Modeling, parameter estimation, and open-loop control of MEMS deformable mirrors Curtis R. Vogel*a , Glenn A. Tylerb , Yang Luc a Department
of Mathematical Sciences, Montana State University, Bozeman, MT 59717-2400 Optical Sciences Company, 1341 South Sunkist Street, Anaheim, CA 92806 Department of Mechanical Engineering, Boston University, Boston, MA 02215
b the c
ABSTRACT We present a model for MEMS deformable mirrors (DMs) that couples a 2-dimensional, linear 4th order partial differential equation for the DM facesheet with linear spring models for the actuators. We estimate the parameters in this model using the method of output least squares, and we demonstrate the effectiveness of this approach with data from a 140-actuator MEMS test mirror produced at Boston University. A scheme for robust, computationally efficient open-loop control, which is based on this model, is also presented. Keywords: adaptive optics; MEMS; deformable mirrors
1. INTRODUCTION In this paper we present a continuum mechanical model for MEMS deformable mirrors (DMs), implemented as a 2-dimensional, 4th order linear partial differential equation for the DM facesheet6 coupled with a system of scalar linear algebraic equations for the DM actuators. This model requires a number of parameters that depend on the flexure and tension of the facesheet and physical characteristics of the actuators. We introduce a fast, robust numerical scheme to estimate these parameters from measurements of facesheet displacement that correspond to known actuator commands, and we validate this scheme with data from a Boston University MEMS test mirror. We also outline a robust open-loop control approach that is based on the model, and we discuss its efficient implementation for DMs with very large numbers of actuators. The DM modeling and open-loop control approaches presented in this paper are quite similar to those first introduced by Vogel and Yang7 for continuous facesheet, point-actuated DMs. Here the model has been specifically tailored for a Boston University MEMS test mirror. It is simpler than the previous model in that the actuators are assumed to behave like linear springs, but it is more complicated in that it contains a linear 2nd order tension term to account for the effects of facesheet stretching and it contains actuator offset terms that account for a non-flat facesheet when no actuation is applied. The model parameters are estimated using an output least squares 8 approach for partial differential equations that was first proposed for DMs by Ellerbroek and Vogel.2 A similar modeling and open-loop control effort have been carried out by Stewart et al5 at Boston University. That work was based on a very similar 4th order partial differential equation, but with a nonlinear 2nd order term to account for the effects of facesheet stretching. The model also has a nonlinearity to account for effects that occur when the (electrostatic) actuator plate displacement deviates significantly from the gap size. Stewart et al take a significantly different approach to parameter estimation, employing what is known in the Inverse Problems community as the Equation Error Method.1 They have also demonstrated the feasibility of their open-loop control scheme on MEMS mirrors. Somewhat similar results by Morsitski et al4 from the Laboratory for Adaptive Optics at the University of Santa Cruz are based on a 4th order linear partial differential equation model for a plate with no stretching. They use the free-space Green’s function (with no boundary conditions imposed) for the plate equation to formulate their open-loop control scheme. *Contact:
[email protected], Telephone 1 406 994 5332
This paper is organized as follows. In section 2 we review our modeling assumptions and present the DM model. In section 3 we introduce the method of output least squares for DM model parameter estimation, and we present experimental results obtained with a 140-actuator Boston University test mirror. Section 4 contains the discussion of open-loop implementation.
2. DEFORMABLE MIRROR MODEL Our model is based on the following assumptions. 1. The (temporal) dynamics of DM facesheet are fast relative to the temporal changes in actuation. 2. The DM facesheet is a thin, elastic, stretched plate. 3. Forces on the facesheet due to actuator loading are balanced by restoring force due to flexure and stretching of the facesheet. 4. Each DM actuator applies a load to the facesheet over a relatively small area. 5. Actuator deflection depends linearly on the load induced by facesheet and it depends linearly on the square of the applied voltage. Evidence for the last assumption is presented in Fig. 1. −7
6
x 10
poke index = (4,4); bias = 141V; gain coef = −2.8698e−11 measured data linear fit−to−data
peak differential DM deflection
4
2
0
−2
−4
−6
0
0.5
1
1.5 2 2.5 poke voltage squared
3
3.5
4 4
x 10
Figure 1. Actuator deflection for a 140-actuator Boston University MEMS test mirror as a function of the square of the applied voltage.
