car_dynamics.py

import math
import numpy as np
import abc
from collections import Iterable

from estimators import sample_nlds

try:
    import torch
    torch_imported = True
except ImportError:
    print("Unable to import pytorch module")
    torch_imported = False


def cal_vxvy_from_coord(state, state_prev, dt, x_ind=0, y_ind=1, theta_ind=2):
    """
    Calculate longitudinal and lateral velocity by rotating current position into the frame of previous position.

    Args:
        state (numpy array [4 x nt]): state of the system at different time instances
        state_prev (numpy array [4 x nt]): corresponding previous states of state
        dt (numpy array [nt] or float): time step/difference between state and state_prev
        x_ind (int): row index for x coordinate of state and state_prev; defaults to 0
        y_ind (int): row index for y coordinate of state and state_prev; defaults to 1
        theta_ind (int): row index for theta coordinate of state and state_prev; defaults to 2

    Returns:
        vxy (numpy array [2 x nt]): linear and lateral velocities at different time instances

    """
    if len(state.shape) == 1:
        state = state[:, np.newaxis]
    if len(state_prev.shape) == 1:
        state_prev = state_prev[:, np.newaxis]

    prev_ang = state_prev[theta_ind, :]
    R = np.array([[np.cos(prev_ang), np.sin(prev_ang)],
                  [-np.sin(prev_ang), np.cos(prev_ang)]])
    xy_dot = (state[[x_ind, y_ind], :] - state_prev[[x_ind, y_ind], :])/dt
    vxy = np.zeros(xy_dot.shape)
    for i in range(xy_dot.shape[1]):
        vxy[:, i:i+1] = np.matmul(R[:, :, i], xy_dot[:, i:i+1])

    return vxy


class AbstractDyn(object):
    """
    Abstract dynamics class with useful base functions for derived classes

    Args:
        param_dict (dict): dictionary of parameters needed for defining the dynamics
        state_keys (list): list of states string that are observed by sensor; defaults to empty
        state_dot_keys (list): list of derivative of states string that are observed by sensor; defaults to empty
        additional_output_keys (list): list of additional things that are observed by sensors; defaults to empty
        use_torch_tensor (bool): whether to use tensor instead of numpy arrays; defaults to False

    """
    __metaclass__ = abc.ABCMeta

    def __init__(self, param_dict, state_keys=[], state_dot_keys=[], additional_output_keys=[], use_torch_tensor=False):
        # store the parameter dictionary for the vehicle dynamics
        self.param_dict = param_dict.copy()

        # store expected keys of param dictionary
        self.expected_keys = self.global_expected_keys.copy()

        # copy state dict from global
        self.state_dict = self.global_state_dict.copy()

        # store whether user will be using torch tensor or not
        self.use_torch_tensor = use_torch_tensor

        # default value of tensor complaince of dynamic object to False
        if 'tensor_compliance' not in self.__dir__():
            self.tensor_compliance = False

        # assert if dynamic class is not tensor compliant
        self.assert_tensor_compliance()

        # check if param dictionary is valid
        assert self.check_param_dict(), "Parameter dictionary does not contain all requried keys"

        # store state its derivative indices for generating output
        self.state_indices = []
        self.state_dot_indices = []
        for key in state_keys:
            if key in self.state_dict:
                self.state_indices.append(self.state_dict[key])
        for key_dot in state_dot_keys:
            if key_dot in self.state_dict:
                self.state_dot_indices.append(self.state_dict[key_dot])

        # store the additional output keys
        self.additional_output_keys = additional_output_keys

        # dimensionality of states, inputs and outputs
        self.num_states = len(self.state_dict.keys())
        self.num_out = len(self.state_indices) + \
            len(self.state_dot_indices) + len(self.additional_output_keys)
        self.num_in = 0

    def check_param_dict(self):
        """
        Basic checking of parameter dictionary. Returns false if any expected keys is not present in param_dict.

        Returns:
            (bool): True if all expected keys are present in param_dict and False otherwise

        """
        for key in self.expected_keys:
            if key not in self.param_dict.keys():
                return False

        return True

    def assert_tensor_compliance(self):
        """
        Check if torch library was successfully imported or if the dynamic library is torch compliant
        """
        if self.use_torch_tensor:
            assert torch_imported, "Pytorch module was not successfully imported which prohibits the use of tensor with this library"

            assert self.tensor_compliance, "Dynamic class is not yet tensor compliant"

    def sample_nlds(self, z0, U, T, Q=None, P0=None, R=None, Qu=None, store_variables=True, overwrite_keys=[], overwrite_vals=[]):
        """
        Retrieve ground truth, initial and output data (SNLDS: Stochastic non-linear dynamic system)

        Args:
            z0 (numpy array [n x 1]): initial ground truth condition
            U (numpy array [nu x nt]): inputs for the process and observation model
            T (numpy array [nt x 1 or 1 x nt]): time instances for retrieving ground truth and output data
            Q (numpy array [nq x nq]): noise covariance matrix involved in the stochastic dynamic model
            P0 (numpy array [n x n]): initial covariance for the initial estimate around the ground truth
            R (numpy array [nr x nr]): covariance matrix of the noise involved in observed sensor data
            Qu (numpy array [nqu x nqu]): noise covariance matrix involved in the input
            store_variables (bool): whether to store ground truth, initial and output arrays within the class
            overwrite_keys (list): list of state keys to be overwritten
            overwrite_vals (list): list of ground truth values to overwrite state propagation

        """
        # check sizes of received matrices
        num_sol = len(T)
        assert len(
            z0) == self.num_states, "True initial condition is of incorrect size"
        assert U.shape == (
            self.num_in, num_sol), "Incorrect size of input matrix"

        def process_model(x, u, noise, input_noise, dt, param_dict): return self.forward_prop(
            x, self.dxdt(x, u + input_noise, param_dict), dt) + noise

        def observation_model(
            x, u, noise, param_dict): return self.output_model(x, u, param_dict) + noise

        # check size of array of parameter dictionary
        for key in self.param_dict:
            if not isinstance(self.param_dict[key], Iterable):
                self.param_dict[key] = [self.param_dict[key]]*num_sol
            else:
                assert len(
                    self.param_dict[key]) == num_sol, "Parameter {} should have the length of nt".format(key)

        # convert param dictionary to additional input for process model
        self.param_list = []
        for i in range(num_sol):
            param_dict = dict()
            for key in self.param_dict:
                param_dict[key] = self.param_dict[key][i]
            self.param_list.append(param_dict)

        # obtain indices of ground truth states to be overwritten
        overwrite_inds = []
        for key in overwrite_keys:
            assert key in self.state_dict.keys(
            ), "State key {} to be overwritten doesn't exist".format(key)
            overwrite_inds.append(self.state_dict[key])

        # get ground truth data, initial and output data
        dts = np.diff(T)
        dts = np.append(dts, dts[-1])
        gt_states, initial_cond, outputs, additional_args_pm_list, additional_args_om_list = sample_nlds(z0, U, num_sol, process_model, observation_model, self.num_out, Q=Q, P0=P0, R=R, Qu=Qu, additional_args_pm=[
                                                                                                         dts, self.param_list], additional_args_om=[self.param_list], overwrite_inds=overwrite_inds, overwrite_vals=overwrite_vals)

        # separately, get the derivative of ground truth
        gt_states_dot = np.zeros(gt_states.shape)
        for i in range(1, num_sol):
            gt_states_dot[:, i -
                          1] = self.dxdt(gt_states[:, i-1], U[:, i-1], self.param_list[i-1])

        gt_states_dot[:, num_sol -
                      1] = self.dxdt(gt_states[:, num_sol-1], U[:, num_sol-1], self.param_list[num_sol-1])

        # store varibles to avoid parsing things around?
        if store_variables:
            self.Q = Q
            self.Qu = Qu
            self.R = R
            self.P0 = P0

            self.U = U
            self.T = T

            self.gt_states = gt_states
            self.gt_states_dot = gt_states_dot
            self.initial_cond = initial_cond
            self.outputs = outputs

            self.process_model = process_model
            self.observation_model = observation_model

            self.additional_args_pm_list = additional_args_pm_list
            self.additional_args_om_list = additional_args_om_list

        return gt_states, gt_states_dot, initial_cond, outputs

    @abc.abstractmethod
    def dxdt(self, state, u, param_dict):
        """
        Abstract Differential model for the dynamic model. Must be overwritten in derived classes.

