An algorithm for efficient decomposition of interface channels between non-convex polygons in a Computer-Aided Design (CAD) environment is presented. This algorithm forms the computational basis for the solution of several design automation problems. In this paper, the channel decomposition algorithm is presented and applied to the problem of interference detection between non-convex polygons. The resulting interference detection algorithm does not require preprocessing of the data and uses a simple data structure. In a companion paper (Ku and Ravani 1988), the channel decomposition algorithm is applied to the problem of model-based rigid-body guidance in presence of geometric constraints.

This paper presents an efficient algorithm for guidance of a convex rigid body in-between non-convex polygonal objects in a Computer-Aided Design (CAD) environment. A shrinking procedure is used that separates the kinematic from the shape constraints by reducing the problem to that of guidance of a line segment in an expanded environment. A slicing technique together with an algorithm for decomposition of the interface channel between the non-convex objects is used to generate the motion program for the line segment. The results can be applied to model based guidance of mobile robots or automatic motion planning for robot manipulators.

This paper deals with automatic generation of motion of a point under both geometric and non-geometric constraints. Optimal point paths are generated which are not only free of collisions with polygonal obstacles representing geometric constraints but also conform to non-geometric constraints such as speed of the motion, a maximum allowable change in the velocity vector and a minimum clearance from the obstacle boundaries. The concept of passage networks and conforming paths on the network are introduced. These are used to provide a new representation of the free space as well as a motion generation algorithm with a computational time complexity of only O ( n 3 .log( n )), where n designates the total number of obstacle vertices. The algorithm finds the shortest or fastest (curved) path that also conforms with preset constraints on the motion of the point. The point paths generated are proved to be optimal while conforming to the constraints.

This paper develops theories for evaluating the efficiency of the ball screw mechanism and additionally, for designing this mechanism. Initially, a quasi-static analysis, which is similar to that of the early work in this area, is employed to evaluate efficiency. Dynamic forces, which are neglected by the quasi-static analysis, will have an effect on efficiency. Thus, an exact theory based on the simultaneous solution of both the Newton-Euler equations of motion and the relevant kinematic equations is employed to determine mechanism efficiency, as well as the steady-state motion of all components within the ball screw. However, the development of design methods based on this exact theory is difficult due to the extensive computation necessary and thus, an approximate closed-form representation, that still accounts for the ball screw dynamics, is derived. The validity of this closed-form solution is proven and it is then used in developing an optimum design methodology for the ball screw mechanism based on efficiency. Additionally, the self-braking condition is examined, as are load capacity considerations.

This paper studies the kinematics of the Ball Screw Mechanism (BSM) with the aim of developing a foundation for understanding the motion of the balls and their contact patterns with the contacting elements. It is shown that there is always slip between the balls and the nut or screw, and therefore, the no slip condition assumed in the BSM literature is not attainable. The effect of contact deformation on the motion of the balls is also studied and is used to develop the pattern of the constant sliding lines of contacts between the ball and the screw or the nut. The results have applications in efficiency analysis, design, wear analysis and finite element modeling of the BSM.

This paper studies continuous computational geometry of motions and develops a method for Computer Aided Geometric Design (CAGD) of motion interpolants. The approach uses a mapping of spatial kinematics to convert the problem of interpolating displacements to point interpolation in the space of the mapping. To facilitate the point interpolation, the previously non-oriented mapping space is made orientable. Methods are then developed for designing spline curves in the mapping space with tangent, curvature and torsion continuities. The results have application in computer animation of three dimensional objects used in computer graphics, computer vision and simulation of mechanical systems.

This paper develops the theoretical foundation for computations of spatial displacements from the simple geometric features of points, lines, planes and their combinations. Using an oriented projective three space with a Clifford Algebra, all these three features are handled in a similar fashion. Furthermore, issues related to uniqueness of computations and minimal number of required features are discussed. It is shown that contrary to the common intuition, specification of a minimum of four points (planes) or three lines (each pair being non-planar) are necessary for computation of a unique displacement. Only when the sense of the orientations of these features are specified then the minimal number of required features reduces to three for points and planes and two for lines. The results, in addition to their theoretical interest in computational geometry of motion, have application in robot calibration.

