420 lines
14 KiB
Python
420 lines
14 KiB
Python
import numpy as np
|
|
from abc import abstractmethod
|
|
|
|
try:
|
|
from abc import ABC
|
|
except:
|
|
# for Python 2.7
|
|
from abc import ABCMeta
|
|
|
|
|
|
class ABC(object):
|
|
__metaclass__ = ABCMeta
|
|
pass
|
|
import matplotlib.pyplot as plt
|
|
import matplotlib.patches as plt_patches
|
|
import math
|
|
|
|
# Colors
|
|
CAR = '#F1C40F'
|
|
CAR_OUTLINE = '#B7950B'
|
|
|
|
|
|
#########################
|
|
# Temporal State Vector #
|
|
#########################
|
|
|
|
class TemporalState:
|
|
def __init__(self, x, y, psi):
|
|
"""
|
|
Temporal State Vector containing car pose (x, y, psi)
|
|
:param x: x position in global coordinate system | [m]
|
|
:param y: y position in global coordinate system | [m]
|
|
:param psi: yaw angle | [rad]
|
|
"""
|
|
self.x = x
|
|
self.y = y
|
|
self.psi = psi
|
|
|
|
self.members = ['x', 'y', 'psi']
|
|
|
|
def __iadd__(self, other):
|
|
"""
|
|
Overload Sum-Add operator.
|
|
:param other: numpy array to be added to state vector
|
|
"""
|
|
for state_id in range(len(self.members)):
|
|
vars(self)[self.members[state_id]] += other[state_id]
|
|
return self
|
|
|
|
|
|
########################
|
|
# Spatial State Vector #
|
|
########################
|
|
|
|
class SpatialState(ABC):
|
|
"""
|
|
Spatial State Vector - Abstract Base Class.
|
|
"""
|
|
|
|
@abstractmethod
|
|
def __init__(self):
|
|
self.members = None
|
|
self.e_y = None
|
|
self.e_psi = None
|
|
|
|
def __getitem__(self, item):
|
|
if isinstance(item, int):
|
|
members = [self.members[item]]
|
|
else:
|
|
members = self.members[item]
|
|
return [vars(self)[key] for key in members]
|
|
|
|
def __setitem__(self, key, value):
|
|
vars(self)[self.members[key]] = value
|
|
|
|
def __len__(self):
|
|
return len(self.members)
|
|
|
|
def __iadd__(self, other):
|
|
"""
|
|
Overload Sum-Add operator.
|
|
:param other: numpy array to be added to state vector
|
|
"""
|
|
|
|
for state_id in range(len(self.members)):
|
|
vars(self)[self.members[state_id]] += other[state_id]
|
|
return self
|
|
|
|
def list_states(self):
|
|
"""
|
|
Return list of names of all states.
|
|
"""
|
|
return self.members
|
|
|
|
|
|
class SimpleSpatialState(SpatialState):
|
|
def __init__(self, e_y=0.0, e_psi=0.0, t=0.0):
|
|
"""
|
|
Simplified Spatial State Vector containing orthogonal deviation from
|
|
reference path (e_y), difference in orientation (e_psi) and velocity
|
|
:param e_y: orthogonal deviation from center-line | [m]
|
|
:param e_psi: yaw angle relative to path | [rad]
|
|
:param t: time | [s]
|
|
"""
|
|
super(SimpleSpatialState, self).__init__()
|
|
|
|
self.e_y = e_y
|
|
self.e_psi = e_psi
|
|
self.t = t
|
|
|
|
self.members = ['e_y', 'e_psi', 't']
|
|
|
|
|
|
####################################
|
|
# Spatial Bicycle Model Base Class #
|
|
####################################
|
|
|
|
class SpatialBicycleModel(ABC):
|
|
def __init__(self, reference_path, length, width, Ts):
|
|
"""
|
|
Abstract Base Class for Spatial Reformulation of Bicycle Model.
