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{{collapse top| Hand-eye calibration problem}}

==Hand-eye calibration problem==

"In robotics, the '''hand-eye calibration problem''', or the '''robot-sensor calibration problem''', is the problem of determining the transformation between the robot end-effector and a camera, as well as the transformation between the robot base and the world coordinate system".<ref>[https://arxiv.org/abs/1907.12425] </ref> It takes the form of AX=ZB, where A and B are two systems, usually a robot base and a camera, and X and Z are unknown transformation matrices. A special case of the problem occurs where X=Z, taking the form of the problem AX=XB. Solutions to the problem take the forms of several types of methods, including "separable closed-form solutions, simultaneous closed-form solutions, and iterative solutions".<ref>[https://www.nist.gov/publications/overview-robot-sensor-calibration-methods-evaluation-perception-systems?pub_id=910651]</ref>

===Methods===

====Separable forms====

Given the equation AX=ZB, it is possible to decompose the equation into a purely rotational and translational part; methods utilizing this are referred to as separable methods. Where '''R'''_A represents a 3x3 rotation matrix and '''t'''_A a 3x1 translation vector, the equation can be broken into two parts:<ref>[https://arxiv.org/pdf/1907.12425.pdf]</ref>

:'''R'''_A'''R'''_X='''R'''_Z'''R'''_B

:'''R'''_A'''t'''_X+'''t'''_A='''R'''_Z'''t'''_B+'''t'''_Z

Equation 2 becomes linear if '''R'''_Z is known. As such, the most frequent approach is to Rx and Rz using the first equation then using it to solve for the second two variables in the second equation. Rotation is represented using [[quaternion]]s, allowing for a linear solution to be found. While separable methods are useful, any error in the estimation for the rotation matrices is compounded when being applied to the translation vector.<ref>[https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=910651], page 3. </ref> A solution to this is a simultaneous method, which calculates both at the same time.

====Simultaneous forms====

By formulating the matrices as [[dual-quaternion]]s, it is possible to get a linear equation by which X is solvable in a linear format.<ref>[https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=910651]</ref>

*[http://math.loyola.edu/~mili/Calibration/]

*[https://ieeexplore.ieee.org/abstract/document/8788685/keywords#keywords] - Octonion solution

*[https://arxiv.org/abs/1706.03498]

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