Harn-Jou Yeh

Geometric representation of the swept volume using Jacobian rank-deficiency conditions

A broadly applicable formulation for representing the boundary of swept geometric entities is presented. Geometric entities of multiple parameters are considered. A constraint function is defined as one entity is swept along another. Boundaries in terms of inequality constraints imposed on each entity are considered which gives rise to an ability of modeling complex solids. A rank-deficiency condition is imposed on the constraint Jacobian of the sweep to determine singular sets. Because of the generality of the rank-deficiency condition, the formulation is applicable to entities of any dimension. The boundary to the resulting swept volume, in terms of enveloping surfaces, is generated by substituting the resulting singularities into the constraint equation. Singular entities (hyperentities) are then intersected to determine sub-entities that may exist on the boundary of the generated swept volume. Physical behavior of singular entities is discussed. A perturbation method is used to identify the boundary envelope. Numerical examples illustrating this formulation are presented. Applications to NC part geometry verification, robotic manipulators, and computer modeling are discussed.

Analytical boundary of the workspace for general 3-DOF mechanisms

An analytical method is presented to obtain all surfaces en veloping the workspace of a general 3-DOF mechanism. The method is applicable to kinematic chains that can be mod eled using the Denavit-Hartenberg representation for serial kinematic chains or its modification for closed-loop kinematic chains. The method developed is based upon analytical crite ria for determining singular behavior of the mechanism. By manipulating the Jacobian of the underlying mechanism, first- order singularities are computed. These singularities are then substituted into the constraint equation to parameterize singu lar surfaces representing barriers to motion. Singular surfaces are those resultihg from a singular behavior of a joint gen eralized coordinate, allowing the manipulator to lose one or more degrees of mobility. These surfaces are then intersected to determine singular curves, which represent the manipulator losing at least two degrees of mobility. Difficulties in sepa rating singular behaviors at points along singular curves are encountered. Also, difficulties in computing tangents at the intersections of singular curves are addressed. These difficul ties are resolved using an analysis of a quadratic form of the intersection of singular surfaces. An example is presented to validate the theory. Although the methods used are numerical, the main result of this work is the ability to analytically define boundary surfaces of the workspace.

On the determination of starting points for parametric surface intersections

Two numerical algorithms for computing starting points on the curve of intersection between two parametric surfaces are presented. The problem of determining intersection curves between two surfaces is analytically formulated by parametrizing inequality constraints into equality constraints and augmenting the constraint function. The first method uses an iterative optimization formulation and an iterative conjugate gradient algorithm to minimize a function comprising the vector of coordinates and a weighted constraint term. The second method uses the Moore-Penrose pseudo inverse of the constraint function to determine a starting point. Numerical examples are presented to validate both methods. Both methods require an initial point on one of the surfaces. Numerical examples illustrating the validity of the presented methods are discussed. The local versus the global views of the intersection problem in terms of iterative and recursive subdivision methods are addressed. Difficulties in determining more than one point are also illustrated using examples. The two algorithms are compared by studying their computational complexity. The Moore-Penrose inverse method has showed superior efficiency in the computational complexity, number of iterations needed, and time for conversion to a starting point. It is also shown that the Moore-Penrose inverse converges to a starting point in cases where the iterative optimization method does not.

Swept volumes: void and boundary identification

A general formulation for determining complex sweeps comprising a multiple of parameters has been presented by the authors in recent work. This paper investigates the boundaries to swept volumes, and in specific, addresses the problem of determination of voids in the volume. The determination of voids has become of major concern in CAD software, where the accurate calculation of the swept volume is used in computing solid properties such as mass and moments of inertia. A mathematical formulation based on the concept of a normal acceleration function on singular surfaces is introduced. Criteria are derived regarding the identification of a boundary from the definiteness properties of the normal acceleration function. Numerical examples are illustrated in detail and represent the first treatment of void identification in complex sweeps.

Interior and exterior boundaries to the workspace of mechanical manipulators

Abstract Analytical methods for identifying the boundary to the workspace of serial mechanical manipulators and the boundary to voids in the workspace are presented. The determination of parametric equations of surface patches that envelop the workspace of serial manipulators was presented elsewhere and is extended in this paper to an analytical method for void identification. Because of the ability to identify closed-form surface patches that exist internal and external to the workspace, a mathematical formulation based on the concept of a normal acceleration function is introduced. Admissible motion in the normal direction to a point on a singular surface is delineated and characterized by definiteness properties of a quadratic form. An enclosure bound by surface patches that do not admit normal motion is identified as a void. Several examples are treated using this formulation to illustrate the method.

