Jerk and Current Limited Flatness-based Open Loop Control of Foveation Scanning Electrostatic Micromirrors

Transcription

1 Preprints of the 19th World Congress The International Federation of Atomatic Control Jerk and Crrent Limited Flatness-based Open Loop Control of Foveation Scanning Electrostatic Micromirrors Richard Schroedter*, Klas Janschek**, Thilo Sandner* * Franhofer Institte for Photonic Microsystems (FhG-IPMS), AMS, Microscanner R&, resden, Germany, ( richard.schroedter@ipms.franhofer.de). ** Institte of Atomation, Faclty of Electrical and Compter Engineering, Technische Universität resden, resden, Germany ( klas.janschek@t-dresden.de) Abstract: This paper describes open loop control measres for performance improvements of electrostatic micromirrors in context with foveation scanning for 3 time-of-flight cameras. The generation of high accracy scanning trajectories with scene dependent variable slope is realized by a flatness-based open loop control scheme. Previos flatness-based control soltions have shown nfavorable residal oscillations excited at slope reverse points and high drive crrents at zero-crossing comb deflection de to sqare-root voltage law for electrostatic mirror control. Jerk limited trajectories are introdced for redction of residal oscillations and their impact on scanning trajectory properties is expressed by design formlas. A considerable redction of zero-crossing drive crrents can be achieved by sing a dal-comb control law at small deflection angles redcing the effective drive voltage slope. The paper addresses the basic micromirror models and describes in detail the jerk and crrent limitation control measres in context with the flatness-based control concept. Experimental reslts prove the adeqacy of the proposed soltions. 1. INTROUCTION The crrent paper describes the open loop control of the scanning fnction of a novel 3 time-of-flight (TOF) laser camera with foveation properties for robotic applications [1]. Foveated imaging denotes a higher image resoltion at specific regions of interest. In the crrent design this is realized by adapting the scanning speed of a micromirror assembly within a range of typ. 1 Hz. The core element of the laser camera is an innovative laser scanning -micromirror assembly, developed at Franhofer Institte for Photonic Microsystems (FhG-IPMS), based on a two-stage gimballed electrostatic comb transdcer [], a schematics view is given in Fig. 1. The inner cardanic axis is operating in resonant-mode at 16 Hz, whereas the oter axis (typ. eigenfreqency 13 Hz) is formed by a vertical comb strctre in a so-called Staggered Vertical Comb (SVC) configration that allows qasistatic operation with large deflection angles, typ. ±1 (cf. Fig. a). A micromirror with elliptic apertre of.6x3.6 mm² is monted on the inner axis silicon plate (cf. Fig. b). An example for typical scanning trajectories with variable slope is shown in Fig. 3. The command tracking of a qasistatic micromirror axis is challenged in general by the inherently nonear transdcer characteristics and the extremely lightly damped mass-spring dynamics. Closed loop control is employed less for sch MEMS-devices (MEMS micro-electro-mechanical systems), mainly for technological simplicity, i.e. for avoiding additional sensing devices [3]. Open loop control concepts in general rely fndamentally on accrate models of MEMS dynamics. A common commanding techniqe sing ear model dynamics is inpt shaping, where the lightly damped eigenmode oscillations are smoothed ot by destrctive interference of plse-shaped command inpts, e.g. [4,5]. Another rather straightforward approach is prefiltering of the commanded trajectory profiles by some compensating prefilter with inverse microscanner dynamics, e.g. [6]. For the crrent application ear soltions show moderate performances with still residal oscillations de to imperfect cancellation of the nonear mass-spring microscanner dynamics [6]. e to the nonear microscanner system dynamics it is worth investigating alternative nonear command tracking techniqes. In this context the flatnessbased design paradigm [7] is a very promising candidate; it has been applied to some MEMS applications, e.g. [8] and it has been sccessflly adopted for the MEMS microscanner investigated here [9]. The reslts in [9] demonstrate, as to be expected, considerable performance improvements w.r.t. the ear approaches described in [6], however nfavorable residal oscillations excited at slope reverse points and high drive crrents at zero-crossing comb deflection de to sqareroot voltage law for electrostatic mirror control are still existing. The contribtions of this paper comprise performance improvements w.r.t. flatness-based control laws given in [9] by applying (i) limited jerk trajectories for redction of residal oscillations and (ii) a dal-comb control law at small deflection angles for redction of drive crrents. The design soltions have been sccessflly verified on a real microscanner assembly. Copyright 14 IFAC 685

