Lie Groups

Complete implementations of common Lie groups for robotics and control with multiple parameterizations.

Supported Groups

Group	Description	Parameters	Common Use Cases
R^n	Euclidean space	n	State vectors, positions, velocities
SO(2)	2D rotations	1 (angle)	Planar robotics, heading estimation
SO(3)Quat	3D rotations (quaternion)	4 (w,x,y,z)	Attitude estimation, spacecraft control
SO(3)EulerB321	3D rotations (Euler)	3 (ψ,θ,φ)	Aircraft dynamics, human-readable angles
SO(3)Mrp	3D rotations (MRP)	3	Singularity-free attitude control
SE(2)	2D rigid transformations	3 (x,y,θ)	Mobile robots, 2D localization
SE(3)Quat	3D rigid transformations	7 (pos + quat)	Manipulators, 3D SLAM
SE_2(3)	Extended pose	10 (p + v + quat)	IMU preintegration, VIO

Features

All groups support:

• Group operations: Composition (*), inverse, identity

• Group actions: Transform vectors/points (@)

• Lie algebra operations: exp, log, adjoint

• Jacobians: Left and right Jacobians for integration

• Conversions: Between different parameterizations

• Numerical stability: Taylor series near singularities

Quick Examples

SE(3): 3D Rigid Transformations

>>> import cyecca.lie as lie
>>> import casadi as ca
>>> # Create SE(3) transformation
>>> position = ca.DM([1, 2, 3])
>>> quaternion = ca.DM([1, 0, 0, 0])
>>> X = lie.SE3Quat.elem(ca.vertcat(position, quaternion))
>>> X.p.param.shape
(3, 1)

SO(3): 3D Rotations

Quaternion to Euler conversion:

>>> q = lie.SO3Quat.elem(ca.DM([1, 0, 0, 0]))
>>> euler = lie.SO3EulerB321.from_Quat(q)
>>> mrp = lie.SO3Mrp.from_Quat(q)

Integration Methods

Lie groups provide methods for numerical integration:

• Exponential map: algebra_element.exp(group)

• Logarithmic map: group_element.log()

• Left/Right Jacobians: For manifold integration

See Lie Groups Module for detailed API reference.