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drone_params.py
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import numpy as np
import sympy as sym
class Drone:
"""Parameters of full drone model controlled with collective thrust and body torques.
Attitude of drone is represented by unit quaternion.
Parameters are based on the papers:
Hummingbird:
Z. Bouček and M. Flídr, "Explicit Interpolating Control of Unmanned Aerial Vehicle,"
2019 24th International Conference on Methods and Models in Automation and Robotics (MMAR),
2019, pp. 384-389, doi: 10.1109/MMAR.2019.8864719.
Crazyflie:
Förster, Julian. “System Identification of the Crazyflie 2.0 Nano Quadrocopter.” (2015).
Landry, B. (2015). Planning and Control for Quadrotor Flight through Cluttered Environments.
Thesis, 2014, 71. http://groups.csail.mit.edu/robotics-center/public_papers/Landry15.pdf
"""
def __init__(self, drone_type: str = 'crazyflie', aerodynics: bool = True):
# drone parameters
self.g = 9.81305
self.aerodynamics = aerodynics
if drone_type == 'crazyflie':
# Bitcraze Crazyflie
self.mass = 0.032
# Matrix of Inertia
# # according to Landry, B. (2015). Planning and Control for Quadrotor Flight through Cluttered Environments.
# # Thesis, 2014, 71. http://groups.csail.mit.edu/robotics-center/public_papers/Landry15.pdf
# self.Ix = 2.3951e-5
# self.Iy = 2.3951e-5
# self.Iz = 3.2347e-5
# according to Förster thesis
self.Ix = 6.410179e-6
self.Iy = 6.410179e-6
self.Iz = 9.860228e-6
self.l = 0.0397
self.proppeler_d = 51*1e-3 # propeller diameter in meters
area_x = 2.0
area_y = 2.0
area_z = 2.0
self.speed_limit = 2.0
self.quat_limit = 1.0
self.arate_limit = np.Inf
# aerodynamic effect
# F_a = K_aero*T*R'*v
# F = R*([0 0 T]+F_a)
self.K_aero = 1e-7*np.array([[-10.2506, -0.3177, -0.4332],
[-0.3177, -10.2506, -0.4332],
[-7.7050, -7.7050, -7.5530]])
else:
# AscTec Hummingbird
self.mass = 0.542
self.Ix = 0.0064
self.Iy = 0.0064
self.Iz = 0.0125
self.l = 0.174
self.proppeler_d = 0
area_x = 10.0
area_y = 10.0
area_z = 10.0
self.speed_limit = 5.0
self.quat_limit = 1.0
self.arate_limit = 5.0
self.K_aero = np.zeros((3, 3))
self.safe_radius = (self.l + self.proppeler_d/2)*1.1
self.area = np.array([area_x, area_y, area_z])
# weight matrices (same for every axis)
if drone_type == 'crazyflie':
Qpos = 1/(0.5*self.area)**2
Qspeed = 1/(0.5*self.speed_limit)**2
Qquat = 100 # quaternion
Qarate = 1/np.deg2rad(100)**2 # angular rate
Rthrust = 1/((4*0.15)-self.mass*self.g)**2
tau_max = 0.15*self.l - 0*self.l
Rtorque_x = 1/tau_max**2
Rtorque_y = 1/tau_max**2
Rtorque_z = 1/(2*(0.005964552*0.15 + 1.563383e-5))**2
Rtorque = np.array([Rtorque_x, Rtorque_y, Rtorque_z])
else:
Qpos = 5000*np.ones(3) # postition
Qspeed = 800 # speed
Qquat = 100 # quaternion
Qarate = 10 # angular rate
Rthrust = 100 # collective thrust
Rtorque = 10*np.ones(3) # body torque
self.Q = np.diag(np.block([Qpos, Qspeed *
np.ones(3), Qquat*np.ones(4), Qarate*np.ones(3)]))
self.R = np.diag(np.block([Rthrust, Rtorque]))