        Args:
            state (numpy array [n x 1 or 1 x n]): current state vector
            u (*): current input
            param_dict (dict): dictionary of current parameters needed for defining the dynamics

        Returns:
            state_dot (numpy array [n x 1 or 1 x n]): current derivative of the state
        """
        return

    def forward_prop(self, state, state_dot, dt):
        """
        Forward propagation of model via first order Euler approximation, i.e. X[k+1] = X[k] + dt*X'[k], where X' is the derivative.

        Args:
            state (numpy array [n x 1 or 1 x n]): current state vector
            state_dot (numpy array [n x 1 or 1 x n]): current derivative of the state
            dt (float): time step/difference between discrete step k and step k+1

        Returns:
            state (numpy array [n x 1 or 1 x n]): next time-step state vector

        """
        return state + dt*state_dot

    def additional_output_model(self, state, u, param_dict):
        """
        Additional output model for the system that does not simply involve the bare states or their derivatives.
        Must be overwritten in derived classes if expected additional outputs.

        Args:
            state (numpy array [n x 1 or 1 x n]): current state vector
            u (*): current input
            param_dict (dict): dictionary of current parameters needed for defining the dynamics

        Returns:
            output (numpy array [len(self.additional_output_keys) x 1): observed additional output
        """
        if self.use_torch_tensor:
            return torch.empty(0, device=state.device)
        else:
            return np.array([])

    def output_model(self, state, u, param_dict):
        """
        Simulate the output of the system.

        Args:
            state (numpy array [n x 1]): current state vector
            u (*): current input
            param_dict (dict): dictionary of current parameters needed for defining the dynamics

        Returns:
            output (numpy array [len(self.state_indices) + len(self.state_dot_indices) + len(self.additional_output_keys) x 1]):
                observed state and state derivatives of the system

        """
        if self.use_torch_tensor:
            if len(self.state_dot_indices):
                dxdt = self.dxdt(state, u, param_dict)
                state_dot_out = dxdt[0, self.state_dot_indices]
            else:
                state_dot_out = torch.empty(0).to(state.device)
            return torch.cat((state[self.state_indices], state_dot_out, self.additional_output_model(state, u, param_dict)))
        else:
            return np.concatenate((state[self.state_indices], self.dxdt(state, u, param_dict)[0, self.state_dot_indices], self.additional_output_model(state, u, param_dict)))

    def cal_vxvy_from_coord(self, output=True):
        """
        Calculate longitudinal and lateral velocity by rotating current position into the frame of previous position
        using the available function on encapsulated data.

        Args:
            output (bool): whether to use ground truth or output data

        Returns:
            vxy (numpy array [2 x nt]): linear and lateral velocities at different time instances

        """
        # error checking
        assert 'x' in self.state_dict, "Function requires dynamic class to contain x coordinate state"
        assert 'y' in self.state_dict, "Function requires dynamic class to contain y coordinate state"
        assert 'theta' in self.state_dict, "Function requires dynamic class to contain theta coordinate state"

        if output:
            # check if x,y and theta are part of the output
            assert self.state_dict['x'] in self.state_indices, "No output for state x available"
            assert self.state_dict['y'] in self.state_indices, "No output for state y available"
            assert self.state_dict['theta'] in self.state_indices, "No output for state theta available"

            state = self.outputs[:, 1:]
            state_prev = self.outputs[:, :-1]
        else:
            state = self.gt_states[:, 1:]
            state_prev = self.gt_states[:, :-1]

        return cal_vxvy_from_coord(state, state_prev, np.diff(self.T), x_ind=self.state_dict['x'], y_ind=self.state_dict['y'], theta_ind=self.state_dict['theta'])


class FrontSteered(AbstractDyn):
    """
    Front Steered model derived from AbstractDyn class. The assumptions of the this model are:
    (i) steering angles of the front left and right wheels are the same. front steered only.
    (ii) steering command is small enough such that sin(steering) = steering
    (iii) vehicle operates in the linear region of the tire-force curve with negligible inclination and bang angles
    (iv) left and right wheels on each axle have the same stiffness
    (v) dominant forces are the tire-road contact forces. influences due to wind and air resistance are ignored.
    
    model from: "Tire-Stiffness and Vehicle-State Estimation Based on Noise-Adaptive Particle Filtering"
    inputs = [steering_angle, wf, wr]; wf -> front wheel rotation rate, wr -> rear wheel rotation rate
    states = [x, y, theta, vx, vy, omega]

    Args:
        param_dict (dict): dictionary of parameters needed for defining the dynamics
        state_keys (list): list of states string that are observed by sensor
        state_dot_keys (list): list of derivative of states string that are observed by sensor; defaults to empty
        additional_output_keys (list): list of additional things that are observed by sensors; defaults to empty
        use_torch_tensor (bool): whether to use tensor instead of numpy arrays; defaults to False

    """
    # state dictionary for this model
    global_state_dict = {'x': 0, 'y': 1,
                         'theta': 2, 'vx': 3, 'vy': 4, 'omega': 5}
    # expected keys/parameters of this model
    global_expected_keys = ["mass", "lr", "lf",
                            "e_wr", "cxf", "cxr", "cyf", "cyr", "iz"]

    def __init__(self, param_dict, state_keys, state_dot_keys=[], additional_output_keys=[], use_torch_tensor=False):
        super(FrontSteered, self).__init__(param_dict, state_keys=state_keys, state_dot_keys=state_dot_keys,
                                           additional_output_keys=additional_output_keys, use_torch_tensor=use_torch_tensor)

        # dimensionality of input
        self.num_in = 3

    def dxdt(self, state, u, param_dict):
        """
        Obtain derivative of the state based on current state and input

        Args:
            state (numpy array [6 x 1]): current state of the system consisting of x, y, theta, vx, vy and omega
            u (numpy array [3 x 1]): current input consisting of steering_angle, wf and wr
            param_dict (dict): dictionary of current parameters needed for defining the dynamics

        Returns:
            state_dot (numpy array [6 x 1]): derivative of the states

        """
        # get the inputs
        steering_angle = u[0]
        wf = u[1]
        wr = u[2]

        # get the states
        heading = state[self.state_dict['theta']]
        vx = state[self.state_dict['vx']]
        vy = state[self.state_dict['vy']]
        omega = state[self.state_dict['omega']]

        # calculate lateral tire forces
        if vx != 0.0:
            slip_ang_f = steering_angle - (vy + param_dict['lf']*omega)/vx
            slip_ang_r = (param_dict['lr']*omega - vy)/vx
        else:
            slip_ang_f = 0.0
            slip_ang_r = 0.0
        fy_f = param_dict['cyf']*slip_ang_f
        fy_r = param_dict['cyr']*slip_ang_r

        # calculate longitudinal force
        wheel_slip_f = (vx - param_dict['e_wr']*wf) / \
            max(vx, param_dict['e_wr']*wf)
        wheel_slip_r = (vx - param_dict['e_wr']*wr) / \
            max(vx, param_dict['e_wr']*wr)
        fx_f = param_dict['cxf']*wheel_slip_f
        fx_r = param_dict['cxr']*wheel_slip_r

        # calculate lateral and longitudinal acceleration
        vx_dot = (fx_f*math.cos(steering_angle) + fx_r + fy_f *
                  math.sin(steering_angle) + param_dict['mass']*vy*omega)/param_dict['mass']
        ax_inertial = vx_dot - vy*omega
        vy_dot = (fy_f*math.cos(steering_angle) + fy_r + fx_f *
                  math.sin(steering_angle) - param_dict['mass']*vx*omega)/param_dict['mass']
        ay_inertial = vy_dot + vx*omega

        # calculate angular acceleration
        omega_dot = (param_dict['lf']*(fy_f*math.cos(steering_angle) + fx_f *
                                       math.sin(steering_angle)) - param_dict['lr']*fy_r)/param_dict['iz']

        # kinematic model based on derived dynamic quantities
        x_dot = vx*math.cos(heading) - vy*math.sin(heading)
        y_dot = vx*math.sin(heading) + vy*math.cos(heading)
        heading_dot = omega

        return np.array([[x_dot, y_dot, heading_dot, vx_dot, vy_dot, omega_dot]])

    def additional_output_model(self, state, u, param_dict):
        """
        Specialisation of inherited method to provide inertial accelerations.

