inertialsim.symbolic ¶

Symbolic trajectories.

The symbolic module is optional. It provides exact symbolic/analytical trajectories (attitude, position, and velocity as functions of time) along with their derivatives. Symbolic pose, angular rate and specific force enable true ground-truth comparisons for testing and analysis.

Singular configurations

Be aware of potential singularities in symbolic expressions, for example when converting between rotation representations near multiples of \(\pi\). These are well handled in the numeric inertialsim.geometry module but are not completely eliminated in these symbolic expressions.

The module requires one additional dependency:

SymPy

Classes:

Name	Description
`SO3`	SO(3) Lie group.
`SE3`	SE(3) Lie group.
`Trajectory`	Base class for symbolic trajectories.
`ConingRotation`	Coning rotation.
`EarthRotation`	Earth rotation.
`ZigZagTrajectory`	Zigzag trajectory.
`GroundVehiclePath`	Simulated ground vehicle path.

Functions:

Name	Description
`lambdify`	Create a lambda function from a symbolic expression.
`integral`	Integrate each matrix component from zero to the time symbol.
`double_integral`	Double integrate each matrix component from zero to the time symbol.
`power_series`	Compute the power series coefficients for a 3D rotation vector.

Attributes:

Name	Type	Description
`TIME_SYMBOL`	`Symbol`

TIME_SYMBOL `module-attribute` ¶

TIME_SYMBOL: Symbol = sympy.symbols(
    "t", real=True, positive=True
)

SO3 ¶

SO(3) Lie group.

Provides a minimal symbolic implementation of SO(3) methods.

Methods:

Name	Description
`Rx`	Rotation around the x-axis.
`Ry`	Rotation around the y-axis.
`Rz`	Rotation around the z-axis.
`exp`	Exponential map.
`log`	Logarithm map.
`jacobian`	Left and right Jacobians.

Rx `staticmethod` ¶

Rx(roll: Expr) -> Matrix

Rotation around the x-axis.

Parameters:

Name	Type	Description	Default
`roll`	`Expr`	Rotation angle around the x-axis.	required

Returns:

Type	Description
`Matrix`	The corresponding rotation matrix in SO(3).

Ry `staticmethod` ¶

Ry(pitch: Expr) -> Matrix

Rotation around the y-axis.

Parameters:

Name	Type	Description	Default
`pitch`	`Expr`	Rotation angle around the y-axis.	required

Returns:

Type	Description
`Matrix`	The corresponding rotation matrix in SO(3).

Rz `staticmethod` ¶

Rz(yaw: Expr) -> Matrix

Rotation around the z-axis.

Parameters:

Name	Type	Description	Default
`yaw`	`Expr`	Rotation angle around the z-axis.	required

Returns:

Type	Description
`Matrix`	The corresponding rotation matrix in SO(3).

exp `staticmethod` ¶

exp(φ: Matrix) -> Matrix

Exponential map.

Computes the exponential map from rotation vector in the Lie algebra, so(3), to the rotation matrix in the Lie group, SO(3).

Parameters:

Name	Type	Description	Default
`φ`	`Matrix`	Rotation vector in so(3).	required

Returns:

Name	Type	Description
`R`	`Matrix`	The corresponding rotation matrix in SO(3).

log `staticmethod` ¶

log(R: Matrix) -> Matrix

Logarithm map.

Computes the logarithm map from the rotation matrix in the Lie group, SO(3), to the rotation vector in the Lie algebra, so(3).

Parameters:

Name	Type	Description	Default
`R`	`Matrix`	Rotation matrix in SO(3).	required

Returns:

Name	Type	Description
`φ`	`Matrix`	The corresponding rotation vector in so(3).

jacobian `staticmethod` ¶

jacobian(
    φ: Matrix, side: str = "left", *, inverse: bool = False
) -> Matrix

Left and right Jacobians.

Parameters:

Name	Type	Description	Default
`φ`	`Matrix`	Rotation vector in so(3).	required
`side`	`str`	Return the "left" or "right" Jacobian.	`'left'`
`inverse`	`bool`	Return the inverse Jacobian.	`False`

Returns:

Type	Description
`Matrix`	The requested Jacobian matrix.

SE3 ¶

SE(3) Lie group.

Provides convenience functions like the exponential and logarithmic maps for 3D rigid body transformations. This class does not contain a complete implementation of the SE(3) Lie group.

Methods:

Name	Description
`exp`	Exponential map.
`log`	Logarithmic map.
`jacobian`	Left and right jacobians.

exp `staticmethod` ¶

exp(φ: Matrix, η: Matrix) -> tuple[Matrix, Matrix]

Exponential map.

Computes the exponential map from the Lie algebra se(3) to the Lie group SE(3). The algebra is represented by the rotation vector, φ, and the generalized translation/velocity vector, η. In many references, the translation vector is denoted as lower-case greek letter, rho. We use the η for generalized velocity from Savage. The output is the conventional rotation matrix and translation/velocity vector.

Parameters:

Name	Type	Description	Default
`φ`	`Matrix`	Rotation vector.	required
`η`	`Matrix`	Generalized translation or velocity vector.	required

Returns:

Type	Description
`tuple[Matrix, Matrix]`	A tuple containing the rotation matrix and the translation vector.

log `staticmethod` ¶

log(R: Matrix, v: Matrix) -> tuple[Matrix, Matrix]

Logarithmic map.

Computes the logarithmic map from the Lie group SE(3) to the Lie algebra se(3). The group is represented by the rotation matrix R and the translation/velocity vector v. We use the v for velocity for consistency with the exponential map. The output is the rotation vector and generalized translation/velocity vector.

Parameters:

Name	Type	Description	Default
`R`	`Matrix`	Rotation matrix.	required
`v`	`Matrix`	Translation/velocity vector.	required

Returns:

Type	Description
`Matrix`	A tuple containing the rotation vector, φ, and the generalized
`Matrix`	translation/velocity vector, η.

jacobian `classmethod` ¶

jacobian(
    φ: Matrix,
    η: Matrix,
    side: str = "left",
    *,
    inverse: bool = False,
) -> Matrix

Left and right jacobians.

Parameters:

Name	Type	Description	Default
`φ`	`Matrix`	Rotation vector.	required
`η`	`Matrix`	Generalized translation or velocity vector.	required
`side`	`str`	Return the "left" or "right" Jacobian.	`'left'`
`inverse`	`bool`	Return the inverse Jacobian.	`False`

Returns:

Type	Description
`Matrix`	The 6x6 Jacobian matrix.

Trajectory ¶

Trajectory(
    attitude: Matrix, position: Matrix, gravity: Matrix
)

Base class for symbolic trajectories.

Define symbolic/analytical trajectories in attitude (orientation) and position. Velocity and IMU data (angular rate and specific force) are analytically derived. Additional useful intermediates (Jacobians, Lie algebra derivatives, etc.) are also analytically derived. Methods to efficiently sample numerical values are provided.

The suggested use is to subclass Trajectory and provide convenient methods to define the symbolic expressions for attitude, position, and gravity in the initializer.

Attitude can be a rotation vector or a rotation matrix. Position must be a Cartesian vector. Inputs must use the symbol t for time.

