Lie¶

MuJoCo does not currently offer a native Lie group interface for rigid body transforms, though it does have a collection of functions for manipulating quaternions and rotation matrices. The goal of this library is to provide this unified interface. Whenever possible, the underlying lie operation leverages the corresponding MuJoCo function. For example, from_matrix() uses mujoco.mju_mat2Quat under the hood.

This library is heavily ported from jaxlie, swapping out JAX for Numpy and adding a few additional features.

MatrixLieGroup¶

class mink.lie.base.MatrixLieGroup¶

Interface definition for matrix Lie groups.

matrix_dim¶

Dimension of square matrix output.

Type:: int

parameters_dim¶

Dimension of underlying parameters.

Type:: int

tangent_dim¶

Dimension of tangent space.

Type:: int

space_dim¶

Dimension of coordinates that can be transformed.

Type:: int

abstract classmethod identity() → Self¶

Returns identity element.

Return type:: Self

abstract classmethod from_matrix(matrix: ndarray) → Self¶

Get group member from matrix representation.

Parameters:: matrix (ndarray)
Return type:: Self

abstract classmethod sample_uniform() → Self¶

Draw a uniform sample from the group.

Return type:: Self

abstract as_matrix() → ndarray¶

Get transformation as a matrix.

Return type:: ndarray

abstract parameters() → ndarray¶

Get underlying representation.

Return type:: ndarray

abstract apply(target: ndarray) → ndarray¶

Applies group action to a point.

Parameters:: target (ndarray)
Return type:: ndarray

abstract multiply(other: Self) → Self¶

Composes this transformation with another.

Parameters:: other (Self)
Return type:: Self

abstract classmethod exp(tangent: ndarray) → Self¶

Computes expm(wedge(tangent)).

Parameters:: tangent (ndarray)
Return type:: Self

abstract log() → ndarray¶

Computes vee(logm(transformation matrix)).

Return type:: ndarray

abstract adjoint() → ndarray¶

Computes the adjoint.

Return type:: ndarray

abstract inverse() → Self¶

Computes the inverse of the transform.

Return type:: Self

abstract normalize() → Self¶

Normalize/projects values and returns.

Return type:: Self

interpolate(other: Self, alpha: float = 0.5) → Self¶

Interpolate between two matrix Lie groups.

Parameters:

other (Self) – The other Lie group, which serves as the end point of interpolation.
alpha (float) – The fraction of interpolation between [self, other]. This must be within [0.0, 1.0]. 0.0 = self, 1.0 = other.

Returns:

The interpolated matrix Lie group.

Return type:

Self

plus(other: ndarray) → Self¶

Alias for rplus.

Parameters:: other (ndarray)
Return type:: Self

minus(other: Self) → ndarray¶

Alias for rminus.

Parameters:: other (Self)
Return type:: ndarray

SO3¶

class mink.lie.so3.SO3¶

Bases: MatrixLieGroup

Special orthogonal group for 3D rotations.

Internal parameterization is (qw, qx, qy, qz). Tangent parameterization is (omega_x, omega_y, omega_z).

parameters() → ndarray¶

Get underlying representation.

Return type:: ndarray

classmethod from_matrix(matrix: ndarray) → SO3¶

Get group member from matrix representation.

Parameters:: matrix (ndarray)
Return type:: SO3

classmethod identity() → SO3¶

Returns identity element.

Return type:: SO3

classmethod sample_uniform() → SO3¶

Draw a uniform sample from the group.

Return type:: SO3

as_matrix() → ndarray¶

Get transformation as a matrix.

Return type:: ndarray

inverse() → SO3¶

Computes the inverse of the transform.

Return type:: SO3

normalize() → SO3¶

Normalize/projects values and returns.

Return type:: SO3

apply(target: ndarray) → ndarray¶

Applies group action to a point.

Parameters:: target (ndarray)
Return type:: ndarray

multiply(other: SO3) → SO3¶

Composes this transformation with another.

Parameters:: other (SO3)
Return type:: SO3

classmethod exp(tangent: ndarray) → SO3¶

Computes expm(wedge(tangent)).

Parameters:: tangent (ndarray)
Return type:: SO3

log() → ndarray¶

Computes vee(logm(transformation matrix)).

Return type:: ndarray

adjoint() → ndarray¶

Computes the adjoint.

Return type:: ndarray

clamp(roll_radians: tuple[float, float] = (-inf, inf), pitch_radians: tuple[float, float] = (-inf, inf), yaw_radians: tuple[float, float] = (-inf, inf)) → SO3¶

Clamp a SO3 within RPY limits.

Parameters:

roll_radians (tuple[float, float]) – The (lower, upper) limits for the roll.
pitch_radians (tuple[float, float]) – The (lower, upper) limits for the pitch.
yaw_radians (tuple[float, float]) – The (lower, upper) limits for the yaw.

Returns:

A SO3 within the RPY limits.

Return type:

SO3

SE3¶

class mink.lie.se3.SE3¶

Bases: MatrixLieGroup

Special Euclidean group for proper rigid transforms in 3D.

Internal parameterization is (qw, qx, qy, qz, x, y, z). Tangent parameterization is (vx, vy, vz, omega_x, omega_y, omega_z).

parameters() → ndarray¶

Get underlying representation.

Return type:: ndarray

classmethod identity() → SE3¶

Returns identity element.

Return type:: SE3

classmethod from_matrix(matrix: ndarray) → SE3¶

Get group member from matrix representation.

Parameters:: matrix (ndarray)
Return type:: SE3

classmethod sample_uniform() → SE3¶

Draw a uniform sample from the group.

Return type:: SE3

as_matrix() → ndarray¶

Get transformation as a matrix.

Return type:: ndarray

classmethod exp(tangent: ndarray) → SE3¶

Computes expm(wedge(tangent)).

Parameters:: tangent (ndarray)
Return type:: SE3

inverse() → SE3¶

Computes the inverse of the transform.

Return type:: SE3

normalize() → SE3¶

Normalize/projects values and returns.

Return type:: SE3

apply(target: ndarray) → ndarray¶

Applies group action to a point.

Parameters:: target (ndarray)
Return type:: ndarray

multiply(other: SE3) → SE3¶

Composes this transformation with another.

Parameters:: other (SE3)
Return type:: SE3

log() → ndarray¶

Computes vee(logm(transformation matrix)).

Return type:: ndarray

adjoint() → ndarray¶

Computes the adjoint.

Return type:: ndarray

clamp(x_translation: tuple[float, float] = (-inf, inf), y_translation: tuple[float, float] = (-inf, inf), z_translation: tuple[float, float] = (-inf, inf), roll_radians: tuple[float, float] = (-inf, inf), pitch_radians: tuple[float, float] = (-inf, inf), yaw_radians: tuple[float, float] = (-inf, inf)) → SE3¶

Clamp a SE3 within translation and RPY limits.

Parameters:

x_translation (tuple[float, float]) – The (lower, upper) limits for translation along the x axis.
y_translation (tuple[float, float]) – The (lower, upper) limits for translation along the y axis.
z_translation (tuple[float, float]) – The (lower, upper) limits for translation along the z axis.
roll_radians (tuple[float, float]) – The (lower, upper) limits for the roll.
pitch_radians (tuple[float, float]) – The (lower, upper) limits for the pitch.
yaw_radians (tuple[float, float]) – The (lower, upper) limits for the yaw.

Returns:

A SE3 within the translation and RPY limits.

Return type:

SE3