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DubinsAirplaneFunctions.py
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# __DUBINSAIRPLANEFUNCTIONS__
# The functions here implement the 3D Dubins Airplane model with totally
# 16 cases of possible trajectories
#
# Authors:
# Kostas Alexis (konstantinos.alexis@mavt.ethz.ch)
from __future__ import division
import numpy as np
from math import tan, sin, cos, atan2, fmod, acos, asin, pow, sqrt, fabs,atan
from ElementaryFunctions import my_max, my_min
pi = np.pi
# create the results dictionary
DubinsAirplaneSolution = { }
DubinsAirplaneSolution['case'] = 0
DubinsAirplaneSolution['p_s'] = np.zeros((3,1))
DubinsAirplaneSolution['angl_s'] = 0
DubinsAirplaneSolution['p_e'] = np.zeros((3,1))
DubinsAirplaneSolution['R'] = 0
DubinsAirplaneSolution['gamma'] = 0
DubinsAirplaneSolution['L'] = 0
DubinsAirplaneSolution['c_s'] = np.zeros((3,1))
DubinsAirplaneSolution['psi_s'] = 0
DubinsAirplaneSolution['lamda_s'] = 0
DubinsAirplaneSolution['lamda_si'] = 0
DubinsAirplaneSolution['k_s'] = 0
DubinsAirplaneSolution['c_ei'] = np.zeros((3,1))
DubinsAirplaneSolution['c_si'] = np.zeros((3,1))
DubinsAirplaneSolution['psi_ei'] = 0
DubinsAirplaneSolution['lamda_ei'] = 0
DubinsAirplaneSolution['psi_si'] = 0
DubinsAirplaneSolution['k_ei'] = 0
DubinsAirplaneSolution['c_e'] = 0
DubinsAirplaneSolution['k_si'] = 0
DubinsAirplaneSolution['psi_e'] = 0
DubinsAirplaneSolution['lamda_e'] = 0
DubinsAirplaneSolution['k_e'] = 0
DubinsAirplaneSolution['w_s'] = np.zeros((3,1))
DubinsAirplaneSolution['q_s'] = np.zeros((3,1))
DubinsAirplaneSolution['w_si'] = np.zeros((3,1))
DubinsAirplaneSolution['q_si'] = np.zeros((3,1))
DubinsAirplaneSolution['w_l'] = np.zeros((3,1))
DubinsAirplaneSolution['q_l'] = np.zeros((3,1))
DubinsAirplaneSolution['w_ei'] = np.zeros((3,1))
DubinsAirplaneSolution['q_ei'] = np.zeros((3,1))
DubinsAirplaneSolution['w_e'] = np.zeros((3,1))
DubinsAirplaneSolution['q_e'] = np.zeros((3,1))
def roty(theta=None):
# Rotation around y
R = np.array([ [cos(theta), 0, sin(theta)],
[0, 1, 0],
[-sin(theta), 0, cos(theta)] ])
return R
def rotz(theta=None):
# Rotation around z
R = np.array([ [cos(theta), -sin(theta), 0],
[sin(theta), cos(theta), 0],
[0, 0, 1] ])
return R
def computeDubinsRSR(R=None, crs=None, cre=None, anglstart=None,anglend=None):
# Compute Dubins RSR
theta = atan2( cre[1]-crs[1], cre[0]-crs[0] )
L = np.linalg.norm( crs[0:2]-cre[0:2],ord=2 ) + R* fmod( 2*pi+fmod(theta-pi/2,2*pi )-fmod( anglstart-pi/2,2*pi),2*pi ) + R*fmod( 2*pi+fmod(anglend-pi/2,2*pi )-fmod( theta-pi/2,2*pi),2*pi )
return L
def computeDubinsLSL(R=None, cls=None, cle=None, anglstart=None, anglend=None):
# Compute Dubins LSL
theta = atan2( cle[1]-cls[1], cle[0]-cls[0] )
L = np.linalg.norm(cls[0:2]-cle[0:2]) + R*fmod( 2*pi-fmod(theta+pi/2,2*pi)+fmod(anglstart+pi/2,2*pi),2*pi ) + R*fmod( 2*pi-fmod(anglend+pi/2,2*pi)+fmod(theta+pi/2,2*pi),2*pi )
return L
def computeDubinsLSR(R=None, cls=None, cre=None, anglstart=None, anglend=None):
# Compute Dubins LSR
ell = np.linalg.norm( cre[0:2]-cls[0:2],ord=2 )
theta = atan2( cre[1]-cls[1],cre[0]-cls[0] )
acos_value = 2 * R / ell
if fabs( acos_value ) > 1:
flag_zero = 1
else:
flag_zero = 0
acos_value = my_max( acos_value,-1 )
acos_value = my_min( acos_value,1 )
if ell == 0:
theta2 = 0
else:
theta2 = acos( acos_value )
if flag_zero == 1:
theta2 = 0
if theta2 == 0:
L = pow(10.0,8)
else:
L = sqrt( pow(ell,2) - 4*pow(R,2) ) + R*fmod( 2*pi-fmod(theta+theta2,2*pi ) + fmod( anglstart+pi/2,2*pi),2*pi ) + R*fmod( 2*pi-fmod(theta+theta2-pi,2*pi)+fmod(anglend-pi/2,2*pi),2*pi )
return L
def computeDubinsRSL(R=None, crs=None, cle=None, anglstart=None, anglend=None):
# Compute Dubins RSL
ell = np.linalg.norm( cle[0:2]-crs[0:2],ord=2 )
theta = atan2( cle[1]-crs[1],cle[0]-crs[0] )
asin_value = 2 * R / ell
if fabs( asin_value ) > 1:
flag_zero = 1
else:
flag_zero = 0
asin_value = my_max( asin_value, -1 )
asin_value = my_min( asin_value, 1 )
if ell == 0:
theta2 = 0
else:
theta2 = theta - pi/2 + asin( asin_value )
if theta2 == 0:
L = pow( 10.0, 8 )
else:
L = sqrt( fabs(pow(ell,2)-4 * pow(R,2)) ) + R * fmod( 2 * pi + fmod( theta2, 2 * pi )-fmod( anglstart-pi/2, 2 * pi ), 2 * pi ) + R * fmod( 2 * pi + fmod( theta2 + pi, 2 * pi )-fmod( anglend + pi / 2, 2 * pi ), 2 * pi )
return L
def computeOptimalRadius(zs=None, anglstart=None, ze=None, anglend=None, R_min=None, gamma_max=None, idx=None, k=None, hdist=None):
# Compute Optimal Radius
R1 = R_min
R2 = 2 * R_min
R = ( R1 + R2 ) / 2
if idx == 1:
error = 1
while fabs( error ) > 0.1:
crs = zs + R * np.dot( rotz( pi / 2 ),np.array( [cos(anglstart), sin(anglstart), 0] ).T )
cre = ze + R * np.dot( rotz( pi / 2 ), np.array( [cos(anglend), sin(anglend), 0] ).T )
L = computeDubinsRSR(R, crs, cre, anglstart, anglend)