Parameters:

Name	Type	Description	Default
`attitude`	`Matrix`	Attitude in rotation vector or rotation matrix form.	required
`position`	`Matrix`	Position in world coordinates.	required
`gravity`	`Matrix`	Gravity vector in world coordinates.	required

Methods:

Name	Description
`compose`	Compose two trajectories.
`rotation_vector`	Return the attitude as a rotation vector.
`rotation_vector_rate`	Return the attitude rate as a rotation vector.
`rotation_matrix`	Return the attitude as a rotation matrix.
`angular_rate`	Return the angular rate.
`rotation`	Return the attitude as a Rotation instance.
`extended_pose`	Return the attitude, position, and velocity.
`velocity`	Return the velocity.
`position`	Return the position.
`acceleration`	Return the acceleration.
`specific_force`	Return the specific force.
`gravity`	Return the gravity vector.
`plot_attitude`	Plot attitude variables.
`plot_translation`	Plot translation variables.
`plot`	Plots trajectory variables.

Attributes:

Name	Type	Description
`time`	`Symbol`

time `instance-attribute` ¶

time: Symbol = TIME_SYMBOL

compose ¶

compose(
    other: Self,
    gravity: Literal["self", "other"] | Matrix = "self",
) -> Trajectory

Compose two trajectories.

Composes the current (self) trajectory with the input (other) such that \(T_{composed} = T_{self} * T_{other}\). Gravity can be inherited from self (default), inherited from other (rotated by self), or can be a new input. Users must ensure that coordinate systems are respected when composing trajectories and that the same time symbol is used for both.

Parameters:

Name	Type	Description	Default
`other`	`Self`	Trajectory object to compose with.	required
`gravity`	`Literal['self', 'other'] \| Matrix`	Gravity source to use. Can be "self" (default), "other", or a SymPy Matrix.	`'self'`

Returns:

Type	Description
`Trajectory`	New Trajectory instance.

rotation_vector ¶

rotation_vector(
    time: ArrayLike | None = None,
) -> Matrix | NDArray

Return the attitude as a rotation vector.

Returns a symbolic representation if the time input is None (default) or a numeric array sampled at the input times if supplied.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike \| None`	Array of sampling times.	`None`

Returns:

Type	Description
`Matrix \| NDArray`	Rotation vector.

rotation_vector_rate ¶

rotation_vector_rate(
    time: ArrayLike | None = None,
) -> Matrix | NDArray

Return the attitude rate as a rotation vector.

Returns a symbolic representation if the time input is None (default) or a numeric array sampled at the input times if supplied.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike \| None`	Array of sampling times.	`None`

Returns:

Type	Description
`Matrix \| NDArray`	Rotation vector rate.

rotation_matrix ¶

rotation_matrix(
    time: ArrayLike | None = None,
) -> Matrix | NDArray

Return the attitude as a rotation matrix.

Returns a symbolic representation if the time input is None (default) or a numeric array sampled at the input times if supplied.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike \| None`	Array of sampling times.	`None`

Returns:

Type	Description
`Matrix \| NDArray`	Rotation matrix.

angular_rate ¶

angular_rate(
    time: ArrayLike | None = None,
) -> Matrix | NDArray

Return the angular rate.

Returns a symbolic representation if the time input is None (default) or a numeric array sampled at the input times if supplied.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike \| None`	Array of sampling times.	`None`

Returns:

Type	Description
`Matrix \| NDArray`	Angular rate.

rotation ¶

rotation(time: ArrayLike) -> Rotation

Return the attitude as a Rotation instance.

Returns a geometry.Rotation instance with data sampled at the input times.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike`	Array of sampling times.	required

Returns:

Type	Description
`Rotation`	Rotation instance.

extended_pose ¶

extended_pose(time: ArrayLike) -> ExtendedPose

Return the attitude, position, and velocity.

Returns a geometry.ExtendedPose instance with data sampled at the input times.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike`	Array of sampling times.	required

Returns:

Type	Description
`ExtendedPose`	ExtendedPose instance.

velocity ¶

velocity(time: ArrayLike | None = None) -> Matrix | NDArray

Return the velocity.

Returns a symbolic representation if the time input is None (default) or a numeric array sampled at the input times if supplied.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike \| None`	Array of sampling times.	`None`

Returns:

Type	Description
`Matrix \| NDArray`	Velocity vector.

position ¶

position(time: ArrayLike | None = None) -> Matrix | NDArray

Return the position.

Returns a symbolic representation if the time input is None (default) or a numeric array sampled at the input times if supplied.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike \| None`	Array of sampling times.	`None`

Returns:

Type	Description
`Matrix \| NDArray`	Position vector.

acceleration ¶

acceleration(
    time: ArrayLike | None = None,
) -> Matrix | NDArray

Return the acceleration.

Returns a symbolic representation if the time input is None (default) or a numeric array sampled at the input times if supplied.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike \| None`	Array of sampling times.	`None`

Returns:

Type	Description
`Matrix \| NDArray`	Acceleration vector in world/navigation coordinates.

specific_force ¶

specific_force(
    time: ArrayLike | None = None,
) -> Matrix | NDArray

Return the specific force.

Returns a symbolic representation if the time input is None (default) or a numeric array sampled at the input times if supplied.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike \| None`	Array of sampling times.	`None`

Returns:

Type	Description
`Matrix \| NDArray`	Specific force vector.

gravity ¶

gravity(time: ArrayLike | None = None) -> Matrix | NDArray

Return the gravity vector.

Returns a symbolic representation if the time input is None (default) or a numeric array sampled at the input times if supplied.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike \| None`	Array of sampling times.	`None`

Returns:

Type	Description
`Matrix \| NDArray`	Gravity vector.

plot_attitude ¶

plot_attitude(
    time: ArrayLike, title_prefix: str = ""
) -> tuple[TimeSeries, TimeSeries, TimeSeries]

Plot attitude variables.

Plots the rotation vector, rotation vector rate, and angular rate sampled at the input times.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike`	Sampling times.	required
`title_prefix`	`str`	Prefix for plot titles.	`''`

Returns:

Name	Type	Description
`plots`	`tuple[TimeSeries, TimeSeries, TimeSeries]`	Tuple containing three plot handles.

plot_translation ¶

plot_translation(
    time: ArrayLike, title_prefix: str = ""
) -> tuple[Scatter, TimeSeries, TimeSeries]

Plot translation variables.

Plots the position, velocity, and specific force sampled at the input times. Position is plotted as a plan-view (x-axis v y-axis). Velocity and specific force are plotted as time series graphs.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike`	Sampling times.	required
`title_prefix`	`str`	Prefix for plot titles.	`''`

Returns:

Name	Type	Description
`plots`	`tuple[Scatter, TimeSeries, TimeSeries]`	Tuple containing three plot handles.

plot ¶

plot(
    time: ArrayLike, title_prefix: str = ""
) -> tuple[
    tuple[TimeSeries, TimeSeries, TimeSeries],
    tuple[Scatter, TimeSeries, TimeSeries],
]

Plots trajectory variables.

