Nonlinear Stabilization of a Spherical Particle Trapped in an Optical Tweezer

Transcription

1 Nonlinear Stabilization of a Spherical Particle Trapped in an Optical Tweezer Aruna Ranaweera ranawera@engineering.ucsb.edu Bassa Baieh baieh@engineering.ucsb.edu Andrew R. Teel teel@ece.ucsb.edu Departent of Mechanical and Environental Engineering, University of California, Santa Barbara, CA Departent of Electrical and Coputer Engineering, University of California, Santa Barbara, CA Abstract We describe the nonlinear equations of otion of a spherical particle trapped in an optical tweezer and outline the use of nonlinear feedback control to achieve global asyptotic stability of the origin. Control algoriths are derived for both an inertial syste odel and for a reduced order non-inertial odel. The latter is well-suited for practical ipleentation and coputer siulations show that the non-inertial control algorith achieves not only global asyptotic stability, but also a faster settling tie than the open loop syste. We also show that the controller is very effective in reducing rando position fluctuations due to theral noise. I. INTRODUCTION The optical tweezer is a device that uses a focused laser bea to trap and anipulate individual dielectric particles in an aqueous ediu. The laser bea is sent through a high nuerical aperture (highly converging) icroscope objective that is used for both trapping and viewing particles of interest. Several illiwatts of laser power at the focus can generate trapping forces on the order of piconewtons. While tiny by conventional standards, this level of force is well suited for bioolecular studies. Although biological olecules are too sall to be trapped at roo teperature, a olecule can be grasped once a trappable handle, such as a polystyrene bead, is (biocheically) attached to that olecule []. as the laser wavelength [], [2]. Since the ost coonly used laser wavelength in biological eperients is 64 n, this paper will investigate the trapping behavior of polystyrene beads of diaeter. For sall enough displaceents fro the center of the trap (up to approiately 2 n) [3], the optical tweezer can be odelled as a Hookeian spring, characterized by a fied trap-stiffness. For larger displaceents fro the center of the trap (up to the aiu trapping radius R of approiately 675 n), the optical tweezer behaves like a nonlinear restoring spring [3]. In the reaining sections of this paper, we describe the nonlinear equations of otion of the optical tweezer and describe both linear control laws for asyptotic stabilization (AS) of the origin and nonlinear control laws for global asyptotic stabilization (GAS) of the origin. II. NONLINEAR DYNAMICS The equation of otion along the -ais for a trapped bead of ass and lateral position is given by ẍ = F ( r ) + F d (ẋ) + F e (t), () where F ( ) is the optical trapping force, F d ( ) is the viscous drag, and F e ( ) is an eternal disturbance force. The relative position r is defined as r := T, (2) Fig.. Optical Tweezer. Although optical tweezers have been used to trap dielectric particles with diaeters in the range of tens of nanoeters to tens of icrons, strongest trapping is epected for particles that are roughly the sae size Research supported in part by NSF under grant CMS where T is the trap (laser focus) position. For relative displaceents within the trapping radius R, the trap behaves like a nonlinear restoring spring: { α3 F = 3 r α r for r < R (3) otherwise Figure 2 shows a typical trapping force odel in which α 3 = 22 pn/ 3, α = pn/, and R =.675. The aiu restoring force of pn occurs at r = R a = The nonlinear spring constants α and α 3 were obtained by fitting a cubic polynoial to eperiental results published by Sions et al. [3]. The drag force can be epressed as F d = βẋ, (4)

