Share This Article:

A Survey of Modeling and Control of Piezoelectric Actuators

DOI: 10.4236/mme.2013.31001    6,082 Downloads   14,314 Views   Citations

ABSTRACT

Piezoelectric actuators (PEAs) have been widely used in micro- and nanopositioning applications due to their fine resolution, fast responses, and large actuating forces. However, the existence of nonlinearities such as hysteresis makes modeling and control of PEAs challenging. This paper reviews the recent achievements in modeling and control of piezoelectric actuators. Specifically, various methods for modeling linear and nonlinear behaviors of PEAs, including vibration dynamics, hysteresis, and creep, are examined; and the issues involved are identified. In the control of PEAs as applied to positioning, a review of various control schemes of both model-based and non-model-based is presented along with their limitations. The challenges associated with the control problem are also discussed. This paper is concluded with the emerging issues identified in modeling and control of PEAs for future research.

 

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

J. Peng and X. Chen, "A Survey of Modeling and Control of Piezoelectric Actuators," Modern Mechanical Engineering, Vol. 3 No. 1, 2013, pp. 1-20. doi: 10.4236/mme.2013.31001.

References

[1] D. Croft, G. Shed and S. Devasia, “Creep, Hysteresis, and Vibration Compensation for Piezoactuators: Atomic Force Microscopy Application,” Journal of Dynamic Systems, Measurement, and Control, Vol. 123, No. 1, 2001, pp. 35-43. doi:10.1115/1.1341197
[2] Q. Zou, K. K. Leang, E. Sadoun, M. J. Reed and S. Devasia, “Control Issues in High-Speed AFM for Biological Applications: Collagen Imaging Example,” Asian Journal of Control, Vol. 6, No. 2, 2004, pp. 164-178. doi:10.1111/j.1934-6093.2004.tb00195.x
[3] R. Kassies, K. O. Van der Werf, A. Lenferink, C. N. Hunter, J. D. Olsen, V. Subramaniam and C. Otto, “Combined AFM and Confocal Fluorescence Microscope for Applications in Bio-Nanotechnology,” Journal of Microscopy, Vol. 217, No. 1, 2005, pp. 109-116. doi:10.1111/j.0022-2720.2005.01428.x
[4] H. Song, G. Vdovin, R. Fraanje, G. Schitter and M. Verhaegen, “Extracting Hysteresis from Nonlinear Measurement of Wavefront-Sensorless Adaptive Optics System,” Optics Letters, Vol. 34, No. 1, 2009, pp. 61-63. doi:10.1364/OL.34.000061
[5] W. Yang, S.-Y. Lee and B.-J. You, “A Piezoelectric Actuator with a Motion-Decoupling Amplifier for Optical Disk Drives,” Smart Materials and Structures, Vol. 19, No. 6, 2010, pp. 065027.1-065027.10. doi:10.1088/0964-1726/19/6/065027
[6] G. Stoppler and S. Douglas, “Adaptronic Gantry Machine Tool with Piezoelectric Actuator for Active Error Compensation of Structural Oscillations at the Tool Centre Point,” Mechatronics, Vol. 18, No. 8, 2008, pp. 426-433. doi:10.1016/j.mechatronics.2008.03.002
[7] S. R. Viswamurthy, A. K. Rao and R. Ganguli, “Dynamic Hysteresis of Piezoceramic Stack Actuators Used in Helicopter Vibration Control: Experiments and Simulations,” Smart Materials and Structures, Vol. 16, No. 4, 2007, pp. 1109-1119. doi:10.1088/0964-1726/16/4/020
[8] M. S. Senousy, F. X. Li, D. Mumford, M. Gadala and R. K. N. D. Rajapakse, “Thermo-Electro-Mechanical Performance of Piezoelectric Stack Actuators for Fuel Injector Applications,” Journal of Intelligent Material Systems and Structures, Vol. 20, No. 4, 2009, pp. 387-399. doi:10.1177/1045389X08095030
[9] J.-J. Wei, Z.-C. Qiu, J.-D. Han and Y.-C. Wang, “Experimental Comparison Research on Active Vibration Control for Flexible Piezoelectric Manipulator Using Fuzzy Controller,” Journal of Intelligent and Robotic Systems, Vol. 59, No. 1, 2010, pp. 31-56. doi:10.1007/s10846-009-9390-2
[10] D. Shu, J. Maser, M. Holt, R. Winarski, C. Preissner, A. Smolyanitskiy, B. Lai, S. Vogt and G. B. Stephenson, “Optomechanical Design of a Hard X-Ray Nanoprobe Instrument with Nanometer-Scale Active Vibration Control,” AIP Conference Proceedings, Vol. 879, 2007, pp. 1321-1324. doi.org/10.1063/1.2436307
[11] H. Zhou and B. Henson, “Analysis of a Diamond-Shaped Mechanical Amplifier for a Piezo Actuator,” The International Journal of Advanced Manufacturing Technology, Vol. 32, No. 1-2, 2007, pp. 1-7. doi:10.1007/s00170-005-0303-7
[12] M. Muraokaa and S. Sanada, “Displacement Amplifier for Piezoelectric Actuator Based on Honeycomb Link Mechanism,” Sensors and Actuators A: Physical, Vol. 157, No. 1, 2010, pp. 84-90. doi:10.1016/j.sna.2009.10.024
[13] Y. Zhang, G. Liu and J. Hesselbach, “On Development of a Rotary-Linear Actuator Using Piezoelectric Translators,” IEEE/ASME Transactions on Mechatronics, Vol. 11, No. 5, 2006, pp. 647-650. doi:10.1109/TMECH.2006.882998
[14] J. Li, R. Sedaghati, J. Dargahi and D. Waechter, “Design and Development of a New Piezoelectric Linear Inchworm Actuator,” Mechatronics, Vol. 15, No. 6, 2005, pp. 651-681. doi:10.1016/j.mechatronics.2005.02.002
[15] R. Merry, R. Van de Molengraft and M. Steinbuch, “Modeling of a Walking Piezo Actuator,” Sensors and Actuators A: Physical, Vol. 162, No. 1, 2010, pp. 51-60. doi:10.1016/j.sna.2010.05.033
[16] R. Merry, M. Uyanik, R. Van de Molengraft, R. Koops, M. Van Veghel and M. Steinbuch, “Identification, Control and Hysteresis Compensation of a 3 DOF Metrological AFM,” Asian Journal of Control, Vol. 11, No. 2, 2009, pp. 130-143. doi:10.1002/asjc.89
[17] T.-F. Lu, D. C. Handley, Y. K. Yong and C. Eales, “A Three-DOF Compliant Micromotion Stage with Flexure Hinges,” Industrial Robot: An International Journal, Vol. 31, No. 4, 2004, pp. 355-361. doi:10.1108/01439910410541873
[18] S. W. Lee, K.-G. Ahn and J. Ni, “Development of a Piezoelectric Multi-Axis Stage Based on Stick-and-Clamping Actuation Technology,” Smart Materials and Structures, Vol. 16, No. 6, 2007, pp. 2354-2367. doi:10.1088/0964-1726/16/6/040
[19] S. Devasia, E. Eleftheriou and S. O. R. Moheimani, “A Survey of Control Issues in Nanopositioning,” IEEE Transactions on Control Systems Technology, Vol. 15, No. 5, 2007. pp. 802-823. doi:10.1109/TCST.2007.903345
[20] P. Instrumente, “The World of Nanopositioning and Micropositioning 2005/2006,” Physik Instrumente, Karlsruhe, 2005.