Our facesheet model takes the form Dfs ∇4 w − T ∇2 w +
na
pi δ(x − xi ) = 0,
(1)
i=1
where w(x) denotes the deflection (out-of-plane displacement) of facesheet at location x = (x, y) in the neutral plane; pi denotes the load due to ith actuator at location xi ; na denotes the number of actuators; δ(·) denotes the Dirac delta; ∇2 denotes the 2-d Laplacian operator; and ∇4 w = (∇2 )2 w =
∂4w ∂4w ∂4w + 2 + . ∂x4 ∂x2 ∂y 2 ∂y 4
(2)
is the 2-d biharmonic operator. The terms on the left-hand-side of the equal sign in model equation (1) represent loads due to flexure, stretching, and actuation, respectively, and the parameters are the flexural rigidity Dfs and the tension T . We employ a scalar linear model for each of the actuators, k (w(xi ) − zi ) = A (pi + γai ),
i = 1, . . . , na ,
(3)
where ai denotes the actuator command, w(xi ) is the facesheet deflection at the ith actuator, and as in Eq. (1), pi is the load on facesheet due to ith actuator. The parameters in this model are the actuator spring stiffness constant k, the area of contact A between actuator and the facesheet, the actuator gain coefficient γ, and the local offset zi in the facesheet deflection. For electrostatically actuated MEMS DMs, we take ai = Vi2 , where the Vi are the applied voltages. The offsets zi account for surface variations in the unactuated DM due, for example, to mirror polishing. If we solve for pi = (k/A) [w(xi ) − γai − zi ] in actuator Eq. (3) and substitute into the facesheet model Eq. (1), we obtain the reduced DM model 4
2
∇ w − β3 ∇ w + β2
na
w(xi ) δ(x − xi ) = β1
i=1
na
ai δ(x − xi ) +
i=1
na
β0,i δ(x − xi )
(4)
i=1
with parameters β3
= T /Dfs
β2 β1
= k/ADfs = γk/ADfs = γβ2
β0,i
= zi k/ADfs = zi β2 ,
i = 1, . . . , na .
Given parameter values, the reduced model (4) can be solved numerically, e.g., using a Galerkin-finite element method.9
3. PARAMETER ESTIMATION In order to estimate the parameters β0,i , i = 1, . . . , na , β1 , β2 , β3 in the reduced model (4), we apply the method of output least squares (OLS).8 We conducted a sequence of nexp experiments, with actuator command vectors ak and corresponding measured DM deflections wobs (x, y; ak ), k = 1, . . . , nexp . Denote the solution to (4) that results from a particular choice of the parameter vector β and the actuator command vector ak by w(x, y; β, ak ). The optimal parameter estimate βˆ is taken to be the minimizer of the OLS fit-to-data cost function
nexp
JOLS (β) =
[w(x, y; β, ak ) − wobs (x, y; ak )]2 dx dy.
(5)
k=1
In practice, measurements are discrete and so we replace the above integral with a discrete sum. In Fig. 2 we present parameter estimation results obtained with data from 4 experiments conducted on a 140-actuator MEMS DM produced for testing purposes by Boston University. In the first experiment, a constant 141 volt bias was applied to all the DM actuators. In the second, a 0-volt poke was applied to a single actuator in the lower left corner of the DM, while the remaining actuators were held at bias voltage. In the third experiments, a 0-volt poke was applied to a 2-by-2 block of corner actuators, while in the fourth experiment, the same poke voltage was applied to a 3-by-3 block of corner actuators. A central actuator was also unactuated in each of these experiments. The results shown in Fig. 2 are from the 2-by-2 poke experiment. The RMS error between the measured and simulated DM deflections, averaged over all 4 experiments, was 23 nanometers. It should be noted that the model presented here is linear. We anticipate being able to achieve better results by incorporating nonlinear effects.
−7
processed DM deflection; experiment no. 3
−7
Simulated DM deflection
x 10
x 10
6
6 2 vertical distance / actuator spacing
vertical distance / actuator spacing
2 4 4 2
6 8
0
10
−2
12
−4
4
4 6
2
8
0
10
−2
12 −4
14
14 −6 0
5 10 horizontal distance / actuator spacing
15
difference between observed and simulated DM deflections
0
10
−8
x 10
4
15
−7 x 10 horz cross sect of DM differential deflection
5 measured modeled 0
6 4
6
−5 −5
0 5 10 horizontal distance / actuator spacing
2
8
−7
0
10
10
x 10
15
vert cross sect of DM differential deflection
−2 12
−4
14
−6 0
5 10 horizontal distance / actuator spacing
meters
vertical distance / actuator spacing
8
meters
10 2
5 10 horizontal distance / actuator spacing
5 measured modeled 0
15 −5
0
5 10 15 vertical distance / actuator spacing
20
Figure 2. Parameter estimation results for 140-actuator Boston University MEMS test mirror. Upper left plot shows measured deflection for 2-by-2 actuator poke experiment; upper right shows corresponding simulated deflection; lower left plot shows difference between measured and simulated deflections; lower right plot shows horizontal and vertical crosssections of measured and simulated deflections.