|
|
:param reference_path: reference path object to follow
|
|
:param length: length of car in m
|
|
:param width: width of car in m
|
|
:param Ts: sampling time of model
|
|
"""
|
|
|
|
# Precision
|
|
self.eps = 1e-12
|
|
|
|
# Car Parameters
|
|
self.length = length
|
|
self.width = width
|
|
self.safety_margin = self._compute_safety_margin()
|
|
|
|
# Reference Path
|
|
self.reference_path = reference_path
|
|
|
|
# Set initial distance traveled
|
|
self.s = 0.0
|
|
|
|
# Set sampling time
|
|
self.Ts = Ts
|
|
|
|
# Set initial waypoint ID
|
|
self.wp_id = 0
|
|
|
|
# Set initial waypoint
|
|
self.current_waypoint = self.reference_path.waypoints[self.wp_id]
|
|
|
|
# Declare spatial state variable | Initialization in sub-class
|
|
self.spatial_state = None
|
|
|
|
# Declare temporal state variable | Initialization in sub-class
|
|
self.temporal_state = None
|
|
|
|
def s2t(self, reference_waypoint, reference_state):
|
|
"""
|
|
Convert spatial state to temporal state given a reference waypoint.
|
|
:param reference_waypoint: waypoint object to use as reference
|
|
:param reference_state: state vector as np.array to use as reference
|
|
:return Temporal State equivalent to reference state
|
|
"""
|
|
|
|
# Compute temporal state variables
|
|
if isinstance(reference_state, np.ndarray):
|
|
x = reference_waypoint.x - reference_state[0] * np.sin(
|
|
reference_waypoint.psi)
|
|
y = reference_waypoint.y + reference_state[0] * np.cos(
|
|
reference_waypoint.psi)
|
|
psi = reference_waypoint.psi + reference_state[1]
|
|
elif isinstance(reference_state, SpatialState):
|
|
x = reference_waypoint.x - reference_state.e_y * np.sin(
|
|
reference_waypoint.psi)
|
|
y = reference_waypoint.y + reference_state.e_y * np.cos(
|
|
reference_waypoint.psi)
|
|
psi = reference_waypoint.psi + reference_state.e_psi
|
|
else:
|
|
print('Reference State type not supported!')
|
|
x, y, psi = None, None, None
|
|
exit(1)
|
|
|
|
return TemporalState(x, y, psi)
|
|
|
|
def t2s(self, reference_waypoint, reference_state):
|
|
"""
|
|
Convert spatial state to temporal state. Either convert self.spatial_
|
|
state with current waypoint as reference or provide reference waypoint
|
|
and reference_state.
|
|
:return Spatial State equivalent to reference state
|
|
"""
|
|
|
|
# Compute spatial state variables
|
|
if isinstance(reference_state, np.ndarray):
|
|
e_y = np.cos(reference_waypoint.psi) * \
|
|
(reference_state[1] - reference_waypoint.y) - \
|
|
np.sin(reference_waypoint.psi) * (reference_state[0] -
|
|
reference_waypoint.x)
|
|
e_psi = reference_state[2] - reference_waypoint.psi
|
|
|
|
# Ensure e_psi is kept within range (-pi, pi]
|
|
e_psi = np.mod(e_psi + math.pi, 2 * math.pi) - math.pi
|
|
elif isinstance(reference_state, TemporalState):
|
|
e_y = np.cos(reference_waypoint.psi) * \
|
|
(reference_state.y - reference_waypoint.y) - \
|
|
np.sin(reference_waypoint.psi) * (reference_state.x -
|
|
reference_waypoint.x)
|
|
e_psi = reference_state.psi - reference_waypoint.psi
|
|
|
|
# Ensure e_psi is kept within range (-pi, pi]
|
|
e_psi = np.mod(e_psi + math.pi, 2 * math.pi) - math.pi
|
|
else:
|
|
print('Reference State type not supported!')
|
|
e_y, e_psi = None, None
|
|
exit(1)
|
|
|
|
# time state can be set to zero since it's only relevant for the MPC
|
|
# prediction horizon
|
|
t = 0.0
|
|
|
|
return SimpleSpatialState(e_y, e_psi, t)
|
|
|
|
def drive(self, u):
|
|
"""
|
|
Drive.