Plots attitude and translation variables. See plot_attitude and plot_translation for details.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike`	Sampling times.	required
`title_prefix`	`str`	Prefix for plot titles.	`''`

Returns:

Name	Type	Description
`attitude_plots`	`tuple[TimeSeries, TimeSeries, TimeSeries]`	Tuple containing three plot handles.
`translation_plots`	`tuple[Scatter, TimeSeries, TimeSeries]`	Tuple containing three plot handles.

ConingRotation ¶

ConingRotation(
    amplitude: float | None = None,
    frequency: float | None = None,
)

Bases: Trajectory

Coning rotation.

Coning motion is rotation of a fixed magnitude about a rotation vector that oscillates out-of-phase around two axes. See Reference [33] for further details.

\[ \mathbf{\phi}_{coning}(t) = \begin{bmatrix} \theta \sin \omega t \\ \theta \cos \omega t \\ 0 \end{bmatrix} \]

The rotation vector inscribes a cone in the world frame. Coning motion results in an angular rate that has a constant component in the third axis. Naive integration schemes will rectify this fixed component into a linear drift. For this reason, it is a standard test case.

As a pure rotation, by default, gravity is set to zero.

Amplitude, \(\theta\), is the amplitude of the rotation in radians and is typically small resulting from vibration. Frequency, \(\omega\) should be the frequency in Hz, which is converted internally to angular frequency. If both or either are None, they are represented symbolically.

Parameters:

Name	Type	Description	Default
`amplitude`	`float \| None`	Amplitude of coning rotation.	`None`
`frequency`	`float \| None`	Frequency of coning rotation.	`None`

Methods:

Name	Description
`compose`	Compose two trajectories.
`rotation_vector`	Return the attitude as a rotation vector.
`rotation_vector_rate`	Return the attitude rate as a rotation vector.
`rotation_matrix`	Return the attitude as a rotation matrix.
`angular_rate`	Return the angular rate.
`rotation`	Return the attitude as a Rotation instance.
`extended_pose`	Return the attitude, position, and velocity.
`velocity`	Return the velocity.
`position`	Return the position.
`acceleration`	Return the acceleration.
`specific_force`	Return the specific force.
`gravity`	Return the gravity vector.
`plot_attitude`	Plot attitude variables.
`plot_translation`	Plot translation variables.
`plot`	Plots trajectory variables.

Attributes:

Name	Type	Description
`time`	`Symbol`

time `instance-attribute` ¶

time: Symbol = TIME_SYMBOL

compose ¶

compose(
    other: Self,
    gravity: Literal["self", "other"] | Matrix = "self",
) -> Trajectory

Compose two trajectories.

Composes the current (self) trajectory with the input (other) such that \(T_{composed} = T_{self} * T_{other}\). Gravity can be inherited from self (default), inherited from other (rotated by self), or can be a new input. Users must ensure that coordinate systems are respected when composing trajectories and that the same time symbol is used for both.

Parameters:

Name	Type	Description	Default
`other`	`Self`	Trajectory object to compose with.	required
`gravity`	`Literal['self', 'other'] \| Matrix`	Gravity source to use. Can be "self" (default), "other", or a SymPy Matrix.	`'self'`

Returns:

Type	Description
`Trajectory`	New Trajectory instance.

rotation_vector ¶

rotation_vector(
    time: ArrayLike | None = None,
) -> Matrix | NDArray

Return the attitude as a rotation vector.

Returns a symbolic representation if the time input is None (default) or a numeric array sampled at the input times if supplied.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike \| None`	Array of sampling times.	`None`

Returns:

Type	Description
`Matrix \| NDArray`	Rotation vector.

rotation_vector_rate ¶

rotation_vector_rate(
    time: ArrayLike | None = None,
) -> Matrix | NDArray

Return the attitude rate as a rotation vector.

Returns a symbolic representation if the time input is None (default) or a numeric array sampled at the input times if supplied.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike \| None`	Array of sampling times.	`None`

Returns:

Type	Description
`Matrix \| NDArray`	Rotation vector rate.

rotation_matrix ¶

rotation_matrix(
    time: ArrayLike | None = None,
) -> Matrix | NDArray

Return the attitude as a rotation matrix.

Returns a symbolic representation if the time input is None (default) or a numeric array sampled at the input times if supplied.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike \| None`	Array of sampling times.	`None`

Returns:

Type	Description
`Matrix \| NDArray`	Rotation matrix.

angular_rate ¶

angular_rate(
    time: ArrayLike | None = None,
) -> Matrix | NDArray

Return the angular rate.

Returns a symbolic representation if the time input is None (default) or a numeric array sampled at the input times if supplied.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike \| None`	Array of sampling times.	`None`

Returns:

Type	Description
`Matrix \| NDArray`	Angular rate.

rotation ¶

rotation(time: ArrayLike) -> Rotation

Return the attitude as a Rotation instance.

Returns a geometry.Rotation instance with data sampled at the input times.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike`	Array of sampling times.	required

Returns:

Type	Description
`Rotation`	Rotation instance.

extended_pose ¶

extended_pose(time: ArrayLike) -> ExtendedPose

Return the attitude, position, and velocity.

Returns a geometry.ExtendedPose instance with data sampled at the input times.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike`	Array of sampling times.	required

Returns:

Type	Description
`ExtendedPose`	ExtendedPose instance.

velocity ¶

velocity(time: ArrayLike | None = None) -> Matrix | NDArray

Return the velocity.

Returns a symbolic representation if the time input is None (default) or a numeric array sampled at the input times if supplied.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike \| None`	Array of sampling times.	`None`

Returns:

Type	Description
`Matrix \| NDArray`	Velocity vector.

position ¶

position(time: ArrayLike | None = None) -> Matrix | NDArray

Return the position.

Returns a symbolic representation if the time input is None (default) or a numeric array sampled at the input times if supplied.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike \| None`	Array of sampling times.	`None`

Returns:

Type	Description
`Matrix \| NDArray`	Position vector.

acceleration ¶

acceleration(
    time: ArrayLike | None = None,
) -> Matrix | NDArray

Return the acceleration.

Returns a symbolic representation if the time input is None (default) or a numeric array sampled at the input times if supplied.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike \| None`	Array of sampling times.	`None`

Returns:

Type	Description
`Matrix \| NDArray`	Acceleration vector in world/navigation coordinates.

specific_force ¶

specific_force(
    time: ArrayLike | None = None,
) -> Matrix | NDArray

Return the specific force.

Returns a symbolic representation if the time input is None (default) or a numeric array sampled at the input times if supplied.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike \| None`	Array of sampling times.	`None`

Returns:

Type	Description
`Matrix \| NDArray`	Specific force vector.

gravity ¶

gravity(time: ArrayLike | None = None) -> Matrix | NDArray

Return the gravity vector.

Returns a symbolic representation if the time input is None (default) or a numeric array sampled at the input times if supplied.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike \| None`	Array of sampling times.	`None`

Returns:

Type	Description
`Matrix \| NDArray`	Gravity vector.

plot_attitude ¶

plot_attitude(
    time: ArrayLike, title_prefix: str = ""
) -> tuple[TimeSeries, TimeSeries, TimeSeries]

Plot attitude variables.

