Hand–eye calibration problem: Difference between revisions

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{{short description|Robotics problem on coordinating two parts of a robot}}

In [[robotics]] and [[mathematics]], the '''hand eyehand–eye calibration problem''' (also called the '''robot-sensorrobot–sensor''' or '''robot-worldrobot–world calibration problem''') is the problem of determining the transformation between a robot [[end-effector]] and a sensor or sensors (camera or laser scanner) 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 is conceptually analogous to biological [[eye–hand coordination|hand–eye coordination]] (hence the name). 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/abs/1706.03498 "On the covariance of X in AX = XB."] 12 June 2017.</ref>

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'''_A}} represents a 3×3 [[rotation matrix]] and {{math|'''t'''_A}} a 3×1 translation vector, the equation can be broken into two parts:<ref>Amy Tabb, Khalil Yousef. [https://arxiv.org/abs/1907.12425 "Solving the Robot-World Hand-Eye(s) Calibration Problem with Iterative Methods."] Machine Vision and Applications, August 2017, Volume 28, Issue 5-6, pp 569-590.</ref>

:{{math|'''R'''_A'''R'''_X{{=}}'''R'''_Z'''R'''_B}}

:{{math|'''R'''_A'''t'''_X+'''t'''_A{{=}}'''R'''_Z'''t'''_B+'''t'''_Z}}

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===Simultaneous solutions===

Simultaneous solutions are based on solving for both {{math|X}} and {{math|Z}} at the same time (rather than basing the solution of one part off of the other as in separable solutions), propagation of error is significantly reduced.<ref name="dual-quaternions">Algo Li, et al. [https://web.archive.org/web/20200209164546/https://pdfs.semanticscholar.org/225d/e4ea2d3f18b7743bfeabf925fa603fc47bcb.pdf "Simultaneous robot-world and hand-eye calibration using dual-quaternions and Kronecker product."] International Journal of the Physical Sciences Vol. 5(10), pp. 1530-1536, 4 September 2010.</ref> By formulating the matrices as [[dual quaternion]]s, it is possible to get a [[linear equation]] by which {{math|X}} is solvable in a linear format.<ref name="tsapps"/> An alternative way applies the [[least squares|least-squares method]] to the [[Kronecker product]] of the matrices {{math|A⊗B}}. As confirmed by experimental results, simultaneous solutions have less error than separable quaternion solutions.<ref name="dual-quaternions"/>

===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{{!}}{{!}}}}. As the program iterates, it will converge on a solution to {{math|X}} independent to the initial robot orientation of {{math|'''R'''_B}}. Solutions can also be two-step iterative processes, and like simultaneous solutions can also decompose the equations into [[dual quaternion]]s.<ref>Zhiqiang Zhang, et al. [https://link.springer.com/article/10.1007/s11548-017-1646-x "A computationally efficient method for hand–eye calibration."] 19 July 2017.</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"/>

===The AX=XB case===

The matrix equation {{math|AX{{=}}XB}}, where {{math|X}} is unknown, has an infinite number of solutions that can be easily studied by a geometrical approach.<ref> Irene Fassi, Giovanni Legnani [https://doi.org/10.1002/rob.20082 "Hand to sensor calibration: A geometrical interpretation of the matrix equation AX =XB."] Journal of Robotic Systems, 28 July 2005</ref> To find {{math|X}} it is necessary to consider a simultaneous set of 2 equations {{math|A₁X{{=}}XB₁}} and {{math|A₂X{{=}}XB₂}}; the matrices {{math|A₁, A₂, B₁, B₂}} have to be dermined by experiments to be performed in an optimized way.

<ref> Giovanni Legnani. [https://doi.org/10.2316/Journal.206.2018.1.206-4974 "Optimization of hand-to-camera calibration using geometrical interpretation of matrix equation AX = XB." ] International Journal of Robotics and Automation - January 2018. </ref>

=== The 2D laser profile scanner case ===

<math>\begin{bmatrix} p_b \\ 1 \end{bmatrix}=

\begin{bmatrix} R_b & T_b \\ 0 & 1 \end{bmatrix}\centerdot

\begin{bmatrix} R_s & T_s \\ 0 & 1 \end{bmatrix}\centerdot

\begin{bmatrix} p_s \\ 1 \end{bmatrix}</math>

where <math>p_b</math> represents the unknown coordinate of the point <math>p</math> in the robot base system, <math>R_b, T_b</math> represent the known relationship between the robot base system and end-effector, <math>R_s,T_s</math> are the unknown relationship between the end-effector and the scanner, and <math>p_s</math> is the known coordinate of the point <math>p</math> in the local scanner system. Methods are as follows,

* Straight edges

There is a method using straight edges for hand-eye calibration.<ref>{{Cite journal|last=Xu|first=Jing|last2=Hoo|first2=Jian Li|last3=Dritsas|first3=Stylianos|last4=Fernandez|first4=Javier Gomez|date=2022-02-01|title=Hand-eye calibration for 2D laser profile scanners using straight edges of common objects|url=https://www.sciencedirect.com/science/article/pii/S0736584521001046|journal=Robotics and Computer-Integrated Manufacturing|language=en|volume=73|pages=102221|doi=10.1016/j.rcim.2021.102221|issn=0736-5845|url-access=subscription}}</ref>

==References==