Advances in Robot Kinematics by A. Gfrerrer (auth.), J. Lenarčič, M. M. Stanišić (eds.)

By A. Gfrerrer (auth.), J. Lenarčič, M. M. Stanišić (eds.)

This publication offers the newest study advances within the conception, layout, keep an eye on and alertness of robot structures, that are meant for various reasons comparable to manipulation, production, automation, surgical procedure, locomotion and biomechanics. the problems addressed are essentially kinematic in nature, together with synthesis, calibration, redundancy, strength keep watch over, dexterity, inverse and ahead kinematics, kinematic singularities, in addition to over-constrained structures. equipment used comprise line geometry, quaternion algebra, screw algebra, and linear algebra. those equipment are utilized to either parallel and serial multi-degree-of-freedom structures. the consequences should still curiosity researchers, lecturers and scholars, in fields of engineering and arithmetic regarding robotic idea, layout, keep watch over and alertness. All articles within the booklet have been said on the 7th foreign symposium on Advances in robotic Kinematics that was once organised in June 2000 within the attractive old Mediterranean city of Piran in Slovenia. The previous symposia of the sequence came about in Ljubljana (1988), Linz (1990), Ferrara (1992), Ljubljana (1994), and Piran (1996), and Salzburg (1998).

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The three unit quaternions qa, qb, and qc lie along a great circle if they are coplanar (Juttler, 1998). A set of 4-vectors are coplanar if they are perpendicular to a pair of mutually perpendicular 4-vectors. Since all the specified q's are zero in their 4th component, the first perpendicular is (0; 0, 0,1). A second 4-vector, perpendicular to the first as well as to qa and qc has the form ( - 2H; 0, 1- H2, 0). This second vector is also perpendicular to qb if the dot product of these two 4-vectors is zero according to h2 + ( 1) h - (1 - b ) H - H 2 = 0, thus determining the value of h.

32) Slll2"=Slll X(slll"2cosr-cos"2s111r 0 ). :. , cOSX=-_I. cosr (34) The local coordinate system for the real parts at the origin with respect to the global one is defined by the unit vectors exLO = el O' ezLO above and eyLO = ezLO xexLO ' We are able now to write the unit vectors for the axes: elO =exLO ' e WIO =cosxcosrexLo+cosxsinreyLo+sgnksinxezLo, (35,36) ew20 =cosXcos(,,-r)exLO +cosxsin(,,-r)eyLO +sgnksinxezLo, (37) (38) Because of the symmetry both frame plane angles are equal: f31 = f32 = f3 .

11 ~ .. 0 " ••• 0" '41- ++ I I 10·" 10'" 10" 10· '1c= lx 10- 1 Figure 1. Comparison of the accuracy of 3 methods for converting a rotation matrix to a set of Euler parameters. efficient method with rt < rtc that provides accurate results is the Shepperd method. This gain in efficiency comes from calculating only 1 of the 4 columns of the matrix A (cf. equation 11). The appropriate column is selected by equation 12. The straightforward determination of the eigenvector from equation 17 results in the determination of all 3 eigenvalues and eigenvectors whereas only 1 is required.

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