error = ( L + 2 * pi * k * R ) - fabs( hdist ) / tan( gamma_max )
if error > 0:
R2 = R
else:
R1 = R
R= ( R1 + R2 ) / 2
elif idx == 2:
error = 1
while fabs( error ) > 0.1:
crs = zs + R * np.dot( rotz( pi / 2 ),np.array( [cos(anglstart), sin(anglstart), 0] ).T )
cle = ze + R * np.dot( rotz( -pi/2 ),np.array( [cos(anglend), sin(anglend), 0] ).T )
L = computeDubinsRSL( R, crs, cle, anglstart, anglend )
error = ( L + 2 * pi * k * R ) * tan( gamma_max ) - fabs( hdist )
if error > 0:
R2 = R
else:
R1 = R
R = ( R1 + R2 ) / 2
elif idx == 3:
error = 1
while fabs( error ) > 0.1:
cls = zs + R * np.dot( rotz( -pi / 2 ), np.array( [cos(anglstart), sin(anglstart), 0] ).T )
cre = ze + R * np.dot( rotz( pi / 2 ), np.array( [cos(anglend), sin(anglend), 0] ).T )
L = computeDubinsLSR( R, cls, cre, anglstart, anglend )
error = ( L + 2 * pi * k * R ) * tan( gamma_max ) - fabs( hdist )
if error > 0:
R2 = R
else:
R1 = R
R = ( R1 + R2 ) / 2
elif idx == 4:
error = 1
while fabs( error ) > 0.1:
cls = zs + R * np.dot( rotz( -pi / 2 ), np.array( [cos(anglstart), sin(anglstart), 0] ).T )
cle = ze + R * np.dot( rotz( -pi / 2 ), np.array( [cos(anglend), sin(anglend), 0] ).T )
L = computeDubinsLSL( R, cls, cle, anglstart, anglend )
error = ( L + 2 * pi * k * R ) * tan( gamma_max ) - fabs( hdist )
if error > 0:
R2 = R
else:
R1 = R
R = ( R1 + R2 ) / 2
return R
def MinTurnRadius_DubinsAirplane(V=None,phi_max=None):
# Compute Minimum Turning Radius
g = 9.8065
Rmin = pow( V,2 ) / (g * tan( phi_max ) )
return Rmin
def addSpiralBeginning(zs=None, anglstart=None, ze=None, anglend=None, R_min=None, gamma_max=None, idx=None, hdist=None):
# Add Spiral in the Dubins Airplane Path beginning
cli = np.zeros((3,1))
cri = np.zeros((3,1))
zi = np.zeros((3,1))
anglinter = 0
L = 0
ci = np.zeros((3,1))
psii = 0
psi1 = 0
psi2 = 2 * pi
psi = ( psi1 + psi2 ) / 2
if idx == 1: # RLSR
crs = zs + R_min * np.dot( rotz( pi/2 ), np.array( [cos(anglstart), sin(anglstart), 0] ).T )
cre = ze + R_min * np.dot( rotz( pi/2 ), np.array( [cos(anglend), sin(anglend), 0] ).T )
L = computeDubinsRSR( R_min, crs, cre, anglstart, anglend )
error = L - fabs( hdist / tan( gamma_max ) )
while fabs( error ) > 0.001:
zi = crs + np.dot( rotz( psi ),( zs-crs ) )
anglinter = anglstart + psi
cli = zi + R_min * np.dot( rotz( -pi/2 ), np.array( [cos(anglinter), sin(anglinter), 0] ).T )
L = computeDubinsLSR( R_min, cli, cre, anglinter, anglend )
error = ( L + fabs( psi ) * R_min ) - fabs( hdist / tan( gamma_max ) )
if error > 0:
psi2 = (179*psi2+psi)/180
else:
psi1 = (179*psi1+psi)/180
psi = ( psi1 + psi2 ) / 2
zi = crs + np.dot( rotz( psi ), ( zs-crs ) )
anglinter = anglstart + psi
L = L + fabs( psi ) * R_min
ci = cli
psii = psi
elif idx == 2: # RLSL
crs = zs + R_min * np.dot( rotz( pi / 2 ), np.array( [cos(anglstart), sin(anglstart), 0] ).T )
cle = ze + R_min * np.dot( rotz( -pi / 2 ), np.array( [cos(anglend), sin(anglend), 0] ).T )
L = computeDubinsRSL( R_min, crs, cle, anglstart, anglend )
error = L - fabs( hdist / tan( gamma_max ) )
while fabs( error ) > 0.001:
zi = crs + np.dot( rotz( psi ), ( zs-crs ) )
anglinter = anglstart + psi
cli = zi + R_min * np.dot( rotz( -pi / 2 ), np.array( [cos(anglinter), sin(anglinter), 0] ).T )
L = computeDubinsLSL( R_min, cli, cle, anglinter, anglend )
error = ( L + fabs( psi ) * R_min ) - fabs( hdist / tan( gamma_max ) )
if error > 0:
psi2 = (179*psi2+psi)/180
else:
psi1 = (179*psi1+psi)/180
psi = ( psi1 + psi2 ) / 2
zi = crs + np.dot( rotz( psi ), ( zs-crs ) )
anglinter = anglstart + psi
L = L + fabs( psi ) * R_min
ci = cli
psii = psi
elif idx == 3: # LRSR
cls = zs + R_min * np.dot( rotz( -pi/2 ), np.array( [cos(anglstart), sin(anglstart), 0] ).T )
cre = ze + R_min * np.dot( rotz( pi/2 ), np.array( [cos(anglend), sin(anglend), 0] ).T )
L = computeDubinsLSR( R_min, cls, cre, anglstart, anglend )
error = L - fabs( hdist / tan( gamma_max ) )
while fabs( error ) > 0.001:
zi = cls + np.dot( rotz( -psi ), ( zs-cls ) )
anglinter = anglstart - psi
cri = zi + R_min * np.dot( rotz( pi / 2 ), np.array( [cos(anglinter), sin(anglinter), 0] ).T )
L = computeDubinsRSR( R_min, cri, cre, anglinter, anglend )
error = ( L + fabs( psi ) * R_min ) - fabs( hdist / tan( gamma_max ) )
if error > 0:
psi2 = (179*psi2+psi)/180
else:
psi1 = (179*psi1+psi)/180
psi = ( psi1 + psi2 ) / 2
zi = cls + np.dot( rotz( -psi ), ( zs-cls ) )
anglinter = anglstart - psi
L = L + fabs( psi ) * R_min
ci = cri
psii = psi
elif idx == 4: # LRSL
cls = zs + R_min * np.dot( rotz( -pi/2 ), np.array( [cos(anglstart), sin(anglstart), 0] ).T )
cle = ze + R_min * np.dot( rotz( -pi/2 ), np.array( [cos(anglend), sin(anglend), 0] ).T )
# above modified by liucz 2015-10-12, fix spell mistake cre -> cle
# origin is "cre = ze + R_min * np.dot( rotz( -pi/2 ), np.array( [cos(anglend), sin(anglend), 0] ).T )"
L = computeDubinsLSL( R_min, cls, cle, anglstart, anglend )
