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@AtsushiSakai @jwdinius
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"""
Frenet optimal trajectory generator
author: Atsushi Sakai (@Atsushi_twi)
Ref:
- [Optimal Trajectory Generation for Dynamic Street Scenarios in a Frenet Frame](https://www.researchgate.net/profile/Moritz_Werling/publication/224156269_Optimal_Trajectory_Generation_for_Dynamic_Street_Scenarios_in_a_Frenet_Frame/links/54f749df0cf210398e9277af.pdf)
- [Optimal trajectory generation for dynamic street scenarios in a Frenet Frame](https://www.youtube.com/watch?v=Cj6tAQe7UCY)
"""
import numpy as np
import matplotlib.pyplot as plt
import copy
import math
import cubic_spline_planner
SIM_LOOP = 500
# Parameter
MAX_SPEED = 50.0 / 3.6 # maximum speed [m/s]
MAX_ACCEL = 2.0 # maximum acceleration [m/ss]
MAX_CURVATURE = 1.0 # maximum curvature [1/m]
MAX_ROAD_WIDTH = 7.0 # maximum road width [m]
D_ROAD_W = 1.0 # road width sampling length [m]
DT = 0.2 # time tick [s]
MAXT = 5.0 # max prediction time [m]
MINT = 4.0 # min prediction time [m]
TARGET_SPEED = 30.0 / 3.6 # target speed [m/s]
D_T_S = 5.0 / 3.6 # target speed sampling length [m/s]
N_S_SAMPLE = 1 # sampling number of target speed
ROBOT_RADIUS = 2.0 # robot radius [m]
# cost weights
KJ = 0.1
KT = 0.1
KD = 1.0
KLAT = 1.0
KLON = 1.0
show_animation = True
class quintic_polynomial:
def __init__(self, xs, vxs, axs, xe, vxe, axe, T):
# calc coefficient of quintic polynomial
self.xs = xs
self.vxs = vxs
self.axs = axs
self.xe = xe
self.vxe = vxe
self.axe = axe
self.a0 = xs
self.a1 = vxs
self.a2 = axs / 2.0
A = np.array([[T**3, T**4, T**5],
[3 * T ** 2, 4 * T ** 3, 5 * T ** 4],
[6 * T, 12 * T ** 2, 20 * T ** 3]])
b = np.array([xe - self.a0 - self.a1 * T - self.a2 * T**2,
vxe - self.a1 - 2 * self.a2 * T,
axe - 2 * self.a2])
x = np.linalg.solve(A, b)
self.a3 = x[0]
self.a4 = x[1]
self.a5 = x[2]
def calc_point(self, t):
xt = self.a0 + self.a1 * t + self.a2 * t**2 + \
self.a3 * t**3 + self.a4 * t**4 + self.a5 * t**5
return xt
def calc_first_derivative(self, t):
xt = self.a1 + 2 * self.a2 * t + \
3 * self.a3 * t**2 + 4 * self.a4 * t**3 + 5 * self.a5 * t**4
return xt
def calc_second_derivative(self, t):
xt = 2 * self.a2 + 6 * self.a3 * t + 12 * self.a4 * t**2 + 20 * self.a5 * t**3
return xt
def calc_third_derivative(self, t):
xt = 6 * self.a3 + 24 * self.a4 * t + 60 * self.a5 * t**2
return xt
class quartic_polynomial:
def __init__(self, xs, vxs, axs, vxe, axe, T):
# calc coefficient of quintic polynomial
self.xs = xs
self.vxs = vxs
self.axs = axs
self.vxe = vxe
self.axe = axe
self.a0 = xs
self.a1 = vxs
self.a2 = axs / 2.0
A = np.array([[3 * T ** 2, 4 * T ** 3],
[6 * T, 12 * T ** 2]])
b = np.array([vxe - self.a1 - 2 * self.a2 * T,