This paper develops a mathematical foundation for Computer Aided Design (CAD) of sculptured shapes based on line geometry. First, a new representation is presented for a line based on Plücker coordinates that would allow specification of a line segment (rather than an infinite line) in an elegant manner and suitable for computational purposes. Then, methods are presented for geometric design of shape patches (here referred to as line constructs) by interpolating or approximating a set of control lines (rather than control points) using ruled surfaces, line congruences and line complexes. The methods presented are general and allow utilization of different basis functions such as Bézier, Cardinal and B-spline bases. As Computer Aided Manufacturing (CAM) application of the theory, methods are developed for generation of offsets of ruled surfaces useful in Numerical Control (NC) milling and tool motion generation in the Wire Cut Electric Discharge Machining (EDM) operation.

This paper deals with discrete computational geometry of motions and develops geometric construction algorithms for interpolating in-between rigid displacements. It combines concepts from the fields of kinematics and computational geometry and develops a computational framework for constructing motion interpolants useful in mechanical systems animation, robot trajectory planning and key framing in computer graphics. A de Casteljau type algorithm is presented for constructing Bézier and rational Bézier motion interpolants. In addition, the problem of achieving higher order of continuity in piecing together motion interpolants is studied. A geometric construction algorithm is presented for designing C 2 continuous motion interpolants.

In this paper we study Computer Aided Geometric Design (CAGD) and Manufacturing (CAM) of developable surfaces. We develop direct representations of developable surfaces in terms of point as well as plane geometries. The point representation uses a Bezier curve, the tangents of which span the surface. The plane representation uses control planes instead of control points and determines a surface which is a Bezier interpolation of the control planes. In this case, a de Casteljau type construction method is presented for geometric design of developable Bezier surfaces. In design of piecewise surface patches, a computational geometric algorithm similar to Farin-Boehm construction used in design of piecewise parametric curves is developed for designing developable surfaces with C 2 continuity. In the area of manufacturing or fabrication of developable surfaces, we present simple methods for both development of a surface into a plane and bending of a flat plane into a desired developable surface. The approach presented uses plane and line geometries and eliminates the need for solving differential equations of Riccatti type used in previous methods. The results are illustrated using an example generated by a CAD/CAM system implemented based on the theory presented.

This paper follows a previous one on the computation of spatial displacements (Ravani and Ge, 1992). The first paper dealt with the problem of computing spatial displacements from a minimum number of simple features of points, lines, planes, and their combinations. The present paper deals with the same problem using a redundant set of the simple geometric features. The problem for redundant information is formulated as a least squares problem which includes all simple features. A Clifford algebra is used to unify the handling of various feature information. An algorithm for determining the best orientation is developed which involves finding the eigenvector associated with the least eigenvalue of a 4 × 4 symmetric matrix. The best translation is found to be a rational cubic function of the best orientation. Special cases are discussed which yield the best orientation in closed form. In addition, simple algorithms are provided for automatic generation of body-fixed coordinate frames from various feature information. The results have applications in robot and world model calibration for off-line programming and computer vision.

This paper studies planar motion approximation problems from a computational geometric perspective and develop a computational geometric structure that can be used for mechanical motion synthesis. This allows for development of computational algorithms and software systems to support the mechanical design activity. The approach uses an orientable kinematic mapping to transform the mechanical design problem into a curve design problem in the space of the mapping. The curve design problem for synthesis of an analytic motion is carried out by Hermite interpolation. In case of a mechanical linkage, however, the Hermite interpolation is combined with a first-order curve fitting procedure for synthesizing the motion.

In this paper we present a method for dynamically interpolating the motion of a rigid body in space between known end-positions. We begin by creating an image space representation of the equations of motion for a rigid body. These equations are written as a system of first-order state equations whose trajectory is optimized based on the minimization of a certain performance index. This leads to a set of boundary-value equations between the end positions, which when solved give the interpolated motion. Finally, a program developed to perform such interpolations numerically is utilized by generating a trajectory between given terminal states.

In this paper, three new representations of a line-segment are introduced that combine Plücker line coordinates with specifications of length and location of a line-segment. For each of the three line-segment representations, a right conoidal interpolant of two arbitrarily disposed line-segments is developed. These interpolants are then combined with deCasteljau-like algorithms for generating ruled surfaces. The results can be extended to develop new algorithms for computer aided geometric design of geometric shapes based on line geometry.