Minimization of the output least squares cost function was carried out using the unconstrained optimization function fminunc from the MATLAB3 Optimization Toolbox. In order to dramatically increase numerical efficiency and robustness, we computed gradients of JOLS using an adjoint, or costate, method.2, 8 This required only two solutions of partial differential equations of the form (4) per gradient evaluation. Had we instead used the standard brute-force finite difference gradient approximation, based on JOLS (β1 , . . . , βj + , . . . , βn ) − JOLS (β1 , . . . , βj − , . . . , βn ) ∂JOLS = + O(2 ), ∂βj 2 we would have needed 2n solutions of Eq. (4) per gradient evaluation, where n = na + 3 is the number of model parameters. n = 143 for the 140-actuator MEMS test mirror, so the adjoint approach yields more than two orders of magnitude reduction in the computational cost of the parameter estimation.
4. OPEN-LOOP CONTROL Given a phase profile φ = φ(x), our goal is to compute an actuator command vector a for which the DM deflection w = w(x; a) closely matches φ. In addition, we would like the dependence of a on φ to be stable. This decreases
the likelihood of “unreasonable” actuator commands that may damage the DM, and it enhances the numerical efficiency of iterative schemes to solve the control problem. To this end, we minimize J(w, a) =
1 1 (w − φ)T Q(w − φ) + aT Ra 2 2
(6)
subject to the constraint Aw = F a + b.
(7)
The R in Eq. (6) is a penalty matrix, typically taken to be a small multiple of the identity, while Q is usually taken to be a diagonal matrix corresponding to a pupil mask, with diagonal entries set to either zero or one. The matrix A in Eq. (7) is obtained from discretization of the operator on the left-hand-side of the reduced DM model (4), while the matrix F and the vector b arise from the two terms on the right-hand-side of the model equation. From the constraint, we obtain
w = A−1 F a + woffset
(8)
woffset = A−1 b.
(9)
where
By substituting (8) into (6) and setting the derivative with respect to a equal to zero, we obtain a = (F T A−1 QA−1 F + R)−1 F T A−1 Q(φ − woffset ).
(10)
Note that we have assumed that the matrix A in (7) is symmetric. This is the case with Galerkin-finite element discretizations of Eq. (4). Let K = F T A−1 QA−1 F + R (11) and
B = F T A−1 Q.
(12)
Note that the size of K is na -by-na , where na is the number of actuators. If na is relatively small, B and the inverse of K can easily be assembled and stored off-line, and (10) can be implemented as a = K −1 B(φ − woffset ). On the other hand, if na is very large, it is more efficient to implement (10) by solving the linear system Kw = c,
where c = B(φ − woffset )
(13)
using an iterative technique like the conjugate gradient method.10 This requires repeated multiplication by K, which in turn requires repeated solutions of linear systems involving the matrix A. These systems can be efficiently solved using a multigrid technique.9
ACKNOWLEDGMENTS We thank Dr. Thomas Bifano, Director of the Boston University Photonics Center, for providing us with access to the Boston University MEMS test mirror used in the experiments described in this paper. Support for Dr. Glenn Tyler came from the National Science Foundation Small Business Innovative Research Program Grant Number 0912623, awarded to the Optical Sciences Company (tOSC). Dr. Curtis Vogel was supported by a grant from tOSC to Montana State University.
REFERENCES [1] H.T. Banks and K. Kunisch, Estimation Techniques for Distributed Parameter Systems, Birkhuser, 1989. [2] B.L. Ellerbroek and C.R. Vogel, “Topical Review: Inverse problems in atmospheric adaptive optics”, Inverse Problems, 25 (2009). [3] MATLAB. The Mathworks, Natick, Massachusetts, USA. [4] K.M. Morzitski, K.B.W. Harpse, D.T. Gavel, and S.M. Ammons, “The open-loop control of MEMS: Modeling and experimental results”, Proc. SPIE 6467, 64670G (2007) [5] J.B. Stewart, A. Doiuf, Y. Zhou, T.G. Bifano, “Open-loop control of a MEMS deformable mirror for largeamplitude wavefront control”, Journal of the Optical Society of America-A, 24 (2007), pp. 3827-3833. [6] S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells, McGraw-Hill, 1959. [7] C.R. Vogel and Q. Yang, “Modelling and simulation of a continuous facesheet MEMS deformable mirror”, JOSA-A, 23 (2006), pp. 1074-1081. [8] C.R. Vogel, Computational Methods for Inverse Problems, Society of Industrial and Applied Mathematics, 2002. [9] C.R. Vogel and J.J. Heys, “Fast multigrid solution of biharmonic operator equations”, SIAM J. Scientific Computing, submitted. [10] Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd Edition, Society of Industrial and Applied Mathematics, 2003.
