Chintien Huang

Analytic expressions for the finite screw systems

Abstract In this paper, we study the screws resulting from incompletely specified finite displacements of a rigid body. In this regard, explicit expressions are derived for the screw systems determined by the finite displacement of two points, or a line, or a point.

The finite screw systems associated with a prismatic-revolute dyad and the screw displacement of a point

Abstract A linear representation of the finite motion of a prismatic-revolute dyad is presented. This representation shows that all screws associated with the finite displacements of the outermost body of a prismatic-revolute dyad form a screw system of the third order (a 3-system). The screw system associated with infinitesimal displacements of the outermost body of the prismatic-revolute dyad is shown to be a limiting case of the 3-system. The analytic expression of the screw system of displacement of a point of a body is also derived from the 3-system.

Design and Kinematic Analysis of a Multiple-Mode 5R2P Closed-Loop Linkage

This paper presents the design and kinematic analysis of a multiple-mode 5R2P linkage, which is a spatial closed-loop linkage built by combining two overconstrained mechanisms: the Bennett and RPRP linkages. In the paper, the construction of the multiple-mode 5R2P linkage and its variations is investigated. The analytical inverse kinematic solution of a 4R2P open chain is adopted to analyze the 5R2P linkage for a given driving joint angle. The analysis is then conducted for the whole revolution of the driving crank. The result suggests that the multiple-mode linkage consists of three operation modes: the 5R2P, Bennett, and RPRP modes. Transitional configurations between different modes are also identified in this paper. The multiple-mode linkage features the simpler motions of the Bennett and RPRP modes as well as the more complicated motion of the 5R2P mode without the need to disconnect the linkage.

A Geometric Interpretation of Finite Screw Systems Using the Bisecting Linear Line Complex

This paper uses the concept of bisecting linear line complex of the two position theory in kinematics to present a geometric foundation for finite displacement screw systems, with an emphasis on incompletely specified displacement of points. It is shown that the bisecting linear line complex arising from the finite displacement of points is subject to a reciprocal condition if a specific definition of pitch of finite screws is used. The screw systems of finite displacements are then characterized in terms of intersections of bisecting linear line complexes. The line varieties corresponding to the two-system and foursystem associated with finite displacements of two points and a point, respectively, are illustrated. This paper demonstrates that the bisecting linear line complex provides a geometric framework for studying finite and infinitesimal kinematics. DOI: 10.1115/1.2965362

On the Line Geometry of Finite Screw Systems and Point Displacements

This paper unveils line-geometric foundations of finite displacement screw systems, with an emphasis on incompletely specified displacement of points. Linear line complexes are basic entities used in this research. Bisecting linear line complexes arising from finite displacements are proved to be subject to a reciprocal condition if a new definition of pitch of finite screws is defined. This definition was the one used to formulate finite screw systems. The relations among intersections of linear line complexes, screw systems, and varieties of lines are established in order to investigate finite screw systems. A novel treatment of point displacements allows us to visualize finite screw systems when they are formed by intersecting linear line complexes. This paper provides geometric insights into finite displacement screws and presents a new framework for the unification of finite and infinitesimal kinematics.© 2006 ASME

Derivation of screw systems for displacing plane elements

This paper sysmetically derives the screw systems for displacing three plane elements, a plane, a unidirectional plane, and a plane-line, by using the linear representation of the screw triangle. The novel three-system for displacing a unidirectional plane is derived from the four-system for displacing a plane by adding a geometrical constraint to the plane. The two-system for displacing a plane-line is obtained by adding another constraint to the undirectional plane.

Spatial Generalizations of Planar Point-Angle and Path Generation Problems

This paper deals with the spatial generalizations of two classical planar synthesis problems: the point-angle and path generation problems. The two planar synthesis problems involve the guidance of a point through specified positions by using planar four-bar linkages. In spatial generalizations, we are concerned with the guidance of a line by using spatial 4C linkages. The equivalent screw triangle is employed to derive the synthesis equations of the spatial 4C linkage. The synthesis of the RCCC linkage is achieved by constraining the translational motion in the driving C joint. Our results show that the synthesis of the 4C linkage for line guidance yields the same maximum number of positions as the planar 4-bar linkage for point guidance. The maximum number of positions of the path generation of a line is nine, while that of the lineangle problem is five. In addition to presenting the spatial generalizations of planar synthesis problems, the results provided in this paper can be used to design spatial four-bar linkages to match line specifications, in which only an infinitely-extended line, such as a laser beam, is of interest.Copyright

On the Regulus Associated With the General Displacement of a Line and Its Application in Determining Displacement Screws

In the two position theory of finite kinematics, we are concerned with not only the displacement of a rigid body, but also with the displacement of a certain element of the body. This paper deals with the displacement of a line and unveils the regulus that corresponds to such a displacement. The regulus is then used as a basic entity to determine the displacements of a rigid body from line specifications. Residing on a special hyperbolic paraboloid, the regulus is obtained by the intersection of three linear line complexes corresponding to a specific set of basis screws of a three-system. When determining the displacements of a rigid body from line specifications, a displacement screw is obtained by fitting a linear line complex to two or more line reguli. When two exact pairs of homologous lines are specified, we obtain a unique linear line complex, which determines the corresponding displacement screw. When more than two pairs of homologous lines with measurement errors are specified, it becomes a redundantly specified problem, and a linear line complex that has the best fit to more than two reguli is determined.

A Closed-Form Solution of Spatial Sliders for Rigid-Body Guidance

This paper presents a closed-form solution to the four-position synthesis problem by using a spatial slider, which is a spatial dyad of two perpendicularly intersected cylindrical joints. We utilize the dialytic elimination method to simplify the synthesis equations and to obtain a univariate ninth degree polynomial equation. Among the nine sets of solutions, two of them are infinite, and one is the displacement screw from the first position to the second position. Therefore, we have at most six real solutions that can be used to design spatial sliders for the four-position synthesis problem. A numerical example is provided in to demonstrate the validity of the solution procedure.

Numerical Solutions of Polynomial Equations for the Eight-Position Synthesis of the Cylindrical-Spherical Dyad

This paper investigates the numerical solutions of equations for the eight-position rigid-body guidance of the cylindrical-spherical (C-S) dyad. We seek to determine the number of finite solutions by using the numerical continuation method. We derive the design equations using the geometric constraints of the C-S dyad and obtain seven quartic polynomial equations and one quadratic equation. We then solve the system of equations by using the software package Bertini. After examining various specifications, including those with random complex numbers, we conclude that there are 804 finite solutions of the C-S dyad for guiding a body through eight prescribed positions. When designing spatial dyads for rigid-body guidance, the C-S dyad is one of the four dyads that result in systems of equal numbers of equations and unknowns if the maximum number of allowable positions is specified. The numbers of finite solutions in the syntheses of the other three dyads have been obtained previously, and this paper provides the computational kinematic result of the last unsolved problem, the eight-position synthesis of the C-S dyad.Copyright