[21] I. Mayergoyz and G.Bertotti (Eds.), “The Science of Hysteresis,” Vol. 3, Elsevier, St. Louis, 2005.
[22] H. J. M. T. S. Adriaens, W. L. De Koning and R. Banning, “Modeling Piezoelectric Actuators,” IEEE/ASME Transactions on Mechatronics, Vol. 5, No. 4, 2000, pp. 331341. doi:10.1109/3516.891044
[23] J. A. Main and E. Garcia, “Piezoelectric Stack Actuators and Control System Design: Strategies and Pitfalls,” Journal of Guidance, Control, and Dynamics, Vol. 20, No. 3, 1997, pp. 479-485. doi:10.2514/2.4066
[24] T. Fett and G. Thun, “Determination of Room-Temperature Tensile Creep of PZT,” Journal of Materials Science Letters, Vol. 17, No. 22, 1998, pp. 1929-1931. doi:10.1023/A:1006608509876
[25] W. T. Ang, F. A. Garmón, P. Khosla and C. N. Riviere, “Modeling Rate-Dependent Hysteresis in Piezoelectric Actuator,” 2003 IEEE/RSJ International Conference on Intelligent Robots and Systems, Las Vegas, 27 October-1 November 2003, pp. 1975-1980. doi:10.1109/IROS.2003.1248937
[26] W. T. Ang, P. K. Khosla and C. N. Riviere, “Feedforward Controller with Inverse Rate-Dependent Model for Piezoelectric Actuators in Trajectory-Tracking Applications,” IEEE/ASME Transactions on Mechatronics, Vol. 12, No. 2, 2007, pp. 134-142. doi:10.1109/TMECH.2007.892824
[27] R. S. Robinson, “Interactive Computer Correction of Piezoelectric Creep in Scanning Tunneling Microscopy Images,” Journal of Computer-Assisted Microscopy, Vol. 2, No. 1, 1996, pp. 53-58.
[28] X. B. Chen, Q. Zhang, D. Kang and W. Zhang, “On the Dynamics of Piezoactuated Positioning Systems,” Review of Scientific Instruments, Vol. 79, No. 11, 2008, pp. 116101.1-116101.3. doi:10.1063/1.2982238
[29] W. Chen and C. S. Lynch, “A Micro-Electro-Mechanical Model for Polarization Switching of Ferroelectric Materials,” Acta Materialia, Vol. 46, No. 15, 1998, pp. 53055311. doi:10.1016/S1359-6454(98)00207-9
[30] W. Seemann, A. Arockiarajan and B. Delibas, “Modeling and Simulation of Piezoceramic Materials Using Micromechanical Approach,” European Congress on Computaional Methods in Applied Sciences and Engineering 2004 (ECCOMAS 2004), Jyvaskyla, 24-28 July 2004.
[31] A. Arockiarajan, B. Delibas, A. Menzel and W. Seemann, “Studies on Nonlinear Electromechanical Behavior of Piezoelectric Materials Using Finite Element Modeling,” IWPMA 2005, 2nd International Workshop on Piezoelectric Materials and Applications in Actuators, Heinz Nixdorf Institute, Paderborn, 22-25 May 2005.
[32] B. Delibas, A. Arockiarajan and W. Seemann, “Intergranular Effects on Domain Switchings in Polycrystalline Piezoceramics,” IWPMA 2005, 2nd International Workshop on Piezoelectric Materials and Applications in Actuators, Heinz Nixdorf Institute, Paderborn, 22-25 May 2005.
[33] M. Kamlah and Q. Jiang, “A Constitutive Model for Ferroelectric PZT Ceramics under Uniaxial Loading,” Smart Materials and Structures, Vol. 8, No. 4, 1999, pp. 441459. doi:10.1088/0964-1726/8/4/302
[34] M. Kamlah and C. Tsakmakis, “Phenomenological Modeling of the Non-Linear Electro-Mechanical Coupling in Ferroelectrics,” International Journal of Solids and Structures, Vol. 36, No. 5, 1999, pp. 669-695. doi:10.1016/S0020-7683(98)00040-7
[35] M. Kamlah and U. Bohle, “Finite Element Analysis of Piezoceramic Components Taking into Account Ferroelectric Hysteresis Behavior,” International Journal of Solids and Structures, Vol. 38, No. 4, 2001, pp. 605-633. doi:10.1016/S0020-7683(00)00055-X
[36] C. M. Landis, J. Wang and J. Sheng, “Microelectromechanical Determination of the Possible Remanent Strain and Polarization States in Polycrystalline Ferroelectrics and the Implications for Phenomenological Constitutive Theories,” Journal of Intelligent Material Systems and Structures, Vol. 15, No. 7, 2004, pp. 513-525. doi:10.1177/1045389X04041653
[37] J. Schroder and H. Romanowski, “A Simple Coordinate Invariant Thermodynamic Consistent Model for Nonlinear Electro-Mechanical Coupled Ferroelectrica,” European Congress on Computaional Methods in Applied Sciences and Engineering 2004 (ECCOMAS 2004), Jyvaskyla, 24-28 July 2004.