2 Fig. 1. Principle of laser light deflection with gimballed micro mirror Fig. 4. Static deflection characteristic crve (a) (b) Fig.. (a) Schematic configration of qasistatic microscanner with staggered vertical comb (SVC) drive, (b) Photograph of qasistatic / resonant -Microscanner developed by FhG-IPMS The qasistatic comb drive capacitance derivatives, shown in Fig. 5a, have been determined by measring the static deflection characteristic and solving (1) separately, i.e. for comb 1 (eqivalent comptations for comb ): (a) single comb actation for positive deflection angles C θ τ ( θ) = θ and, (3) ' spring ( ) for 1 ( θ),1 (b) dal comb actation for negative deflection angles, when the comb is emerged (i.e. opposite comb is immersed), ' sing correspondent C ( θ ) from single comb actation τ ( θ) ( θ) ' spring ', C ( θ) = C ( θ) for θ<. (4) 1 ( θ) ( θ),1,1 Fig. 3. Mirrors deflection scheme sing foveation trajectories for oter qasistatic axis. MICROSCANNER ESIGN MOEL The nonear microscanner system dynamics can be denoted by torqe eqilibrim: 1 1 τ : = Jθ ɺɺ + bθɺ + τ ( θ) = C ( θ) + C ( θ) eff spring 1,1,. (1) with the model parameters J = kgm for mirror inertia, b = Nms/rad for viscos damping and C ' : = C ( θ ) θ. The driving voltage, is i i restricted to ± 15 V. An ANSYS analysis,1 of, the microscanner provides the nonear progressive spring stiffness, shown in Fig. 5b. Conseqently, the nonear spring torqe is denoted by: τ ( θ) = θ k( θ ') dθ '. () spring The static deflection characteristic ( θ ) can be measred at stationary conditions ( θθ= ɺɺɺ, ), measrement reslts are shown in Fig. 4 for single sided actation. The reassembled capacitance derivative crves are shown in Fig. 5a. Therein the singlarity at zero deflection angle is replaced by a ear interpolation. The region of very small vales of capacitance derivative has been exponentially extrapolated. This is the deflection region, where the comb is emerged and the deflection cold not be measred, becase of too small torqes. The important property of this comb arrangement is that even at small angles with emerged comb figres there is still torqe athority with that emerged comb finger. This allows a dal comb actation with opposing (bipolar) torqes arond zero deflection region (cf. section 4). (a) (b) Fig. 5. Comb characteristics as a fnction of the mirror deflection angle: (a) capacitance derivatives, (b) nonear torsional spring stiffness 686

3 3. JERK-LIMITE FLATNESS-BASE TRAJECTORY ESIGN The flatness-based open loop control, presented in [9], allows a systematic command trajectory design for qasistatic microscanner with extremely lightly damped mechanical mass-spring system. The control scheme is sketched in Fig. 6. The deflection angle θ was chosen as flat otpt z, hence the inpt = complies with the nonear driving r fnction law as follows: θ( z) = z r ( ( ). (5) ( zzz, ɺ, ɺɺ) = Jzɺɺ + bzɺ + τ z spring ) C ( z) According to the highest time derivatives of the inpt ( n = ) and otpt ( q= ) in (5), the relative degree is defined as the difference r= n q=. Any desired command trajectory mst be r -times continosly differentiable, and therefore, in this case, the command deflection angle θ reqire continity ntil the second r derivative. Fig. 6. Control scheme for qasistatic axis of microscanner The application of fifth order polynomials for ronding off discontinities of triangle shaped command trajectories gives no inherent restriction to redce residal mirror oscillation, cased by small ncertainties of the microscanner model. It has been shown that the level of jerk, i.e. third time derivative of displacement, has a strong inflence on the joint position errors of robot maniplators, i.e. redced jerk means smaller errors [1]. This leads to minimm jerk trajectory generation schemes, motivated also from biological systems (mscle movements, c.f. [11]) and has been applied sccessflly to mechatronic