self.nx = self.Q.shape[1] # dimension of state
self.nu = self.R.shape[1] # dimension of control
# state and control constraints
# area = int(10/2)
self.x_max = np.block([self.area, self.speed_limit*np.ones(3),
self.quat_limit*np.ones(4), self.arate_limit*np.ones(3)])
self.x_min = -self.x_max.copy()
self.x_min[2] = 0
# set q_w > 0 to lock the attitude representation and avoid flipping between positive and negative rotations
self.x_min[6] = 0
# x_max[2] = 10
if drone_type == 'crazyflie':
self.u_max = np.block(
[4*0.15, tau_max, tau_max, 2*(0.005964552*0.15 + 1.563383e-5)])
self.u_min = -self.u_max.copy()
self.u_min[0] = 0.0
else:
self.u_max = np.block([18.4167, 2.8543*np.ones(3)])
self.u_min = -self.u_max.copy()
self.u_min[0] = 0.2051
self.uf = np.zeros(self.nu)
self.uf[0] = self.mass*self.g
# time constraints
self.t0 = 0.0
self.tf_min = 0.0
if drone_type == 'crazyflie':
self.tf_max = 3.0*60.0
else:
self.tf_max = 15*60.0
# boundary constraints
self.x0 = np.zeros(self.nx)
self.x0[0] = -.8*self.area[0] # x
self.x0[1] = -.8*self.area[1] # y
self.x0[2] = -.8*self.area[2]+self.x_max[2] # z
self.x0[6] = 1.0 # q_w
self.xf = np.zeros(self.nx)
self.xf[0] = .8*self.area[0] # x
self.xf[1] = .8*self.area[1] # y
self.xf[2] = -.8*self.area[2]+self.x_max[2] # z
self.xf[6] = 1.0 # q_w
state = sym.symbols('x0:13')
control = sym.symbols('u0:4')
dynamics = self.__symbolic_dynamics(state, control)
self.eval_dynamics = sym.lambdify((state, control), dynamics)
self.eval_dynamics_state = []
for state_dyn in dynamics:
self.eval_dynamics_state.append(sym.lambdify((state, control), state_dyn))
def __eval_dynamics_old(self, state_t, control_t):
"""
Dynamics evaluation based on calculations in ps_build_dynamics
state vector state_t:
0: rx, 1: ry, 2: rz, 3: vx, 4: vy, 5: vz, 6: qw, 7: qx, 8: qy, 9: qz, 10: wx, 11: wy, 12: wz
control vector control_t:
0: T, 1: taux, 2: tauy, 3: tauz
"""
return np.array([state_t[3], state_t[4], state_t[5],
2*(state_t[7]*state_t[9]+state_t[8] *
state_t[6])*control_t[0]/(self.mass*((state_t[6]**2+state_t[7]**2+state_t[8]
** 2+state_t[9]**2))),
2*(state_t[8]*state_t[9]-state_t[7] *
state_t[6])*control_t[0]/(self.mass*((state_t[6]**2+state_t[7]**2+state_t[8]
** 2+state_t[9]**2))),
-self.g + control_t[0]*(-2*(state_t[7]**2+state_t[8]
** 2)/(state_t[6]**2+state_t[7]**2+state_t[8]
** 2+state_t[9]**2)+1)/self.mass,
- (state_t[6]*state_t[12])/2 - (state_t[8] *
state_t[10])/2 - (state_t[9]*state_t[12])/2,
(state_t[6]*state_t[10])/2 + (state_t[8] *
state_t[12])/2 - (state_t[9]*state_t[11])/2,
(state_t[6]*state_t[11])/2 - (state_t[7] *
state_t[12])/2 + (state_t[9]*state_t[10])/2,
(state_t[6]*state_t[12])/2 + (state_t[7] *
state_t[11])/2 - (state_t[8]*state_t[10])/2,
(control_t[1] + (self.Iy - self.Iz)
* state_t[11]*state_t[12])/self.Ix,
(control_t[2] + (self.Iz - self.Ix)
* state_t[10]*state_t[12])/self.Iy,
(control_t[3] + (self.Ix - self.Iy)*state_t[10]*state_t[11])/self.Iz])
def symbolic_dynamics(self):
"""This function calculates quadrotor dynamics using sympy."""