Plots the rotation vector, rotation vector rate, and angular rate sampled at the input times.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike`	Sampling times.	required
`title_prefix`	`str`	Prefix for plot titles.	`''`

Returns:

Name	Type	Description
`plots`	`tuple[TimeSeries, TimeSeries, TimeSeries]`	Tuple containing three plot handles.

plot_translation ¶

plot_translation(
    time: ArrayLike, title_prefix: str = ""
) -> tuple[Scatter, TimeSeries, TimeSeries]

Plot translation variables.

Plots the position, velocity, and specific force sampled at the input times. Position is plotted as a plan-view (x-axis v y-axis). Velocity and specific force are plotted as time series graphs.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike`	Sampling times.	required
`title_prefix`	`str`	Prefix for plot titles.	`''`

Returns:

Name	Type	Description
`plots`	`tuple[Scatter, TimeSeries, TimeSeries]`	Tuple containing three plot handles.

plot ¶

plot(
    time: ArrayLike, title_prefix: str = ""
) -> tuple[
    tuple[TimeSeries, TimeSeries, TimeSeries],
    tuple[Scatter, TimeSeries, TimeSeries],
]

Plots trajectory variables.

Plots attitude and translation variables. See plot_attitude and plot_translation for details.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike`	Sampling times.	required
`title_prefix`	`str`	Prefix for plot titles.	`''`

Returns:

Name	Type	Description
`attitude_plots`	`tuple[TimeSeries, TimeSeries, TimeSeries]`	Tuple containing three plot handles.
`translation_plots`	`tuple[Scatter, TimeSeries, TimeSeries]`	Tuple containing three plot handles.

EarthRotation ¶

EarthRotation(angular_rate: ArrayLike | None = None)

Bases: Trajectory

Earth rotation.

Earth's rotation relative to inertial space. Earth rotates around a fixed axis at an approximately constant rate. See the geodesy.datums and geodesy.EarthRate modules for more details and to calculate standard rotation rates.

This trajectory defines a pure rotation with translation and gravity components set to zero. Since Earth is in gravitational freefall this is sufficient for the standard definition of an inertial frame in inertial navigation.

If the angular_rate input is not supplied it will be represented by the symbolic values \(\omega_{x}\), \(\omega_{y}\), and \(\omega_{z}\) in the world/navigation frame.

Parameters:

Name	Type	Description	Default
`angular_rate`	`ArrayLike \| None`	Angular rate of Earth (rad/s) relative to inertial space.	`None`

Methods:

Name	Description
`compose`	Compose two trajectories.
`rotation_vector`	Return the attitude as a rotation vector.
`rotation_vector_rate`	Return the attitude rate as a rotation vector.
`rotation_matrix`	Return the attitude as a rotation matrix.
`angular_rate`	Return the angular rate.
`rotation`	Return the attitude as a Rotation instance.
`extended_pose`	Return the attitude, position, and velocity.
`velocity`	Return the velocity.
`position`	Return the position.
`acceleration`	Return the acceleration.
`specific_force`	Return the specific force.
`gravity`	Return the gravity vector.
`plot_attitude`	Plot attitude variables.
`plot_translation`	Plot translation variables.
`plot`	Plots trajectory variables.

Attributes:

Name	Type	Description
`time`	`Symbol`

time `instance-attribute` ¶

time: Symbol = TIME_SYMBOL

compose ¶

compose(
    other: Self,
    gravity: Literal["self", "other"] | Matrix = "self",
) -> Trajectory

Compose two trajectories.

Composes the current (self) trajectory with the input (other) such that \(T_{composed} = T_{self} * T_{other}\). Gravity can be inherited from self (default), inherited from other (rotated by self), or can be a new input. Users must ensure that coordinate systems are respected when composing trajectories and that the same time symbol is used for both.

Parameters:

Name	Type	Description	Default
`other`	`Self`	Trajectory object to compose with.	required
`gravity`	`Literal['self', 'other'] \| Matrix`	Gravity source to use. Can be "self" (default), "other", or a SymPy Matrix.	`'self'`

Returns:

Type	Description
`Trajectory`	New Trajectory instance.

rotation_vector ¶

rotation_vector(
    time: ArrayLike | None = None,
) -> Matrix | NDArray

Return the attitude as a rotation vector.

Returns a symbolic representation if the time input is None (default) or a numeric array sampled at the input times if supplied.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike \| None`	Array of sampling times.	`None`

Returns:

Type	Description
`Matrix \| NDArray`	Rotation vector.

rotation_vector_rate ¶

rotation_vector_rate(
    time: ArrayLike | None = None,
) -> Matrix | NDArray

Return the attitude rate as a rotation vector.

Returns a symbolic representation if the time input is None (default) or a numeric array sampled at the input times if supplied.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike \| None`	Array of sampling times.	`None`

Returns:

Type	Description
`Matrix \| NDArray`	Rotation vector rate.

rotation_matrix ¶

rotation_matrix(
    time: ArrayLike | None = None,
) -> Matrix | NDArray

Return the attitude as a rotation matrix.

Returns a symbolic representation if the time input is None (default) or a numeric array sampled at the input times if supplied.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike \| None`	Array of sampling times.	`None`

Returns:

Type	Description
`Matrix \| NDArray`	Rotation matrix.

angular_rate ¶

angular_rate(
    time: ArrayLike | None = None,
) -> Matrix | NDArray

Return the angular rate.

Returns a symbolic representation if the time input is None (default) or a numeric array sampled at the input times if supplied.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike \| None`	Array of sampling times.	`None`

Returns:

Type	Description
`Matrix \| NDArray`	Angular rate.

rotation ¶

rotation(time: ArrayLike) -> Rotation

Return the attitude as a Rotation instance.

Returns a geometry.Rotation instance with data sampled at the input times.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike`	Array of sampling times.	required

Returns:

Type	Description
`Rotation`	Rotation instance.

extended_pose ¶

extended_pose(time: ArrayLike) -> ExtendedPose

Return the attitude, position, and velocity.

Returns a geometry.ExtendedPose instance with data sampled at the input times.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike`	Array of sampling times.	required

Returns:

Type	Description
`ExtendedPose`	ExtendedPose instance.

velocity ¶

velocity(time: ArrayLike | None = None) -> Matrix | NDArray

Return the velocity.

Returns a symbolic representation if the time input is None (default) or a numeric array sampled at the input times if supplied.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike \| None`	Array of sampling times.	`None`

Returns:

Type	Description
`Matrix \| NDArray`	Velocity vector.

position ¶

position(time: ArrayLike | None = None) -> Matrix | NDArray

Return the position.

Returns a symbolic representation if the time input is None (default) or a numeric array sampled at the input times if supplied.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike \| None`	Array of sampling times.	`None`

Returns:

Type	Description
`Matrix \| NDArray`	Position vector.

acceleration ¶

acceleration(
    time: ArrayLike | None = None,
) -> Matrix | NDArray

Return the acceleration.

