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car_path.py
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1023 lines (935 loc) · 34.3 KB
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# -----------------------------------------------------------------------------
# Copyright (c) 2025 Shaw Wang from XJTU, [email protected]
#
# 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.
# -----------------------------------------------------------------------------
# referenced from https://msp.org/pjm/1990/145-2/pjm-v145-n2-p06-s.pdf
# referenced from https://ompl.kavrakilab.org/ReedsSheppStateSpace_8cpp_source.html
# referenced from https://blog.csdn.net/qq_44339029/article/details/126200191
# referenced from https://blog.csdn.net/jianmo1993/article/details/143200673
# referenced from https://cpb-us-e2.wpmucdn.com/faculty.sites.uci.edu/dist/e/700/files/2014/04/Dubins_Set_Robotics_2001.pdf
# referenced from https://blog.csdn.net/qq_44339029/article/details/126095951
import time
from functools import wraps
CAR_PATH_MODULE_TIMER_ENABLED = False
def timer(enabled=True):
def decorator(func):
@wraps(func)
def wrapper(*args, **kwargs):
if enabled:
start_time = time.time() # 记录开始时间
result = func(*args, **kwargs)
end_time = time.time() # 记录结束时间
elapsed_time = end_time - start_time
print(
f"Function '{func.__name__}' executed in {elapsed_time:.9f} seconds"
)
else:
result = func(*args, **kwargs) # 如果不计时，直接执行
return result
return wrapper
return decorator
import numpy as np
_Pi_2 = 0.5 * np.pi
_Pi = 1.0 * np.pi
_2_Pi = 2.0 * np.pi
def turn_motion(x, y, phi, kappa, delta_phi):
"""
params:
x: x-coordinate of the current position
y: y-coordinate of the current position
phi: current heading angle
kappa: curvature, positive for left turn, negative for right turn
delta_phi: angle of turn,
positive for left forward turn, and right backward turn,
negative for right forward turn and left backward turn
returns:
x_new: x-coordinate of the new position
y_new: y-coordinate of the new position
phi_new: new heading angle
"""
phi_new = (phi + delta_phi) % _2_Pi
x_new = x + (np.sin(phi_new) - np.sin(phi)) / kappa
y_new = y - (np.cos(phi_new) - np.cos(phi)) / kappa
return x_new, y_new, phi_new
def line_motion(x, y, phi, delta_s):
"""
params:
x: x-coordinate of the current position
y: y-coordinate of the current position
phi: current heading angle
delta_s: distance to move, positive for forward, negative for backward
returns:
x_new: x-coordinate of the new position
y_new: y-coordinate of the new position
phi_new: new heading angle
"""
x_new = x + np.cos(phi) * delta_s
y_new = y + np.sin(phi) * delta_s
phi_new = np.ones_like(delta_s) * phi
return x_new, y_new, phi_new
@timer(CAR_PATH_MODULE_TIMER_ENABLED)
def path_interp(types, lens, step, kappa, x=0, y=0, phi=0):
"""
params:
types: types of the path segments
lens: lengths of the path segments
step: step size of the path interpolation
kappa: curvature of reeds-shepp path
x: x-coordinate of the start position
y: y-coordinate of the start position
phi: start heading angle
returns:
path_x: local x-coordinate of the rs path
path_y: local y-coordinate of the rs path
path_phi: local heading angle of the rs path
"""
step = step * kappa
path_x = []
path_y = []
path_phi = []
# 计算每段路径的插值点
for type, len in zip(types, lens):
steps = np.arange(0, len, step)
