to(ca.sum2(states[:, :3]**2)**.5 == states[:, 3])
# Limit changes with respect to the measured position while respecting the measured tether speed as radial velocity
obj = ca.sumsqr(states[:, :3] - mea_pos)
weight_radial_speed = 5
obj += ca.sumsqr(weight_radial_speed*(states[:, 7] - df.ground_tether_reelout_speed.values))
# Penalize large steps in acceleration
weight_control_steps = 1/25
control_steps = controls[1:, :3]-controls[:-1, :3]
obj += ca.sumsqr(weight_control_steps*control_steps)
opti.minimize(obj)
opti.subject_to(opti.bounded(-3, control_steps, 3))
# solve optimization problem
opti.solver('ipopt', {'ipopt': {'max_iter': 50}})
sol = opti.solve()
get_values = sol.value
states_sol = get_values(states)
controls_sol = get_values(controls)
return states_sol, controls_sol
# def setup_integrator_kinematic_model(tf, solver='idas'):
# # ODE for kinematic model
# r = ca.SX.sym('r', 3)
# v = ca.SX.sym('v', 3)
# x = ca.vertcat(r, v)
# a = ca.SX.sym('a', 3)
# dx = ca.vertcat(v, a)
#
# # Create an integrator
# ode = {'x': x, 'ode': dx, 'p': a}
#
# intg = ca.integrator('intg', solver, ode, {'tf': tf})
# return intg
#
#
# def find_acceleration_matching_kite_trajectory(df, solver='idas'):
# n_intervals = df.shape[0]-1
# tf = .1
#
# intg = setup_integrator_kinematic_model(tf, solver)
#
# opti = ca.casadi.Opti()
#
# # Decision variables for states
# states = opti.variable(n_intervals+1, 6)
# # Decision variables for control vector
# controls = opti.variable(n_intervals, 3)
#
# # Gap-closing shooting constraints
# for k in range(n_intervals):
# res = intg(x0=states[k, :], p=controls[k, :])
# opti.subject_to(states[k+1, :].T == res["xf"])
#
# # Initial guesses
# mea_pos = df[['rx', 'ry', 'rz']].values
# opti.set_initial(states[:, :3], mea_pos)
# # opti.set_initial(states[:, 3], df.kite_distance.values)
# mea_speed = df[['vx', 'vy', 'vz']].values
# opti.set_initial(states[:, 3:6], mea_speed)
# # opti.set_initial(states[:, 7], df.ground_tether_reelout_speed.values)
#
# accelerations = df[['kite_1_ax', 'kite_1_ay', 'kite_1_az']].values
# opti.set_initial(controls[:, :3], accelerations[:-1, :3])
#
# # opti.subject_to(ca.sum2(states[:, :3]**2)**.5 == states[:, 3])
# obj = ca.sumsqr(states[:, :3] - mea_pos)
#
# weight_radial_speed = 5
# for i, dl_mea in enumerate(df.ground_tether_reelout_speed.values):
# l = ca.norm_2(states[i, :3])
# dl = ca.dot(states[i, :3], states[i, 3:6]) / l
# obj += ca.sumsqr(weight_radial_speed*(dl - dl_mea))
#
# weight_control_steps = 1/25
# control_steps = controls[1:, :3]-controls[:-1, :3]
# obj += ca.sumsqr(weight_control_steps*control_steps)
# opti.minimize(obj)
#
# opti.subject_to(opti.bounded(-3, control_steps, 3))
#
# # solve optimization problem
# opti.solver('ipopt', {'ipopt': {'max_iter': 50}})
#
# sol = opti.solve()
# get_values = sol.value
#
# states_sol = get_values(states)
# controls_sol = get_values(controls)
#
# return states_sol, controls_sol
def apply_low_pass_filter(data, cutoff_freq=.3, fillna=True):
