#!/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)