Returns a symbolic representation if the time input is None (default) or a numeric array sampled at the input times if supplied.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike \| None`	Array of sampling times.	`None`

Returns:

Type	Description
`Matrix \| NDArray`	Acceleration vector in world/navigation coordinates.

specific_force ¶

specific_force(
    time: ArrayLike | None = None,
) -> Matrix | NDArray

Return the specific force.

Returns a symbolic representation if the time input is None (default) or a numeric array sampled at the input times if supplied.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike \| None`	Array of sampling times.	`None`

Returns:

Type	Description
`Matrix \| NDArray`	Specific force vector.

gravity ¶

gravity(time: ArrayLike | None = None) -> Matrix | NDArray

Return the gravity vector.

Returns a symbolic representation if the time input is None (default) or a numeric array sampled at the input times if supplied.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike \| None`	Array of sampling times.	`None`

Returns:

Type	Description
`Matrix \| NDArray`	Gravity vector.

plot_attitude ¶

plot_attitude(
    time: ArrayLike, title_prefix: str = ""
) -> tuple[TimeSeries, TimeSeries, TimeSeries]

Plot attitude variables.

Plots the rotation vector, rotation vector rate, and angular rate sampled at the input times.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike`	Sampling times.	required
`title_prefix`	`str`	Prefix for plot titles.	`''`

Returns:

Name	Type	Description
`plots`	`tuple[TimeSeries, TimeSeries, TimeSeries]`	Tuple containing three plot handles.

plot_translation ¶

plot_translation(
    time: ArrayLike, title_prefix: str = ""
) -> tuple[Scatter, TimeSeries, TimeSeries]

Plot translation variables.

Plots the position, velocity, and specific force sampled at the input times. Position is plotted as a plan-view (x-axis v y-axis). Velocity and specific force are plotted as time series graphs.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike`	Sampling times.	required
`title_prefix`	`str`	Prefix for plot titles.	`''`

Returns:

Name	Type	Description
`plots`	`tuple[Scatter, TimeSeries, TimeSeries]`	Tuple containing three plot handles.

plot ¶

plot(
    time: ArrayLike, title_prefix: str = ""
) -> tuple[
    tuple[TimeSeries, TimeSeries, TimeSeries],
    tuple[Scatter, TimeSeries, TimeSeries],
]

Plots trajectory variables.

Plots attitude and translation variables. See plot_attitude and plot_translation for details.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike`	Sampling times.	required
`title_prefix`	`str`	Prefix for plot titles.	`''`

Returns:

Name	Type	Description
`attitude_plots`	`tuple[TimeSeries, TimeSeries, TimeSeries]`	Tuple containing three plot handles.
`translation_plots`	`tuple[Scatter, TimeSeries, TimeSeries]`	Tuple containing three plot handles.

ZigZagTrajectory ¶

ZigZagTrajectory(
    attitude_amplitudes: ArrayLike | None = None,
    attitude_frequencies: ArrayLike | None = None,
    position_amplitudes: ArrayLike | None = None,
    position_frequencies: ArrayLike | None = None,
    gravity: float = standard_magnitude,
)

Bases: Trajectory

Zigzag trajectory.

The attitude is defined by independent oscillations in roll, pitch, and yaw. The position is defined by a Figure-8 pattern in the x-y plane and oscillations in the z-axis. This approximates the motion of a holonomic vehicle, like a drone, translating and rotation in all axes independently. The effect of gravity is modeled by a constant vertical value.

\[ \mathbf{\phi}_{zigzag}(t) = \begin{bmatrix} \theta_{x} \sin \omega_{x} t \\ \theta_{y} \cos \omega_{y} t \\ \theta_{z} \sin \omega_{z} t \\ \end{bmatrix} \]

\[ \mathbf{p}_{zigzag}(t) = \begin{bmatrix} A_{x} \sin \omega_{px} t \\ A_{y} \sin \omega_{py} t \cos \omega_{py} t \\ A_{z} \cos \omega_{pz} t \\ \end{bmatrix} \]

Attitude amplitudes must be an array of values for the x, y, z axes in radians. Position amplitudes must be an array of values for the x, y, z axes in meters. Frequencies must be an array of values in Hz, which are converted internally to angular frequencies. If any amplitude or frequency inputs are None, they are represented symbolically.

By default, standard gravity (9.80665 m/s/s) is applied in the vertical axis.

Example

The following example results in an attitude with large changes in yaw angle at low frequency (± 70 degrees once every 50 seconds), with smaller, higher frequency changes in roll and pitch. The position follows a 100m long Figure-8 path in x, y and oscillates ± 5 meters in z.

zigzag = ZigZagTrajectory(
    attitude_amplitudes=np.deg2rad([2.0, 3.0, 70.0]),
    attitude_frequencies=[0.25, 0.4, 0.02],
    position_amplitudes=[100, 75, 5],
    position_frequencies=[0.009, 0.011, 0.034],
)

Parameters:

Name	Type	Description	Default
`attitude_amplitudes`	`ArrayLike \| None`	Attitude (rotation) amplitudes.	`None`
`attitude_frequencies`	`ArrayLike \| None`	Attitude (rotation) frequencies.	`None`
`position_amplitudes`	`ArrayLike \| None`	Position amplitudes.	`None`
`position_frequencies`	`ArrayLike \| None`	Position frequencies.	`None`
`gravity`	`float`	Magnitude of gravity.	`standard_magnitude`

Methods:

Name	Description
`compose`	Compose two trajectories.
`rotation_vector`	Return the attitude as a rotation vector.
`rotation_vector_rate`	Return the attitude rate as a rotation vector.
`rotation_matrix`	Return the attitude as a rotation matrix.
`angular_rate`	Return the angular rate.
`rotation`	Return the attitude as a Rotation instance.
`extended_pose`	Return the attitude, position, and velocity.
`velocity`	Return the velocity.
`position`	Return the position.
`acceleration`	Return the acceleration.
`specific_force`	Return the specific force.
`gravity`	Return the gravity vector.
`plot_attitude`	Plot attitude variables.
`plot_translation`	Plot translation variables.
`plot`	Plots trajectory variables.

Attributes:

Name	Type	Description
`time`	`Symbol`

time `instance-attribute` ¶

time: Symbol = TIME_SYMBOL

compose ¶

compose(
    other: Self,
    gravity: Literal["self", "other"] | Matrix = "self",
) -> Trajectory

Compose two trajectories.

Composes the current (self) trajectory with the input (other) such that \(T_{composed} = T_{self} * T_{other}\). Gravity can be inherited from self (default), inherited from other (rotated by self), or can be a new input. Users must ensure that coordinate systems are respected when composing trajectories and that the same time symbol is used for both.

Parameters:

Name	Type	Description	Default
`other`	`Self`	Trajectory object to compose with.	required
`gravity`	`Literal['self', 'other'] \| Matrix`	Gravity source to use. Can be "self" (default), "other", or a SymPy Matrix.	`'self'`

Returns:

Type	Description
`Trajectory`	New Trajectory instance.

rotation_vector ¶

rotation_vector(
    time: ArrayLike | None = None,
) -> Matrix | NDArray

Return the attitude as a rotation vector.

