UAV — Longitudinal Dynamics¶

An unmanned aerial vehicle (UAV) is a remotely piloted or autonomous aircraft. This page mirrors the ELV layout: quick start, math model, derivatives, and API.

Quick start

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Model API

Python class documentation for the UAV longitudinal dynamics.

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Gymnasium environment

Ready environment for RL agents.

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Theory

State equations and numerical parameters.

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Control object structure¶

The model is defined in the state space:

\[\dot{x} = A x + B u, \quad y = C x + D u\]

where:

\[ x = \begin{bmatrix} u & w & q & \theta \end{bmatrix}^{\top}, \quad u_{in} = \eta \]

The typical matrix structure is:

\[ \begin{bmatrix} \dot{u} \\ \dot{w} \\ \dot{q} \\ \dot{\theta} \end{bmatrix} = \begin{bmatrix} x_u & x_w & x_q & x_{\theta} \\ z_u & z_w & z_q & z_{\theta} \\ m_u & m_w & m_q & m_{\theta} \\ 0 & 0 & 1 & 0 \end{bmatrix} \begin{bmatrix} u \\ w \\ q \\ \theta \end{bmatrix} + \begin{bmatrix} x_{\eta} \\ z_{\eta} \\ m_{\eta} \\ 0 \end{bmatrix} \eta \]

VariablesCoefficients

u: longitudinal speed, m/s
w: vertical speed, m/s
q: pitch rate, rad/s
θ: pitch angle, rad
η: stabilizer control deflection, rad

x_u, x_w, x_q, x_θ — partial derivatives of longitudinal force \(X\) with respect to \(u, w, q, \theta\)
z_u, z_w, z_q, z_θ — partial derivatives of normal force \(Z\)
m_u, m_w, m_q, m_θ — partial derivatives of pitch moment \(M\)
x_η, z_η, m_η — derivatives with respect to the control \(\eta\)

Units

Angles and angular rates are in radians. API methods can produce values in degrees.

Mathematical model¶

\[ \dot{x} = A x + B u, \qquad y = C x + D u \]

Numerical matrices (example linearization):

\[ \begin{bmatrix} \dot{u} \\ \dot{w} \\ \dot{q} \\ \dot{\theta} \end{bmatrix} = \begin{bmatrix} -0.1982 & 0.593 & 1.245 & -9.779 \\ -0.7239 & -3.9848 & 18.7028 & -0.6286 \\ 0.3537 & -5.5023 & -5.4722 & 0 \\ 0 & 0 & 1 & 0 \end{bmatrix} \begin{bmatrix} u \\ w \\ q \\ \theta \end{bmatrix} + \begin{bmatrix} 0.2281 \\ -4.6830 \\ -36.1341 \\ 0.0 \end{bmatrix} \eta \]

Derivatives (numerical values)¶

Matrix A (derivatives):

Coefficient	Value
x_u	-0.1982
x_w	0.593
x_q	1.245
x_θ	-9.779
z_u	-0.7239
z_w	-3.9848
z_q	18.7028
z_θ	-0.6286
m_u	0.3537
m_w	-5.5023
m_q	-5.4722
m_θ	0.0

Input η (column B):

Coefficient	Value
x_η	0.2281
z_η	-4.6830
m_η	-36.1341

Sources¶

A. Rauf, Muhammad Aamir Zafar, Z. Ashraf and H. Akhtar, "Aerodynamic modeling and state-space model extraction of a UAV using DATCOM and Simulink," 2011 3rd International Conference on Computer Research and Development, Shanghai, China, 2011, pp. 88-92, doi: 10.1109/ICCRD.2011.5763860.

Reward¶

The default reward function returns the negative absolute tracking error for the pitch angle:

\[r_t = -|\theta(t) - \theta_{\text{ref}}(t)|\]

Higher reward (closer to 0) indicates better tracking performance. A custom reward function can be passed via the reward_func parameter.

