The Hermite–Obreshkov–Padé (HOP) method of numerical integration is applicable to stiff systems of differential equations, where the linearization has large range of eigenvalues. A practical implementation of HOP requires the ability to determine high-order time derivatives of the system variables. In the case of a constrained multibody dynamical system, the power series solution for the kinematic differential equation is the foundation for an algorithmic differentiation (AD) procedure determining those derivatives. The AD procedure is extended in this paper to determine rates of change in the time derivatives with respect to variation in the position and velocity state variables of the multibody system. The coefficients of this variation form the Jacobian matrix required for Newton–Raphson iteration. That procedure solves the implicit relations for the state variables at the end of each integration time step. The resulting numerical method is applied to the rotation of a dynamically unbalanced constant-velocity (CV) shaft coupling, where the deflection angle of the output shaft is constrained to low levels by springs of high rate and damping.

References

References
1.
Griewank
,
A.
,
1996
, “
ODE Solving via Automatic Differentiation and Rational Prediction
,”
Numerical Analysis 1995
, Vol.
344
,
D. F.
Griffiths
, and
G. A.
Watson
, eds.,
Pitman Research Notes in Mathematics Series
,
Longman, Harlow, United Kingdom
, pp.
36
56
.
2.
Corliss
,
G.
,
Griewank
,
A.
, and
Henneberger
,
P.
,
1997
, “
High-Order Stiff ODE Solvers Via Automatic Differentiation and Rational Prediction
,”
Numerical Analysis and Its Applications
, Vol.
1196
,
L.
Vulkov
,
J.
Wasniewski
, and
P.
Yalamov
, eds., Lecture Notes in Computer Science,
Springer
,
Berlin, Germany
, pp.
114
125
.
3.
Barrio
,
R.
,
2005
, “
Performance of the Taylor Series Method for ODEs/DAEs
,”
Appl. Math. Comput.
,
163
(
2
), pp.
525
545
.10.1016/j.amc.2004.02.015
4.
Milenkovic
,
P.
,
2012
, “
Series Solution for Finite Displacement of Single-Loop Spatial Linkages
,”
ASME J. Mech. Rob.
,
4
(
2
), p.
021016
.10.1115/1.4006193
5.
Milenkovic
,
P.
,
2011
, “
Solution of the Forward Dynamics of a Single-Loop Linkage Using Power Series
,”
ASME J. Dyn. Syst., Meas., Control
,
133
(
6
), p.
061002
.10.1115/1.4004766
6.
Griewank
,
A.
, and
Walther
,
A.
,
2008
,
Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation
,
Society for Industrial and Applied Mathematics (SIAM)
,
Philadelphia, PA
.
7.
Griewank
,
A.
, and
Walther
,
A.
,
2006
, “
On the Efficient Generation of Taylor Expansions for DAE Solutions by Automatic Differentiation
,”
Applied Parallel Computing, State of the Art in Scientific Computing
,
J.
Dongarra
, ,
K.
Madsen
, and
J.
Wasniewski
, eds.,
Springer
,
Berlin, Germany
, pp.
1089
1098
.
8.
Hairer
,
E.
,
Nørsett
,
S. P.
, and
Wanner
,
G.
,
1993
,
Solving Ordinary Differential Equations: Nonstiff Problems
,
Springer
,
Berlin, Germany
.
9.
Hairer
,
E.
, and
Wanner
,
G.
,
1991
,
Solving Ordinary Differential Equations II, Stiff and Differential-Algebraic Problems
,
Springer
,
Berlin, Germany
.
10.
Iserles
,
A.
, and
Nørsett
,
S. P.
,
1991
,
Order Stars
,
Chapman & Hall
,
London, United Kingdom
.
11.
Martín-Vaquero
,
J.
,
2010
, “
A 17th-Order Radau IIA Method for Package RADAU. Applications in Mechanical Systems
,”
Comput. Math. Appl.
,
59
(
8
), pp.
2464
2472
.10.1016/j.camwa.2009.12.025
12.
Gad
,
E.
,
Nakhla
,
M.
, and
Achar
,
R.
,
2009
, “
A-Stable and L-Stable High-Order Integration Methods for Solving Stiff Differential Equations
,”
IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst.
,
28
(
9
), pp.
1359
1372
.10.1109/TCAD.2009.2024712
13.
Zhou
,
Y.
,
Gad
,
E.
, and
Nakhla
,
M. S.
,
2012
, “
Structural Characterization and Efficient Implementation Techniques for-Stable High-Order Integration Methods
,”
IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst.
,
31
(
1
), pp.
101
108
.10.1109/TCAD.2011.2167326
14.
Milenkovic
,
P.
,
2013
, “
Projective Constraint Stabilization for a Power Series Forward Dynamics Solver
,”
ASME J. Dyn. Syst., Meas. Control
,
135
(
3
), p.
031004
.10.1115/1.4023212
15.
Scholz
,
H. E.
,
1998
, “
The Examination of Nonlinear Stability and Solvability of the Algebraic Equations for the Implicit Taylor Series Method
,”
Appl. Num. Math.
,
28
(
2
), pp.
439
458
.10.1016/S0168-9274(98)00059-2
16.
Negrut
,
D.
,
Haug
,
E. J.
, and
German
,
H. C.
,
2003
, “
An Implicit Runge–Kutta Method for Integration of Differential Algebraic Equations of Multibody Dynamics
,”
Multibody Syst. Dyn.
,
9
(
2
), pp.
121
142
.10.1023/A:1022506312444
17.
Eriksson
,
K.
,
Johnson
,
C.
, and
Logg
,
A.
,
2004
, “
Explicit Time-Stepping for Stiff ODEs
,”
SIAM J. Scientific Comput.
,
25
(
4
), pp.
1142
1157
.10.1137/S1064827502409626
18.
Featherstone
,
R.
,
2010
, “
Exploiting Sparsity in Operational-Space Dynamics
,”
Int. J. Rob. Res.
,
29
(
10
), pp.
1353
1368
.10.1177/0278364909357644
19.
Featherstone
,
R.
,
2005
, “
Efficient Factorization of the Joint-Space Inertia Matrix for Branched Kinematic Trees
,”
Int. J. Rob. Res.
,
24
(
6
), pp.
487
500
.10.1177/0278364905054928
20.
R.
Featherstone
, 2011, personal communication.
You do not currently have access to this content.