Returns a symbolic representation if the time input is None (default) or a numeric array sampled at the input times if supplied.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike \| None`	Array of sampling times.	`None`

Returns:

Type	Description
`Matrix \| NDArray`	Rotation vector.

rotation_vector_rate ¶

rotation_vector_rate(
    time: ArrayLike | None = None,
) -> Matrix | NDArray

Return the attitude rate as a rotation vector.

Returns a symbolic representation if the time input is None (default) or a numeric array sampled at the input times if supplied.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike \| None`	Array of sampling times.	`None`

Returns:

Type	Description
`Matrix \| NDArray`	Rotation vector rate.

rotation_matrix ¶

rotation_matrix(
    time: ArrayLike | None = None,
) -> Matrix | NDArray

Return the attitude as a rotation matrix.

Returns a symbolic representation if the time input is None (default) or a numeric array sampled at the input times if supplied.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike \| None`	Array of sampling times.	`None`

Returns:

Type	Description
`Matrix \| NDArray`	Rotation matrix.

angular_rate ¶

angular_rate(
    time: ArrayLike | None = None,
) -> Matrix | NDArray

Return the angular rate.

Returns a symbolic representation if the time input is None (default) or a numeric array sampled at the input times if supplied.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike \| None`	Array of sampling times.	`None`

Returns:

Type	Description
`Matrix \| NDArray`	Angular rate.

rotation ¶

rotation(time: ArrayLike) -> Rotation

Return the attitude as a Rotation instance.

Returns a geometry.Rotation instance with data sampled at the input times.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike`	Array of sampling times.	required

Returns:

Type	Description
`Rotation`	Rotation instance.

extended_pose ¶

extended_pose(time: ArrayLike) -> ExtendedPose

Return the attitude, position, and velocity.

Returns a geometry.ExtendedPose instance with data sampled at the input times.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike`	Array of sampling times.	required

Returns:

Type	Description
`ExtendedPose`	ExtendedPose instance.

velocity ¶

velocity(time: ArrayLike | None = None) -> Matrix | NDArray

Return the velocity.

Returns a symbolic representation if the time input is None (default) or a numeric array sampled at the input times if supplied.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike \| None`	Array of sampling times.	`None`

Returns:

Type	Description
`Matrix \| NDArray`	Velocity vector.

position ¶

position(time: ArrayLike | None = None) -> Matrix | NDArray

Return the position.

Returns a symbolic representation if the time input is None (default) or a numeric array sampled at the input times if supplied.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike \| None`	Array of sampling times.	`None`

Returns:

Type	Description
`Matrix \| NDArray`	Position vector.

acceleration ¶

acceleration(
    time: ArrayLike | None = None,
) -> Matrix | NDArray

Return the acceleration.

Returns a symbolic representation if the time input is None (default) or a numeric array sampled at the input times if supplied.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike \| None`	Array of sampling times.	`None`

Returns:

Type	Description
`Matrix \| NDArray`	Acceleration vector in world/navigation coordinates.

specific_force ¶

specific_force(
    time: ArrayLike | None = None,
) -> Matrix | NDArray

Return the specific force.

Returns a symbolic representation if the time input is None (default) or a numeric array sampled at the input times if supplied.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike \| None`	Array of sampling times.	`None`

Returns:

Type	Description
`Matrix \| NDArray`	Specific force vector.

gravity ¶

gravity(time: ArrayLike | None = None) -> Matrix | NDArray

Return the gravity vector.

Returns a symbolic representation if the time input is None (default) or a numeric array sampled at the input times if supplied.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike \| None`	Array of sampling times.	`None`

Returns:

Type	Description
`Matrix \| NDArray`	Gravity vector.

plot_attitude ¶

plot_attitude(
    time: ArrayLike, title_prefix: str = ""
) -> tuple[TimeSeries, TimeSeries, TimeSeries]

Plot attitude variables.

Plots the rotation vector, rotation vector rate, and angular rate sampled at the input times.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike`	Sampling times.	required
`title_prefix`	`str`	Prefix for plot titles.	`''`

Returns:

Name	Type	Description
`plots`	`tuple[TimeSeries, TimeSeries, TimeSeries]`	Tuple containing three plot handles.

plot_translation ¶

plot_translation(
    time: ArrayLike, title_prefix: str = ""
) -> tuple[Scatter, TimeSeries, TimeSeries]

Plot translation variables.

Plots the position, velocity, and specific force sampled at the input times. Position is plotted as a plan-view (x-axis v y-axis). Velocity and specific force are plotted as time series graphs.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike`	Sampling times.	required
`title_prefix`	`str`	Prefix for plot titles.	`''`

Returns:

Name	Type	Description
`plots`	`tuple[Scatter, TimeSeries, TimeSeries]`	Tuple containing three plot handles.

plot ¶

plot(
    time: ArrayLike, title_prefix: str = ""
) -> tuple[
    tuple[TimeSeries, TimeSeries, TimeSeries],
    tuple[Scatter, TimeSeries, TimeSeries],
]

Plots trajectory variables.

Plots attitude and translation variables. See plot_attitude and plot_translation for details.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike`	Sampling times.	required
`title_prefix`	`str`	Prefix for plot titles.	`''`

Returns:

Name	Type	Description
`attitude_plots`	`tuple[TimeSeries, TimeSeries, TimeSeries]`	Tuple containing three plot handles.
`translation_plots`	`tuple[Scatter, TimeSeries, TimeSeries]`	Tuple containing three plot handles.

GroundVehiclePath ¶

GroundVehiclePath(
    *,
    north: float | None = None,
    east: float | None = None,
    speed: float | None = None,
    altitude_amplitude: float | None = None,
    altitude_frequency: float | None = None,
    roll_amplitude: float | None = None,
    roll_frequency: float | None = None,
    pitch_amplitude: float | None = None,
    pitch_frequency: float | None = None,
    gravity: ArrayLike = _DEFAULT_GRAVITY,
)

Bases: Trajectory

Simulated ground vehicle path.

The position data traces a Lissajous curve in the horizontal plane and is defined by North & East magnitudes (meters) and average speed (meters/second). An independent oscillation can be added to the z-axis.

The North-relative heading is calculated as the tangent of the position curve. Independent oscillations can be added in roll & pitch.

Overall, the motion approximates a non-holonomic vehicle like a car driving smoothly in x, y, and heading while experiencing disturbances in roll, pitch, and vertical position.

Gravity is a constant vector in NED coordinates and defaults to 9.80665m/s/s in the down direction. Earth rotation should be applied by composing this trajectory with the EarthRotation class from inertialsim.symbolic.

\[ \mathbf{p}_(t) = \begin{bmatrix} N * \sin (\omega_{N} t + \delta) \\ E * \sin (\omega_{E} t) \\ D \cos (\omega_{D} t) \\ \end{bmatrix} \]

\(\omega_{N}\) is 3, \(\omega_{E}\) is 2, and \(\delta\) is \(\pi/2\). Attitude rotates NED basis vectors into IMU basis vectors.

\[ R(t) = R_z(\psi(t)) R_y(\theta(t)) R_x(\phi(t)). \]

Attitude amplitudes must be an array of values for the x, y, z axes in radians. Position amplitudes must be an array of values for the x, y, z axes in meters. Frequencies must be an array of values in Hz, which are converted internally to angular frequencies. If any amplitude or frequency inputs are None, they are represented symbolically.