Quick start¶

GymnasiumModel only

import gymnasium as gym 
import numpy as np

from tensoraerospace.envs import LinearLongitudinalUAV
from tensoraerospace.utils import generate_time_period
from tensoraerospace.signals.standard import unit_step

dt = 0.01
tp = generate_time_period(tn=20, dt=dt)
number_time_steps = len(tp)
reference_signals = unit_step(degree=5, tp=tp, time_step=10, output_rad=True).reshape(1, -1)

env = gym.make(
    'LinearLongitudinalUAV-v0',
    number_time_steps=number_time_steps, 
    initial_state=[[0],[0],[0],[0]],
    reference_signal=reference_signals,
)
state, info = env.reset()
for _ in range(200):
    action = np.array([[0.1]])
    state, reward, terminated, truncated, info = env.step(action)
    if terminated or truncated:
        break

import numpy as np
from tensoraerospace.aerospacemodel import LongitudinalUAV

dt = 0.01
number_time_steps = 200

x0 = np.array([0.0, 0.0, 0.0, 0.0])

model = LongitudinalUAV(
    x0=x0,
    number_time_steps=number_time_steps,
    selected_state_output=["u", "w", "q", "theta"],
    dt=dt,
)

for t in range(number_time_steps - 1):
    u = np.array([[0.05]])
    x_next = model.run_step(u)

Python API¶

ModelGymnasium environment

`LongitudinalUAV(x0, number_time_steps, selected_state_output=None, t0=0, dt=0.01)` ¶

Bases: ModelBase

UAV model in longitudinal control channel.

Parameters:

Name	Type	Description	Default
`x0`	`ndarray \| list[float]`	Initial state of the control object.	required
`number_time_steps`	`int`	Number of time steps.	required
`selected_state_output`	`optional`	Selected states of the control object. Defaults to None.	`None`
`t0`	`int`	Initial time. Defaults to 0.	`0`
`dt`	`float`	Discretization frequency. Defaults to 0.01.	`0.01`

Action space

ele: elevator [deg]

State space

u: Longitudinal aircraft velocity [m/s] w: Normal aircraft velocity [m/s] q: Pitch angular velocity [deg/s] theta: Pitch [deg]

Output space

u: Longitudinal aircraft velocity [m/s] w: Normal aircraft velocity [m/s] q: Pitch angular velocity [deg/s] theta: Pitch [deg]

Initialize LongitudinalUAV instance.

Parameters:

Name	Type	Description	Default
`x0`	`ndarray \| list[float]`	Initial state of the control object.	required
`number_time_steps`	`int`	Number of time steps.	required
`selected_state_output`	`list[str] \| None`	Selected states of the control object. Defaults to None.	`None`
`t0`	`float`	Initial time. Defaults to 0.	`0`
`dt`	`float`	Discretization frequency. Defaults to 0.01.	`0.01`

`import_linear_system()` ¶

Load (set) stored linearized system matrices.

`initialise_system(x0, number_time_steps)` ¶

Initialize the system and allocate history buffers.

Parameters:

Name	Type	Description	Default
`x0`	`ndarray \| list[float]`	Initial state.	required
`number_time_steps`	`int`	Number of simulation steps.	required

`run_step(ut_0)` ¶

Run one discrete-time simulation step.

Parameters:

Name	Type	Description	Default
`ut_0`	`ndarray`	Control vector.	required

Returns:

Type	Description
`ndarray`	np.ndarray: Next state at time t+1.

`update_system_attributes()` ¶

Update time-dependent attributes after each simulation step.

`get_state(state_name, to_deg=False, to_rad=False)` ¶

Return the time history of a state.

Parameters:

Name	Type	Description	Default
`state_name`	`str`	State name.	required
`to_deg`	`bool`	Convert radians to degrees.	`False`
`to_rad`	`bool`	Convert degrees to radians.	`False`

Returns:

Type	Description
`ndarray`	np.ndarray: State history array.

`get_control(control_name, to_deg=False, to_rad=False)` ¶

Return the time history of a control input.

Parameters:

Name	Type	Description	Default
`control_name`	`str`	Control signal name.	required
`to_deg`	`bool`	Convert radians to degrees.	`False`
`to_rad`	`bool`	Convert degrees to radians.	`False`

Returns:

Type	Description
`ndarray`	np.ndarray: Control history array.

`get_output(state_name, to_deg=False, to_rad=False)` ¶

Return the time history of an output signal.

Parameters:

Name	Type	Description	Default
`state_name`	`str`	Output name.	required
`to_deg`	`bool`	Convert radians to degrees.	`False`
`to_rad`	`bool`	Convert degrees to radians.	`False`

Returns:

Type	Description
`ndarray`	np.ndarray: Output history array.

`plot_output(output_name, time, lang='rus', to_deg=False, to_rad=False, figsize=(10, 10))` ¶

Plot an output signal over time.

Parameters:

Name	Type	Description	Default
`output_name`	`str`	Output name.	required
`time`	`ndarray`	Time vector.	required
`lang`	`str`	Axis label language ('rus' or 'eng').	`'rus'`
`to_deg`	`bool`	Convert radians to degrees.	`False`
`to_rad`	`bool`	Convert degrees to radians.	`False`
`figsize`	`tuple`	Figure size.	`(10, 10)`

Returns:

Type	Description
`Figure`	matplotlib.figure.Figure: Figure object.