        Args:
            state (numpy array [6 x 1]): current state of the system consisting of x, y, theta, vx, vy and omega
            u (numpy array [3 x 1]): current input consisting of steering_angle, wf and wr

        Returns:
            output (numpy array [len(self.additional_output_keys) x 1): observed additional output

        """
        # get the inputs
        steering_angle = u[0]
        wf = u[1]
        wr = u[2]

        # get the states
        heading = state[2]
        vx = state[3]
        vy = state[4]
        omega = state[5]

        # calculate lateral tire forces
        if vx != 0.0:
            slip_ang_f = steering_angle - (vy + param_dict['lf']*omega)/vx
            slip_ang_r = (param_dict['lr']*omega - vy)/vx
        else:
            slip_ang_f = 0.0
            slip_ang_r = 0.0
        fy_f = param_dict['cyf']*slip_ang_f
        fy_r = param_dict['cyr']*slip_ang_r

        # calculate longitudinal force
        wheel_slip_f = (vx - param_dict['e_wr']*wf) / \
            max(vx, param_dict['e_wr']*wf)
        wheel_slip_r = (vx - param_dict['e_wr']*wr) / \
            max(vx, param_dict['e_wr']*wr)
        fx_f = param_dict['cxf']*wheel_slip_f
        fx_r = param_dict['cxr']*wheel_slip_r

        ax_inertial = (fx_f*math.cos(steering_angle) + fx_r + fy_f *
                       math.sin(steering_angle))/param_dict['mass']
        ay_inertial = (fy_f*math.cos(steering_angle) + fy_r + fx_f *
                       math.sin(steering_angle))/param_dict['mass']

        # create output array to return
        output = np.array([])
        if 'ax' in self.additional_output_keys:
            output = np.append(output, ax_inertial)
        if 'ay' in self.additional_output_keys:
            output = np.append(output, ay_inertial)

        return output


class RoverDyn(AbstractDyn):
    """
    Class derived from AbstractDyn class
    model from: Michigan guys' RTD python repo
    inputs = [steering_angle, commanded velocity]
    states = [x, y, theta, vx]

    Args:
        param_dict (dict): dictionary of parameters needed for defining the dynamics
        state_keys (list): list of states string that are observed by sensor
        state_dot_keys (list): list of derivative of states string that are observed by sensor; defaults to empty
        additional_output_keys (list): list of additional things that are observed by sensors; defaults to empty
        use_torch_tensor (bool): whether to use tensor instead of numpy arrays; defaults to False

    """
    # state dictionary for this model
    global_state_dict = {'x': 0, 'y': 1, 'theta': 2, 'vx': 3}
    # expected keys/parameters of this model
    global_expected_keys = ["c1", "c2", "c3",
                            "c4", "c5", "c6", "c7", "c8", "c9"]

    def __init__(self, param_dict, state_keys, state_dot_keys=[], additional_output_keys=[], use_torch_tensor=False):
        super(RoverDyn, self).__init__(param_dict, state_keys=state_keys, state_dot_keys=state_dot_keys,
                                       additional_output_keys=additional_output_keys, use_torch_tensor=use_torch_tensor)

        # specify expected dimensionality of input
        self.num_in = 2

    def dxdt(self, state, u, param_dict):
        """
        Obtain derivative of the state based on current state and input

        Args:
            state (numpy array [4 x 1]): current state of the system consisting of x, y, theta and vx
            u (numpy array [2 x 1]): current input consisting of steering angle and commanded velocity
            param_dict (dict): dictionary of current parameters needed for defining the dynamics

        Returns:
            state_dot (numpy array [4 x 1]): derivative of the states

        """
        # get the inputs
        steering_angle = u[0]
        vx_cmd = u[1]

        # get the states
        theta = state[self.state_dict['theta']]
        vx = state[self.state_dict['vx']]

        ang_rate = math.tan(param_dict['c1']*steering_angle + param_dict['c2'])*vx/(
            param_dict['c3'] + param_dict['c4']*vx**2)
        vy = ang_rate*(param_dict['c8'] + param_dict['c9']*vx**2)

        # calculate the derivatives
        x_dot = vx*math.cos(theta) - vy*math.sin(theta)
        y_dot = vx*math.sin(theta) + vy*math.cos(theta)
        heading_dot = ang_rate
        vx_dot = param_dict['c5'] + param_dict['c6'] * \
            (vx - vx_cmd) + param_dict['c7']*(vx - vx_cmd)**2

        return np.array([[x_dot, y_dot, heading_dot, vx_dot]])

    def additional_output_model(self, state, u, param_dict):
        """
        Specialisation of inherited method to provide lateral velocity.

        Args:
            state (numpy array [4 x 1]): current state of the system consisting of x, y, theta and vx
            u (numpy array [2 x 1]): current input consisting of steering angle and commanded velocity
            param_dict (dict): dictionary of current parameters needed for defining the dynamics

        Returns:
            output (numpy array [len(self.additional_output_keys) x 1): observed additional output

        """
        # get the inputs
        steering_angle = u[0]

        # get the states
        vx = state[self.state_dict['vx']]

        ang_rate = math.tan(param_dict['c1']*steering_angle + param_dict['c2'])*vx/(
            param_dict['c3'] + param_dict['c4']*vx**2)
        vy = ang_rate*(param_dict['c8'] + param_dict['c9']*vx**2)

        # create output array
        output = np.array([])
        if 'vy' in self.additional_output_keys:
            output = np.append(output, vy)

        return output


def partial_init(obj, class_type, est_params, param_dict, state_keys, simulate_gt, state_dot_keys=[], additional_output_keys=[], use_torch_tensor=False):
    """
    Function to help initialise an Estimate class deriving from the dynamic class.

    Args:
        obj (Est obj): object in which the initialisation is being performed (self)
        class_type (obj type): pass in the class of the object to be used when invoking super
        est_params (list): list of parameter strings to be estimated with the states
        param_dict (dict): dictionary of parameters needed for defining the dynamics
        state_keys (list): list of states string that are observed by sensor
        simulate_gt (bool): specify whether model is during the stage of simulating ground truth
        state_dot_keys (list): list of derivative of states string that are observed by sensor; defaults to empty
        additional_output_keys (list): list of additional things that are observed by sensors; defaults to empty
        use_torch_tensor (bool): whether to use tensor instead of numpy arrays; defaults to False

    """
    # store if we are simulating ground truth or not
    obj.simulate_gt = simulate_gt

    if obj.simulate_gt:
        super(class_type, obj).__init__(param_dict, state_keys, state_dot_keys=state_dot_keys,
                                        additional_output_keys=additional_output_keys, use_torch_tensor=False)
        obj.later_use_torch_tensor = use_torch_tensor
    else:
        # remove parameters to be estimated from dictionary
        pruned_param_dict = param_dict.copy()
        for est_param in est_params:
            if est_param in pruned_param_dict:
                del pruned_param_dict[est_param]

        super(class_type, obj).__init__(pruned_param_dict, state_keys, state_dot_keys=state_dot_keys,
                                        additional_output_keys=additional_output_keys, use_torch_tensor=use_torch_tensor)

        # check if est_params are in expected keys
        for est_param in est_params:
            assert est_param in obj.global_expected_keys, "Parameter {} to be estimated is not in expected keys".format(
                est_param)
            obj.expected_keys.remove(est_param)

    # append state dictionary
    for i, est_param in enumerate(est_params):
        obj.state_dict[est_param] = obj.num_states + i

    # dimensionalities of different things
    obj.est_params = list(est_params)
    obj.num_dyn_states = obj.num_states
    obj.num_states += len(obj.est_params)


def re_initialise(obj):
    """
    Function to help re-initialise an Estimate class deriving from the dynamic class after simulation of ground truth data is completed.

