0
Research Papers

Semi-Analytic Probability Density Function for System Uncertainty

[+] Author and Article Information
Ahmad Bani Younes

Assistant ProfessorMem. ASME Department of Aerospace Engineering, Khalifa University, P. O. Box 127788, Abu Dhabi, UAE e-mail: ahmad.younes@kustar.ac.ae

James Turner

Visiting ProfessorMem. ASME Department of Aerospace Engineering, Khalifa University, P. O. Box 127788, Abu Dhabi, UAE e-mail: james.turner@kustar.ac.ae

Manuscript received January 28, 2016; final manuscript received June 13, 2016; published online August 19, 2016. Assoc. Editor: Athanasios Pantelous.

ASME J. Risk Uncertainty Part B 2(4), 041007 (Aug 19, 2016) (7 pages) Paper No: RISK-16-1041; doi: 10.1115/1.4033886 History: Received January 28, 2016; Accepted June 13, 2016

In general, the behavior of science and engineering is predicted based on nonlinear math models. Imprecise knowledge of the model parameters alters the system response from the assumed nominal model data. One proposes an algorithm for generating insights into the range of variability that can be expected due to model uncertainty. An automatic differentiation tool builds the exact partial derivative models required to develop a state transition tensor series (STTS)-based solution for nonlinearly mapping initial uncertainty models into instantaneous uncertainty models. The fully nonlinear statistical system properties are recovered via series approximations. The governing nonlinear probability distribution function is approximated by developing an inverse mapping algorithm for the forward series model. Numerical examples are presented, which demonstrate the effectiveness of the proposed methodology.

FIGURES IN THIS ARTICLE
<>
Copyright © 2016 by ASME
Your Session has timed out. Please sign back in to continue.