# state/control variables and parameters
qw, qx, qy, qz = sym.symbols('qw, qx, qy, qz', real=True)
wx, wy, wz = sym.symbols('wx, wy, wz', real=True)
x, y, z = sym.symbols('x, y, z', real=True)
vx, vy, vz = sym.symbols('vx, vy, vz', real=True)
m = sym.Symbol('m', positive=True)
g = sym.Symbol('g', positive=True)
Ix, Iy, Iz = sym.symbols('Ix, Iy, Iz', positive=True)
T = sym.Symbol('T', positive=True)
taux, tauy, tauz = sym.symbols('taux, tauy, tauz', real=True)
# vectors
u = sym.Matrix([T, taux, tauy, tauz])
q = sym.Matrix([qw, qx, qy, qz])
qv = sym.Matrix(q[1:])
w = sym.Matrix([wx, wy, wz])
r = sym.Matrix([x, y, z])
v = sym.Matrix([vx, vy, vz])
I = sym.diag(Ix, Iy, Iz)
x = sym.Matrix([r, v, q, w])
tau = sym.Matrix(u[1:])
# rotation matrix
R = sym.Matrix([[qw**2 + qx**2 - qy**2 - qz**2, 2*qx*qy - 2*qw*qz, 2*qw*qy + 2*qx*qz],
[2*qw*qz + 2*qx*qy, qw**2 - qx**2 + qy **
2 - qz**2, 2*qy*qz - 2*qw*qx],
[2*qx*qz - 2*qw*qy, 2*qw*qx + 2*qy*qz, qw**2 - qx**2 - qy**2 + qz**2]])
Ralt2 = R*sym.sqrt(q.dot(q))**(-2)
# alternative rotation matrix
s = sym.sqrt(q.dot(q))**(-2)
Ralt = sym.Matrix([[1-2*s*(q[2]**2+q[3]**2), 2*s*(q[1]*q[2]-q[3]*q[0]), 2*s*(q[1]*q[3]+q[2]*q[0])],
[2*s*(q[1]*q[2]+q[3]*q[0]), 1-2*s *
(q[1]**2+q[3]**2), 2*s*(q[2]*q[3]-q[1]*q[0])],
[2*s*(q[1]*q[3]-q[2]*q[0]), 2*s*(q[2]*q[3]+q[1]*q[0]), 1-2*s*(q[1]**2+q[2]**2)]])
# gamma matrix for rotational dynamics
Omega = sym.Matrix([[0, -qz, qy], [qz, 0, -qx], [-qy, qx, 0]])
Gamma = sym.Matrix.hstack(-qv, qw*sym.eye(3)-Omega)
# equations of motion
dr = v
dvalt2 = -sym.Matrix([0, 0, g]) + Ralt2*sym.Matrix([0, 0, T])/m
dvalt = -sym.Matrix([0, 0, g]) + Ralt*sym.Matrix([0, 0, T])/m
dv = -sym.Matrix([0, 0, g]) + R*sym.Matrix([0, 0, T])/m
dw = -I.inv() * (w.cross(I*w)) + I.inv()*tau
dq = 0.5*Gamma.T*w
# print(dvalt2)
# print(dvalt)
# print('---')
# print(dv)
# print(dq)
# print(dw)
return [dr,dvalt, dq, dw]
def __symbolic_dynamics(self, state, control):
"""Symbolic dynamics for injection to ps_solution evaluation and OCP definition in ps_control"""
# Unpack vectors
qw, qx, qy, qz = state[6:10]
q = sym.Matrix([qw, qx, qy, qz])
qv = sym.Matrix(q[1:])
w = sym.Matrix(state[10:])
v = sym.Matrix(state[3:6:])
thrust = control[0]
torque = sym.Matrix(control[1:])
I = np.diag([self.Ix, self.Iy, self.Iz])
# get rotational matrix
s = sym.sqrt(q.dot(q))**(-2)
R = sym.Matrix([[1-2*s*(q[2]**2+q[3]**2), 2*s*(q[1]*q[2]-q[3]*q[0]), 2*s*(q[1]*q[3]+q[2]*q[0])],
[2*s*(q[1]*q[2]+q[3]*q[0]), 1-2*s *
(q[1]**2+q[3]**2), 2*s*(q[2]*q[3]-q[1]*q[0])],
[2*s*(q[1]*q[3]-q[2]*q[0]), 2*s*(q[2]*q[3]+q[1]*q[0]), 1-2*s*(q[1]**2+q[2]**2)]])
# gamma matrix for rotational dynamics
Omega = sym.Matrix([[0, -qz, qy], [qz, 0, -qx], [-qy, qx, 0]])
Gamma = sym.Matrix.hstack(-qv, qw*sym.eye(3)-Omega)
# equations of motion
dr = v
if self.aerodynamics:
F_a = self.K_aero*thrust*R.T*v
else:
F_a = np.zeros(3)
dv = -sym.Matrix([0, 0, self.g]) + (R*(sym.Matrix([0, 0, thrust])+F_a))/self.mass
invI = np.linalg.pinv(I)
dq = 0.5*Gamma.T*w
dw = -invI * (w.cross(I*w)) + invI*torque
return sym.Matrix([dr, dv, dq, dw])[:]
class AttitudeController:
"""
Attitude controller class.
