The Problem of Scheduling Parallel Computations of Multibody Dynamic Analysis Is NP-Hard

A formulation of a graph problem for scheduling parallel computations of multibody dynamic analysis is presented. The complexity of scheduling parallel computations for a multibody dynamic analysis is studied. The problem of finding a shortest critical branch spanning tree is described and transformed to a minimum radius spanning tree, which is solved by an algorithm of polynomial complexity. The problems of shortest critical branch minimum weight spanning tree (SCBMWST) and the minimum weight shortest critical branch spanning tree (MWSCBST) are also presented. Both problems are shown to be NP-hard by proving that the bounded critical branch bounded weight spanning tree (BCBBWST) problem is NP-complete. It is also shown that the minimum computational cost spanning tree (MCCST) is at least as hard as SCBMWST or MWSCBST problems, hence itself an NP-hard problem. A heuristic approach to solving these problems is developed and implemented, and simulation results are discussed.

Parametric Deflection Approximations for Initially Curved, Large-Deflection Beams in Compliant Mechanisms

The analysis of systems containing highly flexible members is made difficult by the nonlineararities caused by large deflections of the flexible members. The analysis and design of many such systems may be simplified by using pseudo-rigid-body approximations in modeling the flexible members. The pseudo-rigid-body model represents flexible members as rigid links, joined at pin joints with torsional springs. Appropriate values for link lengths and torsional spring stiffnesses are determined such that the deflection path and force-deflection relationships are modeled accurately. Pseudo-rigid-body approximations have been developed for initially straight beams with externally applied forces at the beam end. This work develops approximations for another fundamental type of flexible member, the initially curved beam with applied force at the beam end. This type of flexible member is commonly used in compliant mechanisms. An example of the use of the resulting pseudo-rigid-body approximations in compliant mechanisms is included.

Alternative Inverse Kinematic Equations for General RRP and RRR Regional Structures of Manipulators

Many manipulators with six degrees of freedom are constructed with two distinct sections, a regional structure for spatial positioning, and an orientational structure having a common intersection point for the joint axes. With this arrangement, inverse kinematic solutions for position and orientation may be found separately. While solutions for general three link manipulators have been available since the work of Pieper in 1969, this paper presents new forms of the inverse kinematic equations for general RRP and RRR regional structures. Cartesian coordinates of the F-surface (generated by movement of the outer two joints) together with the outer joint angle are used as the equation variables. In addition, a second degree polynomial approxiamation of the equation may be used for quick iteration to a solution. It is hoped that these new equations will be useful by themselves and in workspace regions where solutions using equations in terms of the joint variables are numerically inaccurate or impossible.

An Efficient Algorithm to Determine the Intersection of Simple Polygons for Robot Path Planning

This paper presents an efficient algorithm for determining the intersection of two simple polygons. The proposed algorithm is based on the idea of searching for the vertices of the intersection polygon vertex by vertex along the boundary in a clockwise direction. This method finds the intersection polygon vertices and their order in one pass. The algorithm almost eliminates the need for testing whether candidate vertices are inside both polygons and the sorting stage is no longer needed.

Minimizing the Strain Energy for Flexible-Link Cooperating Manipulators

In control of cooperating manipulator systems, force distribution is a central problem. For rigid manipulators, solutions are based on various criteria such as minimum potential damage to the grasped object, minimum energy consumption and optimal load distribution in actuators. However, when considering manipulators with structural flexibility, minimizing the deflections of the mechanical components becomes a highly desirable objective. In this paper, we propose a force distribution scheme based on a local minimum of the strain energy stored in the elastic links of the cooperating manipulators. Numerical example are given to demonstrate the effectiveness of the scheme.

A Synthesis Method for Placing Workpieces in RPR Planar Robotic Workcells in Which Large Rotations Are Required at the Workpiece

A new method is developed for determining both a satisfactory location of a workpiece and a suitable mounting-angle of the tool for planar RPR robots that can provide dexterous workspace. The method is an analytical representation of the geometry of the robot and the task, and is particularly well suited to applications in which the task requires large rotations of the end-effector. It is determined that, when the task requires that the end-effector rotate a full turn at just two locations and when the first or third joint in the robot is rotatable by one turn, then the radial location of the workpiece is fixed in the workcell but its angular location is not fixed. When the mounting-angle of the tool is also a variable, the method accommodates tasks in which the tool must rotate a full turn at three locations on the workpiece. The results are presented as coordinates of points in a two-dimensional Cartesian reference frame attached to the workcell. Consequently, a technician or an engineer can determine the location for the workpiece by laying out these coordinates directly in the workcell. Example problems illustrate the method. Practical applications include welding and deposition of adhesives.

Workspace Cusp Locations, and Simplified Workspace Boundary Equations for General RRR Regional Structures of Manipulators

General, complex geometry forms of RRR regional structures are often avoided due to the presence of inner boundaries within the workspace which tend to complicate robot guidance. Despite the added complexity, certain RRR geometries may have useful applications as they contain large workspace regions where four alternate configurations may be used to reach a given spatial location. Cusp points often appear on the workspace boundaries of general RRR regional structures, and determining their precise location may be useful for both design and guidance purposes. A twelfth degree polynomial equation in the outer joint variable is derived which defines the location of non-trivial cusps in the workspace. A new closed form workspace boundary equation is derived in the outer joint variable and x coordinate of the toroidal surface generated by rotation of the two outer revolutes. If the outer joint variable is incremented, a quadratic in x is formed at each step which enables a very efficient determination of the workspace boundaries while also providing the coordinates of the boundary on the toroidal surface.

Orientation Capability and Robot Wrists

This paper introduces a measure of the orientation capability of manipulator wrists, and a means of representing orientation capability. Wrists with the various combinations of roll, pitch and yaw joints, with one, two or three joints are studied. It is shown that for all non-degenerate orthogonal three joint wrists, the image space representing the orientation capability is doubly-covered, and the possibilities for restricting joint angles to make it singly-covered are given. Degenerate postures for each configuration are also given, and singularity-free wrists with three joints are discussed.

Constraint Manifold Synthesis of Planar Linkages

This paper formulates the design theory of planar four-bar linkages using the planar form of dual quaternions known as planar quaternions. The set of positions reachable by the floating link of a dyad is a quadratic algebraic surface called a constraint manifold. Determining the coefficients of the quadratic form defining this manifold is equivalent to setting the design parameters of the linkage. If the task of the linkage is specified as geometric constraints on the location of the floating link, then algebraic constraints are obtained on the quaternion components. We seek the coefficients of the constraint manifold that satisfies these constraints. The result is an algebraic formulation that is symmetric in its characterization of the linkage and task, and provides a versatile tool for the formulation and solution of linkage design problems.

A New Circuit Rectification Method in Four-Bar Linkages

A circuit defect exists in four-bar linkages when a linkage cannot pass through all the design positions without being disassembled and reassembled in a different configuration. This paper describes a new and efficient procedure to rectify circuit defects in four-bar linkages at the synthesis level prior to the analysis level. It is achieved by controlling the change of sign of one of the angles opposite to the shortest link. The procedure utilizes the algebraic method where special points are employed to eliminate the solutions with circuit defects as a function of the design positions. Complete rectification is accomplished at the mid-synthesis level by controlling the critical angles of the linkage that identify the existence of circuit defects. This accounts for the rectification of circuit defects at the synthesis stage.