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python_2_12_parallel_quadcopter_quaternion_feedback_robustness.py
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#!/usr/bin/env python3
# -*- coding: utf-8 -*-
"""
MIT License
Copyright (c) 2022 Jongrae.K
Permission is hereby granted, free of charge, to any person obtaining a copy
of this software and associated documentation files (the "Software"), to deal
in the Software without restriction, including without limitation the rights
to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
copies of the Software, and to permit persons to whom the Software is
furnished to do so, subject to the following conditions:
The above copyright notice and this permission notice shall be included in all
copies or substantial portions of the Software.
THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
SOFTWARE.
"""
import numpy as np
from numpy import linspace
from scipy.integrate import solve_ivp
init_time = 0 # [s]
final_time = 120.0 # [s]
num_data = 1200
tout = linspace(init_time, final_time, num_data)
J_inertia = np.array([[0.005, -0.001, 0.004],
[-0.001, 0.006, -0.002],
[0.004, -0.002, 0.004]])
J_inv = np.linalg.inv(J_inertia)
C_T = 8.8e-7 # motor thruster coefficient [N/(rad/s)^2]
C_D = 11.3e-8 # motor drag coefficient [Nm/(rad/s)^2]
L_arm = 0.127 # length from the centre of quadcopter to the motor [m]
q0 = np.array([1,1,-1,1])/np.sqrt(4)
w0 = np.array([0.1,-0.2,0.1])
r0 = np.array([0,0,-50])
v0 = np.array([0,0,0])
state_0 = np.hstack((q0,w0,r0,v0))
num_MC = 300
#ts_all = np.inf*np.ones(num_MC)
#dJ_norm_all = np.inf*np.ones(num_MC)
#-----------------------------------------------------------------------------
# propellar & motor
#-----------------------------------------------------------------------------
def propeller_motor_actuator(C_T,C_D,L_arm,w_command):
# assume perfect motor angular velocity control
w_motor = w_command;
F_fblr = C_T*(w_motor**2)
tau_fblr = C_D*(w_motor**2)
F_motor = -np.sum(F_fblr)
M_motor = np.array([ L_arm*(F_fblr[2]-F_fblr[3]),
L_arm*(F_fblr[0]-F_fblr[1]),
np.sum(tau_fblr[0:2])-np.sum(tau_fblr[2::])])
FM_w_Motor = np.hstack([F_motor, M_motor, w_motor])
return FM_w_Motor
def propeller_motor_FM2w_conversion(F_M_Desired, C_T, C_D, L_arm):
inv_C_T = 1/C_T;
inv_C_D = 1/C_D;
inv_2_L_C_T = 2/(L_arm*C_T);
Conv_Mat = 0.25*np.array([
[-inv_C_T, 0, inv_2_L_C_T, inv_C_D],
[-inv_C_T, 0, -inv_2_L_C_T, inv_C_D],
[-inv_C_T, inv_2_L_C_T, 0, -inv_C_D],
[-inv_C_T, -inv_2_L_C_T, 0, -inv_C_D]])
w_motor_fblr_squared_desired = Conv_Mat@F_M_Desired
w_motor_fblr_squared_desired[w_motor_fblr_squared_desired<0] = 0.0
return w_motor_fblr_squared_desired
#-----------------------------------------------------------------------------
# quaternion feedback & altitude control
#-----------------------------------------------------------------------------
def quaternion_feedback_and_altitude_control(q_current, w_current, rv_current,
J_inertia, C_T, C_D, L_arm, C_BR,
mass_quadcopter, grv_acce):
zR_desired = -30 #[m]
zdotR_desired = 0 #[m/s]
K_qf = 0.01*np.eye(3);
C_qf = 0.001*np.eye(3)
k1 = 0.1
