  The hand-eye calibration problem is an important application problem in robot
research. Based on the 2-norm of dual quaternion vectors, we propose a new dual
quaternion optimization method for the hand-eye calibration problem. The dual
quaternion optimization problem is decomposed to two quaternion optimization
subproblems. The first quaternion optimization subproblem governs the rotation
of the robot hand. It can be solved efficiently by the eigenvalue decomposition
or singular value decomposition. If the optimal value of the first quaternion
optimization subproblem is zero, then the system is rotationwise noiseless,
i.e., there exists a ``perfect'' robot hand motion which meets all the testing
poses rotationwise exactly. In this case, we apply the regularization technique
for solving the second subproblem to minimize the distance of the translation.
Otherwise we apply the patching technique to solve the second quaternion
optimization subproblem. Then solving the second quaternion optimization
subproblem turns out to be solving a quadratically constrained quadratic
program. In this way, we give a complete description for the solution set of
hand-eye calibration problems. This is new in the hand-eye calibration
literature. The numerical results are also presented to show the efficiency of
the proposed method.
