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Update spatial_bicycle_models.py 

Add comments. SimpleBicycleModel checked.

New Convention:
each function call with control signals
(D, delta) according to order in control signal returned by MPC
master
matssteinweg 2019-11-23 22:29:25 +01:00
parent eabf75c9ad
commit 53d30d46b4
1 changed files with 199 additions and 182 deletions
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381
spatial_bicycle_models.py
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@ -1,7 +1,4 @@
import numpy as np import numpy as np
import math
import time
import cvxpy as cp
from abc import ABC, abstractmethod from abc import ABC, abstractmethod
@ -10,9 +7,9 @@ from abc import ABC, abstractmethod
######################### #########################
class TemporalState: class TemporalState:
def __init__(self, x, y, psi, v_x=0, v_y=0): def __init__(self, x, y, psi, v_x, v_y):
""" """
Temporal State Vector containing x, y coordinates and heading psi Temporal State Vector containing car pose (x, y, psi) and velocity
:param x: x position in global coordinate system | [m] :param x: x position in global coordinate system | [m]
:param y: y position in global coordinate system | [m] :param y: y position in global coordinate system | [m]
:param psi: yaw angle | [rad] :param psi: yaw angle | [rad]
@ -31,6 +28,10 @@ class TemporalState:
######################## ########################
class SpatialState(ABC): class SpatialState(ABC):
"""
Spatial State Vector - Abstract Base Class.
"""
@abstractmethod @abstractmethod
def __init__(self): def __init__(self):
pass pass
@ -41,19 +42,28 @@ class SpatialState(ABC):
def __len__(self): def __len__(self):
return len(vars(self)) return len(vars(self))
def list_states(self):
return list(vars(self).keys())
def __iadd__(self, other): def __iadd__(self, other):
"""
Overload Sum-Add operator.
:param other: numpy array to be added to state vector
"""
for state_id, state in enumerate(vars(self).values()): for state_id, state in enumerate(vars(self).values()):
vars(self)[list(vars(self).keys())[state_id]] += other[state_id] vars(self)[list(vars(self).keys())[state_id]] += other[state_id]
return self return self
def list_states(self):
"""
Return list of names of all states.
"""
return list(vars(self).keys())
class SimpleSpatialState(SpatialState): class SimpleSpatialState(SpatialState):
def __init__(self, e_y, e_psi, v): def __init__(self, e_y, e_psi, v):
""" """
Temporal State Vector containing x, y coordinates and heading psi 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_y: orthogonal deviation from center-line | [m]
:param e_psi: yaw angle relative to path | [rad] :param e_psi: yaw angle relative to path | [rad]
:param v: absolute velocity | [m/s] :param v: absolute velocity | [m/s]
@ -68,10 +78,14 @@ class SimpleSpatialState(SpatialState):
class ExtendedSpatialState(SpatialState): class ExtendedSpatialState(SpatialState):
def __init__(self, e_y, e_psi, v_x, v_y, omega, t): def __init__(self, e_y, e_psi, v_x, v_y, omega, t):
""" """
Temporal State Vector containing x, y coordinates and heading psi Extended Spatial State Vector containing separate velocities in x and
y direction, angular velocity and time
:param e_y: orthogonal deviation from center-line | [m] :param e_y: orthogonal deviation from center-line | [m]
:param e_psi: yaw angle relative to path | [rad] :param e_psi: yaw angle relative to path | [rad]
:param v: absolute velocity | [m/s] :param v_x: velocity in x direction (car frame) | [m/s]
:param v_y: velocity in y direction (car frame) | [m/s]
:param omega: anglular velocity of the car | [rad/s]
:param t: simulation time| [s]
""" """
super(ExtendedSpatialState, self).__init__() super(ExtendedSpatialState, self).__init__()
@ -90,8 +104,8 @@ class ExtendedSpatialState(SpatialState):
class SpatialBicycleModel(ABC): class SpatialBicycleModel(ABC):
def __init__(self, reference_path): def __init__(self, reference_path):
""" """
Construct spatial bicycle model. Abstract Base Class for Spatial Reformulation of Bicycle Model.