"""
def __init__(self, k_quat, k_rate, inertia_matrix):
"""
Initialize the attitude controller.
Args:
k_quat (float): Quaternion gain.
k_rate (float): Angular velocity gain.
inertia_matrix (ndarray): Inertia matrix of drone.
"""
self.k_quat = k_quat
self.k_rate = k_rate
self.inertia_matrix = inertia_matrix
self.prev_err = np.array([1, 0, 0, 0])
self.prev_time = 0.0
def update(self, quat_des, quat_cur, ang_vel_cur, ang_vel_des, curr_time):
"""
Calculates the control torque for the attitude controller.
Args:
quat_des (ndarray): Desired quaternion.
quat_cur (ndarray): Current quaternion.
ang_vel_cur (ndarray): Current angular velocity.
ang_vel_des (ndarray): Desired angular velocity.
curr_time (float): Current time.
Returns:
ndarray: Control torque.
"""
# Calculate quaternion error
q_err = quat_err(quat_cur, quat_des)
# Calculate derivative of quaternion error
d_err = quat_mult(0.5*q_err, np.concatenate([[0], ang_vel_cur]) - quat_mult(
quat_mult(quat_inv(q_err), np.concatenate([[0], ang_vel_des])), q_err))
norm = np.linalg.norm(d_err, 2)
if norm == 0.0:
d_err = np.zeros(4)
d_err[0] = 1
else:
d_err /= norm
d_err = (d_err - self.prev_err)/(curr_time - self.prev_time)
norm = np.linalg.norm(d_err, 2)
if norm == 0.0:
d_err = np.zeros(4)
d_err[0] = 1
else:
d_err /= norm
# Update previous error and time
self.prev_err = d_err
self.prev_time = curr_time
# Calculate control torque
torque = self.inertia_matrix@quat_dyn_matrix(
q_err) @ (-2*self.k_quat@self.k_rate@d_err - self.k_rate**2 @ (q_err - np.array([1, 0, 0, 0])))
torque += np.cross(ang_vel_cur, self.inertia_matrix@ang_vel_cur)
return torque
class PositionController:
"""
Position controller class.
"""
def __init__(self, k_position, k_speed, mass, gravity):
"""
Initialize the PositionController class.
Args:
k_position (np.ndarray): The position gain matrix.
k_speed (np.ndarray): The speed gain matrix.
mass (float): The mass of the drone.
gravity (float): The gravitational acceleration.
"""
self.k_position = k_position
self.k_speed = k_speed
self.mass = mass
self.gravity = gravity
def update(self, pos_des, pos_cur, vel_des, vel_cur):
"""
Calculate the control force based on desired and current positions and velocities.
Args:
pos_des (np.ndarray): The desired position.
pos_cur (np.ndarray): The current position.
vel_des (np.ndarray): The desired velocity.
vel_cur (np.ndarray): The current velocity.
Returns:
np.ndarray: The calculated control force.