    Args:
        obj (Est obj): object in which the re-initialisation is being performed (self)

    """
    if obj.simulate_gt:
        # turn off simulation mode
        obj.simulate_gt = False

        # check if est_params are in expected keys
        for est_param in obj.est_params:
            assert est_param in obj.global_expected_keys, "Parameter {} to be estimated is not in expected keys".format(
                est_param)
            obj.expected_keys.remove(est_param)

        # remove parameters to be estimated from dictionary
        for est_param in obj.est_params:
            if est_param in obj.param_dict:
                del obj.param_dict[est_param]

        # check if param dictionary is valid
        assert obj.check_param_dict(), "Parameter dictionary does not contain all requried keys"

        # retrieve initially proved direction on usage of torch tensor
        obj.use_torch_tensor = obj.later_use_torch_tensor
        del obj.later_use_torch_tensor

        # assert if dynamic class is not tensor compliant
        obj.assert_tensor_compliance()


def partial_dxdt(obj, class_type, state, u, param_dict):
    """
    Function to help calculate derivative of an Estimate class deriving from dynamic class. It does the calculation by the getting
    the states' derivatives from its parent's class and assuming that the estimated parameters are stationary (not drifting over time).

    Args:
        obj (Est obj): object in which the initialisation is being performed (self)
        class_type (obj type): pass in the class of the object to be used when invoking super
        state (numpy array [num_states x 1]): current state of the system
        u (numpy array [num_in x 1]): current input
        param_dict (dict): dictionary of current non-estimated parameters needed for defining the dynamics

    """
    # put the parameters into the dictionary
    if not obj.simulate_gt:
        for i, est_param in enumerate(obj.est_params):
            param_dict[est_param] = state[obj.num_dyn_states+i]

    state_dot = super(class_type, obj).dxdt(state, u, param_dict)
    if obj.use_torch_tensor:
        for i, est_param in enumerate(obj.est_params):
            param_dict[est_param] = param_dict[est_param].cpu().item()
        return torch.cat((state_dot, torch.zeros((1, len(obj.est_params)), device=state_dot.device, dtype=state_dot.dtype)), dim=1)
    else:
        return np.concatenate((state_dot, np.zeros((1, len(obj.est_params)))), axis=1)


class RoverPartialDynEst(RoverDyn):
    """
    Class derived from RoverDyn class
    model from: same as RoverDyn but some parameters are the states themselves to be estimated online
    inputs = [steering_angle, commanded velocity]
    states = [x, y, theta, vx, ...]

    Args:
        param_dict (dict): dictionary of parameters needed for defining the dynamics
        est_params (list): list of parameter strings to be estimated with the states
        state_keys (list): list of states string that are observed by sensor
        state_dot_keys (list): list of derivative of states string that are observed by sensor; defaults to empty
        simulate_gt (bool): specify whether model is during the stage of simulating ground truth
        additional_output_keys (list): list of additional things that are observed by sensors; defaults to empty
        use_torch_tensor (bool): whether to use tensor instead of numpy arrays; defaults to False

    """

    def __init__(self, param_dict, est_params, state_keys, state_dot_keys=[], simulate_gt=False, additional_output_keys=[], use_torch_tensor=False):
        partial_init(self, RoverPartialDynEst, est_params, param_dict, state_keys, simulate_gt,
                     state_dot_keys, additional_output_keys, use_torch_tensor=use_torch_tensor)

    def re_initialise(self):
        """
        re-initialise after ground truth data has been generated using its parent's derivative and raw possibly time-varying parameters
        """
        re_initialise(self)

    def dxdt(self, state, u, param_dict):
        """
        Obtain derivative of the state based on current state and input similar to RoverDyn. Derivatives of
        estimated parameters are zero, i.e. assuming they are constant and not drifting over time.

        Args:
            state (numpy array [4+len(self.est_params) x 1]): current state of the system consisting of x, y, theta, vx 
                and estimated parameters
            u (numpy array [2 x 1]): current input consisting of steering angle and commanded velocity
            param_dict (dict): dictionary of current non-estimated parameters needed for defining the dynamics

        Returns:
            state_dot (numpy array [4+len(self.est_params) x 1]): derivative of the states

        """
        return partial_dxdt(self, RoverPartialDynEst, state, u, param_dict)


class OneWheelFriction(AbstractDyn):
    """
    Class derived from AbstractDyn class
    model from: "Application of Extended Kalman Filter for Road Condition Estimation" paper
    inputs = [wheel torque]
    states = [v, w, z]

    The assumptions of this model are:
    (i) velocity is longitudinal only, i.e. no lateral component
    (ii) Tire model taken LuGre dynamic model with steady-state error compensation

    The model parameters are:
        sigma_0: rubber longitudinal lumped stiffness (1/m)
        sigma_1: rubber longitudinal lumped damping (s/m)
        sigma_2: viscous relative damping (Ns/m)
        sigma_w: coefficient of viscous friction (Nm/s)
        L: contact patch length (m)
        mu_c: normalised Coulomb friction
        mu_s: normalised static friction
        vs: Stribeck relative velocity (m/s)
        theta: road condition coefficient
        r: wheel radius
        m: wheel mass (kg)
        J: moment of inertia of the wheel (kgm^2)
        k: lumped model constant (usually between 1.2 and 1.4)
        Fn: normal force

    Args:
        param_dict (dict): dictionary of parameters needed for defining the dynamics
        state_keys (list): list of states string that are observed by sensor
        state_dot_keys (list): list of derivative of states string that are observed by sensor; defaults to empty
        additional_output_keys (list): list of additional things that are observed by sensors; defaults to empty
        use_torch_tensor (bool): whether to use tensor instead of numpy arrays; defaults to False

    """
    # state dictionary for this model
    global_state_dict = {'v': 0, 'w': 1, 'z': 2}
    # expected keys/parameters of this model
    global_expected_keys = ["sigma_0", "sigma_1", "sigma_2", "sigma_w",
                            "L", "mu_c", "mu_s", "vs", "theta", "r", "m", "J", "k", "Fn"]
    tensor_compliance = True

    def __init__(self, param_dict, state_keys, state_dot_keys=[], additional_output_keys=[], use_torch_tensor=False):
        super(OneWheelFriction, self).__init__(param_dict, state_keys=state_keys, state_dot_keys=state_dot_keys,
                                               additional_output_keys=additional_output_keys, use_torch_tensor=use_torch_tensor)
        # specify expected dimensionality of input
        self.num_in = 1

    def dxdt(self, state, u, param_dict):
        """
        Obtain derivative of the state based on current state and input

        Args:
            state(numpy array[3 x 1]): current state of the system consisting of v, w and z
            u(numpy array[1 x 1]): current input consisting of torque acting on wheel axis
            param_dict(dict): dictionary of current parameters needed for defining the dynamics

        Returns:
            state_dot(numpy array[3 x 1]): derivative of the states

        """
        # get the parameters
        mu_c = param_dict['mu_c']
        mu_s = param_dict['mu_s']
        vs = param_dict['vs']
        Fn = param_dict['Fn']
        m = param_dict['m']
        sigma_0 = param_dict['sigma_0']
        sigma_1 = param_dict['sigma_1']
        sigma_2 = param_dict['sigma_2']
        sigma_w = param_dict['sigma_w']
        r = param_dict['r']
        theta = param_dict['theta']
        k = param_dict['k']
        L = param_dict['L']
        J = param_dict['J']

        # get the inputs
        if isinstance(u, Iterable):
            ur = u[0]
        else:
            ur = u

        # get the states
        v = state[self.state_dict['v']]
        z = state[self.state_dict['z']]
        w = state[self.state_dict['w']]

        # define function
        def gvr(vr):
            if self.use_torch_tensor:
                return mu_c + (mu_s - mu_c)*torch.exp(-torch.sqrt(torch.abs(vr/vs)))
            else:
                return mu_c + (mu_s - mu_c)*math.exp(-math.sqrt(math.fabs(vr/vs)))

        # calculate the state derivatives
        vr = r*w - v
        if self.use_torch_tensor:
            zdot = vr - z*(theta*sigma_0*torch.abs(vr) /
                           gvr(vr) + k*r*torch.abs(w)/L)
        else:
            zdot = vr - z*(theta*sigma_0*math.fabs(vr) /
                           gvr(vr) + k*r*math.fabs(w)/L)
        vdot = Fn/m*(sigma_0*z + sigma_1*zdot + sigma_2*vr)
        wdot = (-r*Fn*(sigma_0*z + sigma_1*zdot) - sigma_w*w + ur)/J

        if self.use_torch_tensor:
            return torch.stack([vdot, wdot, zdot]).unsqueeze(0)
        else:
            return np.array([[vdot, wdot, zdot]])


class OneWheelFrictionEst(OneWheelFriction):
    """
    Class derived from OneWheelFriction class
    model from: same as OneWheelFriction but some parameters are the states themselves to be estimated online
    inputs = [wheel torque]
    states = [v, w, z, ...]

