An axiomatic approach for solving geometric problems symbolically

technical reportThis paper describes a new approach for solving geometric constraint problems and problems in geometry theorem proving. We developed a rewrite-rule mechanism operating on geometric predicates. Termination and completeness of the problem solving algorithm can be obtained through well foundedness and confluence of the set of rewrite rules. To guarantee these properties we adapted the Knuth-Bendix completion algorithm to the specific requirements of the geometric problem. A symbolic, geometric solution has the advantage over the usual algebraic approach that it speaks the language of geometry. Therefore, it has the potential to be used in many practical applications in interactive Computer Aided Design

Interaction with constraints in 3D modeling

Journal ArticleInteractive geometric modeling is an important part of the industrial product design process. This paper describes how constraints can be used to facilitate the interactive definition of geometric objects and assemblies. We have implemented a geometric modeling system that combines the definition of objects by interactive construction operations and specification of geometric constraints. The modeling operations automatically generate constraints to maintain the properties intended by their invocation, and constraints, in turn, determine the degrees of freedom for further interactive mouse-controlled modeling operations. A symbolic geometry constraint solver is employed for solving systems of constraints

Robot task specification and execution through relational positioning

This paper presents a relational positioning methodology for flexibly and intuitively specifying offline programmed robot tasks, and for assisting the execution of teleoperated tasks featuring precise or repetitive movements. By formulating an object positioning problem in terms of symbolic geometric constraints, the movements of an object can be restricted totally or partially, independently of its initial configuration. To solve the problem, a 3D sequential geometric constraint solver called PMF –Positioning Mobile with respect to Fixed– has been developed. PMF exploits the fact that in geometric constraint sets the rotational component can often be decoupled from the translational one and solved independently.Peer Reviewe

Moving into higher dimensions of geometric constraint solving

Journal ArticleIn this paper, we present an approach to geometric constraint solving, based on degree of freedom analysis. Any geometric primitive (point, line, circle, plane, etc.) possesses an intrinsic degree of freedom in its embedding space which is usually two or three dimensional. Constraints reduce the degrees of freedom of an object (or a set of objects). We use graph algorithms to determine upper and lower bounds for the degrees of freedom of a set of constrained objects, symbolically. This analysis is then used to establish dependency graphs and evaluation schemes for symbolic or numeric solutions to constraint problems. The approach has been generalized for n-dimensional space, which, among other things, allows for a uniform handling of 2-D and 3-D constraint problems or algebraic constraints between scalar dimension. Also, higher than three dimensional solutions can be interpreted as approaches to over- and under- constrained problems. In this paper, we will present the theoretical background of the approach, and demonstrate how it can be applied within an interactive design environment

Symbolic Automation and Numerical Synthesis for Robot Kinematics

This research analyzes three topics in robot arm kinematics. First, the direct kinematics which determines the Cartesian position and orientation of the end effector for the specified values of joint parameters is analyzed. Second, the differential motions concerning the differential relationships between the command variables in position and orientation of the end effector and the joint-controlled variables are studied. Finally, the inverse kinematics which determines the joint variables for a specified Cartesian position and orientation of the end effector is considered. This dissertation presents a methodology for incorporating the artificial intelligence types of knowledge into automating solutions for the direct kinematics problem and the manipulator Jacobian matrix. Furthermore, the dissertation utilizes the backward recursive techniques, the trigonometric identity rules, and a set of heuristic rules for implementing this methodology. To expedite computation efforts, a new algorithm is developed to obtain a differential relationship of a robotic manipulator via the vector kinematics method. Moreover, the speed control model for general robotic manipulators, together with the inverse Jacobian regarding cases of under-determined and over-determined of joint-controlled variables, are also discussed. Three mathematical approaches are proposed for solving the inverse kinematics problem: the inverse homogeneous transformation matrices approach, the geometric approach, and the arm-wrist partitioned synthesis approach. The first two approaches yield the symbolic closed-form solutions; the last approach, based on the iterative technique, provides a maximum of 16 distinct solutions of joint motion variables for any given position and orientation of the end effector in the workspace

Description of a robotics-oriented relational positioning methodology

This paper presents a relational positioning methodology for flexibly and intuitively specifying offline programmed robot tasks, as well as for assisting the execution of teleoperated tasks demanding precise movements. In relational positioning, the movements of an object can be restricted totally or partially by specifying its allowed positions in terms of a set of geometric constraints. These allowed positions are found by means of a 3D sequential geometric constraint solver called PMF – Positioning Mobile with respect to Fixed. PMF exploits the fact that in a set of geometric constraints, the rotational component can often be separated from the translational one and solved independently

Alignment study of Kentucky\u27s mathematics placement examinations and entry level credit-bearing mathematics course examinations.

This research alignment study compares content assessed on course finals from Kentucky public universities in highest level remedial mathematics courses and content assessed on college placement examinations. These assessments are used to determine if a student is ready for credit-bearing courses at a university. The study addressed the following four research questions: (1) What mathematical prerequisite knowledge do state universities consider necessary to be college ready? Specifically 1a) What content domains do the state universities emphasize in their remediation courses?; 1b) Is there consistency across the state public universities with regard to the content domains?; and (2) How well do Kentucky\u27s mathematics placement assessments (ACT, COMPASS, and KYOTE) align in both content and cognitive demand with four-year universities’ Kentucky Mathematics College Readiness Expectations (KM-CRE)? The study was implemented in two phases. In Phase 1, course finals in the highest mathematics remediation class offered at five Kentucky universities were analyzed using Common Core State Standards (CCSS). Phase 2 of the study involved an alignment analysis between the universities’ identified KM-CRE and Kentucky\u27s approved college placement examinations: ACT, KYOTE, and COMPASS. The study is framed using Webb\u27s alignment modeling the areas of (1) categorical concurrence, (2) balance of representation, (3) range of knowledge, and (4) depth of knowledge. Findings suggested that consistency across universities in content emphasis exists. Examinations were heavily weighted in Algebra readiness Expressions and Equations, Functions, and Algebra). Findings in the alignment study suggested some content alignment existed but more alignment is needed through intentional assessment of college ready content. Additionally, all placement examinations revealed a strong cognitive complexity alignment to KM-CRE. Implications of this study suggest the redesign of the placement examinations to assess the content knowledge necessary for college success