A new 6-degree-of-freedom measurement method of X-Y stages based on additional information

In some occasions, high accuracy and real-time 6-degree-of-freedom (6-DOF) displacements should be measured so that the control system can regulate the X-Y stages’ attitude in real time. The accuracy and real-time property of measurement's result depend on not only the sensors itself, but also the accuracy and effort of the computational algorithm using the sensors’ measured data. As the relation between sensors’ measured data and displacements is immensely complex and usually described as strong nonlinear coupling equation when 6-DOF displacements are all considered, computational accuracy and effort are difficult to be ensured simultaneously. This paper designs a 6-DOF displacements’ measurement setup for X-Y stages based on nine interferometers’ additional information, and derives the corresponding computational algorithm. For rotation, its range is usually very small and high accuracy computational results can be obtained using two interferometers’ differential computation; for translation, the closed form solutions without rotational displacement's computational error's transmission are derived by making full use of all additional interferometers’ information, so that the computational accuracy can be ensured. In addition, the algorithm has simple form and doesn’t involve iteration and transcendental function's computation, so that it helps real-time computation. This algorithm can acquire the computational accuracy of 10−15 rad and 10−13 mm for rotational and translational displacement separately, which can be seen in the simulation result for a lithography's wafer stage.

► A new 6-DOF's measurement program for X-Y stage based on nine interferometers is proposed.
► The computational algorithm with high accuracy and simple form is derived.
► The direction cosines can be determined without an assumed stacked slide model.