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 sensor or sensors (camera or laser scanner) or between a robot base and the world coordinate system.
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.
[2] The covariance of X in the equation can be calculated for any randomly perturbed matrices A and B.
[3] 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.
Each type of solution has specific advantages and disadvantages as well as formulations and applications to the problem.
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 RA represents a 3×3 rotation matrix and tA a 3×1 translation vector, the equation can be broken into two parts:[4] The second equation becomes linear if RZ is known.
Rotation is represented using quaternions, 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.
[5] An alternative way applies the least-squares method to the Kronecker product of the matrices A⊗B.
[6] 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 ||AX−XB||.
As the program iterates, it will converge on a solution to X independent to the initial robot orientation of RB.
[5] The matrix equation AX=XB, where X is unknown, has an infinitive number of solutions that can be easily studied by a geometrical approach.
[8] To find X it is necessary to consider a simultaneous set of 2 equations A1X=XB1 and A2X=XB2; the matrices A1, A2, B1, B2 have to be dermined by experiments to be performed in an optimized way.
represent the known relationship between the robot base system and end-effector,