446 lines
14 KiB
Python
446 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) and velocity
|
|
:param x: x position in global coordinate system | [m]
|
|
:param y: y position in global coordinate system | [m]
|
|
:param psi: yaw angle | [rad]
|
|
:param v_x: velocity in x direction (car frame) | [m/s]
|
|
:param v_y: velocity in y direction (car frame) | [m/s]
|
|
"""
|
|
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
|
|
pass
|
|
|
|
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, e_psi, t):
|
|
"""
|
|
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):
|
|
"""
|
|
Abstract Base Class for Spatial Reformulation of Bicycle Model.
|
|
:param reference_path: reference path object to follow
|
|
"""
|
|
|
|
# Precision
|
|
self.eps = 1e-12
|
|
|
|
# Car Parameters
|
|
self.l = length
|
|
self.w = 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 to None (Initialization required)
|
|
self.sampling_time = None
|
|
|
|
# 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
|
|
|
|
# Declare system matrices of linearized model | Used for MPC
|
|
self.A, self.B = None, None
|
|
|
|
def s2t(self, reference_waypoint=None, reference_state=None):
|
|
"""
|
|
Convert spatial state to temporal state. Either convert self.spatial_
|
|
state with current waypoint as reference or provide reference waypoint
|
|
and reference_state.
|
|
:return x, y, psi
|
|
"""
|
|
|
|
# Compute spatial state for current waypoint if no waypoint given
|
|
if reference_waypoint is None and reference_state is None:
|
|
|
|
# compute temporal state variables
|
|
x = self.current_waypoint.x - self.spatial_state.e_y * np.sin(
|
|
self.current_waypoint.psi)
|
|
y = self.current_waypoint.y + self.spatial_state.e_y * np.cos(
|
|
self.current_waypoint.psi)
|
|
psi = self.current_waypoint.psi + self.spatial_state.e_psi
|
|
|
|
else:
|
|
|
|
# compute temporal state variables
|
|
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]
|
|
|
|
return x, y, psi
|
|
|
|
def t2s(self):
|
|
"""
|
|
Convert spatial state to temporal state. Either convert self.spatial_
|
|
state with current waypoint as reference or provide reference waypoint
|
|
and reference_state.
|
|
:return x, y, psi
|
|
"""
|
|
|
|
# compute temporal state variables
|
|
e_y = np.cos(self.current_waypoint.psi) * \
|
|
(self.temporal_state.y - self.current_waypoint.y) - \
|
|
np.sin(self.current_waypoint.psi) * (self.temporal_state.x -
|
|
self.current_waypoint.x)
|
|
e_psi = self.temporal_state.psi - self.current_waypoint.psi
|
|
e_psi = np.mod(e_psi + math.pi, 2*math.pi) - math.pi
|
|
t = 0
|
|
|
|
return SimpleSpatialState(e_y, e_psi, t)
|
|
|
|
def set_sampling_time(self, Ts):
|
|
"""
|
|
Set sampling time of bicycle model.
|
|
:param Ts: sampling time in s
|
|
"""
|
|
self.Ts = Ts
|
|
|
|
def drive(self, u):
|
|
"""
|
|
Drive.
|
|
:param u: input vector
|
|
:return: numpy array with spatial derivatives for all state variables
|
|
"""
|
|
|
|
# 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.l * 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
|
|
length = self.l / np.sqrt(2)
|
|
width = self.w / np.sqrt(2)
|
|
|
|
return length, width
|
|
|
|
def get_current_waypoint(self):
|
|
"""
|
|
Create waypoint on reference path at current location of car by
|
|
interpolation information from given path waypoints.