[38] J. Schroder and H. Romanowski, “A Thermodynamically Consistent Mesoscopic Model for Transversely Isotropic Ferroelectric Ceramics in a Coordinate-Invariant Setting,” Achive of Applied Mechanics, Vol. 74, No. 11-12, 2005, pp. 863-877. doi:10.1007/s00419-005-0412-7
[39] S. Klinkel, “A Thermodynamic Consistent 1D Model for Ferroelastic and Ferroelectric Hysteresis Effects in Piezoceramics,” Communications in Numerical Methods in Engineering, Vol. 22, No. 7, 2006, pp. 727-739. doi:10.1002/cnm.845
[40] T. Hegewald, B. Kaltenbacher, M. Kaltenbacher and R. Lerch, “Efficient Modeling of Ferroelectric Behavior for the Analysis of Piezoceramic Actuators,” Journal of Intelligent Material Systems and Structures, Vol. 19, No. 10, 2008, pp. 1117-1129. doi:10.1177/1045389X07083608
[41] IEEE, “ANSI/ IEEE Std. 176-1987: IEEE Standard on Piezoelectricity,” The Institute of Electrical and Electronics Engineers, New York, 1988.
[42] P. G. Harper, “Kinematic Theory of Piezoelectric Hysteresis,” Journal of Applied Physics, Vol. 52, No. 11, 1981, pp. 6851-6855. doi:10.1063/1.328677
[43] D. C. Jiles and D. L. Atherton, “Theory of the Magnetisation Process in Ferromagnets and Its Application to the Magnetomechanical Effect,” Journal of Physics D: Applied Physics, Vol. 17, No. 6, 1984, pp. 1265-1281. doi:10.1088/0022-3727/17/6/023
[44] R. C. Smith and C. L. Hom, “Domain Wall Model for Ferroelectric Hysteresis,” Technical Report CRSC-TR9907, Center for Research in Scientific Computation, New York, 1999.
[45] T. Hegewald, M. Kaltenbacher and R. Lerch, “Characterization and Modeling of Piezoelectric Stack Actuators,” IWPMA 2005, 2nd International Workshop on Piezoelectric Materials and Applications in Actuators, Heinz Nixdorf Institute, Paderborn, 22-25 May 2005, pp. 99106.
[46] E. Preisach, “On the Magnetic Aftereffect,” Zeitschrift für Physik A Hadrons and Nuclei, Vol. 94, No. 5-6, 1935, pp. 277-302.
[47] I. D. Mayergoyz, “Mathematical Models of Hysteresis,” Physical Review Letters, Vol. 56, No. 15, 1986, pp. 15181521. doi:10.1103/PhysRevLett.56.1518
[48] I. D. Mayergoyz and G. Friedman, “Generalized Preisach model of Hysteresis,” IEEE Transactions on Magnetics, Vol. 24, No. 1, 1988, pp. 212-217. doi:10.1109/20.43892
[49] P. Hejda and T. Zelinka, “Generalized Preisach Model of Hysteresis—Theory and Experiment,” Czechoslovak Journal of Physics, Vol. 40, No. 1, 1990, pp. 57-68. doi:10.1007/BF01598355
[50] M.-J. Jang, C.-L. Chen and J.-R. Lee, “Modeling and Control of a Piezoelectric Actuator Driven System with Asymmetric Hysteresis,” Journal of the Franklin Institute, Vol. 346, No. 1, 2009, pp. 17-32. doi:10.1016/j.jfranklin.2008.06.005
[51] P. N. Sreeram, G. Salvady and N. G. Naganathan, “Hysteresis Prediction for a Piezoceramic Material System,” The ASME Winter Annual Meeting, New Orleans, 28 November-3 December 3 1993, pp. 35-42.
[52] D. C. Hughes and J. T. Wen, “Preisach Modeling and Compensation for Smart Material Hysteresis,” Proceedings of SPIE, Vol. 2427, 1995, pp. 50-64.
[53] P. Ge and M. Jouaneh, “Generalized Preisach Model for Hysteresis Nonlinearity of Piezoceramic Actuators,” Precision Engineering, Vol. 20, No. 2, 1997, pp. 99-111. do:10.1016/S0141-6359(97)00014-7
[54] H. Hu and R. B. Mrad, “On the Classical Preisach Model for Hysteresis in Piezoceramic Actuators,” Mechatronics, Vol. 13, No. 2, 2002, pp. 85-94. doi:10.1016/S0957-4158(01)00043-5
[55] G. Robert, D. Damjanovic and N. Setter, “Preisach Modeling of Piezoelectric Nonlinearity in Ferroelectric Ceramics,” Journal of Applied Physics, Vol. 89, No. 9, 2001, pp. 5067-5074. doi:10.1063/1.1359166
[56] G. Song, J. Zhao, X. Zhou and J. A. De Abreu-Garcia, “Tracking Control of a Piezoceramic Actuator with Hysteresis Compensation Using Inverse Preisach Model,” IEEE/ASME Transactions on Mechatronics, Vol. 10, No. 2, 2005, pp. 198-209. doi:10.1109/TMECH.2005.844708
[57] X. Zhao and Y. Tan, “Neural Network based Identification of Preisach-Type Hysteresis in Piezoelectric Actuator Using Hysteretic Operator,” Sensors and Actuators A: Physical, Vol. 126, No. 2, 2006, pp. 306-311. doi:10.1016/j.sna.2005.10.023
[58] X. Zhao and Y. Tan, “Modeling Hysteresis in Piezo Actuator based on Neural Networks,” Advances in Computation and Intelligence, Lecture Notes in Computer Science, Vol. 5370, 2008, pp. 290-296.
[59] K. Kuhnen and H. Janocha, “Adaptive Inverse Control of Piezoelectric Actuators with Hysteresis Operators,” Proceedings of European Control Conference (ECC), Karsruhe, 31 August-3 September 1999, paper F 0291.
[60] P. Krejci and K. Kuhnen, “Inverse Control of Systems with Hysteresis and Creep,” IEE Proceedings—Control Theory and Applications, Vol. 148, No. 3, 2001, pp. 185192. doi:10.1049/ip-cta:20010375
[61] K. Kuhnen and H. Janocha, “Complex Hysteresis Modeling of a Broad Class of Hysteretic Nonlinearities,” Proceedings of the 8th international conference on New Actuators, Bremen, 10-12 June 2002, pp. 688-691.