error = L - fabs( hdist / tan( gamma_max ) )
while fabs( error ) > 0.001:
zi = cls + np.dot( rotz( -psi ), ( zs-cls ) )
anglinter = anglstart - psi
cri = zi + R_min * np.dot( rotz( pi / 2 ), np.array( [cos(anglinter), sin(anglinter), 0 ] ).T )
# above is modified by licz 2015-10-12, fix written mistake np.array -> np.dot
# origin is "cri = zi + R_min * np.array( rotz( pi / 2 ), np.array( [cos(anglinter), sin(anglinter), 0 ] ).T )"
L = computeDubinsRSL( R_min, cri, cle, anglinter, anglend )
error = ( L + fabs( psi ) * R_min ) - fabs( hdist / tan( gamma_max) )
if error > 0:
psi2 = (179*psi2+psi)/180
else:
psi1 = (179*psi1+psi)/180
psi = ( psi1 + psi2 ) / 2
zi = cls + np.dot( rotz( -psi ), ( zs-cls ) )
anglinter = anglstart - psi
L = L + fabs( psi ) * R_min
ci = cri
psii = psi
return zi, anglinter, L, ci, psii
def addSpiralEnd(zs=None, anglstart=None, ze=None, anglend=None, R_min=None, gamma_max=None, idx=None, hdist=None):
# Add Spiral at the end of the Dubins Airplane path
cli = np.zeros((3,1))
cri = np.zeros((3,1))
zi = np.zeros((3,1))
anglinter = 0
L = 0
ci = np.zeros((3,1))
psii = 0
psi1 = 0
psi2 = 2 * pi
psi = ( psi1 + psi2 ) / 2
if idx == 1: # RSLR
crs = zs + R_min * np.dot( rotz( pi / 2 ), np.array( [cos(anglstart), sin(anglstart), 0] ).T )
cre = ze + R_min * np.dot( rotz( pi/2 ), np.array( [cos(anglend), sin(anglend), 0] ).T )
L = computeDubinsRSR( R_min, crs, cre, anglstart, anglend )
error = L - fabs( hdist / tan( gamma_max ) )
while fabs( error ) > 0.001:
zi = cre + np.dot( rotz( -psi ), ( ze-cre ) )
anglinter = anglend - psi
cli = zi + R_min * np.dot( rotz( -pi / 2 ), np.array( [cos(anglinter), sin(anglinter), 0] ).T )
L = computeDubinsRSL( R_min, crs, cli, anglstart, anglinter )
error = ( L + fabs( psi ) * R_min ) - fabs( hdist / tan( gamma_max ) )
if error > 0:
psi2 = (179*psi2+psi)/180
else:
psi1 = (179*psi1+psi)/180
psi = ( psi1 + psi2 ) / 2
zi = cre + np.dot( rotz( -psi ), ( ze-cre ) )
anglinter = anglend - psi
L = L + abs( psi ) * R_min
ci = cli
psii = psi
elif idx == 2: # RSRL
crs = zs + R_min * np.dot( rotz( pi / 2 ), np.array( [cos(anglstart), sin(anglstart), 0] ).T )
cle = ze + R_min * np.dot( rotz( -pi/2 ), np.array( [cos(anglend), sin(anglend), 0] ).T )
L = computeDubinsRSL( R_min, crs, cle, anglstart, anglend )
error = L - fabs( hdist / tan( gamma_max ) )
while fabs( error ) > 0.001:
zi = cle + np.dot( rotz( psi ), ( ze-cle ) )
anglinter = anglend + psi
cri = zi + R_min * np.dot( rotz( pi / 2 ), np.array( [cos(anglinter), sin(anglinter), 0] ).T )
L = computeDubinsRSR( R_min, crs, cri, anglstart, anglinter )
error = ( L + fabs( psi ) * R_min ) - fabs( hdist / tan( gamma_max ) )
if error > 0:
psi2 = (179*psi2+psi)/180
else:
psi1 = (179*psi1+psi)/180
psi = ( psi1 + psi2 ) / 2
zi = cle + np.dot( rotz( psi ), ( ze-cle ) )
anglinter = anglend + psi
cri = zi + R_min * np.dot( rotz( pi / 2 ), np.array( [cos(anglinter), sin(anglinter), 0] ).T )
L = L + fabs( psi ) * R_min
ci = cri
psii = psi
elif idx == 3: # LSLR
cls = zs + R_min * np.dot( rotz( -pi / 2 ), np.array( [cos(anglstart), sin(anglstart), 0] ).T )
cre = ze + R_min * np.dot( rotz( pi / 2 ), np.array( [cos(anglend), sin(anglend), 0] ).T )
L = computeDubinsLSR( R_min, cls, cre, anglstart, anglend )
error = L - fabs( hdist / tan( gamma_max ))
while fabs( error ) > 0.001:
zi = cre + np.dot( rotz( -psi ), ( ze-cre ) )
anglinter = anglend - psi
cli = zi + R_min * np.dot( rotz( -pi / 2 ), np.array( [cos(anglinter), sin(anglinter), 0] ).T )
L = computeDubinsLSL( R_min, cls, cli, anglstart, anglinter )
error = ( L + fabs( psi ) * R_min ) - fabs( hdist / tan( gamma_max ) )
if error > 0:
psi2 = (179*psi2+psi)/180
else:
psi1 = (179*psi1+psi)/180
psi = ( psi1 + psi2 ) / 2
zi = cre + np.dot( rotz( -psi ), ( ze-cre ) )
anglinter = anglend - psi
L = L + fabs( psi ) * R_min
ci = cli
psii = psi
elif idx == 4:
cls = zs + R_min * np.dot( rotz( -pi/2 ), np.array( [cos(anglstart), sin(anglstart), 0] ).T )
cle = ze + R_min * np.dot( rotz( -pi/2 ), np.array( [cos(anglend), sin(anglend), 0] ).T )
L = computeDubinsLSL( R_min, cls, cle, anglstart, anglend )
error = L - fabs( hdist / tan( gamma_max ) )
while fabs( error ) > 0.001:
zi = cle + np.dot( rotz( psi ), ( ze-cle ) )
anglinter = anglend + psi
cri = zi + R_min * np.dot( rotz( pi / 2 ), np.array( [cos(anglinter), sin(anglinter), 0] ).T )
L = computeDubinsLSR( R_min, cls, cri, anglstart, anglinter )
error = ( L + fabs( psi ) * R_min ) - fabs( hdist / tan( gamma_max ) )
if error > 0:
psi2 = (179*psi2+psi)/180
else:
psi1 = (179*psi1+psi)/180
psi = ( psi1 + psi2 ) / 2
zi = cle + np.dot( rotz( psi ), ( ze-cle ))
anglinter = anglend + psi
L = L + fabs( psi ) * R_min
ci = cri
psii = psi