from scipy.signal import butter, filtfilt
fs = 10.0 # sample rate, Hz
nyq = 0.5 * fs # Nyquist Frequency
order = 2 # Trend can be well captured with order of 2
normal_cutoff = cutoff_freq / nyq
# Get the filter coefficients
b, a = butter(order, normal_cutoff, btype='low', analog=False)
if fillna:
data = data.interpolate()
if np.isnan(data.iloc[0]):
data.iloc[0] = data.iloc[1]
filtered_data = filtfilt(b, a, data)
return filtered_data
def plot_cartesian_results(flight_data_raw, flight_data_rec):
import matplotlib.pyplot as plt
fig, ax = plt.subplots(3, 4, sharex=True)
plot_interval = (29.9, 51.2)
plot_idx = (
(flight_data_raw['time'] == plot_interval[0]).idxmax(),
(flight_data_raw['time'] == plot_interval[1]).idxmax()
)
plt.xlim([flight_data_raw.loc[plot_idx[0], 'time'], flight_data_raw.loc[plot_idx[1], 'time']])
norm = lambda fd, p: (fd.loc[plot_idx[0]:plot_idx[1], f'{p}x']**2 + fd.loc[plot_idx[0]:plot_idx[1], f'{p}y']**2 +
fd.loc[plot_idx[0]:plot_idx[1], f'{p}z']**2)**.5
ax[0, 0].plot(flight_data_raw.loc[plot_idx[0]:plot_idx[1], 'time'], flight_data_raw.loc[plot_idx[0]:plot_idx[1], 'rx'])
ax[0, 1].plot(flight_data_raw.loc[plot_idx[0]:plot_idx[1], 'time'], flight_data_raw.loc[plot_idx[0]:plot_idx[1], 'ry'])
ax[0, 2].plot(flight_data_raw.loc[plot_idx[0]:plot_idx[1], 'time'], flight_data_raw.loc[plot_idx[0]:plot_idx[1], 'rz'])
ax[0, 3].plot(flight_data_raw.loc[plot_idx[0]:plot_idx[1], 'time'], norm(flight_data_raw, 'r'))
ax[0, 0].plot(flight_data_rec.loc[plot_idx[0]:plot_idx[1], 'time'], flight_data_rec.loc[plot_idx[0]:plot_idx[1], 'rx'], '-.')#, color='C0')
ax[0, 1].plot(flight_data_rec.loc[plot_idx[0]:plot_idx[1], 'time'], flight_data_rec.loc[plot_idx[0]:plot_idx[1], 'ry'], '-.')#, color='C1')
ax[0, 2].plot(flight_data_rec.loc[plot_idx[0]:plot_idx[1], 'time'], flight_data_rec.loc[plot_idx[0]:plot_idx[1], 'rz'], '-.')#, color='C2')
ax[0, 3].plot(flight_data_rec.loc[plot_idx[0]:plot_idx[1], 'time'], norm(flight_data_rec, 'r'))
ax[1, 0].plot(flight_data_raw.loc[plot_idx[0]:plot_idx[1], 'time'], flight_data_raw.loc[plot_idx[0]:plot_idx[1], 'vx'])
ax[1, 1].plot(flight_data_raw.loc[plot_idx[0]:plot_idx[1], 'time'], flight_data_raw.loc[plot_idx[0]:plot_idx[1], 'vy'])
ax[1, 2].plot(flight_data_raw.loc[plot_idx[0]:plot_idx[1], 'time'], flight_data_raw.loc[plot_idx[0]:plot_idx[1], 'vz'])
ax[1, 3].plot(flight_data_raw.loc[plot_idx[0]:plot_idx[1], 'time'], norm(flight_data_raw, 'v'))
