Research Papers

A Comparison of Ordinary Differential Equation Solvers for Dynamical Systems With Impacts

[+] Author and Article Information
Spyridon Dallas

Department of Mechanical Engineering,
National Technical University of Athens,
9 Heroon Polytechniou Street,
Athens 15780, Greece
e-mail: spyro.d.mechs@gmail.com

Konstantinos Machairas

Department of Mechanical Engineering,
National Technical University of Athens,
9 Heroon Polytechniou Street,
Athens 15780, Greece
e-mail: kmach@central.ntua.gr

Evangelos Papadopoulos

Department of Mechanical Engineering,
National Technical University of Athens,
9 Heroon Polytechniou Street,
Athens 15780, Greece
e-mail: egpapado@central.ntua.gr

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received December 4, 2016; final manuscript received May 24, 2017; published online September 7, 2017. Assoc. Editor: Stefano Lenci.

J. Comput. Nonlinear Dynam 12(6), 061016 (Sep 07, 2017) (8 pages) Paper No: CND-16-1600; doi: 10.1115/1.4037074 History: Received December 04, 2016; Revised May 24, 2017

In this paper, a method is developed that results in guidelines for selecting the best Ordinary Differential Equation (ODE) solver and its parameters, for a class of nonlinear hybrid system were impacts are present. A monopod interacting compliantly with the ground is introduced as a new benchmark problem, and is used to compare the various solvers available in the widely used matlab ode suite. To provide result generality, the mathematical description of the hybrid system is brought to a dimensionless form, and its dimensionless parameters are selected in a range taken from existing systems and corresponding to different levels of numerical stiffness. The effect of error tolerance and phase transition strategy is taken into account. The obtained system responses are evaluated using solution speed and accuracy criteria. It is shown that hybrid systems represent a class of problems that cycle between phases in which the system of the equations of motion (EoM) is stiff (interaction with the ground), and phases in which it is not (flight phases); for such systems, the appropriate type of solver was an open question. Based on this evaluation, both general and case-specific guidelines are provided for selecting the most appropriate ODE solver. Interestingly, the best solver for a realistic test case turned out to be a solver recommended for numerically nonstiff ODE problems.