|
|
"""
|
|
|
|
# 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 for interpolation
|
|
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]
|
|
#
|
|
# # Weight for next waypoint
|
|
# w = (s_next - self.s) / (s_next - s_prev)
|
|
#
|
|
# # Interpolate between the two waypoints
|
|
# prev_wp = self.reference_path.waypoints[prev_wp_id]
|
|
# next_wp = self.reference_path.waypoints[next_wp_id]
|
|
# x = w * next_wp.x + (1 - w) * prev_wp.x
|
|
# y = w * next_wp.y + (1 - w) * prev_wp.y
|
|
# psi = w * next_wp.psi + (1 - w) * prev_wp.psi
|
|
# kappa = w * next_wp.kappa + (1 - w) * prev_wp.kappa
|
|
|
|
|
|
|
|
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.l, height=self.w,
|
|
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.l/2 * np.cos(self.temporal_state.psi) -
|
|
self.w/2 * np.sin(self.temporal_state.psi)))
|
|
car.set_y(car.get_y() - (self.w/2 * np.cos(self.temporal_state.psi) +
|
|
self.l/2 * np.sin(self.temporal_state.psi)))
|
|
|
|
# Show safety margin
|
|
safety_margin = plt_patches.Ellipse(cog, width=2*self.safety_margin[0],
|
|
height=2*self.safety_margin[1],
|
|
angle=yaw,
|
|
fill=False, edgecolor=CAR, zorder=20)
|
|
|
|
# Add rectangle to current axis
|
|
ax = plt.gca()
|
|
ax.add_patch(safety_margin)
|
|
ax.add_patch(car)
|
|
|
|
@abstractmethod
|
|
def get_spatial_derivatives(self, state, input, kappa):
|
|
pass
|
|
|
|
@abstractmethod
|
|
def linearize(self):
|
|
pass
|
|
|
|
|
|
#################
|
|
# Bicycle Model #
|
|
#################
|
|
|
|
class BicycleModel(SpatialBicycleModel):
|
|
def __init__(self, length, width, reference_path, e_y, e_psi, t):
|
|
"""
|
|
Simplified Spatial Bicycle Model. Spatial Reformulation of Kinematic
|
|
Bicycle Model. Uses Simplified Spatial State.
|
|
:param length: length of the car in m
|
|
:param width: with of the car in m
|
|
:param reference_path: reference path model is supposed to follow
|
|
:param e_y: deviation from reference path | [m]
|
|
:param e_psi: heading offset from reference path | [rad]
|
|
"""
|
|
|
|
# Initialize base class
|
|
super(BicycleModel, self).__init__(reference_path, length=length,
|
|
width=width)
|
|
|
|
# Initialize spatial state
|
|
self.spatial_state = SimpleSpatialState(e_y, e_psi, t)
|
|
|
|
# Number of spatial state variables
|
|
self.n_states = len(self.spatial_state)
|
|
|
|
# Initialize temporal state
|
|
self.temporal_state = self.s2t()
|
|
|
|
def s2t(self, reference_waypoint=None, reference_state=None):
|
|
"""
|
|
Convert spatial state to temporal state. Either convert self.spatial_
|
|
state with current waypoint as reference or provide reference waypoint
|
|
and reference_state.
|
|
:return temporal state equivalent to self.spatial_state or provided
|
|
reference state
|
|
"""
|
|
|
|
if reference_state is None and reference_waypoint is None:
|
|
# Get pose information from base class implementation
|
|
x, y, psi = super(BicycleModel, self).s2t()
|
|
# Compute simplified velocities
|
|
else:
|
|
# Get pose information from base class implementation
|
|
x, y, psi = super(BicycleModel, self).s2t(reference_waypoint,
|
|
reference_state)
|
|
|
|
return TemporalState(x, y, psi)
|
|
|
|
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
|
|
"""
|
|
|
|
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.l * 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
|
|
"""
|
|
|
|
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=None, kappa_r=None, delta_s=None):
|
|
"""
|
|
Linearize the system equations around the current state and waypoint.
|
|
:param kappa_r: kappa of waypoint around which to linearize
|
|
"""
|
|
|
|
# Get linearization state
|
|
if kappa_r is None and delta_s is None:
|
|
# Get curvature of linearization waypoint
|
|
kappa_r = self.reference_path.waypoints[self.wp_id].kappa
|
|
# Get delta_s
|
|
next_waypoint = self.reference_path.waypoints[self.wp_id + 1]
|
|
delta_s = next_waypoint - self.current_waypoint
|
|
|
|
###################
|
|
# System Matrices #
|
|
###################
|
|
|
|
# Construct Jacobian Matrix
|
|
a_1 = np.array([1, delta_s, 0])
|
|
a_2 = np.array([-kappa_r**2*delta_s, 1, 0])
|
|
a_3 = np.array([-kappa_r/v*delta_s, 0, 0])
|
|
|
|
b_1 = np.array([0, ])
|
|
b_2 = np.array([delta_s, ])
|
|
b_3 = np.array([0, ])
|
|
|
|
A = np.stack((a_1, a_2, a_3), axis=0)
|
|
B = np.stack((b_1, b_2, b_3), axis=0)
|
|
|
|
return A, B |