if type == "Lp":
seg_x, seg_y, seg_phi = turn_motion(x, y, phi, kappa, steps)
x, y, phi = turn_motion(x, y, phi, kappa, len)
elif type == "Sp":
seg_x, seg_y, seg_phi = line_motion(x, y, phi, steps / kappa)
x, y, phi = line_motion(x, y, phi, len / kappa)
elif type == "Rp":
seg_x, seg_y, seg_phi = turn_motion(x, y, phi, -kappa, -steps)
x, y, phi = turn_motion(x, y, phi, -kappa, -len)
elif type == "Rm":
seg_x, seg_y, seg_phi = turn_motion(x, y, phi, -kappa, steps)
x, y, phi = turn_motion(x, y, phi, -kappa, len)
elif type == "Sm":
seg_x, seg_y, seg_phi = line_motion(x, y, phi, -steps / kappa)
x, y, phi = line_motion(x, y, phi, -len / kappa)
elif type == "Lm":
seg_x, seg_y, seg_phi = turn_motion(x, y, phi, kappa, -steps)
x, y, phi = turn_motion(x, y, phi, kappa, -len)
path_x.append(seg_x)
path_y.append(seg_y)
path_phi.append(seg_phi)
path_x.append([x])
path_y.append([y])
path_phi.append([phi])
path_x = np.concatenate(path_x)
path_y = np.concatenate(path_y)
path_phi = np.concatenate(path_phi)
return path_x, path_y, path_phi
def _corrd_trans(x, y, phi, x_, y_, phi_, kappa):
"""
params:
x: x-coordinate of the current position
y: y-coordinate of the current position
phi: current heading angle
x_: x-coordinate of the target position
y_: y-coordinate of the target position
phi_: target heading angle
kappa: curvature of reeds-shepp path
returns:
delta_x: x-coordinate of the local corrdinate
delta_y: y-coordinate of the local corrdinate
delta_phi: heading angle of the local corrdinate
"""
delta_x = (x_ - x) * kappa
delta_y = (y_ - y) * kappa
delta = np.array([delta_x, delta_y])
rot_mat = np.array([[np.cos(phi), np.sin(phi)], [-np.sin(phi), np.cos(phi)]])
delta = rot_mat @ delta
delta_phi = (phi_ - phi) % _2_Pi
return *delta, delta_phi
def _polar_trans(x, y):
"""
params:
x: x-coordinate of the current position
y: y-coordinate of the current position
returns:
rh: distance from the origin
theta: angle from the positive x-axis
"""
r = np.sqrt(x**2 + y**2)
theta = np.arctan2(y, x)
return r, theta % _2_Pi
# dubins path
def _dubins_LpSpLp(x, y, phi):
"""
x = u*cos(t) + sin(t + v)
y = u*sin(t) - cos(t + v) + 1
phi = t + v
"""
u, t = _polar_trans(x - np.sin(phi), y + np.cos(phi) - 1)
t = t % _2_Pi
v = (phi - t) % _2_Pi
return True, [t, u, v]
def _dubins_RpSpRp(x, y, phi):
"""
x = u*cos(t) + sin(t + v)
y = -u*sin(t) + cos(t + v) - 1
phi = -t - v
"""
u, theta = _polar_trans(y + 1 - np.cos(phi), x + np.sin(phi))
t = (theta - _Pi_2) % _2_Pi
v = (-t - phi) % _2_Pi
return True, [t, u, v]
def _dubins_LpSpRp(x, y, phi):
"""
x = u*cos(t) + 2*sin(t) - sin(t - v)
y = u*sin(t) - 2*cos(t) + cos(t - v) + 1
phi = t - v
"""
u1, t1 = _polar_trans(x + np.sin(phi), y - np.cos(phi) - 1)
if u1 >= 2:
u = np.sqrt(u1**2 - 4)
t = (t1 + np.arctan2(2, u)) % _2_Pi
v = (t - phi) % _2_Pi
return True, [t, u, v]
return False, []
def dubins_RpSpLp(x, y, phi):
"""
x = u*cos(t) + 2*sin(t) - sin(t - v)
y = -u*sin(t) + 2*cos(t) - cos(t - v) - 1
phi = -t + v
"""
u1, t1 = _polar_trans(y + 1 + np.cos(phi), x - np.sin(phi))
if u1 >= 2:
u = np.sqrt(u1**2 - 4)
t = (t1 - np.arctan2(u, 2)) % _2_Pi
v = (phi + t) % _2_Pi
return True, [t, u, v]
return False, []
def _dubins_LpRpLp(x, y, phi):
"""
x = 2*sin(t) - 2*sin(t - u) + sin(t - u + v)
y = -2*cos(t) + 2*cos(t - u) - cos(t - u + v) + 1
phi = t - u + v
"""
u1, t1 = _polar_trans(x - np.sin(phi), y + np.cos(phi) - 1)
