Research Papers

Projective Constraint Stabilization for a Power Series Forward Dynamics Solver

[+] Author and Article Information
Paul Milenkovic

Department of Electrical and Computer Engineering,
University of Wisconsin—Madison,
1415 Engineering Drive,
Madison, WI 53706
e-mail: phmilenk@wisc.edu

The top panel of Fig. 4 from the earlier paper [5] is inverted in relation to Table 1 in that paper. The corrected orientation is given in Fig. 4 of the current paper.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received March 8, 2012; final manuscript received November 30, 2012; published online February 21, 2013. Assoc. Editor: Nariman Sepehri.

J. Dyn. Sys., Meas., Control 135(3), 031004 (Feb 21, 2013) (9 pages) Paper No: DS-12-1079; doi: 10.1115/1.4023212 History: Received March 08, 2012; Revised November 30, 2012

A power series expression for the forward dynamics of a closed kinematic chain provides an explicit time-step update of the system state. The resulting numerical differential equation solver applies kinematic constraints to the power series terms for acceleration and higher derivatives of motion. Integrating acceleration determines velocity and position time histories that approximate the constraints to a high degree of precision when using a high order of the expansion. When high precision is not required, a lower order achieves shorter computation times, but that condition results in violation of the constraints in the absence of any correction. Projecting the velocities and positions onto the constraint manifold after each time step produces step changes. This paper determines which choices of linear subspace for this projection give step changes that are equal to the residues of truncating the power series solution for the kinematic portion of the problem. The limit of that power series gives position and velocity time histories that approximate the dynamics while giving an exact kinematic solution. Thus projection onto the constraints in this procedure determines sample values of an underlying solution for the motion trajectories, where that underlying solution is continuous in both velocity and position and also satisfies the kinematic constraints at all times. This property is confirmed by numerical simulation of a Clemens constant-velocity coupling.