return zi, anglinter, L, ci, psii
def DubinsAirplanePath(init_conf=None, final_conf=None, R_min=None, gamma_max=None):
# Compute the Dubins Airplane path
zs = (init_conf[0:3]).T
anglstart = init_conf[3]
ze = (final_conf[0:3]).T
anglend = final_conf[3]
DubinsAirplaneSolution['p_s'] = zs
DubinsAirplaneSolution['angl_s'] = anglstart
DubinsAirplaneSolution['p_e'] = ze
DubinsAirplaneSolution['angl_e'] = anglend
crs = zs + R_min*np.dot(rotz(pi/2), np.array([cos(anglstart), sin(anglstart), 0]).T)
cls = zs + R_min*np.dot(rotz(-pi/2),np.array([cos(anglstart), sin(anglstart), 0]).T)
cre = ze + R_min*np.dot(rotz(pi/2),np.array([cos(anglend), sin(anglend), 0]).T)
cle = ze + R_min*np.dot(rotz(-pi/2),np.array([cos(anglend), sin(anglend), 0]).T)
# compute L1, L2, L3, L4
L1 = computeDubinsRSR(R_min, crs, cre, anglstart, anglend)
L2 = computeDubinsRSL(R_min, crs, cle, anglstart, anglend)
L3 = computeDubinsLSR(R_min, cls, cre, anglstart, anglend)
L4 = computeDubinsLSL(R_min, cls, cle, anglstart, anglend)
# L is the minimum distance
L = np.amin(np.array([L1, L2, L3, L4]))
idx = np.where(np.array([L1,L2,L3,L4])==L)[0][0] + 1
hdist = -(ze[2] - zs[2])
if fabs(hdist) <= L*tan(gamma_max):
gam = atan(hdist/L)
DubinsAirplaneSolution['case'] = 1
DubinsAirplaneSolution['R'] = R_min
DubinsAirplaneSolution['gamma'] = gam
DubinsAirplaneSolution['L'] = L/cos(gam)
DubinsAirplaneSolution['k_s'] = 0
DubinsAirplaneSolution['k_e'] = 0
elif fabs(hdist) >= (L+2*pi*R_min)*tan(gamma_max):
k = np.floor( (fabs(hdist)/tan(gamma_max) - L)/(2*pi*R_min))
if hdist >= 0:
DubinsAirplaneSolution['k_s'] = k
DubinsAirplaneSolution['k_e'] = 0
else:
DubinsAirplaneSolution['k_s'] = 0
DubinsAirplaneSolution['k_e'] = k
# find optimal turning radius
R = computeOptimalRadius(zs, anglstart, ze, anglend, R_min, gamma_max, idx, k, hdist)
# recompute the centers of spirals and Dubins path length with new R
crs = zs + R*np.dot(rotz(pi/2), np.array( [cos(anglstart), sin(anglstart), 0] ).T )
cls = zs + R*np.dot(rotz(-pi/2), np.array( [cos(anglstart), sin(anglstart), 0] ).T )
cre = ze + R*np.dot(rotz(pi/2), np.array( [cos(anglend), sin(anglend), 0] ).T )
cle = ze + R*np.dot(rotz(-pi/2), np.array( [cos(anglend), sin(anglend), 0] ).T )
if idx == 1:
L = computeDubinsRSR( R, crs, cre, anglstart, anglend )
elif idx == 2:
L = computeDubinsRSL( R, crs, cle, anglstart, anglend )
elif idx == 3:
L = computeDubinsLSR( R, cls, cre, anglstart, anglend )
elif idx == 4:
L = computeDubinsLSL( R, cls, cle, anglstart, anglend )
DubinsAirplaneSolution['case'] = 1
DubinsAirplaneSolution['R'] = R
gam = np.sign( hdist ) * gamma_max
DubinsAirplaneSolution['gamma'] = gam
DubinsAirplaneSolution['L'] = ( L + 2 * pi * k * R ) / cos( gamma_max )
else:
gam = np.sign( hdist ) * gamma_max
if hdist > 0:
zi, chii, L, ci, psii = addSpiralBeginning( zs, anglstart, ze, anglend, R_min, gam, idx, hdist )
DubinsAirplaneSolution['case'] = 2
else:
zi, chii, L, ci, psii = addSpiralEnd( zs, anglstart, ze, anglend, R_min, gam, idx, hdist )
DubinsAirplaneSolution['case'] = 3
DubinsAirplaneSolution['R'] = R_min
DubinsAirplaneSolution['gamma'] = gam
DubinsAirplaneSolution['L'] = L / cos( gamma_max )
e1 = np.array( [1, 0, 0] ).T
R = DubinsAirplaneSolution['R']
if np.isscalar(DubinsAirplaneSolution['case']):
pass
else:
print('### Error')
if DubinsAirplaneSolution['case'] == 1: # spiral-line-spiral
if idx == 1: # right-straight-right
theta = atan2( cre[1]-crs[1], cre[0]-crs[0])
dist1 = R*fmod(2*pi+fmod(theta-pi/2,2*pi)-fmod(anglstart-pi/2,2*pi),2*pi) + 2*pi*R*DubinsAirplaneSolution['k_s']
dist2 = R*fmod(2*pi+fmod(anglend-pi/2,2*pi)-fmod(theta-pi/2,2*pi),2*pi) + 2*pi*R*DubinsAirplaneSolution['k_e']
w1 = crs + DubinsAirplaneSolution['R']*np.dot(rotz(theta-pi/2),e1.T).T + np.array([0,0,-dist1*tan(gam)]).T
w2 = cre + DubinsAirplaneSolution['R']*np.dot(rotz(theta-pi/2),e1.T).T - np.array([0,0,-dist2*tan(gam)]).T
q1 = (w2-w1)/np.linalg.norm(w2-w1,ord=2) # direction of line
DubinsAirplaneSolution['c_s'] = crs
DubinsAirplaneSolution['psi_s'] = anglstart-pi/2
DubinsAirplaneSolution['lamda_s'] = 1
# end spiral
DubinsAirplaneSolution['c_e'] = cre-np.array([0,0,-dist2*tan(gam)])
DubinsAirplaneSolution['psi_e'] = theta-pi/2
DubinsAirplaneSolution['lamda_e'] = 1
# hyperplane H_s: switch from first spiral to line
DubinsAirplaneSolution['w_s'] = w1
DubinsAirplaneSolution['q_s'] = q1
# hyperplane H_l: switch from line to last spiral
DubinsAirplaneSolution['w_l'] = w2
DubinsAirplaneSolution['q_l'] = q1
# hyperplane H_e: end of Dubins path
DubinsAirplaneSolution['w_e'] = ze
DubinsAirplaneSolution['q_e'] = np.dot(rotz(anglend), np.array([1,0,0]).T)
elif idx == 2: # right-straight-left
ell = np.linalg.norm(cle[0:2] - crs[0:2],ord=2)
theta = atan2(cle[1]-crs[1], cle[0]-crs[0])
theta2 = theta - pi/2 + asin(2*R/ell)
