jaxlie

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jaxlie is a library containing implementations of Lie groups commonly used for rigid body transformations, targeted at computer vision & robotics applications written in JAX. Heavily inspired by the C++ library Sophus.

We implement Lie groups as high-level (data)classes:

Group	Description	Parameterization
jaxlie.SO2	Rotations in 2D.	(real, imaginary): unit complex (∈ S2)
jaxlie.SE2	Proper rigid transforms in 2D.	(real, imaginary, x, y): unit complex & translation
jaxlie.SO3	Rotations in 3D.	(qw, qx, qy, qz): wxyz quaternion (∈ S4)
jaxlie.SE3	Proper rigid transforms in 3D.	(qw, qx, qy, qz, x, y, z): wxyz quaternion & translation

Where each group supports:

• Forward- and reverse-mode AD-friendly exp(), log(), adjoint(), apply(), multiply(), inverse(), identity(), from_matrix(), and as_matrix() operations.
• Helpers + analytical Jacobians for manifold optimization (jaxlie.manifold).
• (Un)flattening as pytree nodes.
• Serialization using flax.
• Compatibility with standard JAX function transformations. (we've included some examples for use with jax.vmap)

We also implement various common utilities for things like uniform random sampling (sample_uniform()) and converting from/to Euler angles (in the SO3 class).

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Install (Python >=3.7)

# Python 3.6 releases also exist, but are no longer being updated.
pip install jaxlie

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Example usage for SE(3)

import numpy as onp

from jaxlie import SE3

#############################
# (1) Constructing transforms.
#############################

# We can compute a w<-b transform by integrating over an se(3) screw, equivalent
# to `SE3.from_matrix(expm(wedge(twist)))`.
twist = onp.array([1.0, 0.0, 0.2, 0.0, 0.5, 0.0])
T_w_b = SE3.exp(twist)

# We can print the (quaternion) rotation term; this is an `SO3` object:
print(T_w_b.rotation())

# Or print the translation; this is a simple array with shape (3,):
print(T_w_b.translation())

# Or the underlying parameters; this is a length-7 (quaternion, translation) array:
print(T_w_b.wxyz_xyz)  # SE3-specific field.
print(T_w_b.parameters())  # Helper shared by all groups.

# There are also other helpers to generate transforms, eg from matrices:
T_w_b = SE3.from_matrix(T_w_b.as_matrix())

# Or from explicit rotation and translation terms:
T_w_b = SE3.from_rotation_and_translation(
    rotation=T_w_b.rotation(),
    translation=T_w_b.translation(),
)

# Or with the dataclass constructor + the underlying length-7 parameterization:
T_w_b = SE3(wxyz_xyz=T_w_b.wxyz_xyz)


#############################
# (2) Applying transforms.
#############################

# Transform points with the `@` operator:
p_b = onp.random.randn(3)
p_w = T_w_b @ p_b
print(p_w)

# or `.apply()`:
p_w = T_w_b.apply(p_b)
print(p_w)

# or the homogeneous matrix form:
p_w = (T_w_b.as_matrix() @ onp.append(p_b, 1.0))[:-1]
print(p_w)


#############################
# (3) Composing transforms.
#############################

# Compose transforms with the `@` operator:
T_b_a = SE3.identity()
T_w_a = T_w_b @ T_b_a
print(T_w_a)

# or `.multiply()`:
T_w_a = T_w_b.multiply(T_b_a)
print(T_w_a)


#############################
# (4) Misc.
#############################

# Compute inverses:
T_b_w = T_w_b.inverse()
identity = T_w_b @ T_b_w
print(identity)

# Compute adjoints:
adjoint_T_w_b = T_w_b.adjoint()
print(adjoint_T_w_b)

# Recover our twist, equivalent to `vee(logm(T_w_b.as_matrix()))`:
twist_recovered = T_w_b.log()
print(twist_recovered)