"""
force = self.mass * (
self.k_position @ (pos_des - pos_cur)
+ self.k_speed @ (vel_des - vel_cur)
+ np.array([0, 0, self.gravity])
)
return force
def connect_controllers(pos_controller, att_controller, pos_des, pos_cur, vel_des, vel_cur, quat_cur, ang_vel_cur, ang_vel_des, psi_des, prev_quat_des, curr_time):
# Calculate desired force in local frame
force_L_des = pos_controller.update(pos_des, pos_cur, vel_des, vel_cur)
force_L_des_normed = force_L_des/np.linalg.norm(force_L_des, 2)
# Calculate desired attitude quaternion
force_b_des_normed = np.array([0, 0, 1])
q1 = 1/np.sqrt((2*(1+force_b_des_normed@force_L_des_normed)))*np.concatenate(
[[1+force_b_des_normed@force_L_des_normed], np.cross(force_b_des_normed, force_L_des_normed)])
q2 = np.array([np.cos(psi_des/2), 0, 0, np.sin(psi_des/2)])
q_des_ = quat_mult(q1, q2)
dot_prod = prev_quat_des@q_des_
if dot_prod < 0:
q_des = -q_des_/np.linalg.norm(q_des_, 2)
else:
q_des = q_des_/np.linalg.norm(q_des_, 2)
# Update previous desired quaternion
prev_quat_des = q_des
# R = quat2rotmat(q_des)
# force_b_des_ = (R)@force_L_des
force_b_des_ = quat_mult(
quat_mult(quat_inv(q_des), np.concatenate([[0], force_L_des])), q_des)
return force_b_des_[-1], prev_quat_des
def quat_dyn_matrix(q):
"""Matrix for quaternion dynamics described in Appendix equation (A.18) in dissertation"""
return np.array([[-q[1], q[0], -q[3], q[2]],
[-q[2], q[3], q[0], -q[1]],
[-q[3], -q[2], q[1], q[0]]])
def quat2rotmat(q):
"""
Converts a quaternion to a rotation matrix.
Parameters:
q (numpy.ndarray): The quaternion to be converted.
Returns:
numpy.ndarray: The resulting rotation matrix.
"""
s = np.sqrt(q.dot(q))**(-2)
R = np.array([[1-2*s*(q[2]**2+q[3]**2), 2*s*(q[1]*q[2]-q[3]*q[0]), 2*s*(q[1]*q[3]+q[2]*q[0])],
[2*s*(q[1]*q[2]+q[3]*q[0]), 1-2*s *
(q[1]**2+q[3]**2), 2*s*(q[2]*q[3]-q[1]*q[0])],
[2*s*(q[1]*q[3]-q[2]*q[0]), 2*s*(q[2]*q[3]+q[1]*q[0]), 1-2*s*(q[1]**2+q[2]**2)]])
return R
def quaternion2euler(q):
"""
Convert a quaternion to Euler angles.
Parameters:
q (numpy.ndarray): A 1-D array representing the quaternion in the order (w, x, y, z).
Returns:
tuple: A tuple containing the roll, pitch, and yaw angles in degrees.
"""
# Normalize the quaternion
q = q / np.linalg.norm(q)
# Extract the components of the quaternion
w, x, y, z = q
# Calculate roll (x-axis rotation)
sinr_cosp = 2 * (w * x + y * z)
cosr_cosp = 1 - 2 * (x**2 + y**2)
roll = np.arctan2(sinr_cosp, cosr_cosp)
# Calculate pitch (y-axis rotation)
sinp = 2 * (w * y - z * x)
sinp = np.clip(sinp, -1, 1)
pitch = np.arcsin(sinp)
# Calculate yaw (z-axis rotation)
siny_cosp = 2 * (w * z + x * y)
cosy_cosp = 1 - 2 * (y**2 + z**2)
yaw = np.arctan2(siny_cosp, cosy_cosp)
# Convert to degrees
roll = np.degrees(roll)
pitch = np.degrees(pitch)
yaw = np.degrees(yaw)
return roll, pitch, yaw
def quat_mult(q1, q2):
"""Return quaternion product of quaternion q1 and q2"""
s1, v1 = q1[0], q1[1:]
s2, v2 = q2[0], q2[1:]
s = s1*s2 - v1@v2.T
v = s1*v2 + s2*v1 + np.cross(v1, v2)
return np.hstack([[s], v])
def quat_err(quat_cur, quat_des):
# Calculate quaternion error
q_err = quat_mult(quat_inv(quat_des), quat_cur)
if q_err[0] < 0:
q_err = -q_err
return q_err/np.linalg.norm(q_err, 2)
def quat_distance(q1, q2):
"""
Calculate the distance between two quaternions.
Parameters:
q1 (numpy.ndarray): The first quaternion.
q2 (numpy.ndarray): The second quaternion.
Returns:
float: The distance between the two quaternions.
"""
q1 = q1/np.linalg.norm(q1)
q2 = q2/np.linalg.norm(q2)
dot_product = np.dot(q1, q2)
return abs(1 - dot_product)
def quat_distance_angle(q1, q2):
"""
Calculate the angle between two quaternions.
Parameters:
q1 (array-like): The first quaternion.
q2 (array-like): The second quaternion.
Returns:
float: The angle between the two quaternions.
