448 lines
15 KiB
Python
448 lines
15 KiB
Python
import numpy as np
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from abc import abstractmethod
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try:
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from abc import ABC
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except:
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# for Python 2.7
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from abc import ABCMeta
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class ABC(object):
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__metaclass__ = ABCMeta
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pass
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import matplotlib.pyplot as plt
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import matplotlib.patches as plt_patches
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import math
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# Colors
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CAR = '#F1C40F'
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CAR_OUTLINE = '#B7950B'
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#########################
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# Temporal State Vector #
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#########################
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class TemporalState:
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def __init__(self, x, y, psi):
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"""
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Temporal State Vector containing car pose (x, y, psi) and velocity
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:param x: x position in global coordinate system | [m]
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:param y: y position in global coordinate system | [m]
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:param psi: yaw angle | [rad]
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:param v_x: velocity in x direction (car frame) | [m/s]
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:param v_y: velocity in y direction (car frame) | [m/s]
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"""
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self.x = x
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self.y = y
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self.psi = psi
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self.members = ['x', 'y', 'psi']
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def __iadd__(self, other):
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"""
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Overload Sum-Add operator.
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:param other: numpy array to be added to state vector
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"""
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for state_id in range(len(self.members)):
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vars(self)[self.members[state_id]] += other[state_id]
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return self
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########################
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# Spatial State Vector #
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########################
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class SpatialState(ABC):
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"""
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Spatial State Vector - Abstract Base Class.
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"""
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@abstractmethod
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def __init__(self):
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self.members = None
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pass
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def __getitem__(self, item):
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if isinstance(item, int):
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members = [self.members[item]]
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else:
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members = self.members[item]
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return [vars(self)[key] for key in members]
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def __setitem__(self, key, value):
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vars(self)[self.members[key]] = value
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def __len__(self):
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return len(self.members)
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def __iadd__(self, other):
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"""
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Overload Sum-Add operator.
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:param other: numpy array to be added to state vector
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"""
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for state_id in range(len(self.members)):
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vars(self)[self.members[state_id]] += other[state_id]
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return self
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def list_states(self):
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"""
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Return list of names of all states.
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"""
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return self.members
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class SimpleSpatialState(SpatialState):
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def __init__(self, e_y, e_psi, t):
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"""
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Simplified Spatial State Vector containing orthogonal deviation from
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reference path (e_y), difference in orientation (e_psi) and velocity
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:param e_y: orthogonal deviation from center-line | [m]
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:param e_psi: yaw angle relative to path | [rad]
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:param t: time | [s]
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"""
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super(SimpleSpatialState, self).__init__()
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self.e_y = e_y
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self.e_psi = e_psi
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self.t = t
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self.members = ['e_y', 'e_psi', 't']
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####################################
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# Spatial Bicycle Model Base Class #
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####################################
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class SpatialBicycleModel(ABC):
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def __init__(self, reference_path, length, width):
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"""
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Abstract Base Class for Spatial Reformulation of Bicycle Model.
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:param reference_path: reference path object to follow
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"""
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# Precision
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self.eps = 1e-12
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# Car Parameters
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self.l = length
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self.w = width
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self.safety_margin = self._compute_safety_margin()
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# Reference Path
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self.reference_path = reference_path
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# Set initial distance traveled
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self.s = 0.0
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# Set sampling time to None (Initialization required)
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self.sampling_time = None
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# Set initial waypoint ID
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self.wp_id = 0
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# Set initial waypoint
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self.current_waypoint = self.reference_path.waypoints[self.wp_id]
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# Declare spatial state variable | Initialization in sub-class
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self.spatial_state = None
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# Declare temporal state variable | Initialization in sub-class
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self.temporal_state = None
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# Declare system matrices of linearized model | Used for MPC
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self.A, self.B = None, None
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def s2t(self, reference_waypoint=None, reference_state=None):
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"""
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Convert spatial state to temporal state. Either convert self.spatial_
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state with current waypoint as reference or provide reference waypoint
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and reference_state.
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:return x, y, psi
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"""
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# Compute spatial state for current waypoint if no waypoint given
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if reference_waypoint is None and reference_state is None:
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# compute temporal state variables
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x = self.current_waypoint.x - self.spatial_state.e_y * np.sin(
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self.current_waypoint.psi)
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y = self.current_waypoint.y + self.spatial_state.e_y * np.cos(
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self.current_waypoint.psi)
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psi = self.current_waypoint.psi + self.spatial_state.e_psi
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else:
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# compute temporal state variables
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x = reference_waypoint.x - reference_state[0] * np.sin(
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reference_waypoint.psi)
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y = reference_waypoint.y + reference_state[0] * np.cos(
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reference_waypoint.psi)
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psi = reference_waypoint.psi + reference_state[1]
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return x, y, psi
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def t2s(self):
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"""
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Convert spatial state to temporal state. Either convert self.spatial_
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state with current waypoint as reference or provide reference waypoint
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and reference_state.
