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.

References

References
1.
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
.
2.
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
.
3.
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.
4.
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
5.
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.
6.
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
.
7.
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
8.
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
.
9.
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
.
10.
Krogh
,
F. T.
,
1973
, “
On Testing a Subroutine for the Numerical Integration of Ordinary Differential Equations
,”
J. ACM
,
20
(
4
), pp.
545
562
.
11.
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
.
12.
Shampine
,
L. F.
,
1981
, “
Evaluation of a Test Set for Stiff ODE Solvers
,”
ACM Trans. Math. Software
,
7
(
4
), pp.
409
420
.
13.
Nowak
,
U.
, and
Gebauer
,
S.
,
1997
, “
A New Test Frame for Ordinary Differential Equation Solvers
,” Zuse Institute Berlin (ZIB), Berlin.
14.
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
15.
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
16.
Werder
,
M.
,
2017
, “Differential Equation (Ode & Dae) Solver Test Suite,” GitHub, San Francisco, CA, accessed June 26, 2017, https://github.com/mauro3/IVPTestSuite.jl
17.
Auer
,
E.
, and
Rauh
,
A.
,
2012
, “
VERICOMP: A System to Compare and Assess Verified IVP Solvers
,”
Computing
,
94
(
2–4
), pp.
163
172
.
18.
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
19.
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
.
20.
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
21.
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
.
22.
Nejad
,
L. A.
,
2005
, “
A Comparison of Stiff ODE Solvers for Astrochemical Kinetics Problems
,”
Astrophys. Space Sci.
,
299
(
1
), pp.
1
29
.
23.
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
.
24.
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
.
25.
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
.
26.
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
.
27.
Shampine
,
L.
, and
Thompson
,
S.
,
2000
, “
Event Location for Ordinary Differential Equations
,”
Comput. Math. Appl.
,
39
(
5–6
), pp.
43
54
.
28.
Shampine
,
L.
, and
Reichelt
,
M.
,
1997
, “
The MATLAB ODE Suite
,”
SIAM J. Sci. Comput.
,
18
(
1
), pp.
1
22
.
29.
Shampine
,
L. F.
,
Gladwell
,
I.
, and
Thompson
,
S.
,
2003
,
Solving ODEs With MATLAB
,
Cambridge University Press
,
Cambridge, UK
.
30.
Ashino
,
R.
,
Nagase
,
M.
, and
Vaillancourt
,
R.
,
2000
, “
Behind and Beyond the MATLAB ODE Suite
,”
Comput. Math. Appl.
,
40
(
4
), pp.
491
512
.
31.
Gilardi
,
G.
, and
Sharf
,
I.
,
2002
, “
Literature Survey of Contact Dynamics Modeling
,”
Mech. Mach. Theory
,
37
(
10
), pp.
1213
1239
.
32.
Khulief
,
Y. A.
,
2012
, “
Modeling of Impact in Multibody Systems: An Overview
,”
ASME. J. Comput. Nonlinear Dyn.
,
8
(
2
), p.
021012
.
33.
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
.
34.
Buckingham
,
E.
,
1914
, “
On Physically Similar Systems; Illustrations of the Use of Dimensional Equations
,”
Phys. Rev.
,
4
(
4
), pp.
345
376
.
35.
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
.
36.
Ahmadi
,
M.
, and
Buehler
,
M.
,
2006
, “
Controlled Passive Dynamic Running Experiments With the ARL-Monopod II
,”
IEEE Trans. Rob.
,
22
(
5
), pp.
974
986
.
37.
Raibert
,
M.
,
1986
,
Legged Robots That Balance
,
The MIT Press
, Cambridge, MA, pp.
145
146
.
38.
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.
39.
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.
40.
Spijker
,
M.
,
1996
, “
Stiffness in Numerical Initial-Value Problems
,”
J. Comput. Appl. Math.
,
72
(
2
), pp.
393
406
.
41.
MathWorks, 2016, “ODE Event Location,” MathWorks, Natick, MA, accessed June 26, 2017, https://www.mathworks.com/help/matlab/math/ode-event-location.html
42.
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
.
43.
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.
44.
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.
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