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Merge pull request #152 from jwdinius/eta3_traj2

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polishing up continuous velocity profile overlay on path
This commit is contained in:
Atsushi Sakai
2019-01-13 08:17:37 +09:00
committed by GitHub
parent 5a6970ecc6 3938079d0a
commit 8bacdd4e72
2 changed files with 450 additions and 4 deletions
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23
PathPlanning/Eta3SplinePath/eta3_spline_path.py
View File

@@ -152,10 +152,10 @@ class eta3_path_segment(object):
+ (10. * eta[1] - 2. * eta[3] + 1. / 6 * eta[5]) * sb \
- (2. * eta[1]**2 * kappa[2] - 1. / 6 * eta[1]**3 *
kappa[3] - 1. / 2 * eta[1] * eta[3] * kappa[2]) * cb
def s_dot(u): return np.linalg.norm(self.coeffs[:, 1:].dot(
np.array([1, 2. * u, 3. * u**2, 4. * u**3, 5. * u**4, 6. * u**5, 7. * u**6])))
self.segment_length = quad(lambda u: s_dot(u), 0, 1)[0]
self.s_dot = lambda u : max(np.linalg.norm(self.coeffs[:, 1:].dot(np.array([1, 2.*u, 3.*u**2, 4.*u**3, 5.*u**4, 6.*u**5, 7.*u**6]))), 1e-6)
self.f_length = lambda ue: quad(lambda u: self.s_dot(u), 0, ue)
self.segment_length = self.f_length(1)[0]
"""
eta3_path_segment::calc_point
@@ -170,6 +170,21 @@ class eta3_path_segment(object):
assert(u >= 0 and u <= 1)
return self.coeffs.dot(np.array([1, u, u**2, u**3, u**4, u**5, u**6, u**7]))
"""
eta3_path_segment::calc_deriv
input
u - parametric representation of a point along the segment, 0 <= u <= 1
returns
(d^nx/du^n,d^ny/du^n) of point along the segment, for 0 < n <= 2
"""
def calc_deriv(self, u, order=1):
assert(u >= 0 and u <= 1)
assert(order > 0 and order <= 2)
if order == 1:
return self.coeffs[:, 1:].dot(np.array([1, 2.*u, 3.*u**2, 4.*u**3, 5.*u**4, 6.*u**5, 7.*u**6]))
else:
return self.coeffs[:, 2:].dot(np.array([2, 6.*u, 12.*u**2, 20.*u**3, 30.*u**4, 42.*u**5]))
def test1():

431
PathPlanning/Eta3SplineTrajectory/eta3_spline_trajectory.py Normal file
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@@ -0,0 +1,431 @@
"""
\eta^3 polynomials trajectory planner
author: Joe Dinius, Ph.D (https://jwdinius.github.io)
Atsushi Sakai (@Atsushi_twi)
Refs:
- https://jwdinius.github.io/blog/2018/eta3traj
- [\eta^3-Splines for the Smooth Path Generation of Wheeled Mobile Robots](https://ieeexplore.ieee.org/document/4339545/)
"""
import numpy as np
import matplotlib.pyplot as plt
from matplotlib.collections import LineCollection
import sys
import os
sys.path.append(os.path.relpath("../Eta3SplinePath"))
from eta3_spline_path import eta3_path, eta3_path_segment
show_animation = True
class MaxVelocityNotReached(Exception):
def __init__(self, actual_vel, max_vel):
self.message = 'Actual velocity {} does not equal desired max velocity {}!'.format(actual_vel, max_vel)
class eta3_trajectory(eta3_path):
"""
eta3_trajectory
input
segments: list of `eta3_trajectory_segment` instances defining a continuous trajectory
"""
def __init__(self, segments, max_vel, v0=0.0, a0=0.0, max_accel=2.0, max_jerk=5.0):
# ensure that all inputs obey the assumptions of the model
assert max_vel > 0 and v0 >= 0 and a0 >= 0 and max_accel > 0 and max_jerk > 0 \
and a0 <= max_accel and v0 <= max_vel
super(eta3_trajectory, self).__init__(segments=segments)
self.total_length = sum([s.segment_length for s in self.segments])
self.max_vel = float(max_vel)
self.v0 = float(v0)
self.a0 = float(a0)
self.max_accel = float(max_accel)
self.max_jerk = float(max_jerk)
length_array = np.array([s.segment_length for s in self.segments])
# add a zero to the beginning for finding the correct segment_id
self.cum_lengths = np.concatenate((np.array([0]), np.cumsum(length_array)))
## compute velocity profile on top of the path
self.velocity_profile()
self.ui_prev = 0
self.prev_seg_id = 0
def velocity_profile(self):
''' /~~~~~----------------~~~~~\
/ \
/ \
/ \
/ \
(v=v0, a=a0) ~~~~~ \
\
\~~~~~ (vf=0, af=0)
pos.|pos.|neg.| cruise at |neg.| neg. |neg.
