Both parametric and implicit representations can be used to model 2D curves and 3D surfaces. Each has certain advantages compared to the other. Implicit polynomial (IP) methods are not as popular as parametric procedures because the lack of general procedures for obtaining IP models of higher degree has prevented their general use in many practical applications. In most cases today, parametric equations are used to model curves and surfaces. One such parametric representation, elliptic Fourier descriptors (EFD) has been widely used to represent 2D and 3D curves, as well as 3D surfaces. Although EFDs can represent nearly all curves, it is often convenient to have an implicit algebraic description, F(x,y)=0, especially for determining whether given points lie on the curve. Algebraic curves and surfaces have proven very useful also in many model-based applications. Various algebraic and geometric invariants obtained from these implicit models have been studied rather extensively. We present a new non-symbolic implicitization technique called the matrix annihilation method, for converting parametric Fourier representations to implicit polynomial form.

Presents novel techniques to modify free-form shapes defined by implicit algebraic models. These models are first simplified by introducing a decomposition in terms of conic-line primitives, and then the desired local/global shape changes are produced by varying the parameters of these primitives. A decomposed model provides efficient control of each primitive using its canonical frame. To make the analysis simpler, we restrict our attention to free-form curves modeled by a 4th-degree (quartic) equation. However, all of the analysis can be directly extended to higher-degree models. We also provide some 3D interpolation procedures based on the resulting primitives to illustrate the potential of our methods in 3D.

Many free-form object boundaries can be modeled by quartics with bounded zero sets. The fact that any nondegenerate closed-bounded algebraic curve of even degree n=2p can be expressed as the product of p conics, which are real ellipses, plus a remaining polynomial of degree n-2,12 can be utilized to express a nondegenerate quartic as the product of two leading ellipses plus a third conic which might be either a closed curve (an ellipse) or an open curve (a hyperbola). However, it can be shown that the leading ellipses can be modified with appropriate constants by constraining the third conic to be a circle, thus implying a 2-ellipse and 1-circle; i.e. an elliptical-circular(E2C)representation of the quartic. The use of such representations is to simplify the analysis of quartics by exploiting the well-known properties of conics and to develop a set of functionally independent geometric invariants for recognition purposes. Also, it is shown that the underlying Euclidean transformation between two configurations of the same quartic can be determined using the centers of the three conics.

This paper outlines a geometric parameterization of 2D curves where the parameterization is in terms of geometric invariants and terms that determine an intrinsic coordinate system. Thus, we present a new approach to handle two fundamental problems: single-computation alignment and recognition of 2D shapes under affine transformations. The approach is model-based, and every shape is first fit by an implicit fourth degree (quartic) polynomial. Based on the decomposition of this equation into three covariant conics, we are able to define a unique intrinsic reference system that incorporates usable alignment information contained in the implicit polynomial representation, a complete set of geometric invariants, and thus an associated canonical form for a quartic. This representation permits shape recognition based on 8 affine invariants. This is illustrated in experiments with real data sets.

We employ a complex representation for an algebraic curve, and illustrate how the algebraic transformation which relates two Euclidean equivalent curves can be determined using this representation. The idea is based on a complex representation of 2D points expressed in terms of the orthogonal x and y variables, with rotations of the complex numbers described using Euler's identity. We develop a simple formula for integer multiples of the rotation angle of the Euclidean transformation in terms of the real coefficients of implicit polynomial equations that are used to model various 2D free-form objects. When there is a translation, it can be determined in a straightforward manner using an estimation of the rotation angle and some new results on conic-line decompositions.

A new method is presented for identifying and comparing closed, bounded, free-form curves that are defined by even implicit polynomial (IP) equations in the X-Y Cartesian coordinates. The method provides a new expression for an IP involving a product of conic factors with unique conic factor centers. The critical points for an IP curve are also defined. The conic factor centers and the critical points are shown to be useful related points that directly map to one another under affine transformations. In particular, the explicit determination of such points implies both a canonical form for the curves and the transformation matrix which relates affine equivalent curves.

A method for mobile robot motion planning is presented for both two- and three-dimensional robots. For the two-dimensional case, the algorithm is based on complex variable theory and conformal mapping. The method gives the robot the capability of avoiding obstacles more efficiently, smoothly, and at a constant curvilinear velocity. In order to establish the structure of the proposed velocity feedback obstacle avoidance scheme, several issues are addressed and successfully resolved. First, an analytical solution is derived for the case of circular obstacles. Second, the algorithm uses conformal mapping to establish the solution for an arbitrarily shaped obstacle based on the derived solution for a circular obstacle. The use of complex variable methodology with the objectives of exploiting the powerful technique of uniformal mapping constitutes the fundamental characteristic of the proposed technique. Third, by analogy, a solution to 3-D space-flying and subsea robots is also derived.>

The authors present the dual-drive control concept, which is a form of hybrid force/velocity control in which the constraint frame is automatically determined from feedback information. This allows compliant tasks to be conveniently specified in terms of a desired (normal) force and a desired (tangential) velocity. The proposed control algorithm is appropriate for compliant tasks that require motion orthogonal to the contact force; for example, turning a crank or tracking a surface. These basic tasks can be performed without the continuous involvement and corresponding overhead of a high-level planner. This is accomplished by the definition of a plane on which compliant motion is to occur and a point with which force and velocity directions are determined. The dual drive controller has been implemented on the SIERA system and is being used to control an IBM 7565 robot. Experimental results for crank-turning and surface-tracking problems are provided to illustrate the algorithm.>

SIERA (System for Implementing and Evaluating Robotic Algorithms) is a multiprocessor system that has been developed at the Laboratory for Engineering Man/Machine Systems (LEMS) at Brown University. It incorporates a tightly coupled bus-based system (the Real Time Servo System) and a loosely coupled point-to-point network (the Armstrong Multiprocessor System). SIERA is capable of controlling many types of commercially available robots because the modular construction of its hardware and software has minimized robot dependencies. Three programming levels are introduced to classify the different ways SIERA can be utilized-for simple robot operation, for robotics research, and for system enhancements. The environment associated with each of these levels provides the necessary programming support and interactive commands. The concepts of servo loops and controller modifications (cmods) are introduced to facilitate programming on the Real Time Servo System. A simple example is presented to illustrate operation of the system.