dist1 = R*fmod(2*pi+fmod(theta2,2*pi)-fmod(anglstart-pi/2,2*pi),2*pi) + 2*pi*R*DubinsAirplaneSolution['k_s']
dist2 = R*fmod(2*pi+fmod(theta2+pi,2*pi)-fmod(anglend+pi/2,2*pi),2*pi) + 2*pi*R*DubinsAirplaneSolution['k_e']
w1 = crs + R*np.dot(rotz(theta2), e1.T).T + np.array([0, 0, -dist1*tan(gam)]).T
w2 = cle + R*np.dot(rotz(theta2+pi),e1.T).T - np.array([0,0,-dist2*tan(gam)]).T
q1 = (w2-w1)/np.linalg.norm(w2-w1,ord=2)
# start spiral
DubinsAirplaneSolution['c_s'] = crs
DubinsAirplaneSolution['psi_s'] = anglstart-pi/2
DubinsAirplaneSolution['lamda_s'] = 1
# end spiral
DubinsAirplaneSolution['c_e'] = cle - np.array([0,0,-dist2*tan(gam)]).T
DubinsAirplaneSolution['psi_e'] = theta2+pi
DubinsAirplaneSolution['lamda_e'] = -1
# hyperplane H_s: switch from first spiral to line
DubinsAirplaneSolution['w_s'] = w1
DubinsAirplaneSolution['q_s'] = q1
# hyperplane H_l: switch from line to end spiral
DubinsAirplaneSolution['w_l'] = w2
DubinsAirplaneSolution['q_l'] = q1
# hyperplane H_e: end of Dubins path
DubinsAirplaneSolution['w_e'] = ze
DubinsAirplaneSolution['q_e'] = np.dot(rotz(anglend),np.array([1,0,0]).T)
elif idx == 3: # left-straight-right
ell = np.linalg.norm(cre[0:2]-cls[0:2],ord=2)
theta = atan2( cre[1]-cls[1],cre[0]-cls[0])
theta2 = acos(2*R/ell)
dist1 = R*fmod(2*pi-fmod(theta+theta2,2*pi) + fmod(anglstart+pi/2,2*pi),2*pi) + 2*pi*R*DubinsAirplaneSolution['k_s']
dist2 = R*fmod(2*pi-fmod(theta+theta2-pi,2*pi)+fmod(anglend-pi/2,2*pi),2*pi) + 2*pi*R*DubinsAirplaneSolution['k_e']
w1 = cls + R*np.dot(rotz(theta+theta2),e1.T).T + np.array([0, 0, -dist1*tan(gam)]).T
w2 = cre + R*np.dot(rotz(-pi+theta+theta2),e1.T).T - np.array([0, 0, -dist2*tan(gam)]).T
q1 = (w2-w1)/np.linalg.norm(w2-w1,ord=2)
# start spiral
DubinsAirplaneSolution['c_s'] = cls
DubinsAirplaneSolution['psi_s'] = anglstart+pi/2
DubinsAirplaneSolution['lamda_s'] = -1
# end spiral
DubinsAirplaneSolution['c_e'] = cre - np.array([0,0,-dist2*tan(gam)]).T
DubinsAirplaneSolution['psi_e'] = fmod(theta+theta2-pi,2*pi)
DubinsAirplaneSolution['lamda_e'] = 1
# hyperplane H_s: switch from first spiral to line
DubinsAirplaneSolution['w_s'] = w1
DubinsAirplaneSolution['q_s'] = q1
# hyperplane H_l: switch from line to end spiral
DubinsAirplaneSolution['w_l'] = w2
DubinsAirplaneSolution['q_l'] = q1
# hyperplane H_e: end of Dubins path
DubinsAirplaneSolution['w_e'] = ze
DubinsAirplaneSolution['q_e'] = np.dot(rotz(anglend), np.array([1, 0, 0]).T)
elif idx ==4: # left-straight-left
theta = atan2(cle[1] -cls[1], cle[0] - cls[0])
dist1 = R*fmod(2*pi-fmod(theta+pi/2,2*pi)+fmod(anglstart+pi/2,2*pi),2*pi) + 2*pi*R*DubinsAirplaneSolution['k_s']
dist2 = R*fmod(2*pi-fmod(anglend+pi/2,2*pi)+fmod(theta+pi/2,2*pi),2*pi) + 2*pi*R*DubinsAirplaneSolution['k_e']
w1 = cls + DubinsAirplaneSolution['R']*np.dot(rotz(theta+pi/2),e1.T).T + np.array([0,0,-dist1*tan(gam)]).T
w2 = cle + DubinsAirplaneSolution['R']*np.dot(rotz(theta+pi/2),e1.T).T - np.array([0,0,-dist2*tan(gam)]).T
q1 = (w2-w1)/np.linalg.norm(w2-w1,ord=2)
# start spiral
DubinsAirplaneSolution['c_s'] = cls
DubinsAirplaneSolution['psi_s'] = anglstart+pi/2
DubinsAirplaneSolution['lamda_s'] = -1
# end spiral
DubinsAirplaneSolution['c_e'] = cle - np.array([0,0,-dist2*tan(gam)]).T
DubinsAirplaneSolution['psi_e'] = theta+pi/2
DubinsAirplaneSolution['lamda_e'] = -1
# hyperplane H_s: switch from first spiral to line
DubinsAirplaneSolution['w_s'] = w1
DubinsAirplaneSolution['q_s'] = q1
# hyperplane H_l: switch from line to end spiral
DubinsAirplaneSolution['w_l'] = w2
DubinsAirplaneSolution['q_l'] = q1
# hyperplane H_e: end of Dubins path
DubinsAirplaneSolution['w_e'] = ze
DubinsAirplaneSolution['q_e'] = np.dot(rotz(anglend), np.array([1,0,0]).T)
elif DubinsAirplaneSolution['case'] == 2:
if idx == 1: # right-left-straight-right
# start spiral
DubinsAirplaneSolution['c_s'] = crs
DubinsAirplaneSolution['psi_s'] = anglstart-pi/2
DubinsAirplaneSolution['lamda_s'] = 1
DubinsAirplaneSolution['k_s'] = 0
ell = np.linalg.norm(cre[0:2]-ci[0:2],ord=2)
theta = atan2(cre[1] - ci[1], cre[0] - ci[0])
theta2 = acos(2*R/ell)
dist1 = R_min*psii + R*fmod(2*pi-fmod(theta+theta2,2*pi) + fmod(chii+pi/2,2*pi),2*pi)
dist2 = R*fmod(2*pi-fmod(theta+theta2-pi,2*pi)+fmod(anglend-pi/2,2*pi),2*pi)
w1 = ci + R*np.dot(rotz(theta+theta2),e1.T).T + np.array([0, 0, -dist1*tan(gam)]).T
w2 = cre + R*np.dot(rotz(-pi+theta+theta2),e1.T).T - np.array([0, 0, -dist2*tan(gam)]).T
q1 = (w2-w1)/np.linalg.norm(w2-w1,ord=2)
# intermediate-start spiral
DubinsAirplaneSolution['c_si'] = ci + np.array([0, 0, -R_min*psii*tan(gam)]).T
DubinsAirplaneSolution['psi_si'] = chii + pi/2
DubinsAirplaneSolution['lamda_si'] = -1
DubinsAirplaneSolution['k_si'] = 0
# end spiral
DubinsAirplaneSolution['c_e'] = cre - np.array([0,0,-dist2*tan(gam)]).T
