Typical Rocket — Longitudinal Dynamics¶

Canonical missile model in the longitudinal channel. The page mirrors the ELV layout: quick start, math model, derivative tables, and API.

Quick start

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Model API

Python class documentation for the typical rocket.

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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 return values in degrees.

Mathematical model¶

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

Numerical matrices (example linearization):

\[ \begin{bmatrix} \dot{u} \\ \dot{w} \\ \dot{q} \\ \dot{\theta} \end{bmatrix} = \begin{bmatrix} -0.0089 & -0.1474 & 0 & -9.75 \\ -0.0216 & -0.3601 & 5.9470 & 0.01958 \\ 0 & -0.0015 & -0.0224 & 0.0006 \\ 0 & 0 & 1 & 0 \end{bmatrix} \begin{bmatrix} u \\ w \\ q \\ \theta \end{bmatrix} + \begin{bmatrix} 9.748 \\ 3.77 \\ -0.034 \\ 0.01 \end{bmatrix} \eta \]

Derivatives (numerical values)¶

Matrix A (derivatives):

Coefficient	Value
x_u	-0.0089
x_w	-0.1474
x_q	0
x_θ	-9.75
z_u	-0.0216
z_w	-0.3601
z_q	5.9470
z_θ	0.01958
m_u	0.0
m_w	-0.0015
m_q	-0.0224
m_θ	0.0006

Input η (column B):

Coefficient	Value
x_η	9.748
z_η	3.77
m_η	-0.034
θ_η	0.01

Actuator limits

The default control limits are:

Maximum magnitude: \(\pm 25^\circ\)
Maximum rate: \(60^\circ/\text{s\)

Internal computations use radians; the limits are converted accordingly.

Sources¶

Arikapalli V. S. N. et al. Missile Longitudinal Dynamics Control Design using Pole Placement and LQR Methods — A Critical Analysis // Defence Science Journal. 2021. 71(5). Link

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 LinearLongitudinalMissileModel
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(
    'LinearLongitudinalMissileModel-v0',
    number_time_steps=number_time_steps, 
    initial_state=[[0],[0],[0],[0]],  # u, w, q, theta
    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 MissileModel

dt = 0.01
number_time_steps = 200

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

model = MissileModel(
    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

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

Bases: ModelBase

Missile 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 MissileModel 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[int] \| 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 stored linearized matrices.

`initialise_system(x0, number_time_steps)` ¶

Initialize the system.

Parameters:

Name	Type	Description	Default
`x0`	`ndarray \| list[float]`	Initial state of the control object.	required
`number_time_steps`	`int`	Number of time steps in the iteration.	required

`run_step(ut_0)` ¶

Execute one time step iteration.

Parameters:

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

Returns:

Name	Type	Description
`xt1`	`ndarray`	Control object state at step t+1.

`update_system_attributes()` ¶

Update attributes that change with each time step.

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

Get state array history.

Parameters:

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

Returns:

Type	Description
`ndarray`	np.ndarray: Array of selected state history.

Example

state_hist = model.get_state('alpha', to_deg=True)

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

Get control signal array history.

Parameters:

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

Returns:

Type	Description
`ndarray`	np.ndarray: Array of selected control signal history.

Example

control_hist = model.get_control('stab', to_deg=True)

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

Get output signal array history.

Parameters:

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

Returns:

Type	Description
`ndarray`	np.ndarray: Array of selected output signal history.

Example

output_hist = model.get_output('alpha', to_deg=True)

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

Plot output signal.

Parameters:

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

Returns:

Name	Type	Description
`Figure`	`Figure`	Matplotlib figure object.

Raises:

Type	Description
`Exception`	If both to_rad and to_deg are specified, or if output_name is invalid.

Example

fig = model.plot_output('alpha', time_array, lang='rus', to_deg=True)

`LinearLongitudinalMissileModel(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 MissileModel in Gym 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 linear missile model environment.

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

Evaluate control performance.

Parameters:

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

Returns:

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

`step(action)` ¶

Execute a simulation step.

Parameters:

Name	Type	Description	Default
`action`	`ndarray`	Array of control signals for selected control surfaces.	required

Returns:

Name	Type	Description
`next_state`	`ndarray`	Next state of the control object.
`reward`	`ndarray`	Evaluation of control algorithm actions.
`done`	`bool`	Simulation status, whether 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 the environment. Status: WIP.

Raises:

Type	Description
`NotImplementedError`	Rendering is not implemented.

Typical Rocket — Longitudinal Dynamics¶

Control object structure¶

Mathematical model¶

Derivatives (numerical values)¶

Sources¶

Reward¶

Quick start¶

Python API¶

MissileModel(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)) ¶

LinearLongitudinalMissileModel(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() ¶

`MissileModel(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))` ¶

`LinearLongitudinalMissileModel(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()` ¶