:param reference_path: reference path model is supposed to follow :param reference_path: reference path object to follow
""" """
# Precision # Precision
@ -100,28 +114,34 @@ class SpatialBicycleModel(ABC):
# Reference Path # Reference Path
self.reference_path = reference_path self.reference_path = reference_path
# set initial distance traveled # Set initial distance traveled
self.s = 0.0 self.s = 0.0
# set initial waypoint ID # Set initial waypoint ID
self.wp_id = 0 self.wp_id = 0
# set initial waypoint # Set initial waypoint
self.current_waypoint = self.reference_path.waypoints[self.wp_id] self.current_waypoint = self.reference_path.waypoints[self.wp_id]
# initialize spatial state # Declare spatial state variable | Initialization in sub-class
self.spatial_state = None self.spatial_state = None
def s2t(self, reference_waypoint=None, predicted_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 Convert spatial state to temporal state. Either convert self.spatial_
state with current waypoint as reference or provide reference waypoint state with current waypoint as reference or provide reference waypoint
and (e_y, e_psi). and reference_state.
:return x, y, psi :return x, y, psi
""" """
# Compute spatial state for current waypoint if no waypoint given # Compute spatial state for current waypoint if no waypoint given
if reference_waypoint is None and predicted_state is None: if reference_waypoint is None and reference_state is None:
# compute temporal state variables # compute temporal state variables
x = self.current_waypoint.x - self.spatial_state.e_y * np.sin( x = self.current_waypoint.x - self.spatial_state.e_y * np.sin(
@ -133,52 +153,53 @@ class SpatialBicycleModel(ABC):
else: else:
# compute temporal state variables # compute temporal state variables
x = reference_waypoint.x - predicted_state[0] * np.sin( x = reference_waypoint.x - reference_state[0] * np.sin(
reference_waypoint.psi) reference_waypoint.psi)
y = reference_waypoint.y + predicted_state[0] * np.cos( y = reference_waypoint.y + reference_state[0] * np.cos(
reference_waypoint.psi) reference_waypoint.psi)
psi = reference_waypoint.psi + predicted_state[1] psi = reference_waypoint.psi + reference_state[1]
return x, y, psi return x, y, psi
def drive(self, delta, D): def drive(self, D, delta):
""" """
Update states of spatial bicycle model. Update states of spatial bicycle model. Model drive to the next
:param delta: angular velocity | [rad] waypoint on the reference path.
:param D: acceleration command | [-1, 1] :param D: acceleration command | [-1, 1]
:param delta: angular velocity | [rad]
""" """
# get spatial derivatives # Get spatial derivatives
spatial_derivatives = self.get_spatial_derivatives(delta, D) spatial_derivatives = self.get_spatial_derivatives(D, delta)
# get delta_s # Get delta_s | distance to next waypoint
next_waypoint = self.reference_path.waypoints[self.wp_id+1] next_waypoint = self.reference_path.waypoints[self.wp_id+1]
delta_s = next_waypoint - self.current_waypoint delta_s = next_waypoint - self.current_waypoint
# update spatial state (euler method) # Update spatial state (Forward Euler Approximation)
self.spatial_state += spatial_derivatives * delta_s self.spatial_state += spatial_derivatives * delta_s
# assert that unique projections exists # Assert that unique projections of car pose onto path exists
assert self.spatial_state.e_y < (1 / (self.current_waypoint.kappa + assert self.spatial_state.e_y < (1 / (self.current_waypoint.kappa +
self.eps)) self.eps))
# increase waypoint ID # Increase waypoint ID
self.wp_id += 1 self.wp_id += 1
# update current waypoint # Update current waypoint
self.current_waypoint = self.reference_path.waypoints[self.wp_id] self.current_waypoint = self.reference_path.waypoints[self.wp_id]
# update temporal_state to match spatial state # Update temporal_state to match spatial state
self.temporal_state = self.s2t() self.temporal_state = self.s2t()
# update s # Update s | total driven distance along path
self.s += delta_s self.s += delta_s
# linearize model around new operating point # Linearize model around new operating point
self.A, self.B = self.linearize() self.A, self.B = self.linearize()
@abstractmethod @abstractmethod
def get_spatial_derivatives(self, delta, D): def get_spatial_derivatives(self, D, delta):
pass pass
@abstractmethod @abstractmethod
@ -193,12 +214,15 @@ class SpatialBicycleModel(ABC):
class SimpleBicycleModel(SpatialBicycleModel): class SimpleBicycleModel(SpatialBicycleModel):
def __init__(self, reference_path, e_y, e_psi, v): def __init__(self, reference_path, e_y, e_psi, v):
""" """
Construct spatial bicycle model. Simplified Spatial Bicycle Model. Spatial Reformulation of Kinematic
:param e_y: initial deviation from reference path | [m] Bicycle Model. Uses Simplified Spatial State.