applications, e.g. [1]. For the crrent micro mirror system we will not follow the minimm jerk approach, bt rather specify a hard jerk limit jerk imm magnitde j as a design parameter. This jerk-limited trajectory design approach allows king the imm trajectory jerk with the tracking errors of the mirror deflection. The following third-order polynomial describes the deflection trajectory approach with the allowed imm jerk magnitde j and the constants θ (initial deflection), θ ɺ (initial velocity): j θ( t) = θ + θt+ t 6 ɺ 3. (6) The triangle shaped movement in Fig. 7 is a typical command trajectory for qasistatic microscanners with the region of interest (ROI) incorporated in the ear scanning area ±θ. The imm mirror deflection (imm angle of the ideal triangle) is denoted by ±θ. Fig. 7. Jerk limited triangle trajectory The imm jerk magnitde j can be parametrized sing the general relation between the imm deflection θ, the ratio of ear scanning area k : = θ / θ and the trajectory freqency f : j = 18 ( 1 k ) θ f 3. (7) Figre 8 shows the shaped trajectory for a 1 Hz triangle trajectory in the range of θ = 7 with variation of the ear scanning area; the eqivalent jerk limitations and tracking errors (experimental reslts) are given in Table 1. The reslts show a proportional relationship between imm jerk magnitde and peak-to-peak tracking error. Ths the expression (7) together with the experimental reslts from Table 1 simplify the triangle trajectory design and allow a transparent parametric prediction of the mirror residal oscillations according to the given jerk limitation or the ear scanning area. Fig. 8. Variation of ROI ear area k and jerk-limitation for a typical triangle command trajectory with 1 Hz and θ = 7 4. LIMITE RIVE CURRENT TRAJECTORY ESIGN The flatness-based command tracking design, as presented in the previos paper [9], implements only a single sided comb actation. Eqation (8) expresses the flatness-based command voltage for both combs, where only the positive deflection side of the capacitance derivative C and negative 1 deflection side of C are tilized for torqe generation: τ, θ eff C1 = τ, θ eff < C. (8) 687

4 At zero crossing point the drive torqe becomes almost zero τ ( θ= ) de to the static deflection characteristics eff ( θ= ) =, (Fig. 9). Therefore, the root fnction in (8) leads to very steep voltage slope, which cases large crrent peaksi = C ɺ de to stray capacitance C, as shown in Fig. 9. As visible from the system model (1) each comb can generate nominally only a nidirectional torqe according to the sqare-law for voltage control. However the special characteristics of the comb capacitance derivative fnction implies a simltaneos dal torqe athority with opposing torqes by both combs arond zero mirror deflection (cf. Fig. 5a : negative deflections for comb 1, positive deflections for comb ). Therefore opposing torqes can be applied by both combs offering the advantage of two-sided actation and incorporating an additional degree of freedom for control design. Crrent limitation for capacitive loads can be achieved in general by limiting the command voltage slope. For a given imm slope ɺ the original command voltage trajectory can be extrapolated early at a imm voltage slope (cf. Fig. 1). Starting at the initial voltage, that, first exceeds the slope limit ɺ, the ear command voltage law can be stated as: = ɺ t+,, d t t*, t* : ( t*) > ɺ dt. (9) For a ear command voltage at comb 1 the opposing,,1 comb drive voltage mst be applied as, 1 τ, eff,,1 C C C =. (1) to retain the torqe balance of the mechanical system (1) and to follow the demanded deflection trajectory (eqivalent comptations for comb ). The reslting command voltage trajectory, shown in Fig. 1, can comprehend larger voltage derivatives for the opposite comb drive cased by different capacitance characteristics for both combs. Nevertheless the final command voltage trajectory is bonded in contrast to the original single-sided comb actation with infinity (very large) slope at the zero crossing point. Fig. 1. Crrent and jerk limited trajectory design with limitation ɺ = 11 kv/s : (a) voltage trajectory with/withot crrent limitation (c.l.), the ear extrapolated parts of trajectory are indicated with bold e; (b) voltage derivatives of pper voltage trajectories The benefits of limiting the drive crrent can be acconted for (i) redcing the power amplifier (voltage generator) reqirements and (ii) diminishing electrical cross-talk, which indicates a problem for on-chip piezoresistive position sensors that are planned to be sed for feedback control. 