DubinsAirplaneSolution['psi_e'] = fmod(theta+theta2-pi,2*pi)
DubinsAirplaneSolution['lamda_e'] = 1
DubinsAirplaneSolution['k_e'] = 0
# hyperplane H_s: switch from first to second spiral
DubinsAirplaneSolution['w_s'] = zi - np.array([0, 0, -psii*R_min*tan(gam)]).T
DubinsAirplaneSolution['q_s'] = np.array([cos(chii), sin(chii), 0]).T
# hyperplane H_si: switch from second spiral to straight line
DubinsAirplaneSolution['w_si'] = w1
DubinsAirplaneSolution['q_si'] = q1
# hyperplane H_l: switch from straight-line to end spiral
DubinsAirplaneSolution['w_l'] = w2
DubinsAirplaneSolution['q_l'] = q1
# hyperplane H_e: end of Dubins path
DubinsAirplaneSolution['w_e'] = ze
DubinsAirplaneSolution['q_e'] = np.dot(rotz(anglend), np.array([1,0,0]).T)
elif idx == 2: # right-left-straight-left
theta = atan2(cle[1]-ci[1],cle[0]-ci[0])
dist1 = R*fmod(2*pi-fmod(theta+pi/2,2*pi)+fmod(chii+pi/2,2*pi),2*pi)
dist2 = psii*R
dist3 = R*fmod(2*pi-fmod(anglend+pi/2,2*pi)+fmod(theta+pi/2,2*pi),2*pi)
w1 = ci + DubinsAirplaneSolution['R']*np.dot(rotz(theta+pi/2),e1.T).T + np.array([0, 0, -(dist1+dist2)*tan(gam)]).T
w2 = cle + DubinsAirplaneSolution['R']*np.dot(rotz(theta+pi/2),e1.T).T - np.array([0,0,-dist3*tan(gam)]).T
q1 = (w2-w1)/np.linalg.norm(w2-w1,ord=2) # direction of line
# start spiral
DubinsAirplaneSolution['c_s'] = crs
DubinsAirplaneSolution['psi_s'] = anglstart-pi/2
DubinsAirplaneSolution['lamda_s'] = 1
DubinsAirplaneSolution['k_s'] = 0
# intermediate-start spiral
DubinsAirplaneSolution['c_si'] = ci + np.array([0, 0, -dist2*tan(gam)]).T
DubinsAirplaneSolution['psi_si'] = chii+pi/2
DubinsAirplaneSolution['lamda_si'] = -1
DubinsAirplaneSolution['k_si'] = 0
# end spiral
DubinsAirplaneSolution['c_e'] = cle - np.array([0, 0, -dist3*tan(gam)]).T
DubinsAirplaneSolution['psi_e'] = theta+pi/2
DubinsAirplaneSolution['lamda_e'] = -1
DubinsAirplaneSolution['k_e'] = 0
# hyperplane H_s: switch from first to second spiral
DubinsAirplaneSolution['w_s'] = zi - np.array([0, 0, -dist2*tan(gam)]).T
DubinsAirplaneSolution['q_s'] = np.array([cos(chii), sin(chii), 0]).T
# hyperplane H_si: switch from second spiral to straight line
DubinsAirplaneSolution['w_si'] = w1
DubinsAirplaneSolution['q_si'] = q1
# hyperplane H_l: switch from straight-line to end spiral
DubinsAirplaneSolution['w_l'] = w2
DubinsAirplaneSolution['q_l'] = q1
# hyperplane H_e: end of Dubins path
DubinsAirplaneSolution['w_e'] = ze
DubinsAirplaneSolution['q_e'] = np.dot(rotz(anglend), np.array([1, 0, 0]).T)
elif idx == 3: # left-right-straight-right
theta = atan2(cre[1]-ci[1], cre[0] - ci[0])
dist1 = R*fmod(2*pi+fmod(theta-pi/2,2*pi)-fmod(chii-pi/2,2*pi),2*pi)
dist2 = psii*R
dist3 = R*fmod(2*pi+fmod(anglend-pi/2,2*pi)-fmod(theta-pi/2,2*pi),2*pi)
w1 = ci + DubinsAirplaneSolution['R']*np.dot(rotz(theta-pi/2),e1.T).T + np.array([0, 0, -(dist1+dist2)*tan(gam)]).T
w2 = cre + DubinsAirplaneSolution['R']*np.dot(rotz(theta-pi/2),e1.T).T - np.array([0, 0, -dist3*tan(gam)]).T
q1 = (w2-w1)/np.linalg.norm(w2-w1,ord=2) # direction of line
# start spiral
DubinsAirplaneSolution['c_s'] = cls
DubinsAirplaneSolution['psi_s'] = anglstart+pi/2
DubinsAirplaneSolution['lamda_s'] = -1
DubinsAirplaneSolution['k_s'] = 0
# intermediate-start spiral
DubinsAirplaneSolution['c_si'] = ci + np.array([0, 0, -dist2*tan(gam)]).T
DubinsAirplaneSolution['psi_si'] = chii-pi/2
DubinsAirplaneSolution['lamda_si'] = 1
DubinsAirplaneSolution['k_si'] = 0
# end spiral
DubinsAirplaneSolution['c_e'] = cre - np.array([0, 0, -dist3*tan(gam)]).T
DubinsAirplaneSolution['psi_e'] = theta-pi/2
DubinsAirplaneSolution['lamda_e'] = 1
DubinsAirplaneSolution['k_e'] = 0
# hyperplane H_s: switch from first to second spiral
DubinsAirplaneSolution['w_s'] = zi - np.array([0, 0, -dist2*tan(gam)]).T
DubinsAirplaneSolution['q_s'] = np.array([cos(chii), sin(chii), 0]).T
# hyperplane H_si: switch from second spiral to straight line
DubinsAirplaneSolution['w_si'] = w1
DubinsAirplaneSolution['q_si'] = q1
# hyperplane H_l: switch from straight-line to end spiral
DubinsAirplaneSolution['w_l'] = w2
DubinsAirplaneSolution['q_l'] = q1
# hyperplane H_e: end of Dubins path
DubinsAirplaneSolution['w_e'] = ze
DubinsAirplaneSolution['q_e'] = np.dot(rotz(anglend), np.array([1,0,0]).T)
elif idx == 4: # left-right-straight-left
ell = np.linalg.norm(cle[0:2]-ci[0:2],ord=2)
theta = atan2(cle[1] - ci[1], cle[0] - ci[0])
theta2 = theta - pi/2 + asin(2*R/ell)
dist1 = R*fmod(2*pi+fmod(theta2,2*pi) - fmod(chii-pi/2,2*pi),2*pi)
dist2 = R*psii
dist3 = R*fmod(2*pi+fmod(theta2+pi,2*pi) - fmod(anglend+pi/2,2*pi),2*pi)
w1 = ci + R*np.dot(rotz(theta2),e1.T).T + np.array([0, 0, -(dist1+dist2)*tan(gam)]).T
w2 = cle + R*np.dot(rotz(theta2+pi),e1.T).T - np.array([0, 0, -dist3*tan(gam)]).T
q1 = (w2-w1)/np.linalg.norm(w2-w1,ord=2)
# start spiral
DubinsAirplaneSolution['c_s'] = cls