:param e_psi: initial heading offset from reference path | [rad]
:param v: initial velocity | [m/s]
:param reference_path: reference path model is supposed to follow :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]
:param v: initial velocity | [m/s]
""" """
# Initialize base class
super(SimpleBicycleModel, self).__init__(reference_path) super(SimpleBicycleModel, self).__init__(reference_path)
# Constants # Constants
@ -209,221 +233,224 @@ class SimpleBicycleModel(SpatialBicycleModel):
self.Cr2 = 0.1 self.Cr2 = 0.1
self.Cr0 = 0.6 self.Cr0 = 0.6
# Spatial state # Initialize spatial state
self.spatial_state = SimpleSpatialState(e_y, e_psi, v) self.spatial_state = SimpleSpatialState(e_y, e_psi, v)
# Temporal state # Initialize temporal state
self.temporal_state = self.s2t() self.temporal_state = self.s2t()
# Linear System Matrices # Compute linear system matrices | Used for MPC
self.A, self.B = self.linearize() self.A, self.B = self.linearize()
def s2t(self, reference_waypoint=None, predicted_state=None): def s2t(self, reference_waypoint=None, reference_state=None):
""" """
Convert spatial state to temporal state Convert spatial state to temporal state. Either convert self.spatial_
:return temporal state equivalent to self.spatial_state 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
""" """
# compute velocity information if reference_state is None and reference_waypoint is None:
if predicted_state is None and reference_waypoint is None: # Get pose information from base class implementation
# get information from base class
x, y, psi = super(SimpleBicycleModel, self).s2t() x, y, psi = super(SimpleBicycleModel, self).s2t()
# Compute simplified velocities
v_x = self.spatial_state.v v_x = self.spatial_state.v
v_y = 0 v_y = 0
else: else:
# get information from base class # Get pose information from base class implementation
x, y, psi = super(SimpleBicycleModel, self).s2t(reference_waypoint, x, y, psi = super(SimpleBicycleModel, self).s2t(reference_waypoint,
predicted_state) reference_state)
v_x = predicted_state[2] v_x = reference_state[2]
v_y = 0 v_y = 0
return TemporalState(x, y, psi, v_x, v_y) return TemporalState(x, y, psi, v_x, v_y)
def get_velocities(self, delta): def get_temporal_derivatives(self, D, delta):
""" """
Compute relevant velocity components for current update. Compute relevant temporal derivatives needed for state update.
:param delta: steering command :param D: duty-cycle of DC motor | [-1, 1]
:return: velocities in x, y and waypoint direction :param delta: steering command | [rad]
"""
# approximation for small delta
v_x = self.spatial_state.v
v_y = self.spatial_state.v * delta * self.C1
# velocity along waypoint direction
v_sigma = v_x * np.cos(self.spatial_state.e_psi) - v_y * np.sin(
self.spatial_state.e_psi)
return v_x, v_y, v_sigma
def get_temporal_derivatives(self, v_sigma, delta, D):
"""
Compute temporal derivatives needed for state update.
