Summary: Many kinetic problems arising from robotics are modeled in optimization of real function in dual quaternion variables. For example, hand-eye calibration fits data to find dual quaternion variables, which integrate the relative position and orientation between a robot gripper and a camera mounted rigidly on the gripper. But now derivatives of real function in dual quaternion variables are unwieldy. To solve optimization of real function in dual quaternion variables, we utilize eight-dimensional vectors to represent dual quaternions. Then the unit dual quaternion constraint is turned into two quadratic equations. Using optimization theory, we derive an algorithm for computing the projection of dual quaterions onto a unit dual quaternion set, which is closed, nonconvex and unbounded. Based on this projection, we propose a proximal linearized algorithm for optimization of real function in dual quaternion variables. The global convergence of the proximal linearized algorithm is analyzed. Finally, numerical experiments on hand-eye calibration problems and dual quaternion-based regressions illustrate the effectiveness of the proposed proximal linearized algorithm.