max |max.|max.| max. |max.| max. |max.
jerk|acc.|jerk| velocity |jerk| acc. |jerk
index 0 1 2 3 (optional) 4 5 6
'''
# delta_a: accel change from initial position to end of maximal jerk section
delta_a = self.max_accel - self.a0
# t_s1: time of traversal of maximal jerk section
t_s1 = delta_a / self.max_jerk
# v_s1: velocity at the end of the maximal jerk section
v_s1 = self.v0 + self.a0 * t_s1 + self.max_jerk * t_s1**2 / 2.
# s_s1: length of the maximal jerk section
s_s1 = self.v0 * t_s1 + self.a0 * t_s1**2 / 2. + self.max_jerk * t_s1**3 / 6.
# t_sf: time of traversal of final section, which is also maximal jerk, but has final velocity 0
t_sf = self.max_accel / self.max_jerk
# v_sf: velocity at beginning of final section
v_sf = self.max_jerk * t_sf**2 / 2.
# s_sf: length of final section
s_sf = self.max_jerk * t_sf**3 / 6.
# solve for the maximum achievable velocity based on the kinematic limits imposed by max_accel and max_jerk
# this leads to a quadratic equation in v_max: a*v_max**2 + b*v_max + c = 0
a = 1 / self.max_accel
b = 3. * self.max_accel / (2. * self.max_jerk) + v_s1 / self.max_accel - (self.max_accel**2 / self.max_jerk + v_s1) / self.max_accel
c = s_s1 + s_sf - self.total_length - 7. * self.max_accel**3 / (3. * self.max_jerk**2) \
- v_s1 * (self.max_accel / self.max_jerk + v_s1 / self.max_accel) \
+ (self.max_accel**2 / self.max_jerk + v_s1 / self.max_accel)**2 / (2. * self.max_accel)
v_max = ( -b + np.sqrt(b**2 - 4. * a * c) ) / (2. * a)
# v_max represents the maximum velocity that could be attained if there was no cruise period
# (i.e. driving at constant speed without accelerating or jerking)
# if this velocity is less than our desired max velocity, the max velocity needs to be updated
# XXX the way to handle this `if` condition needs to be more thoroughly worked through
if self.max_vel > v_max:
# when this condition is tripped, there will be no cruise period (s_cruise=0)
self.max_vel = v_max
# setup arrays to store values at END of trajectory sections
self.times = np.zeros((7,))
self.vels = np.zeros((7,))
self.seg_lengths = np.zeros((7,))
# Section 0: max jerk up to max acceleration
index = 0
self.times[0] = t_s1
self.vels[0] = v_s1
self.seg_lengths[0] = s_s1
# Section 1: accelerate at max_accel
index = 1
# compute change in velocity over the section
delta_v = (self.max_vel - self.max_jerk * (self.max_accel / self.max_jerk)**2 / 2.) - self.vels[index-1]
self.times[index] = delta_v / self.max_accel
self.vels[index] = self.vels[index-1] + self.max_accel * self.times[index]
self.seg_lengths[index] = self.vels[index-1] * self.times[index] + self.max_accel * self.times[index]**2 / 2.
# Section 2: decrease acceleration (down to 0) until max speed is hit
index = 2
self.times[index] = self.max_accel / self.max_jerk
self.vels[index] = self.vels[index-1] + self.max_accel * self.times[index] \
- self.max_jerk * self.times[index]**2 / 2.
# as a check, the velocity at the end of the section should be self.max_vel
if not np.isclose(self.vels[index], self.max_vel):
raise MaxVelocityNotReached(self.vels[index], self.max_vel)
self.seg_lengths[index] = self.vels[index-1] * self.times[index] + self.max_accel * self.times[index]**2 / 2. \
- self.max_jerk * self.times[index]**3 / 6.