k2 = 0.5
q_13 = q_current[0:3]
w = w_current
Fmg_R = grv_acce*mass_quadcopter #[N]
Falt_R = k1*(zR_desired-rv_current[2])+k2*(zdotR_desired-rv_current[5])
F_desired_R = np.array([0,0,-Fmg_R+Falt_R])
F_desired_B = C_BR@F_desired_R;
u_qf = -K_qf@q_13 - C_qf@w - np.cross(w,J_inertia@w);
M_Desired = u_qf;
F_M_desired = np.hstack((F_desired_B[2], M_Desired))
w_motor_fblr_squared_desired = propeller_motor_FM2w_conversion(F_M_desired, C_T, C_D, L_arm)
w_motor_fblr_desired = np.sqrt(w_motor_fblr_squared_desired)
return w_motor_fblr_desired
#-----------------------------------------------------------------------------
# translational & rotational kinematics & dynamics
#-----------------------------------------------------------------------------
def drdt_linear_dynamics(rv_true, mass, grv_const, motor_force_in_R):
v = rv_true[3::]
drdt = v
dvdt = np.array([0,0,grv_const]) + motor_force_in_R/mass
drv_dt = np.hstack((drdt, dvdt))
return drv_dt
def dqdt_attitude_kinematics(q_true, w_true):
quat=q_true
wx=np.array([[0, -w_true[2], w_true[1]],
[w_true[2], 0, -w_true[0]],
[-w_true[1], w_true[0], 0]])
Omega_13 = np.hstack((-wx,np.resize(w_true,(3,1))))
Omega_4 = np.hstack((-w_true,0))
Omega = np.vstack((Omega_13, Omega_4))
dqdt = 0.5*(Omega@quat)
return dqdt
def dwdt_attitude_dynamics(w_true,J_inertia,inv_J_inertia, M_torque):
Jw = J_inertia@w_true
Jw_dot = -np.cross(w_true,Jw) + M_torque
dwdt = inv_J_inertia@Jw_dot
return dwdt
def dqdt_dwdt_drvdt(time,state,quadcopter_uav):
m_quadcopter = 0.49 #[kg]
grv_acce = 9.81 #[m/s^2]
q_current = state[0:4]
q_current = q_current/np.linalg.norm(q_current)
w_current = state[4:7]
rv_current = state[7::]
J_inertia = quadcopter_uav[0]
J_inv = quadcopter_uav[1]
C_T = quadcopter_uav[2]
C_D = quadcopter_uav[3]
L_arm = quadcopter_uav[4]
q_13 = q_current[0:3]
q4 = q_current[3]
q13x = np.array([[0, -q_13[2], q_13[1]],
[q_13[2], 0, -q_13[0]],
[-q_13[1], q_13[0], 0]])
C_BR = (q4**2-q_13@q_13)*np.eye(3) + 2*(q_13.reshape((3,1))*q_13.reshape((1,3)))-2*q4*q13x
#--------------------------------
# Begin: this part is controller
#--------------------------------
w_motor_fblr_desired = quaternion_feedback_and_altitude_control(q_current,
w_current, rv_current, J_inertia, C_T, C_D, L_arm, C_BR, m_quadcopter, grv_acce)
#--------------------------------
# End: this part is controller
#--------------------------------
# Motor Force & Torque
FM_w_Motor = propeller_motor_actuator(C_T, C_D, L_arm, w_motor_fblr_desired)
F_motor = np.array([0,0,FM_w_Motor[0]])
M_torque = FM_w_Motor[1:4]
motor_force_in_R = C_BR.transpose()@F_motor
dqdt = dqdt_attitude_kinematics(q_current, w_current)
dwdt = dwdt_attitude_dynamics(w_current, J_inertia, J_inv, M_torque)
drvdt = drdt_linear_dynamics(rv_current, m_quadcopter, grv_acce, motor_force_in_R)
dstate_dt = np.hstack((dqdt,dwdt,drvdt))
return dstate_dt
#-----------------------------------------------------------------------------
# Make it a function for Monte-Carlo Simulation
#-----------------------------------------------------------------------------
def robustness_analysis_MC(MC_id):
J_inertia = np.array([[0.005, -0.001, 0.004],
