Pywayne Vio Se3

SE(3) rigid body transformation library for 3D rotation and translation operations. Use when working with robot poses, camera transformations, SLAM systems, or any 3D rigid body motion tasks. Supports SE(3) matrix operations, Lie group/algebra mappings (log/Log, exp/Exp), representation conversions (quaternion, axis-angle, Euler angles), and batch processing of trajectories.

SE3 Rigid Body Transformations

Quick Start

python

import numpy as np
from pywayne.vio.SE3 import *

# Create SE(3) transformation from rotation and translation
R = np.eye(3)
t = np.array([1, 2, 3])
T = SE3_from_Rt(R, t)

# Lie algebra operations
xi = np.array([0.1, 0.2, 0.3, 0.05, 0.1, 0.15])  # [rho, theta]
T_from_xi = SE3_Exp(xi)  # se(3) vector -> SE(3)
xi_recovered = SE3_Log(T_from_xi)  # SE(3) -> se(3) vector

Core Operations

Basic Matrix Operations

Create/Verify SE(3) matrices:

• check_SE3(T) - Validate 4x4 matrix is valid SE(3)
• SE3_from_Rt(R, t) - Construct from rotation matrix and translation
• SE3_to_Rt(T) - Extract rotation matrix and translation vector

Combine/invert transformations:

• SE3_mul(T1, T2) - Matrix multiplication (compose transforms)
• SE3_inv(T) - Matrix inverse
• SE3_diff(T1, T2, from_1_to_2=True) - Compute relative transform

Lie Group/Lie Algebra Mappings

Vector form (preferred):

• SE3_Exp(xi) - se(3) 6D vector -> SE(3) matrix, xi = [rho, theta]
• SE3_Log(T) - SE(3) matrix -> se(3) 6D vector

Matrix form (theoretical):

• SE3_exp(xi_hat) - se(3) 4x4 matrix -> SE(3) matrix
• SE3_log(T) - SE(3) matrix -> se(3) 4x4 matrix
• SE3_skew(xi) - 6D vector -> 4x4 Lie algebra matrix
• SE3_unskew(xi_hat) - 4x4 matrix -> 6D vector

Naming convention: Uppercase = vector, lowercase = matrix

Representation Conversions

Quaternion + translation:

• SE3_from_quat_trans(q, t) - q is wxyz quaternion
• SE3_to_quat_trans(T) - Returns (quaternion, translation)

Axis-angle + translation:

• SE3_from_axis_angle_trans(axis, angle, t)
• SE3_to_axis_angle_trans(T) - Returns (axis, angle, translation)

Euler angles + translation:

• SE3_from_euler_trans(euler_angles, t, axes='zyx', intrinsic=True)
• SE3_to_euler_trans(T, axes='zyx', intrinsic=True)

Statistical Operations

• SE3_mean(T_batch) - Compute mean of multiple SE(3) matrices (Nx4x4 -> 4x4)

Input/Output Formats

Single transformation:

• Input: 4x4 or 3x3/3 arrays
• Output: 4x4 or scalar vectors

Batch operations:

• Input: Nx4x4 or Nx3x3/Nx3 arrays
• Output: Same batched format
• All functions support both single and batch inputs

6D vector format: [rho_1, rho_2, rho_3, theta_1, theta_2, theta_3]

• First 3: translation (linear velocity)
• Last 3: rotation (angular velocity)

Common Patterns

Trajectory Processing

python

# Batch process robot trajectory
poses = np.array([...])  # Nx4x4
log_poses = SE3_Log(poses)  # Nx6 Lie algebra space
mean_pose = SE3_Exp(np.mean(log_poses, axis=0))  # Intrinsic mean

Relative Motion

python

# Relative transform between two poses
T_rel = SE3_diff(T_world_keyframe1, T_world_keyframe2)
# T_rel transforms points from frame2 to frame1

Camera Pose Estimation

python

# Camera to world transformation
R_cam = np.column_stack([right, up, forward])  # Camera axes
t_cam = camera_position
T_cam2world = SE3_from_Rt(R_cam, t_cam)
T_world2cam = SE3_inv(T_cam2world)

Notes

• All angles in radians
• Right-multiply convention: P' = T @ P
• Numerically stable for large angles and displacements
• Batch operations use vectorized NumPy for efficiency
• Performance reference (1000 transforms): Exp ~2.5ms, Log ~0.8ms