Abstract
The basis for any model-based control of dynamical systems is a numerically efficient formulation of the motion equations, preferably expressed in terms of a minimal set of independent coordinates. To this end the coordinates of a constrained system are commonly split into a set of dependent and independent ones. The drawback of such coordinate partitioning is that the splitting is not globally valid since an atlas of local charts is required to globally parameterize the configuration space. Therefore different formulations in redundant coordinates have been proposed. They usually involve the inverse of the mass matrix and are computationally rather complex. In this paper an efficient formulation of the motion equations in redundant coordinates is presented for general non-holonomic systems that is valid in any regular configuration. This gives rise to a globally valid system of redundant differential equations. It is tailored for solving the inverse dynamics problem, and an explicit inverse dynamics solution is presented for general full-actuated systems. Moreover, the proposed formulation gives rise to a non-redundant system of motion equations for non-redundantly full-actuated systems that do not exhibit input singularities.
Article PDF
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.Avoid common mistakes on your manuscript.
References
Aghili, F.: A unified approach for inverse and direct dynamics of constrained multibody systems based on linear projection operator: applications to control and simulation. IEEE Trans. Robot. 21(5), 834–849 (2005)
Agrawal, O.P., Saigal, S.: Dynamic analysis of multi-body systems using tangent coordinates. Comput. Struct. 31(3), 349–355 (1989)
Angles, J., Lee, S.: The formulation of dynamical equations of holonomic mechanical systems using a natural orthogonal complement. ASME J. Appl. Mech. 55, 243–244 (1988)
Ascher, U.M., Chin, H., Reich, S.: Stabilization of DAEs and invariant manifolds. Numer. Math. 67, 131–149 (1994)
Bayo, E., Avello, A.: Singularity-free augmented Lagrangian algorithms for constrained multibody dynamics. Nonlinear Dyn. 5, 209–231 (1994)
Blajer, W., Schiehlen, W., Schirm, W.: A projective criterion to the coordinate partitioning method for multibody dynamics. Arch. Appl. Mech. 64, 86–98 (1994)
Bloch, A.M.: Nonholonomic mechanics and control. In: Interdisciplinary Applied Mathematics, vol. 24. Springer, Berlin (2003)
Bloch, A.M., Krishnaprasad, P.S., Marsden, J.E., Murray, R.M.: Nonholonomic mechanical systems with symmetry. Arch. Ration. Mech. Anal. 136(1), 21–99 (1996)
Brauchli, H.: Mass-orthogonal formulation of equations of motion for multibody systems. Z. Angew. Math. Phys. 42(2), 169–182 (1991)
Bullo, F., Lewis, A.D.: Geometric control of mechanical systems. In: Texts in Applied Mathematics, vol. 49. Springer, Berlin (2004)
Fisette, P., Vaneghem, B.: Numerical integration of multibody system dynamic equations using the coordinate partitioning method in an implicit Newmark scheme. Comput. Methods Appl. Mech. Eng. 135(1–2), 85–105 (1996)
Galicki, M.: Task space control of mobile manipulators. Robotica 29, 221–232 (2011)
García De Jalón, J., Unda, J., Avello, A.: Natural coordinates for the computer analysis of multibody systems. Comput. Methods Appl. Mech. Eng. 56(3), 309–327 (1986)
Haug, J.E., Yen, J.: Generalized coordinate partitioning methods for numerical integration of differential/algebraic equations of dynamics. In: Haug, E.J., Deyo, R.C. (eds.) NATO ASI Series, vol. F69, pp. 97–114. Springer, Berlin (1990)
Hufnagel, T., Müller, A.: A realtime coordinate switching method for model-based control of parallel manipulators. In: ECCOMAS Thematic Conference on Multibody Dynamics, Brussels, Belgium, 4–7 July (2011)
Hufnagel, T., Schramm, D.: Consequences of the use of decentralized controllers for redundantly actuated parallel manipulators. In: 13th World Congress in Mechanism and Machine Science, Guanajuato, Mexico, 19–25 June (2011)
Huston, R.L., Passerello, C.E.: Nonholonomic systems with non-linear constraint equations. Int. J. Non-Linear Mech. 11(5), 331–336 (1976)
