Reference Trajectory Generation for an Atomic Force Microscope

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1 Mechatronics 2010, June 28-30, Swiss Federal Institute of Technology ETH, Zurich Switzerland 1 Reference Trajectory Generation for an Atomic Force Microscope Ivan Furlan and Silvano Balemi 1 Department of innovative technologies, University of Applied Sciences of Southern Switzerland (SUPSI), 6928 Lugano-Manno, Switzerland. ABSTRACT This paper addresses the design of the scanning trajectory for a portable Atomic Force Microscope with large scanning range and high resolution. The proposed reference scanning trajectory presents a low motion bandwidth and is well suited for a discrete-time implementation. The dependence of the trajectory parameters in function of the scanning speed for different scaling policies, has been analyzed. An optimal choice of scaling policy is then proposed. 1. Introduction Atomic force microscopy is typically performed by scanning the considered surface forwards and backwards along many parallel lines while a constant distance between the scanning head and the surface is maintained (see Fig. 1) y ais (m) ais (m) 10 4 Figure. 1: Trajectory for scanning a surface (units in meters). In contact mode, the constant distance between the scanning head and the surface is obtained by controlling the deflection ξ of the cantilever, which remains in contact with the surface. A sketch of the motion in vertical direction (when travelling along a line on the surface) is shown in Fig. 2. The Atomic Force Microscope (AFM) considered in this work moves along the trajectory thanks to a two-degree-offreedom platform based on fleures and actuated by voice-coil motors [1, 2]. In the desired trajectory the scanning head accelerates until the beginning of the first scanning line, then proceeds forwards at a constant scanning speed until the end of the scanning line, thes direction and travels again at the same speed along the same scanning line but backwards until the beginning of the scanning line, where the heads performs simultaneously a turn and a move sidewards to the net scanning line and so on until the last scanning line. This trajectory differs from those considered in other works (e.g. [3]) in that only the transitions between scanning lines can be designed, while the trajectory along the scanning lines is determined by the constant scanning speed. The acceleration/deceleration and the move to the net line must occur rapidly (in order to reduce overall measurement time) but also smoothly (in order to reduce the perturbations of the motion along the scanning line). Moreover, 1 Corresponding author: silvano.balemi@supsi.ch

2 2 12th Mechatronics Forum Biennial International Conference.. z ξ = z w. Cantilever tip. Cantilever w Figure. 2: Atomic force microscope scanning the surface along a line: ξ cantilever deflection, w surface profile, z head elevation. the motion between scanning lines must be easily computable and easily scaled both in size (scanning of surfaces of different sizes) as well as in speed (scanning with different speeds). In section 2 a simple trajectory is proposed which guarantees a smooth transition between scanning lines. The values of the different trajectory parameters are given as a function of the scanning speed v and of a parameter ω. In section 3 the dependence of the trajectory parameters as a function of the scanning speed for different scaling policies is analyzed. An optimal choice of scaling policy is proposed. In section 4 implementation issues are discussed. Conclusions are given in section Trajectory definition and design 2.1. Trajectory definition In order to guarantee smooth transitions between scanning lines the trajectory and its derivatives should be continuous. The ideal choice would include a continuous jerk (fourth derivative of the trajectory). However, the realization of such a trajectory would require a large amount of computations, both off-line (computation of the trajectory parameters) and on-line (eecution of the trajectory). The trajectory chosen in this work only guarantees a continuous acceleration but not a continuous jerk. The transition between two lines is described by the following equation:» (t) y(t)» = R sin (ω t) v side t r sin(2 ω t)» o + y o The trajectory begins at the position ( o, y o) (here for a simpler analysis at time t = 0) and ends at position ( o, y o + v side t turn) where t turn = /ω is the time needed to turn and to move to the net line. In the sequel the transition from one scanning line to the net one is analyzed. Later the other motions before or after a scanning line will be considered Turn and move sidewards When the head changes direction and moves to another scanning line, the speed at the end of the scanning line is v while the initial speed along the net scanning line should be v. Thus the following conditions must be satisfied (1) (0) = o, (t turn) = o ẋ(0) = v, ẋ(t turn) = v y(0) = y o, y(t turn) = y o + ẏ(0) = 0, ẏ(t turn) = 0 (2) where is the distance between the two scanning lines. With the motion equation (1) and conditions (2), the following relations are obtained From equation (3) and with t turn = /ω the mean speed v side is v side t turn = (3) R ω = v (4) v side 2 ω r = 0 (5) v side = = ω t turn (6)

