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A recursive, numerically stable, and efficient simulation algorithm for serial robots

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Abstract

Traditionally, the dynamic model, i.e., the equations of motion, of a robotic system is derived from Euler–Lagrange (EL) or Newton–Euler (NE) equations. The EL equations begin with a set of generally independent generalized coordinates, whereas the NE equations are based on the Cartesian coordinates. The NE equations consider various forces and moments on the free body diagram of each link of the robotic system at hand, and, hence, require the calculation of the constrained forces and moments that eventually do not participate in the motion of the coupled system. Hence, the principle of elimination of constraint forces has been proposed in the literature. One such methodology is based on the Decoupled Natural Orthogonal Complement (DeNOC) matrices, reported elsewhere. It is shown in this paper that one can also begin with the EL equations of motion based on the kinetic and potential energies of the system, and use the DeNOC matrices to obtain the independent equations of motion. The advantage of the proposed approach is that a computationally more efficient forward dynamics algorithm for the serial robots having slender rods is obtained, which is numerically stable. The typical six-degree-of-freedom PUMA robot is considered here to illustrate the advantages of the proposed algorithm.

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References

  1. Schiehlen, W.O.: Multibody system dynamics: roots and perspectives. Multibody Syst. Dyn. 1, 149–188 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  2. Garcia, J.: Improved dynamic formulations for the dynamic simulation of multibody systems. Available: http://mat21.etsii.upm.es/mbs/matlabcode/Mbs3dv1/papers/2003GarciaDeJalonEtAl.pdf

  3. Vereshchagin, A.F.: Gauss principle of least constraint for modeling the dynamics of automatic manipulators using a digital computer. Sov. Phys. — Dokl. 20(1), 33–34 (1975)

    MATH  Google Scholar 

  4. Armstrong, W.W.: Recursive solution of the equations of motion of an n-link manipulator. In: Proceedings of the Fifth World Congress on the Theory of Machines and Mechanisms, Montreal, Canada, vol. 2, pp. 1342–1346 (1979)

  5. Walker, M.W., Orin, D.E.: Efficient dynamic computer simulation of robotic mechanisms. ASME J. Dyn. Syst. Meas. Control 104, 205–211 (1982)

    Article  MATH  Google Scholar 

  6. Hollerbach, J.M.: A recursive Lagrangian formulation of manipulator dynamics and a comparative study of dynamics formulation complexity. IEEE Trans. Syst. Man Cybern. SMC-10(11), 730–736 (1980)

    MathSciNet  Google Scholar 

  7. Anderson, K.S.: An order-N formulation for motion simulation of general constrained multi-rigidbody systems. Comput. Struct. 43(3), 565–572 (1992)

    Article  MATH  Google Scholar 

  8. Rosenthal, D.E., Sherman, M.A.: High performance multibody simulations via symbolic equation manipulation and Kane's method. J. Astronaut. Sci. 34(3), 223–239 (1986)

    Google Scholar 

  9. Rosenthal, D.E.: An order N formulation for robotic systems. J. Astronaut. Sci. 38(4), 511–529 (1990)

    Google Scholar 

  10. Banerjee, A.K.: Block-diagonal equations for multibody elastodynamics with geometric stiffness and constraints. J. Guid. Control Dyn. 16(6), 1092–1100 (1993)

    Article  MATH  Google Scholar 

  11. Bae, D.S., Haug, E.J.: A recursive formation for constrained mechanical systems dynamics: part I, open loop systems. Mech. Struct. Mach. 15(3), 359–382 (1987)

    Google Scholar 

  12. Bae, D.S., Haug, E.J.: A recursive formation for constrained mechanical systems dynamics: part II, closed loop systems. Mech. Struct. Mach. 15(4), 481–506 (1987)

    Google Scholar 

  13. Featherstone, R.: The calculation of robotic dynamics using articulated body inertias. Int. J. Robot. Res. 2(1), 13–30 (1983)

    Article  Google Scholar 

  14. Featherstone, R.: Robotic Dynamics Algorithms. Kluwer Academic, Dordrecht, The Netherlands (1987)

    Google Scholar 

  15. Stejskal, V., Valasek, M.: Kinematics and Dynamics of Machinery. M. Dekkar, New York (1996)

    Google Scholar 

  16. Jain, A.: Unified formulation of dynamics for serial rigid multibody systems. J. Guid. Control Dyn. 14(3), 531–542 (1991)

    MATH  Google Scholar 

  17. Saha, S.K.: Dynamic modeling of serial multibody systems using decoupled natural orthogonal complement matrices. ASME J. Appl. Mech. 29(2), 986–996 (1999)