[62] K. Kuhnen and F. Previdi, “Modeling, Identification and Compensation of Complex Hysteretic Nonlinearities: A Modified Prandtl-Ishlinskii Spproach,” European Journal of Control, Vol. 9, No. 4, 2003, pp. 407-421. doi:10.3166/ejc.9.407-418
[63] K. Kuhnen, “Modeling, Identification and Compensation of Complex Hysteretic Nonlinearities and Log(t)-Type Creep Dynamics,” Control and Intelligent System, Vol. 33, No. 2, 2005, pp. 134-147. doi:10.2316/Journal.201.2005.2.201-1420
[64] M. Goldfarb and N. Celanovic, “Modeling Piezoelectric Stack Actuators for Control of Micromanipulation,” IEEE Control Systems Magazine, Vol. 17, No. 3, 1997, pp. 6979. doi:10.1109/37.588158
[65] S.-H. Lee, T. J. Royston and G. Friedman, “Modeling and Compensation of Hysteresis in Piezoceramic Transducers for Vibration Control,” Journal of Intelligent Material Systems and Structures, Vol. 11, No. 10, 2000, pp. 781790. doi:10.1106/GQLJ-JGEU-MHG1-7JDF
[66] T.-J. Yeh, S.-W. Lu and T.-Y. Wu, “Modeling and Identification of Hysteresis in Piezoelectric Actuators,” Journal of Dynamic Systems, Measurement, and Control, Vol. 128, No. 2, 2006, pp. 189-196. doi:10.1115/1.2192819
[67] T.-J. Yeh, R.-F. Hung and S.-W. Lu, “An Integrated Physical Model that Characterizes Creep and Hysteresis in Piezoelectric Actuators,” Simulation Modelling Practice and Theory, Vol. 16, No. 1, 2008, pp. 93-110. doi:10.1016/j.simpat.2007.11.005
[68] R. B. Mrad and H. Hu, “A Model for Voltage to Displacement Dynamics in Piezoceramic Actuators Subject to Dynamic Voltage Excitations,” IEEE/ASME Transactions on Mechatronics, Vol. 7, No. 4, 2002, pp. 479-489. doi:10.1109/TMECH.2002.802724
[69] D. Song and C. J. Li, “Modeling of Piezo Actuator’s Nonlinear and Frequency Dependent Dynamics,” Mechatronics, Vol. 9, No. 4, 1999, pp. 391-410. doi:10.1016/S0957-4158(99)00005-7
[70] Y. Yu, Z. Xiaob, N. G. Naganathan and R. V. Dukkipati, “Dynamic Preisach Modeling of Hysteresis for the Piezoceramic Actuator System,” Mechanism and Machine Theory, Vol. 37, No. 1, 2002, pp. 75-89. doi:10.1016/S0094-114X(01)00065-9
[71] X. Tan, R. Venkataraman and P. S. Krishnaprasad, “Control of Hysteresis: Theory and Experimental Results,” Pentagon Technical Report A687934, 2001.
[72] H. Hu and R. Ben Mrad, “A Discrete-Time Compensation Algorithm for Hysteresis in Piezoceramic Actuators,” Mechanical Systems and Signal Processing, Vol. 18, No. 1, 2004, pp. 169-185. doi:10.1016/S0888-3270(03)00021-9
[73] S. Bashash and N. Jalili, “A Polynomial-Based Linear Mapping Strategy for Feedforward Compensation of Hysteresis in Piezoelectric Actuators,” Journal of Dynamic Systems, Measurement, and Control, Vol. 130, No. 3, 2008, pp. 031008.1-031008.10. doi:10.1115/1.2907372
[74] R. Venkataraman and P. S. Krishnaprasad, “Novel Algorithm for the Inversion of the Preisach Operator Smart Structures and Materials,” Proceedings of SPIE 3984, Smart Structures and Materials 2000: Mathematics and Control in Smart Structures, 2000, pp. 404-414. doi:10.1117/12.388785
[75] R. Venkataraman and P. S. Krishnaprasad, “Approximate Inversion of Hysteresis: Theory and Numerical Results,” Proceedings of the 39th IEEE Conference on Decision and Control, Sydney, Australia, 12-15 December 2000, pp. 4448-4454. doi:10.1109/CDC.2001.914608
[76] J. Y. Peng and X. B. Chen, “H2-Optimal Digital Control of Piezoelectric Actuators,” Proceedings of the 8th World Congress on Intelligent Control and Automation, WCICA2010, Jinan, 7-9 July 2010, pp. 3684-3690. doi:10.1109/WCICA.2010.5553921
[77] U.-X. Tan, W. T. Latt, C. Y. Shee, C. N. Riviere and W. T. Ang, “Feedforward Controller of Ill-Conditioned Hysteresis Using Singularity-Free Prandtl-Ishlinskii Model,” IEEE/ASME Transactions on Mechatronics, Vol. 14, No. 5, 2009, pp. 598-605. doi:10.1109/TMECH.2008.2009936
[78] B. D. Coleman and M. L. Hodgdon, “A Constitutive Relation for Rate-Independent Hysteresis in Ferromagnetically Soft Materials,” International Journal of Engineering Science, Vol. 24, No. 6, 1986, pp. 897-919. doi:10.1016/0020-7225(86)90023-6
[79] R. Bouc, “Forced Vibrations of a Mechanical System with Hysteresis,” Proceedings of the 4th Conference on Non-linear Oscillations, Prague, 5-9 September 1967, pp. 315.
[80] Y.-K. Wen, “Method of Random Vibration of Hysteretic Systems,” Journal of the Engineering Mechanics Division, Vol. 102, No. 2, 1976, pp. 249-263.
