Ultrastick‑25e — Longitudinal Dynamics¶

The Ultrastick‑25e UAV is a lightweight experimental platform. This page mirrors the ELV layout: quick start, math model, derivatives, and API.

Quick start

Launch the environment or the model within minutes.

See example
Model API

Python class documentation for the Ultrastick.

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Gymnasium environment

Ready environment for RL agents.

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Theory

State equations and numerical parameters.

Learn more

Control object structure¶

The model is defined in the state space:

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

where:

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

Variables

u: longitudinal speed, m/s
w: vertical speed, m/s
θ: pitch angle, rad
q: pitch rate, rad/s
h: altitude, m
η: stabilizer deflection, rad
δ_t: throttle deflection, rad

Units

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

Mathematical model¶

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

Numerical matrices (example linearization):

\[ \begin{bmatrix} \dot{u} \\ \dot{w} \\ \dot{\theta} \\ \dot{q} \\ \dot{h} \end{bmatrix} = \begin{bmatrix} -0.5944 & 0.8008 & -9.791 & -0.8747 & 5.077\times 10^{-5} \\ -0.744 & -7.56 & -0.5294 & 15.72 & -0.000939 \\ 0 & 0 & 0 & 1 & 0 \\ 1.041 & -7.406 & 0 & -15.81 & -7.284\times 10^{-18} \\ -0.05399 & 0.9985 & -17 & 0 & 0 \end{bmatrix} \begin{bmatrix} u \\ w \\ \theta \\ q \\ h \end{bmatrix} + \begin{bmatrix} 0.4669 & 0 \\ -2.703 & 0 \\ 0 & 0 \\ -133.7 & 0 \\ 0 & 0 \end{bmatrix} \begin{bmatrix} \eta \\ \delta_t \end{bmatrix} \]

Key derivatives¶

\(\dot{\theta} = q\) (element A[3,4] = 1)
Influence of \(\eta\) on \(\dot{q}\): A[4,*], B[4,1] = −133.7
Influence of \(\eta\) on \(\dot{u}\), \(\dot{w}\): B[1,1] = 0.4669, B[2,1] = −2.703

Sources¶

Ahmed EA, Hafez A, Ouda AN, Ahmed HEH, Abd‑Elkader HM. Modelling of a Small Unmanned Aerial Vehicle. Adv Robot Autom 4:126, 2015.

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¶

Gymnasium

import gymnasium as gym 
import numpy as np

from tensoraerospace.envs import LinearLongitudinalUltrastick
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(
    'LinearLongitudinalUltrastick-v0',
    number_time_steps=number_time_steps, 
    initial_state=[[0],[0],[0],[0],[0]],
    reference_signal=reference_signals,
)
state, info = env.reset()
for _ in range(200):
    action = np.array([[1.0, 0.1]])  # [η, δ_t]
    state, reward, terminated, truncated, info = env.step(action)
    if terminated or truncated:
        break

Model only

import numpy as np
from tensoraerospace.aerospacemodel import Ultrastick

dt = 0.01
number_time_steps = 200

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

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

for t in range(number_time_steps - 1):
    u = np.array([1.0, 0.1])  # [η, δ_t]
    x_next = model.run_step(u)

Python API¶

Model

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

Bases: ModelBase

UAV Ultrastick-25e 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 [rad] delta_t: Dimensionless value, 0 — off, 1 — max thrust

State space

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

Output space

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

`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: System output at the current step (via C/D).

`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.

Gymnasium environment

`LinearLongitudinalUltrastick(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 Ultrastick-25e in Gym environment for training AI agents.

Parameters:

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

Initialize legacy Ultrastick environment.

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

Control evaluation.

Parameters:

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

Returns:

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

`step(action)` ¶

Execute simulation step.

Parameters:

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

Returns:

Name	Type	Description
`next_state`	`ndarray`	Next state of control object.
`reward`	`ndarray`	Evaluation of control algorithm actions.
`done`	`bool`	Simulation status, completed or not.
`logging`	`any`	Additional information (not used).

`reset(seed=None, options=None)` ¶

Reset simulation environment to initial conditions.

Parameters:

Name	Type	Description	Default
`seed`	`int`	Seed for random number generator.	`None`
`options`	`dict`	Additional options for initialization.	`None`

`render()` ¶

Visual display of actions in environment (not implemented).

Ultrastick‑25e — Longitudinal Dynamics¶

Control object structure¶

Mathematical model¶

Key derivatives¶

Sources¶

Reward¶

Quick start¶

Python API¶

Ultrastick(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)) ¶

LinearLongitudinalUltrastick(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() ¶

`Ultrastick(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))` ¶

`LinearLongitudinalUltrastick(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()` ¶