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Baumgarte, J., 1972, “Stabilization of Constraints and Integrals of Motion in Dynamical Systems,” Comput. Methods Appl. Mech. Eng., 1(1) pp. 1–16. [CrossRef]
Gear, C., Leimkuhler, B., and Gupta, G., 1985, “Automatic Integration of Euler-Lagrange Equations With Constraints,” J. Comput. Appl. Math., 12-13, pp. 77–90. [CrossRef]
Führer, C., and Leimkuhler, B., 1991, “Numerical Solution of Differential-Algebraic Equations for Constrained Mechanical Motion,” Numer. Math., 59(1), pp. 55–69. [CrossRef]
Ascher, U. M., Chin, H., and Reich, S., 1994, “Stabilization of DAEs and Invariant Manifolds,” Numer. Math., 67(2), pp. 131–149. [CrossRef]
Milenkovic, P., 2011, “Solution of the Forward Dynamics of a Single-Loop Linkage Using Power Series,” ASME J. Dyn. Sys., Meas., Control, 133(6), p. 061002. [CrossRef]
Milenkovic, P., 2012, “Series Solution for Finite Displacement of Single-Loop Spatial Linkages,” ASME J. Mechanisms Robotics, 4(2), p. 021016. [CrossRef]
Sheth, P. N., and Uicker, J. J., 1972, “IMP (Integrated Mechanisms Program), A Computer-Aided Design Analysis System for Mechanisms and Linkage,” ASME J. Eng. Industry, 94(2), pp. 454–464. [CrossRef]
Elmqvist, H., and Otter, M., 1994, “ Methods for Tearing Systems of Equations in Object-Oriented Modeling,” Proceedings ESM’94, European Simulation Multiconference, Barcelona, Spain, pp. 1–3.
Kecskemethy, A., Krupp, T., and Hiller, M., 1997, “Symbolic Processing of Multiloop Mechanism Dynamics Using Closed-Form Kinematics Solutions,” Multibody Syst. Dyn., 1(1), pp. 23–45. [CrossRef]
Anderson, K. S., and Critchley, J., 2003, “Improved Order-N' Performance Algorithm for the Simulation of Constrained Multi-Rigid-Body Dynamic Systems,” Multibody Syst. Dyn., 9(2), pp. 185–212. [CrossRef]
Schiehlen, W., 1997, “Multibody System Dynamics: Roots and Perspectives,” Multibody Syst. Dyn., 1(2), pp. 149–188. [CrossRef]
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. [CrossRef]
Blajer, W., 2002, “Elimination of Constraint Violation and Accuracy Aspects in Numerical Simulation of Multibody Systems,” Multibody Syst. Dyn., 7(3), pp. 265–284. [CrossRef]
Han, H. S., and Seo, J. H., 2004, “Design of a Multi-Body Dynamics Analysis Program Using the Object-Oriented Concept,” Adv. Eng. Software, 35(2), pp. 95–103. [CrossRef]
Denavit, J., 1955, “A Kinematic Notation for Lower-Pair Mechanisms Based on Matrices,” ASME J. Appl. Mech., 22, pp. 215–221.
Uicker, J. J., Jr., Denavit, J., and Hartenberg, R., 1964, “An Iterative Method for the Displacement Analysis of Spatial Mechanisms,” ASME J. Appl. Mech., 31(2), pp. 309–314. [CrossRef]
Paul, R. P., 1981, Robot Manipulators: Mathematics, Programming, and Control: The Computer Control of Robot Manipulators, The MIT Press, Cambridge, MA.
Park, F. C., and Martin, B. J., 1994, “Robot Sensor Calibration: Solving AX= XB on the Euclidean Group,” IEEE Trans. Rob. Autom., 10(5), pp. 717–721. [CrossRef]
Selig, J., 2005, Geometric Fundamentals of Robotics, 2nd ed., Springer, New York.
Iowa Department of Transportation, Office of Design, 2010, “Horizontal Curves,” Design Manual, Chap. 2(2A-1).
McCarthy, J. M., 2000, Geometric Design of Linkages, Springer, New York.
Dopico, D., Gonzalez, M., and Lugris, U., 2006, “Energy Conserving and Projection Methods for the Real-Time Dynamics of Multibody Systems,” Proceedings of ACMD06, Paper No. A00589.
Milenkovic, P., 2009, “Triangle Pseudocongruence in Constraint Singularity of Constant-Velocity Couplings,” ASME J. Mechanisms Robotics, 1(2), p. 021006. [CrossRef]
Milenkovic, P., 2011, “Nonsingular Spherically Constrained Clemens Linkage Wrist,” ASME J. Mechanisms Robotics, 3(1), p. 011014. [CrossRef]
Skokos, C., 2001, “Alignment Indices: A New, Simple Method for Determining the Ordered or Chaotic Nature of Orbits,” J. Phys. A, 34, pp. 10029–10043. [CrossRef]
Xie, Y., and Steven, G. P., 1994, “Instability, Chaos, and Growth and Decay of Energy of Time-Stepping Schemes for Non-Linear Dynamic Equations,” Commun. Numer. Methods Eng., 10(5), pp. 393–401. [CrossRef]


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Fig. 1

Clemens CV coupling at initial posture: direction vectors ω1,…,ω9 are placed along the joint axis lines

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Fig. 2

Planar projection of the screw motion of a point following an arc of radius R and angle θ

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Fig. 3

Dynamic response of Clemens CV coupling input shaft rotation angle

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Fig. 4

Dynamic response of Clemens CV coupling output shaft pitch and yaw deflection angles

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Fig. 5

Projective correction to the order 2 solution of the input shaft rotation angle

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Fig. 6

Projective correction to the order 2 solution of the input shaft rotation rate

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Fig. 7

Projective correction to the order 4 solution of the input shaft rotation rate

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Fig. 8

Relative computation times for the order 8 forward-dynamics solution




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