:param v_sigma: velocity along the path
:param delta: steering command
:param D: dutycycle of DC motor
:return: temporal derivatives of distance, angle and velocity :return: temporal derivatives of distance, angle and velocity
""" """
# velocity along path # Compute velocity components | Approximation for small delta
v_x = self.spatial_state.v
v_y = self.spatial_state.v * delta * self.C1
# Compute velocity along waypoint direction
v_sigma = v_x * np.cos(self.spatial_state.e_psi) - v_y * np.sin(
self.spatial_state.e_psi)
# Compute velocity along path
s_dot = 1 / (1 - (self.spatial_state.e_y * self.current_waypoint.kappa)) * v_sigma s_dot = 1 / (1 - (self.spatial_state.e_y * self.current_waypoint.kappa)) * v_sigma
# angle rate of change # Compute yaw angle rate of change
psi_dot = self.spatial_state.v * delta * self.C2 psi_dot = self.spatial_state.v * delta * self.C2
# acceleration # Compute acceleration
v_dot = (self.Cm1 - self.Cm2 * self.spatial_state.v) * D - self.Cr2 * ( v_dot = (self.Cm1 - self.Cm2 * self.spatial_state.v) * D - self.Cr2 * (
self.spatial_state.v ** 2) - self.Cr0 - ( self.spatial_state.v ** 2) - self.Cr0 - (
self.spatial_state.v * delta) ** 2 * self.C2 * self.C1 ** 2 self.spatial_state.v * delta) ** 2 * self.C2 * self.C1 ** 2
return s_dot, psi_dot, v_dot return s_dot, psi_dot, v_dot
def get_spatial_derivatives(self, delta, D): def get_spatial_derivatives(self, delta, D):
""" """
Compute spatial derivatives of all state variables for update. Compute spatial derivatives of all state variables for update.
:param delta: steering angle :param delta: steering angle | [rad]
:param D: duty-cycle :param D: duty-cycle of DC motor | [-1, 1]
:return: spatial derivatives for all state variables :return: numpy array with spatial derivatives for all state variables
""" """
# Compute velocities # Compute temporal derivatives
v_x, v_y, v_sigma = self.get_velocities(delta) s_dot, psi_dot, v_dot = self.get_temporal_derivatives(D, delta)
# Compute state derivatives # Compute spatial derivatives
s_dot, psi_dot, v_dot = self.get_temporal_derivatives(v_sigma, delta, d_e_y_d_s = (self.spatial_state.v * np.sin(self.spatial_state.e_psi)
D)
d_e_y = (self.spatial_state.v * np.sin(self.spatial_state.e_psi)
+ self.spatial_state.v * delta * self.C1 * np.cos( + self.spatial_state.v * delta * self.C1 * np.cos(
self.spatial_state.e_psi)) \ self.spatial_state.e_psi)) / s_dot
/ (s_dot + self.eps) d_e_psi_d_s = psi_dot / s_dot - self.current_waypoint.kappa
d_e_psi = (psi_dot / (s_dot + self.eps) - self.current_waypoint.kappa) d_v_d_s = v_dot / s_dot
d_v = v_dot / (s_dot + self.eps)
d_t = 1 / (s_dot + self.eps)
return np.array([d_e_y, d_e_psi, d_v]) return np.array([d_e_y_d_s, d_e_psi_d_s, d_v_d_s])
def linearize(self, delta=0, D=0): def linearize(self, D=0, delta=0):
""" """
Linearize the system equations around the current state and waypoint. Linearize the system equations around the current state and waypoint.
:param delta: reference steering angle :param delta: reference steering angle | [rad]
