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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 thea robot [[end-effector]] and a camera, as well asor the transformation between thea 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 highly studied 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> The covariance of X in the equation can be calculated for any randomly perturbed matrices A and B.<ref>https://arxiv.org/pdf/1706.03498.pdf</ref>

===Methods===

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:'''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 name="tsapps">[https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=910651], page 3. </ref> AOther solutionsolutions toavoid this is a simultaneous method, which calculates both at the same timeproblem.

====Simultaneous solutions====

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:// name="tsapps.nist.gov/publication/get_pdf.cfm?pub_id=910651]<"/ref>

====Iterative solutions====

Iterative solutions are another method used to solve the problem of error propagation. One example of an iterative solution is a program based on minimizing <math>||AX-XB||</math>. As the program iterates, it will converge on a solution to X independent to the initial robot orientation of R_B. Solutions can also be two-step iterative processes, and like simultaneous solutions can also decompose the equations into [[dual quaternion]]s.<ref>https://link.springer.com/article/10.1007/s11548-017-1646-x</ref> However, while iterative solutions to the problem are generally simultaneous and accurate, they can be computationally taxing to carry out and may not always converge on the optimal solution.<ref name="tsapps"/>

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