By default, standard gravity (9.80665 m/s/s) is applied in the vertical axis.

Example

The following example results in a trajectory 50m x 100m at an average speed of 8.0 m/s. Vertical position, roll, and pitch vibrations are simulated at small amplitudes (0.3m, 3.0°, 12.0° respectively) and moderate, unrelated frequencies.

path = GroundVehiclePath(
    north=50.0, east=100.0, speed=8.0,
    altitude_amplitude=0.3, altitude_frequency=0.17,
    roll_amplitude=3.0, roll_frequency=0.26,
    pitch_amplitude=12.0, pitch_frequency=0.08
)

Parameters:

Name	Type	Description	Default
`north`	`float \| None`	North displacement in meters.	`None`
`east`	`float \| None`	East displacement in meters.	`None`
`speed`	`float \| None`	Average speed in m/s.	`None`
`altitude_amplitude`	`float \| None`	Amplitude of vertical position oscillation in meters.	`None`
`altitude_frequency`	`float \| None`	Frequency of vertical position oscillation in Hz.	`None`
`roll_amplitude`	`float \| None`	Amplitude of roll oscillation in degrees.	`None`
`roll_frequency`	`float \| None`	Frequency of roll oscillation in Hz.	`None`
`pitch_amplitude`	`float \| None`	Amplitude of pitch oscillation in degrees.	`None`
`pitch_frequency`	`float \| None`	Frequency of pitch oscillation in Hz.	`None`
`gravity`	`ArrayLike`	Gravity vector.	`_DEFAULT_GRAVITY`

Methods:

Name	Description
`scale`	Scale path to average speed.
`compose`	Compose two trajectories.
`rotation_vector`	Return the attitude as a rotation vector.
`rotation_vector_rate`	Return the attitude rate as a rotation vector.
`rotation_matrix`	Return the attitude as a rotation matrix.
`angular_rate`	Return the angular rate.
`rotation`	Return the attitude as a Rotation instance.
`extended_pose`	Return the attitude, position, and velocity.
`velocity`	Return the velocity.
`position`	Return the position.
`acceleration`	Return the acceleration.
`specific_force`	Return the specific force.
`gravity`	Return the gravity vector.
`plot_attitude`	Plot attitude variables.
`plot_translation`	Plot translation variables.
`plot`	Plots trajectory variables.

Attributes:

Name	Type	Description
`a`
`b`
`δ`	`Pi`
`time`	`Symbol`

a `instance-attribute` ¶

a = 3

b `instance-attribute` ¶

b = 2

δ `instance-attribute` ¶

δ: Pi = sympy.pi / 2

time `instance-attribute` ¶

time: Symbol = TIME_SYMBOL

scale ¶

scale(north: float, east: float, speed: float) -> float

Scale path to average speed.

Calculate a scale factor for the North and East position variables that will make the average speed equal to the input.

Parameters:

Name	Type	Description	Default
`north`	`float`	Amplitude of the northward motion.	required
`east`	`float`	Amplitude of the eastward motion.	required
`speed`	`float`	Desired average speed.	required

Returns:

Name	Type	Description
`float`	`float`	Scale factor to apply to the North and East position variables.

compose ¶

compose(
    other: Self,
    gravity: Literal["self", "other"] | Matrix = "self",
) -> Trajectory

Compose two trajectories.

Composes the current (self) trajectory with the input (other) such that \(T_{composed} = T_{self} * T_{other}\). Gravity can be inherited from self (default), inherited from other (rotated by self), or can be a new input. Users must ensure that coordinate systems are respected when composing trajectories and that the same time symbol is used for both.

Parameters:

Name	Type	Description	Default
`other`	`Self`	Trajectory object to compose with.	required
`gravity`	`Literal['self', 'other'] \| Matrix`	Gravity source to use. Can be "self" (default), "other", or a SymPy Matrix.	`'self'`

Returns:

Type	Description
`Trajectory`	New Trajectory instance.

rotation_vector ¶

rotation_vector(
    time: ArrayLike | None = None,
) -> Matrix | NDArray

Return the attitude as a rotation vector.

Returns a symbolic representation if the time input is None (default) or a numeric array sampled at the input times if supplied.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike \| None`	Array of sampling times.	`None`

Returns:

Type	Description
`Matrix \| NDArray`	Rotation vector.

rotation_vector_rate ¶

rotation_vector_rate(
    time: ArrayLike | None = None,
) -> Matrix | NDArray

Return the attitude rate as a rotation vector.

Returns a symbolic representation if the time input is None (default) or a numeric array sampled at the input times if supplied.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike \| None`	Array of sampling times.	`None`

Returns:

Type	Description
`Matrix \| NDArray`	Rotation vector rate.

rotation_matrix ¶

rotation_matrix(
    time: ArrayLike | None = None,
) -> Matrix | NDArray

Return the attitude as a rotation matrix.

Returns a symbolic representation if the time input is None (default) or a numeric array sampled at the input times if supplied.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike \| None`	Array of sampling times.	`None`

Returns:

Type	Description
`Matrix \| NDArray`	Rotation matrix.

angular_rate ¶

angular_rate(
    time: ArrayLike | None = None,
) -> Matrix | NDArray

Return the angular rate.

Returns a symbolic representation if the time input is None (default) or a numeric array sampled at the input times if supplied.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike \| None`	Array of sampling times.	`None`

Returns:

Type	Description
`Matrix \| NDArray`	Angular rate.

rotation ¶

rotation(time: ArrayLike) -> Rotation

Return the attitude as a Rotation instance.

Returns a geometry.Rotation instance with data sampled at the input times.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike`	Array of sampling times.	required

Returns:

Type	Description
`Rotation`	Rotation instance.

extended_pose ¶

extended_pose(time: ArrayLike) -> ExtendedPose

Return the attitude, position, and velocity.

Returns a geometry.ExtendedPose instance with data sampled at the input times.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike`	Array of sampling times.	required

Returns:

Type	Description
`ExtendedPose`	ExtendedPose instance.

velocity ¶

velocity(time: ArrayLike | None = None) -> Matrix | NDArray

Return the velocity.

Returns a symbolic representation if the time input is None (default) or a numeric array sampled at the input times if supplied.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike \| None`	Array of sampling times.	`None`

Returns:

Type	Description
`Matrix \| NDArray`	Velocity vector.

position ¶

position(time: ArrayLike | None = None) -> Matrix | NDArray

Return the position.

Returns a symbolic representation if the time input is None (default) or a numeric array sampled at the input times if supplied.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike \| None`	Array of sampling times.	`None`

Returns:

Type	Description
`Matrix \| NDArray`	Position vector.

acceleration ¶

acceleration(
    time: ArrayLike | None = None,
) -> Matrix | NDArray

Return the acceleration.