# Section 3: will be done last
# Section 4: apply min jerk until min acceleration is hit
index = 4
self.times[index] = self.max_accel / self.max_jerk
self.vels[index] = self.max_vel - self.max_jerk * self.times[index]**2 / 2.
self.seg_lengths[index] = self.max_vel * self.times[index] - self.max_jerk * self.times[index]**3 / 6.
# Section 5: continue deceleration at max rate
index = 5
# compute velocity change over sections
delta_v = self.vels[index-1] - v_sf
self.times[index] = delta_v / self.max_accel
self.vels[index] = self.vels[index-1] - self.max_accel * self.times[index]
self.seg_lengths[index] = self.vels[index-1] * self.times[index] - self.max_accel * self.times[index]**2 / 2.
# Section 6(final): max jerk to get to zero velocity and zero acceleration simultaneously
index = 6
self.times[index] = t_sf
self.vels[index] = self.vels[index-1] - self.max_jerk * t_sf**2 / 2.
try:
assert np.isclose(self.vels[index], 0)
except AssertionError as e:
print('The final velocity {} is not zero'.format(self.vels[index]))
raise e
self.seg_lengths[index] = s_sf
if self.seg_lengths.sum() < self.total_length:
index = 3
# the length of the cruise section is whatever length hasn't already been accounted for
# NOTE: the total array self.seg_lengths is summed because the entry for the cruise segment is
# initialized to 0!
self.seg_lengths[index] = self.total_length - self.seg_lengths.sum()
self.vels[index] = self.max_vel
self.times[index] = self.seg_lengths[index] / self.max_vel
# make sure that all of the times are positive, otherwise the kinematic limits
# chosen cannot be enforced on the path
assert(np.all(self.times >= 0))
self.total_time = self.times.sum()
def get_interp_param(self, seg_id, s, ui, tol=0.001):
f = lambda u: self.segments[seg_id].f_length(u)[0] - s
fprime = lambda u: self.segments[seg_id].s_dot(u)
while (ui >= 0 and ui <= 1) and abs(f(ui)) > tol:
ui -= f(ui) / fprime(ui)
ui = max(0, min(ui, 1))
return ui
def calc_traj_point(self, time):
# compute velocity at time
if time <= self.times[0]:
linear_velocity = self.v0 + self.max_jerk * time**2 / 2.
s = self.v0 * time + self.max_jerk * time**3 / 6
linear_accel = self.max_jerk * time
elif time <= self.times[:2].sum():
delta_t = time - self.times[0]
linear_velocity = self.vels[0] + self.max_accel * delta_t
s = self.seg_lengths[0] + self.vels[0] * delta_t + self.max_accel * delta_t**2 / 2.
linear_accel = self.max_accel
elif time <= self.times[:3].sum():
delta_t = time - self.times[:2].sum()
linear_velocity = self.vels[1] + self.max_accel * delta_t - self.max_jerk * delta_t**2 / 2.
s = self.seg_lengths[:2].sum() + self.vels[1] * delta_t + self.max_accel * delta_t**2 /2. \
- self.max_jerk * delta_t**3 / 6.
linear_accel = self.max_accel - self.max_jerk * delta_t
elif time <= self.times[:4].sum():
delta_t = time - self.times[:3].sum()
linear_velocity = self.vels[3]
s = self.seg_lengths[:3].sum() + self.vels[3] * delta_t
linear_accel = 0.
elif time <= self.times[:5].sum():
delta_t = time - self.times[:4].sum()
linear_velocity = self.vels[3] - self.max_jerk * delta_t**2 / 2.
s = self.seg_lengths[:4].sum() + self.vels[3] * delta_t - self.max_jerk * delta_t**3 / 6.
linear_accel = -self.max_jerk * delta_t
elif time <= self.times[:-1].sum():
delta_t = time - self.times[:5].sum()
linear_velocity = self.vels[4] - self.max_accel * delta_t
s = self.seg_lengths[:5].sum() + self.vels[4] * delta_t - self.max_accel * delta_t**2 / 2.