"""
dot_product = np.dot(q1, q2)
angle = 2 * np.arccos(abs(dot_product))
return angle
def quat_inv(q):
"""Return inverse quaternion"""
return np.array([q[0], -q[1], -q[2], -q[3]])
def example():
drone = Drone()
k_quat = 0.7*np.eye(4)
k_rate = 3.5*np.eye(4)
k_pos_lat = 0.2
k_speed_lat = 7.0
k_position = np.diag([k_pos_lat, k_pos_lat, 1.0])
k_speed = np.diag([k_speed_lat, k_speed_lat, 0.8])
att_controller = AttitudeController(
k_quat, k_rate, np.diag([drone.Ix, drone.Iy, drone.Iz]))
pos_controller = PositionController(
k_position, k_speed, drone.mass, drone.g)
pos_des = drone.xf[:3]
pos_cur = drone.x0[:3]
# pos_des[1:] = pos_cur[1:]
vel_des = drone.xf[3:6]
vel_cur = drone.x0[3:6]
quat_des = drone.xf[6:10]
quat_cur = drone.x0[6:10]
# quat_cur = np.array([0.3, 0.8, 0.001, 0.002])
quat_cur = np.array([1,0,0,0])
quat_cur = quat_cur/np.linalg.norm(quat_cur, 2)
ang_vel_cur = drone.x0[10:13]
ang_vel_des = drone.xf[10:13]
psi_des = 0.0
prev_quat_des = quat_des
curr_time = 0.01
t_sample = 0.1
t_sample_fast = 0.01
t_max = 120
N_steps = int(t_max/t_sample)
state = np.concatenate([pos_cur, vel_cur, quat_cur, ang_vel_cur])
time = np.linspace(t_sample, t_max, N_steps)
state_mem = np.zeros([13, N_steps])
euler_mem = np.zeros([3, N_steps])
for i in range(N_steps):
pos_cur, vel_cur, quat_cur, ang_vel_cur = np.split(state, [3, 6, 10])
# torque = att_controller.update(np.array([1,0,0,0]), quat_cur, ang_vel_cur, [0,0,0],curr_time = curr_time)
thrust, quat_des = connect_controllers(
pos_controller, att_controller, pos_des, pos_cur, vel_des, vel_cur, quat_cur, ang_vel_cur, ang_vel_des, psi_des, prev_quat_des, curr_time)
# print(thrust, torque)
# thrust = 1.1*drone.mass*drone.g
for iter in range(int(round(t_sample/t_sample_fast))):
pos_cur, vel_cur, quat_cur, ang_vel_cur = np.split(state, [3, 6, 10])
# Calculate control torque
torque = att_controller.update(
quat_des, quat_cur, ang_vel_cur, ang_vel_des, curr_time+iter*t_sample_fast)
control = np.concatenate([[thrust], torque])
# control = np.clip(control, drone.u_min, drone.u_max)
dstate = np.array(drone.eval_dynamics(state, control))
state += t_sample_fast*dstate
state[6:10] = state[6:10]/np.linalg.norm(state[6:10], 2)
euler_mem[:, i] = quaternion2euler(state[6:10])
# state = np.clip(state, drone.x_min, drone.x_max)
state_mem[:, i] = state
curr_time += t_sample
print(np.linalg.norm(pos_cur-pos_des))
import matplotlib.pyplot as plt
plt.figure()
for i in range(0, 3):
plt.plot(time, state_mem[i, :])
plt.xlabel('t[s]')
plt.ylabel('$r[m]$')
plt.legend(['$r_x$', '$r_y$', '$r_z$'])
plt.figure()
for i in range(6, 10):
plt.plot(time, state_mem[i, :])
plt.xlabel('t[s]')
plt.ylabel('$q$')
plt.legend(['$q_w$', '$q_x$', '$q_y$', '$q_z$'])
plt.figure()
for i in range(0, 3):
plt.plot(time, euler_mem[i, :])
plt.xlabel('t[s]')
plt.ylabel('$\phi[deg]$')
plt.legend(['$\phi$', '$\theta$', '$\psi$'])
plt.show()
if __name__ == '__main__':
example()
# drone = Drone()
# Qdiag = np.diag(drone.Q)
# Qxq = np.diag(np.block([Qdiag[:6],Qdiag[-3:]]))
# print(Qdiag)
# print(Qxq)
# print(drone.R)
# print(drone.uf)