5. EXPERIMENTAL RESULTS The experimental validation has been condcted with a microscanner [1,] in a test setp shown in Fig. 11. Command voltage trajectories were created with a two channel programmable voltage generator and amplifier. An external optical reference measrement of the mirror deflection angle has been performed with a position sensitive detector (PS). The measrement eqation (11) for the mirror deflection angle is given as: θˆ 1 arcsi k I I d I + I, (11) PS a b = n with the displacement crrents I, I, the calibration factor a b k and the distance d between PS and micro mirror. PS The drive crrent i was measred in the grond wire. a b (a) (b) Fig. 9. Command trajectories: (a) voltage trajectory ( t ), (b) crrent trajectory i ( t ) showing large displacement crrent throgh capacitance C at zero crossing points Fig. 11. Experimental setp for measrement of mirror deflection angle 688

5 5.1. PERFORMANCE OF JERK LIMITATION The performance of jerk-limited trajectory design is verified with triangle trajectories and a variation of the ear scanning area according to Fig. 8. The measrement reslts, demonstrated in Table 1 prove the feasibility of the proposed design approach for decreasing the mirror tracking error. By redcing the imm jerk magnitde the mirror deflection tracking error can be redced considerably. At the same time the repeatability, defined as the deviation from the mean deflection of 1 periods, persists approximately constant at a very small error level. Comparing the jerk-limited design with the polynomial design shown in [9], the jerk-limited trajectory design improves the performance by redcing the imm jerk by 37%, cf. third to last with last row in Table 1. This leads to a smaller error and a better repeatability, mentioning, that different mirror devices were sed for both measrements. Some residal oscillations at the micro mirror oter eigenfreqency of 13 Hz de to navoidable inaccracies of the design model can be recognized in the measred mirror deflections for k = 4% (cf. Fig. 1). Best performances with jerk limited trajectories can be achieved by minimizing the jerk magnitde and as an navoidable side effect the ear scan area. This is in conflict with a imal admissible ear scanning area for the microscanner that is restricted by the geometric micro mirror design. Ths a compromise mst be fond. Relation (7) can be sed for a transparent parameterization of the command deflection trajectory with respect to the admissible error tolerances of the mirror motion. Table 1. Measrement reslts for 1 Hz jerk-limited triangle trajectory at θ = 7 with 1 periods (cf. Fig. 8): peak-topeak error to reference and peak-to-peak repeatability (deviation to mean at each time) ear area jerk limitation 3 peak-to-peak error [deg] peak-to-peak repeatability [deg] **.33 *.75 * * [9, Table 1] (withot standardization to. deflection) ** denotes the imm jerk of trajectory design from [9] 5.. PERFORMANCE OF CURRENT LIMITATION The drive crrent limiting trajectory design is verified by crrent measrement in the grond wire. The exemplary crrent measrement for a jerk-limited, 8% ear, 1 Hz triangle trajectory with the design parameter ɺ = 11 kv/s (imm voltage slope) is shown in Fig. 13. It indicates that the drive crrent is mainly inflenced by the constant stray capacitance C. Ths, the drive crrent can be expressed by: i Cɺ. (1) Figre 13 illstrates the crrent peaks at mirror deflection zero crossing time at t=.5s. A reloading of the comb capacitances is reqired reslting in an nfavorable electrical copg from the spply condctor to the integrated piezoresistive sensors. Fig. 13. Crrent measrement for jerk-limited 8% ear 1 Hz triangle trajectory with limited ɺ = 11 kv/s, θ = 7 and calclated command voltage derivative (1), with C = 43 pf Fig. 1. Measrement reslt of jerk-limited 1 Hz triangle trajectory with 4% ear area and θ = 7 : (a) reference and measrement, (b) error to reference, (c) minimm and imm repeatability of 1 periods The proposed crrent limiting trajectory design achieves the predicted crrent depression as an almost ear fnction of the design parameter ɺ, see measred reslts in Fig. 14. Here, the measrement eqation for the imm crrent is determined by the pper and lower limit of the mean of 1 periods as follows: 1 1 i =, ( i ) min( i ), 1 i = i. (13), i 1 i= 1 689