linear_accel = -self.max_accel
elif time < self.times.sum():
delta_t = time - self.times[:-1].sum()
linear_velocity = self.vels[5] - self.max_accel * delta_t + self.max_jerk * delta_t**2 / 2.
s = self.seg_lengths[:-1].sum() + self.vels[5] * delta_t - self.max_accel * delta_t**2 / 2. \
+ self.max_jerk * delta_t**3 / 6.
linear_accel = -self.max_accel + self.max_jerk*delta_t
else:
linear_velocity = 0.
s = self.total_length
linear_accel = 0.
seg_id = np.max(np.argwhere(self.cum_lengths <= s))
# will happen at the end of the segment
if seg_id == len(self.segments):
seg_id -= 1
ui = 1
else:
# compute interpolation parameter using length from current segment's starting point
curr_segment_length = s - self.cum_lengths[seg_id]
ui = self.get_interp_param(seg_id=seg_id, s=curr_segment_length, ui=self.ui_prev)
if not seg_id == self.prev_seg_id:
self.ui_prev = 0
else:
self.ui_prev = ui
self.prev_seg_id = seg_id
# TODO(jwd): normalize!
# compute angular velocity of current point= (ydd*xd - xdd*yd) / (xd**2 + yd**2)
d = self.segments[seg_id].calc_deriv(ui, order=1)
dd = self.segments[seg_id].calc_deriv(ui, order=2)
# su - the rate of change of arclength wrt u
su = self.segments[seg_id].s_dot(ui)
if not np.isclose(su, 0.) and not np.isclose(linear_velocity, 0.):
# ut - time-derivative of interpolation parameter u
ut = linear_velocity / su
# utt - time-derivative of ut
utt = linear_accel / su - (d[0] * dd[0] + d[1] * dd[1]) / su**2 * ut
xt = d[0] * ut
yt = d[1] * ut
xtt = dd[0] * ut**2 + d[0] * utt
ytt = dd[1] * ut**2 + d[1] * utt
angular_velocity = (ytt * xt - xtt * yt) / linear_velocity**2
else:
angular_velocity = 0.
# combine path point with orientation and velocities
pos = self.segments[seg_id].calc_point(ui)
state = np.array([pos[0], pos[1], np.arctan2(d[1], d[0]), linear_velocity, angular_velocity])
return state
def test1(max_vel=0.5):
for i in range(10):
trajectory_segments = []
# segment 1: lane-change curve
start_pose = [0, 0, 0]
end_pose = [4, 3.0, 0]
# NOTE: The ordering on kappa is [kappa_A, kappad_A, kappa_B, kappad_B], with kappad_* being the curvature derivative
kappa = [0, 0, 0, 0]
eta = [i, i, 0, 0, 0, 0]
trajectory_segments.append(eta3_path_segment(
start_pose=start_pose, end_pose=end_pose, eta=eta, kappa=kappa))
traj = eta3_trajectory(trajectory_segments, max_vel=max_vel, max_accel=0.5)
# interpolate at several points along the path
times = np.linspace(0, traj.total_time, 101)
state = np.empty((5, times.size))
for j, t in enumerate(times):
state[:, j] = traj.calc_traj_point(t)
if show_animation:
# plot the path
plt.plot(state[0, :], state[1, :])
plt.pause(1.0)
plt.show()
if show_animation:
plt.close("all")
def test2(max_vel=0.5):
for i in range(10):
trajectory_segments = []
# segment 1: lane-change curve
start_pose = [0, 0, 0]
end_pose = [4, 3.0, 0]
# NOTE: The ordering on kappa is [kappa_A, kappad_A, kappa_B, kappad_B], with kappad_* being the curvature derivative
kappa = [0, 0, 0, 0]
