Phase-Shift Master-Slave Mechanisms for High Angular-Speed Wedge-Prism Systems

Transcription

1 International Journal of Optomechatronics ISSN: (Print) (Online) Journal homepage: Phase-Shift Master-Slave Mechanisms for High Angular-Speed Wedge-Prism Systems Olivier Chappuis & Reymond Clavel To cite this article: Olivier Chappuis & Reymond Clavel (2013) Phase-Shift Master-Slave Mechanisms for High Angular-Speed Wedge-Prism Systems, International Journal of Optomechatronics, 7:1, 15-32, DOI: / To link to this article: Published online: 13 Feb Submit your article to this journal Article views: 203 Full Terms & Conditions of access and use can be found at

2 International Journal of Optomechatronics, 7: 15 32, 2013 Copyright # Taylor & Francis Group, LLC ISSN: print= online DOI: / PHASE-SHIFT MASTER-SLAVE MECHANISMS FOR HIGH ANGULAR-SPEED WEDGE-PRISM SYSTEMS Olivier Chappuis and Reymond Clavel Laboratory of Robotics Systems (LSRO), Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland Laser micro-machining requires high dynamic laser spot trajectory and accuracy to control the laser beam scanning. A well-known technique to scan a laser beam over a specimen is to use a pair of wedge-prisms. However, it is difficult to master-slave the phase-shift between two rotating prisms at high angular-speed. We present here two drive mechanisms that decouple the phase-shift control and the angular-speed control. These mechanisms simplify the required control architecture and are suitable to achieve a high-dynamic trajectory. These concepts are based on differential timing belts and gears mechanisms that modify the phase-shift between the prisms without interrupting the rotation. This article focuses on the kinematic and mechanical design aspects of such mechanisms. Keywords: differential mechanisms, master-slave mechanisms, wedge-prisms 1. INTRODUCTION Laser welding, engraving, or machining require to raster a beam over a specimen. These processes are based on spot trajectories that have to meet ever increasing dynamics and accuracy requirements, which generate more complex motion control. It is sometime possible to reduce the controller architecture complexity by introducing new mechanical designs. This article explores various mechanical designs for continuous scanning of a laser beam. More specifically, we explore the generation of continuous circular spot trajectories. Of particular interest is the generation of variable-diameter circles at high angular-speed without machining interruption (Ashkenasi et al. 2011). A simple method to achieve these particular trajectories consists in using paired-wedge-prism configuration and in controlling the phase-shift between the prisms. This article first introduces the optical model of the paired-wedge-prism system emphasizing the motion control issues. In the second part, two phase-shift masterslave mechanisms that simplify the motion control are presented. These mechanisms are alternatives to complex high angular-speed phase-shift controllers. Conclusions and outlook will be presented at the end of this work. Address correspondence to Olivier Chappuis, Laboratory of Robotics Systems (LSRO), Ecole Polytechnique Fédérale de Lausanne (EPFL), Station 9, CH-1015, Lausanne, Switzerland. olivier.chappuis@epfl.ch 15

3 16 O. CHAPPUIS AND R. CLAVEL NOMENCLATURE D E pl E u f i G i H i 0 i p1 i p2 n r r enc R T T s W i Wedge-prism diameter Estimation of maximal angular error due to controller position loop Maximal measured phase-shift error Encoder interpolation rate Designation of mechanism Gears Number i, for i ¼ 1,...,4 Distance between double-prism system output and focal plane Speed ratio between Shift Motor and differential planetary gear Speed ratio between Speed Motor and Prism 1 Speed ratio between differential planetary gear and Prism 2 Refractive index Offset at the output of double-prism system Encoder resolution Radius of the laser spot circular trajectory Wedge-prism thickness PID position loop sampling period Designation of mechanism Pulley Number i, for i ¼ 1,...,6 Z 0 Distance between parallel planes of prisms Z i Teeth number of Gears Number i, for i ¼ 1,...,4 a Wedge angle b Laser beam parasitic polar angle c Laser beam parasitic azimuthal angle D x Displacement of linear cart D X Laser beam offset along X-axis D Y Laser beam offset along Y-axis h Nominal deflection angle of wedge-prism h s Deflection angle at the output of paired-wedge-prism system q Radius of differential timing belt mechanism pulleys u Angular phase-shift between prisms of paired-wedge-prism system x Prism angular-speed x 0 Speed Motor angular-speed x 4 Angular-speed of Gears 4 of differential planetary gear x enc Encoder angular-speed x mot Angular-speed of Brushless DC Motor used for tests Prism 2 angular-speed x p2 2. PAIRED-WEDGE-PRISM MODEL 2.1. Beam Deflection A wedge-prism is an optical component, which deflects a laser beam with a constant angle h. It is well-adapted for beam steering applications. Figure 1 shows Figure 1. Wedge-prism with dimensions used in this work (color figure available online).

4 PHASE-SHIFT MASTER-SLAVE MECHANISMS 17 Table 1. Wedge-prism dimensions Dimension Thickness Diameter Deflection angle Prism angle Value T ¼ 3mm D ¼ 22.4 mm h ¼ 2 a ¼ 3.86 the type of wedge-prisms used in this work. Corresponding wedge-prism dimensions are available in Table 1. If the wedge angle a is small, the deflection angle h for normal incidence is given by the following approximation where n is the refractive index (typically for glass for a 632 nm wavelength): h ¼ðn 1Þa ð1þ When the wedge-prism rotates, the laser beam describes a circle on the focal plane. The deflection angle can be modified by using a paired-wedge-prism configuration (Ostaszewski et al. 2006), as presented in Figure 2. Introducing a second wedge-prism modifies the deflection angle h S (Figure 2). The angle h S (u) can be modified by controlling the phase-shift u between the two wedges prisms. Table 2 presents two particular configurations of relative positioning between the wedge-prisms. The deflection angle h S is calculated when the slanted planes of both prisms are parallel (phase-shift u ¼ 0 ) and anti-parallel (phase-shift u ¼ 180 ). Between these two configurations, the deflection angle at the output of the paired-wedge-prism system is given by a numerical model, which is established by using three-dimensional Snell s law (Ostaszewski et al. 2006). Deflection angle at the double-prism system output (h S ) versus angular phase-shift (u) is shown in Figure 3. Figure 2. Paired-wedge-prism configuration. Output deflection angle h S depends on angular phase-shift u between the two wedges prisms (color figure available online).

5 18 O. CHAPPUIS AND R. CLAVEL Table 2. Prisms angular position and deflection angle Phase-shift u Relative positioning Sketch Angle h s h S ¼ 0 Phase-shift u ¼ 0 General phase-shift u ¼]0 :180 [ See theoretical model h S ¼ 2 h Phase-shift u ¼ 180 Nominal wedge-prism deflection angle h is calculated according to Equation (1). The radius R(u) of the circular trajectory on the focal plane is directly linked to the deflection angle h S by the equation (2): R ð/ Þ ¼ H tanðh s ð/ ÞÞþr ð/ Þ ð2þ where H is the distance between the output of the double-prism system and the focal plane (Figure 2). The function r(u) is the offset at the output of the paired-wedgeprism system induced by beam deflection generated by the first prism. Notice that r(u) is proportional to the distance Z 0 between both prisms (Figure 2) Beam Deflection Experiment This section presents the experiment and the corresponding methodology to validate the theoretical model of the laser beam deflection derived in the previous section (Figure 3).

6 PHASE-SHIFT MASTER-SLAVE MECHANISMS 19 Figure 3. Laser beam deflection angle h s versus angular phase-shift u (for a wedge-prism deflection angle h ¼ 2 ) (color figure available online) Experimental setup. The setup design is shown in Figure 4. The spot displacement induced by the two prisms compared to the spot position without displacement (i.e., without prisms) is measured on a CMOS sensor. The angular position of the prism holder 1 (which holds the first prism) can be modified compared to the angular position of the prism holder 2. The phase-shift angle between the prisms can be set with known angle by inserting an indexing pin in one of the 19 indexing holes, separated by steps of 10. The theoretical model (Figure 3) considers that the incident laser beam is perfectly perpendicular to the prism surface and exactly centered. These alignment conditions are difficult to achieve, in particular with respect to the incident angle. The manual alignment of optical components like mirrors and laser generates parasitic angles and deviation of the incident beam, which is no longer perpendicular and centered compared to the first prism. In order to take into account these errors in the Figure 4. Test setup for the laser beam deflection by wedge-prisms. The wedge-prisms are mounted on Prism holder 1 and 2 (color figure available online).