3 Mechatronics 2010, June 28-30, Swiss Federal Institute of Technology ETH, Zurich Switzerland 3 From equations (4), (5) and (6) the radii R and r are R = v ω r = 2 (7) (8) This means that r only depends on the distance between two scanning lines. A sample trajectory during a turn/move motion and the respective speed are shown in Fig. 3. Note that the acceleration at the beginning and at the end of the motion is zero ,y [m] d/dt, dy/dt [m/s] t [s] t [s] Figure. 3: Left: trajectory between two lines: ais (continuous line) and y ais (dashed line). Right: speed between two lines: ais (continuous line) and y ais (dashed line) Turn At the end of a scanning line the head must be decelerated and accelerated backwards. The motion is similar to that of the case Turn and move sidewards, but with the parameters r = 0 and R being negative (no motion along the y ais) Accelerate At the beginning of the first scanning line the head must be accelerated. The motion is similar to that of the case Turn and move sidewards, but with the parameter r = 0 and with half of the motion duration (from t = t turn/2 to t = t turn) Decelerate At the end of the last scanning line the head must be stopped (decelerated). The motion is similar to that of the case Turn and move sidewards, but with the parameter r = 0 and with half of the motion duration (from t = 0 to t = t turn/2). The whole trajectory can thus be generated in a simple manner: the head travels at a constant speed on the scanning lines, the transitions are all governed by the same formula (1) with the necessary parametrizations Trajectory design In the trajectory design the following limitation on the maimal acceleration along the /y aes must be considered ẍ(t) a ma ÿ(t) a ma where a ma is the maimal available acceleration provided by the head actuators. From this limitation the following conditions are obtained ẍ(t) = R ω 2 a ma ÿ(t) = 4 r ω 2 (9) a ma

4 4 12th Mechatronics Forum Biennial International Conference 3. Analysis of the Scaling policies as a function of the scanning speed In this section the evolution of the different trajectory parameters is analyzed for different values of the scanning speed v. Of course the constraints of equation (9) must be satisfied. Note that the maimal acceleration a ma in equation (9) is proportional to the magnetic field generated by the voice-coil motors and thus proportional to the motor currents. However, the currents are limited by the power supply voltage. Therefore, the relation between the power supply voltage and a ma resp. ω has also to be determined Power supply voltage limitations In this section the actuation voltage of the drivers necessary for following the desired trajectory is determined. In order to do this the motion equations for the different parts of the trajectory are determined (accelerate: ACC; forward movement: FW; turn direction: TD; backward movement: BW and move sidewards: MS). Only the first scanning line is considered here. ACC(t) = R cos (ω t) + first FW(t) = R ω `t ω + first TD(t) = R sin `ω `t t ` ω line + FW + t ω line BW(t) = R ω `t 3 t ` ω line + FW + t ω line MS(t) = R sin `ω `t 3 2 t ω line + first y ACC(t) = y FW(t) = y TD(t) = y BW(t) = y first y MS(t) = v side t sin (2 ω t MS) + y first where first, y first = first point of the scanning surface [m] t line = time for covering a scanning line [s] t MS = t 3 ω line If the damping is assumed to be negligible, the relationship between the trajectory in (and similarly in y) and the driver actuation voltage is given by U (s) = X (s) K F2i `s 2 m + k f (s LM + R M) (10) where K F2i = Force to current constant [A/N] 8>< k f = Fleure spring constant [N/m] m = Moving mass [Kg] L M = inductance of the motor [H] >: R M = resistance of the motor [Ω] With equation (10) and the equations for the whole trajectory along the and y aes, the applied voltage at the voice-coil actuators as a function of time can be determined. The maimal supply voltage is needed when the scanning head turns and moves sidewards. The equations for the maimal voltages in this case are u (t) = K F2i R `k f m ω 2 (L M ω cos (ω t) + R M sin(ω t)) + K F2i k f R M u y(t) = K F2i R `4 m ω 2 k f (2 LM ω cos (ω t) + R M sin(ω t)) +K F2i R M k f v side t + y K F2i k f R M + L M K F2i k f v side From these equations, approimated formulas showing the general dependence of the supply voltage from the variable ω can be derived u (t) K F2i L M m R ω 3 u y(t) 8 m KF2i ω Use of maimal acceleration The atomic force microscope must be able to scan large and small surfaces at different scanning speeds (fast speed for smooth surfaces, slow speed for rough surfaces). Then, in order to uniquely determine the trajectory for a given v, one more variable, for instance the parameter ω, must be specified. From (9) and equation (7) the new epression v ω = a