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Park, H. W. , Wensing, P. M. , and Kim, S. , 2017, “ High-Speed Bounding With the MIT Cheetah 2: Control Design and Experiments,” Int. J. Rob. Res., 36(2), pp. 167–192. [CrossRef]
Paraskevas, I. , and Papadopoulos, E. , 2016, “ Parametric Sensitivity and Control of On-Orbit Manipulators During Impacts Using the Centre of Percussion Concept,” Control Eng. Pract., 47, pp. 48–59. [CrossRef]
Vasilopoulos, V. , Paraskevas, I. , and Papadopoulos, E. , 2014, “ All-Terrain Legged Locomotion Using a Terradynamics Approach,” International Conference on Intelligent Robots and Systems (IROS), Chicago, IL, Sept. 14–18, pp. 4849–4854.
Blum, Y. , Lipfert, S. W. , Rummel, J. , and Seyfarth, A. , 2010, “ Swing Leg Control in Human Running,” Bioinspiration Biomimetics, 5(2), p. 026006. https://doi.org/10.1088/1748-3182/5/2/026006
Koutsoukis, K. , and Papadopoulos, E. , 2016, “ On Passive Quadrupedal Bounding With Translational Spinal Joint,” IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), Deajeon, South Korea, Oct. 9–14, pp. 3406–3411.
Byrne, G. D. , and Hindmarsh, A. C. , 1987, “ Stiff ODE Solvers: A Review of Current and Coming Attractions,” J. Comput. Phys., 70(1), pp. 1–62. [CrossRef]
Petcu, D. , 2004, “ Software Issues in Solving Initial Value Problems for Ordinary Differential Equations,” Creat. Math, 13, pp. 97–110. http://www.creative-mathematics.ubm.ro/issues/16_13.pdf
Hull, T. E. , Enright, B. M. , and Sedgwick, A. E. , 1972, “ Comparing Numerical Methods for Ordinary Differential Equations,” SIAM J. Numer. Anal., 9(4), pp. 603–637. [CrossRef]
Enright, H. W. , Hull, T. E. , and Lindberg, B. , 1975, “ Comparing Numerical Methods for Stiff Systems of ODE's,” BIT Numer. Math., 15(1), pp. 10–48. [CrossRef]
Krogh, F. T. , 1973, “ On Testing a Subroutine for the Numerical Integration of Ordinary Differential Equations,” J. ACM, 20(4), pp. 545–562. [CrossRef]
Enright, W. H. , and Pryce, J. D. , 1987, “ Two FORTRAN Packages for Assessing Initial Value Methods,” ACM Trans. Math. Software, 13(1), pp. 1–27. [CrossRef]
Shampine, L. F. , 1981, “ Evaluation of a Test Set for Stiff ODE Solvers,” ACM Trans. Math. Software, 7(4), pp. 409–420. [CrossRef]
Nowak, U. , and Gebauer, S. , 1997, “ A New Test Frame for Ordinary Differential Equation Solvers,” Zuse Institute Berlin (ZIB), Berlin.
Mazzia, F. , Iavernaro, F. , and Magherini, C. , 2008, “ Test Set for Initial Value Problem Solvers,” Department of Mathematics, University of Bari, Bari, Italy, Report No. 40. http://www.ii.uni.wroc.pl/~asz/Magazyn/ksiazkidysk/TESTSET.PDF
Soetaert, K., Cash, J., Mazzia, F., and LAPACK, 2017, “deTestSet: Test Set for Differential Equations,” University of California Berkeley, Berkeley, CA, accessed June 26, 2017, https://cran.r-project.org/web/packages/deTestSet/index.html
Werder, M., 2017, “Differential Equation (Ode & Dae) Solver Test Suite,” GitHub, San Francisco, CA, accessed June 26, 2017, https://github.com/mauro3/IVPTestSuite.jl
Auer, E. , and Rauh, A. , 2012, “ VERICOMP: A System to Compare and Assess Verified IVP Solvers,” Computing, 94(2–4), pp. 163–172. [CrossRef]
MathWorks, 2017, “Chose an ODE Solver,” MathWorks, Natick, MA, accessed June 26, 2017, https://www.mathworks.com/help/matlab/math/choose-an-ode-solver.html
Gattwald, B. A. , and Wanner, G. , 1982, “ Comparison of Numerical Methods for Stiff Differential Equations in Biology and Chemistry,” Simulation, 38(2), pp. 61–66. [CrossRef]
Radhakrishnan, K. , 1984, “ Comparison of Numerical Techniques for Integration of Stiff Ordinary Differential Equations Arising in Combustion Chemistry,” NASA Lewis Research Center, Cleveland, OH, NASA Technical Paper No. 2372. https://ntrs.nasa.gov/search.jsp?R=19850001758
Sandu, A. , Verwer, J. G. , Van Loon, M. , Carmichael, G. R. , Potra, F. A. , Dabdub, D. , and Seinfeld, J. H. , 1997, “ Benchmarking Stiff ODE Solvers for Atmospheric Chemistry Problems—I: Implicit vs Explicit,” Atmos. Environ., 31(19), pp. 3151–3166. [CrossRef]
Nejad, L. A. , 2005, “ A Comparison of Stiff ODE Solvers for Astrochemical Kinetics Problems,” Astrophys. Space Sci., 299(1), pp. 1–29. [CrossRef]
Petzold, L. , 1983, “ Automatic Selection of Methods for Solving Stiff and Nonstiff Systems of Ordinary Differential Equations,” SIAM J. Sci. Stat. Comput., 4(1), pp. 136–148. [CrossRef]