ax[1, 0].plot(flight_data_raw.loc[plot_idx[0]:plot_idx[1], 'time'], flight_data_raw.loc[plot_idx[0]:plot_idx[1], 'kite_1_vx'], color='C2')
ax[1, 1].plot(flight_data_raw.loc[plot_idx[0]:plot_idx[1], 'time'], flight_data_raw.loc[plot_idx[0]:plot_idx[1], 'kite_1_vy'], color='C2')
ax[1, 2].plot(flight_data_raw.loc[plot_idx[0]:plot_idx[1], 'time'], flight_data_raw.loc[plot_idx[0]:plot_idx[1], 'kite_1_vz'], color='C2')
ax[1, 3].plot(flight_data_raw.loc[plot_idx[0]:plot_idx[1], 'time'], norm(flight_data_raw, 'kite_1_v'), color='C2')
ax[1, 0].plot(flight_data_rec.loc[plot_idx[0]:plot_idx[1], 'time'], flight_data_rec.loc[plot_idx[0]:plot_idx[1], 'vx'], '-.')#, color='C0')
ax[1, 1].plot(flight_data_rec.loc[plot_idx[0]:plot_idx[1], 'time'], flight_data_rec.loc[plot_idx[0]:plot_idx[1], 'vy'], '-.')#, color='C1')
ax[1, 2].plot(flight_data_rec.loc[plot_idx[0]:plot_idx[1], 'time'], flight_data_rec.loc[plot_idx[0]:plot_idx[1], 'vz'], '-.')#, color='C2')
ax[1, 3].plot(flight_data_rec.loc[plot_idx[0]:plot_idx[1], 'time'], norm(flight_data_rec, 'v'), '-.')
ax[2, 0].plot(flight_data_raw.loc[plot_idx[0]:plot_idx[1], 'time'], flight_data_raw.loc[plot_idx[0]:plot_idx[1], 'kite_1_ax'])
ax[2, 1].plot(flight_data_raw.loc[plot_idx[0]:plot_idx[1], 'time'], flight_data_raw.loc[plot_idx[0]:plot_idx[1], 'kite_1_ay'])
ax[2, 2].plot(flight_data_raw.loc[plot_idx[0]:plot_idx[1], 'time'], flight_data_raw.loc[plot_idx[0]:plot_idx[1], 'kite_1_az'])
ax[2, 3].plot(flight_data_raw.loc[plot_idx[0]:plot_idx[1], 'time'], norm(flight_data_raw, 'kite_1_a'))
ax[2, 0].plot(flight_data_rec.loc[plot_idx[0]:plot_idx[1], 'time'], flight_data_rec.loc[plot_idx[0]:plot_idx[1], 'ax'])#, color='C0')
ax[2, 1].plot(flight_data_rec.loc[plot_idx[0]:plot_idx[1], 'time'], flight_data_rec.loc[plot_idx[0]:plot_idx[1], 'ay'])#, color='C1')
ax[2, 2].plot(flight_data_rec.loc[plot_idx[0]:plot_idx[1], 'time'], flight_data_rec.loc[plot_idx[0]:plot_idx[1], 'az'])#, color='C2')
ax[2, 3].plot(flight_data_rec.loc[plot_idx[0]:plot_idx[1], 'time'], norm(flight_data_rec, 'a'))
def plot_tau_results(flight_data_raw, flight_data_rec):
import matplotlib.pyplot as plt
from utils import add_panel_labels
fig, ax = plt.subplots(3, 3, sharex=True, figsize=[8, 5.4])
wspace = 0.5
plt.subplots_adjust(top=0.93, bottom=0.085, left=0.115, right=0.99, hspace=0.1, wspace=wspace)
plot_interval = (29.9, 51.2)
plot_idx = (
(flight_data_raw['time'] == plot_interval[0]).idxmax(),
(flight_data_raw['time'] == plot_interval[1]).idxmax()
)
plt.xlim([flight_data_raw.loc[plot_idx[0], 'time'], flight_data_raw.loc[plot_idx[1], 'time']])
norm = lambda fd, p: (fd.loc[plot_idx[0]:plot_idx[1], f'{p}x']**2 + fd.loc[plot_idx[0]:plot_idx[1], f'{p}y']**2 +