[81] M. Ismail, F. Ikhouane and J. Rodellar, “The Hysteresis Bouc-Wen Model, a Survey,” Archives of Computational Methods in Engineering, Vol. 16, No. 2, 2009, pp. 161188. doi:10.1007/s11831-009-9031-8
[82] O. Gomis-Bellmunt, F. Ikhouane and D. MontesinosMiracle, “Control of Bouc-Wen Hysteretic Dystems: Application to a Piezoelectric Actuator,” The 13th Power Electronics and Motion Control Conference, 2008. EPEPEMC 2008, Poznan, 1-3 September 2008, pp. 16701675. doi:10.1109/EPEPEMC.2008.4635507
[83] M. Rakotondrabe, “Bouc-Wen Modeling and Inverse Multiplicative Structure to Compensate Hysteresis Nonlinearity in Piezoelectric Actuators,” IEEE Transactions on Automation Science and Engineering, Vol. 8, No. 2, 2011, pp. 428-431. doi:0.1109/TASE.2010.2081979
[84] D. H. Wang, W. Zhu and Q. Yang, “Linearization of Stack Piezoelectric Ceramic Actuators Based on BoucWen Model,” Journal of Intelligent Material Systems and Structures, Vol. 22, No. 5, 2011, pp. 401-413. doi:10.1177/1045389X10386132
[85] L. Deng and Y. Tan, “Modeling Hysteresis in Piezoelectric Actuators Using NARMAX Models,” Sensors and Actuators A: Physical, Vol. 149, No. 1, 2009, pp. 106112. doi:10.1016/j.sna.2008.09.022
[86] A. Badel, J. Qiu and T. Nakano, “A New Simple Asymmetric Hysteresis Operator and Its Application to Inverse Control of Piezoelectric Actuators,” IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control, Vol. 55, No. 5, 2008, pp. 1086-1094. doi:10.1109/TUFFC.2008.761
[87] H. Jung, J. Y. Shim and D. Gweon, “New Open-Loop Actuating Method of Piezoelectric Actuators for Removing Hysteresis and Creep,” Review of Scientific Instruments, Vol. 71, No. 9, 2000, pp. 3436-3440. doi:10.1063/1.1287627
[88] K. Kuhnen and H. Janocha, “Compensation of Creep and Hysteresis Effects of Piezoelectric Actuators with Inverse Systems,” 6th International Conference on New actuators, Actuator’98, Bremen, 17-19 June 1998, pp. 309-312.
[89] H. Janocha and K. Kuhnen, “Real-Time Compensation of Hysteresis and Creep in Piezoelectric Actuators,” Sensors Actuators A: Physical, Vol. 79, No. 2, 2000, pp. 83-89. doi:10.1016/S0924-4247(99)00215-0
[90] J. S. Dewey, K. Leang and S. Devasia, “Experimental and Theoretical Results in Output-Trajectory Redesign for Flexible Structures,” Journal of Dynamic Systems, Measurement, and Control, Vol. 120, No. 4, 1998, pp. 456-461. doi:10.1115/1.2801486
[91] K. S. Abidi and A. Sabanovic, “Sliding Mode Control for High-Precision Motion of a Piezostage,” IEEE Transactions on Industrial Electronics, Vol. 54, No. 1, 2007, pp. 629-637. doi:10.1109/TIE.2006.885477
[92] H. C. Liaw, B. Shirinzadeh and J. Smith, “Enhanced Sliding Mode Motion Tracking Control of Piezoelectric Actuators,” Sensors and Actuators A: Physical, Vol. 138, No. 1, 2007, pp. 194-202. doi:10.1016/j.sna.2007.04.062
[93] G. M. Clayton, S. Tien, A. J. Fleming, S. O. R. Moheimani and S. Devasia, “Inverse-Feedforward of ChargeControlled Piezopositioners,” Mechatronics, Vol. 18, No. 5-6, 2008, pp. 273-281. doi:10.1016/j.mechatronics.2007.07.006
[94] J. W. Li, X. B. Chen and W. J. Zhang, “A New Approach to Modeling System Dynamics—In the Case of a Piezoelectric Actuator with a Host System,” IEEE/ASME Transactions on Mechatronics, Vol. 15, No. 3, 2010, pp. 371380. doi:10.1109/TMECH.2009.2026473
[95] J. W. Li, X. B. Chen and W. J. Zhang, “Axiomatic-Design-Theory-Based Approach to Modeling Linear High Order System Dynamics,” IEEE/ASME Transactions on Mechatronics, Vol. 16, No. 2, 2011, pp. 341-350. doi:10.1109/TMECH.2010.2043535
[96] K. Kuhnen, H. Janocha, D. Thull and A. Kugi, “A New Drive Concept for High-Speed Positioning of Piezoelectric Actuators,” Proceedings of the 10th International Conference on New Actuators, Bremen, 14-16 June 2006, pp. 82-85.
[97] K. K. Leang and S. Devasia, “Feedback-Linearized Inverse Feedforward for Creep, Hysteresis, and Vibration Compensation in AFM Piezoactuators,” IEEE Transactions on Control Systems Technology, Vol. 15, No. 5, 2007, pp. 927-935. doi:10.1109/TCST.2007.902956
[98] X. Dang and Y. Tan, “An Inner Product-Based Dynamic Neural Network Hysteresis Model for Piezoceramic Actuators,” Sensors and Actuators A: Physical, Vol. 121, No. 2, 2005, pp. 535-542. doi:10.1016/j.sna.2005.04.003
[99] C. Ru, L. Chen, B. Shao, W. Rong and L. Sun, “A Hysteresis Compensation Method of Piezoelectric Actuator: Model, Identification and Control,” Control Engineering Practice, Vol. 17, No. 9, 2009, pp. 1107-1114. doi:10.1016/j.conengprac.2009.04.013
[100] L. Sun, C. Ru, W. Rong, L. Chen and M. Kong, “Tracking Control of Piezoelectric Actuator Based on a New Mathematical Model,” Journal of Micromechanics and Microengineering, Vol. 14, No. 11, 2004, pp. 1439-1444. doi:10.1088/0960-1317/14/11/001
[101] C. Ru and L. Sun, “Hysteresis and Creep Compensation for Piezoelectric Actuator in Open-Loop Operation,” Sensors and Actuators A: Physical, Vol. 122, No. 1, 2005, pp. 124-130. doi:10.1016/j.sna.2005.03.056
[102] H. H. Najafabadi, S. M. Rezaei, S. S. Ghidary, M. Zareinejad, K. Razi and R. Seifabadi, “Hysteresis Compensation of Piezoelectric Actuators under Dynamic Load Condition,” IEEE/RSJ International Conference on Intelligent Robots and Systems, 2007. IROS 2007, San Diego, 29 October-2 November 2007, pp. 1166-1171. doi:10.1109/IROS.2007.4399048
[103] E. Kouno, “A Fast Response Piezoelectric Actuator for Servo Correction of Systematic Errors in Precision Machining,” CIRP Annals-Manufacturing Technology, Vol. 33, No. 1, 1982, pp. 369-372. doi: 10.1016/S0007-8506(07)61444-9
[104] J. F. Cuttino, A. C. Miller and D. E. Schinstock, “Performance Optimization of a Fast Tool Servo for Single-Point Diamond Turning Machines,” IEEE/ASME Transactions on Mechatronics, Vol. 4, No. 2, 1999, pp. 169-179. doi:10.1109/3516.769543
[105] J. Lin, H. Chiang and C. C. Lin, “Tuning PID Control Gains for Micro Piezo-Stage in using Grey Relational Analysis,” 2008 International Conference on Machine Learning and Cybernetics, Kunming, 12-15 July 2008, pp. 3863-3868. doi: 10.1109/ICMLC.2008.4621078
[106] H.-J. Shieh, Y.-J. Chiu and Y.-T. Chen, “Optimal PID Control System of a Piezoelectric Microospitioner,” 2008 IEEE/SICE International Symposium on System Integration, Nagoya, 4 December 2008, pp. 1-5. doi:10.1109/SI.2008.4770417
[107] D. Y. Abramovitch, S. Hoen and R. Workman, “SemiAutomatic Tuning of PID Gains for Atomic Force Microscopes,” Asian Journal of Control, Vol. 11, No. 2, 2009, pp. 188-195. doi:10.1002/asjc.95
[108] K. K. Tan, T. H. Lee and H. X. Zhou, “Micro-Positioning of Linear-Piezoelectric Motors Based on a Learning Nonlinear PID Controller,” IEEE/ASME Transactions on Mechatronics, Vol. 6, No. 4, 2001, pp. 428-436. doi:10.1109/3516.974856
[109] W. S. Oates and R. C. Smith, “Nonlinear Control Design for a Piezoelectric-Driven Nanopositioning Stage,” Pentagon Technical Report A591444, 2005.