2 Trapping Force F (pn) Relative Displaceent r () Fig. 2. Typical nonlinear trapping force odel for a - diaeter polystyrene bead. where β > is the viscous daping factor given by Stoke s equation, β = 6πη f r, in which r is the bead radius and η f is the fluid viscosity. For a - bead trapped in water at roo teperature, β. pns/. Equations (), (3), and (4) can be cobined to obtain the nonlinear equation of otion for a trapped particle: in which ẍ = δ( r )(α 3 3 r α r ) βẋ + F e (t), (5) δ( r ) := { for r < R otherwise. (6) In practice, the ass of the trapped particle is sall enough that it can be ignored. For eaple, for a - diaeter polystyrene bead, 5.5 g. Therefore, (5) can be siplified to obtain the nonlinear equation of otion for a trapped particle at low Reynolds nuber (i.e., assuing viscous drag doinates inertia): = δ( r )(α 3 3 r α r ) βẋ + F e (t). (7) III. CONTROL DESIGN FOR NON-INERTIAL SYSTEM MODEL By defining the trap position as the control input, u := T, we can epress (7) in state space for as ẋ = δ( u) [ α3 ( u) 3 α ( u) ] + F e β β y =. (8) In the reainder of this section, we will derive feedback control laws to stabilize the origin of the first order syste described by (8) under the assuption of zero eternal disturbance, F e (t) =. A. Asyptotic Stability As long as the particle is not outside of the trapping radius R, the scalar syste ẋ = β F ( r) will be stabilized by the trapping force F ( r ). We will eploit the nature of this trapping force to stabilize the particle. As shown in Figure 3, for an appropriate choice of µ > and λ >, the cubic trapping force F ( ) can be approiated by a hyperbolic tangent function f t ( ): Trapping Force F (pn) F ( r ) f t ( r ) = µ tanh( λ r ) Relative Displaceent r () Fig. 3. Approiation of trapping force F ( r) (solid line) using a hyperbolic tangent function f t( r) (dashed line) for µ = pn and λ = 5. Although F ( r ) behaves like f t ( r ) for r < R a < R, what is iportant is that both functions eert a restoring force within the trapping radius R. Therefore, we can apply a siplified version of the saturation analysis fro [4]. Specifically, for any p >, we can choose a Lyapunov function V () = p 2 and set λ r = bp, r = bp, (9) λ in which the control gain b := β >, according to (8). Then, V () = 2µbp tanh( bp) = 2µy tanh(y), which shows that the origin of the syste is asyptotically stable in the doain r < R [5]. Note that the proof only requires that µ be positive, so it is not necessary for the tanh function f t ( r ) to eactly atch the force profile F ( r ). What atters is that the restoring force for both functions is of the sae sign, which is true within the trapping radius R. Fro (2) and (9), the asyptotically stabilizing linear feedback control law is given by u = ( p λβ ). () The linear control algorith given by () was siulated for syste paraeters β =. pns/, α 3 = 22 pn/ 3, α = pn/, λ = 5, and control paraeter p =., as shown in the left coluns of Figures 4, 5, and 6. According to (9), when () λβ p R, r R =.675. The particle reains otionless (ẋ = ),

3 but away fro the origin ( ) and out of the controller s reach (Figures 5 and 6). It follows that, for the chosen value of p, the AS controller is asyptotically stable for () < λβ p R = The basin of attraction can be increased by decreasing p, but this will result in a slower rate of convergence AS Controller r r u =.3 u T N = T Fig. 4. Siulation of non-inertial controllers with initial position within the linear region, () =.. For coparison, the open loop position is shown as a dotted line. The GAS controller achieves faster settling tie than both the AS controller and the open loop syste AS Controller r 2 4 u = T r = T 2 4 Fig. 5. Siulation of non-inertial controllers with initial position within the nonlinear trapping region, () =.4. The AS controller does not achieve asyptotic stability because the control law drives the relative position outside of the trapping radius. The GAS controller achieves faster settling tie than the open loop syste. B. Global Asyptotic Stability To achieve global asyptotic stability (GAS), we suggest setting the relative position equal to a hyperbolic.5 AS Controller.5 r u = T r = T 2 4 Fig. 6. Siulation of non-inertial controllers with initial position outside of the trapping radius, () =.8. Only the GAS controller achieves asyptotic stability. tangent function: r = ω tanh ( ) λ bp, () in which, < ω < R =.675. By choosing ω within this range, GAS is guaranteed because r will always eist within the region in which the nonlinear restoring force F ( r ) is never zero (ecept at the origin). Furtherore, by picking ω = R a =.3893, in addition to achieving GAS, the particle will also be driven (restricted) into the region in which the nonlinear restoring force F ( r ) is aiized. Fro (2) and (), the globally asyptotically stabilizing nonlinear feedback control law is given by ( ) p = ω tanh λβ. (2) Using the sae syste paraeters as in Section III-A, the nonlinear control algorith given by (2) was siulated for control paraeters p = and ω =.3893, as shown in the right coluns of Figures 4, 5, and 6. The figures show that the GAS controller achieves a faster settling tie than both the open loop syste and the AS controller. In each figure, it is clear that when the particle is far fro the origin, the GAS controller drives the initial relative position to r = R a, which we specified by our choice of ω. This results in the aiu possible restoring force and therefore, iniu settling tie (for a given p). Once the particle has been brought closer to the origin, the relative position is driven towards zero according to the tanh function in (). In practice, the trap position T is actuated using an acousto-optic deflector (AOD) that has position res-