Jungnickel, U., Kielau, G., Maißer, P. Müller, A.: A passivity-based control of Euler–Lagrange systems with a non-quadratic Lagrangian. Z. Angew. Math. Mech. 88(12), 982–992 (2008)
Kamman, J.W., Huston, R.L.: Dynamics of constrained multibody systems. J. Appl. Mech. 51, 899–903 (1984)
Laulusa, A., Bauchau, O.A.: Review of classical approaches for constraint enforcement in multibody systems. J. Comput. Nonlinear Dyn. 3, 011004 (2008)
Maggi, G.A.: Da alcune nuove forma della equazioni della dinamica applicabile ai sistemi anolonomi. Atti Reale Acad. Naz. Lincei. Rend. Cl. fis. e math, Ser. 5 10(2), 287–291 (1901)
Maisser, P.: A differential geometric approach to the multi body system dynamics. Z. Angew. Math. Mech. 71(4), T116–T119 (1991)
Mani, N.K., Haug, E.J., Atkinson, K.E.: Application of SVD for analysis of mechanical systems dynamics. J. Mech. Transm. Autom. Des. 107, 82–87 (1985)
Merlet, J.P.: Redundant parallel manipulators. Lab. Robot. Autom. 8(1), 17–24 (1996)
Müller, A.: On the concept of mobility used in robotics. In: 33rd Mechanisms & Robotics Conference ASME 2009 International Design Engineering Technical Conferences, San Diego, CA, 30 August–2 September (2009)
Müller, A.: A robust inverse dynamics formulation for redundantly actuated PKM. In: 13th World Congress in Mechanism and Machine Science, Guanajuato, Mexico, 19–25 June (2011)
Müller, A.: Semialgebraic regularization of kinematic loop constraints in multibody system models. J. Comput. Nonlinear Dyn. 6, 041010 (2011)
Müller, A., Hufnagel, T.: Adaptive and singularity-free inverse dynamics models for control of parallel manipulators with actuation redundancy. In: 8th International Conference on Multibody Systems, Nonlinear Dynamics, and Control, ASME 2011 International Design Engineering Technical Conferences, Washington, DC, 28–31 August (2011)
Murray, R.M., Li, Z., Sastry, S.S.: A mathematical Introduction to Robotic Manipulation. CRC Press, Boca Raton (1993)
Negrut, D., Haug, E.J., German, H.C.: An implicit Runge-Kutta method for integration of differential algebraic equations of multibody dynamics. Multibody Syst. Dyn. 9, 121–142 (2003)
Nikravesh, P.E.: Some methods for dynamic analysis of constrained mechanical systems: a survey. In: Haug, E.J. (ed.) Computer Aided Analysis and Optimization of Mechanical Systems Dynamics, pp. 351–367. Springer, Berlin, Heidelberg (1984)
Nikravesh, P.E.: Computer Aided Analysis of Mechanical Systems. Prentice-Hall, London (1988)
Nikravesh, P.E., Skinivasan, M.: Generalized co-ordinate partitioning in static equilibrium analysis of large-scale mechanical systems. Int. J. Numer. Methods Eng. 21, 451–464 (1985)
Ortega, R., Loria, A., Nicklasson, P.J., Sira-Ramirez, H.: Passivity-based control of Euler–Lagrange-systems. In: Mechanical, Electrical and Electromechanical Applications. Springer, London (1998)
Passerello, C.E., Huston, R.L.: Another look at nonholonomic systems. J. Appl. Mech. 40, 101–104 (1973)
Sing, R.P., Likings, P.W.: Singular Value decomposition for constrained dynamical systems. J. Appl. Mech. 52, 943–948 (1985)
Terze, Z., Naudet, J.: Structure of optimized generalized coordinates partitioned vectors for holonomic and non-holonomic systems. Multibody Syst. Dyn. 24(2), 203–218 (2010)
Udwadia, F.E., Kalaba, R.E.: On the foundations of analytical dynamics. Int. J. Non-Linear Mech. 37, 1079–1090 (2002)
Voronets, P.V.: Equations of motion for nonholonomic systems. Mat. Sb. 22(4) (1901)
Wehage, R.A., Haug, E.J.: Generalized coordinates partitioning for dimension reduction in analysis of constrained dynamic systems. J. Mech. Des. 104, 247–255 (1982)
Wojtyra, M.: Joint reactions in rigid body mechanisms with dependent constraints. Mech. Mach. Theory 44(12), 2265–2278 (2009)
Zlatanov, D., Fenton, R.G., Benhabib, B.: Identification and classification of the singular configurations of mechanisms. Mech. Mach. Theory 33(6), 743–760 (1998)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Müller, A. Motion equations in redundant coordinates with application to inverse dynamics of constrained mechanical systems. Nonlinear Dyn 67, 2527–2541 (2012). https://doi.org/10.1007/s11071-011-0165-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-011-0165-5