Returns a symbolic representation if the time input is None (default) or a numeric array sampled at the input times if supplied.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike \| None`	Array of sampling times.	`None`

Returns:

Type	Description
`Matrix \| NDArray`	Acceleration vector in world/navigation coordinates.

specific_force ¶

specific_force(
    time: ArrayLike | None = None,
) -> Matrix | NDArray

Return the specific force.

Returns a symbolic representation if the time input is None (default) or a numeric array sampled at the input times if supplied.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike \| None`	Array of sampling times.	`None`

Returns:

Type	Description
`Matrix \| NDArray`	Specific force vector.

gravity ¶

gravity(time: ArrayLike | None = None) -> Matrix | NDArray

Return the gravity vector.

Returns a symbolic representation if the time input is None (default) or a numeric array sampled at the input times if supplied.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike \| None`	Array of sampling times.	`None`

Returns:

Type	Description
`Matrix \| NDArray`	Gravity vector.

plot_attitude ¶

plot_attitude(
    time: ArrayLike, title_prefix: str = ""
) -> tuple[TimeSeries, TimeSeries, TimeSeries]

Plot attitude variables.

Plots the rotation vector, rotation vector rate, and angular rate sampled at the input times.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike`	Sampling times.	required
`title_prefix`	`str`	Prefix for plot titles.	`''`

Returns:

Name	Type	Description
`plots`	`tuple[TimeSeries, TimeSeries, TimeSeries]`	Tuple containing three plot handles.

plot_translation ¶

plot_translation(
    time: ArrayLike, title_prefix: str = ""
) -> tuple[Scatter, TimeSeries, TimeSeries]

Plot translation variables.

Plots the position, velocity, and specific force sampled at the input times. Position is plotted as a plan-view (x-axis v y-axis). Velocity and specific force are plotted as time series graphs.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike`	Sampling times.	required
`title_prefix`	`str`	Prefix for plot titles.	`''`

Returns:

Name	Type	Description
`plots`	`tuple[Scatter, TimeSeries, TimeSeries]`	Tuple containing three plot handles.

plot ¶

plot(
    time: ArrayLike, title_prefix: str = ""
) -> tuple[
    tuple[TimeSeries, TimeSeries, TimeSeries],
    tuple[Scatter, TimeSeries, TimeSeries],
]

Plots trajectory variables.

Plots attitude and translation variables. See plot_attitude and plot_translation for details.

Parameters:

Name	Type	Description	Default
`time`	`ArrayLike`	Sampling times.	required
`title_prefix`	`str`	Prefix for plot titles.	`''`

Returns:

Name	Type	Description
`attitude_plots`	`tuple[TimeSeries, TimeSeries, TimeSeries]`	Tuple containing three plot handles.
`translation_plots`	`tuple[Scatter, TimeSeries, TimeSeries]`	Tuple containing three plot handles.

lambdify ¶

lambdify(fn: Expr, time: Symbol) -> Callable

Create a lambda function from a symbolic expression.

Parameters:

Name	Type	Description	Default
`fn`	`Expr`	Symbolic expression to be converted into a lambda function.	required
`time`	`Symbol`	Symbolic variable representing time.	required

Returns:

Type	Description
`Callable`	A callable that evaluates the symbolic expression at input time values.

integral ¶

integral(matrix: Matrix, time: Symbol) -> Matrix

Integrate each matrix component from zero to the time symbol.

Parameters:

Name	Type	Description	Default
`matrix`	`Matrix`	Matrix whose components are integrated.	required
`time`	`Symbol`	Symbol used as the integration variable and upper bound.	required

Returns:

Type	Description
`Matrix`	Matrix containing the integrated components.

double_integral ¶

double_integral(matrix: Matrix, time: Symbol) -> Matrix

Double integrate each matrix component from zero to the time symbol.

Parameters:

Name	Type	Description	Default
`matrix`	`Matrix`	Matrix whose components are double integrated.	required
`time`	`Symbol`	Symbol used as the integration variable and upper bound.	required

Returns:

Type	Description
`Matrix`	Matrix containing the double integrated components.

power_series ¶

power_series(
    φ: Matrix,
) -> tuple[
    Integer | Expr,
    Rational | Expr,
    Rational | Expr,
    Rational | Expr,
    Rational | Expr,
    Rational | Expr,
]

Compute the power series coefficients for a 3D rotation vector.

This function computes the coefficients of the power series expansion used in the SO(3) and SE(3) Lie groups.

Returns:

Type	Description
`Integer \| Expr`	A tuple containing six coefficients (f1, f2, f3, f4, f5, fn) used in the
`Rational \| Expr`	power series expansion. The types are sympy.Int (f1) or sympy.Rational
`Rational \| Expr`	(f2, ...) for exact values when the rotation vector is zero, and
`Rational \| Expr`	sympy.Expr otherwise.

inertialsim.symbolic ¶

TIME_SYMBOL module-attribute ¶

SO3 ¶

Rx staticmethod ¶

Ry staticmethod ¶

Rz staticmethod ¶

exp staticmethod ¶

log staticmethod ¶

jacobian staticmethod ¶

SE3 ¶

exp staticmethod ¶

log staticmethod ¶

jacobian classmethod ¶

Trajectory ¶

time instance-attribute ¶

compose ¶

rotation_vector ¶

rotation_vector_rate ¶

rotation_matrix ¶

angular_rate ¶

rotation ¶

extended_pose ¶

velocity ¶

position ¶

acceleration ¶

specific_force ¶

gravity ¶

plot_attitude ¶

plot_translation ¶

plot ¶

ConingRotation ¶

time instance-attribute ¶

compose ¶

rotation_vector ¶

rotation_vector_rate ¶

rotation_matrix ¶

angular_rate ¶

rotation ¶

extended_pose ¶

velocity ¶

position ¶

acceleration ¶

specific_force ¶

gravity ¶

plot_attitude ¶

plot_translation ¶

plot ¶

EarthRotation ¶

time instance-attribute ¶

compose ¶

rotation_vector ¶

rotation_vector_rate ¶

rotation_matrix ¶

angular_rate ¶

rotation ¶

extended_pose ¶

velocity ¶

position ¶

acceleration ¶

specific_force ¶

gravity ¶

plot_attitude ¶

plot_translation ¶

plot ¶

ZigZagTrajectory ¶

time instance-attribute ¶

compose ¶

rotation_vector ¶

rotation_vector_rate ¶

rotation_matrix ¶

angular_rate ¶

rotation ¶

extended_pose ¶

velocity ¶

position ¶

acceleration ¶

specific_force ¶

gravity ¶

plot_attitude ¶

plot_translation ¶

TIME_SYMBOL `module-attribute` ¶

Rx `staticmethod` ¶

Ry `staticmethod` ¶

Rz `staticmethod` ¶

exp `staticmethod` ¶

log `staticmethod` ¶

jacobian `staticmethod` ¶

exp `staticmethod` ¶

log `staticmethod` ¶

jacobian `classmethod` ¶

time `instance-attribute` ¶

time `instance-attribute` ¶

time `instance-attribute` ¶

time `instance-attribute` ¶

a `instance-attribute` ¶

b `instance-attribute` ¶

δ `instance-attribute` ¶

time `instance-attribute` ¶