from __future__ import annotations
import dataclasses
from typing import Tuple, cast
import numpy as np
import numpy.typing as npt
from typing_extensions import override
from . import _base, hints
from ._so2 import SO2
from .utils import broadcast_leading_axes, get_epsilon
[docs]
@dataclasses.dataclass(frozen=True)
class SE2(
_base.SEBase[SO2],
matrix_dim=3,
parameters_dim=4,
tangent_dim=3,
space_dim=2,
):
"""Special Euclidean group for proper rigid transforms in 2D. Broadcasting
rules are the same as for numpy.
Ported to numpy from `jaxlie.SE2`.
Internal parameterization is `(cos, sin, x, y)`. Tangent parameterization is `(vx,
vy, omega)`.
"""
# SE2-specific.
unit_complex_xy: npt.NDArray[np.floating]
"""Internal parameters. `(cos, sin, x, y)`. Shape should be `(*, 4)`."""
@override
def __repr__(self) -> str:
unit_complex = np.round(self.unit_complex_xy[..., :2], 5)
xy = np.round(self.unit_complex_xy[..., 2:], 5)
return f"{self.__class__.__name__}(unit_complex={unit_complex}, xy={xy})"
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@staticmethod
def from_xy_theta(x: hints.Scalar, y: hints.Scalar, theta: hints.Scalar) -> SE2:
"""Construct a transformation from standard 2D pose parameters.
This is not the same as integrating over a length-3 twist.
"""
cos = np.cos(theta)
sin = np.sin(theta)
return SE2(unit_complex_xy=np.stack([cos, sin, x, y], axis=-1))
# SE-specific.
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@classmethod
@override
def from_rotation_and_translation(
cls,
rotation: SO2,
translation: npt.NDArray[np.floating],
) -> SE2:
assert translation.shape[-1:] == (2,)
rotation, translation = broadcast_leading_axes((rotation, translation))
return SE2(
unit_complex_xy=np.concatenate(
[rotation.unit_complex, translation], axis=-1
)
)
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@override
def rotation(self) -> SO2:
return SO2(unit_complex=self.unit_complex_xy[..., :2])
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@override
def translation(self) -> npt.NDArray[np.floating]:
return self.unit_complex_xy[..., 2:]
# Factory.
[docs]
@classmethod
@override
def identity(
cls, batch_axes: Tuple[int, ...] = (), dtype: npt.DTypeLike = np.float64
) -> SE2:
return SE2(
unit_complex_xy=np.broadcast_to(
np.array([1.0, 0.0, 0.0, 0.0], dtype=dtype), (*batch_axes, 4)
)
)
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@classmethod
@override
def from_matrix(cls, matrix: npt.NDArray[np.floating]) -> SE2:
assert matrix.shape[-2:] == (3, 3) or matrix.shape[-2:] == (2, 3)
# Currently assumes bottom row is [0, 0, 1].
return SE2.from_rotation_and_translation(
rotation=SO2.from_matrix(matrix[..., :2, :2]),
translation=matrix[..., :2, 2],
)
# Accessors.
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@override
def parameters(self) -> npt.NDArray[np.floating]:
return self.unit_complex_xy
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@override
def as_matrix(self) -> npt.NDArray[np.floating]:
cos, sin, x, y = np.moveaxis(self.unit_complex_xy, -1, 0)
out = np.stack(
[
cos,
-sin,
x,
sin,
cos,
y,
np.zeros_like(x),
np.zeros_like(x),
np.ones_like(x),
],
axis=-1,
).reshape((*self.get_batch_axes(), 3, 3))
return out
# Operations.
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@classmethod
@override
def exp(cls, tangent: npt.NDArray[np.floating]) -> SE2:
# Reference:
# > https://github.com/strasdat/Sophus/blob/a0fe89a323e20c42d3cecb590937eb7a06b8343a/sophus/se2.hpp#L558
# Also see:
# > http://ethaneade.com/lie.pdf
assert tangent.shape[-1:] == (3,)
theta = tangent[..., 2]
use_taylor = np.abs(theta) < get_epsilon(tangent.dtype)
# Shim to avoid NaNs in np.where branches, which cause failures for
# reverse-mode AD in JAX. This isn't needed for vanilla numpy.
safe_theta = cast(
np.ndarray,
np.where(
use_taylor,
np.ones_like(theta), # Any non-zero value should do here.
theta,
),
)
theta_sq = theta**2
sin_over_theta = np.where(
use_taylor,
1.0 - theta_sq / 6.0,
np.sin(safe_theta) / safe_theta,
)
one_minus_cos_over_theta = np.where(
use_taylor,
0.5 * theta - theta * theta_sq / 24.0,
(1.0 - np.cos(safe_theta)) / safe_theta,
)
V = np.stack(
[
sin_over_theta,
-one_minus_cos_over_theta,
one_minus_cos_over_theta,
sin_over_theta,
],
axis=-1,
).reshape((*tangent.shape[:-1], 2, 2))
return SE2.from_rotation_and_translation(
rotation=SO2.from_radians(theta),
translation=np.einsum("...ij,...j->...i", V, tangent[..., :2]).astype(
tangent.dtype
),
)
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@override
def log(self) -> npt.NDArray[np.floating]:
# Reference:
# > https://github.com/strasdat/Sophus/blob/a0fe89a323e20c42d3cecb590937eb7a06b8343a/sophus/se2.hpp#L160
# Also see:
# > http://ethaneade.com/lie.pdf
theta = self.rotation().log()[..., 0]
cos = np.cos(theta)
cos_minus_one = cos - 1.0
half_theta = theta / 2.0
use_taylor = np.abs(cos_minus_one) < get_epsilon(theta.dtype)
# Shim to avoid NaNs in np.where branches, which cause failures for
# reverse-mode AD in JAX. This isn't needed for vanilla numpy.
safe_cos_minus_one = np.where(
use_taylor,
np.ones_like(cos_minus_one), # Any non-zero value should do here.
cos_minus_one,
)
half_theta_over_tan_half_theta = np.where(
use_taylor,
# Taylor approximation.
1.0 - theta**2 / 12.0,
# Default.
-(half_theta * np.sin(theta)) / safe_cos_minus_one,
)
V_inv = np.stack(
[
half_theta_over_tan_half_theta,
half_theta,
-half_theta,
half_theta_over_tan_half_theta,
],
axis=-1,
).reshape((*theta.shape, 2, 2))
tangent = np.concatenate(
[
np.einsum("...ij,...j->...i", V_inv, self.translation()),
theta[..., None],
],
axis=-1,
)
return tangent.astype(self.unit_complex_xy.dtype)
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@override
def adjoint(self: SE2) -> npt.NDArray[np.floating]:
cos, sin, x, y = np.moveaxis(self.unit_complex_xy, -1, 0)
return np.stack(
[
cos,
-sin,
y,
sin,
cos,
-x,
np.zeros_like(x),
np.zeros_like(x),
np.ones_like(x),
],
axis=-1,
).reshape((*self.get_batch_axes(), 3, 3))