axe - 2 * self.a2])
x = np.linalg.solve(A, b)
self.a3 = x[0]
self.a4 = x[1]
def calc_point(self, t):
xt = self.a0 + self.a1 * t + self.a2 * t**2 + \
self.a3 * t**3 + self.a4 * t**4
return xt
def calc_first_derivative(self, t):
xt = self.a1 + 2 * self.a2 * t + \
3 * self.a3 * t**2 + 4 * self.a4 * t**3
return xt
def calc_second_derivative(self, t):
xt = 2 * self.a2 + 6 * self.a3 * t + 12 * self.a4 * t**2
return xt
def calc_third_derivative(self, t):
xt = 6 * self.a3 + 24 * self.a4 * t
return xt
class Frenet_path:
def __init__(self):
self.t = []
self.d = []
self.d_d = []
self.d_dd = []
self.d_ddd = []
self.s = []
self.s_d = []
self.s_dd = []
self.s_ddd = []
self.cd = 0.0
self.cv = 0.0
self.cf = 0.0
self.x = []
self.y = []
self.yaw = []
self.ds = []
self.c = []
def calc_frenet_paths(c_speed, c_d, c_d_d, c_d_dd, s0):
frenet_paths = []
# generate path to each offset goal
for di in np.arange(-MAX_ROAD_WIDTH, MAX_ROAD_WIDTH, D_ROAD_W):
# Lateral motion planning
for Ti in np.arange(MINT, MAXT, DT):
fp = Frenet_path()
lat_qp = quintic_polynomial(c_d, c_d_d, c_d_dd, di, 0.0, 0.0, Ti)
fp.t = [t for t in np.arange(0.0, Ti, DT)]
fp.d = [lat_qp.calc_point(t) for t in fp.t]
fp.d_d = [lat_qp.calc_first_derivative(t) for t in fp.t]
fp.d_dd = [lat_qp.calc_second_derivative(t) for t in fp.t]
fp.d_ddd = [lat_qp.calc_third_derivative(t) for t in fp.t]
# Loongitudinal motion planning (Velocity keeping)
for tv in np.arange(TARGET_SPEED - D_T_S * N_S_SAMPLE, TARGET_SPEED + D_T_S * N_S_SAMPLE, D_T_S):
tfp = copy.deepcopy(fp)
lon_qp = quartic_polynomial(s0, c_speed, 0.0, tv, 0.0, Ti)
tfp.s = [lon_qp.calc_point(t) for t in fp.t]
tfp.s_d = [lon_qp.calc_first_derivative(t) for t in fp.t]
tfp.s_dd = [lon_qp.calc_second_derivative(t) for t in fp.t]
tfp.s_ddd = [lon_qp.calc_third_derivative(t) for t in fp.t]
Jp = sum(np.power(tfp.d_ddd, 2)) # square of jerk
Js = sum(np.power(tfp.s_ddd, 2)) # square of jerk
# square of diff from target speed
ds = (TARGET_SPEED - tfp.s_d[-1])**2
tfp.cd = KJ * Jp + KT * Ti + KD * tfp.d[-1]**2
tfp.cv = KJ * Js + KT * Ti + KD * ds
tfp.cf = KLAT * tfp.cd + KLON * tfp.cv
frenet_paths.append(tfp)
return frenet_paths
def calc_global_paths(fplist, csp):
for fp in fplist:
# calc global positions
for i in range(len(fp.s)):
ix, iy = csp.calc_position(fp.s[i])
if ix is None:
break
iyaw = csp.calc_yaw(fp.s[i])
di = fp.d[i]
fx = ix + di * math.cos(iyaw + math.pi / 2.0)
fy = iy + di * math.sin(iyaw + math.pi / 2.0)
fp.x.append(fx)
fp.y.append(fy)
# calc yaw and ds
for i in range(len(fp.x) - 1):
dx = fp.x[i + 1] - fp.x[i]
dy = fp.y[i + 1] - fp.y[i]
fp.yaw.append(math.atan2(dy, dx))
fp.ds.append(math.sqrt(dx**2 + dy**2))
fp.yaw.append(fp.yaw[-1])
fp.ds.append(fp.ds[-1])