Determining intersection curves between surfaces of two solids

Intersection curves between two parametric surfaces are numerically computed using continuation methods. A starting point to initiate the algorithm is determined using the Moore-Penrose pseudo-inverse. Singularities along the curve are detected using a row-rank deficiency of the Jacobian. At singular points where two or more curves intersect, bifurcation points are calculated. To numerically compute a multiple of curves at a bifurcation point, a 2nd-order expansion method is used to render the equation into a quadratic form, such that the tangents are computed. The solution is then switched to a bifurcation branch. The method is demonstrated for two intersecting surfaces having two intersecting curves. The method is also validated for special cases through a number of examples.

NC Verification of Up to 5 Axis Machining Processes Using Manifold Stratification

A numerically controlled machining verification method is developed based on a formulation for delineating the volume generated by the motion of a cutting tool on the workpiece (stock). Varieties and subvarieties that are subsets of some Eucledian space defined by the zeros of a finite number of analytic functions are computed and are characterized as closed form equations of surface patches of this volume. The motion of a cutter tool is modeled as a surface undergoing a sweep operation along another geometric entity. A topological space describing the swept volume will be built as a stratified space with corners. Singularities of the variety are loci of points where the Jacobian of the manifold has lower rank than maximal. It is shown that varieties appearing inside the manifold representing the removed material are due to a lower degree strata of the Jacobian. Some of the varieties are complicated and will be shown to be reducible because of their parametrization and are addressed. Benefits of this method are evident in its ability to depict the manifold and to compute a value for the volume.

Workspace, Void, and Volume Determination of the General 5DOF Manipulator∗

ABSTRACT Algorithms for identifying closed form surface patches on the boundary of five-degree of freedom (DOF) manipulator workspaces are developed and illustrated. Numerical algorithms for the determination of three- and four-DOF manipulator workspaces are available, but formulations for determining equations of surface patches bounding the workspace of five-DOF manipulators have not been presented. In this work, constant singular sets, in terms of generalized variables, are determined. When substituted into the constraint vector function, they yield hyper-entities that exist internal and external to the workspace envelope. The appearance of surfaces parametrized in three variables within the workspace pertaining to coupled singular behavior is also addressed Previous results pertaining to bifurcation points that were unexplained are now addressed and clarified.Examples illustrating results obtained are presented.

Local dexterity analysis for open kinematic chains

Two methods for determining the dexterity of manipulator arms are presented. A numerical method is illustrated for determining all possible orientations of the end-effector of an open kinematic chain based upon a row-rank deficiency of the sub-Jacobian of the kinematic constraint equations. This numerical method has been adapted from a theory for determining the accessible output set for planar and spatial manipulators. Boundaries of regions representing orientation solutions are mapped using continuation algorithms. One-dimensional curves in three-dimensional space are traced. Orientations of the manipulator end-effector at a point are studied (local dexterity). Only local dexterity solutions can be found. A second analytical method is also presented which determines the global dexterity of the system. This method relies upon determining all singularities of the system and identifying singular surfaces and singular curves. Regions of singular surfaces that are boundaries are identified and intersected with a service sphere. This method, however, although analytical and accurate, has many limitations. The two methods are compared via implementation into a six degree of freedom manipulator.

Path Trajectory Verification for Robot Manipulators in a Manufacturing Environment

Abstract In a manufacturing environment where a robotic arm is programmed to follow a specified trajectory such as in welding, painting and soldering, it is often the case that the arm reaches a singular configu- ration, where the programme is stopped, the arm is switched to a new configuration and the motion is continued. This difficulty has been a long-standing problem. This paper presents a mathematical formulation for verifying whether a trajectory can be completed, uninterrupted, avoiding halting of a planned path. In some cases, this formulation also allows the selection of an initial configuration to ensure a smooth path trajectory. The paper presents an analytical formulation for determining barriers to motion inside the workspace of manipulator arms. Crossability analysis of the end-effector on a barrier is addressed. A criterion for selecting an initial configuration that would result in an uninterrupted motion is introduced. The mathematical theory is validated through numerical examples of planar and spatial manipulator arms.