:param D: reference duty-cycle :param D: reference duty-cycle of DC-motor | [-1, 1]
""" """
# get current state # Get current state | operating point to linearize around
e_y = self.spatial_state.e_y e_y = self.spatial_state.e_y
e_psi = self.spatial_state.e_psi e_psi = self.spatial_state.e_psi
v = self.spatial_state.v v = self.spatial_state.v
# get information about current waypoint # Get curvature of current waypoint
kappa = self.reference_path.waypoints[self.wp_id].kappa kappa = self.reference_path.waypoints[self.wp_id].kappa
# get delta_s # Get delta_s
next_waypoint = self.reference_path.waypoints[self.wp_id+1] next_waypoint = self.reference_path.waypoints[self.wp_id+1]
delta_s = next_waypoint - self.current_waypoint delta_s = next_waypoint - self.current_waypoint
# set helper variables
v_x = v
v_y = v * delta * self.C1
############################## ##############################
# Helper Partial Derivatives # # Helper Partial Derivatives #
############################## ##############################
# Compute velocity components
v_x = v
v_y = v * delta * self.C1
# Compute partial derivatives of s_dot w.r.t. each state variable,
# input variable and kappa
s_dot = 1 / (1 - e_y*kappa) * (v_x * np.cos(e_psi) - v_y * np.sin(e_psi)) s_dot = 1 / (1 - e_y*kappa) * (v_x * np.cos(e_psi) - v_y * np.sin(e_psi))
d_s_dot_d_e_y = kappa / (1-e_y*kappa)**2 * (v_x * np.cos(e_psi) - v_y * np.sin(e_psi)) d_s_dot_d_e_y = kappa / (1-e_y*kappa)**2 * (v_x * np.cos(e_psi) - v_y * np.sin(e_psi))
d_s_dot_d_e_psi = 1 / (1 - e_y*kappa) * (-v_x * np.sin(e_psi) - v_y * np.cos(e_psi)) d_s_dot_d_e_psi = 1 / (1 - e_y*kappa) * (-v_x * np.sin(e_psi) - v_y * np.cos(e_psi))
d_s_dot_d_v = 1 / (1 - e_y*kappa) * (np.cos(e_psi) - delta * self.C1 * np.sin(e_psi)) d_s_dot_d_v = 1 / (1 - e_y*kappa) * (np.cos(e_psi) - delta * self.C1 * np.sin(e_psi))
d_s_dot_d_t = 0 # d_s_dot_d_D = 0
d_s_dot_d_delta = 1 / (1 - e_y*kappa) * (- v * self.C1 * np.sin(e_psi)) d_s_dot_d_delta = 1 / (1 - e_y*kappa) * (- v * self.C1 * np.sin(e_psi))
d_s_dot_d_D = 0
d_s_dot_d_kappa = e_y / (1-e_y*kappa)**2 * (v_x * np.cos(e_psi) - v_y * np.sin(e_psi)) d_s_dot_d_kappa = e_y / (1-e_y*kappa)**2 * (v_x * np.cos(e_psi) - v_y * np.sin(e_psi))
# Check
c_1 = (v_x * np.sin(e_psi) + v_y * np.cos(e_psi)) # Compute partial derivatives of v_psi w.r.t. each state variable,
d_c_1_d_e_y = 0 # input variable and kappa
d_c_1_d_e_psi = v_x * np.cos(e_psi) - v_y * np.sin(e_psi) v_psi = (v_x * np.sin(e_psi) + v_y * np.cos(e_psi))
d_c_1_d_v = np.sin(e_psi) + self.C1 * delta * np.cos(e_psi) # d_v_psi_d_e_y = 0
d_c_1_d_t = 0 d_v_psi_d_e_psi = v_x * np.cos(e_psi) - v_y * np.sin(e_psi)
d_c_1_d_delta = self.C1 * v * np.cos(e_psi) d_v_psi_d_v = np.sin(e_psi) + self.C1 * delta * np.cos(e_psi)
d_c_1_d_D = 0 # d_v_psi_d_D = 0
# Check d_v_psi_d_delta = self.C1 * v * np.cos(e_psi)
# d_v_psi_d_kappa = 0
# Compute partial derivatives of psi_dot w.r.t. each state variable,
# input variable and kappa
psi_dot = v * delta * self.C2 psi_dot = v * delta * self.C2
d_psi_dot_d_e_y = 0 # d_psi_dot_d_e_y = 0