7 20 O. CHAPPUIS AND R. CLAVEL Figure 5. Parasitic angles and displacement offsets of the incident beam on the input prism (color figure available online). theoretical model, offsets (DX and DY induced by centering error) and parasitic angles (c and b induced by mirror orientation errors) are introduced in the model. Displacement offsets and parasitic angles are shown in Figure Experimental procedure. The spot displacement corresponds to the distance between the center of the spot without deflection, and the centre of the spot with deflection induced by phase-shifted prisms. The spot position is measured on a CMOS sensor as shown in Figure 6. The optical spot center is determined by calculating the maximum of a theoretical perfect Gaussian, which best fits the acquired spot intensity data. Calculation of R(u) requires a Gaussian fitting needing the acquisition of the spot pixels intensity and the coordinates on the CMOS sensor. Figure 6. Displacement measuring procedure for H ¼ 14 mm and Z 0 ¼ 20 mm (color figure available online).

8 PHASE-SHIFT MASTER-SLAVE MECHANISMS 21 Figure 7. Measured intensity of each spot pixel (color figure available online). Figures 7 and 8 show the measured intensity value of each spot pixel and the fitted Gaussian respectively Laser spot deflection measurement. The spot displacement R(u) versus prisms angular phase-shift according to setup shown in Figure 4 is presented in Figure 9. The adjusted theoretical model plotted in Figure 9 integrates the following offsets and parasitic angles: DX ¼ 2 mm, DY ¼ 2 mm, c ¼ 0.2 and b ¼ 0.1 This set of parameters, which minimizes the RMS error between the measured curves and the adjusted theoretical model, is calculated by iterations using 3-D Snell s law model. With these error parameters, we can observe in Figure 9 that the adjusted theoretical model fits well the measurements. Furthermore, the value of each error parameter is consistent with the setup capabilities where the laser and the mirror are manipulated with a micrometer resolution. Figure 10 presents the error in mm measured between the adjusted theoretical model and the two sets of measurements. Figure 8. Perfect Gaussian fitted on measured data (color figure available online).

9 22 O. CHAPPUIS AND R. CLAVEL Figure 9. Adjusted theoretical model of the spot displacement and measurements versus angular phaseshift between prisms (color figure available online). The error for a phase-shift between 30 and 180, is less than 4 mm, which is about the dimension of 1 CMOS pixel (dimension : 4.2 mm 4.2 mm). The error for a phase-shift between 0 and 30 increases up to 16 mm (about 4 pixels). This deviation can easily be explained by interference fringes observed when the prisms phase-shift is small (Figure 11, right). These interference fringes are generated by internal reflections between the prisms slanted planes, when they are nearly parallel (i.e., when the phase-shift is close to 0 ). To minimize these interference fringes, prisms with anti-reflective coating adjusted for the Laser wavelength can be used. This interference effect decreases the Gaussian fitting accuracy, which in turn decreases the accuracy of the spot center calculation. Figure 10. Error between adjusted theoretical model and measurements (color figure available online).

10 PHASE-SHIFT MASTER-SLAVE MECHANISMS 23 Figure 11. Spot pictures acquired by the CMOS sensor. Links: no interference fringes. Right: interference fringes (color figure available online). 3. PRISMS MOTION CONTROL The radius R of the circular trajectory can be modified by changing the angular phase-shift between the prisms. For high angular-speed and high accuracy requirements on the angular phase-shift and the angular-speed stability, several drive mechanism can be proposed. As case study, we consider the specifications in Table 3. A straightforward approach to drive a system of prism (Ostaszewski et al. 2006) is shown in Figure 12. In this configuration, each prism is actuated by a single motor. For master-slave applications, where both position and speed have to be controlled, the commonly used controller architecture is shown in Figure 13. This type of architecture is well adapted to control a paired-wedge-prism configuration at low angular-speed (Ostaszewski et al. 2006). In this case, the two motors are position-controlled. A trajectory generator applies a position profile with the desired phase-shift to both prisms. The motors drivers are synchronized in order to apply the position profiles nearly at the same time (jitter time under 100 ns). Such architecture exhibits some limitations for high angular-speed applications. To illustrate this, we consider two high-accuracy state-of-the-art encoders (Heidenhain ERO1480 sin=cos signals) mounted on the motors. The main encoder specifications are listed in Table 4. The estimation of the maximal angular position error between the two prisms is detailed in Table 5. This estimation takes into account the error due to position loop (E pl ), which quantifies the influence of all external perturbations (frictions, pulleys eccentricities, different bearings and timing belts preloads, different motors, unbalances due to prisms, etc.). Considering the PID position loop sampling period of T s ¼ 20 ms and an encoder angular-speed of x enc ¼ rpm, we estimate for each Table 3. Case study specifications Dimension Prism angular-speed range Prism angular-speed stability Prism phase-shift range Prisms phase-shift accuracy Value Up to rpm 2 rpm 0 to