4 olution of better than. n, a useful range of approiately, and bandwidth on the order of several tens of kilohertz [6], [7]. Therefore, the control values shown in the figures lie well within the range of practically achievable trap dynaics. Hence, actuator saturation is not a practical concern for this syste. IV. CONTROL DESIGN FOR INERTIAL SYSTEM MODEL In the previous section, we derived feedback control laws to stabilize the origin of the noninertial syste described by (7). In this section, we will derive stabilizing control laws assuing the ass is large enough that it cannot be ignored, in which case the inertial equation of otion (5) ust be used. Our hope is that these derivations will prove useful in the future when the technology behind optical tweezers evolves such that larger asses can be trapped than is currently possible. By setting u := T and :=, we can epress (5) in state space for as ẋ = 2 ẋ 2 = δ( u) [ α3 ( u) 3 α ( u) ] β 2 + F e y =. (3) As in the noninertial case, we will derive control laws under the assuption of zero eternal disturbance, F e (t) =. A. Asyptotic Stability As before, we can use the restoring force F ( r ) to stabilize the syste, η = Aη + BF ( r ), in which η := [ 2 ] T, and [ ] [ ] A = β, B =. (4) Notice that A is stable, but not Hurwitz and (A,B) is reachable, and therefore stabilizable. Therefore, we can apply the saturation analysis fro [4]. Specifically, if we can find P = P T >, such that A T P + P A = Q, (5) we can choose a Lyapunov function V (η) = η T P η and set Then, λ r = B T P η r = λ BT P η. (6) V (η) = η T Qη + 2µB T P η tanh( B T P η) = η T Qη 2µy tanh(y), which shows that the origin of the η syste is asyptotically stable in the doain r < R [5]. Substituting (4) into (5), it can be shown that the positive [ seidefinite ] atri Q ust take the for Q =, in which q q >. This iplies that the[ positive definite atri β P ust take the for P = p p ] p β ( 2 q + p), in which p >. Therefore, fro (4), λ BT P η = [ p λ + ( ) ] β 2 q + p 2. (7) The epression given by (7) is difficult to apply in practice because is typically any orders of agnitude saller than β. However, by choosing p = k and q = 2(βk 2 p), we can re-paraetrize (7) in a ore convenient for as λ BT P η = λ (k + k 2 2 ), (8) in which k > k 2 > β k to preserve p, q >. Fro (2), (6), and (8), the asyptotically stabilizing linear feedback control law is given by ( u = k ) k 2 λ λ 2. (9) According to (8), for a particle starting at rest, 2 () =, when absolute position () λ k R, r R for all tie and therefore the particle will reain otionless ( 2 = ), but away fro the origin ( ) and out of the controller s reach. B. Global Asyptotic Stability As in the noninertial case, to achieve global asyptotic stability (GAS), we suggest setting the relative position equal to a hyperbolic tangent function: ( ) r = ω tanh λ BT P η, (2) in which, < ω < R. Fro (2), (8), and (2), the globally asyptotically stabilizing nonlinear feedback control law is given by ( ) = ω tanh λ BT P η [ ] = ω tanh λ (k + k 2 2 ). (2) The nonlinear control algorith given by (2) was siulated for syste paraeters = 5.5 g, β =. pns/, α 3 = 22 pn/ 3, α = pn/, λ = 5, and control paraeters k =,

5 k 2 = 2 β k. 4, and ω =.3893 as shown in Figure 7. Since k k 2, this corresponds to position feedback. In fact, the control gains chosen here are roughly equivalent to the GAS control gain (p = ) that was used in Figure 6. It should be noted that since our syste siulation paraeters reflect values that we epect to encounter in an actual eperient, the ass is negligible copared to the drag. Therefore, the inertial syste odel and noninertial syste odel are essentially equivalent, as can be seen by the siilarity between Figures 6 and 7. /s.5 r u = N T Fig. 7. Global asyptotic stabilization of the origin of the inertial syste odel for () =.8 and start fro rest 2 () =. V. STABILIZATION IN THE PRESENCE OF THERMAL NOISE In practice, a particle trapped in an optical tweezer is subject to an eternal Langevin (rando theral) force F e (t) with an average value of zero and a constant power spectru (i.e., ideal white noise force) given by 4βk B T, where k B is Boltzann s constant and T is absolute teperature [8]. For a - bead trapped at roo teperature, the power spectru is approiately Figure 8 shows a siulation of the perforance of the non-inertial GAS controller (2) subject to eternal theral noise. The sapling tie is s, which is typical for optical tweezer eperients. Clearly, the controller is very effective in reducing rando position fluctuations (Brownian otion) due to theral (Langevin) noise. The net effect of the controller is to increase the effective stiffness of the trap. The non-inertial GAS controller is designed to provide global asyptotic stability to the origin assuing the eternal disturbance has an epected value of zero, as in the case of theral noise. According to (7), a sall constant eternal disturbance F e will result in stabilization to soe constant steady state value r α F e = T Fig. 8. Siulation of non-inertial GAS controller subject to eternal theral noise for () = and p =. For coparison, open loop position is shown as a dotted line. Therefore, according to (), the steady state position is given by λβ ( ) p tanh F e, (22) ωα which can be greatly reduced by increasing p. For eaple, Figure 9 shows the perforance of the non-inertial GAS controller (2) subject to eternal theral noise and a constant eternal disturbance of pn. Clearly, the GAS controller is very effective in stabilizing the origin, even with the constant disturbance. According to (22), the steady state position error is.3 n. In theory, the aiu constant disturbance that the controller can reject is slightly less than the aiu restoring force, which is pn for this syste, according to Figure 2. A ore thorough treatent of constant disturbance rejection would require integral feedback, but we will not investigate that in this paper. Although we have built a coplete optical tweezer syste [6], the control algoriths described in this paper have not yet been ipleented. Since the photodetectors used to detect particle position in our eperiental setup cannot efficiently detect relative displaceents of greater than approiately, we will not be able to investigate global stability of the controllers. However, the local asyptotic stabilization properties of the controllers can be investigated within a range of ±.5 of the origin. A coprehensive eperiental verification of GAS would require sophisticated realtie iage processing and video feedback, which is currently unavailable in our laboratory. VI. CONCLUSION This paper describes the global asyptotic stabilization of a spherical particle trapped in an optical tweezer. We have outlined a ethod for achieving GAS for both

