593 Copyright © 2004 by Marcel Dekker, Inc. Software for Controller Simulation594 Copyright © 2004 by Marcel Dekker, Inc. 595 Figure B.1–1: Program TRESP for time response of nonlinear continuous systems. For digital control simulation, TRESP needs subroutine DIG(IK, T, x), which contains the discrete controller equations; it is called once in every sample period T. The time T R should be selected as an integral divisor of T. Five or 10 Runge-Kutta periods within each sample period is usually sufficient. The program also allows digital filtering (e.g., for reconstruction of ve- locity estimates from joint position encoder measurements). Note that for Copyright © 2004 by Marcel Dekker, Inc. Software for Controller Simulation596 digital controls purposes, subroutine DIG is called before the Runge-Kutta routine, while for digital filtering, DIG is called after the call to Runge-Kutta. For some systems the Runge-Kutta integrator in the figure may not work; then an adaptive step-size Runge-Kutta routine (e.g., Runge-KuttaFehlburg) can be used [Press et al. 1986]. (Note: The program given here works for all examples in the book.) Copyright © 2004 by Marcel Dekker, Inc. 597 REFERENCE [Press et al. 1986] Press, W.H., Flannery, B.P., Teukolsky, S.A., and Vetterling, W.T., Numerical Recipes. New York: Cambridge University Press, 1986. Copyright © 2004 by Marcel Dekker, Inc. 599 Appendix C Dynamics of Some Common Robot Arms In this appendix we give the dynamics of some common robot arms. We assume that the robot dynamics are given by (C.1.1) where the matrix M(q) is symmetric and positive definite with elements m ij (q), that is, and N (q, q) is an n×1 vector with elements n i , that is, Note in particular that the gravity terms are indentified in the expressions of n i by the gravity constant g=9.8 meters/s 2 . We will also adopt the following notation: Length of link i is L i in meters Mass of link i is m i in kilograms Mass moment of inertia of Link i about axis u is I uui in kg-m-m S i =sin q i and C i =cos q i S ij =sin(q i +q j ) and C ij =cos(q i +q j ) S ijk =sin(q i +q j +q k ) and C ijk =cos(q i +q j +q k ) SS i =sin 2 q i , CC i =cos 2 q i , and CS i =cos q i sin q i ; SS ij =sin 2 (q i +q j ) Copyright © 2004 by Marcel Dekker, Inc. Dynamics of Some Common Robot Arms600 C.1 SCARA ARM The first robot we consider is a general SCARA configuration robot shown in Figure C.1.1. These equations will apply to the AdeptOne and AdeptTwo robots. The dynamics include the first four degrees of freedom and are symbolically given by Figure C.1.1: SCARA manipulator. Copyright © 2004 by Marcel Dekker, Inc. 601 C.2 Stanford Manipulator The Stanford manipulator shown in Figure C.2.1 has the following dynamics [Bejczy 1974], [Paul 1981]: C.2 Stanford Manipulator Copyright © 2004 by Marcel Dekker, Inc. 603 C.3 PUMA 560 Manipulator The PUMA 560 is shown in Figure C.3.1. Many simplifications can be made for this particular structure in order to obtain the following dynamics which appeared in [Armstrong et al. 1986]. Figure C.3.1: PUMA 560 manipulator. C.3 PUMA 560 Manipulator Copyright © 2004 by Marcel Dekker, Inc. [...]... Common Robot Arms C.3 PUMA 560 Manipulator Copyright © 2004 by Marcel Dekker, Inc 605 REFERENCES [Armstrong et al 1986] Armstrong, B., O.Khatib, and J.Burdick, “The explicit dynamic model and inertial parameters of the PUMA 560 arm,” Proc 1986 IEEE Conf Robot Autom., pp 510–518, San Francisco, Apr 7– 10, 1986 [Bejczy 1974] Bejczy, A.K., Robot arm dynamics and control, ” NASA-JPL Technical Memorandum... Autom., pp 510–518, San Francisco, Apr 7– 10, 1986 [Bejczy 1974] Bejczy, A.K., Robot arm dynamics and control, ” NASA-JPL Technical Memorandum 33–669, 1974 [Paul 1981] Paul, R.P., Robot Manipulators: Mathematics, Programming and Control Cambridge, MA: MIT Press, 1981 607 Copyright © 2004 by Marcel Dekker, Inc . by Marcel Dekker, Inc. Software for Controller Simulation596 digital controls purposes, subroutine DIG is called before the Runge-Kutta routine, while for digital filtering, DIG is called after. 1974] Bejczy, A.K., Robot arm dynamics and control, ” NASA-JPL Technical Memorandum 33–669, 1974. [Paul 1981] Paul, R.P., Robot Manipulators: Mathematics, Programming and Control. Cambridge, MA:. will also adopt the following notation: Length of link i is L i in meters Mass of link i is m i in kilograms Mass moment of inertia of Link i about axis u is I uui in kg-m-m S i =sin q i and