if u1 <= 4:
u = 2 * np.arcsin(u1 / 4)
t = (t1 + u / 2) % _2_Pi
v = (phi + u - t) % _2_Pi
return True, [t, u, v]
return False, []
def _dubins_RpLpRp(x, y, phi):
"""
x = 2*sin(t) - 2*sin(t - u) + sin(t - u + v)
y = 2*cos(t) - 2*cos(t - u) + cos(t - u + v) - 1
phi = -t + u - v
"""
u1, t1 = _polar_trans(y - np.cos(phi) + 1, x + np.sin(phi))
if u1 <= 4:
u = 2 * np.arcsin(u1 / 4)
t = (t1 + u / 2 - _Pi_2) % _2_Pi
v = (u - t - phi) % _2_Pi
return True, [t, u, v]
return False, []
def _dubins_CSC(x, y, phi, sols):
flag, lens = _dubins_LpSpLp(x, y, phi)
if flag:
sols.append({"cost": np.sum(lens), "type": ["Lp", "Sp", "Lp"], "lens": lens})
flag, lens = _dubins_RpSpRp(x, y, phi)
if flag:
sols.append({"cost": np.sum(lens), "type": ["Rp", "Sp", "Rp"], "lens": lens})
flag, lens = _dubins_LpSpRp(x, y, phi)
if flag:
sols.append({"cost": np.sum(lens), "type": ["Lp", "Sp", "Rp"], "lens": lens})
flag, lens = dubins_RpSpLp(x, y, phi)
if flag:
sols.append({"cost": np.sum(lens), "type": ["Rp", "Sp", "Lp"], "lens": lens})
return sols
def _dubins_CCC(x, y, phi, sols):
flag, lens = _dubins_LpRpLp(x, y, phi)
if flag:
sols.append({"cost": np.sum(lens), "type": ["Lp", "Rp", "Lp"], "lens": lens})
flag, lens = _dubins_RpLpRp(x, y, phi)
if flag:
sols.append({"cost": np.sum(lens), "type": ["Rp", "Lp", "Rp"], "lens": lens})
return sols
@timer(CAR_PATH_MODULE_TIMER_ENABLED)
def dubins_all_paths(x, y, phi, x_, y_, phi_, kappa):
"""
params:
x: x-coordinate of the current position
y: y-coordinate of the current position
phi: current heading angle
x_: x-coordinate of the target position
y_: y-coordinate of the target position
phi_: target heading angle
kappa: curvature of path
returns:
dubins_paths: a list of dubins paths, each path is a dictionary with keys "cost", "type", "lens"
"""
sols = []
loc = _corrd_trans(x, y, phi, x_, y_, phi_, kappa)
_dubins_CSC(*loc, sols)
_dubins_CCC(*loc, sols)
return sols
@timer(CAR_PATH_MODULE_TIMER_ENABLED)
def dubins_shortest_path(x, y, phi, x_, y_, phi_, kappa):
return min(
dubins_all_paths(x, y, phi, x_, y_, phi_, kappa), key=lambda sol: sol["cost"]
)
@timer(CAR_PATH_MODULE_TIMER_ENABLED)
def dubins_shortest_path_length(x, y, phi, x_, y_, phi_, kappa):
return dubins_shortest_path(x, y, phi, x_, y_, phi_, kappa)["cost"]
# reeds-shepp path
def _LpSpLp(x, y, phi):
"""
init: [0, 0, 0], motion: ['L+', 'S+', 'L+'], lens: [t, u, v]
x = u*cos(t) + sin(t + v) -> u*cos(t) = x - sin(phi)
y = u*sin(t) - cos(t + v) + 1 -> u*sin(t) = y + cos(phi) - 1
phi = t + v
u, t = polar(x - sin(phi), y + cos(phi) - 1)
-> v = phi -t
"""
u, t = _polar_trans(x - np.sin(phi), y + np.cos(phi) - 1)
v = (phi - t) % _2_Pi
if t <= _Pi and v <= _Pi:
return True, t, u, v
return False, None, None, None
def _LpSpRp(x, y, phi):
"""
init: [0, 0, 0], motion: ['L+', 'S+', 'R+'], lens: [t, u, v]
x = u*cos(t) + 2*sin(t) - sin(t - v)
y = u*sin(t) - 2*cos(t) + cos(t - v) + 1 -> u*sin(t) = y - cos(phi) - 1
phi = t - v
-> u*cos(t) + 2*cos(t - pi/2) = x + sin(phi)
-> u*sin(t) + 2*sin(t - pi/2) = y - cos(phi) - 1
u1, t1 = polar(x + np.sin(phi), y - np.cos(phi) - 1)
-> u = sqrt(u1**2 - 4)
-> t = t1 + atan(2/u1)
-> v = t - phi
"""
u1, t1 = _polar_trans(x + np.sin(phi), y - np.cos(phi) - 1)
if u1 > 2.0:
u = np.sqrt(u1**2 - 4)
beta = np.arctan2(2, u)
t = (t1 + beta) % _2_Pi
v = (t - phi) % _2_Pi
if t <= _Pi and v <= _Pi:
return True, t, u, v
return False, None, None, None
def _CSC(x, y, phi, sols):
flag, t, u, v = _LpSpLp(x, y, phi)
if flag: # LpSpLp
sols.append({"type": ["Lp", "Sp", "Lp"], "lens": [t, u, v], "cost": t + u + v})
flag, t, u, v = _LpSpLp(-x, y, -phi) # timeflip
if flag: # LmSmLm
sols.append({"type": ["Lm", "Sm", "Lm"], "lens": [t, u, v], "cost": t + u + v})
flag, t, u, v = _LpSpLp(x, -y, -phi) # reflect
if flag: # RpSpRp
sols.append({"type": ["Rp", "Sp", "Rp"], "lens": [t, u, v], "cost": t + u + v})
flag, t, u, v = _LpSpLp(-x, -y, phi) # timeflip + reflect
if flag: # RmSmRm
sols.append({"type": ["Rm", "Sm", "Rm"], "lens": [t, u, v], "cost": t + u + v})
flag, t, u, v = _LpSpRp(x, y, phi)
if flag: # LpSpRp
sols.append({"type": ["Lp", "Sp", "Rp"], "lens": [t, u, v], "cost": t + u + v})
flag, t, u, v = _LpSpRp(-x, y, -phi) # timeflip
if flag: # LmSmRm
sols.append({"type": ["Lm", "Sm", "Rm"], "lens": [t, u, v], "cost": t + u + v})
flag, t, u, v = _LpSpRp(x, -y, -phi) # reflect
if flag: # RpSpLp
sols.append({"type": ["Rp", "Sp", "Lp"], "lens": [t, u, v], "cost": t + u + v})
flag, t, u, v = _LpSpRp(-x, -y, phi) # timeflip + reflect
if flag: # RmSmLm
sols.append({"type": ["Rm", "Sm", "Lm"], "lens": [t, u, v], "cost": t + u + v})
def _LpRmLp(x, y, phi):
"""
init: [0, 0, 0], motion: ['L+', 'R-', 'L+'], lens: [t, u, v]
x = 2*sin(t) - 2*sin(t + u) + sin(t + u + v)
y = -2*cos(t) + 2*cos(t + u) - cos(t + u + v) + 1
phi = t + u + v
-> 2*sin(t+u) - 2*sin(t) = -(x - sin(phi))
-> 4*cos(u/2)*cos(t + u/2) = -(x - sin(phi))
-> 2*cos(t+u) - 2*cos(t) = y + cos(phi) - 1
-> 4*sin(u/2)*sin(t + u/2) = -(y + cos(phi) - 1)
u1, t1 = polar(-(x - sin(phi)), -(y + cos(phi) - 1))
-> u = 2*arcsin(u1/4)
-> t = theta - u/2
-> v = phi - t - u
"""
u1, t1 = _polar_trans(-(x - np.sin(phi)), -(y + np.cos(phi) - 1))
if u1 <= 4.0:
u = 2.0 * np.arcsin(0.25 * u1)
t = (t1 - 0.5 * u) % _2_Pi
v = (phi - t - u) % _2_Pi
if t <= _Pi and u <= _Pi and v <= _Pi:
return True, t, u, v
return False, None, None, None
def _LpRmLm(x, y, phi):
"""
init: [0, 0, 0], motion: ['L+', 'R-', 'L-'], lens: [t, u, v]
x = 2*sin(t) - 2*sin(t + u) + sin(t + u - v)
y = -2*cos(t) + 2*cos(t + u) - cos(t + u - v) + 1
phi = t + u - v
-> 2*sin(t+u) - 2*sin(t) = x - sin(phi)
-> 4*cos(u/2)*cos(t + u/2) = -(x - sin(phi))
-> 2*cos(t+u) - 2*cos(t) = y + cos(phi) - 1
-> 4*sin(u/2)*sin(t + u/2) = -(y + cos(phi) - 1)
u1, t1 = polar(-(x - np.sin(phi)), -(y + np.cos(phi) - 1))
-> u = 2*arcsin(u1/4)
-> t = theta - u/2
-> v = t + u - phi
"""
u1, theta = _polar_trans(-(x - np.sin(phi)), -(y + np.cos(phi) - 1))
if u1 <= 4.0:
u = 2.0 * np.arcsin(0.25 * u1)
t = (theta - 0.5 * u) % _2_Pi
v = (t + u - phi) % _2_Pi
if t <= _Pi and u <= _Pi and v <= _Pi:
return True, t, u, v
return False, None, None, None
def _CCC(x, y, phi, sols):
flag, t, u, v = _LpRmLp(x, y, phi)
if flag: # LpRmLp
sols.append({"type": ["Lp", "Rm", "Lp"], "lens": [t, u, v], "cost": t + u + v})
flag, t, u, v = _LpRmLp(-x, y, -phi) # timeflip
if flag: # LmRpLm
sols.append({"type": ["Lm", "Rp", "Lm"], "lens": [t, u, v], "cost": t + u + v})
flag, t, u, v = _LpRmLp(x, -y, -phi) # reflect
if flag: # RpLmRp
sols.append({"type": ["Rp", "Lm", "Rp"], "lens": [t, u, v], "cost": t + u + v})
flag, t, u, v = _LpRmLp(-x, -y, phi) # timeflip + reflect
if flag: # RmLpRm
sols.append({"type": ["Rm", "Lp", "Rm"], "lens": [t, u, v], "cost": t + u + v})