[110] C. Edwards and S. K. Spurgeon, “Sliding Mode Control: Theory and Applications,” Taylor & Francis, Abingdon, 1998.
[111] P.-K. Huang, P.-H. Shieh, F.-J. Lin and H.-J. Shieh, “Sliding-Mode Control for a Two-Dimensional PiezoPositioning Stage,” Control Theory & Applications, IET, Vol. 1, No. 4, 2007, pp. 1104-1113. doi:0.1049/iet-cta:20060371
[112] J.-C. Shen, W.-Y. Jywe1, C.-H. Liu, Y.-T. Jian and J. Yang, “Sliding-Mode Control of a Three-Degrees-ofFreedom Nanopositioner,” Asian Journal of Control, Vol. 10, No. 3, 2008, pp. 267-276. doi:10.1002/asjc.33
[113] J.-C. Shen, J.-C. Shen, H.-K. Chiang and Y.-L. Shu, “Precision Tracking Control of a Piezoelectric-Actuated System,” Precision Engineering, Vol. 32, No. 2, 2008, pp. 71-78. doi:10.1016/j.precisioneng.2007.04.002
[114] Q. Xu and Y. Li, “Dynamics Modeling and Sliding Mode Control of an XY Micropositioning Stage,” The 9th International Symposium on Robot Control (SYROCO’09), Gifu, 9-12 September 2009, pp. 781-786.
[115] Y. Li and Q. Xu, “Adaptive Sliding Mode Control with Perturbation Estimation and PID Sliding Surface for Motion Tracking of a Piezo-Driven Micromanipulator,” IEEE Transactions on Control Systems Technology, Vol. 18, No. 4, 2010, pp. 798-810. doi:10.1109/TCST.2009.2028878
[116] H. C. Liaw, B. Shirinzadeh and J. Smith, “Sliding-Mode Enhanced Adaptive Motion Tracking Control of Piezoelectric Actuation Systems for Micro/Nano Manipulation,” IEEE Transactions on Control Systems Technology, Vol. 16, No. 4, 2008, pp. 826-833. doi:10.1109/TCST.2007.916301
[117] X. Chen and T. Hisayama, “Adaptive Sliding-Mode Position Control for Piezo-Actuated Stage,” IEEE Transactions on Industrial Electronics, Vol. 55, No. 11, 2008, pp. 3927-3934. doi:10.1109/TIE.2008.926768
[118] H. C. Liaw and B. Shirinzadeh, “Enhanced Sliding-Mode Constrained Motion Tracking Control of Piezo-Actuated Flexure-Based Mechanisms,” IEEE/ASME International Conference on Advanced Intelligent Mechatronics, 2009. AIM 2009, Singapore, 14-17 July 2009 2009, pp. 18791884. doi:10.1109/AIM.2009.5229763
[119] G. Schitter, P. Menold, H. F. Knapp, F. Allgower and A. Stemmer, “High Performance Feedback for Fast Scanning Atomic Force Microscopes,” Review of Scientific Instruments, Vol. 72, No. 8, 2001, pp. 3320-3327. doi:10.1063/1.1387253
[120] M. S. Tsai and J. S. Chen, “Robust Tracking Control of a Piezoactuator Using a New Approximate Hysteresis Model,” Journal of Dynamic Systems, Measurement, and Control, Vol. 125, No. 1, 2003, pp. 96-102. doi:10.1115/1.1540114
[121] G. Schitter, A. Stemmer, and F. Allgower, “Robust TwoDegree-of-Freedom Control of an Atomic Force Microscope,” Asian Journal of Control, Vol. 6, No. 2, 2004, pp. 156-163. doi: 10.1111/j.1934-6093.2004.tb00194.x
[122] A. Sebastian and S. M. Salapaka, “Design Methodologies for Robust Nano-Positioning,” IEEE Transactions on Control Systems Technology, Vol. 13, No. 6, 2005, pp. 868-876. doi:10.1109/TCST.2005.854336
[123] Y. Okazaki, “A Micro-Positioning Tool Post Using a Piezoelectric Actuator for Diamond Turning Machines,” Precision Engineering, Vol. 12, No. 3, 1990, pp. 151-156. doi:10.1016/0141-6359(90)90087-F
[124] D. Croft, S. Stilson and S. Devasia, “Optimal Tracking of Piezo-Based Nanopositioners,” Nanotechnology, Vol. 10, No. 2, 1999, pp. 201-208. doi:10.1088/0957-4484/10/2/316
[125] C. J. Li, H. S. M. Beigi, S. Li and J. Liang, “Nonlinear Piezo-Actuator Control by Learning Self Tuning Regulator,” Journal of Dynamic Systems, Measurement, and Control, Vol. 115, No. 4, 1993, pp. 720-723. doi:10.1115/1.2899203
[126] H.-J. Shieh, F.-J. Lin, P.-K. Huang and L.-T. Teng, “Adaptive Tracking Control Solely Using Displacement Feedback for a Piezo-Positioning Mechanism,” IEE Proceedings——Control Theory and Applications, Vol. 151, No. 5, 2004, pp. 653-660. doi:10.1049/ip-cta:20040795
[127] X. Tan and J. S. Baras, “Adaptive Identification and Control of Hysteresis in Smart Materials,” IEEE Transactions on Automatic Control, Vol. 50, No. 6, 2005, pp. 827-839. doi:10.1109/TAC.2005.849215
[128] H. C. Liaw and B. Shirinzadeh, “Enhanced Adaptive Motion Tracking Control of Piezo-Actuated FlexureBased Four-Bar Mechanisms for Micro/Nano Manipulation,” Sensors and Actuators A: Physical, Vol. 147, No. 1, 2008, pp. 254-262. doi:10.1016/j.sna.2008.03.020