[-0.001, 0.006, -0.002],
[0.004, -0.002, 0.004]])
not_find_dJ = True
np.random.seed()
while not_find_dJ:
dJ = np.diag(0.002*np.random.randn(3))
J_inertia_perturbed = J_inertia + dJ
pd_cond = np.min(np.linalg.eig(J_inertia_perturbed)[0])>0
j3_cond = J_inertia_perturbed[0,0]+J_inertia_perturbed[1,1] > J_inertia_perturbed[2,2]
j2_cond = J_inertia_perturbed[0,0]+J_inertia_perturbed[2,2] > J_inertia_perturbed[1,1]
j1_cond = J_inertia_perturbed[1,1]+J_inertia_perturbed[2,2] > J_inertia_perturbed[0,0]
if pd_cond and j1_cond and j2_cond and j3_cond:
not_find_dJ = False
dJ_norm_all = np.linalg.norm(dJ)
J_inv_perturbed = np.linalg.inv(J_inertia_perturbed)
quadcopter_uav=(J_inertia_perturbed, J_inv_perturbed, C_T, C_D, L_arm)
sol = solve_ivp(dqdt_dwdt_drvdt, (init_time, final_time), state_0, t_eval=tout,
atol=1e-9, rtol=1e-6, max_step=0.01,
args=(quadcopter_uav,))
qout = sol.y[0:4,:]
q13=qout[0:3,:]
q13_norm = np.sqrt((np.sum(q13**2,axis=0)))
q13_ts = (q13_norm>0.01)*np.ones(num_data)
q13_ts = np.cumsum(q13_ts)
q13_ts = tout[q13_ts==q13_ts[-1]]
q13_ts = q13_ts[0]
ts_all = q13_ts
print(f'#{MC_id}: {np.linalg.norm(dJ):6.5f}, {q13_ts:4.2f}\n')
return ts_all, dJ_norm_all
#-----------------------------------------------------------------------------
# Run Parallel Monte-Carlo Simulation
#-----------------------------------------------------------------------------
from multiprocessing import Pool
import time
num_core = 4
start = time.time()
with Pool(num_core) as p:
result = p.map(robustness_analysis_MC, range(num_MC))
ts_all = np.array([i[0] for i in result])
dJ_norm_all = np.array([i[1] for i in result])
end = time.time()
print(f'{end-start} [seconds]')
#-----------------------------------------------------------------------------
# Plot Results
#-----------------------------------------------------------------------------
import matplotlib.pyplot as plt
fig0, (ax0) = plt.subplots(nrows=1,ncols=1)
ax0.plot(dJ_norm_all*1e3, ts_all,'.')
ax0.set_ylabel(r'$t_s$ [s]',fontsize=14)
ax0.set_xlabel(r'$||\Delta J||\times 10^{-3}$ [kg m$^2$]',fontsize=14)
fig0.set_figheight(6) # size in inches
fig0.set_figwidth(8) # size in inches
# xtick_list = np.array([0,2,4,6,8,10,12])*10
# ax.set_xticks(xtick_list)
# ax.set_xticklabels(xtick_list,fontsize=14)
# ytick_list = np.array([-1.0,0.0,1.0])
# ax.set_yticks(ytick_list)
# ax.set_yticklabels(ytick_list,fontsize=14)
# ax.legend(('$q_1$','$q_2$','$q_3$','$q_4$'),fontsize=14, loc='upper right')
# ax.axis((init_time,final_time,-1.1,1.1))
# ax.set_ylabel('quaternion',fontsize=14)
# ax1.plot(tout,wout[0,:],'r-',tout,wout[1,:],'b--',tout,wout[2,:],'m-.')
# ax1.set_xticks(xtick_list)
# ax1.set_xticklabels(xtick_list,fontsize=14)
# ax1.set_xlabel('time [s]',fontsize=14)
# ytick_list = np.array([-3.0,-2.0,-1.0,0.0,1.0,2.0,3.0])
# ax1.set_yticks(ytick_list)
# ax1.set_yticklabels(ytick_list,fontsize=14)
# ax1.legend(('$\omega_1$','$\omega_2$','$\omega_3$'),fontsize=14, loc='upper right')
# ax1.axis((init_time,final_time,-3,3))
# ax1.set_xlabel('time [s]',fontsize=14)
# ax1.set_ylabel('$\omega$ [rad/s]',fontsize=14)