`LinearLongitudinalUAV(initial_state, reference_signal, number_time_steps, tracking_states=None, state_space=None, control_space=None, output_space=None, reward_func=None)` ¶

Bases: Env

Simulation of LongitudinalUAV control object in OpenAI Gym environment for training AI agents.

Parameters:

Name	Type	Description	Default
`initial_state`	`ndarray \| list[float]`	Initial state.	required
`reference_signal`	`ndarray \| Callable`	Reference signal.	required
`number_time_steps`	`int`	Number of simulation steps.	required
`tracking_states`	`list[str] \| None`	Tracked states.	`None`
`state_space`	`list[str] \| None`	State space.	`None`
`control_space`	`list[str] \| None`	Control space.	`None`
`output_space`	`list[str] \| None`	Full output space (including noise).	`None`
`reward_func`	`Callable \| None`	Reward function (WIP status).	`None`

Initialize UAV longitudinal environment.

`reward(state, ref_signal, ts)` `staticmethod` ¶

Evaluate control performance.

Parameters:

Name	Type	Description	Default
`state`	`ndarray`	Current state.	required
`ref_signal`	`ndarray`	Reference signal.	required
`ts`	`int`	Time step.	required

Returns:

Name	Type	Description
`float`	`float`	Control evaluation reward.

`step(action)` ¶

Execute one simulation step.

Parameters:

Name	Type	Description	Default
`action`	`ndarray`	Control signal array for selected actuators.	required

Returns:

Name	Type	Description
`tuple`	`tuple[ndarray, float, bool, bool, dict[str, float]]`	Tuple containing: - next_state (np.ndarray): Next state of the control object. - reward (np.ndarray): Evaluation of control algorithm actions. - done (bool): Simulation status, whether completed or not. - truncated (bool): Whether episode was truncated. - info (dict): Additional information.

`reset(seed=None, options=None)` ¶

Reset simulation environment to initial conditions.

Parameters:

Name	Type	Description	Default
`seed`	`int`	Random seed. Defaults to None.	`None`
`options`	`dict`	Additional initialization options. Defaults to None.	`None`

Returns:

Name	Type	Description
`tuple`	`tuple[ndarray, dict[str, float]]`	Tuple containing: - observation (np.ndarray): Initial observation. - info (dict): Additional information.

`render()` ¶

Visual rendering of actions in the environment. Work in progress.

Raises:

Type	Description
`NotImplementedError`	Rendering is not yet implemented.

UAV — Longitudinal Dynamics¶

Control object structure¶

Mathematical model¶

Derivatives (numerical values)¶

Sources¶

Reward¶

Quick start¶

Python API¶

LongitudinalUAV(x0, number_time_steps, selected_state_output=None, t0=0, dt=0.01) ¶

import_linear_system() ¶

initialise_system(x0, number_time_steps) ¶

run_step(ut_0) ¶

update_system_attributes() ¶

get_state(state_name, to_deg=False, to_rad=False) ¶

get_control(control_name, to_deg=False, to_rad=False) ¶

get_output(state_name, to_deg=False, to_rad=False) ¶

plot_output(output_name, time, lang='rus', to_deg=False, to_rad=False, figsize=(10, 10)) ¶

LinearLongitudinalUAV(initial_state, reference_signal, number_time_steps, tracking_states=None, state_space=None, control_space=None, output_space=None, reward_func=None) ¶

reward(state, ref_signal, ts) staticmethod ¶

step(action) ¶

reset(seed=None, options=None) ¶

render() ¶

`LongitudinalUAV(x0, number_time_steps, selected_state_output=None, t0=0, dt=0.01)` ¶

`import_linear_system()` ¶

`initialise_system(x0, number_time_steps)` ¶

`run_step(ut_0)` ¶

`update_system_attributes()` ¶

`get_state(state_name, to_deg=False, to_rad=False)` ¶

`get_control(control_name, to_deg=False, to_rad=False)` ¶

`get_output(state_name, to_deg=False, to_rad=False)` ¶

`plot_output(output_name, time, lang='rus', to_deg=False, to_rad=False, figsize=(10, 10))` ¶

`LinearLongitudinalUAV(initial_state, reference_signal, number_time_steps, tracking_states=None, state_space=None, control_space=None, output_space=None, reward_func=None)` ¶

`reward(state, ref_signal, ts)` `staticmethod` ¶

`step(action)` ¶

`reset(seed=None, options=None)` ¶

`render()` ¶