11 24 O. CHAPPUIS AND R. CLAVEL Figure 12. Two motors drive solution cut view. The prisms are mounted on toothed pulleys (Prism 1 and Prism 2). Each toothed pulley is driven by a timing belt (color figure available online). Figure 13. Typical control architecture for phase-shift slaving at low angular speed (Ostaszewski et al. 2006) (color figure available online). Table 4. Encoder specifications Encoder parameter Value Resolution r enc ¼ 512 inc=turn Accuracy Interpolation rate f i ¼ 1024

12 PHASE-SHIFT MASTER-SLAVE MECHANISMS 25 Table 5. Maximal angular position error estimation Error sources Estimation of maximal error E pl due to position loop (following error and instability : 2x 250 interpolated increments in 20 ms at rpm) Maximal error due to jitter on PWM (100 ns at rpm) Maximal error due to encoder accuracy ( ) Total Value motor the maximal error E pl generated by external perturbations as 10% of the angular displacement performed by the motor during the sampling period without any perturbation as shown in the following: E pl ¼0:1 x enc 60 T s r enc f i ffi250 interpolated increments ð3þ We consider here the worst case for the error estimation. However, at high angular-speed, the inertias will filter the external perturbations and reduce the dynamical errors. Nevertheless, it is difficult to estimate theoretically this effect. Estimation of Table 5 shows that this control architecture is not really adapted to very high angular-speed applications. In order to validate this error estimation, we measure the phase-shift error between two Brushless DC motors (Maxon EC-max30, 60 W) coupled with Heidenhain encoders (see parameters in Table 4) and rotating with load (Figure 12). Phase-shift is controlled by the controller architecture detailed in Figure 13. Phase-shift error is measured with numerical oscilloscope and calculated by subtracting measured phase-shift (extracted from encoder s signals) from phase-shift set point. Figure 14 shows the results of phase-shift error measurement between motors rotating at constant angular-speed x mot ¼ rpm. Figure 14. Measurement of phase-shift error between two Brushless DC motor with load, rotating at x mot ¼ rpm. The phase-shift is controlled by the controller architecture presented in Figure 13. The blue curve is calculated by subtracting measured phase-shift from phase-shift setpoint. This measurement has been performed at the CSEM (Centre Suisse d Electronique et de Microtechnique, Neuchâtel) (color figure available online).

13 26 O. CHAPPUIS AND R. CLAVEL By analyzing results in Figure 14, we observe a maximal phase-shift error E u of 0.315, which is consistent with error estimation presented in Table 5. In this case, the maximal angular position error does not meet the specifications on phase-shift accuracy given in Table 3. Although this solution is the easiest one regarding the mechanical design, it is not well adapted for high-speed trajectories. 4. PHASE-SHIFT MASTER-SLAVE MECHANISMS A master-slave mechanism to control the phase-shift between both prisms may be implemented to simplify the controller. In this case, the control loop only controls the angular-speed of the paired prisms, whose angular phase-shift is mechanically slaved. An advantage of this approach is that phase-shift and angular-speed controls are decoupled. The controller architecture is significantly simplified as shown in Figure 15. Two phase-shift master-slave mechanisms for high-angular-speed applications are presented in the following sections. The approach considered in these mechanisms permits a dynamic modification of the phase-shift Differential Timing Belt Mechanism This mechanism is inspired by the differential drive mechanism in Figure 16, patented by Gary R. Paulson (Paulson 1975). If motors 13 and 15 turn at equal angular-speed in the same direction, the output shaft 11 turns in the opposite direction with the same angular-speed (Figure 15, bottom). If the motors 13 and 15 turn at equal angular-speed but in opposite directions, the output shaft 11 follows a translational motion without rotating (Figure 15, top). If we consider now the case where the motor 13 is replaced by a pulley like 17 or 19 and where the output shaft is actuated in the X direction, we obtain exactly the configuration of the differential timing belt mechanism. The corresponding principle is shown in Figure 17. Pulley 1 (W 1 ) is driven by a motor and a timing belt bind the pulley 1 (W 1 ) to the pulley 2 (W 2 ). The timing belt circuit between W 1 and W 2 is Figure 15. Control architecture with phase-shift master-slave mechanism (color figure available online).