|
|
:param u: input vector containing [v, delta]
|
|
"""
|
|
|
|
# Get input signals
|
|
v, delta = u
|
|
|
|
# Compute temporal state derivatives
|
|
x_dot = v * np.cos(self.temporal_state.psi)
|
|
y_dot = v * np.sin(self.temporal_state.psi)
|
|
psi_dot = v / self.length * np.tan(delta)
|
|
temporal_derivatives = np.array([x_dot, y_dot, psi_dot])
|
|
|
|
# Update spatial state (Forward Euler Approximation)
|
|
self.temporal_state += temporal_derivatives * self.Ts
|
|
|
|
# Compute velocity along path
|
|
s_dot = 1 / (1 - self.spatial_state.e_y * self.current_waypoint.kappa) \
|
|
* v * np.cos(self.spatial_state.e_psi)
|
|
|
|
# Update distance travelled along reference path
|
|
self.s += s_dot * self.Ts
|
|
|
|
def _compute_safety_margin(self):
|
|
"""
|
|
Compute safety margin for car if modeled by its center of gravity.
|
|
"""
|
|
|
|
# Model ellipsoid around the car
|
|
safety_margin = self.width / np.sqrt(2)
|
|
|
|
return safety_margin
|
|
|
|
def get_current_waypoint(self):
|
|
"""
|
|
Get closest waypoint on reference path based on car's current location.
|
|
"""
|
|
|
|
# Compute cumulative path length
|
|
length_cum = np.cumsum(self.reference_path.segment_lengths)
|
|
# Get first index with distance larger than distance traveled by car
|
|
# so far
|
|
greater_than_threshold = length_cum > self.s
|
|
next_wp_id = greater_than_threshold.searchsorted(True)
|
|
# Get previous index
|
|
prev_wp_id = next_wp_id - 1
|
|
|
|
# Get distance traveled for both enclosing waypoints
|
|
s_next = length_cum[next_wp_id]
|
|
s_prev = length_cum[prev_wp_id]
|
|
|
|
if np.abs(self.s - s_next) < np.abs(self.s - s_prev):
|
|
self.wp_id = next_wp_id
|
|
self.current_waypoint = self.reference_path.waypoints[next_wp_id]
|
|
else:
|
|
self.wp_id = prev_wp_id
|
|
self.current_waypoint = self.reference_path.waypoints[prev_wp_id]
|
|
|
|
def show(self):
|
|
"""
|
|
Display car on current axis.
|
|
"""
|
|
|
|
# Get car's center of gravity
|
|
cog = (self.temporal_state.x, self.temporal_state.y)
|
|
# Get current angle with respect to x-axis
|
|
yaw = np.rad2deg(self.temporal_state.psi)
|
|
# Draw rectangle
|
|
car = plt_patches.Rectangle(cog, width=self.length, height=self.width,
|
|
angle=yaw, facecolor=CAR,
|
|
edgecolor=CAR_OUTLINE, zorder=20)
|
|
|
|
# Shift center rectangle to match center of the car
|
|
car.set_x(car.get_x() - (self.length / 2 *
|
|
np.cos(self.temporal_state.psi) -
|
|
self.width / 2 *
|
|
np.sin(self.temporal_state.psi)))
|
|
car.set_y(car.get_y() - (self.width / 2 *
|
|
np.cos(self.temporal_state.psi) +
|
|
self.length / 2 *
|
|
np.sin(self.temporal_state.psi)))
|
|
|
|
# Add rectangle to current axis
|
|
ax = plt.gca()
|
|
ax.add_patch(car)
|
|
|
|
@abstractmethod
|
|
def get_spatial_derivatives(self, state, input, kappa):
|
|
pass
|
|
|
|
@abstractmethod
|
|
def linearize(self, v_ref, kappa_ref, delta_s):
|
|
pass
|
|
|
|
|
|
#################
|
|
# Bicycle Model #
|
|
#################
|
|
|
|
class BicycleModel(SpatialBicycleModel):
|
|
def __init__(self, reference_path, length, width, Ts):
|
|
"""
|
|
Simplified Spatial Bicycle Model. Spatial Reformulation of Kinematic
|
|
Bicycle Model. Uses Simplified Spatial State.