6 = T Fig. 9. Siulation of non-inertial GAS controller subject to eternal theral noise and a constant disturbance of pn for () = and p =. a full-order inertial syste odel and a reduced order non-inertial syste odel. For the non-inertial odel, which is well-suited for practical ipleentation, we have obtained a nonlinear control law that achieves not only GAS, but also a faster settling tie than the open loop syste. Global asyptotic stabilization is iportant in biological or icro-iing eperients in which a located particle needs to be oved to a very specific location. We have also shown that the non-inertial GAS controller is very effective in reducing rando position fluctuations due to theral noise. This is especially useful for biological eperients in which the position of a trapped particle needs to be held constant in spite of eternal (theral) disturbances. For a practical optical tweezer syste, the inertial controllers derived in Section IV are unnecessarily coplicated because the trappable ass is tiny. However, these controllers will prove useful in the future if the laser technology behind optical tweezers evolves such that larger asses can be trapped. The inertial analysis outlines a general ethod to obtain a GAS controller for any syste that has the inertial equation of otion given by (5). We have also derived asyptotically stabilizing linear controllers for both the inertial syste odel and the noninertial syste odel. Although these AS controllers have a liited basin of attraction, they are easier to ipleent than the nonlinear GAS controllers. For alost a decade, the linear trapping behavior of optical tweezers has been used to quantify forces acting on a trapped particle for sall displaceents fro the center of the trap. For the ost part, the nonlinear trapping region has not been used quantitatively, ecept to obtain a rough estiate of the aiu trapping force [7]. The control laws presented in this paper represent an attept to quantitatively eploit the entire nonlinear profile of the restoring force of an optical tweezer. Since the nonlinear trapping region for a - polystyrene bead is approiately three ties as large as the linear region [3], it is our hope that the algoriths described in this paper will be of value to the any users of optical tweezers. VII. REFERENCES [] S. Chu, Laser anipulation of atos and particles, Science, vol. 253, pp , 99. [2] A. D. Mehta, J. T. Finer, and J. A. Spudich, Reflections of a lucid dreaer: optical trap design considerations, in Methods in Cell Biology (M. P. Sheetz, ed.), vol. 55, ch. 4, pp , Acadeic Press, 998. [3] R. M. Sions, J. T. Finer, S. Chu, and J. Spudich, Quantitative easureents of force and displaceent using an optical trap, Biophys. J., vol. 7, pp , 996. [4] E. Liu, Y. Chitour, and E. Sontag, On finite gain stabilizability of linear systes subject to input saturation, SIAM Journal on Control and Optiization, vol. 34, pp. 9 29, July 996. [5] H. K. Khalil, Nonlinear Systes. Prentice Hall, 2nd ed., 996. [6] A. Ranaweera and B. Baieh, Calibration of the characteristic frequency of an optical tweezer using an adaptive noralized gradient approach, in Proceedings of the 23 Aerican Control Conference, (Denver, CO), pp , IEEE, June 23. [7] K. Visscher, S. P. Gross, and S. M. Block, Construction of ultiple-bea optical traps with nanoeter-resolution position sensing, IEEE J. Select. Topics Quantu Electronics, vol. 2, no. 4, pp , 996. [8] F. Gittes and C. H. Schidt, Signals and noise in icroechanical easureents, in Methods in Cell Biology (M. P. Sheetz, ed.), vol. 55, ch. 8, pp , Acadeic Press, 998.