14 PHASE-SHIFT MASTER-SLAVE MECHANISMS 27 Figure 16. Differential drive proposed by Gary R. Paulson (Paulson 1975). closed through four guide pulleys (W 3,W 4,W 5 and W 6 ). Two of these four guide pulleys (W 3 and W 4 ) are mounted on a motorized cart which is driven by a second motor. Prism 1 is then bound to the pulley 1 by a timing belt. Prism 2 is bound to the pulley 2 by another timing belt. If the motorized cart does not move, pulley 2 (W 2 ), which drives the prism 2, turns at the same angular-speed as the pulley 1 (W 1 ), which drives the prism 1. It means that both prisms turn with the same angular-speed and with a known angular phase-shift, given by the initial condition before drive. In the case where Figure 17. Differential timing belt principle (color figure available online).

15 28 O. CHAPPUIS AND R. CLAVEL Figure 18. Conventional preloaded linear stage with motion screw (color figure available online). the motorized cart performs a displacement of Dx while both prisms turn with the same angular-speed. Pulley 2 phase will be shifted by an angle u given by the equation (4), where q is the pulley radius (Figure 17) as shown in the following: u ¼ 2Dx q ð4þ According to equation (4), for an angular phase-shift in a range of 0 to 180 and a pulley radius q equal to 20 mm, the stroke Dx is in a range from 0 to 31.4 mm. The required accuracy on the linear stage to reach the specified accuracy on the phase-shift is 8 mm. This accuracy can be easily achieved with a conventional preloaded linear stage as shown in Figure 18. The control of such a linear stage is well known in the literature (Kafader 2007; Awaby et al. 1998). Speed and phase-shift control are mechanically decoupled. Figure 19 and 20 present an example of mechanical design based on this concept. In this case, the speed motor drives Prism 1 and the Prism 2 at the same angular-speed, by means of the transmission pulley in Figure 19. The angular phaseshift between the prisms is controlled by the cart linear position. The cart position is determined by the phase-shift motor, which drives the preloaded lead-screw (Figure 19 and 20). The main advantage of the mechanism proposed in Figure 19 and 20 is the dynamical modification of the angular phase-shift. The cart can moves and modify the angular phase-shift while the speed motor is in operation. As seen previously, the control complexity is reduced. However, the number of components is high increasing the manufacturing costs and the mechanical complexity Differential Planetary Gear Mechanism This second mechanism is based on the epicyclic gearing theory (Henriod 1999). Epicyclic gearings are currently used in the automotive industry, due to its

16 PHASE-SHIFT MASTER-SLAVE MECHANISMS 29 Figure 19. Topside CAD view of the proposed timing belt master-slave mechanism. Prisms are assembled on toothed pulleys (Prism 1 and Prism 2). The cart is mounted on two guideways. The Speed motor torque is transmitted to prism 1 and 2 by timing belts 1 and 2 through the transmission pulley (color figure available online). compactness compared to the high achievable gear ratios. These types of gearings are two DOF mechanisms. The output shaft speed is determined by the speed of two input shafts. The principle of the differential planetary gear mechanism is shown in Figure 21. When the shift motor is not active, the coupling part which supports the satellite gears G 2 and G 3 does not rotate around axis 1. In this case, if the speed Figure 20. Underside CAD view of the proposed timing belt master-slave mechanism. Phase-shift motor drives the preloaded lead-screw, which positions the cart. Micromectric resolution on the cart position can be easily reached by means of such lead-screw drive mechanism (color figure available online).