Shampine, L. F. , 1977, “ Stiffness and Nonstiff Differential Equation Solvers—II: Detecting Stiffness With Runge–Kutta Methods,” ACM Trans. Math. Software, 3(1), pp. 44–53. [CrossRef]
Liu, L. , Felgner, F. , and Frey, G. , 2010, “ Comparison of 4 Numerical Solvers for Stiff and Hybrid Systems Simulation,” IEEE 15th Conference on Emerging Technologies and Factory Automation (ETFA), Bilbao, Spain, Sept. 13–16, pp. 1–8.
Abelman, S. , and Patidar, K. C. , 2008, “ Comparison of Some Recent Numerical Methods for Initial-Value Problems for Stiff Ordinary Differential Equations,” Comput. Math. Appl., 55(4), pp. 733–744. [CrossRef]
Shampine, L. , and Thompson, S. , 2000, “ Event Location for Ordinary Differential Equations,” Comput. Math. Appl., 39(5–6), pp. 43–54. [CrossRef]
Shampine, L. , and Reichelt, M. , 1997, “ The MATLAB ODE Suite,” SIAM J. Sci. Comput., 18(1), pp. 1–22. [CrossRef]
Shampine, L. F. , Gladwell, I. , and Thompson, S. , 2003, Solving ODEs With MATLAB, Cambridge University Press, Cambridge, UK. [CrossRef]
Ashino, R. , Nagase, M. , and Vaillancourt, R. , 2000, “ Behind and Beyond the MATLAB ODE Suite,” Comput. Math. Appl., 40(4), pp. 491–512. [CrossRef]
Gilardi, G. , and Sharf, I. , 2002, “ Literature Survey of Contact Dynamics Modeling,” Mech. Mach. Theory, 37(10), pp. 1213–1239. [CrossRef]
Khulief, Y. A. , 2012, “ Modeling of Impact in Multibody Systems: An Overview,” ASME. J. Comput. Nonlinear Dyn., 8(2), p. 021012. [CrossRef]
Hunt, K. H. , and Crossley, F. E. , 1975, “ Coefficient of Restitution Interpreted as Damping in Vibroimpact,” ASME J. Appl. Mech., 42(2), pp. 440–445. [CrossRef]
Buckingham, E. , 1914, “ On Physically Similar Systems; Illustrations of the Use of Dimensional Equations,” Phys. Rev., 4(4), pp. 345–376. [CrossRef]
Cherouvim, N. , and Papadopoulos, E. , 2008, “ The SAHR Robot—Controlling Hopping Speed and Height Using a Single Actuator,” Appl. Bionics Biomech., 5(3), pp. 149–156.
Ahmadi, M. , and Buehler, M. , 2006, “ Controlled Passive Dynamic Running Experiments With the ARL-Monopod II,” IEEE Trans. Rob., 22(5), pp. 974–986. [CrossRef]
Raibert, M. , 1986, Legged Robots That Balance, The MIT Press, Cambridge, MA, pp. 145–146.
Okubo, H. , Nakano, E. , and Handa, M. , 1996, “ Design of a Jumping Machine Using Self-Energizing Spring,” IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), Osaka, Japan, Nov. 4–8, pp. 186–191.
Vasilopoulos, V. , Paraskevas, I. , and Papadopoulos, E. , 2015, “ Control and Energy Considerations for a Hopping Monopod on Rough Compliant Terrains,” IEEE International Conference on Robotics and Automation (ICRA), Seattle, WA, May 26–30, pp. 4570–4575.
Spijker, M. , 1996, “ Stiffness in Numerical Initial-Value Problems,” J. Comput. Appl. Math., 72(2), pp. 393–406. [CrossRef]
MathWorks, 2016, “ODE Event Location,” MathWorks, Natick, MA, accessed June 26, 2017, https://www.mathworks.com/help/matlab/math/ode-event-location.html
Papetti, S. , Avanzini, F. , and Rocchesso, D. , 2011, “ Numerical Methods for a Nonlinear Impact Model: A Comparative Study With Closed-Form Corrections,” IEEE Trans. Audio Speech Lang. Process., 19(7), pp. 2146–2158. [CrossRef]
Machairas, K. , and Papadopoulos, E. , 2016, “ An Active Compliance Controller for Quadruped Trotting,” 24th Mediterranean Conference on Control and Automation (MED), Athens, Greece, June 21–24, pp. 743–748.
Saha, S. , Fiorini, P. , and Shah, S. , 2006, “ Landing Mechanisms for Hopping Robots: Considerations and Prospects,” Ninth ESA Workshop on Advanced Space Technologies for Robotics and Automation (ASTRA), Noordwijk, The Netherlands, Nov. 28–30, pp. 1–8.


Grahic Jump Location
Fig. 1

Two snapshots of a vertically hopping monopod at different phases: (a) flight phase and (b) stance phase

Grahic Jump Location
Fig. 2

Response of hopping monopod by various solvers, for error tolerance (1 × 10−3) and transition by “if…else…end” statement

Grahic Jump Location
Fig. 3

Absolute error between the reference toe displacement and the toe displacement provided by an ODE solver

Grahic Jump Location
Fig. 4

Correlation of absolute and relative error from reference with DT and IAE criteria




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