In [1]:
%matplotlib inline
%config InlineBackend.figure_format = 'retina'
import pdb
import time
import pylab as pl
from IPython import display
import matplotlib
import numpy as np
import matplotlib.pyplot as plt
import copy
import math
from cubic_spline_planner import *
In [2]:
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
In [3]:
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
In [4]:
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 = []
In [5]:
# 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
In [6]:
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
In [7]:
faTrajX = []
faTrajY = []
def calc_global_paths(fplist, csp):
#faTrajX = []
#faTrajY = []
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)
# Just for plotting
faTrajX.append(fp.x)
faTrajY.append(fp.y)
# 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
In [8]:
faTrajCollisionX = []
faTrajCollisionY = []
faObCollisionX = []
faObCollisionY = []
def check_collision(fp, ob):
#pdb.set_trace()
for i in range(len(ob[:, 0])):
# Calculate the distance for each trajectory point to the current object
d = [((ix - ob[i, 0])**2 + (iy - ob[i, 1])**2)
for (ix, iy) in zip(fp.x, fp.y)]
# Check if any trajectory point is too close to the object using the robot radius
collision = any([di <= ROBOT_RADIUS**2 for di in d])
if collision:
#plot(ft.x, ft.y, 'rx')
faTrajCollisionX.append(fp.x)
faTrajCollisionY.append(fp.y)
#plot(ox, oy, 'yo');
#pdb.set_trace()
if ob[i, 0] not in faObCollisionX or ob[i, 1] not in faObCollisionY:
faObCollisionX.append(ob[i, 0])
faObCollisionY.append(ob[i, 1])
return True
return False
In [9]:
#faTrajOkX = []
#faTrajOkY = []
def check_paths(fplist, ob):
okind = []
for i in range(len(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 check_collision(fplist[i], ob):
continue
okind.append(i)
return [fplist[i] for i in okind]
In [10]:
fpplist = []
def frenet_optimal_planning(csp, s0, c_speed, c_d, c_d_d, c_d_dd, ob):
#pdb.set_trace()
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)
#fpplist = deepcopy(fplist)
fpplist.extend(fplist)
# find minimum cost path
mincost = float("inf")
bestpath = None
for fp in fplist:
if mincost >= fp.cf:
mincost = fp.cf
bestpath = fp
return bestpath
In [11]:
from cubic_spline_planner import *
def generate_target_course(x, y):
csp = 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
In [12]:
show_animation = True
#show_animation = False
In [19]:
# 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]
fig = plt.figure()
faTx = tx
faTy = ty
faObx = ob[:, 0]
faOby = ob[:, 1]
faPathx = []
faPathy = []
faRobotx = []
faRoboty = []
faSpeed = []
for i in range(100):
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
faPathx.append(path.x[1:])
faPathy.append(path.y[1:])
faRobotx.append(path.x[1])
faRoboty.append(path.y[1])
faSpeed.append(c_speed)
if show_animation:
plt.cla()
plt.plot(tx, ty, animated=True)
plt.plot(ob[:, 0], ob[:, 1], "xk")
plt.plot(path.x[1], path.y[1], "vc")
for (ix, iy) in zip(faTrajX, faTrajY):
#pdb.set_trace()
plt.plot(ix[1:], iy[1:], '-', color=[0.5, 0.5, 0.5])
faTrajX = []
faTrajY = []
for (ix, iy) in zip(faTrajCollisionX, faTrajCollisionY):
#pdb.set_trace()
plt.plot(ix[1:], iy[1:], 'rx')
faTrajCollisionX = []
faTrajCollisionY = []
#pdb.set_trace()
for fp in fpplist:
#pdb.set_trace()
plt.plot(fp.x[1:], fp.y[1:], '-g')
fpplist = []
#pdb.set_trace()
for (ix, iy) in zip(faObCollisionX, faObCollisionY):
#pdb.set_trace()
plt.plot(ix, iy, 'oy')
faObCollisionX = []
faObCollisionY = []
plt.plot(path.x[1:], path.y[1:], "-ob")
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)
display.clear_output(wait=True)
display.display(pl.gcf())
#print("Finish")
#if show_animation:
#plt.grid(True)
#plt.pause(0.0001)
#plt.show()
In [20]:
from matplotlib import animation
fig, ax = plt.subplots();
ax.set_xlim((faPathx[1][1] - area, faPathx[1][1] + area))
ax.set_ylim((faPathy[1][1] - area, faPathy[1][1] + area))
h_ref = ax.plot([], [], "-b") # reference path
h_obstacles = ax.plot([], [], "xk") #obstacles to animate
h_path = ax.plot([], [], "-or") # trajectory to animate
h_car = ax.plot([], [], "vc")
patches = list(h_obstacles) + list(h_path) + list(h_ref) + list(h_car) #things to animate
In [21]:
def init():
#init lines
h_ref[0].set_data(faTx, faTy)
h_obstacles[0].set_data(faObx, faOby)
h_path[0].set_data([], [])
h_car[0].set_data([], [])
ax.set_xlabel('x')
ax.set_ylabel('y')
ax.grid(True)
return patches #return everything that must be updated
In [22]:
# animation function. This is called sequentially
def animate(i):
ax.set_xlim((faPathx[i][1] - area, faPathx[i][1] + area))
ax.set_ylim((faPathy[i][1] - area, faPathy[i][1] + area))
h_path[0].set_data(faPathx[i], faPathy[i])
h_car[0].set_data(faRobotx[i], faRoboty[i])
ax.set_title("v[km/h]:" + str(faSpeed[i] * 3.6)[0:4], fontsize=12)
return patches #return everything that must be updated
In [23]:
nFrames = len(faPathx)
anim = animation.FuncAnimation(fig, animate, init_func=init,
frames=nFrames, interval=50, blit=True)
plt.show()
In [24]:
from IPython.display import HTML
HTML(anim.to_html5_video())
Out[24]:
In [ ]: