Create spatial_bicycle_models.py

Two different bicycle models (simple, extended)

spatial_bicycle_models.py

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+import numpy as np
+import math
+import time
+import cvxpy as cp
+from abc import ABC, abstractmethod
+
+
+#########################
+# Temporal State Vector #
+#########################
+
+class TemporalState:
+    def __init__(self, x, y, psi, v_x=0, v_y=0):
+        """
+        Temporal State Vector containing x, y coordinates and heading psi
+        :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.v_x = v_x
+        self.v_y = v_y
+
+
+########################
+# Spatial State Vector #
+########################
+
+class SpatialState(ABC):
+    @abstractmethod
+    def __init__(self):
+        pass
+
+    def __getitem__(self, item):
+        return list(vars(self).values())[item]
+
+    def __len__(self):
+        return len(vars(self))
+
+    def list_states(self):
+        return list(vars(self).keys())
+
+    def __iadd__(self, other):
+        for state_id, state in enumerate(vars(self).values()):
+            vars(self)[list(vars(self).keys())[state_id]] += other[state_id]
+        return self
+
+
+class SimpleSpatialState(SpatialState):
+    def __init__(self, e_y, e_psi, v):
+        """
+        Temporal State Vector containing x, y coordinates and heading psi
+        :param e_y: orthogonal deviation from center-line | [m]
+        :param e_psi: yaw angle relative to path | [rad]
+        :param v: absolute velocity | [m/s]
+        """
+        super(SimpleSpatialState, self).__init__()
+
+        self.e_y = e_y
+        self.e_psi = e_psi
+        self.v = v
+
+
+class ExtendedSpatialState(SpatialState):
+    def __init__(self, e_y, e_psi, v_x, v_y, omega, t):
+        """
+        Temporal State Vector containing x, y coordinates and heading psi
+        :param e_y: orthogonal deviation from center-line | [m]
+        :param e_psi: yaw angle relative to path | [rad]
+        :param v: absolute velocity | [m/s]
+        """
+        super(ExtendedSpatialState, self).__init__()
+
+        self.e_y = e_y
+        self.e_psi = e_psi
+        self.v_x = v_x
+        self.v_y = v_y
+        self.omega = omega
+        self.t = t
+
+
+####################################
+# Spatial Bicycle Model Base Class #
+####################################
+
+class SpatialBicycleModel(ABC):
+    def __init__(self, reference_path):
+        """
+        Construct spatial bicycle model.
+        :param reference_path: reference path model is supposed to follow
+        """
+
+        # Precision
+        self.eps = 1e-12
+
+        # Reference Path
+        self.reference_path = reference_path
+
+        # set initial distance traveled
+        self.s = 0.0
+
+        # set initial waypoint ID
+        self.wp_id = 0
+
+        # set initial waypoint
+        self.current_waypoint = self.reference_path.waypoints[self.wp_id]
+
+        # initialize spatial state
+        self.spatial_state = None
+
+    def s2t(self, reference_waypoint=None, predicted_state=None):
+        """
+        Convert spatial state to temporal state. Either convert self.spatial
+        state with current waypoint as reference or provide reference waypoint
+        and (e_y, e_psi).
+        :return x, y, psi
+        """
+
+        # Compute spatial state for current waypoint if no waypoint given
+        if reference_waypoint is None and predicted_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 - predicted_state[0] * np.sin(
+                reference_waypoint.psi)
+            y = reference_waypoint.y + predicted_state[0] * np.cos(
+                reference_waypoint.psi)
+            psi = reference_waypoint.psi + predicted_state[1]
+
+        return x, y, psi
+
+    def drive(self, delta, D):
+        """
+        Update states of spatial bicycle model.
+        :param delta: angular velocity | [rad]
+        :param D: acceleration command | [-1, 1]
+        """
+
+        # get spatial derivatives
+        spatial_derivatives = self.get_spatial_derivatives(delta, D)
+
+        # get delta_s
+        next_waypoint = self.reference_path.waypoints[self.wp_id+1]
+        delta_s = next_waypoint - self.current_waypoint
+
+        # update spatial state (euler method)
+        self.spatial_state += spatial_derivatives * delta_s
+
+        # assert that unique projections exists
+        assert self.spatial_state.e_y < (1 / (self.current_waypoint.kappa +
+                                              self.eps))
+
+        # increase waypoint ID
+        self.wp_id += 1
+
+        # update current waypoint
+        self.current_waypoint = self.reference_path.waypoints[self.wp_id]
+
+        # update temporal_state to match spatial state
+        self.temporal_state = self.s2t()
+
+        # update s
+        self.s += delta_s
+
+        # linearize model around new operating point
+        self.A, self.B = self.linearize()
+
+    @abstractmethod
+    def get_spatial_derivatives(self, delta, D):
+        pass
+
+    @abstractmethod
+    def linearize(self):
+        pass
+
+
+########################
+# Simple Bicycle Model #
+########################
+
+class SimpleBicycleModel(SpatialBicycleModel):
+    def __init__(self, reference_path, e_y, e_psi, v):
+        """
+        Construct spatial bicycle model.
+        :param e_y: initial deviation from reference path | [m]
+        :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
+        """
+        super(SimpleBicycleModel, self).__init__(reference_path)
+
+        # Constants
+        self.C1 = 0.5
+        self.C2 = 17.06
+        self.Cm1 = 12.0
+        self.Cm2 = 2.17
+        self.Cr2 = 0.1
+        self.Cr0 = 0.6
+
+        # Spatial state
+        self.spatial_state = SimpleSpatialState(e_y, e_psi, v)
+
+        # Temporal state
+        self.temporal_state = self.s2t()
+
+        # Linear System Matrices
+        self.A, self.B = self.linearize()
+
+    def s2t(self, reference_waypoint=None, predicted_state=None):
+        """
+        Convert spatial state to temporal state
+        :return temporal state equivalent to self.spatial_state
+        """
+
+        # compute velocity information
+        if predicted_state is None and reference_waypoint is None:
+            # get information from base class
+            x, y, psi = super(SimpleBicycleModel, self).s2t()
+            v_x = self.spatial_state.v
+            v_y = 0
+        else:
+            # get information from base class
+            x, y, psi = super(SimpleBicycleModel, self).s2t(reference_waypoint,
+                                                            predicted_state)
+            v_x = predicted_state[2]
+            v_y = 0
+
+        return TemporalState(x, y, psi, v_x, v_y)
+
+    def get_velocities(self, delta):
+        """
+        Compute relevant velocity components for current update.
+        :param delta: steering command
+        :return: velocities in x, y and waypoint direction
+        """
+
+        # 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
+        """
+
+        # velocity along path
+        s_dot = 1 / (1 - (self.spatial_state.e_y * self.current_waypoint.kappa)) * v_sigma
+
+        # angle rate of change
+        psi_dot = self.spatial_state.v * delta * self.C2
+
+        # acceleration
+        v_dot = (self.Cm1 - self.Cm2 * self.spatial_state.v) * D - self.Cr2 * (
+                self.spatial_state.v ** 2) - self.Cr0 - (
+                        self.spatial_state.v * delta) ** 2 * self.C2 * self.C1 ** 2
+
+        return s_dot, psi_dot, v_dot
+
+    def get_spatial_derivatives(self, delta, D):
+        """
+        Compute spatial derivatives of all state variables for update.
+        :param delta: steering angle
+        :param D: duty-cycle
+        :return: spatial derivatives for all state variables
+        """
+
+        # Compute velocities
+        v_x, v_y, v_sigma = self.get_velocities(delta)
+
+        # Compute state derivatives
+        s_dot, psi_dot, v_dot = self.get_temporal_derivatives(v_sigma, delta,
+                                                              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.e_psi)) \
+                / (s_dot + self.eps)
+        d_e_psi = (psi_dot / (s_dot + self.eps) - self.current_waypoint.kappa)
+        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])
+
+    def linearize(self, delta=0, D=0):
+        """
+        Linearize the system equations around the current state and waypoint.
+        :param delta: reference steering angle
+        :param D: reference duty-cycle
+        """
+
+        # get current state
+        e_y = self.spatial_state.e_y
+        e_psi = self.spatial_state.e_psi
+        v = self.spatial_state.v
+
+        # get information about current waypoint
+        kappa = 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
+
+        # set helper variables
+        v_x = v
+        v_y = v * delta * self.C1
+
+        ##############################
+        # Helper Partial Derivatives #
+        ##############################
+
+        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_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_t = 0
+        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))
+        # Check
+
+        c_1 = (v_x * np.sin(e_psi) + v_y * np.cos(e_psi))
+        d_c_1_d_e_y = 0
+        d_c_1_d_e_psi = v_x * np.cos(e_psi) - v_y * np.sin(e_psi)
+        d_c_1_d_v = np.sin(e_psi) + self.C1 * delta * np.cos(e_psi)
+        d_c_1_d_t = 0
+        d_c_1_d_delta = self.C1 * v * np.cos(e_psi)
+        d_c_1_d_D = 0
+        # Check
+
+        psi_dot = v * delta * self.C2
+        d_psi_dot_d_e_y = 0
+        d_psi_dot_d_e_psi = 0
+        d_psi_dot_d_v = delta * self.C2
+        d_psi_dot_d_t = 0
+        d_psi_dot_d_delta = v * self.C2
+        d_psi_dot_d_D = 0
+        # Check
+
+        v_dot = (self.Cm1 - self.Cm2 * v) * D - self.Cr2 * (v ** 2) - self.Cr0 - (
+                        v * delta) ** 2 * self.C2 * (self.C1 ** 2)
+        d_v_dot_d_e_y = 0
+        d_v_dot_d_e_psi = 0
+        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_t = 0
+        d_v_dot_d_delta = -2 * (v ** 2) * delta * self.C2 * self.C1 ** 2
+        d_v_dot_d_D = self.Cm1 - self.Cm2 * v
+        # Check
+
+        #######################
+        # Partial Derivatives #
+        #######################
+
+        # derivatives for E_Y
+        d_e_y_d_e_y = -c_1 * d_s_dot_d_e_y / (s_dot**2)
+        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)
+        d_e_y_d_t = 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_kappa = -d_s_dot_d_kappa * c_1 / (s_dot**2)
+
+        # derivatives for E_PSI
+        d_e_psi_d_e_y = - psi_dot * d_s_dot_d_e_y / (s_dot**2)
+        d_e_psi_d_e_psi = - psi_dot * d_s_dot_d_e_psi / (s_dot**2)
+        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_t = 0
+        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_D = 0
+        d_e_psi_d_kappa = -d_s_dot_d_kappa * psi_dot / (s_dot**2) - 1
+
+        # derivatives for V
+        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_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_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)
+        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
+        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_3 = np.array([d_v_d_e_y, d_v_d_e_psi, d_v_d_v, d_v_d_kappa]) * delta_s
+        a_4 = np.array([0, 0, 0, 1])
+        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_2 = np.array([d_e_psi_d_D, d_e_psi_d_delta]) * delta_s
+        b_3 = np.array([d_v_d_D, d_v_d_delta]) * delta_s
+        b_4 = np.array([0, 0])
+        B = np.stack((b_1, b_2, b_3, b_4), axis=0)
+
+        # set system matrices
+        return A, B
+
+
+##########################
+# Extended Bicycle Model #
+##########################
+
+class ExtendedBicycleModel(SpatialBicycleModel):
+    def __init__(self, reference_path, e_y, e_psi, v_x, v_y, omega, t):
+        """
+        Construct spatial bicycle model.
+        :param e_y: initial deviation from reference path | [m]
+        :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
+        """
+        super(ExtendedBicycleModel, self).__init__(reference_path)
+
+        # Constants
+        self.m = 0.041
+        self.Iz = 27.8e-6
+        self.lf = 0.029
+        self.lr = 0.033
+
+        self.Cm1 = 0.287
+        self.Cm2 = 0.0545
+        self.Cr2 = 0.0518
+        self.Cr0 = 0.00035
+
+        self.Br = 3.3852
+        self.Cr = 1.2691
+        self.Dr = 0.1737
+        self.Bf = 2.579
+        self.Cf = 1.2
+        self.Df = 0.192
+
+        # Spatial state
+        self.spatial_state = ExtendedSpatialState(e_y, e_psi, v_x, v_y, omega, t)
+
+        # Temporal state
+        self.temporal_state = self.s2t()
+
+        # Linear System Matrices
+        self.A, self.B = self.linearize()
+
+    def s2t(self, reference_waypoint=None, predicted_state=None):
+        """
+        Convert spatial state to temporal state
+        :return temporal state equivalent to self.spatial_state
+        """
+
+        # compute velocity information
+        if predicted_state is None and reference_waypoint is None:
+            # get information from base class
+            x, y, psi = super(ExtendedBicycleModel, self).s2t()
+            v_x = self.spatial_state.v_x
+            v_y = self.spatial_state.v_y
+        else:
+            # get information from base class
+            x, y, psi = super(ExtendedBicycleModel, self).s2t(reference_waypoint,
+                                                            predicted_state)
+            v_x = predicted_state[2]
+            v_y = predicted_state[3]
+
+        return TemporalState(x, y, psi, v_x, v_y)
+
+    def get_forces(self, delta, D):
+        """
+        Compute forces required for temporal derivatives of v_x and v_y
+        :param delta:
+        :param D:
+        :return:
+        """
+
+        F_rx = (self.Cm1 - self.Cm2 * self.spatial_state.v_x) * D - self.Cr0 - self.Cr2 * self.spatial_state.v_x ** 2
+
+        alpha_f = - np.arctan2(self.spatial_state.omega*self.lf + self.spatial_state.v_y, self.spatial_state.v_x) + delta
+        F_fy = self.Df * np.sin(self.Cf*np.arctan(self.Bf*alpha_f))
+
+        alpha_r = np.arctan2(self.spatial_state.omega*self.lr - self.spatial_state.v_y, self.spatial_state.v_x)
+        F_ry = self.Dr * np.sin(self.Cr * np.arctan(self.Br*alpha_r))
+
+        return F_rx, F_fy, F_ry, alpha_f, alpha_r
+
+    def get_temporal_derivatives(self, delta, F_rx, F_fy, F_ry):
+        """
+        Compute temporal derivatives needed for state update.
+        :param delta: steering command
+        :param D: duty-cycle of DC motor
+        :return: temporal derivatives of distance, angle and velocity
+        """
+
+        # velocity along path
+        s_dot = 1 / (1 - (self.spatial_state.e_y * self.current_waypoint.kappa)) \
+                * (self.spatial_state.v_x * np.cos(self.spatial_state.e_psi)
+                   + self.spatial_state.v_y * np.sin(self.spatial_state.e_psi))
+
+        # velocity in x and y direction
+        v_x_dot = (F_rx - F_fy * np.sin(delta) + self.m * self.spatial_state.
+                   v_y * self.spatial_state.omega) / self.m
+        v_y_dot = (F_ry + F_fy * np.cos(delta) - self.m * self.spatial_state.
+                   v_x * self.spatial_state.omega) / self.m
+
+        # omega dot
+        omega_dot = (F_fy * self.lf * np.cos(delta) - F_ry * self.lr) / self.Iz
+
+        return s_dot, v_x_dot, v_y_dot, omega_dot
+
+    def get_spatial_derivatives(self, delta, D):
+        """
+        Compute spatial derivatives of all state variables for update.
+        :param delta: steering angle
+        :param psi_dot: heading rate of change
+        :param s_dot: velocity along path
+        :param v_dot: acceleration
+        :return: spatial derivatives for all state variables
+        """
+
+        # get required forces
+        F_rx, F_fy, F_ry, _, _ = self.get_forces(delta, D)
+
+        # Compute state derivatives
+        s_dot, v_x_dot, v_y_dot, omega_dot = \
+            self.get_temporal_derivatives(delta, F_rx, F_fy, F_ry)
+
+
+        d_e_y = (self.spatial_state.v_x * np.sin(self.spatial_state.e_psi)
+                 + self.spatial_state.v_y * np.cos(self.spatial_state.e_psi)) \
+                / (s_dot + self.eps)
+        d_e_psi = (self.spatial_state.omega / (s_dot + self.eps) - self.current_waypoint.kappa)
+
+        d_v_x = v_x_dot / (s_dot + self.eps)
+        d_v_y = v_y_dot / (s_dot + self.eps)
+        d_omega = omega_dot / (s_dot + self.eps)
+        d_t = 1 / (s_dot + self.eps)
+
+        return np.array([d_e_y, d_e_psi, d_v_x, d_v_y, d_omega, d_t])
+
+    def linearize(self, delta=0, D=0):
+        """
+        Linearize the system equations around the current state and waypoint.
+        :param delta: reference steering angle
+        :param D: reference dutycycle
+        """
+
+        # get current state
+        e_y = self.spatial_state.e_y
+        e_psi = self.spatial_state.e_psi
+        v_x = self.spatial_state.v_x
+        v_y = self.spatial_state.v_y
+        omega = self.spatial_state.omega
+        t = self.spatial_state.t
+
+        # get information about current waypoint
+        kappa = 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
+
+        # get temporal derivatives
+        F_rx, F_fy, F_ry, alpha_f, alpha_r = self.get_forces(delta, D)
+        s_dot, v_x_dot, v_y_dot, omega_dot = self.\
+            get_temporal_derivatives(delta, F_rx, F_fy, F_ry)
+
+        ##############################
+        # Forces Partial Derivatives #
+        ##############################
+
+        d_alpha_f_d_v_x = 1 / (1 + ((omega * self.lf + v_y) / v_x)**2) * (omega * self.lf + v_y) / (v_x**2)
+        d_alpha_f_d_v_y = - 1 / (1 + ((omega * self.lf + v_y) / v_x)**2) / v_x
+        d_alpha_f_d_omega = - 1 / (1 + ((omega * self.lf + v_y) / v_x)**2) * (self.lf / v_x)
+        d_alpha_f_d_delta = 1
+
+        d_alpha_r_d_v_x = - 1 / (1 + ((omega * self.lr - v_y) / v_x)**2) * (omega * self.lr - v_y) / (v_x**2)
+        d_alpha_r_d_v_y = - 1 / (1 + ((omega * self.lr - v_y) / v_x)**2) / v_x
+        d_alpha_r_d_omega = 1 / (1 + ((omega * self.lr - v_y) / v_x)**2) * (self.lr * v_x)
+
+        d_F_fy_d_v_x = self.Df * np.cos(self.Cf * np.arctan(self.Bf * alpha_f)) * self.Cf / (1 + (self.Bf * alpha_f)**2) * self.Bf * d_alpha_f_d_v_x
+        d_F_fy_d_v_y = self.Df * np.cos(self.Cf * np.arctan(self.Bf * alpha_f)) * self.Cf / (1 + (self.Bf * alpha_f)**2) * self.Bf * d_alpha_f_d_v_y