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:return x, y, psi
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"""
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# compute temporal state variables
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e_y = np.cos(self.current_waypoint.psi) * \
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(self.temporal_state.y - self.current_waypoint.y) - \
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np.sin(self.current_waypoint.psi) * (self.temporal_state.x -
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self.current_waypoint.x)
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e_psi = self.temporal_state.psi - self.current_waypoint.psi
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e_psi = np.mod(e_psi + math.pi, 2*math.pi) - math.pi
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t = 0
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return SimpleSpatialState(e_y, e_psi, t)
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def set_sampling_time(self, Ts):
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"""
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Set sampling time of bicycle model.
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:param Ts: sampling time in s
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"""
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self.Ts = Ts
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def drive(self, u):
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"""
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Drive.
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:param u: input vector
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:return: numpy array with spatial derivatives for all state variables
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"""
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# Get input signals
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v, delta = u
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# Compute temporal state derivatives
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x_dot = v * np.cos(self.temporal_state.psi)
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y_dot = v * np.sin(self.temporal_state.psi)
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psi_dot = v / self.l * np.tan(delta)
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temporal_derivatives = np.array([x_dot, y_dot, psi_dot])
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# Update spatial state (Forward Euler Approximation)
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self.temporal_state += temporal_derivatives * self.Ts
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# Compute velocity along path
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s_dot = 1 / (1 - self.spatial_state.e_y * self.current_waypoint.kappa) \
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* v * np.cos(self.spatial_state.e_psi)
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# Update distance travelled along reference path
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self.s += s_dot * self.Ts
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def _compute_safety_margin(self):
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"""
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Compute safety margin for car if modeled by its center of gravity.
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"""
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# Model ellipsoid around the car
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length = self.l / np.sqrt(2)
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width = self.w / np.sqrt(2) + 0.02
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return length, width
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def get_current_waypoint(self):
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"""
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Create waypoint on reference path at current location of car by
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interpolation information from given path waypoints.
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"""
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# Compute cumulative path length
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length_cum = np.cumsum(self.reference_path.segment_lengths)
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# Get first index with distance larger than distance traveled by car
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# so far
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greater_than_threshold = length_cum > self.s
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next_wp_id = greater_than_threshold.searchsorted(True)
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# Get previous index for interpolation
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prev_wp_id = next_wp_id - 1
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# Get distance traveled for both enclosing waypoints
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s_next = length_cum[next_wp_id]
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s_prev = length_cum[prev_wp_id]
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if np.abs(self.s - s_next) < np.abs(self.s - s_prev):
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self.wp_id = next_wp_id
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self.current_waypoint = self.reference_path.waypoints[next_wp_id]
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else:
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self.wp_id = prev_wp_id
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self.current_waypoint = self.reference_path.waypoints[prev_wp_id]
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#
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# # Weight for next waypoint
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# w = (s_next - self.s) / (s_next - s_prev)
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#
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# # Interpolate between the two waypoints
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# prev_wp = self.reference_path.waypoints[prev_wp_id]
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# next_wp = self.reference_path.waypoints[next_wp_id]
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# x = w * next_wp.x + (1 - w) * prev_wp.x
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# y = w * next_wp.y + (1 - w) * prev_wp.y
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# psi = w * next_wp.psi + (1 - w) * prev_wp.psi
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# kappa = w * next_wp.kappa + (1 - w) * prev_wp.kappa
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def show(self):
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"""
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Display car on current axis.
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"""
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# Get car's center of gravity
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cog = (self.temporal_state.x, self.temporal_state.y)
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# Get current angle with respect to x-axis
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yaw = np.rad2deg(self.temporal_state.psi)
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# Draw rectangle
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car = plt_patches.Rectangle(cog, width=self.l, height=self.w,
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angle=yaw, facecolor=CAR, edgecolor=CAR_OUTLINE, zorder=20)
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# Shift center rectangle to match center of the car
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car.set_x(car.get_x() - (self.l/2 * np.cos(self.temporal_state.psi) -
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self.w/2 * np.sin(self.temporal_state.psi)))
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car.set_y(car.get_y() - (self.w/2 * np.cos(self.temporal_state.psi) +
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self.l/2 * np.sin(self.temporal_state.psi)))
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# Show safety margin
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safety_margin = plt_patches.Ellipse(cog, width=2*self.safety_margin[0],
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height=2*self.safety_margin[1],
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angle=yaw,
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fill=False, edgecolor=CAR, zorder=20)
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# Add rectangle to current axis
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ax = plt.gca()
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ax.add_patch(safety_margin)
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ax.add_patch(car)
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@abstractmethod
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def get_spatial_derivatives(self, state, input, kappa):
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pass
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@abstractmethod
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def linearize(self):
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pass
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#################
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# Bicycle Model #
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#################
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class BicycleModel(SpatialBicycleModel):
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def __init__(self, length, width, reference_path, e_y, e_psi, t):
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"""
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Simplified Spatial Bicycle Model. Spatial Reformulation of Kinematic
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Bicycle Model. Uses Simplified Spatial State.
