SE3 Rigid Body Transformations

Quick Start

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

# 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

# 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

# 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