17 30 O. CHAPPUIS AND R. CLAVEL Figure 21. Differential planetary gear principle. Prism 1 and planetary gear G 1 are driven by the Speed motor. Prism 2 is coupled to the planetary gear G 4, which is also driven by the Speed motor through satellites gears G 2 and G 3. The shift motor drives the coupling parts, which holds the satellites gears G 2 and G 3 (color figure available online). motor turns with an angular-speed of x 0, Prism 1, which is directly driven by speed motor, turns at x 0. Prism 2 also rotates with an angular-speed x 0 due to the gear transmission through G 1,G 2,G 3, and G 4. Consequently, both prisms turn at the same angular-speed and with an angular phase-shift given by the initial position. The speed ratio between speed motor output and prism 2 is given by the equation (5) (Henriod 1999), where Z i are the teeth number of each gear: x 0 x P2 ¼ x 0 x 4 i P2 ¼ Z 2 Z 1 Z 4 Z 3 i P2 ð5þ Obviously, the desired speed ratio between speed motor and prism 2 is 1. In order to get this value, the speed ratio i P2 must be as demonstrated in the following: i P2 ¼ Z 1 Z 2 Z 3 Z 4 ð6þ Considering the case where the shift motor performs an angular displacement of u 0, the coupling part supporting the satellite gears G 2 and G 3 also moves, which shifts the satellite gears around G 1 and G 4. This displacement generates an angular phase-shift between G1 and G4, due to the angular ratio different from 1. The angular ratio between shift motor and prism 2 is given by the following equation (7) (Henriod 1999): u 0 u ¼ i Z 2 Z 4 Z 1 Z 3 0 i P2 ¼ i 0 Z 2 Z 4 Z 1 Z 3 Z 2 Z 4 Z 1 Z 3 ð7þ

18 PHASE-SHIFT MASTER-SLAVE MECHANISMS 31 Figure 22. Differential drive mechanism patented by Kamprath and Hoover (Kamprath and Hoover 1994). This angular ratio can be very high (typically 10 to 100), which improves the resolution of the system. Indeed, it is possible to obtain a small angular displacement of the prism with a big angular displacement of the shift motor output shaft. The required angular accuracy on the shift motor is thus decreased by a factor corresponding to this angular ratio. Kamprath and Hoover (1994) patented a similar mechanism designed to change the relative angular position between two discs (Figure 22). The global mechanical complexity of the differential planetary gear solution is high. Gear backlash is a source of drawbacks and needs to be eliminated. The driven inertia is also high due to the gears. The noise induced by the gears at high angular-speed might also be a drawback. Nevertheless, the control complexity is low compared to the design presented in section 3. The high achievable gear ratio decreases the required accuracy on the motor which controls the phase-shift. As the differential timing belt solution, this mechanism also permits a dynamical modification of the angular phase-shift. 5. ANALYSIS AND DISCUSSION A comparison of the designs performances is presented in Table 6. The simplest mechanical design is the two motors one. However, we demonstrated that this configuration is not easy to control at high angular-speed. The Table 6. Comparison of the proposed solutions Two motors Belt Planetary gear Overall dimensions Small High Medium Dynamic phase-shift Yes Yes Yes Driven inertia Low High High Backlash elimination Not required Required Required Control complexity High Low Low Mechanical complexity Low Medium High

19 32 O. CHAPPUIS AND R. CLAVEL differential timing belt mechanism is considered less complex than the differential planetary gear design from a mechanical point of view. 6. CONCLUSION This work focused on novel mechanical designs to simplify the motion controller of a high angular-speed paired-wedge-prisms system. These designs decouple the phase-shift and the velocity controls. An advantage of the proposed designs is the possibility to modify dynamically the phase-shift between the prisms without system interruption. In addition, they can be adapted to other applications that require a phase-shift control between two rotating parts at high angular-speed. The coming step in this investigation is the implementation of dynamical tests to assess and compare the performances of the master-slave mechanisms for the proposed designs. The outcome of this work will help to make decisions in potential applications in systems of paired-wedge-prisms. ACKNOWLEDGMENT This work is supported by the European Commission through the Seventh Framework program, FemtoPrint ( NMP, project no REFERENCES Ashkenasi, D., N. Müller, T. Kaszemeikat, and G. Illing Advanced laser micro machining using a novel trepanning system. JLMN Journal of Laser Micro=Nanoengineering 6(1): 1 5 Awaby, B. A., W.-C. Shih, and D. M. Auslander Nanometer positioning of a linear motion stage under static loads. IEEE=ASME Transactions on Mechatronics 3(2): Henriod, Georges Engrenages conception fabrication mise en œuvre. 7th ed. Paris: Dunod Kafader, Urs The selection of high-precision microdrives. Switzerland: Maxon Academy. Kamprath, D. R. and E. Hoover Differential drive for sheet registration drive rolls woth skew detection. United States Patent. Ostaszewski, M., S. Harford, N. Doughty, C. Hoffman, M. Sanchez, D. Gutow, and R. Pierce Risley prism beam pointer. Proceedings of SPIE Vol Paulson, G. L Differential drive rotating disc impact printer. United States Patent.