The article presents a unified algebraic approach for the modeling of the instantaneous motions of all linear elements, such as points, lines, and planes, embedded in a rigid body. The paper first addresses the Clifford algebra based displacement operator and its higher derivatives from which the coordinate-independent characteristic numbers with simple geometric meaning are defined. With Clifford algebra, the paper also presents the computation method and examples to demonstrate the process of obtaining the displacement operator and the characteristic numbers. Because of the coordinate independent feature, no tedious coordinate transformation typically found in the conventional instantaneous invariants method is needed.

Issue Section:

Technical Papers
1.

Bottema, O., 1961, “On Instantaneous Invariants,”

*Proceedings of International Conference for Teachers of Mechanisms*, Yale University, New Haven, Connecticut, pp. 159–164.2.

Veldkamp

, G. R.

, 1967

, “Canonical Systems and Instantaneous Invariants in Spatial Kinematics

,” J. Mec.

, 2

, pp. 329

–388

.3.

Veldkamp, G. R., 1979, “Curvature Theory in Spatial Kinematics,”

*Proceedings of Fifth World Congress on the Theory of Machines and Mechanisms*, Montreal, Canada, July, pp. 565–570.4.

McCarthy

, J. M.

, and Roth

, B.

, 1981

, “The Curvature Theory of Line Trajectories in Spatial Kinematics

,” ASME J. Mech. Des.

, 103

, pp. 718

–724

.5.

Ting

, K. L.

, and Soni

, A. H.

, 1983

, “Instantaneous Kinematics of a Plane in Space Motion

,” ASME J. Mech., Transm., Autom. Des.

, 105

(3

), pp. 552

–559

.6.

Kirson

, Y.

, and Yang

, A. T.

, 1978

, “Instantaneous Invariants in Three-Dimensional Kinematics

,” ASME J. Appl. Mech.

, 45

, pp. 409

–414

.7.

Dimentberg, F. M., 1965,

*The Screw Calculus and Its Applications in Mechanics*(in Russian), Moscow. (English translation: AD680993, Clearinghouse for Federal Technical and Scientific Information, Virginia).8.

Kirson, Y., 1975, “Higher Order Curvature Theory in Space Kinematics,” Ph.D. Dissertation, University of California at Berkeley, 140 pp.

9.

Yang A. T., Kirson, Y., and Roth, B., 1975 “On a Kinematic Theory for Ruled Surfaces,”

*Proceedings of Fourth World Congress on the Theory of Machines and Mechanisms*, Newcastle Upon Tyne, England, Sept., pp. 737–742.10.

Hsia

, L. M.

, and Yang

, A. T.

, 1981

, “On the Principle of Transference in Three-Dimensional Kinematics

,” ASME J. Mech. Des.

, 103

, pp. 652

–656

.11.

Bottema, O., and Roth, B., 1979,

*Theoretical Kinematics*, North-Holland Publishing Company, New York, ISBN: 0-486-66346-9, pp. 558.12.

McCarthy

, J. M.

, and Ravani

, B.

, 1986

, “Differential Kinematics of Spherical and Spatial Motions Using Kinematic Mapping

,” ASME J. Appl. Mech.

, 53

, pp. 15

–22

.13.

McCarthy

, J. M.

, 1987

, “The Instantaneous Kinematics of Line Trajectories in Terms of a Kinematic Mapping of Spatial Rigid Motion

,” ASME J. Mech., Transm., Autom. Des.

, 109

, pp. 95

–100

.14.

Lee

, C.

, Yang

, A. T.

, and Ravani

, B.

, 1993

, “Coordinate System Invariant Form of Instantaneous Invariants in Spatial Kinematics

,” ASME J. Mech. Des.

, 115

, pp. 946

–952

.15.

Kumar

, V.

, 1992

, “Instantaneous Kinematics of Parallel-Chain Robotic Mechanisms

,” ASME J. Mech. Des.

, 114

(3

), pp. 349

–358

.16.

Etemadi-Zanganeh

, K.

, and Angeles

, J.

, 1995

, “Instantaneous Kinematics of General Hybrid Parallel Manipulators

,” ASME J. Mech. Des.

, 117

(4

), pp. 581

–588

.17.

Wolf

, A.

, and Shoham

, M.

, 2003

, “Investigation of Parallel Manipulators Using Linear Complex Approximation

,” ASME J. Mech. Des.

, 125

(3

), pp. 564

–572

.18.

Ghosal

, A.