    Args:
        param_dict (dict): dictionary of parameters needed for defining the dynamics
        est_params (list): list of parameter strings to be estimated with the states
        state_keys (list): list of states string that are observed by sensor
        state_dot_keys (list): list of derivative of states string that are observed by sensor; defaults to empty
        simulate_gt (bool): specify whether model is during the stage of simulating ground truth
        additional_output_keys (list): list of additional things that are observed by sensors; defaults to empty
        use_torch_tensor (bool): whether to use tensor instead of numpy arrays; defaults to False

    """

    def __init__(self, param_dict, est_params, state_keys, state_dot_keys=[], simulate_gt=False, additional_output_keys=[], use_torch_tensor=False):
        partial_init(self, OneWheelFrictionEst, est_params, param_dict, state_keys, simulate_gt,
                     state_dot_keys, additional_output_keys, use_torch_tensor=use_torch_tensor)

    def re_initialise(self):
        """
        re-initialise after ground truth data has been generated using its parent's derivative and raw possibly time-varying parameters
        """
        re_initialise(self)

    def dxdt(self, state, u, param_dict):
        """
        Obtain derivative of the state based on current state and input similar to OneWheelFriction. Derivatives of
        estimated parameters are zero, i.e. assuming they are constant and not drifting over time.

        Args:
            state (numpy array [3+len(self.est_params) x 1]): current state of the system consisting of v, w, z and estimated parameters
            u (numpy array [1 x 1]): current input consisting of torque acting on wheel axis
            param_dict (dict): dictionary of current non-estimated parameters needed for defining the dynamics

        Returns:
            state_dot (numpy array [3+len(self.est_params) x 1]): derivative of the states

        """
        return partial_dxdt(self, OneWheelFrictionEst, state, u, param_dict)


class FrontDriveFrontSteer(AbstractDyn):
    """
    Class derived from AbstractDyn class
    model from:
    inputs = [linear acceleration, steering angle]
    states = [x, y, theta, vx, vy, w]

    The assumptions of this model are:
    (i) The radius of the turn, R, must be large compared to the vehicle wheelbase, L = lf + lr, and the vehicle track, t.
    (ii) The left and right steering angles of the front wheels must be approximately the same.
    (iii) The sideslip angles of the front wheels are approximately the same, as are the sideslip angles of the rear wheels.
    (iv) The sideslip angle at the CG is beta = tan-1(v/u) which is approximated by v/u with small angle assumption
    (v) The radius, R, sideslip angle, beta, and yaw rate, w, are fixed in a steady turn so that the instantaneous speed
        tangent to the path at the CG is u = Rw.
    (vi) Tractive force is limited by the coefficient of friction between the tire and the road: f_x = mue*f_z. The calculation
        of f_z must consider fore-aft weight shift (acceleration and braking) as well as transverse weight shift due to cornering.
        To simplify, we may assume small (i.e. no) lateral and yaw motion. Note that tractive force is also limited by power
        limits of the vehicle.
    (vii) The gradient of the road theta = 0. The vehicle drawbar force in the x-axis Rhx = 0. Wind velocity uw = 0. Additionally,
        since the vehicle is front-wheel-drive, f_xr = 0.
    (viii) The front and rear tire forces are a linear function of tire slip.
    (ix) Assumptions regarding use of a bicycle model - "steady turning behavior, at a constant forward speed, due to a small
        and steady steer displacement at the front wheels".

    The model parameters are:
        fx: linear constant for converting input acceleration into tractive force of the front wheels (N/A)
        cf: front cornering stiffness coeffient
        cr: back cornering stiffness coeffient
        lf: distance from CG to front axle (m)
        lr: distance from CG to rear axle (m)
        m: mass of vehicle (kg)
        iz: rotational moment of inertia (kg m^2)
        rc: resistance coeffient due to motion prevelant at high speed, given by 0.5*rho*A
        fr: coefficient of rolling resistance (N/A)
        g: gravity

    Args:
        param_dict (dict): dictionary of parameters needed for defining the dynamics
        state_keys (list): list of states string that are observed by sensor
        state_dot_keys (list): list of derivative of states string that are observed by sensor; defaults to empty
        additional_output_keys (list): list of additional things that are observed by sensors; defaults to empty
        use_torch_tensor (bool): whether to use tensor instead of numpy arrays; defaults to False

    """
    # state dictionary for this model
    global_state_dict = {'x': 0, 'y': 1, 'theta': 2, 'vx': 3, 'vy': 4, 'w': 5}
    # expected keys/parameters of this model
    global_expected_keys = ["fx", "cf", "cr",
                            "lf", "lr", "m", "iz", "rc", "fr", "g"]

    def __init__(self, param_dict, state_keys, state_dot_keys=[], additional_output_keys=[], use_torch_tensor=False):
        super(FrontDriveFrontSteer, self).__init__(param_dict, state_keys=state_keys, state_dot_keys=state_dot_keys,
                                                   additional_output_keys=additional_output_keys, use_torch_tensor=use_torch_tensor)

        # specify expected dimensionality of input
        self.num_in = 2

    def dxdt(self, state, u, param_dict):
        """
        Obtain derivative of the state based on current state and input

        Args:
            state (numpy array [6 x 1]): current state of the system consisting of x, y, theta vx, vy and w
            u (numpy array [2 x 1]): current input consisting of linear acceleration and steering angle
            param_dict (dict): dictionary of current parameters needed for defining the dynamics

        Returns:
            state_dot (numpy array [6 x 1]): derivative of the states

        """
        # get the parameters
        m = param_dict['m']
        lr = param_dict['lr']
        lf = param_dict['lf']
        cr = param_dict['cr']
        cf = param_dict['cf']
        fx = param_dict['fx']
        rc = param_dict['rc']
        iz = param_dict['iz']
        fr = param_dict['fr']
        g = param_dict['g']

        # get the inputs
        a_req = u[0]
        steering_angle = u[1]

        # get the states
        theta = state[self.state_dict['theta']]
        vx = state[self.state_dict['vx']]
        vy = state[self.state_dict['vy']]
        w = state[self.state_dict['w']]

        # avoid division by zero
        if np.isclose(vy, 0.0, atol=1e-6) and np.isclose(vx, 0.0, atol=1e-6):
            vy_by_vx = 0.0
        else:
            vy_by_vx = vy/vx
        if np.isclose(w, 0.0, atol=1e-6) and np.isclose(vx, 0.0, atol=1e-6):
            w_by_vx = 0.0
        else:
            w_by_vx = w/vx

        # calculate the derivatives
        x_dot = vx*math.cos(theta) - vy*math.sin(theta)
        y_dot = vy*math.cos(theta) + vx*math.sin(theta)
        theta_dot = w
        vx_dot = (fx*a_req + cf*(-steering_angle**2 + steering_angle *
                                 vy_by_vx) + lf*cf*steering_angle*w_by_vx - 0.5*rc*vx**2 - fr*m*g + m*vy*w)/m
        vy_dot = (-(cf + cr)*vy_by_vx + (lr*cr - lf*cf) *
                  w_by_vx + cf*steering_angle + m*vx*w)/m
        w_dot = ((lr*cr - lf*cf)*vy_by_vx - (cf*lf**2 + cr*lr**2)
                 * w_by_vx + lf*cf*steering_angle)/iz

        return np.array([[x_dot, y_dot, theta_dot, vx_dot, vy_dot, w_dot]])


class FrontDriveFrontSteerEst(FrontDriveFrontSteer):
    """
    Class derived from FrontDriveFrontSteer class
    model from: same as FrontDriveFrontSteer but some parameters are the states themselves to be estimated online
    inputs = [linear acceleration, steering angle]
    states = [x, y, theta, vx, vy, w, ...]