# calc curvature
for i in range(len(fp.yaw) - 1):
fp.c.append((fp.yaw[i + 1] - fp.yaw[i]) / fp.ds[i])
return fplist
def check_collision(fp, ob):
for i in range(len(ob[:, 0])):
d = [((ix - ob[i, 0])**2 + (iy - ob[i, 1])**2)
for (ix, iy) in zip(fp.x, fp.y)]
collision = any([di <= ROBOT_RADIUS**2 for di in d])
if collision:
return False
return True
def check_paths(fplist, ob):
okind = []
for i, _ in enumerate(fplist):
if any([v > MAX_SPEED for v in fplist[i].s_d]): # Max speed check
continue
elif any([abs(a) > MAX_ACCEL for a in fplist[i].s_dd]): # Max accel check
continue
elif any([abs(c) > MAX_CURVATURE for c in fplist[i].c]): # Max curvature check
continue
elif not check_collision(fplist[i], ob):
continue
okind.append(i)
return [fplist[i] for i in okind]
def frenet_optimal_planning(csp, s0, c_speed, c_d, c_d_d, c_d_dd, ob):
fplist = calc_frenet_paths(c_speed, c_d, c_d_d, c_d_dd, s0)
fplist = calc_global_paths(fplist, csp)
fplist = check_paths(fplist, ob)
# find minimum cost path
mincost = float("inf")
bestpath = None
for fp in fplist:
if mincost >= fp.cf:
mincost = fp.cf
bestpath = fp
return bestpath
def generate_target_course(x, y):
csp = cubic_spline_planner.Spline2D(x, y)
s = np.arange(0, csp.s[-1], 0.1)
rx, ry, ryaw, rk = [], [], [], []
for i_s in s:
ix, iy = csp.calc_position(i_s)
rx.append(ix)
ry.append(iy)
ryaw.append(csp.calc_yaw(i_s))
rk.append(csp.calc_curvature(i_s))
return rx, ry, ryaw, rk, csp
def main():
print(__file__ + " start!!")
# way points
wx = [0.0, 10.0, 20.5, 35.0, 70.5]
wy = [0.0, -6.0, 5.0, 6.5, 0.0]
# obstacle lists
ob = np.array([[20.0, 10.0],
[30.0, 6.0],
[30.0, 8.0],
[35.0, 8.0],
[50.0, 3.0]
])
tx, ty, tyaw, tc, csp = generate_target_course(wx, wy)
# initial state
c_speed = 10.0 / 3.6 # current speed [m/s]
c_d = 2.0 # current lateral position [m]
c_d_d = 0.0 # current lateral speed [m/s]
c_d_dd = 0.0 # current latral acceleration [m/s]
s0 = 0.0 # current course position
area = 20.0 # animation area length [m]
for i in range(SIM_LOOP):
path = frenet_optimal_planning(
csp, s0, c_speed, c_d, c_d_d, c_d_dd, ob)
s0 = path.s[1]
c_d = path.d[1]
c_d_d = path.d_d[1]
c_d_dd = path.d_dd[1]
c_speed = path.s_d[1]
if np.hypot(path.x[1] - tx[-1], path.y[1] - ty[-1]) <= 1.0:
print("Goal")
break
if show_animation: # pragma: no cover
plt.cla()
plt.plot(tx, ty)
plt.plot(ob[:, 0], ob[:, 1], "xk")
plt.plot(path.x[1:], path.y[1:], "-or")
plt.plot(path.x[1], path.y[1], "vc")
plt.xlim(path.x[1] - area, path.x[1] + area)
plt.ylim(path.y[1] - area, path.y[1] + area)
plt.title("v[km/h]:" + str(c_speed * 3.6)[0:4])
plt.grid(True)
plt.pause(0.0001)
print("Finish")
if show_animation: # pragma: no cover
plt.grid(True)
plt.pause(0.0001)
plt.show()
if __name__ == '__main__':
main()