flag, t, u, v = _LpRmLm(x, y, phi)
if flag: # LpRmLm
sols.append({"type": ["Lp", "Rm", "Lm"], "lens": [t, u, v], "cost": t + u + v})
flag, t, u, v = _LpRmLm(-x, y, -phi) # timeflip
if flag: # LmRpLp
sols.append({"type": ["Lm", "Rp", "Lp"], "lens": [t, u, v], "cost": t + u + v})
flag, t, u, v = _LpRmLm(x, -y, -phi) # reflect
if flag: # RpLmRm
sols.append({"type": ["Rp", "Lm", "Rm"], "lens": [t, u, v], "cost": t + u + v})
flag, t, u, v = _LpRmLm(-x, -y, phi) # timeflip + reflect
if flag: # RmLpRp
sols.append({"type": ["Rm", "Lp", "Rp"], "lens": [t, u, v], "cost": t + u + v})
# backwards
xb = x * np.cos(phi) + y * np.sin(phi)
yb = x * np.sin(phi) - y * np.cos(phi)
flag, t, u, v = _LpRmLm(xb, yb, phi)
if flag: # LmRmLp
sols.append({"type": ["Lm", "Rm", "Lp"], "lens": [v, u, t], "cost": t + u + v})
flag, t, u, v = _LpRmLm(-xb, yb, -phi) # timeflip
if flag: # LpRpLm
sols.append({"type": ["Lp", "Rp", "Lm"], "lens": [v, u, t], "cost": t + u + v})
flag, t, u, v = _LpRmLm(xb, -yb, -phi) # reflect
if flag: # RmLmRp
sols.append({"type": ["Rm", "Lm", "Rp"], "lens": [v, u, t], "cost": t + u + v})
flag, t, u, v = _LpRmLm(-xb, -yb, phi) # timeflip + reflect
if flag: # RpLpRm
sols.append({"type": ["Rp", "Lp", "Rm"], "lens": [v, u, t], "cost": t + u + v})
def _LpRupLumRm(x, y, phi):
"""
init: [0, 0, 0], motion: ['L+', 'R+', 'L-', 'R-'], lens: [t, u, u, v]
x = 2*sin(t) + 2*sin(t - 2*u) - 2*sin(t - u) - sin(t - 2*u + v)
y = -2*cos(t) - 2*cos(t - 2*u) + 2*cos(t - u) + cos(t - 2*u + v) + 1
phi = t - 2*u + v
-> 2*sin(t-2*u) - 2*sin(t-u) + 2*sin(t) = x + sin(phi)
-> -2*cos(t-2*u) + 2*cos(t-u) - 2*cos(t) = y - cos(phi) - 1
-> 4*cos(u)*sin(t - u) - 2*sin(t - u) = x + sin(phi)
-> -4*cos(u)*cos(t-u) + 2*cos(t-u) = y - cos(phi) - 1
-> (4*cos(u) - 2)*cos(t - u - pi/2) = x + sin(phi)
-> (4*cos(u) - 2)*sin(t - u - pi/2) = y - cos(phi) - 1
u1, t1 = polar(x + np.sin(phi), y - np.cos(phi) - 1)
-> u = arccos((u1+2)/4)
-> t = theta + u + pi/2
-> v = phi + 2*u - t
"""
u1, t1 = _polar_trans(x + np.sin(phi), y - np.cos(phi) - 1)
if u1 <= 2.0:
u = np.arccos((u1 + 2.0) / 4.0)
t = (t1 + u + _Pi_2) % _2_Pi
v = (phi + 2 * u - t) % _2_Pi
if t <= _Pi and v <= _Pi:
return True, t, u, v
return False, None, None, None
def _LpRumLumRp(x, y, phi):
"""
init: [0, 0, 0], motion: ['L+', 'R-', 'L-', 'R+'], lens: [t, u, u, v]
x = 4*sin(t) - 2*sin(t + u) - sin(t - v)
y = -4*cos(t) + 2*cos(t + u) + cos(t - v) + 1
phi = t - v
-> 4*sin(t) - 2*sin(t + u) = x + sin(phi)
-> -4*cos(t) + 2*cos(t + u) = y - cos(phi) - 1
-> 4*cos(t-pi/2) + 2*cos(t+u+pi/2) = x + sin(phi)
-> 4*sin(t-pi/2) + 2*sin(t+u+pi/2) = y - cos(phi) - 1
u1, t1 = polar(x + np.sin(phi), y - np.cos(phi) - 1)
作图分析，可以得到：
cos(u) = (4^2+2^2-u1^2)/(2*4*2)
t - pi/2 - beta = t1, 其中beta = arcsin((2*sin(u1)/u1)
v = t - phi
"""
u1, theta = _polar_trans(x + np.sin(phi), y - np.cos(phi) - 1)
u2 = 20 - u1**2
if 0 <= u2 <= 16:
u = np.arccos(u2 / 16) # [0, pi]
t = (theta + np.arcsin((2 * np.sin(u)) / u1) + _Pi / 2) % _2_Pi
v = (t - phi) % _2_Pi
if t <= _Pi and v <= _Pi:
return True, t, u, v
return False, None, None, None
def _CCCC(x, y, phi, sols):
flag, t, u, v = _LpRupLumRm(x, y, phi)
if flag: # LpRupLumRm
sols.append(
{
"type": ["Lp", "Rp", "Lm", "Rm"],
"lens": [t, u, u, v],
"cost": t + u + u + v,
}
)
flag, t, u, v = _LpRupLumRm(-x, y, -phi) # timeflip
if flag: # LmRumLupRp
sols.append(