fd.loc[plot_idx[0]:plot_idx[1], f'{p}z']**2)**.5
ax[0, 0].plot(flight_data_raw.loc[plot_idx[0]:plot_idx[1], 'time'], flight_data_raw.loc[plot_idx[0]:plot_idx[1], 'rx'], label='Sensor 0')
ax[0, 1].plot(flight_data_raw.loc[plot_idx[0]:plot_idx[1], 'time'], flight_data_raw.loc[plot_idx[0]:plot_idx[1], 'ry'])
ax[0, 2].plot(flight_data_raw.loc[plot_idx[0]:plot_idx[1], 'time'], flight_data_raw.loc[plot_idx[0]:plot_idx[1], 'rz'])
ax[1, 0].plot(flight_data_raw.loc[plot_idx[0]:plot_idx[1], 'time'], norm(flight_data_raw, 'r'))
ax[0, 0].plot(np.nan, np.nan, label='Sensor 1', color='C2')
ax[0, 0].plot(flight_data_rec.loc[plot_idx[0]:plot_idx[1], 'time'], flight_data_rec.loc[plot_idx[0]:plot_idx[1], 'rx'], '-.', label='Reconstructed')#, color='C0')
ax[0, 1].plot(flight_data_rec.loc[plot_idx[0]:plot_idx[1], 'time'], flight_data_rec.loc[plot_idx[0]:plot_idx[1], 'ry'], '-.')#, color='C1')
ax[0, 2].plot(flight_data_rec.loc[plot_idx[0]:plot_idx[1], 'time'], flight_data_rec.loc[plot_idx[0]:plot_idx[1], 'rz'], '-.')#, color='C2')
ax[1, 0].plot(flight_data_rec.loc[plot_idx[0]:plot_idx[1], 'time'], norm(flight_data_rec, 'r'), '-.')
ax[0, 0].set_ylim([200, 260])
ax[0, 0].set_ylabel('$r_x$ [m]')
ax[0, 1].set_ylim([-85, 85])
ax[0, 1].set_ylabel('$r_y$ [m]')
ax[0, 2].set_ylim([145, 190])
ax[0, 2].set_ylabel('$r_z$ [m]')
ax[1, 0].set_ylim([275, 305])
ax[1, 0].set_ylabel('r [m]')
ax[1, 1].plot(flight_data_raw.loc[plot_idx[0]:plot_idx[1], 'time'], flight_data_raw.loc[plot_idx[0]:plot_idx[1], 'vt'])
# ax[1, 1].plot(flight_data_raw.loc[plot_idx[0]:plot_idx[1], 'time'], flight_data_raw.loc[plot_idx[0]:plot_idx[1], 'kite_course']*180/np.pi)
ax[1, 2].plot(flight_data_raw.loc[plot_idx[0]:plot_idx[1], 'time'], flight_data_raw.loc[plot_idx[0]:plot_idx[1], 'vr'])
# ax[1, 3].plot(flight_data_raw.loc[plot_idx[0]:plot_idx[1], 'time'], norm(flight_data_raw, 'v'))
ax[1, 1].plot(flight_data_raw.loc[plot_idx[0]:plot_idx[1], 'time'], flight_data_raw.loc[plot_idx[0]:plot_idx[1], 'kite_1_vt'], color='C2')
# ax[1, 1].plot(flight_data_raw.loc[plot_idx[0]:plot_idx[1], 'time'], flight_data_raw.loc[plot_idx[0]:plot_idx[1], 'kite_course1']*180/np.pi, color='C2')
ax[1, 2].plot(flight_data_raw.loc[plot_idx[0]:plot_idx[1], 'time'], flight_data_raw.loc[plot_idx[0]:plot_idx[1], 'kite_1_vr'], color='C2')
# ax[1, 3].plot(flight_data_raw.loc[plot_idx[0]:plot_idx[1], 'time'], norm(flight_data_raw, 'kite_1_v'), color='C2')
ax[1, 1].plot(flight_data_rec.loc[plot_idx[0]:plot_idx[1], 'time'], flight_data_rec.loc[plot_idx[0]:plot_idx[1], 'vt'], '-.')#, color='C0')
# ax[1, 1].plot(flight_data_rec.loc[plot_idx[0]:plot_idx[1], 'time'], flight_data_rec.loc[plot_idx[0]:plot_idx[1], 'kite_course']*180/np.pi, '-.')