DubinsAirplaneSolution['psi_s'] = anglstart+pi/2
DubinsAirplaneSolution['lamda_s'] = -1
DubinsAirplaneSolution['k_s'] = 0
# intermediate-start spiral
DubinsAirplaneSolution['c_si'] = ci + np.array([0, 0, -dist2*tan(gam)]).T
DubinsAirplaneSolution['psi_si'] = chii-pi/2
DubinsAirplaneSolution['lamda_si'] = 1
DubinsAirplaneSolution['k_si'] = 0
# end spiral
DubinsAirplaneSolution['c_e'] = cle - np.array([0, 0, -dist3*tan(gam)]).T
DubinsAirplaneSolution['psi_e'] = theta2+pi
DubinsAirplaneSolution['lamda_e'] = -1
DubinsAirplaneSolution['k_e'] = 0
# hyperplane H_s: switch from first to second spiral
DubinsAirplaneSolution['w_s'] = zi - np.array([0, 0, -dist2*tan(gam)]).T
DubinsAirplaneSolution['q_s'] = np.array([cos(chii), sin(chii), 0]).T
# hyperplane H_si: switch from second spiral to straight line
DubinsAirplaneSolution['w_si'] = w1
DubinsAirplaneSolution['q_si'] = q1
# hyperplane H_l: switch from straight-line to end spiral
DubinsAirplaneSolution['w_l'] = w2
DubinsAirplaneSolution['q_l'] = q1
# hyperplane H_e: end of Dubins path
DubinsAirplaneSolution['w_e'] = ze
DubinsAirplaneSolution['q_e'] = np.dot(rotz(anglend), np.array([1, 0, 0]).T)
elif DubinsAirplaneSolution['case'] == 3:
if idx == 1: # right-straight-left-right
# path specific calculations
ell = np.linalg.norm(ci[0:2] - crs[0:2],ord=2)
theta = atan2(ci[1] - crs[1], ci[0] - crs[0])
theta2 = theta-pi/2 + asin(2*R/ell)
dist1 = R*fmod(2*pi+fmod(theta2,2*pi) - fmod(anglstart-pi/2,2*pi),2*pi)
dist2 = R*fmod(2*pi+fmod(theta2+pi,2*pi)-fmod(chii+pi/2,2*pi),2*pi)
dist3 = fabs(R_min*psii)
w1 = crs + R*np.dot(rotz(theta2),e1.T).T + np.array([0, 0, -dist1*tan(gam)]).T
w2 = ci + R*np.dot(rotz(theta2+pi),e1.T).T - np.array([0, 0, -(dist2+dist3)*tan(gam)]).T
q1 = (w2-w1)/np.linalg.norm(w2-w1,ord=2)
# start spiral
DubinsAirplaneSolution['c_s'] = crs
DubinsAirplaneSolution['psi_s'] = anglstart-pi/2
DubinsAirplaneSolution['lamda_s'] = 1
DubinsAirplaneSolution['k_s'] = 0
# intermediate-end spiral
DubinsAirplaneSolution['c_ei'] = ci - np.array([0, 0, -(dist2+dist3)*tan(gam)]).T
DubinsAirplaneSolution['psi_ei'] = theta2+pi
DubinsAirplaneSolution['lamda_ei'] = -1
DubinsAirplaneSolution['k_ei'] = 0
# end spiral
DubinsAirplaneSolution['c_e'] = cre - np.array([0, 0, -dist3*tan(gam)]).T
DubinsAirplaneSolution['psi_e'] = anglend-pi/2-psii
DubinsAirplaneSolution['lamda_e'] = 1
DubinsAirplaneSolution['k_e'] = 0
# hyperplane H_s: switch from first to second spiral
DubinsAirplaneSolution['w_s'] = w1
DubinsAirplaneSolution['q_s'] = q1
# hyperplane H_l: switch from straight-line to intermediate spiral
DubinsAirplaneSolution['w_l'] = w2
DubinsAirplaneSolution['q_l'] = q1
# hyperplane H_ei: switch from intermediate spiral to
# end spiral
DubinsAirplaneSolution['w_ei'] = zi - np.array([0, 0, -dist3*tan(gam)]).T
DubinsAirplaneSolution['q_ei'] = np.array([cos(chii), sin(chii), 0]).T
# hyperplane H_e: end of Dubins path
DubinsAirplaneSolution['w_e'] = ze
DubinsAirplaneSolution['q_e'] = np.dot(rotz(anglend),np.array([1,0,0]).T)
elif idx == 2: # right-straight-right-left
# path specific calculations
theta = atan2(ci[1] - crs[1], ci[0] - crs[0])
dist1 = R*fmod(2*pi+fmod(theta-pi/2,2*pi) - fmod(anglstart-pi/2,2*pi),2*pi)
dist2 = R*fmod(2*pi+fmod(chii-pi/2,2*pi) - fmod(theta-pi/2,2*pi),2*pi)
dist3 = fabs(R_min*psii)
w1 = crs + R*np.dot(rotz(theta-pi/2),e1.T).T + np.array([0, 0, -dist1*tan(gam)]).T
w2 = ci + R*np.dot(rotz(theta-pi/2),e1.T).T - np.array([0, 0, -(dist2+dist3)*tan(gam)]).T
q1 = (w2-w1)/np.linalg.norm(w2-w1,ord=2)
# start spiral
DubinsAirplaneSolution['c_s'] = crs
DubinsAirplaneSolution['psi_s'] = anglstart-pi/2
DubinsAirplaneSolution['lamda_s'] = 1
DubinsAirplaneSolution['k_s'] = 0
# intermediate-end spiral
DubinsAirplaneSolution['c_ei'] = ci - np.array([0, 0, -(dist2+dist3)*tan(gam)]).T
DubinsAirplaneSolution['psi_ei'] = theta - pi/2
DubinsAirplaneSolution['lamda_ei'] = 1
DubinsAirplaneSolution['k_ei'] = 0
# end spiral
DubinsAirplaneSolution['c_e'] = cle - np.array([0, 0, -dist3*tan(gam)]).T
DubinsAirplaneSolution['psi_e'] = anglend+pi/2+psii
DubinsAirplaneSolution['lamda_e'] = -1
DubinsAirplaneSolution['k_e'] = 0
# hyperplane H_s: switch from first to second spiral
DubinsAirplaneSolution['w_s'] = w1
DubinsAirplaneSolution['q_s'] = q1
# hyperplane H_l: switch from straight-line to intermediate spiral
DubinsAirplaneSolution['w_l'] = w2
DubinsAirplaneSolution['q_l'] = q1
# hyperplane H_ei: switch from intermediate spiral to
# end spiral
DubinsAirplaneSolution['w_ei'] = zi - np.array([0, 0, -dist3*tan(gam)]).T
DubinsAirplaneSolution['q_ei'] = np.array([cos(chii), sin(chii), 0]).T
# hyperplane H_e: end of Dubins path
DubinsAirplaneSolution['w_e'] = ze
DubinsAirplaneSolution['q_e'] = np.dot(rotz(anglend), np.array([1, 0, 0]).T)