{
"type": ["Lm", "Rm", "Lp", "Rp"],
"lens": [t, u, u, v],
"cost": t + u + u + v,
}
)
flag, t, u, v = _LpRupLumRm(x, -y, -phi) # reflect
if flag: # RpLupRumLm
sols.append(
{
"type": ["Rp", "Lp", "Rm", "Lm"],
"lens": [t, u, u, v],
"cost": t + u + u + v,
}
)
flag, t, u, v = _LpRupLumRm(-x, -y, phi) # timeflip + reflect
if flag: # RmLumRupLp
sols.append(
{
"type": ["Rm", "Lm", "Rp", "Lp"],
"lens": [t, u, u, v],
"cost": t + u + u + v,
}
)
flag, t, u, v = _LpRumLumRp(x, y, phi)
if flag: # LpRumLumRp
sols.append(
{
"type": ["Lp", "Rm", "Lm", "Rp"],
"lens": [t, u, u, v],
"cost": t + u + u + v,
}
)
flag, t, u, v = _LpRumLumRp(-x, y, -phi) # timeflip
if flag: # LmRupLupRm
sols.append(
{
"type": ["Lm", "Rp", "Lp", "Rm"],
"lens": [t, u, u, v],
"cost": t + u + u + v,
}
)
flag, t, u, v = _LpRumLumRp(x, -y, -phi) # reflect
if flag: # RpLumRumLp
sols.append(
{
"type": ["Rp", "Lm", "Rm", "Lp"],
"lens": [t, u, u, v],
"cost": t + u + u + v,
}
)
flag, t, u, v = _LpRumLumRp(-x, -y, phi) # timeflip + reflect
if flag: # RmLupRupLm
sols.append(
{
"type": ["Rm", "Lp", "Rp", "Lm"],
"lens": [t, u, u, v],
"cost": t + u + u + v,
}
)
def _LpRmSmRm(x, y, phi):
"""
init: [0, 0, 0], motion: ['L+', 'R-', 'S-', 'R-'], lens: [t, pi_2, u, v]
x = -u*cos(pi_2 + t) + 2*sin(t) - sin(pi_2 + t + v)
y = -u*sin(pi_2 + t) - 2*cos(t) + cos(pi_2 + t + v) + 1
phi = pi_2 + t + v
-> -u*cos(pi_2 + t) + 2*sin(t) = x + sin(phi)
-> -u*sin(pi_2 + t) - 2*cos(t) = y - cos(phi) - 1
-> u*cos(t - pi_2) + 2*cos(t - pi_2) = x + sin(phi)
-> u*sin(t - pi_2) + 2*sin(t - pi_2) = y - cos(phi) - 1
->(u + 2)*cos(t - pi_2) = x + sin(phi)
->(u + 2)*sin(t - pi_2) = y - cos(phi) - 1
u1, t1 = polar(x + np.sin(phi), y - np.cos(phi) - 1)
-> u = u1 - 2
-> t = t1 + pi_2
-> v = phi - t - pi_2
"""
u1, t1 = _polar_trans(x + np.sin(phi), y - np.cos(phi) - 1)
if u1 >= 2.0:
u = u1 - 2.0
t = (t1 + _Pi_2) % _2_Pi
v = (phi - t - _Pi_2) % _2_Pi
if t <= _Pi and v <= _Pi:
return True, t, u, v
return False, None, None, None
def _LpRmSmLm(x, y, phi):
"""
init: [0, 0, 0], motion: ['L+', 'R-', 'S-', 'L-'], lens: [t, pi_2, u, v]
x = -u*cos(pi_2 + t) + 2*sin(t) - 2*sin(pi_2 + t) + sin(pi_2 + t - v)
y = -u*sin(pi_2 + t) - 2*cos(t) + 2*cos(pi_2 + t) - cos(pi_2 + t - v) + 1
phi = pi_2 + t - v
-> -u*cos(pi_2 + t) + 2*sin(t) - 2*sin(pi_2 + t) = x - sin(phi)
-> -u*sin(pi_2 + t) - 2*cos(t) + 2*cos(pi_2 + t) = y + cos(phi) - 1
-> u*cos(t - pi_2) + 2*cos(t - pi_2) - 2*cos(t) = x - sin(phi)
-> u*sin(t - pi_2) + 2*sin(t - pi_2) - 2*sin(t) = y + cos(phi) - 1
->(u + 2)*cos(t - pi_2) - 2*cos(t) = x - sin(phi)
->(u + 2)*sin(t - pi_2) - 2*sin(t) = y + cos(phi) - 1
u1, t1 = polar(x - np.sin(phi), y + np.cos(phi) - 1)
作图分析，可以得到：
u = sqrt(u1^2 - 4) - 2
t = t1 + arctan(2/(u+2)) + pi/2
v = t - phi + pi/2
"""
u1, t1 = _polar_trans(x - np.sin(phi), y + np.cos(phi) - 1)
if u1 >= 2.0 * np.sqrt(2.0):
u = np.sqrt(u1**2 - 4.0) - 2.0
t = (t1 + np.arctan(2.0 / (u + 2.0)) + _Pi_2) % _2_Pi
v = (t - phi + _Pi_2) % _2_Pi
if t <= _Pi and v <= _Pi:
return True, t, u, v
return False, None, None, None
def _CCSC(x, y, phi, sols):
flag, t, u, v = _LpRmSmRm(x, y, phi)
if flag: # LpRmSmRm
sols.append(
{
"type": ["Lp", "Rm", "Sm", "Rm"],
"lens": [t, _Pi_2, u, v],
"cost": t + _Pi_2 + u + v,
}
)
flag, t, u, v = _LpRmSmRm(-x, y, -phi) # timeflip
if flag: # LmRpSpRp
sols.append(
{
"type": ["Lm", "Rp", "Sp", "Rp"],