, and Ravani

, B.

, 2001

, “A Differential-Geometric Analysis of Singularities of Point Trajectories of Serial and Parallel Manipulators

,” ASME J. Mech. Des.

, 123

(1

), pp. 80

–89

.19.

Ball, R. S., 1900,

*A Treatise on the Theory of Screws*, Cambridge University Press, Cambridge, UK, 544 pp., Chap. 26.20.

Hunt, K. H., 1978,

*Kinematic Geometry of Mechanisms*, Clarendon Press, Oxford, 465 pp.21.

Karger, A., and Novak, J., 1985,

*Space Kinematics and Lie Groups*, Gordon and Breach Science Publishers, Montreux, ISBN: 2-88124-023-2, 422 pp.22.

Selig, J. M., 1996,

*Geometrical Method in Robotics*, Springer, ISBN: 0-387-94728-0, pp. 269.23.

Denavit

, J.

, and Hartenberg

, R. S.

, 1955

, “A Kinematic Notation for Lower-Pair Mechanisms Based on Matrices

,” ASME J. Appl. Mech.

, 22

, pp. 215

–221

.24.

Yang

, A. T.

, 1969

, “Displacement Analysis of Spatial Five Link Mechanisms Using 3×3 Matrices With Dual Number Elements

,” ASME J. Mech., Transm., Autom. Des.

, 91

, pp. 152

–157

.25.

Samuel

, A. E.

, McAree

, P. R.

, and Hunt

, K. H.

, 1991

, “Unifying Screw Geometry and Matrix Transformations

,” Int. J. Robot. Res.

, 10

, pp. 454

–472

.26.

Clifford

, W. K.

, 1873

, “Preliminary Sketch of Biquaternions

,” Proc. London Math. Soc.

, iv

(64/65

), pp. 381

–395

.27.

Yang, A. T., 1963, “Application of Quaternion Algebra and Dual Numbers to the Analysis of Spatial Mechanims,” Doctoral dissertation, Columbia University, New York, N.Y., 241 pp.

28.

McCarthy, J. M., 1990,

*Introduction to Theoretical Kinematics*, MIT Press, ISBN: 0-262-13252-4, pp. 160.29.

McCarthy, J. M., 2000,

*Geometric Design of Linkages*, Springer Verlag, ISBN: 0-387-98983-8, pp. 311.30.

Selig

, J. M.

, 2000

, “Clifford Algebra of Points, Lines and Planes

,” Robotica

, 18

, pp. 545

–556

.31.

Selig, J. M., 2001, “Robot Kinematics and Flags,”

*Geometric Algebra with Applications in Science and Engineering*(Corrochano, E. B. and Sobczyk, G., eds.), Birkha¨user, ISBN: 0-8176-4199-8, pp. 211–234 & 553.32.

Roth

, B.

, 1967

, “On the Screw Axes and Other Special Lines Associated With Spatial Displacements of a Rigid Body

,” ASME J. Eng. Ind.

, 89

, pp. 102

–110

.33.

Papantonopoulou, Aigli, 2002,

*Algebra: Pure and Applied*, Prentice-Hall, Inc., New Jersey, ISBN 0-13-088254-2, pp. 550.34.

McCarthy

, J. M.

, 1987

, “On the Scalar and Dual Formulations of the Curvature Theory of Line Trajectories

,” ASME J. Mech. Transm. Autom. Des.

, 109

, pp. 101

–106

.35.

Veldkamp

, G. R.

, 1976

, “On the Use of Dual Numbers, Vectors and Matrices in Instantaneous, Spatial Kinematics

,” Mech. Mach. Theory

, 11

(2

), pp. 141

–156

.36.

Ting, K. L., 1982, “Three-Dimensional Kinematic Analysis of Tangent-Plane Motion,” Ph.D. dissertation, Oklahoma State University, Stillwater, OK.

37.

Tsai

, L. W.

, and Roth

, B.

, 1973

, “Incompletely Specified Displacements: Geometry and Spatial Linkage Synthesis

,” ASME J. Eng. Ind.

, 95

(3

), pp. 725

–736

.38.

Ablamowicz, R., 1996, “Clifford Algebra Computations With Maple,”

*Clifford (Geometric) Algebra with Applications Physics, Mathematics, and Engineering*(Baylis, William E., ed.), Birkha¨user, Boston, ISBN: 0-8176-3868-7, pp. 463–501.Copyright © 2004

by ASME

You do not currently have access to this content.