Robot Manipulator Control Theory and Practice - Frank L.Lewis Part 16 doc

Software for Controller Simulation 594 Copyright © 2004 by Marcel Dekker, Inc... For digital control simulation, TRESP needs subroutine DIGIK, T, x, which contains the discrete controlle

593

Software for Controller Simulation 594

Copyright © 2004 by Marcel Dekker, Inc.

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

Software for Controller Simulation 596

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.

[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

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/s2 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=sin2 q i , CC i=cos2 q i , and CS i =cos q i sin q i ;

SS ij=sin2(q i +q j )

Copyright © 2004 by Marcel Dekker, Inc.

Dynamics of Some Common Robot Arms 600

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.

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

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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

Dynamics of Some Common Robot Arms 604

Copyright © 2004 by Marcel Dekker, Inc.

605 C.3 PUMA 560 Manipulator

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 33–669, 1974.

[Paul 1981] Paul, R.P., Robot Manipulators: Mathematics, Programming and Control Cambridge, MA: MIT Press, 1981.

Copyright © 2004 by Marcel Dekker, Inc.