References

Park, R. S., and Scheeres, D. J., 2007, “Nonlinear Semi-Analytic Methods for Trajectory Estimation,” J. Guid. Control Dyn., 30(6), pp. 1668–1676. 0731-5090 10.2514/1.29106
Park, R. S., and Scheeres, D. J., 2006, “Nonlinear Mapping of Gaussian Statistics: Theory and Applications to Spacecraft Trajectory Design,” J. Guid. Control Dyn., 29(6), pp. 1367–1375. 0731-5090 10.2514/1.20177
Younes, A. B., and Turner, J., 2012, “High-Order Uncertainty Propagation Using State Transition Tensor Series,” Jer-Nan Juang Astrodynamics Symposium, Univelt, Inc., San Diego, CA, No. AAS 12-636.
Turner, J., and Younes, A. B., 2013, “On the Expected Value of Sensed Data,” 23rd AAS/AIAA Space Flight Mechanics Meeting, Univelt, Inc., San Diego, CA, No. AAS 13-377.
Younes, A. B., Turner, J., Majji, M., and Junkins, J., 2012, “Recent Advances in Algorithmic Differentiation,” High-Order Uncertainty Propagation Enabled by Computational Differentiation, S. Forth, P. Hovland, E. Phipps, J. Utke, and A. Walther, eds., Springer, Berlin/Heidelberg, pp. 251–260.
Younes, A. B., and Turner, J., 2015, “System Uncertainty Propagation Using Automatic Differentiation,” Proceedings of the ASME 2015 International Mechanical Engineering Congress and Exposition: Dynamics, Vibration, and Control, ASME, Houston, TX, Nov. 13–19, Vol. 4A, Paper No. IMECE2015-51412.
Griffith, D. T., Turner, J. D., and Junkins, J. L., 2004, “An Embedded Function Tool for Modeling and Simulating Estimation Problems in Aerospace Engineering,” AAS/AIAA Spaceflight Mechanics Meeting, Univelt, Inc., San Diego, CA, No. AAS 04-148.
Griffith, D. T., Sinclair, A., Turner, J. D., Hurtado, J., and John, J., 2004, “Automatic Generation and Integration of Equations of Motion by Operator Overloading Techniques,” AAS/AIAA Spaceflight Mechanics Meeting, Univelt, Inc., San Diego, CA, No. AAS 04-242.
Majji, M., Junkins, J., and Turner, J., 2008, “A High Order Method for Estimation of Dynamic Systems,” J. Astronaut. Sci., 56(3), pp. 401–440. 0021-9142 10.1007/BF03256560
Majji, M., Junkins, J., and Turner, J., 2010, “A Perturbation Method for Estimation of Dynamic Systems,” Nonlinear Dyn., 60(3), pp. 303–325. 0924-090X 10.1007/s11071-009-9597-6
Xiu, D., 2010, Numerical Methods for Stochastic Computations: A Spectral Method Approach, Princeton University Press, Princeton, NJ.
Fujimoto, K., Scheeres, D., and Alfriend, K., 2012, “Analytical Nonlinear Propagation of Uncertainty in the Two-Body Problem,” J. Guid. Control Dyn., 35(2), pp. 497–509. 0731-5090 10.2514/1.54385
Majji, M., Weisman, R., and Alfriend, K., 2012, “Solution of the Liouvilles Equation for Keplerian Motion: Application to Uncertainty Calculations,” 22nd AAS/AIAA Space Flight Mechanics Meeting, Univelt, Inc., San Diego, CA, Vol. 143.
Turner, J. D., Majji, M., and Junkins, J. L., 2011, “Keynote Paper: High Accuracy Trajectory and Uncertainty Propagation Algorithm for Long-Term Asteroid Motion Prediction,” Proceedings of the International Conference on Computational and Experimental Engineering and Sciences, Nanjing, China, Apr. 17–21.
Bischof, C. H., Carle, A., Hovland, P. D., Khademi, P., and Mauer, A., 1998, “ADIFOR 2.0 User’s Guide (Revision D),” Mathematics and Computer Science Division, Technical Memorandum No. 192, and Center for Research on Parallel Computation, Technical Report CRPC-95516-S.
Griewank, A., 1989, “On Automatic Differentiation,” Mathematical Programming, M. Iri and K. Tanabe, eds., Kluwer Academic Publishers, Dordrecht, The Netherlands, pp. 83–108.
Wengert, R. E., 1964, “A Simple Automatic Derivative Evaluation Program,” Commun. ACM, 7(8), pp. 463–464. 0001-0782 10.1145/355586.364791
Wilkins, R. D., 1964, “Investigation of a New Analytical Method for Numerical Derivative Evaluation,” Commun. ACM, 7(8), pp. 465–471. 0001-0782 10.1145/355586.364792
Bischof, C., and Eberhard, P., 1996, “Automatic Differentiation of Numerical Integration Algorithms,” Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL, Tech. Rep. ANL/MCS-P621-1196.
Bischof, C., Carle, A., Corliss, G., Griewank, A., and Hovland, P., 1992, “Adifor: Generating Derivative Codes From Fortran Programs,” Sci. Programm., 1(1), pp. 1–29.
Turner, J. D., 2003, “Automated Generation of High-Order Partial Derivative Models,” AIAA J., 41(8), pp. 1590–1598. 0001-1452 10.2514/2.2112
Griffith, D. T., Turner, J. D., and Junkins, J. L., 2005, “Automatic Generation and Integration of Equation of Motion for Flexible Multibody Dynamical Systems,” AAS J. Astronaut. Sci., 53(3), pp. 251–279.
Younes, A. B., Turner, J., Majji, M., and Junkins, J., 2010, “An Investigation of State Feedback Gain Sensitivity Calculations,” AIAA/AAS Astrodynamics Specialist Conference: Guidance, Navigation, and Control, No. AIAA-2010-8274.
Younes, A. B., and Turner, J., 2012, “Numerical Integration of Constrained Multi-Body Dynamics Using 5th Order Exact Analytic Continuation Algorithm,” Jer-Nan Juang Astrodynamics Symposium, Univelt, Inc., San Diego, CA, No. AAS 12-638.
Younes, A. B., Turner, J., Majji, M., and Junkins, J., 2012, “High-Order State Feedback Gain Sensitivity Calculations: Using Computational Differentiation,” Jer-Nan Juang Astrodynamics Symposium, Univelt, Inc., San Diego, CA, No. AAS 12-637.
Younes, A. B., Turner, J., Majji, M., and Junkins, J., 2011, “Nonlinear Tracking Control of Maneuvering Rigid Spacecraft,” No. AAS 11-168, Advances in the Astronautical Sciences, Univelt, Inc., San Diego, CA, Vol. 140.
Younes, A. B., and Turner, J., 2015, “Generalized Least Squares and Newton’s Method Algorithms for Nonlinear Root-Solving Applications,” J. Astronaut. Sci., 60(3), pp. 517–540. 0021-9142
Younes, A. B., and Turner, J., 2015, “Feedback Control Sensitivity Calculations Using Computational Differentiation,” Proceedings of the ASME 2015 International Mechanical Engineering Congress and Exposition: Dynamics, Vibration, and Control, ASME, Houston, TX, Vol. 4B, No. IMECE2015-51439.
Younes, A. B., and Turner, J., 2014, “An Analytic Continuation Method to Integrate Constrained Multibody Dynamical Systems,” Proceedings of the ASME 2014 International Mechanical Engineering Congress and Exposition: Dynamics, Vibration, and Control, ASME, Montreal, QC, Nov. 14–20, Vol. 4B, No. IMECE2014-37809.
Younes, A. B., Turner, J., and Junkins, J., 2013, “Higher-Order Optimal Tracking Feedback Gain Sensitivity Calculations: Using Computational Differentiation,” Proceedings of the 36th Annual AAS Rocky Mountain Section Guidance and Control Conference, Univelt, Inc., San Diego, CA, No. AAS 13-017.
Macsyma, Inc., 1995, Macsyma, Symbolic/Numeric/Graphical Mathematics Software: Mathematics and System Reference Manual, 15th ed, Macsyma, Inc.
Turner, J., 2006, OCEA User Manual, AMDYN SYSTEMS Inc., Plano, TX.
Hahn, T., 2005, “Cubaa Library for Multidimensional Numerical Integration,” Comput. Phys. Commun., 168(2), pp. 78–95. 0010-4655 10.1016/j.cpc.2005.01.010

Figures

Grahic Jump Location
Fig. 1

Performance impact of the STTS over MC

Grahic Jump Location
Fig. 2

Block diagram shows the partial derivative execution of a predefined model

Grahic Jump Location
Fig. 3

Schematic diagram demonstrates the application of Liouvilles theorem to compute truncation error

Grahic Jump Location
Fig. 4

Phase portrait and uncertainty propagation about the nominal trajectory using fourth-order state transition tensor. (a) Phase portrait of the dynamic motion (for ϵ=1) and the uncertainty propagation about the nominal trajectory using fourth-order state transition tensor. (b) STTS solution validated by MC and exact integration along the nominal trajectory (for ϵ=1) using fourth-order state transition tensor. (c) Fourth-order STTS 3σ solution predicts a good match with the exact propagation.

Grahic Jump Location
Fig. 5

Propagation of PDF over time for ϵ={0,0.2,0.4}: (a) ϵ=0, (b) ϵ=0.2, and (c) ϵ=0.4

Grahic Jump Location
Fig. 6

Propagation of PDF over time for ϵ={0.6,0.8,1.0}: (a) ϵ=0.6, (b) ϵ=0.8, and (c) ϵ=1.0

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Articles from Part A: Civil Engineering
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In