    Args:
        param_dict (dict): dictionary of parameters needed for defining the dynamics
        est_params (list): list of parameter strings to be estimated with the states
        state_keys (list): list of states string that are observed by sensor
        state_dot_keys (list): list of derivative of states string that are observed by sensor; defaults to empty
        simulate_gt (bool): specify whether model is during the stage of simulating ground truth
        additional_output_keys (list): list of additional things that are observed by sensors; defaults to empty
        use_torch_tensor (bool): whether to use tensor instead of numpy arrays; defaults to False

    """

    def __init__(self, param_dict, est_params, state_keys, state_dot_keys=[], simulate_gt=False, additional_output_keys=[], use_torch_tensor=False):
        partial_init(self, FrontDriveFrontSteerEst, est_params, param_dict, state_keys,
                     simulate_gt, state_dot_keys, additional_output_keys, use_torch_tensor=use_torch_tensor)

    def re_initialise(self):
        """
        re-initialise after ground truth data has been generated using its parent's derivative and raw possibly time-varying parameters
        """
        re_initialise(self)

    def dxdt(self, state, u, param_dict):
        """
        Obtain derivative of the state based on current state and input similar to RoverDyn. Derivatives of
        estimated parameters are zero, i.e. assuming they are constant and not drifting over time.

        Args:
            state (numpy array [6+len(self.est_params) x 1]): current state of the system consisting of x, y, theta, vx,
                vy, w and estimated parameters
            u (numpy array [2 x 1]): current input consisting of linear acceleration and steering angle
            param_dict (dict): dictionary of current non-estimated parameters needed for defining the dynamics

        Returns:
            state_dot (numpy array [6+len(self.est_params) x 1]): derivative of the states

        """
        return partial_dxdt(self, FrontDriveFrontSteerEst, state, u, param_dict)


class RearDriveFrontSteer(AbstractDyn):
    """
    Class derived from AbstractDyn class
    model from: Modeling discussion presentation
    inputs = [front wheel speed, rear wheel speed, front wheel angle (steering angle)]
    states = [x, y, theta, vx, vy, w]

    The assumptions of this model are:
    (i) Bicycle model assumption- treat two front wheels as one (same tire properties & steering angles), similarly for two rear wheels
    (ii) Only in-plane motions are considered - pitch and roll dynamics are neglected, load changes neglected.
    (iii) Road is flat (zero gradient)
    (iv) Aerodynamic drag is considered but wind speed is ignored
    (v) Longitudinal and lateral tire forces are a linear function of slip ratio and slip angle, i.e. small slip ratio and slip angle
    (vi) Small tire velocity angle, tan(theta) ~ theta

    The model parameters are:
        m: mass of the vehicle (kg)
        iz: rotational moment of inertia (kg m^2)
        lf: distance from CG to front axle (m)
        lr: distance from CG to rear axle (m)
        ref: front effective wheel radius (m)
        rer: rear effective wheel radius (m)
        fr: coefficient of rolling resistance (N/A)
        g: gravitational acceleration (m/s^2)
        sc_f: cornering stiffness coefficient of the front wheel
        sc_r: cornering stiffness coefficient of the rear wheel
        sl_f: longitudinal stiffness coefficient of the front wheel
        sl_r: longitudinal stiffness coefficient of the rear wheel
        cd: aero dynamic drag coefficient

    Args:
        param_dict (dict): dictionary of parameters needed for defining the dynamics
        state_keys (list): list of states string that are observed by sensor
        state_dot_keys (list): list of derivative of states string that are observed by sensor; defaults to empty
        additional_output_keys (list): list of additional things that are observed by sensors; defaults to empty
        use_torch_tensor (bool): whether to use tensor instead of numpy arrays; defaults to False

    """
    # state dictionary for this model
    global_state_dict = {'x': 0, 'y': 1, 'theta': 2, 'vx': 3, 'vy': 4, 'w': 5}
    # expected keys/parameters of this model
    global_expected_keys = ["m", "iz", "lf", "lr", "ref", "rer",
                            "fr", "g", "sc_f", "sc_r", "sl_f", "sl_r", "af", "rho", "cd"]

    def __init__(self, param_dict, state_keys, state_dot_keys=[], additional_output_keys=[], use_torch_tensor=False):
        super(RearDriveFrontSteer, self).__init__(param_dict, state_keys=state_keys, state_dot_keys=state_dot_keys,
                                                  additional_output_keys=additional_output_keys, use_torch_tensor=use_torch_tensor)
        # specify expected dimensionality of input
        self.num_in = 3

    def dxdt(self, state, u, param_dict):
        """
        Obtain derivative of the state based on current state and input

        Args:
            state (numpy array [6 x 1]): current state of the system consisting of x, y, theta vx, vy and w
            u (numpy array [3 x 1]): current input consisting of front wheel speed, rear wheel speed, steering angle
            param_dict (dict): dictionary of current parameters needed for defining the dynamics

        Returns:
            state_dot (numpy array [6 x 1]): derivative of the states

        """
        # get the parameters
        m = param_dict["m"]
        iz = param_dict["iz"]
        lf = param_dict["lf"]
        lr = param_dict["lr"]
        ref = param_dict["ref"]
        rer = param_dict["rer"]
        fr = param_dict["fr"]
        g = param_dict["g"]
        sc_r = param_dict["sc_r"]
        sc_f = param_dict["sc_f"]
        sl_r = param_dict["sl_r"]
        sl_f = param_dict["sl_f"]
        af = param_dict["af"]
        rho = param_dict["rho"]
        cd = param_dict["cd"]

        # get the inputs
        wf = u[0]
        wr = u[1]
        steering_angle = u[2]

        # get the states
        theta = state[self.state_dict['theta']]
        vx = state[self.state_dict['vx']]
        vy = state[self.state_dict['vy']]
        w = state[self.state_dict['w']]

        # compute aero dynamic force
        f_aero = 0.5*rho*cd*af*(vx**2)

        # calculate longitudinal slip ratio
        sigma_xf = ref*wf - vx
        if sigma_xf < 0.0:
           # braking phase
           if vx != 0.0:
               sigma_xf /= vx
           else:
               sigma_xf = 0.0
        else:
            # acceleration phase
            if wf != 0.0:
                sigma_xf /= (ref*wf)
            else:
                sigma_xf = 0.0

        sigma_xr = rer*wr - vx
        if sigma_xr < 0.0:
           # braking phase
           if vx != 0.0:
               sigma_xr /= vx
           else:
               sigma_xr = 0.0
        else:
            # acceleration phase
            if wr != 0.0:
                sigma_xr /= (rer*wr)
            else:
                sigma_xr = 0.0

        # calculate logitudinal tire forces
        fxf = sl_f*sigma_xf
        fxr = sl_r*sigma_xr

        # calculate rolling resistance
        rx = fr*m*g

        # calculate slip angle
        if vx == 0.0:
            if vy + lf*w == 0.0:
                theta_vf = 0.0
            else:
                theta_vf = 0.5*math.pi
            if vy - lr*w == 0.0:
                theta_vr = 0.0
            else:
                theta_vr = 0.5*math.pi
        else:
            theta_vf = (vy + lf*w)/vx
            theta_vr = (vy - lr*w)/vx
        alpha_f = steering_angle - theta_vf
        alpha_r = -theta_vr

        # calculate lateral tire force
        fyf = sc_f*alpha_f
        fyr = sc_r*alpha_r

        # calculate the state derivatives
        x_dot = vx*math.cos(theta) - vy*math.sin(theta)
        y_dot = vy*math.cos(theta) + vx*math.sin(theta)
        theta_dot = w
        vx_dot = (fxr + fxf*math.cos(steering_angle) - fyf *
                  math.sin(steering_angle) - rx - f_aero + m*vy*w)/m
        vy_dot = (fyr + fyf*math.cos(steering_angle) +
                  fxf*math.sin(steering_angle) - m*vx*w)/m
        w_dot = (lf*fxf*math.sin(steering_angle) + lf *
                 fyf*math.cos(steering_angle) - lr*fyr)/iz

        return np.array([[x_dot, y_dot, theta_dot, vx_dot, vy_dot, w_dot]])

    def output_model(self, state, u, param_dict):
        """
        Simulate the output of the system similar to the base class but with a special treatment for acceleration outputs.

