Line Geometry of the Finite Displacement Screws of Points and Lines

• Main Authors: Wu-Chang Kuo, 郭武彰
• Other Authors: Chin-Tien Huang
• Format: Others
• Language: zh-TW
• Published: 2009
• Online Access: http://ndltd.ncl.edu.tw/handle/42137394033893357974

博士 === 國立成功大學 === 機械工程學系碩博士班 === 97 === The research of line geometry stems from Plücker’s unique 6-tuple vector representation of a line in the three-dimensional space. In line geometry, we are concerned with line varieties, which are composed of linearly dependent lines of certain ranks. In kinematics, a screw can be thought of as a line with an associated pitch, and a screw is also represented by a 6-tuple vector form similar to the Plücker coordinates of a line. Screw theory has been shown to be a great tool in studying the instantaneous kinematics and statics of rigid-body systems because instantaneous screws and wrenches have linear properties and form screw systems. These linear features provide efficient methods for investigating spatial mechanisms, such as analyzing instantaneous motion, singularities and stiffness of serial and parallel manipulators. A line variety and a screw system are constructed similarly by the linear combinations of linearly independent lines and screws, respectively. Furthermore, there is a one-to-one correspondence between a screw and a line variety. In the context of instantaneous kinematics, all the lines normal to the velocity of every point of a rigid body undergoing a screw motion constitute a line variety called linear line complex. By using the intersections of line varieties and the notion of reciprocity, we can also obtain the correspondence between line varieties and screw systems. Therefore lines and line varieties are the basic entities in the research of screw geometry. In this dissertation, line geometry will be employed to investigate the finite kinematics of a rigid body. Research in the past two decades had introduced new definition of pitch and obtained screw systems pertaining to finite displacements. However, there was a lack of investigation into the linearity of finite screws using line geometry. This dissertation extends the research of line geometry in instantaneous screws to that in finite screws. As a result, the nullplane and helicoidal vector field are found in the displacements of point and line elements of a rigid body. The linear line complex associated with a finite displacement screw, which is composed of the pencils of lines on all the nullplanes in the three-dimensional space, is also found. Furthermore, line varieties corresponding to the finite screw systems associated with point and line displacements are obtained by the intersections of linear line complexes. For the finite displacement of a point, the nullplane passes through the midpoint of the pair of homologous points. The nullplane is perpendicular to the line segment connecting the homologous points. All the pencils of lines on the nullplanes constitute the linear line complex of the finite screw of the point displacement. According to the helicoidal vector field associated with the screw, the pitch is half-translation divided by the tangent of half-rotation. Moreover, the line varieties correspond to the two-point and one-point incompletely specified displacements are, respectively, a congruence and a pencil of lines. For the finite displacement of a line, the nullplane passes through the intersection of a pair of intersecting homologous lines. The direction of nullplane is the same as the internal bisector the pair of homologous lines. All the pencils of lines on the nullplanes constitute the linear line complex of the finite screw of the line displacement. According to the helicoidal vector field of the screw, the pitch is the translation divided by the sine of rotation. Furthermore, the line variety corresponds to the one-line incompletely specified displacement is a regulus. In this dissertation, the developed line geometry was also used to solve rigid body registration problems. It is well known that the displacement of a rigid body can be determined by three pairs of homologous points. Three pairs of homologous points of a finite displacement provide three pencils of lines that uniquely determine a linear line complex. We select five lines from the three pencils of lines to calculate the linear line complex, which in term gives a unique screw. Another way to specify a displacement is by using two pairs of homologous lines. Two pairs of homologous lines of a finite displacement provide two reguli that uniquely determine a linear line complex. We select five lines from the two reguli to calculate the linear line complex and thus the screw. In addition to exact specifications, we also introduce an approximate algorithm to determine the linear line complex based on the specifications of more than three points or two lines. This dissertation has proved, based on line geometry, that the new definition of pitch of finite displacement screws provides linear properties of finite displacements. It gives geometric insight into line varieties which correspond to the finite screw systems associated with point and line displacements. The research presented in this dissertation also helps unify the study of instantaneous kinematics and finite kinemtics by using screws.