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"Rotation matrices" and "quaternions" can both represent "rotations" but "quaternions" do not represent "rotation matrices". |
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{{short description|Robotics problem on coordinating two parts of a robot}}
In [[robotics]] and [[mathematics]], the '''hand eye calibration problem''' (also called the '''robot-sensor''' or '''robot-world calibration problem''') is the problem of determining the transformation between a robot [[end-effector]] and a camera or between a robot base and the world coordinate system.<ref>Amy Tabb, Khalil M. Ahmad Yousef. [https://arxiv.org/abs/1907.12425 "Solving the Robot-World Hand-Eye(s) Calibration Problem with Iterative Methods."] 29 Jul 2019.</ref> It takes the form of {{math|AX{{=}}ZB}}, where ''A'' and ''B'' are two systems, usually a robot base and a camera, and {{math|X}} and {{math|Z}} are unknown transformation matrices. A highly studied special case of the problem occurs where {{math|X{{=}}Z}}, taking the form of the problem {{math|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>Mili I. Shah, Roger D. Eastman, Tsai Hong Hong. [https://www.nist.gov/publications/overview-robot-sensor-calibration-methods-evaluation-perception-systems?pub_id=910651 "An Overview of Robot-Sensor Calibration Methods for Evaluation of Perception Systems."] 22 March 2012</ref> The covariance of {{math|X}} in the equation can be calculated for any randomly perturbed matrices {{math|A}} and {{math|B}}.<ref>Huy Nguyen, Quang-Cuong Pham. [https://arxiv.org/
The problem is an important part of [[robot calibration]], with efficiency and accuracy of the solutions determining the speed accuracy of the calibrations of robots.
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===Separable solutions===
Given the equation {{math|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 {{math|'''R'''<sub>A</sub>}} represents a 3×3 rotation matrix and {{math|'''t'''<sub>A</sub>}} a 3×1 translation vector, the equation can be broken into two parts:<ref>Amy Tabb, Khalil Yousef. [https://arxiv.org/
:{{math|'''R'''<sub>A</sub>'''R'''<sub>X</sub>{{=}}'''R'''<sub>Z</sub>'''R'''<sub>B</sub>}}
:{{math|'''R'''<sub>A</sub>'''t'''<sub>X</sub>+'''t'''<sub>A</sub>{{=}}'''R'''<sub>Z</sub>'''t'''<sub>B</sub>+'''t'''<sub>Z</sub>}}
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