5 Mechatronics 2010, June 28-30, Swiss Federal Institute of Technology ETH, Zurich Switzerland 5 Table 1: Dependencies of quantities in function of scanning speed v and of the maimal acceleration a = k v n. a t turn ω R û û y a y v y k v n k v n+1 k v n 1 1 k v n+2 k v 2n 1 k v 3n 3 2 k v 2n 2 2 k v n 1 indicates that, as an alternative, the maimal acceleration a = ẍ(t) can be specified (which i determines the angular speed ω). A simple choice consists in always eploiting the maimal acceleration/deceleration (or equivalently, the maimal force) in order to reduce the overall measurement time. If the maimal acceleration a = ẍ(t) is assumed to be the same for all scanning speeds v, then ω = ẍ(t) /v increases for decreasing scanning speed v. Also, it means that the side motion speed v side = 2 ẍ(t) /(v ) increases linearly with decreasing v (see equation (6)) but also that the necessary voltage û y = ẍ(t) 3 /v 3 for the side motion increases with a power of three! This policy is obviously useless in practice Policy analysis In order to define uniquely the trajectory for a given value of v, different dependences of the acceleration a in function of the speed v can be analyzed. In particular the parametrization a = k v n in function of k and n is investigated. In Table 1 the epression for various quantities in function of v and of the parameters k and n are given. Different cases can be identified: n = 0: Constant maimal acceleration ẍ(t), n = 1/2: Constant maimal voltage û, n = 1: Constant angular speed ω, constant turn time, constant maimal lateral speed and acceleration for all speeds v, n = 2: Constant radius R. Because the scanning speed may be arbitrarily high, no quantity (voltage) should increase with decreasing speed, that is no quantity in Table 1 may show negative eponent of v. The conditions on ω, û, û y, a y, v y, a are all satisfied with n 1, the conditions on t turn and R are satisfied for n 1, which leads to n = Implementation The generation of the scanning trajectory was implemented in a digital controller at high sampling frequencies (40kHz). Thus an efficient discrete-time implementation had to be developed Generation of the trajectory In order to generate the parts of trajectory describing the transition between two scanning lines or the other motions before or after a scanning line, equation (1) was implemented using the discrete-time state-space equations of the two second order oscillator systems s s = Φ s y s k+1 y k sq = Φ 2 sq s qy s qy and the epression» y k+1» s = C turn s y k k+1» sq + C side s qy k k» 0 + q o k + y o

6 6 12th Mechatronics Forum Biennial International Conference where Φ = C turn = C side = cos(ω ts) sin(ω t s) sin(ω t s) cos(ω t s) R r and where t s is the chosen sampling time. Ideally, after a given number of samples starting from ( o, y o), the correct initial point of the net scanning line should be reached eactly. This implies that t turn for the sideways and for the simple turn motion but also that t turn/2 for the deceleration and the acceleration motion be a multiple of the sampling time t s (see Fig. 4) 2 R = r ( o,y o ) Then t turn = t s with Figure. 4: Eample of resulting trajectory with discretization of the trajectory. tturn = 2 1 = 2 2 t s 2 ω t s Note that also t turn = t 2 2 s is also a multiple of t s since by construction is even. Then, the circular frequency ω becomes ω = t s and the radius R Finally, the rotation matri Φ becomes 4.2. Specific cases Φ = R = v ω 2 4 cos sin = v ts nturn sin cos For the different cases presented above it suffices to correctly initialize the initial conditions of the states [s, s y] T, [s q, s qy] T to generate the trajectory. Turn and move sidewards 3 5 Turn: Accelerate: [s, s y] T = [0, 1] T [s q, s qy] T = [1, 0] T q = [s, s y] T = [0, 1] T [s q, s qy] T = [0, 0] T q = 0 [s, s y] T = [1, 0] T [s q, s qy] T = [0, 0] T q = 0

7 Mechatronics 2010, June 28-30, Swiss Federal Institute of Technology ETH, Zurich Switzerland 7 Decelerate: [s, s y] T = [0, 1] T [s q, s qy] T = [0, 0] T q = 0 Note that C turn and C side do not change (R and r remain positive) Results The method has been etended in a straightforward manner to the case when then trajectory is rotated with respect to the and y orthogonal aes. A real case is presented in Fig. (5) where the reference trajectory is shown together with the actual trajectory obtained with the AFM. The deviation is due to fairly strong couplings between the aes and to unmodeled plant dynamics. The influence of the non-ideal reference trajectory is not relevant y ais (m) ais (m) Figure. 5: Measurement of the scanning motion of the AFM with the new reference trajectory generation method. Units in meters, scanning time per line about 50ms. 5. Conclusions In this paper a reference trajectory generation method and a scaling policy for adaptations to changes in the scanning speed have been proposed. The method can be easily implemented and requires only simple computations, making it suitable also for hardware implementations (e.g. in FPGAs). References [1] S. Henein, C. Aymon, S. Bottinelli, and R. Clavel. Articulated structures with fleible joints dedicated to high precision robotics. In Proc. of International Advanced Robotics Program; Workshop on Micro Robots, Micro Machines and Systems. Moscow, Russia, November, 1999.

8 [2] S. Balemi, J. Moerschell, J-M. Breguet, D. Brändlin, S. Bottinelli, and I. Beltrami. Surface inspection system for industrial applications. In Proc. of Conf. on Robotics and Mechatronics, pages , Aachen, Germany, September [3] A.J. Fleming and A.G. Wills. Optimal input scaling for bandlimited scanning systems. In Proc. of 17th IFAC World Congress, Seoul, Korea, July [4] S. Devasia, E. Eleftheriou, and S.O.R. Moheimani. A survey of control issues in nanopositioning. IEEE Transactions on Control Systems Technology, 15(5):802823, September

Santosh Devasia Mechanical Eng. Dept., UW

Nano-positioning Santosh Devasia Mechanical Eng. Dept., UW http://faculty.washington.edu/devasia/ Outline of Talk 1. Why Nano-positioning 2. Sensors for Nano-positioning 3. Actuators for Nano-positioning