Abstract
A general methodology to design open loop controllers for nonlinear, dynamic, continuous systems is presented and applied to control a single flexible link (SFL). In this application, the partial differential equations that describe the beam system are first analyzed via the finite element method (FEM) and Newmark integration method. Two open loop control inputs to achieve specified system performance criteria are then computed by posing and solving inverse dynamics problems. These analyses use nonlinear programming (NLP) algorithms and analytical gradients that are computed by the direct sensitivity method. The open loop control is verified experimentally.
Closed loop controller synthesis for linear time invariant (LTI) and linear time varying systems (LTV) is relatively well understood. To apply this knowledge base to the control of the SFL, the nonlinear finite element plant model is linearized and recast in standard state space form.
Similar content being viewed by others
References
Bayo, E.; Movaghar, R; Medus, M. 1988: Inverse Dynamics of a Single-Link Flexible Robot. Analytical and Experimental Results. International Journal of Robotics and Automation 3.3, 150–157
Bayo, E. 1987: A Finite Element Approach to Control the End-Point Motion of a Single-Link Flexible Robot. Journal of Robotic Systems 4.1, 63–75
Book, W.J. 1993: Controlled Motion in an Elastic World. Journal of Dynamic Systems, Measurement and Control 115, 252–261
Boutaghou, Z.-E.; Erdman, A.G. 1991: A unified Approach for the Dynamics of Beams Undergoing Arbitrary Spatial Motion. Transactions of the ASME 113, 494–502
Cannon, R.H., Jr.; Schmitz, E. 1984: Initial Experiments on the End-Point Control of a Flexible One-Link Robot. International Journal of Robotics Research 3.3, 62–75
Ciarlet, P.G. 1988: Mathematical Elasticity, Volume 1: Three-Dimensional Elasticity. North-Holland
Cook, R.D; Malkus, D.S.; Plesha, M.E. 1989: Concepts and Applications of Finite Element Analysis. 3rd Edition. New York: John Wiley & Sons
Gevarter, W.B. 1970: Basic Relations for Control of Flexible Vehicles. AIAA Journal 8.4, 666–672
Haug, E.J.; Choi, K.K.; Komkov, V. 1986: Design Sensitivity Analysis of Structural Systems. Orlando, FL: Harcourt Brace Jovanovich Publishers
Hjelmstad, K.D. 1997: Fundamentals of Structural Mechanics. New Jeresy: Prentice Hall Inc.
Laskin, R.A.; Likins, P.W.; Longman, R.W. 1983: Dynamical Equations of a Free-Free Beam Subject to Large Overall Motions. The Journal of the Astronautical Sciences 31.4, 507–528
Pirie, C.L.; Okubo, S.; Dullerud, G.E.; Tortorelli, D.A. 2001: Robust Nonlinear Trajectory Tracking on a Single Flexible Link. Submitted to ASME Journal of Dynamic Systems, Measurement, and Control
Rattan, K.S.; Feliu, V. 1992: Feed forward Control of Flexible Manipulators. Proceedings of the International Conferences on Robotics and Automation, 778–793
Reissner, E. 1973: On One Dimensional Large Displacement Finite Strain Beam Theory. Studies in Applied Mathematics 52.2, 87–95
Rogers, D.F.; Adams, J.A. 1990: Mathematical Elements for Computer Graphics. 2nd Edition. New York: McGraw-Hill Publishing Company
Serna, M.; Bayo, E. 1990: Off-line Trajectory Planning for Flexible Manipulators. Modelling the Innovation: Communications, Automation and Information Systems, 150–157
Simo, J.C.; Vu-Quoc, L. 1986: On the Dynamics of Flexible Beams Under Large Overall Motions – The Plane Case: Part 1. Journal of Applied Mechanics 53, 849–854
Simo, J.C.; Vu-Quoc, L. 1986: On the Dynamics of Flexible Beams Under Large Overall Motions – The Plane Case: Part 2. Journal of Applied Mechanics 53, 855–863
Simo, J.C.; Vu-Quoc, L. 1985: A Finite Strain Beam Formulation, Part I: The Three Dimensional Dynmaic Problem. Computer Methods in Applied Mechanics and Engineering 49, 55–70
Simo, J.C.; Vu-Quoc, L. 1986: A Three Dimensional Finite Strain Rod Model, Part II: Computational Aspsects. Computer Methods in Applied Mechanics and Engineering 58, 79–116
Singer, N.C.; Seering, W.P. 1990: Preshaping Command Inputs to Reduce System Vibration. Journal of dynamic Systems, Measurement and Control 112, 76–82
Spector, V.A.; Flashner, H. 1990: Modeling and Design Implications of Noncollocated Control in Flexible Systems. Journal of Dynamic Systems, Measurement and Control 112, 186–193
Spivey, C.; Tortorelli, D.A. 1994: Tangent Operators, Sensitivity Expressions and Optimal Design of Nonlinear Elastica in Contact with Applications to Beams. International Journal of Numerical Methods in Engineering 37, 49–73
Timoshenko, S.P.; Gere, J.M. 1990: Mechanics of Materials. 3rd Edition. Boston: PWS Publishing Company
VMA Engineering 1992: DOT Users’ Manual. Version 3. Goleta, CA: Vanderplaats, Miura & Associates, Inc. text.
Yuan, K. 1995: Regulation of Single-Link Flexible Manipulator Involving Large Elastic Deflections. J. Guidance 18.3, 635–637
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Okubo , S., Tortorelli , D. Control of nonlinear, continuous, dynamic systems via finite elements, sensitivity analysis, and optimization. Struct Multidisc Optim 26, 183–199 (2004). https://doi.org/10.1007/s00158-003-0338-z
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00158-003-0338-z