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:param length: length of the car in m
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:param width: with of the car in m
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:param reference_path: reference path model is supposed to follow
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:param e_y: deviation from reference path | [m]
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:param e_psi: heading offset from reference path | [rad]
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"""
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# Initialize base class
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super(BicycleModel, self).__init__(reference_path, length=length,
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width=width)
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# Initialize spatial state
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self.spatial_state = SimpleSpatialState(e_y, e_psi, t)
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# Number of spatial state variables
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self.n_states = len(self.spatial_state)
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# Initialize temporal state
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self.temporal_state = self.s2t()
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def s2t(self, reference_waypoint=None, reference_state=None):
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"""
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Convert spatial state to temporal state. Either convert self.spatial_
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state with current waypoint as reference or provide reference waypoint
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and reference_state.
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:return temporal state equivalent to self.spatial_state or provided
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reference state
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"""
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if reference_state is None and reference_waypoint is None:
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# Get pose information from base class implementation
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x, y, psi = super(BicycleModel, self).s2t()
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# Compute simplified velocities
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else:
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# Get pose information from base class implementation
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x, y, psi = super(BicycleModel, self).s2t(reference_waypoint,
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reference_state)
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return TemporalState(x, y, psi)
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def get_temporal_derivatives(self, state, input, kappa):
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"""
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Compute relevant temporal derivatives needed for state update.
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:param state: state vector for which to compute derivatives
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:param input: input vector
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:param kappa: curvature of corresponding waypoint
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:return: temporal derivatives of distance, angle and velocity
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"""
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e_y, e_psi, t = state
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v, delta = input
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# Compute velocity along path
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s_dot = 1 / (1 - (e_y * kappa)) * v * np.cos(e_psi)
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# Compute yaw angle rate of change
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psi_dot = v / self.l * np.tan(delta)
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return s_dot, psi_dot
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def get_spatial_derivatives(self, state, input, kappa):
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"""
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Compute spatial derivatives of all state variables for update.
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:param state: state vector for which to compute derivatives
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:param input: input vector
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:param kappa: curvature of corresponding waypoint
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:return: numpy array with spatial derivatives for all state variables
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"""
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e_y, e_psi, t = state
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v, delta = input
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# Compute temporal derivatives
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s_dot, psi_dot = self.get_temporal_derivatives(state, input, kappa)
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# Compute spatial derivatives
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d_e_y_d_s = v * np.sin(e_psi) / s_dot
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d_e_psi_d_s = psi_dot / s_dot - kappa
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d_t_d_s = 1 / s_dot
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return np.array([d_e_y_d_s, d_e_psi_d_s, d_t_d_s])
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def linearize(self, v=None, kappa_r=None, delta_s=None):
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"""
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Linearize the system equations around the current state and waypoint.
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:param kappa_r: kappa of waypoint around which to linearize
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"""
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# Get linearization state
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if kappa_r is None and delta_s is None:
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# Get curvature of linearization waypoint
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kappa_r = self.reference_path.waypoints[self.wp_id].kappa
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# Get delta_s
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next_waypoint = self.reference_path.waypoints[self.wp_id + 1]
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delta_s = next_waypoint - self.current_waypoint
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###################
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# System Matrices #
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###################
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# Construct Jacobian Matrix
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a_1 = np.array([1, delta_s, 0])
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a_2 = np.array([-kappa_r**2*delta_s, 1, 0])
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a_3 = np.array([-kappa_r/v*delta_s, 0, 1])
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b_1 = np.array([0, 0])
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b_2 = np.array([0, delta_s])
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b_3 = np.array([-1/(v**2)*delta_s, 0])
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f = np.array([0.0, 0.0, 1/v*delta_s])
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A = np.stack((a_1, a_2, a_3), axis=0)
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B = np.stack((b_1, b_2, b_3), axis=0)
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return f, A, B |