[129] J.-H. Xu, “Neural Network Control of a Piezo Tool Positioner,” Canadian Conference on Electrical and Computer Engineering, Vancouver, 14-17 September 1993, pp. 333-336. doi:0.1109/CCECE.1993.332324
[130] F.-J. Lin, H.-J. Shieh and P.-K. Huang, “Adaptive Wavelet Neural Network Control with Hysteresis Estimation for Piezo-Positioning Mechanism,” IEEE Transactions on Neural Networks, Vol. 17, No. 2, 2006, pp. 432-444. doi:10.1109/TNN.2005.863473
[131] F.-J. Lin, H.-J. Shieh, P.-K. Huang and L.-T. Teng, “Adaptive Control with Hysteresis Estimation and Compensation Using RFNN for Piezo-Actuator,” IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control, Vol. 53, No. 9, 2006, pp. 1649-1661. doi:10.1109/TUFFC.2006.1678193
[132] H. Numasato and M. Tomizuka, “Settling Control and Performance of a Dual-Actuator System for Hard Disk Drives,” IEEE/ASME Transactions on Mechatronics, Vol. 8, No. 4, 2003, pp. 431-438. doi:10.1109/TMECH.2003.819999
[133] K. Leang and S. Devasia, “Hysteresis, Creep, and Vibration Compensation for Piezoactuators: Feedback and Feedforward Control,” Proceedings of the 2nd IFAC Conference on Mechatronic Systems, Berkley, 9-11 December 2002, pp. 283-289.
[134] L. Y. Pao, J. A. Butterworth and D. Y. Abramovitch, “Combined Feedforward/Feedback Control of Atomic Force Microscopes,” Proceedings of the 2007 American Control Conference, New York City, 9-13 July 2007, pp. 3509-3515. doi:10.1109/ACC.2007.4282338
[135] D. Croft and S. Devasia, “Vibration Compensation for High Speed Scanning Tunneling Microscopy,” Review of Scientific Instruments, Vol. 70, No. 12, 1999, pp. 46004605. doi:10.1063/1.1150119
[136] H. Hu, H. M. S. Georgiou and R. Ben-Mrad, “Enhancement of Tracking Ability in Piezoceramic Actuators Subject to Dynamic Excitation Conditions,” IEEE/ASME Transactions on Mechatronics, Vol. 10, No. 2, 2005, pp. 230-239. doi:10.1109/TMECH.2005.844705
[137] C.-J. Lin and S.-R. Yang, “Modeling of a Piezo-Actuated Positioning Stage Based on a Hysteresis Observer,” Asian Journal of Control, Vol. 7, No. 1, 2005, pp. 73-80. doi:10.1111/j.1934-6093.2005.tb00230.x
[138] S. S. Aphale, S. Devasia and S. O. Reza Moheimani, “High-Bandwidth Control of a Piezoelectric Nanopositioning Stage in the Presence of Plant Uncertainties,” Nanotechnology, Vol. 19, No. 12, 2008, pp. 125503.1125503.9. doi:10.1088/0957-4484/19/12/125503
[139] D. Croft and S. Devasia, “Vibration Compensation for High Speed Scanning Tunneling Microscopy,” Review of Scientific Instruments, Vol. 70, No. 12, 1999, pp. 46004605. doi:10.1063/1.1150119
[140] S. Khan, M. Elitas, E. D. Kunt and A. Sabanovic, “Discrete Sliding Mode Control of Piezo Actuator in NanoScale Range,” IEEE International Conference on Industrial Technology, 2006. ICIT 2006, Mumbai, 15-17 December 2006, pp. 1454-1459. doi:10.1109/ICIT.2006.372418
[141] J. Yi, S. Chang and Y. Shen, “Disturbance-ObserverBased Hysteresis Compensation for Piezoelectric Actuators,” IEEE/ASME Transactions on Mechatronics, Vol. 14, No. 4, 2009, pp. 4196-4201. doi:10.1109/TMECH.2009.2023986
[142] S. Chang and S. Li, “A High Resolution Long Travel Friction-Drive Micropositioner with Programmable Step Size,” Review of Scientific Instruments, Vol. 70, No. 6, 1999, pp. 2276-2782. doi:10.1063/1.1149794
[143] A. Bergander and J. Breguet, “Performance Improvements for Stick-Slip Positioners,” Proceedings of 2003 International Symposium on Micromechatronics and Human Science, Nagoya, 19-22 October 2003, pp. 59-66. doi:10.1109/MHS.2003.1249910
[144] F. Castanos and L. Fridman, “Analysis and Design of Integral Sliding Manifolds for Systems with Unmatched Perturbations,” IEEE Transactions on Automatic Control, Vol. 51, No. 5, 2006, pp. 853-858. doi:10.1109/TAC.2006.875008
[145] J. H. Painter, D. Kerstetter and S. Jowers, “Reconciling Steady-State Kalman and Alpha-Beta Filter Design,” IEEE Transactions on Aerospace and Electronic Systems, Vol. 26, No. 6, 1990, 1990, pp. 986-991. doi:10.1109/7.62250
[146] X. Wang, “High-Order Integral-Chain Differentiator and Application to Acceleration Feedback,” Submitted to Computer Science for Possible Publication, 2011.