# NOTE: INTEGRATOR ERROR EXPLODES WHEN eta[:1] IS ZERO!
# was: eta = [0, 0, (i - 5) * 20, (5 - i) * 20, 0, 0], now is:
eta = [0.1, 0.1, (i - 5) * 20, (5 - i) * 20, 0, 0]
trajectory_segments.append(eta3_path_segment(
start_pose=start_pose, end_pose=end_pose, eta=eta, kappa=kappa))
traj = eta3_trajectory(trajectory_segments, max_vel=max_vel, max_accel=0.5)
# interpolate at several points along the path
times = np.linspace(0, traj.total_time, 101)
state = np.empty((5, times.size))
for j, t in enumerate(times):
state[:, j] = traj.calc_traj_point(t)
if show_animation:
# plot the path
plt.plot(state[0, :], state[1, :])
plt.pause(1.0)
plt.show()
if show_animation:
plt.close("all")
def test3(max_vel=2.0):
trajectory_segments = []
# segment 1: lane-change curve
start_pose = [0, 0, 0]
end_pose = [4, 1.5, 0]
# NOTE: The ordering on kappa is [kappa_A, kappad_A, kappa_B, kappad_B], with kappad_* being the curvature derivative
kappa = [0, 0, 0, 0]
eta = [4.27, 4.27, 0, 0, 0, 0]
trajectory_segments.append(eta3_path_segment(
start_pose=start_pose, end_pose=end_pose, eta=eta, kappa=kappa))
# segment 2: line segment
start_pose = [4, 1.5, 0]
end_pose = [5.5, 1.5, 0]
kappa = [0, 0, 0, 0]
# NOTE: INTEGRATOR ERROR EXPLODES WHEN eta[:1] IS ZERO!
#was: eta = [0, 0, 0, 0, 0, 0], now is:
eta = [0.5, 0.5, 0, 0, 0, 0]
trajectory_segments.append(eta3_path_segment(
start_pose=start_pose, end_pose=end_pose, eta=eta, kappa=kappa))
# segment 3: cubic spiral
start_pose = [5.5, 1.5, 0]
end_pose = [7.4377, 1.8235, 0.6667]
kappa = [0, 0, 1, 1]
eta = [1.88, 1.88, 0, 0, 0, 0]
trajectory_segments.append(eta3_path_segment(
start_pose=start_pose, end_pose=end_pose, eta=eta, kappa=kappa))
# segment 4: generic twirl arc
start_pose = [7.4377, 1.8235, 0.6667]
end_pose = [7.8, 4.3, 1.8]
kappa = [1, 1, 0.5, 0]
eta = [7, 10, 10, -10, 4, 4]
trajectory_segments.append(eta3_path_segment(
start_pose=start_pose, end_pose=end_pose, eta=eta, kappa=kappa))
# segment 5: circular arc
start_pose = [7.8, 4.3, 1.8]
end_pose = [5.4581, 5.8064, 3.3416]
kappa = [0.5, 0, 0.5, 0]
eta = [2.98, 2.98, 0, 0, 0, 0]
trajectory_segments.append(eta3_path_segment(
start_pose=start_pose, end_pose=end_pose, eta=eta, kappa=kappa))
# construct the whole path
traj = eta3_trajectory(trajectory_segments, max_vel=max_vel, max_accel=0.5, max_jerk=1)
# interpolate at several points along the path
times = np.linspace(0, traj.total_time, 1001)
state = np.empty((5, times.size))
for i, t in enumerate(times):
state[:, i] = traj.calc_traj_point(t)
# plot the path
if show_animation:
fig, ax = plt.subplots()
x, y = state[0, :], state[1, :]
points = np.array([x, y]).T.reshape(-1, 1, 2)
segs = np.concatenate([points[:-1], points[1:]], axis=1)
lc = LineCollection(segs, cmap=plt.get_cmap('inferno'))
ax.set_xlim(np.min(x)-1, np.max(x)+1)
ax.set_ylim(np.min(y)-1, np.max(y)+1)
lc.set_array(state[3, :])
lc.set_linewidth(3)
ax.add_collection(lc)
axcb = fig.colorbar(lc)
axcb.set_label('velocity(m/s)')
ax.set_title('Trajectory')
plt.xlabel('x')
plt.ylabel('y')
plt.pause(1.0)
fig1, ax1 = plt.subplots()
ax1.plot(times, state[3, :], 'b-')
ax1.set_xlabel('time(s)')
ax1.set_ylabel('velocity(m/s)', color='b')
ax1.tick_params('y', colors='b')
ax1.set_title('Control')
ax2 = ax1.twinx()
ax2.plot(times, state[4, :], 'r-')
ax2.set_ylabel('angular velocity(rad/s)', color='r')
ax2.tick_params('y', colors='r')
fig.tight_layout()
plt.show()
def main():
"""
recreate path from reference (see Table 1)
"""
#test1()
#test2()
test3()
if __name__ == '__main__':
main()