|
|
:param reference_path: reference path model is supposed to follow
|
|
:param length: length of the car in m
|
|
:param width: with of the car in m
|
|
:param Ts: sampling time of model in s
|
|
"""
|
|
|
|
# Initialize base class
|
|
super(BicycleModel, self).__init__(reference_path, length=length,
|
|
width=width, Ts=Ts)
|
|
|
|
# Initialize spatial state
|
|
self.spatial_state = SimpleSpatialState()
|
|
|
|
# Number of spatial state variables
|
|
self.n_states = len(self.spatial_state)
|
|
|
|
# Initialize temporal state
|
|
self.temporal_state = self.s2t(reference_state=self.spatial_state,
|
|
reference_waypoint=self.current_waypoint)
|
|
|
|
def get_temporal_derivatives(self, state, input, kappa):
|
|
"""
|
|
Compute relevant temporal derivatives needed for state update.
|
|
:param state: state vector for which to compute derivatives
|
|
:param input: input vector
|
|
:param kappa: curvature of corresponding waypoint
|
|
:return: temporal derivatives of distance, angle and velocity
|
|
"""
|
|
|
|
# Get state and input variables
|
|
e_y, e_psi, t = state
|
|
v, delta = input
|
|
|
|
# Compute velocity along path
|
|
s_dot = 1 / (1 - (e_y * kappa)) * v * np.cos(e_psi)
|
|
|
|
# Compute yaw angle rate of change
|
|
psi_dot = v / self.length * np.tan(delta)
|
|
|
|
return s_dot, psi_dot
|
|
|
|
def get_spatial_derivatives(self, state, input, kappa):
|
|
"""
|
|
Compute spatial derivatives of all state variables for update.
|
|
:param state: state vector for which to compute derivatives
|
|
:param input: input vector
|
|
:param kappa: curvature of corresponding waypoint
|
|
:return: numpy array with spatial derivatives for all state variables
|
|
"""
|
|
|
|
# Get state and input variables
|
|
e_y, e_psi, t = state
|
|
v, delta = input
|
|
|
|
# Compute temporal derivatives
|
|
s_dot, psi_dot = self.get_temporal_derivatives(state, input, kappa)
|
|
|
|
# Compute spatial derivatives
|
|
d_e_y_d_s = v * np.sin(e_psi) / s_dot
|
|
d_e_psi_d_s = psi_dot / s_dot - kappa
|
|
d_t_d_s = 1 / s_dot
|
|
|
|
return np.array([d_e_y_d_s, d_e_psi_d_s, d_t_d_s])
|
|
|
|
def linearize(self, v_ref, kappa_ref, delta_s):
|
|
"""
|
|
Linearize the system equations around provided reference values.
|
|
:param v_ref: velocity reference around which to linearize
|
|
:param kappa_ref: kappa of waypoint around which to linearize
|
|
:param delta_s: distance between current waypoint and next waypoint
|
|
"""
|
|
|
|
###################
|
|
# System Matrices #
|
|
###################
|
|
|
|
# Construct Jacobian Matrix
|
|
a_1 = np.array([1, delta_s, 0])
|
|
a_2 = np.array([-kappa_ref ** 2 * delta_s, 1, 0])
|
|
a_3 = np.array([-kappa_ref / v_ref * delta_s, 0, 1])
|
|
|
|
b_1 = np.array([0, 0])
|
|
b_2 = np.array([0, delta_s])
|
|
b_3 = np.array([-1 / (v_ref ** 2) * delta_s, 0])
|
|
|
|
f = np.array([0.0, 0.0, 1 / v_ref * delta_s])
|
|
|
|
A = np.stack((a_1, a_2, a_3), axis=0)
|
|
B = np.stack((b_1, b_2, b_3), axis=0)
|
|
|
|
return f, A, B
|