d_psi_dot_d_e_psi = 0 # d_psi_dot_d_e_psi = 0
d_psi_dot_d_v = delta * self.C2 d_psi_dot_d_v = delta * self.C2
d_psi_dot_d_t = 0 # d_psi_dot_d_D = 0
d_psi_dot_d_delta = v * self.C2 d_psi_dot_d_delta = v * self.C2
d_psi_dot_d_D = 0 # d_psi_dot_d_kappa = 0
# Check
v_dot = (self.Cm1 - self.Cm2 * v) * D - self.Cr2 * (v ** 2) - self.Cr0 - ( # Compute partial derivatives of v_dot w.r.t. each state variable,
v * delta) ** 2 * self.C2 * (self.C1 ** 2) # input variable and kappa
d_v_dot_d_e_y = 0 v_dot = (self.Cm1 - self.Cm2 * v) * D - self.Cr2 * (v ** 2) - self.Cr0 \
d_v_dot_d_e_psi = 0 - (v * delta) ** 2 * self.C2 * (self.C1 ** 2)
d_v_dot_d_v = -self.Cm2 * D - 2 * self.Cr2 * v - 2 * v * (delta ** 2) * self.C2 * (self.C1 ** 2) # d_v_dot_d_e_y = 0
d_v_dot_d_t = 0 # d_v_dot_d_e_psi = 0
d_v_dot_d_delta = -2 * (v ** 2) * delta * self.C2 * self.C1 ** 2 d_v_dot_d_v = -self.Cm2 * D - 2 * self.Cr2 * v - 2 * v * (delta ** 2) \
* self.C2 * (self.C1 ** 2)
d_v_dot_d_D = self.Cm1 - self.Cm2 * v d_v_dot_d_D = self.Cm1 - self.Cm2 * v
# Check d_v_dot_d_delta = -2 * (v ** 2) * delta * self.C2 * self.C1 ** 2
# d_v_dot_d_kappa = 0
####################### #############################
# Partial Derivatives # # State Partial Derivatives #
####################### #############################
# derivatives for E_Y # Use pre-computed helper derivatives to compute spatial derivatives of
d_e_y_d_e_y = -c_1 * d_s_dot_d_e_y / (s_dot**2) # all state variables using Quotient Rule
d_e_y_d_e_psi = (d_c_1_d_e_psi * s_dot - d_s_dot_d_e_psi * c_1) / (s_dot**2)
d_e_y_d_v = (d_c_1_d_v * s_dot - d_s_dot_d_v * c_1) / (s_dot**2) # Compute partial derivatives of e_y w.r.t. each state variable,
d_e_y_d_t = 0 # input variable and kappa
# e_y = v_psi / s_dot
d_e_y_d_e_y = - d_s_dot_d_e_y * v_psi / (s_dot**2)
d_e_y_d_e_psi = (d_v_psi_d_e_psi * s_dot - d_s_dot_d_e_psi * v_psi) / (s_dot**2)
d_e_y_d_v = (d_v_psi_d_v * s_dot - d_s_dot_d_v * v_psi) / (s_dot**2)
d_e_y_d_D = 0 d_e_y_d_D = 0
d_e_y_d_delta = (d_c_1_d_delta * s_dot - d_s_dot_d_delta * c_1) / (s_dot**2) d_e_y_d_delta = (d_v_psi_d_delta * s_dot - d_s_dot_d_delta * v_psi) / (s_dot**2)
d_e_y_d_kappa = -d_s_dot_d_kappa * c_1 / (s_dot**2) d_e_y_d_kappa = - d_s_dot_d_kappa * v_psi / (s_dot**2)
# derivatives for E_PSI # Compute partial derivatives of e_psi w.r.t. each state variable,
d_e_psi_d_e_y = - psi_dot * d_s_dot_d_e_y / (s_dot**2) # input variable and kappa
d_e_psi_d_e_psi = - psi_dot * d_s_dot_d_e_psi / (s_dot**2) # e_psi = psi_dot / s_dot - kappa
d_e_psi_d_v = (d_psi_dot_d_v * s_dot - psi_dot * d_s_dot_d_v) / (s_dot**2) d_e_psi_d_e_y = - d_s_dot_d_e_y * psi_dot / (s_dot**2)
d_e_psi_d_t = 0 d_e_psi_d_e_psi = - d_s_dot_d_e_psi * psi_dot / (s_dot**2)
d_e_psi_d_delta = (d_psi_dot_d_delta * s_dot - psi_dot * d_s_dot_d_delta) / (s_dot**2) d_e_psi_d_v = (d_psi_dot_d_v * s_dot - d_s_dot_d_v * psi_dot) / (s_dot**2)
d_e_psi_d_D = 0 d_e_psi_d_D = 0
d_e_psi_d_kappa = -d_s_dot_d_kappa * psi_dot / (s_dot**2) - 1 d_e_psi_d_delta = (d_psi_dot_d_delta * s_dot - d_s_dot_d_delta * psi_dot) / (s_dot**2)