    Article  Google Scholar 

  18. Saha, S.K.: A decomposition of manipulator inertia matrix. IEEE Trans. Robot. Autom. 13(2), 301–304 (1997)

    Article  Google Scholar 

  19. Fijany, A., Inna, S., D'Eleuterios, G.M.T.: Parallel O(log N) algorithms for the computation of manipulator forward dynamics. In: Proceedings of the IEEE ICRA, San Diego, CA, USA, pp. 1547–1553 (1994)

  20. Naudet, J., Lefeber, D.: General formulation of an efficient recursive algorithm based on canonical momenta for forward dynamics of closed-loop multibody systems. In: Proceedigns of the XI DINAME, Ouro Preto, Brazil (2005)

  21. Lee, K., Chirikjain, S.G.: A new perspective on O(n) mass-matrix inversion for serial revolute manipulators. In: Proceedings of IEEE ICRA, Barcelona, Spain, pp. 4733–4737 (2005)

  22. Critchley, J.S., Anderson, K.S.: A generalized recursive coordinate reduction method for multibody system dynamics. Multibody Syst. Dyn. 9, 185–212 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  23. Ascher, U.M., Pai, D.K., Cloutier, B.P.: Forward dynamics, elimination methods, and formulation stiffness in robot simulation. Int. J. Robot. Res. 16(6), 747–758 (1997)

    Article  Google Scholar 

  24. Ellis, R.E., Ismaeil, O.M., Carmichael, I.H.: Numerical stability of forward-dynamics algorithms. In: Proceedings of IEEE Conference on Robotics and Automation, Nice, France, pp. 305–311 (1992)

  25. Nikravesh P.E.: Computer-Aided Analysis of Mechanical Systems. Prentice-Hall, Englewood Cliffs, NJ (1988)

    Google Scholar 

  26. Wehage R.A., Haug, E.J.: Generalized coordinate partitioning for dimension reduction in analysis of constrained dynamic systems. ASME J. Mech. Des. 104, 247–255 (1982)

    Article  Google Scholar 

  27. Kamman, J.W., Huston, R.L.: Dynamics of constrained multibody systems. ASME J. Appl. Mech. 51, 899–903 (1984)

    Article  MATH  Google Scholar 

  28. Angeles, J., Lee, S.K.: The formulation of dynamical equations of holonomic mechanical systems using a natural orthogonal complement. ASME J. Appl. Mech. 55, 243–244 (1988)

    Article  MATH  Google Scholar 

  29. Saha, S.K., Schiehlen, W.O.: Recursive kinematics and dynamics for closed loop multibody systems. Int. J. Mech. Struct. Mach. 29(2), 143–175 (2001)

    Article  Google Scholar 

  30. Khan, W.A., Krovi, V.N., Saha, S.K., Angeles, J.: Modular and recursive kinematics and dynamics for parallel manipulators. Multibody Syst. Dyn. 14, 419–455 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  31. Chaudhary, H., Saha, S.K.: Constraint force formulation for closed-loop multibody systems. Trans. ASME J. Mech. Des. (in press)

  32. Denavit, J., Hartenberg, R.S.: A kinematic notation for lower-pair mechanisms based on matrices. ASME J. Appl. Mech. 77, 445–450 (1955)

    Google Scholar 

  33. Meirovitch, L.: Analytical Methods in Vibrations. Macmillan, New York (1967)

    MATH  Google Scholar 

  34. Pratap, R.: MATLAB 6: A Quick Introduction for Scientists and Engineers. Oxford University Press, New York (2002)

    MATH  Google Scholar 

  35. Strang, G.: Linear Algebra and its Applications. Harcourt, Brace, Jovanovich, Florida (1988)

    Google Scholar 

  36. Bhangale, P.: Dynamics Based Modeling, Computational Complexity and Architecture Selection of Robot Manipulators. Ph.D. Thesis, Indian Institute of Technology, New Delhi (2004)

  37. Luh, J.Y.S., Walker, M.W., Paul, R.P.: On-line computational scheme for mechanical manipulators. Trans. ASME J. Dyn. Syst. Meas. Control 102, 64–76 (1980)

    MathSciNet  Google Scholar 

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  1. Mechanical Engineering Department, Indian Institute of Technology Delhi, New Delhi, 110016, India

    Ashish Mohan & S. K. Saha

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  1. Ashish Mohan
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  2. S. K. Saha
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Correspondence to S. K. Saha.

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Mohan, A., Saha, S.K. A recursive, numerically stable, and efficient simulation algorithm for serial robots. Multibody Syst Dyn 17, 291–319 (2007). https://doi.org/10.1007/s11044-007-9044-8

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