+        d_F_fy_d_omega = self.Df * np.cos(self.Cf * np.arctan(self.Bf * alpha_f)) * self.Cf / (1 + (self.Bf * alpha_f)**2) * self.Bf * d_alpha_f_d_omega
+        d_F_fy_d_delta = self.Df * np.cos(self.Cf * np.arctan(self.Bf * alpha_f)) * self.Cf / (1 + (self.Bf * alpha_f)**2) * self.Bf * d_alpha_f_d_delta
+
+        d_F_ry_d_v_x = self.Dr * np.cos(self.Cr * np.arctan(self.Br * alpha_r)) * self.Cr / (1 + (self.Br * alpha_r)**2) * self.Br * d_alpha_r_d_v_x
+        d_F_ry_d_v_y = self.Dr * np.cos(self.Cr * np.arctan(self.Br * alpha_r)) * self.Cr / (1 + (self.Br * alpha_r)**2) * self.Br * d_alpha_r_d_v_y
+        d_F_ry_d_omega = self.Dr * np.cos(self.Cr * np.arctan(self.Br * alpha_r)) * self.Cr / (1 + (self.Br * alpha_r)**2) * self.Br * d_alpha_r_d_omega
+
+        d_F_rx_d_v_x = - self.Cm2 * D - 2 * self.Cr2 * v_x
+        d_F_rx_d_D = self.Cm1 - self.Cm2 * v_x
+
+        ##############################
+        # Helper Partial Derivatives #
+        ##############################
+
+        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_v_x = 1 / (1 - e_y*kappa) * np.cos(e_psi)
+        d_s_dot_d_v_y = -1 / (1 - e_y*kappa) * np.sin(e_psi)
+        d_s_dot_d_omega = 0
+        d_s_dot_d_t = 0
+        d_s_dot_d_delta = 0
+        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))
+        # Check
+
+        c_1 = (v_x * np.sin(e_psi) + v_y * np.cos(e_psi))
+        d_c_1_d_e_y = 0
+        d_c_1_d_e_psi = v_x * np.cos(e_psi) - v_y * np.sin(e_psi)
+        d_c_1_d_v_x = np.sin(e_psi)
+        d_c_1_d_v_y = np.cos(e_psi)
+        d_c_1_d_omega = 0
+        d_c_1_d_t = 0
+        d_c_1_d_delta = 0
+        d_c_1_d_D = 0
+        d_c_1_d_kappa = 0
+        # Check
+
+        d_v_x_dot_d_e_y = 0
+        d_v_x_dot_d_e_psi = 0
+        d_v_x_dot_d_v_x = (d_F_rx_d_v_x - d_F_fy_d_v_x * np.sin(delta)) / self.m
+        d_v_x_dot_d_v_y = - (d_F_fy_d_v_y * np.sin(delta) + self.m * omega) / self.m
+        d_v_x_dot_d_omega = - (d_F_fy_d_omega * np.sin(delta) + self.m * v_y) / self.m
+        d_v_x_dot_d_t = 0
+        d_v_x_dot_d_delta = - (F_fy * np.cos(delta) + d_F_fy_d_delta * np.sin(delta)) / self.m
+        d_v_x_dot_d_D = d_F_rx_d_D / self.m
+        d_v_x_dot_d_kappa = 0
+
+        d_v_y_dot_d_e_y = 0
+        d_v_y_dot_d_e_psi = 0
+        d_v_y_dot_d_v_x = (d_F_ry_d_v_x + d_F_fy_d_v_x * np.cos(delta) - self.m * omega) / self.m
+        d_v_y_dot_d_v_y = (d_F_ry_d_v_y + d_F_fy_d_v_y * np.cos(delta)) / self.m
+        d_v_y_dot_d_omega = (d_F_ry_d_omega + d_F_fy_d_omega * np.cos(delta) - self.m * v_x) / self.m
+        d_v_y_dot_d_t = 0
+        d_v_y_dot_d_delta = d_F_fy_d_delta * np.cos(delta) / self.m
+        d_v_y_dot_d_D = 0
+        d_v_y_dot_d_kappa = 0
+
+        d_omega_dot_d_e_y = 0
+        d_omega_dot_d_e_psi = 0
+        d_omega_dot_d_v_x = (d_F_fy_d_v_x * self.lf * np.cos(delta) - d_F_ry_d_v_x * self.lr) / self.Iz
