# On the covariance of X in AX = XB

## TL;DR

This paper derives a rigorous method to compute the covariance of the hand-eye calibration transformation in robot vision, enhancing the understanding of uncertainty in robot perception tasks.

## Contribution

It introduces a covariance propagation approach in SE(3) for the AX=XB problem, providing precise uncertainty estimates for the transformation.

## Key findings

- Accurately predicts covariance of hand-eye transformation.
- Validated with synthetic and real data.
- Offers a tool for high-precision robot perception applications.

## Abstract

Hand-eye calibration, which consists in identifying the rigid- body transformation between a camera mounted on the robot end-effector and the end-effector itself, is a fundamental problem in robot vision. Mathematically, this problem can be formulated as: solve for X in AX = XB. In this paper, we provide a rigorous derivation of the covariance of the solution X, when A and B are randomly perturbed matrices. This fine-grained information is critical for applications that require a high degree of perception precision. Our approach consists in applying covariance propagation methods in SE(3). Experiments involving synthetic and real calibration data confirm that our approach can predict the covariance of the hand-eye transformation with excellent precision.