Keith L. Doty

A theory of generalized inverses applied to robotics

Robokcs research has made extensive use of techniques based on me Moore-Penrose inverse, or generalized inverse, of matri als. Recently it has been pointed out how noninvariant results may, in general, be obtained by applying these techniques to other areas of robotics, namely hybrid control and inverse ve locity kinematics. Unfortunately, the problems are not restricted to just these particular areas in robotics but are connected with misleading definitions of the metric properties of the six- dimensional wrench and twist vector spaces used in robotics. The current definitions lead to inconsistent results (i.e., results that are not invariant with respect to changes in the reference frame andlor changes in the dimensional units used to express the problem. As a matter of fact, given a linear system u = Ax, where the matrix A may be singular, the Moore-Penrose theory of generalized inverses may be properly and directly applied only when the vector space U of vector u and the vector space X of vector x are inner product spaces. Arbitrary assignment of Euclidean inner products to the space U and X when the vectors u and x have elements with different physical units can lead to inconsistent and noninvariant results. In this article the problem of inconsistent, noninvariant solutions X s to u = Ax in robotics is briefly reviewed and a general theory for com puting consistent, gauge-invariant solutions to nonhomogeneous systems of the form u = Ax is developed. In addition, the dual relationship between rigid-body kinematics and statics is de fined formally as a particular, linear algebraic system whose solution system is also a dual system. Examples illustrate the theory.

On-Line Polynomial Trajectories for Robot Manipulators

Robot motion is commonly specified as a Cartesian trajectory of its end-effector. For executing the end-effector trajectory on-line, a look-ahead of a few points on the trajectory is used to generate the joint trajectories. This paper describes the construction of on-line cubic spline joint trajectories,for a limited-point look-ahead on a specified end-effector trajectory, not necessarily Cartesian. An analytical derivation for the ideal number of look-ahead points on the end-effector trajectory_for cubic spline interpolation is given. Experimental and simulation results of the on-line spline interpolation schemes show improved end-effector path tracking and smooth motion for a wide range of sampling rates on the effector trajectory.

The architecture of a context addressed segment-sequential storage

This paper presents a new approach to the problem of searching large data bases. It describes an architecture in which a cellular structure is adapted to the use of sequential-access bulk storage. This organization combines most of the advantages of a distributed processor with that of inexpensive bulk storage.

A fast algorithm for inverse kinematic analysis of robot manipulators

To solve the inverse kinematics problem, we obtain with little effort a reduced and complete set of equations by a conve nient choice of end-effector frame and application of rotation orthogonality. This approach does not require computation of the forward kinematics and can be used with manipulators of any geometry, although it is most efficient when applied to orthogonal manipulators, a class of robot arms defined in this paper. For manipulators requiring numerical techniques, but for which knowledge of one joint variable allows closed- form solutions of the remaining joint variables, an iterative inverse kinematic method, simple and fast enough to be suitable for real-time manipulator control, has been devel oped. The concepts and techniques presented in this paper are illustrated with two examples. The iterative method devel oped here performs a kinematic inversion of a 6-degree-of- freedom manipulator with no closed-form solutions in less than 30 ms using a desktop computer, an order of magnitude faster than times found in the literature.

A Robot Manipulator With 16 Real Inverse Kinematic Solution Sets

A solution search algorithm based on a one-dimensional numerical approach to the inverse kinematic problem (pre sented in an earlier paper) led to the discovery of a six-DOF manipulator able to position and orient its end-effector in 16 distinct configurations for a given end-effector pose (position and orientation). This paper discusses the consequences of such a discovery and presents a description of the manipula tor, the end-effector pose, and the 16 kinematic solutions.

Fast inverse kinematics of five-revolute-axis robot manipulators

Abstract Based on the kinematic properties of 4-DOF arms presented in a companion paper, we show that all 5-DOF robot manipulators can be solved by use of a simple 1-D iterative technique or simply in closed-form. The 1-D technique is computationally fast and suitable for real-time control of robot manipulators. We are also able to establish a number of sufficient structural conditions that allow closed-form solutions for most 5-revolute-DOF robot manipulators.

Structural kinematics of 6-revolute-axis robot manipulators

Abstract The complexity of the inverse kinematics problem for 6-revolute-axix robots is highly dependent on the mechanical structure of the robot. The study of the kinematics of 6-revolute-axis manipulators, based on results of previous analyses of 4- and 5-axis manipulators, leads to an efficient inverse kinematics method that reduces the problem to solving a system of two nonlinear equations in two variables. We also show that 6R manipulators with at least two intersecting or parallel joint axes can be solved by a faster one-dimensional technique. The numerical techniques are illustrated by computer simulations.

A complete kinematic analysis of four-revolute-axis robot manipulators

Abstract The approach taken in this research is to study the kinematics of low-number-of-axes robots and apply new results to progressively more complex geometries. The kinematic analysis in this paper provides valuable insight in the inverse kinematic study of all-revolute-4-DOF robot and arms leads to special 4-DOF structures with more than one inverse kinematics solution set. These special structures can be of valuable assistance in the design of 4-DOF robot manipulators. This paper also provides an essential theoretical foundation for solving 5- and possibly 6-DOF manipulators. A companion paper builds on this foundation to prove important properties and provide an efficient inverse kinematic technique suitable for real-time control of fast 5-DOF robot manipulators.

Extending object-oriented concepts to support engineering applications

A presentation is made of a structure-function (S-F) paradigm for representing engineering designs. The S-F paradigm treats structures and functions as first-class objects by modeling the physical configuration of the design as structural objects and the behavior of the design as functional objects. The paradigm supports engineering designs by using the associative knowledge between structures and functions to help in the design process and to support simulation by providing for a run-time interaction between structures and functions. The S-F paradigm would be generally applicable to model the passive and active information in any complex system. It may be implemented as a layer on top of any object-oriented model.<<ETX>>

A structure-function-control paradigm for knowledge-based modeling and design of manufacturing workcells

This paper discusses the integration of structural, functional and control knowledge in manufacturing workcell modeling, simulation and design. After an overview of applications of semantic and object-oriented data models in the manufacturing domain, issues relating to the control synthesis for manufacturing workcells are presented. In particular, a data model encompassing functional and control features, along with application domain structural knowledge, is developed. This model assists in explicitly representing the control aspects of engineering design within an object-oriented database and supports a task-level, functionality-driven, manufacturing workcell design. Since manufacturing workcells consist of a number of elements interacting in a complex manner, workcell control design is one of the most difficult steps in the workcell design procedure. Message passage, commonly used in object-oriented databases, provides no explicit modeling of the database behavior. Hence, it can not serve as a tool for the design of system control. On the other hand, Petrinets (PN) have proven successful in describing complex interaction among active agents. This paper will explore the incorporation of Petri nets as a basis for describing application control knowledge within a structure-function-control data model.