"lens": [t, _Pi_2, u, v],
"cost": t + _Pi_2 + u + v,
}
)
flag, t, u, v = _LpRmSmRm(x, -y, -phi) # reflect
if flag: # RpLmSmLm
sols.append(
{
"type": ["Rp", "Lm", "Sm", "Lm"],
"lens": [t, _Pi_2, u, v],
"cost": t + _Pi_2 + u + v,
}
)
flag, t, u, v = _LpRmSmRm(-x, -y, phi) # timeflip + reflect
if flag: # RmLpSpLp
sols.append(
{
"type": ["Rm", "Lp", "Sp", "Lp"],
"lens": [t, _Pi_2, u, v],
"cost": t + _Pi_2 + u + v,
}
)
flag, t, u, v = _LpRmSmLm(x, y, phi)
if flag: # LpRmSmLm
sols.append(
{
"type": ["Lp", "Rm", "Sm", "Lm"],
"lens": [t, _Pi_2, u, v],
"cost": t + _Pi_2 + u + v,
}
)
flag, t, u, v = _LpRmSmLm(-x, y, -phi) # timeflip
if flag: # LmRpSpLp
sols.append(
{
"type": ["Lm", "Rp", "Sp", "Lp"],
"lens": [t, _Pi_2, u, v],
"cost": t + _Pi_2 + u + v,
}
)
flag, t, u, v = _LpRmSmLm(x, -y, -phi) # reflect
if flag: # RpLmSmRm
sols.append(
{
"type": ["Rp", "Lm", "Sm", "Rm"],
"lens": [t, _Pi_2, u, v],
"cost": t + _Pi_2 + u + v,
}
)
flag, t, u, v = _LpRmSmLm(-x, -y, phi) # timeflip + reflect
if flag: # RmLpSpRp
sols.append(
{
"type": ["Rm", "Lp", "Sp", "Rp"],
"lens": [t, _Pi_2, u, v],
"cost": t + _Pi_2 + u + v,
}
)
def _CSCC(x, y, phi, sols):
xb = x * np.cos(phi) + y * np.sin(phi)
yb = x * np.sin(phi) - y * np.cos(phi)
flag, t, u, v = _LpRmSmRm(xb, yb, phi)
if flag: # RmSmRmLp
sols.append(
{
"type": ["Rm", "Sm", "Rm", "Lp"],
"lens": [v, u, _Pi_2, t],
"cost": t + u + _Pi_2 + v,
}
)
flag, t, u, v = _LpRmSmRm(-xb, yb, -phi) # timeflip
if flag: # RpSpRpLm
sols.append(
{
"type": ["Rp", "Sp", "Rp", "Lm"],
"lens": [v, u, _Pi_2, t],
"cost": t + u + _Pi_2 + v,
}
)
flag, t, u, v = _LpRmSmRm(xb, -yb, -phi) # reflect
if flag: # LmSmLmRp
sols.append(
{
"type": ["Lm", "Sm", "Lm", "Rp"],
"lens": [v, u, _Pi_2, t],
"cost": t + u + _Pi_2 + v,
}
)
flag, t, u, v = _LpRmSmRm(-xb, -yb, phi) # timeflip + reflect
if flag: # LpSpLpRm
sols.append(
{
"type": ["Lp", "Sp", "Lp", "Rm"],
"lens": [v, u, _Pi_2, t],
"cost": t + u + _Pi_2 + v,
}
)
flag, t, u, v = _LpRmSmLm(xb, yb, phi)
if flag: # LmSmRmLp
sols.append(
{
"type": ["Lm", "Sm", "Rm", "Lp"],
"lens": [v, u, _Pi_2, t],
"cost": t + u + _Pi_2 + v,
}
)
flag, t, u, v = _LpRmSmLm(-xb, yb, -phi) # timeflip
if flag: # LpSpRpLm
sols.append(
{
"type": ["Lp", "Sp", "Rp", "Lm"],
"lens": [v, u, _Pi_2, t],
"cost": t + u + _Pi_2 + v,
}
)
flag, t, u, v = _LpRmSmLm(xb, -yb, -phi) # reflect
if flag: # RmSmLmRp
sols.append(
{
"type": ["Rm", "Sm", "Lm", "Rp"],
"lens": [v, u, _Pi_2, t],
"cost": t + u + _Pi_2 + v,
}
)
flag, t, u, v = _LpRmSmLm(-xb, -yb, phi) # timeflip + reflect
if flag: # RpSpLpRm
sols.append(
{
"type": ["Rp", "Sp", "Lp", "Rm"],
"lens": [v, u, _Pi_2, t],
"cost": t + u + _Pi_2 + v,
}
)
def _LpRmSmLmRp(x, y, phi):
"""
init: [0, 0, 0], motion: ['L+', 'R-', 'S-', 'L-', 'R+'], lens: [t, pi_2, u, pi_2, v]
x = -u*cos(pi_2 + t) + 4*sin(t) - 2*sin(pi_2 + t) - sin(t - v)
y = -u*sin(pi_2 + t) - 4*cos(t) + 2*cos(pi_2 + t) + cos(t - v) + 1
phi = t - v
-> u*cos(t - pi_2) + 4*cos(t - pi_2) - 2*cos(t) = x + sin(phi)
-> u*sin(t - pi_2) + 4*sin(t - pi_2) - 2*sin(t) = y - cos(phi) - 1
->(u + 4)*cos(t - pi_2) - 2*cos(t) = x + sin(phi)
->(u + 4)*sin(t - pi_2) - 2*sin(t) = y - cos(phi) - 1
u1, t1 = polar(x + np.sin(phi), y - np.cos(phi) - 1)
作图分析，可以得到：
u = sqrt(u1^2 - 4) - 4
t = t1 + arctan(2/(u+4)) + pi/2
v = t - phi
"""
u1, t1 = _polar_trans(x + np.sin(phi), y - np.cos(phi) - 1)
if u1 >= 2 * np.sqrt(5.0):
