In this article we derive the equations of motion of a kinematic chain using concepts naturally associated with the special euclidean group. A simple and coordinate-free formula for the kinetic energy of a rigid body is used to express the kinetic energy of an open chain. From this, the dynamical equations in the Lagrangian form follow easily. The structure of the equations of motion is sufficiently transparent so that one may gain insight into nontrivial problems involving analysis and design. We illustrate the applicability of this formalism by investigating the design of manipulators with configuration- independent and separable kinetic energy.
Get full access to this article
View all access options for this article.
References
1.
Abraham, R., and Marsden, J.1978. Foundations of Mechanics. Reading, MA: Benjamin Cummings.
2.
Baillieul, J.1991. The behaviour of super-articulated mechanisms subject to periodic forcing. In Bonnard, B., Bride, B., Gauthier, J., and Kupka, I. (eds.): Analysis of Controlled Dynamical Systems. Boston, MA: Birkhauser.
3.
Baillieul, J., and Levi, M.1991. Constrained relative motions in rotational mechanics. Arch. Rational Mech. Anal., 115:101-135.
4.
Bogoyavlenskij, O.I.1993. General integrable problems of classical mechanics. Commun. Math. Phys,. 153:24-47.
5.
Brockett, R.1970. Finite Dimensional Linear Systems. New York: Wiley.
6.
Brockett, R., Stokes, A., and Park, F.1993 (Atlanta, May). A geometrical formulation of the dynamical equations describing kinematic chains. Proc. IEEE Conf. Robotics and Automation, IEEE.
7.
Brockett, R.W.1984. Robotic manipulators and the product exponentials formulae . In Lecture Notes in Control and Information Sciences , Proceedings of the International Symposium on Mathematical Theory of Networks and Systems. Berlin: Springer-Verlag, pp. 120-127.
8.
Krishnaprasad, P.S.1990. Geometric phases and optimal reconfiguration of multibody systems. Proc. 1990 American Control Conf, pp. 2440-2444.
9.
McCarthy, J.M.1990. Introduction to Theoretical Kinematics. Cambridge, MA: MIT Press.
10.
Mishchenko, A., and Fomenko, A.1982. Integrability of euler equations on semisimple Lie algebras . Selecta Mathematica Sovietica2(3):207-292.
11.
Murray, R., Li, Z., and Sastry, S.1994. A Mathematical Introduction to Robotic Manipulation. Boca Raton, FL: CRC Press.
12.
Paden, B. and Sastry, S.1988. Optimal kinematic design of 6R manipulators. Int. J. Robot. Res.7(2):43-61.
13.
Park, F.1994. Computational aspects of the product-of-exponentials formula for robot kinematics. IEEE Trans. Auto. Control, 39(3):643-647.
14.
Park, F. and Brockett, R.1994. Kinematic dexterity of robotic mechanisms. Int. J. Robot. Res.13(1):1-15.
15.
Ratiu, T.1982. Euler-poisson equations on Lie algebras and the n- dimensional heavy rigid body. Am. J. Math.104:409-448.
16.
Selig, J.1991. Applying screw theory to robot dynamics. J. Appl. Math. Mechanics55(2):159-167.
17.
Stokes, A.1993. Geometric methods for the design of mechanisms. Ph.D. thesis, Harvard University.
18.
Youcef-Toumi, K., and Asada, H.1987. The design of open-loop manipulator arms with decoupled and configuration-invariant inertia tensors. J. Dynam. Sys. Meas. Control109(3):268-275.
19.
Zhuang, H., Roth, Z., and Hamano, F.1992. A complete and parametrically continuous kinematic model for robot manipulators. IEEE Trans. Robot. Automat.8(4):451-463.