[147] A. Levant, “Robust Exact Differentiation via Sliding Mode Technique,” Automatica, Vol. 34, No. 3, 1998, pp. 379-384. doi:10.1016/S0005-1098(97)00209-4
[148] Q. Xu and Y. Li, “Sliding Mode Control of a PiezoDriven Micropositioning System Using Extended Kalman Filter,” IEEE International Conference on Automation and Logistics (ICAL), 2010, Hong Kong and Macau, 16-20 August 2010, pp. 427-432. doi:10.1109/ICAL.2010.5585322
[149] J. L. Minase, T.-F. Lu and F. Wornle, “State Estimation of Nonlinear Piezoelectric Stack Actuator Hysteresis Model,” Proceedings of SPIE, Vol. 6414, 2006, pp. 641403.1-641403.10.
[150] J. Minase, T.-F. Lu, B. Cazzolato and S. Grainger, “Adaptive Identification of Hysteresis and Creep in Piezoelectric Stack Actuators,” The International Journal of Advanced Manufacturing Technology, Vol. 46, No. 9-12, 2009, pp. 913-921. doi:10.1007/s00170-009-2033-8
[151] F. Yang and R. W. Wilde, “Observers for Linear Systems with Unknown Inputs,” IEEE Transactions on Automatic Control, Vol. 33, No. 7, 1988, pp. 667-681. doi:10.1109/9.1278
[152] M. Hou and P. C. Muller, “Design of Observers for Linear Systems with Unknown Inputs,” IEEE Transactions on Automatic Control, Vol. 37, No. 6, 1992, pp. 871-875. doi:10.1109/9.256351
[153] S. Hui and S. Zak, “Observer Design for Systems with Unknown Inputs,” International Journal of Applied Mathematics and Computer Science, Vol. 15, No. 4, 2005, pp. 431-546.
[154] S. H. Zak and S. Hui, S. “Output Feedback Variable Structure Controllers and State Estimators for Uncertain/Nonlinear Dynamic Systems,” IEE Proceedings D: Control Theory and Applications, Vol. 140, No. 1, 1993, pp. 41-50. doi:10.1049/ip-d.1993.0006
[155] J.-L. Chang and T.-C. Wu, “Robust Disturbance Attenuation with Unknown Input Observer and Sliding Mode Controller,” Electrical Engineering, Vol. 90, No. 7, 2008, pp. 493-502. doi:10.1007/s00202-008-0099-1
[156] T. Floquet, C. Edwards and S. K. Spurgeon, “On Sliding Mode Observers for Systems with Unknown Inputs,” International Journal of Adaptive Control and Signal Processing, Vol. 21, No. 8-9, 2007, pp. 638-656. doi:10.1002/acs.958
[157] K. Kalsi, J. Lian, S. Hui and S. H. Zak, “Sliding-Mode Observers for Systems with Unknown Inputs: A HighGain Approach,” Automatica, Vol. 46, No. 2, 2010, pp. 347-353. doi: 10.1016/j.automatica.2009.10.040
[158] X. Liu, J. Jeong and J. Kim, “A Three Translational DoFS Parallel Cube-Manipulator,” Robotica, Vol. 21, No. 6, 2003, pp. 645-653. doi:10.1017/S0263574703005198
[159] Q. Xu, and Y. Li, “A Novel Design of a 3-PRC Translational Compliant Parallel Micromanipulator for Nanomanipulation,” Robotica, Vol. 24, No. 4, 2006, pp. 527-528. doi:10.1017/S0263574705002559
[160] Y. Yue, F. Gao, X. Zhao and Q. Jeffrey Ge, “Relationship among Input-Force, Payload, Stiffness and Displacement of a 3-DOF Perpendicular Parallel Micro-Manipulator,” Mechanism and Machine Theory, Vol. 45, No. 5, 2010, pp. 756-771. doi:10.1016/j.mechmachtheory.2009.12.006
[161] S. Awtar and A. Slocum, “Constraint-Based Design of Parallel Kinematic XY Flexure Mechanisms,” Journal of Mechanical Design, Vol. 129, No. 8, 2007, pp. 816-830. doi:10.1115/1.2735342
[162] Q. Yao, J. Dong and P. Ferreira, “A Novel ParallelKinematics Mechanisms for Integrated, Multi-Axis Nanopositioning: Part 1. Kinematics and Design for Fabrication,” Precision Engineering, Vol. 32, No. 1, 2008, pp. 719. doi:10.1016/j.precisioneng.2007.03.001
[163] J. Dong, Q. Yao and P. Ferreira, “A Novel ParallelKinematics Mechanism for Integrated, Multi-Axis Nanopositioning: Part 2: Dynamics, Control and Performance Analysis,” Precision Engineering, Vol. 32, No. 1, 2008, pp. 20-33. doi:10.1016/j.precisioneng.2007.03.002
[164] J. Dong, S. Salapaka and P. Ferreira, “Robust Control of a Parallel-Kinematic Nanopositioner,” Journal of Dynamic Systems, Measurement, and Control, Vol. 130, No. 4, 2008, pp. 041007.1-041007.15. doi:10.1115/1.2936861
[165] Y. Tian, B. Shirinzadeh and D. Zhang, “A Flexure-Based Five-Bar Mechanism for Micro/Nano Manipulation,” Sensors and Actuators A: Physical, Vol. 153, No. 1, 2009, pp. 96-104. doi: 10.1016/j.sna.2009.04.022
[166] Y. Tian, B. Shirinzadeh and D. Zhang, “Design and Dynamics of a 3-DOF Flexure-Based Parallel Mechanism for Micro/Nano Manipulation,” Microelectronic Engineering, Vol. 87, No. 2, 2010, pp. 230-241. doi:10.1016/j.mee.2009.08.001
[167] K. Hu, J. Kim, J. Schmiedeler and C. Menq, “Design, Implementation, and Control of a Six-Axis Compliant Stage,” Review of Scientific Instruments, Vol. 79, No. 2, 2008, pp. 025108.2-025105.11. doi:10.1063/1.2841804
[168] T. W. Seo, D. S. Kang and J. Kim, “Synthesis and Comparison of Fine Actuator Controllers for a 3-DOF Micro Parallel Positioning Platform,” Proceedings of the 10th International Conference on Control, Automation, Robotics and Vision, Hanoi, 17-20 December 2008, pp. 16061611. doi:10.1109/ICARCV.2008.4795765
[169] M. Grotjahn, B. Heimann and H. Abdellatif, “Identification of Friction and Rigid-Body Dynamics of Parallel Kinematic Structures for Model-Based Control,” Multibody System Dynamics, Vol. 11, No. 3, 2004, pp. 273-294. doi:10.1023/B:MUBO.0000029426.05860.c2

  
comments powered by Disqus

Copyright © 2018 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.