elif idx == 3: # left-straight-left-right
# path specific calculations
theta = atan2(ci[1]-cls[1],ci[0]-cls[0])
dist1 = R*fmod(2*pi-fmod(theta+pi/2,2*pi)+fmod(anglstart+pi/2,2*pi),2*pi)
dist2 = R*fmod(2*pi-fmod(chii+pi/2,2*pi)+fmod(theta+pi/2,2*pi),2*pi)
dist3 = fabs(R_min*psii)
w1 = cls + DubinsAirplaneSolution['R']*np.dot(rotz(theta+pi/2),e1.T).T + np.array([0, 0, -dist1*tan(gam)]).T
w2 = ci + DubinsAirplaneSolution['R']*np.dot(rotz(theta+pi/2),e1.T).T - np.array([0, 0, -(dist2+dist3)*tan(gam)]).T
q1 = (w2-w1)/np.linalg.norm(w2-w1,ord=2) # direction of line
# start spiral
DubinsAirplaneSolution['c_s'] = cls
DubinsAirplaneSolution['psi_s'] = anglstart+pi/2
DubinsAirplaneSolution['lamda_s'] = -1
DubinsAirplaneSolution['k_s'] = 0
# intermediate-end spiral
DubinsAirplaneSolution['c_ei'] = ci - np.array([0, 0, -(dist2+dist3)*tan(gam)]).T
DubinsAirplaneSolution['psi_ei'] = theta+pi/2
DubinsAirplaneSolution['lamda_ei'] = -1
DubinsAirplaneSolution['k_ei'] = 0
# end spiral
DubinsAirplaneSolution['c_e'] = cre - np.array([0, 0, -dist3*tan(gam)]).T
DubinsAirplaneSolution['psi_e'] = anglend-pi/2-psii
DubinsAirplaneSolution['lamda_e'] = 1
DubinsAirplaneSolution['k_e'] = 0
# hyperplane H_s: switch from first to second spiral
DubinsAirplaneSolution['w_s'] = w1
DubinsAirplaneSolution['q_s'] = q1
# hyperplane H_l: switch from straight-line to intermediate spiral
DubinsAirplaneSolution['w_l'] = w2
DubinsAirplaneSolution['q_l'] = q1
# hyperplane H_ei: switch from intermediate spiral to
# end spiral
DubinsAirplaneSolution['w_ei'] = zi - np.array([0, 0, -dist3*tan(gam)]).T
DubinsAirplaneSolution['q_ei'] = np.array([cos(chii), sin(chii), 0]).T
# hyperplane H_e: end of Dubins path
DubinsAirplaneSolution['w_e'] = ze
DubinsAirplaneSolution['q_e'] = np.dot(rotz(anglend),np.array([1,0,0]).T)
elif idx == 4: # left-straight-right-left
# path specific calculations
ell = np.linalg.norm(ci[0:2] - cls[0:2],ord=2)
theta = atan2( ci[1] - cls[1], ci[0] - cls[0])
theta2 = acos(2*R/ell)
dist1 = R*fmod(2*pi-fmod(theta+theta2,2*pi) + fmod(anglstart+pi/2,2*pi),2*pi)
dist2 = R*fmod(2*pi-fmod(theta+theta2-pi,2*pi)+fmod(chii-pi/2,2*pi),2*pi)
dist3 = fabs(R_min*psii)
w1 = cls + R*np.dot(rotz(theta+theta2),e1.T).T + np.array([0, 0, -dist1*tan(gam)]).T
w2 = ci + R*np.dot(rotz(-pi+theta+theta2),e1.T).T - np.array([0, 0, -(dist2+dist3)*tan(gam)]).T
q1 = (w2-w1)/np.linalg.norm(w2-w1,ord=2)
# start spiral
DubinsAirplaneSolution['c_s'] = cls
DubinsAirplaneSolution['psi_s'] = anglstart+pi/2
DubinsAirplaneSolution['lamda_s'] = -1
DubinsAirplaneSolution['k_s'] = 0
# intermediate-end spiral
DubinsAirplaneSolution['c_ei'] = ci - np.array([0, 0, -(dist2+dist3)*tan(gam)]).T
DubinsAirplaneSolution['psi_ei'] = fmod(theta+theta2-pi,2*pi)
DubinsAirplaneSolution['lamda_ei'] = 1
DubinsAirplaneSolution['k_ei'] = 0
# end spiral
DubinsAirplaneSolution['c_e'] = cle - np.array([0, 0, -dist3*tan(gam)]).T
DubinsAirplaneSolution['psi_e'] = anglend+pi/2+psii
DubinsAirplaneSolution['lamda_e'] = -1
DubinsAirplaneSolution['k_e'] = 0
# hyperplane H_s: switch from first to second spiral
DubinsAirplaneSolution['w_s'] = w1
DubinsAirplaneSolution['q_s'] = q1
# hyperplane H_l: switch from straight-line to intermediate spiral
DubinsAirplaneSolution['w_l'] = w2
DubinsAirplaneSolution['q_l'] = q1
# hyperplane H_ei: switch from intermediate spiral to
# end spiral
DubinsAirplaneSolution['w_ei'] = zi - np.array([0, 0, -dist3*tan(gam)]).T
DubinsAirplaneSolution['q_ei'] = np.array([cos(chii), sin(chii), 0]).T
# hyperplane H_e: end of Dubins path
DubinsAirplaneSolution['w_e'] = ze
DubinsAirplaneSolution['q_e'] = np.dot(rotz(anglend), np.array([1, 0, 0]).T)
if DubinsAirplaneSolution['case'] == 4:
print ('### Not Implemented Case')
return DubinsAirplaneSolution
def drawline(w1=None, q1=None, w2=None, q2=None, step=None):
# extract line path
r = w1
# propagate line until cross half plane
s = 0
NrNc = r.shape
if len(NrNc) == 1:
NrNc_ind = NrNc[0]
last_col = r[:]
else:
NrNc_ind = NrNc[1]
last_col = r[:,NrNc[1]-1]
r.shape = (3,1)
while np.dot( (last_col - w2).T,q2 ) <= 0:
s = s + step
w1.shape = (3,1)
q1.shape = (3,1)
new_col = w1+s*q1
new_col.shape = (3,1)
r = np.hstack( (r, new_col) )
NrNc = r.shape
if len(NrNc) == 1:
NrNc_ind = NrNc[0]
last_col = r[:]
else:
NrNc_ind = NrNc[1]
last_col = r[:,NrNc[1]-1]
return r
def drawspiral(R=None, gam=None, c=None, psi=None, lam=None, k=None, w=None, q=None, step=None):
# extract spiral path
r = np.zeros((1,1))
r = c.T + R*np.array( [cos(psi), sin(psi), 0] ).T
r = r.T
# determine number of required crossings of half plane
NrNc = r.shape
if len(NrNc) ==1 :
NrNc_ind = NrNc[0]
halfplane = np.dot( (r[0:2]-w[0:2].T),q[0:2] )
else:
NrNc_ind = NrNc[1]
halfplane = np.dot( (r[0:2,NrNc_ind-1]-w[0:2].T),q[0:2] )
if (halfplane > 0).all() :