u = np.sqrt(u1**2 - 4.0) - 4.0
t = (t1 + np.arctan(2.0 / (u + 4.0)) + _Pi_2) % _2_Pi
v = (t - phi) % _2_Pi
if t <= _Pi and v <= _Pi:
return True, t, u, v
return False, None, None, None
def _CCSCC(x, y, phi, sols):
flag, t, u, v = _LpRmSmLmRp(x, y, phi)
if flag: # LpRmSmLmRp
sols.append(
{
"type": ["Lp", "Rm", "Sm", "Lm", "Rp"],
"lens": [t, _Pi_2, u, _Pi_2, v],
"cost": t + u + v + _Pi,
}
)
flag, t, u, v = _LpRmSmLmRp(-x, y, -phi) # timeflip
if flag: # LmRpSpLpRm
sols.append(
{
"type": ["Lm", "Rp", "Sp", "Lp", "Rm"],
"lens": [t, _Pi_2, u, _Pi_2, v],
"cost": t + u + v + _Pi,
}
)
flag, t, u, v = _LpRmSmLmRp(x, -y, -phi) # reflect
if flag: # RpLmSmRmLp
sols.append(
{
"type": ["Rp", "Lm", "Sm", "Rm", "Lp"],
"lens": [t, _Pi_2, u, _Pi_2, v],
"cost": t + u + v + _Pi,
}
)
flag, t, u, v = _LpRmSmLmRp(-x, -y, phi) # timeflip + reflect
if flag: # RmLpSpRpLm
sols.append(
{
"type": ["Rm", "Lp", "Sp", "Rp", "Lm"],
"lens": [t, _Pi_2, u, _Pi_2, v],
"cost": t + u + v + _Pi,
}
)
@timer(enabled=CAR_PATH_MODULE_TIMER_ENABLED)
def rs_all_paths(x, y, phi, x_, y_, phi_, kappa):
loc = _corrd_trans(x, y, phi, x_, y_, phi_, kappa)
sols = []
_CSC(*loc, sols)
_CCC(*loc, sols)
_CCSC(*loc, sols)
_CSCC(*loc, sols)
_CCCC(*loc, sols)
_CCSCC(*loc, sols)
return sols
@timer(enabled=CAR_PATH_MODULE_TIMER_ENABLED)
def rs_shortest_path(x, y, phi, x_, y_, phi_, kappa):
"""
params:
x: x-coordinate of the current position
y: y-coordinate of the current position
phi: current heading angle
x_: x-coordinate of the target position
y_: y-coordinate of the target position
phi_: target heading angle
kappa: curvature of reeds-shepp path
step: step size of the path interpolation
returns:
a dictionary containing the shortest reeds-shepp path
and the corresponding x, y, and phi coordinates
"""
return min(
rs_all_paths(x, y, phi, x_, y_, phi_, kappa),
key=lambda sol: sol["cost"],
)
@timer(enabled=CAR_PATH_MODULE_TIMER_ENABLED)
def rs_shortest_path_length(x, y, phi, x_, y_, phi_, kappa):
"""
params:
x: x-coordinate of the current position
y: y-coordinate of the current position
phi: current heading angle
x_: x-coordinate of the target position
y_: y-coordinate of the target position
phi_: target heading angle
kappa: curvature of reeds-shepp path
returns:
cost: the cost of the path from the current position to the target position
"""
return rs_shortest_path(x, y, phi, x_, y_, phi_, kappa)["cost"]
def path_plot(x, y, phi, x_, y_, phi_, kappa, step=0.1, ax=None, path_type="dubins"):
if ax is None:
ax = plt.gca()
if path_type == "dubins":
sols = dubins_all_paths(x, y, phi, x_, y_, phi_, kappa)
elif path_type == "reeds-shepp":
sols = rs_all_paths(x, y, phi, x_, y_, phi_, kappa)
else:
raise ValueError("Invalid path_type")
for sol in sols:
path_x, path_y, path_phi = path_interp(
sol["type"], sol["lens"], step, kappa, x, y, phi
)
ax.plot(path_x, path_y, label=sol["type"])
opt_sol = min(sols, key=lambda sol: sol["cost"])
opt_path_x, opt_path_y, opt_path_phi = path_interp(
opt_sol["type"], opt_sol["lens"], step, kappa, x, y, phi
)
ax.scatter(
opt_path_x,
opt_path_y,
label=f"Shortest Path {opt_sol['type']})",
marker="x",
color="green",
)
ax.arrow(
x,
y,
np.cos(phi),
np.sin(phi),
head_width=0.1,
head_length=0.1,
color="red",
label="start",
)
ax.arrow(
x_,
y_,
np.cos(phi_),
np.sin(phi_),
head_width=0.1,
head_length=0.1,
color="green",
label="goal",
)