+        d_omega_dot_d_v_y = (d_F_fy_d_v_y * self.lf * np.cos(delta) - d_F_fy_d_v_y * self.lr) / self.Iz
+        d_omega_dot_d_omega = (d_F_fy_d_omega * self.lf * np.cos(delta) - d_F_fy_d_omega * self.lr) / self.Iz
+        d_omega_dot_d_t = 0
+        d_omega_dot_d_delta = (- F_fy * np.sin(delta) + d_F_fy_d_delta * np.cos(delta)) / self.Iz
+        d_omega_dot_d_D = 0
+        d_omega_dot_d_kappa = 0
+
+        #######################
+        # Partial Derivatives #
+        #######################
+
+        # derivatives for E_Y
+        d_e_y_d_e_y = -c_1 * d_s_dot_d_e_y / (s_dot**2)
+        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_x = (d_c_1_d_v_x * s_dot - d_s_dot_d_v_x * c_1) / (s_dot**2)
+        d_e_y_d_v_y = (d_c_1_d_v_y * s_dot - d_s_dot_d_v_y * c_1) / (s_dot**2)
+        d_e_y_d_omega = (d_c_1_d_omega * s_dot - d_s_dot_d_omega * c_1) / (s_dot**2)
+        d_e_y_d_t = 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_kappa = -d_s_dot_d_kappa * c_1 / (s_dot**2)
+
+        # derivatives for E_PSI
+        d_e_psi_d_e_y = - omega * d_s_dot_d_e_y / (s_dot**2)
+        d_e_psi_d_e_psi = - omega * d_s_dot_d_e_psi / (s_dot**2)
+        d_e_psi_d_v_x = (- omega * d_s_dot_d_v_x) / (s_dot**2)
+        d_e_psi_d_v_y = (- omega * d_s_dot_d_v_y) / (s_dot**2)
+        d_e_psi_d_omega = (s_dot - omega * d_s_dot_d_omega) / (s_dot**2)
+        d_e_psi_d_t = 0
+        d_e_psi_d_delta = (- omega * d_s_dot_d_delta) / (s_dot**2)
+        d_e_psi_d_D = (- omega * d_s_dot_d_D) / (s_dot**2)
+        d_e_psi_d_kappa = -d_s_dot_d_kappa * omega / (s_dot**2) - 1
+
+        # derivatives for V_X
+        d_v_x_d_e_y = - d_s_dot_d_e_y * v_x_dot / (s_dot**2)
+        d_v_x_d_e_psi = - d_s_dot_d_e_psi * v_x_dot / (s_dot**2)
+        d_v_x_d_v_x = (d_v_x_dot_d_v_x * s_dot - d_s_dot_d_v_x * v_x_dot) / (s_dot**2)
+        d_v_x_d_v_y = (d_v_x_dot_d_v_y * s_dot - d_s_dot_d_v_y * v_x_dot) / (s_dot**2)
+        d_v_x_d_omega = (d_v_x_dot_d_omega * s_dot - d_s_dot_d_omega * v_x_dot) / (s_dot**2)
+        d_v_x_d_t = 0
+        d_v_x_d_delta = (d_v_x_dot_d_delta * s_dot - d_s_dot_d_delta * v_x_dot) / (s_dot**2)
+        d_v_x_d_D = d_v_x_dot_d_D * s_dot / (s_dot**2)
+        d_v_x_d_kappa = -d_s_dot_d_kappa * v_x_dot / (s_dot**2)
+
+        # derivatives for V_Y
+        d_v_y_d_e_y = - d_s_dot_d_e_y * v_y_dot / (s_dot ** 2)
+        d_v_y_d_e_psi = - d_s_dot_d_e_psi * v_y_dot / (s_dot ** 2)
+        d_v_y_d_v_x = (d_v_y_dot_d_v_x * s_dot - d_s_dot_d_v_x * v_y_dot) / (
+                    s_dot ** 2)
+        d_v_y_d_v_y = (d_v_y_dot_d_v_y * s_dot - d_s_dot_d_v_y * v_y_dot) / (
+                    s_dot ** 2)
+        d_v_y_d_omega = (d_v_y_dot_d_omega * s_dot - d_s_dot_d_omega * v_y_dot) / (
+                                    s_dot ** 2)
+        d_v_y_d_t = 0
