Minimium Thermal Deformation

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(Image by Ara Oshagan)

  1. Mechatronics and Control of a Precision Stage

  2. Optimal Commutation of Multiple Linear-Motor Systems

  3. Embedded (Real-Time) Optimization

  4. Model Predictive Control

Mechatronics and Control of a Precision Stage

Transistor 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.

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The multi-scale alignment and positioning system (MAPS) is composed of a 6 degree-of-freedom (DOF) wafer holder and a 3 DOF module holder, both of which move relative to the base. The module can be chosen for a desired task, such as atomic force microscopy, plasmonic lithography, or imprint lithography. (solid model by Ronnie Fesperman and Ozkan Ozturk at the University of North Carolina at Charlotte)

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. The control algorithm receives the position estimate, calculated from the sensor data, X=left[x y  theta_z  z  theta_x  theta_yright]^T and outputs a desired global force vector F=left[F_x F_y  T_z F_z  T_x  T_yright]^T. The global force vector is then transformed into the local motor forces f=left[f_{x1} f_{z1}  cdots f_{x4}  f_{z4}right]^T and finally must be converted into a current vector i=left[i_{11} i_{12} i_{13} cdots i_{41} i_{42} i_{43}right]^T that will be supplied to the four linear motors. This process is reversed when applied to the platen. Feedback inputs enter the UMAC through 12-Bit AD’s for most of the sensors and 4096X SINCOS interpolators for the interferometers. Voltage outputs are sent to the amplifiers through the Power PMAC’s 16-Bit D/A’s. The Power PMAC’s default servo update is 442 microseconds while the phase update is 110 microseconds. See the control diagram below.

 


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Optimal Commutation of Multiple Linear-Motor Systems

The platen (shown below) is propelled in x,y, and theta_z by four linear motors. These four linear motors, using a Halbach magnet array, produce two orthogonal forces (f_x,f_z) each. The relationship between the global force vector F and the local force vector f is F=A^{6times8}f where A is a constant matrix dependent on the placement of the linear motors.

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The force that each motor provides is dependent on the current supplied by each coil as:
 left[ begin{array}{c} f_x f_z end{array} right] =a left[ begin{array}{ccc} cos(omega x - phi) & cos(omega x - phi+frac{pi}{3}) & cos(omega x - phi+frac{2pi}{3}) cos(omega x - phi+frac{pi}{2}) & cos(omega x - phi+frac{pi}{3}+frac{pi}{2} )& cos(omega x - phi+frac{2pi}{3}+frac{pi}{2}) end{array} right] left[ begin{array}{c} i_1 i_2 i_3 end{array} right]

Combining all four motors, the motor law for this system is defined as f=B^{8times12}i giving us the final relationship between the global force vector (F) and the current vector input the the motor (i) F = ABi. our controller outputs F and we need to find the current vector that will be sent to the motors. Since the matrix AB is “fat” (under defined), we have an infinite set of solutions i 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):
 begin{array}{ll} mbox{minimize} & ||i||^2 mbox{subject to} & F=ABi end{array} label{eqn:minpower}

The solution to this problem is given as i^star = (AB)^dag F. It is possible to attack this problem in two stages, a kinematic stage where the desired local force vector is found f=A^dag F and an electro-magnetic stage, where the minimum power current vector, given the kinematic stage, is found i=B^dag f giving us i=B^dag A^dag F. In general this two-stage method does not give the same solution since (AB)^dag neq B^dag A^dag . 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 (||(AB)^dag F||leq ||B^dag A^dag F||). 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 BB^T=(na^2/2)I), the one stage solution turns out to be equal to the two stage solution (||(AB)^dag F||= ||B^dag A^dag F||). 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 (||i_k||^2), separately, is equal to one another:
 begin{array}{ll} mbox{minimize} & ||i||^2 mbox{subject to} & F=ABi &||i_k||^2=p; end{array}

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.

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Here you see that if the optimal commutation commands only 3 of the motors to generate power than relative deformations of greater than 300 nanometers. But most importantly is the deformation at the platen top (Analysis by Laila Asheghian)

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Here it shown that the platen top has relative deformations of more than 10 nanometers due to the anti-symmetric power generation shown above. (Analysis by Laila Asheghian)

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 (min max ||i^k||^2). The power for the motor with the greatest power is defined as P. The second step is to project the solution of the min max problem onto the set of solutions that each motor has power equal to P. This projection is possible by adding a non-zero current in the nullspace of B. The following table compares the minimum power and heat symmetric solution to a given global force vector F.

Method i_1^Ti_1 i_2^Ti_2 i_3^Ti_3 i_4^Ti_4 i^T i
Minimum Power .6 .27 .06 .27 1.2
Equal Power .4 .4 .4 .4 1.6


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Embedded (Real-Time) Optimization

To 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:
 begin{array}{ll} mbox{minimize} & max_k{(||f_k||)} mbox{subject to} & F=A f end{array}
Which can be reformulated as an SOCP:
 begin{array}{ll} mbox{minimize} & q mbox{subject to} & F=A f &||f_k||leq q end{array}
Current coommercial packages that solve general SOCP problems are not fast enough to solve under 110 microseconds. An interior-point method SOCP solver was written to take advantage of this problem and solve the problem as fast as 12 microseconds! The following are implementation results.

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Here the minimum power commutation was implemented while an external disturbance was introduced at motor #1 (making motor #3 a pivot point).

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Here the power symmetric commutation was implemented while an external disturbance was introduced at motor #1 (making motor #3 a pivot point).


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Model Predictive Control