        Args:
            state (numpy array [n x 1]): current state vector
            u (*): current input
            param_dict (dict): dictionary of current parameters needed for defining the dynamics

        Returns:
            output (numpy array [len(self.state_indices) + len(state_dot_indices) x 1]): observed state and
                state derivatives of the system

        """
        state_dot = self.dxdt(state, u, param_dict)
        state_dot[0, self.state_dict['vx']
                  ] -= state[self.state_dict['vy']]*state[self.state_dict['w']]
        state_dot[0, self.state_dict['vy']
                  ] += state[self.state_dict['vx']]*state[self.state_dict['w']]

        return np.concatenate((state[self.state_indices], state_dot[0, self.state_dot_indices]))


class RearDriveFrontSteerEst(RearDriveFrontSteer):
    """
    Class derived from RearDriveFrontSteer class
    model from: same as RearDriveFrontSteer but some parameters are the states themselves to be estimated online
    inputs = [linear acceleration, steering angle]
    states = [x, y, theta, vx, vy, w, ...]

    Args:
        param_dict (dict): dictionary of parameters needed for defining the dynamics
        est_params (list): list of parameter strings to be estimated with the states
        state_keys (list): list of states string that are observed by sensor
        state_dot_keys (list): list of derivative of states string that are observed by sensor; defaults to empty
        simulate_gt (bool): specify whether model is during the stage of simulating ground truth
        additional_output_keys (list): list of additional things that are observed by sensors; defaults to empty
        use_torch_tensor (bool): whether to use tensor instead of numpy arrays; defaults to False

    """

    def __init__(self, param_dict, est_params, state_keys, state_dot_keys=[], simulate_gt=False, additional_output_keys=[], use_torch_tensor=False):
        partial_init(self, RearDriveFrontSteerEst, est_params, param_dict, state_keys, simulate_gt,
                     state_dot_keys, additional_output_keys, use_torch_tensor=use_torch_tensor)

    def re_initialise(self):
        """
        re-initialise after ground truth data has been generated using its parent's derivative and raw possibly time-varying parameters
        """
        re_initialise(self)

    def dxdt(self, state, u, param_dict):
        """
        Obtain derivative of the state based on current state and input similar to RoverDyn. Derivatives of
        estimated parameters are zero, i.e. assuming they are constant and not drifting over time.

        Args:
            state (numpy array [6+len(self.est_params) x 1]): current state of the system consisting of x, y, theta, vx,
                vy, w and estimated parameters
            u (numpy array [3 x 1]): current input consisting of front wheel speed, rear wheel speed, steering angle
            param_dict (dict): dictionary of current non-estimated parameters needed for defining the dynamics

        Returns:
            state_dot (numpy array [6+len(self.est_params) x 1]): derivative of the states

        """
        return partial_dxdt(self, RearDriveFrontSteerEst, state, u, param_dict)


class RearDriveFrontSteerSubStateVel(AbstractDyn):
    """
    Class with the same dynamics & parameters as RearDriveFrontSteer but considers only subset of states
    model from: Modeling discussion presentation
    inputs = [front wheel speed, rear wheel speed, front wheel angle(steering angle), yaw rate]
    states = [vx, vy, ax, ay]

    Args:
        param_dict (dict): dictionary of parameters needed for defining the dynamics
        state_keys (list): list of states string that are observed by sensor
        state_dot_keys (list): list of derivative of states string that are observed by sensor; defaults to empty
        additional_output_keys (list): list of additional things that are observed by sensors; defaults to empty
        use_torch_tensor (bool): whether to use tensor instead of numpy arrays; defaults to False

    """
    # state dictionary for this model
    global_state_dict = {'vx': 0, 'vy': 1, 'ax': 2, 'ay': 3}
    # expected keys/parameters of this model
    global_expected_keys = ["m", "iz", "lf", "lr", "ref",
                            "rer", "fr", "g", "sc_f", "sc_r", "sl_f", "sl_r", "da"]

    def __init__(self, param_dict, state_keys, state_dot_keys=[], additional_output_keys=[], use_torch_tensor=False):
        super(RearDriveFrontSteerSubStateVel, self).__init__(param_dict, state_keys=state_keys,
                                                             state_dot_keys=state_dot_keys, additional_output_keys=additional_output_keys, use_torch_tensor=use_torch_tensor)
        # specify expected dimensionality of input
        self.num_in = 4

    def dxdt(self, state, u, param_dict):
        """
        Obtain derivative of the state based on current state and input

        Args:
            state (numpy array [2 x 1]): current state of the system consisting of vx and vy
            u (numpy array [4 x 1]): current input consisting of front wheel speed, rear wheel speed, steering angle and yaw rate
            param_dict (dict): dictionary of current parameters needed for defining the dynamics

        Returns:
            state_dot (numpy array [2 x 1]): derivative of the states

        """
        # get the parameters
        m = param_dict["m"]
        iz = param_dict["iz"]
        lf = param_dict["lf"]
        lr = param_dict["lr"]
        ref = param_dict["ref"]
        rer = param_dict["rer"]
        fr = param_dict["fr"]
        g = param_dict["g"]
        sc_r = param_dict["sc_r"]
        sc_f = param_dict["sc_f"]
        sl_r = param_dict["sl_r"]
        sl_f = param_dict["sl_f"]
        da = param_dict["da"]

        # get the inputs
        wf = u[0]
        wr = u[1]
        steering_angle = u[2]
        w = u[3]

        # get the states
        vx = state[self.state_dict['vx']]
        vy = state[self.state_dict['vy']]

        # compute aero dynamic force
        f_aero = da*(vx**2)

        # calculate longitudinal slip ratio
        sigma_xf = ref*wf - vx
        if sigma_xf < 0.0:
           # braking phase
           if vx != 0.0:
               sigma_xf /= vx
           else:
               sigma_xf = 0.0
               print("vx = 0 detected")
        else:
            # acceleration phase
            if wf != 0.0:
                sigma_xf /= (ref*wf)
            else:
                sigma_xf = 0.0
                print("wf = 0 detected")

        sigma_xr = rer*wr - vx
        if sigma_xr < 0.0:
           # braking phase
           if vx != 0.0:
               sigma_xr /= vx
           else:
               sigma_xr = 0.0
        else:
            # acceleration phase
            if wr != 0.0:
                sigma_xr /= (rer*wr)
            else:
                sigma_xr = 0.0
                print("wr = 0 detected")

        # calculate logitudinal tire forces
        fxf = sl_f*sigma_xf
        fxr = sl_r*sigma_xr

        # calculate rolling resistance
        rx = fr*m*g

        # calculate slip angle
        if vx == 0.0:
            if vy + lf*w == 0.0:
                theta_vf = 0.0
                print("vy + lf*w = 0 detected")
            else:
                theta_vf = 0.5*math.pi
            if vy - lr*w == 0.0:
                theta_vr = 0.0
                print("vy - lr*w = 0 detected")
            else:
                theta_vr = 0.5*math.pi
        else:
            theta_vf = (vy + lf*w)/vx
            theta_vr = (vy - lr*w)/vx
        alpha_f = steering_angle - theta_vf
        alpha_r = -theta_vr

        # calculate lateral tire force
        fyf = sc_f*alpha_f
        fyr = sc_r*alpha_r

        # calculate the state derivatives
        ax = (fxr + fxf*math.cos(steering_angle) - fyf *
              math.sin(steering_angle) - rx - f_aero)/m
        vx_dot = ax + vy*w
        ay = (fyr + fyf*math.cos(steering_angle) +
              fxf*math.sin(steering_angle))/m
        vy_dot = ay - vx*w

        return np.array([[vx_dot, vy_dot, ax, ay]])

    def forward_prop(self, state, state_dot, dt):
        """
        Forward propagation of model via first order Euler approximation, i.e. X[k+1] = X[k] + dt*X'[k], where X' is the derivative
        for some states while the rest are retrieved from dxdt function directly.