d_e_psi_d_kappa = - d_s_dot_d_kappa * psi_dot / (s_dot**2) - 1
# derivatives for V # Compute partial derivatives of v w.r.t. each state variable,
# input variable and kappa
# v = v_dot / s_dot
d_v_d_e_y = - d_s_dot_d_e_y * v_dot / (s_dot**2) d_v_d_e_y = - d_s_dot_d_e_y * v_dot / (s_dot**2)
d_v_d_e_psi = - d_s_dot_d_e_psi * v_dot / (s_dot**2) d_v_d_e_psi = - d_s_dot_d_e_psi * v_dot / (s_dot**2)
d_v_d_v = (d_v_dot_d_v * s_dot - d_s_dot_d_v * v_dot) / (s_dot**2) d_v_d_v = (d_v_dot_d_v * s_dot - d_s_dot_d_v * v_dot) / (s_dot**2)
d_v_d_t = 0
d_v_d_delta = (d_v_dot_d_delta * s_dot - d_s_dot_d_delta * v_dot) / (s_dot**2)
d_v_d_D = d_v_dot_d_D * s_dot / (s_dot**2) d_v_d_D = d_v_dot_d_D * s_dot / (s_dot**2)
d_v_d_kappa = -d_s_dot_d_kappa * v_dot / (s_dot**2) d_v_d_delta = (d_v_dot_d_delta * s_dot - d_s_dot_d_delta * v_dot) / (s_dot**2)
d_v_d_kappa = - d_s_dot_d_kappa * v_dot / (s_dot**2)
# derivatives for T #############
d_t_d_e_y = - d_s_dot_d_e_y / (s_dot**2) # Jacobians #
d_t_d_e_psi = - d_s_dot_d_e_psi / (s_dot ** 2) #############
d_t_d_v = - d_s_dot_d_v / (s_dot ** 2)
d_t_d_t = 0
d_t_d_delta = - d_s_dot_d_delta / (s_dot ** 2)
d_t_d_D = 0
d_t_d_kappa = - d_s_dot_d_kappa / (s_dot ** 2)
a_1 = np.array([d_e_y_d_e_y, d_e_y_d_e_psi, d_e_y_d_v, d_e_y_d_kappa]) * delta_s # Construct Jacobian Matrix
a_2 = np.array([d_e_psi_d_e_y, d_e_psi_d_e_psi, d_e_psi_d_v, d_e_psi_d_kappa]) * delta_s a_1 = np.array([d_e_y_d_e_y, d_e_y_d_e_psi, d_e_y_d_v, d_e_y_d_kappa])
a_3 = np.array([d_v_d_e_y, d_v_d_e_psi, d_v_d_v, d_v_d_kappa]) * delta_s a_2 = np.array([d_e_psi_d_e_y, d_e_psi_d_e_psi, d_e_psi_d_v, d_e_psi_d_kappa])
a_4 = np.array([0, 0, 0, 1]) a_3 = np.array([d_v_d_e_y, d_v_d_e_psi, d_v_d_v, d_v_d_kappa])
A = np.stack((a_1, a_2, a_3, a_4), axis=0)
A[0, 0] += 1
A[1, 1] += 1
A[2, 2] += 1
b_1 = np.array([d_e_y_d_D, d_e_y_d_delta]) * delta_s b_1 = np.array([d_e_y_d_D, d_e_y_d_delta])
b_2 = np.array([d_e_psi_d_D, d_e_psi_d_delta]) * delta_s b_2 = np.array([d_e_psi_d_D, d_e_psi_d_delta])
b_3 = np.array([d_v_d_D, d_v_d_delta]) * delta_s b_3 = np.array([d_v_d_D, d_v_d_delta])
# Add extra row for kappa | Allows for updating kappa during MPC
# optimization
a_4 = np.array([0, 0, 0, 0])
b_4 = np.array([0, 0]) b_4 = np.array([0, 0])
B = np.stack((b_1, b_2, b_3, b_4), axis=0)
# set system matrices Ja = np.stack((a_1, a_2, a_3, a_4), axis=0)
Jb = np.stack((b_1, b_2, b_3, b_4), axis=0)
###################
# System Matrices #
###################
# Construct system matrices from Jacobians. Multiply by sampling
# distance delta_s + add identity matrix (Forward Euler Approximation)
A = Ja * delta_s + np.identity(Ja.shape[1])
B = Jb * delta_s
return A, B return A, B
@ -770,13 +797,3 @@ class ExtendedBicycleModel(SpatialBicycleModel):
# set system matrices # set system matrices
return A, B return A, B
if __name__ == '__main__':
state = ExtendedSpatialState(0, 1, 2, 3, 4, 5)
print(state[0:2])
print(len(state))
print(state.list_states())
state += np.array([1, 1, 1, 2, 1, 1])
print(vars(state))