ax[1, 2].plot(flight_data_rec.loc[plot_idx[0]:plot_idx[1], 'time'], flight_data_rec.loc[plot_idx[0]:plot_idx[1], 'vr'], '-')#, color='C2')
# ax[1, 3].plot(flight_data_rec.loc[plot_idx[0]:plot_idx[1], 'time'], norm(flight_data_rec, 'v'), '-.')
ax[1, 1].set_ylim([10, 30])
ax[1, 1].set_ylabel('$v_t$ [m s$^{-1}$]')
ax[1, 2].set_ylim([-.5, 4])
ax[1, 2].set_ylabel('$v_r$ [m s$^{-1}$]')
ax[2, 0].plot(flight_data_raw.loc[plot_idx[0]:plot_idx[1], 'time'], flight_data_raw.loc[plot_idx[0]:plot_idx[1], 'kite_1_at'], color='C2')
ax[2, 1].plot(flight_data_raw.loc[plot_idx[0]:plot_idx[1], 'time'], flight_data_raw.loc[plot_idx[0]:plot_idx[1], 'kite_1_an'], color='C2')
ax[2, 2].plot(flight_data_raw.loc[plot_idx[0]:plot_idx[1], 'time'], flight_data_raw.loc[plot_idx[0]:plot_idx[1], 'kite_1_ar'], color='C2')
# ax[2, 3].plot(flight_data_raw.loc[plot_idx[0]:plot_idx[1], 'time'], norm(flight_data_raw, 'kite_1_a'))
ax[2, 0].plot(flight_data_rec.loc[plot_idx[0]:plot_idx[1], 'time'], flight_data_rec.loc[plot_idx[0]:plot_idx[1], 'at'], color='C1')
ax[2, 1].plot(flight_data_rec.loc[plot_idx[0]:plot_idx[1], 'time'], flight_data_rec.loc[plot_idx[0]:plot_idx[1], 'an'], color='C1')
ax[2, 2].plot(flight_data_rec.loc[plot_idx[0]:plot_idx[1], 'time'], flight_data_rec.loc[plot_idx[0]:plot_idx[1], 'ar'], color='C1')
# ax[2, 3].plot(flight_data_rec.loc[plot_idx[0]:plot_idx[1], 'time'], norm(flight_data_rec, 'a'))
# ax[2, 1].plot(flight_data_rec.loc[plot_idx[0]:plot_idx[1], 'time'], norm(flight_data_rec, 'a'), ':')
ax[2, 0].set_ylim([-25, 25])
ax[2, 0].set_ylabel('$a_t$ [m s$^{-2}$]')
ax[2, 1].set_ylim([-50, 50])
ax[2, 1].set_ylabel('$a_n$ [m s$^{-2}$]')
ax[2, 2].set_ylim([-10, 10])
ax[2, 2].set_ylabel('$a_r$ [m s$^{-2}$]')
for a in ax[2, :]: a.set_xlabel('Time [s]')
for a in ax.reshape(-1):
plot_flight_sections2(a, flight_data_raw)
a.grid()
ax[0, 0].legend(bbox_to_anchor=(.5, 1.07, 2 + wspace*2, .5), loc="lower left", mode="expand",
borderaxespad=0, ncol=3)
add_panel_labels(ax, [.5, .485, .46])
def plot_reconstruction():
from scipy.integrate import cumtrapz
import matplotlib.pyplot as plt
from utils import read_and_transform_flight_data, add_panel_labels, plot_flight_sections2
flight_data_raw = read_and_transform_flight_data(False, 65) # Read flight data.
flight_data_rec = read_and_transform_flight_data(True, 65) # Read flight data.