        Args:
            state (numpy array [n x 1 or 1 x n]): current state vector
            state_dot (numpy array [n x 1 or 1 x n]): current derivative of the state
            dt (float): time step/difference between discrete step k and step k+1

        Returns:
            state (numpy array [n x 1 or 1 x n]): next time-step state vector

        """
        next_state = state[:]
        for key in self.state_dict:
            if key != 'ax' and key != 'ay':
                next_state[self.state_dict[key]] += dt * \
                    state_dot[0, self.state_dict[key]]
            else:
                next_state[self.state_dict[key]
                           ] = state_dot[0, self.state_dict[key]]

        return next_state


class RearDriveFrontSteerSubStateVelEst(RearDriveFrontSteerSubStateVel):
    """
    Class derived from RearDriveFrontSteerSubStateVel class
    model from: same as RearDriveFrontSteer but some parameters are the states themselves to be estimated online
    inputs = [linear acceleration, steering angle]
    states = [vx, vy, ...]

    Args:
        param_dict (dict): dictionary of parameters needed for defining the dynamics
        est_params (list): list of parameter strings to be estimated with the states
        state_keys (list): list of states string that are observed by sensor
        state_dot_keys (list): list of derivative of states string that are observed by sensor; defaults to empty
        simulate_gt (bool): specify whether model is during the stage of simulating ground truth
        additional_output_keys (list): list of additional things that are observed by sensors; defaults to empty
        use_torch_tensor (bool): whether to use tensor instead of numpy arrays; defaults to False

    """

    def __init__(self, param_dict, est_params, state_keys, state_dot_keys=[], simulate_gt=False, additional_output_keys=[], use_torch_tensor=False):
        partial_init(self, RearDriveFrontSteerSubStateVelEst, est_params, param_dict, state_keys,
                     simulate_gt, state_dot_keys, additional_output_keys, use_torch_tensor=use_torch_tensor)

    def re_initialise(self):
        """
        re-initialise after ground truth data has been generated using its parent's derivative and raw possibly time-varying parameters
        """
        re_initialise(self)

    def dxdt(self, state, u, param_dict):
        """
        Obtain derivative of the state based on current state and input similar to RoverDyn. Derivatives of
        estimated parameters are zero, i.e. assuming they are constant and not drifting over time.

        Args:
            state (numpy array [2+len(self.est_params) x 1]): current state of the system consisting of vx, vy and estimated parameters
            u (numpy array [4 x 1]): current input consisting of front wheel speed, rear wheel speed, steering angle and yaw rate
            param_dict (dict): dictionary of current non-estimated parameters needed for defining the dynamics

        Returns:
            state_dot (numpy array [6+len(self.est_params) x 1]): derivative of the states

        """
        return partial_dxdt(self, RearDriveFrontSteerSubStateVelEst, state, u, param_dict)


def sample_linear(T, cruise_time, decrease_flag, *args):
    """
    Create an example input vector for the dynamic model in which each phsical entity is linearly increased to the
    corresponding maximum in args then cruise at the value for the required time before decreasing back to zero

    Args:
        T (numpy array [nt x 1]): time instances for generating the inputs
        cruise_time (float): number of seconds to maintain maximum of each physical entity before linearly
            decreasing back to zero
        decrease_flag (bool): whether to hold maximum value constant or ramp back to zero
        args (list): list of the maximum of each physical entity

    Returns:
        U (numpy array [len(args) x nt]): input consisting of each physical entity at different time instances
    """
    # create input vector
    U = np.zeros((len(args), len(T)))
    if decrease_flag:
        t_after_accel = (T[-1] - cruise_time)/2.0
    else:
        t_after_accel = (T[-1] - cruise_time)/1.0
    t_before_deaccel = t_after_accel + cruise_time

    # fill in the input for each physical entity
    for index, maximum in enumerate(args):
        slope = maximum/t_after_accel

        # linearly increase to maximum phase
        U[index, T <= t_after_accel] = slope*T[T <= t_after_accel]

        # cruise phase
        U[index, (T > t_after_accel) & (T < t_before_deaccel)] = maximum

        # deacceleration phase
        U[index, T >= t_before_deaccel] = -slope * \
            (T[T >= t_before_deaccel] - t_before_deaccel) + maximum

    return U


def sample_input_rover(T, max_steering=30*math.pi/180.0, max_speed=5.0, cruise_time=2.0):
    """
    Create an example input vector for the rover dynamic model in which:
    (i) steering angle linearly increase to max and decrease back to zero 
    (ii) velocity linearly increase to max and decrease back to zero

    Args:
        T (numpy array [nt x 1]): time instances for generating the inputs
        max_steering (float): maximum steering angle in radians to linearly increase from zero to; defaults to pi/6 radians
        max_speed (float): maximum speed in m/s to linearly increase from zero to; defaults to 5 m/s
        cruise_time (float): number of seconds to maintain maximum steering and speed before linearly decreasing 
            back to zero; defaults to 2 s

    Returns:
        U (numpy array [2 x nt]): input consisting of steering angle and velocity at different time instances

    """
    return sample_linear(T, cruise_time, *[max_steering, max_speed])


def sample_input_front_steered(T, max_w=45*math.pi/180.0, max_steering=20*math.pi/180.0, cruise_time=2.0):
    """
    Create an example input vector for the front_steered dynamic model in which:
    (i) wheel rate linearly increase to max and decrease back to zero
    (ii) steering angle linearly increase to max and decrease back to zero 

    Args:
        T (numpy array [nt x 1]): time instances for generating the inputs
        max_w (float): maximum wheel rate in rad/s to linearly increase from zero to; defaults to pi/4 rad/s
        max_steering (float): maximum steering angle in radians to linearly increase from zero to; defaults to pi/9 radians
        cruise_time (float): number of seconds to maintain maximum steering and speed before linearly decreasing 
            back to zero; defaults to 2 s

    Returns:
        U (numpy array [3 x nt]): input consisting of steering angle, left and right wheel rates (assumed equal) at 
            different time instances

    """
    return sample_linear(T, cruise_time, *[max_steering, max_w, max_w])


def test_rover_dyn():
    """
    Test function for the rover dynamics with trajectory plots and assert equivalence of RoverPartialDynEst and
    RoverDyn in producing the same ground truth data.
    """
    # assume parameter values in accordance to Michigan's rover
    param_dict = {'c1': 1.6615e-5, 'c2': -1.9555e-07, 'c3': 3.6190e-06, 'c4': 4.3820e-07,
                  'c5': -0.0811, 'c6': -1.4736, 'c7': 0.1257, 'c8': 0.0765, 'c9': -0.0140}

    # timing information
    dt = 0.05
    t_f = 20.0
    T = np.arange(0, t_f, dt)

    # create input vector for rover dynamics
    U = sample_input_rover(T)

    # get the ground truth states for propagating rover model
    z0 = np.zeros((4, 1))
    state_keys = ['x', 'y', 'theta', 'vx']
    dynamic_obj = RoverDyn(param_dict, state_keys=state_keys)
    gt_states, _, initial_cond, outputs = dynamic_obj.sample_nlds(z0, U, T)

    assert np.allclose(
        gt_states, outputs), "Gt not the same as output when no noises are present"
    assert np.allclose(
        z0, initial_cond), "Initial condition is not the same as first ground truth despite confident initial condition"

    # check result from the same model with unknown parameters
    z0 = np.zeros((13, 1))
    for i in range(1, 10):
        z0[3+i] = param_dict['c'+str(i)]
    est_params = ["c1", "c2", "c3", "c4", "c5", "c6", "c7", "c8", "c9"]
    est_param_dynamic_obj = RoverPartialDynEst({}, est_params, state_keys)
    gt_states1, _, initial_cond1, outputs1 = est_param_dynamic_obj.sample_nlds(
        z0, U, T)

    assert np.allclose(np.matlib.repmat(z0[4:, :], 1, len(
        T)), gt_states1[4:, :]), "States should remain the same, stationary"
    assert np.allclose(
        gt_states, gt_states1[:4, :]), "Gt should be the same given no uncertainty in parameters"
    assert np.allclose(
        outputs, outputs1), "Outputs should also be the same given no uncertainty in parameters"

    # do some plots of the resultant trajectory (sanity check)
    import matplotlib.pyplot as plt
    plt.plot(gt_states[0, :], gt_states[1, :], marker='x')
    plt.show()


if __name__ == '__main__':
    test_rover_dyn()