- Volume 11 Issue 1
DOI QR Code
Finding Optimal Controls for Helicopter Maneuvers Using the Direct Multiple-Shooting Method
- Kim, Min-Jae (Department of Aerospace Information System Eng, Konkuk University) ;
- Hong, Ji-Seung (Department of Aerospace Information System Eng, Konkuk University) ;
- Kim, Chang-Joo (Department of Aerospace Information System Eng, Konkuk University)
- Published : 2010.03.01
The purpose of this paper deals with direct multiple-shooting method (DMS) to resolve helicopter maneuver problems of helicopters. The maneuver problem is transformed into nonlinear problems and solved DMS technique. The DMS method is easy in handling constraints and it has large convergence radius compared to other strategies. When parameterized with piecewise constant controls, the problems become most effectively tractable because the search direction is easily estimated by solving the structured Karush-Kuhn-Tucker (KKT) system. However, generally the computation of function, gradients and Hessian matrices has considerably time-consuming for complex system such as helicopter. This study focused on the approximation of the KKT system using the matrix exponential and its integrals. The propose method is validated by solving optimal control problems for the linear system where the KKT system is exactly expressed with the matrix exponential and its integrals. The trajectory tracking problem of various maneuvers like bob up, sidestep near hovering flight speed and hurdle hop, slalom, transient turn, acceleration and deceleration are analyzed to investigate the effects of algorithmic details. The results show the matrix exponential approach to compute gradients and the Hessian matrix is most efficient among the implemented methods when combined with the mixed time integration method for the system dynamics. The analyses with the proposed method show good convergence and capability of tracking the prescribed trajectory. Therefore, it can be used to solve critical areas of helicopter flight dynamic problems.
Supported by : Korea Aerospace Research Institute (KARI)
- Cervantes, L., and Biegler, L. T., "Optimization Strategies for Dynamic Systems", Encyclopedia of Optimization, edited by Floudas, C., and Pardalos, P., Vol. 4, Kluwer, 2001, pp. 216-227.
- Betts, J. T., “Survey of Numerical Methods for Trajectory Optimization”, Journal of Guidance, Control, and Dynamics, Vol. 20, No. 2, Mar.-Apr. 1998, pp. 193-207.
- Bryson, A. E., Jr., and Ho, Y. C., Applied Optimal Control, Hemisphere Publishing, 1975.
- Kirk, D. E., Optimal Control Theory; An Introduction, Dover, New York, 1970.
- Fraser-Andrews, G., “A Multiple-Shooting Technique for Optimal Control”, Journal of Optimization Theory and Applications, Vol. 102, No. 2, Aug. 1999, pp. 299-313. https://doi.org/10.1023/A:1021728407151
- Oberle, H. J., and Grimm, W., “BNDSCO; A Program for the Numerical Solution of Optimal Control Problems”, DFVLR Report No. 515, Institute for Flight Systems Dynamics, Oberpfaffenhofen, German Aerospace Research Establishment DLR, 1989.
- Steibach, M., Fast Recursive SQP Methods for Large Scale Optimal ControlProblems, Ph. D. dissertation, University of Heidelberg, 1995.
- Betts, J. T., “Practical Methods for Optimal Control Optimal Control Using Nonlinear Programming”, Society for Industrial and Applied Mathematics Press, 2001.
- Huntington, G. T., Advancement and Analysis of a Gauss Pseudospectral Transcription for Optimal Control Problems, PH. D. Dissertation, Massachusetts Institute of Technology, June 2007.
- Kim, C.-J., Sung, S. K., Park, S. H., S.-N. Jung, and Yee, K., “Selection of Rotorcraft Models for Application to Optimal Control Problems”, Journal of Guidance, Control, and Dynamics, Vol. 20, No. 2, Mar.-Apr. 1998, pp. 193-207.
- Cloutier, J. R., “State-Dependent Riccati Equation Techniques: An Overview”, Proceeding of the American Control Conference, June 1997, pp. 932-936. https://doi.org/10.1109/ACC.1997.609663
- Cloutier, J. R. and Stansbery, D. T., “The capabilities and Art of State-Dependent Riccati Equation-Based Design”, Proceeding of the American Control Conference, May 2002, pp86-91. https://doi.org/10.1109/ACC.2002.1024785
- Menon, P. K., Lam, T., Crawford, L. S., and Cheng, V. H. L., “Real-Time Computational Methods for SDRE Nonlinear Control of Missiles”, Proceeding of the American Control Conference, May 2002. https://doi.org/10.1109/ACC.2002.1024809
- Kim, C.-J., Sung, S.-K., Yang C. D., and Yu, Y. H., “Rotorcraft Trajectory Tracking Using the State-Dependent Riccati Equation Controller”, Transactions of the Japan Society for Aeronautical and Space Science, accepted and to be pubilished, Vol. 51, No-173, November, 2008. https://doi.org/10.2322/tjsass.51.184
- Moler C. and Loan, C. F. V., "Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later", Society for Industrial and Applied Mathematics, Vol45, No.1, 2003 https://doi.org/10.1137/S00361445024180
- Loan, C. F. V., “Computing Integrals Involving the Matrix Exponential”, IEEE Transactions on Automatic Control, Vol. AC-23, No. 3, pp. 395-404,June, 1978.
- Kim, C.-J., “Numerical Stability Investigation of Integration Inverse Simulation Method for the Analysis of Helicopter Flight during Aggressive Maneuver”, Spring Meeting of Korean Society for Aeronautical and Space Sciences, Apr. 2002.
- Kim, C.-J., Yun, C. Y., and Choi, S., “Fully Implicit Formulation and Its Solution for Rotor Dynamics by Using Differential Algebraic Equation (DAE) Solver and Partial Periodic Trimming Algorithm (PPTA)”, 31st European Rotorcraft Forum, Florence, Italy, Sept. 13-16, 2005.
- Kim, C.-J., Jung, S.-N., Lee J., Byun, Y. H., and Yu, Y. H., “Analysis of Helicopter Mission Task Elements by Using Nonlinear Optimal Control Method”, 33rd European Rotorcraft Forum, Russia, Kazan, Sept. 11-13, 2007.
- Chen, R. T. N., “Effects of Primary Rotor Parameters on Flapping Dynamics”, NASA TP-1431, 1980.
- Rutherford, S., and Thomson, D. G., “Improved methodology for Inverse Simulation”, Aeronautical Journal, Vol. 100, No. 993, Mar. 1996, pp. 79-86.
- Bradley, R., and Thomson, D. G., “The Use of Inverse Simulation for Preliminary Assessment of Helicopter Handling Qualities”, Aeronautical Journal, Vol. 101, No. 1007, Sept. 1997, pp. 287-294.
- Nocedal, J., and Wright, S., J., Numerical Optimization, Springer-Verlag, New York, 1999.
- Leineweber, D., B., “The theory of MUSCO in a Nutshell”, IWR technical Report 96-16, University of Heidelberg, 1996.
- Chang-Joo Kim, Soo Hyung Park, Sang Kyung Sung, and Sung-Nam Jung, “ Nonlinear Optimal Control Analysis Using State-Dependent Matrix Exponential and Its Integrals”, Journal of Guidance, Control, and Dynamics, Vol. 32, No. 1, Jan.. Feb. 2009, pp. 309-313. https://doi.org/10.2514/1.38886
- Numerical Time-Scale Separation for Rotorcraft Nonlinear Optimal Control Analyses vol.37, pp.2, 2014, https://doi.org/10.2514/1.59557