r = np.sum(flight_data_raw[['rx', 'ry', 'rz']].values**2, axis=1)**.5
r_c = np.sum(flight_data_rec[['rx', 'ry', 'rz']].values**2, axis=1)**.5
radial_speed = np.sum(flight_data_raw[['rx', 'ry', 'rz']].values*flight_data_raw[['vx', 'vy', 'vz']].values, axis=1)/r
radial_speed_c = np.sum(flight_data_rec[['rx', 'ry', 'rz']].values*flight_data_rec[['vx', 'vy', 'vz']].values, axis=1)/r_c
fig, ax = plt.subplots(4, 1, sharex=True, gridspec_kw={'height_ratios': [3, 1, 2, 1]}, figsize=[6.4, 6.4])
plt.subplots_adjust(left=.17, bottom=.08, right=.98, top=.98, hspace=.1)
ax[0].set_ylabel('Length / distance [m]')
lt = cumtrapz(flight_data_raw.ground_tether_reelout_speed.values, initial=0)*.1
ax[0].plot(flight_data_raw.time, lt, ':', color='k', label='L') #$\int \hat{\dot{l}}_\mathrm{t}$')
ax[0].plot(flight_data_raw.time, r - r[0], label='abcdef') #$\hat{r}_\mathrm{c} - \hat{r}_\mathrm{c,0}$')
ax[0].plot(flight_data_raw.time, r_c - r_c[0], '--', label='abcdef') #$r_\mathrm{c} - r_\mathrm{c,0}$')
ax[0].legend()
ax[1].set_ylabel('Residual\nlength [m]')
ax[1].plot(flight_data_raw.time, lt-(r - r[0]), label='ab') #$\Delta \hat{l}_\mathrm{t}$')
ax[1].plot(flight_data_raw.time, lt-(r_c - r_c[0]), label='ab') #$\Delta l_\mathrm{t}$')
ax[1].legend(ncol=2)
drdt = apply_low_pass_filter(np.diff(r), cutoff_freq=1, fillna=False)/.1
ax[2].plot(flight_data_raw.time[:-1], drdt, label='abc') #$\hat{\dot{r}}_\mathrm{c}$')
ax[2].plot(flight_data_raw.time, radial_speed, color='r', label='abc') #r'$\hat{v}_\mathrm{c,r}$', linewidth=.7)
ax[2].plot(flight_data_raw.time, radial_speed_c, '-', color='C1', label='abc') #r'$v_\mathrm{c,r}$')
ax[2].plot(flight_data_raw.time, flight_data_raw.ground_tether_reelout_speed, 'k:', label='ab') #$\hat{\dot{l}}_\mathrm{t}$')
ax[2].legend(ncol=4)
ax[2].set_ylabel('Speed [m/s]')
ax[3].plot(flight_data_raw.time, radial_speed_c-flight_data_raw.ground_tether_reelout_speed, label='abcdef') #$v_\mathrm{c,r} - \hat{\dot{l}}_\mathrm{t}$')
ax[3].legend(loc=4)
ax[3].set_ylabel('Residual speed [m/s]')
plot_flight_sections2(ax, flight_data_raw, False)
add_panel_labels(ax, offset_x=.2)
ax[-1].set_xlabel('Time [s]')
ax[-1].set_xlim([flight_data_raw['time'].iloc[0], flight_data_raw['time'].iloc[-1]])
# plot_interval = (29.9, 51.2)
# plot_interval_idx = (
# (flight_data_raw['time'] == plot_interval[0]).idxmax(),
# (flight_data_raw['time'] == plot_interval[1]).idxmax()
# )
# plt.xlim([flight_data_raw.loc[plot_interval_idx[0], 'time'], flight_data_raw.loc[plot_interval_idx[1], 'time']])
# plot_cartesian_results(flight_data_raw, flight_data_rec)
plot_tau_results(flight_data_raw, flight_data_rec)
plt.show()
if __name__ == "__main__":
plot_reconstruction() # Plots figures 4 and A1