## Minimium Thermal Deformation
Model Predictive Control
## Mechatronics and Control of a Precision StageTransistor density has been increasing steadily, as has been described by “Moore’s Law”, with feature sizes of 45 nanometers currently in production. The most common form of nano-manufacturing in the semiconductor industry is photo lithography. Reducing feature size, using photo lithography, is limited in part by diffraction of light. One way of overcoming the diffraction limitation is by using a mask design technique to correct for diffraction known as optical proximity correction (OPC). Alternative nano-manufacturing techniques such as nano-imprint lithography, immersion lithography and plasmonic imaging lithography, to name a few, are being studied. Imprint lithography is a technique where pattern transfer occurs by physically pressing the mask and wafer together and therefore not limited by diffraction. Precision motion stages are vital to the nano-manufacturing industry. The multi-DOF stage used in our research, Multi-Alignment and Positioning System (MAPS), was designed as a low cost, long range, flexible tool for nano-manufacturing.
The MAPS stage is integrated with a Delta Tau Power PMAC real-time controller running on a 800MHz Power PC. Many features of the Power PMAC, such as trajectory generation, brushless motor commutation and phase finding, watchdog timer, built-in PID with feedforward, and a rugged compact chassis, reduce the development of control software and hardware needed for manufacturing tools. In addition, it is an option to write your own servo and phase alogorithms in C ## Optimal Commutation of Multiple Linear-Motor SystemsThe platen (shown below) is propelled in ,, and by four linear motors. These four linear motors, using a Halbach magnet array, produce two orthogonal forces (,) each. The relationship between the global force vector and the local force vector is where is a constant matrix dependent on the placement of the linear motors. The force that each motor provides is dependent on the current supplied by each coil as:
Combining all four motors, the motor law for this system is defined as giving us the final relationship between the global force vector () and the current vector input the the motor () . our controller outputs and we need to find the current vector that will be sent to the motors. Since the matrix is “fat” (under defined), we have an infinite set of solutions that will produce the desired global force. It is natural to choose the solution that will minimize power and thus minimize motor wear etc (there is a bonus that the solution is close-form):
The solution to this problem is given as . It is possible to attack this problem in two stages, a kinematic stage where the desired local force vector is found and an electro-magnetic stage, where the minimum power current vector, given the kinematic stage, is found giving us . In general this two-stage method does not give the same solution since . In fact, the power solved by the one stage method is less than or equal, and therefore optimal, to the solution of the two stage method (). Therefore separating the kinematic stage and the electro-magnetic stage when minimizing the power is not in general minimal. Although due to the structure of the ideal linear motors (namely that ), the one stage solution turns out to be equal to the two stage solution (). It is important to note that using the two stage method, and separating the kinematic and electro-magnetic stages, in real systems won’t give the minimum power solution because it is not possible to manufacture each motor and each coil to be exactly the same. Now, a new commutation problem is proposed, by adding the constraints that each motor’s power (), separately, is equal to one another:
This problem is motivated by the temperature gradient, and subsequent non-symmetric deformation of the platen (see Ansys figures below), produced by the motors generating different power profiles. By each motor generating the same power we force heat symmetry on the platen and therefore symmetric deformation about the center of the platen. This problem is non-convex due to the added quadratic equality constraints.
This problem is difficult to solve as is. We solve the exact solution of this problem by a 2-stage method. The first step of this method involves solving a convex problem that minimizes the maximum power that any of the four motors can have (). The power for the motor with the greatest power is defined as . The second step is to project the solution of the problem onto the set of solutions that each motor has power equal to . This projection is possible by adding a non-zero current in the nullspace of . The following table compares the minimum power and heat symmetric solution to a given global force vector .
## Embedded (Real-Time) OptimizationTo solve the above stated “Equal Power” commmutation problem takes solving a convex optimization problem in real-time (within 110 microseconds for our system running at 9KHz sampling rate). Under ideal conditions the convex optimization problem can be simplified to:
## Model Predictive Control |