6 Fig. 14. Measrement reslt of imm crrent i (13), at zero crossing point for variation of imm voltage derivative ɺ Above all the mechanical scanning performance shold not be affected or even worsened. An experimental analysis demonstrates (reslts are shown in Fig. 15) that the mirror tracking error for a jerk-limited 8% ear 1 Hz triangle trajectory with crrent limitation persists in the range of ±.1 at a repeatability of ±.1 (cf. Table 1, third to last row). Finally the dal-comb actation scheme has been implemented sccessflly to limit the drive crrent, bt shows also prospective optimization potentials for ftre feedback control schemes. Fig. 15. Measrement errors for 1 periods of a jerk-limited 8% ear 1 Hz triangle trajectory with variation of. voltage derivative ɺ : (a) peak-to-peak error to reference, (b) peak-to-peak repeatability 6. CONCLUSIONS This paper presented performance improvements for open loop electrostatic micromirror control sing a flatness-based trajectory design with limited jerk and limited drive crrents. The experimental verification proves the adeqacy of the proposed soltions. Althogh the model-based open loop control shows rather satisfying performances there are still some open isses for improving the system performances (redction of tracking errors) and enhancing the robstness. Improving the micromirror model accracy wold be a rather straightforward approach, bt wold be beneficial eventally only for optimized design of a specific micromirror assembly reqiring comprehensive experiments for model parameter extraction (cf. sect. ). In order to minimize individal experiments we are more interested in robst system concepts allowing certain model and parameter ncertainties. To gain robstness for the weakly damped mass-spring eigenmode an artificial damping, either passively throgh resistive impedance feedback [6,9] or actively sing piezoresistive deflection sensors (already integrated on-chip, bt not sed so far) will be investigated frther. For both variants the soltions presented in this paper for lowering of drive crrents are of crcial importance. REFERENCES [1] Thielemann, J., Sandner, T., Schwarzer, S., Cpcic, U., Schmann-Olsen, H., Kirkhs, T., 1. TACO: Threedimensional Camera with Object etection and Foveation, EC FP7 grant no 4863, [] Jng,., Sandner, T., Kallweit,., Schenk, H., 1. Vertical comb drive microscanners for beam steering, ear scanning, and laser projection applications. In: Proc. SPIE 85, MOEMS and Miniatrized Systems XI, 85U (Febrary 9, 1); doi:1.1117/ [3] Ferreira, A., Aphale, S.S., 11. A Srvey of Modeg and Control Techniqes for Micro- and Nanoelectromechanical Systems, IEEE Transactions on Systems, Man, and Cybernetics Part C: Applications and Reviews, Vol. 41, No. 3, pp [4] Singer, N.C.; Seering, W.P., 199. Preshaping Command Inpts to Redce System Vibration. Trans. ASME, J. yn. Syst. Meas. Control, vol. 11, no. 1, 199, pp [5] O, K.-S.; Chen, K.-S.; Yang, T.-S.; Lee, S.-Y., 11. Fast Positioning and Impact Minimizing of MEMS evices by Sppression of Motion-Indced Vibration by Command- Shaping Method. Jornal of Microelectro mechanical Systems, Vol. (11), No.1, pp [6] Janschek, K.; Sandner, T.; Schroedter, R.; Roth, M., 13. Adaptive Prefilter esign for Control of Qasistatic Microscanners. In: Proceedings of 6th IFAC Symposim on Mechatronic Systems Mechatronics 13, April 1-1, 13, Hangzho, China, doi:1.318/ cn-34.11, pp [7] Fliess, M., Lévine, J. and Rochon, P., Flatness and defect of nonear systems: Introdctory theory and examples, International Jornal of Control, 1995, vol. 61, pp [8] Zh, G.; Levine, J.; Praly, L.; Peter, Y.-A., 6. Flatness- Based Control of Electrostatically Actated MEMS With Application to Adaptive Optics: A Simlation Stdy. J.of Microelectromechanical Systems, Vol. 15 (6), Isse 5, pp [9] Janschek, K.; Schroedter, R., Sandner, T.; 13. Flatness- Based Open Loop Command Tracking for Qasistatic Microscanners. Paper WeAT3.1, 13 ASME ynamic Systems & Control Conference (SCC), October 1-3, 13, Stanford University, CA, USA. [1] Kyriakopolos, K. J.; Saridis, G. N.; Minimm jerk path generation, in Proc. IEEE Int. Conf. Robotics and Atomation, vol. 1, Philadelphia, PA, 1988, pp [11] Hogan, N.; An organizing principle for a class of volntary movements. The Jornal of Neroscience, November 1984, Vol. 4, No. 11, pp [1] Nmasato, H.; Tomizka, M.; 3. Settg Control and Performance of a al-actator System for Hard isk rives. IEEE/ASME Transactions on Mechatronics, Vol. 8, No. 4, ecember 3, pp