+        d_v_y_d_delta = (d_v_y_dot_d_delta * s_dot - d_s_dot_d_delta * v_y_dot) / (
+                                    s_dot ** 2)
+        d_v_y_d_D = d_v_y_dot_d_D * s_dot / (s_dot ** 2)
+        d_v_y_d_kappa = -d_s_dot_d_kappa * v_y_dot / (s_dot ** 2)
+
+        # derivatives for Omega
+        d_omega_d_e_y = (d_omega_dot_d_e_y * s_dot - omega_dot * d_s_dot_d_e_y) / (s_dot**2)
+        d_omega_d_e_psi = (d_omega_dot_d_e_psi * s_dot - omega_dot * d_s_dot_d_e_psi) / (s_dot**2)
+        d_omega_d_v_x = (d_omega_dot_d_v_x * s_dot - omega_dot * d_s_dot_d_v_x) / (s_dot**2)
+        d_omega_d_v_y = (d_omega_dot_d_v_y * s_dot - omega_dot * d_s_dot_d_v_y) / (s_dot**2)
+        d_omega_d_omega = (d_omega_dot_d_omega * s_dot - omega_dot * d_s_dot_d_omega) / (s_dot**2)
+        d_omega_d_t = (d_omega_dot_d_t * s_dot - omega_dot * d_s_dot_d_t) / (s_dot**2)
+        d_omega_d_delta = (d_omega_dot_d_delta * s_dot - omega_dot * d_s_dot_d_delta) / (s_dot**2)
+        d_omega_d_D = (d_omega_dot_d_D * s_dot - omega_dot * d_s_dot_d_D) / (s_dot**2)
+        d_omega_d_kappa = (d_omega_dot_d_kappa * s_dot - omega_dot * d_s_dot_d_kappa) / (s_dot**2)
+
+        # derivatives for T
+        d_t_d_e_y = - d_s_dot_d_e_y / (s_dot**2)
+        d_t_d_e_psi = - d_s_dot_d_e_psi / (s_dot ** 2)
+        d_t_d_v_x = - d_s_dot_d_v_x / (s_dot ** 2)
+        d_t_d_v_y = - d_s_dot_d_v_y / (s_dot ** 2)
+        d_t_d_omega = - d_s_dot_d_omega / (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_x, d_e_y_d_v_y, d_e_y_d_omega, d_e_y_d_t, d_e_y_d_kappa])
+        a_2 = np.array([d_e_psi_d_e_y, d_e_psi_d_e_psi, d_e_psi_d_v_x, d_e_psi_d_v_y, d_e_psi_d_omega, d_e_psi_d_t, d_e_psi_d_kappa])
+        a_3 = np.array([d_v_x_d_e_y, d_v_x_d_e_psi, d_v_x_d_v_x, d_v_x_d_v_y, d_v_x_d_omega, d_v_x_d_t, d_v_x_d_kappa])
+        a_4 = np.array([d_v_y_d_e_y, d_v_y_d_e_psi, d_v_y_d_v_x, d_v_y_d_v_y, d_v_y_d_omega, d_v_y_d_t, d_v_y_d_kappa])
+        a_5 = np.array([d_omega_d_e_y, d_omega_d_e_psi, d_omega_d_v_x, d_omega_d_v_y, d_omega_d_omega, d_omega_d_t, d_omega_d_kappa])
+        a_6 = np.array([d_t_d_e_y, d_t_d_e_psi, d_t_d_v_x, d_t_d_v_y, d_t_d_omega, d_t_d_t, d_t_d_kappa])
+        a_7 = np.array([0, 0, 0, 0, 0, 0, 1])
+        A = np.stack((a_1, a_2, a_3, a_4, a_5, a_6, a_7), axis=0) * delta_s
+        A[0, 0] += 1
+        A[1, 1] += 1
+        A[2, 2] += 1
+        A[3, 3] += 1
+        A[4, 4] += 1
+        A[5, 5] += 1
+        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])
+        b_3 = np.array([d_v_x_d_D, d_v_x_d_delta])
+        b_4 = np.array([d_v_y_d_D, d_v_y_d_delta])
+        b_5 = np.array([d_omega_d_D, d_omega_d_delta])
+        b_6 = np.array([d_t_d_D, d_t_d_delta])
+        b_7 = np.array([0, 0])
+        B = np.stack((b_1, b_2, b_3, b_4, b_5, b_6, b_7), axis=0) * delta_s
+
+        # set system matrices
+        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))