Paul J. Zsombor-Murray

Fast algorithm for the computation of moment invariants

Abstract Moment invariants have been used as feature descriptors in a variety of object recognition applications. When assuming a continuous image function, moments calculated using a double-integral formulation, are invariant to variations in translation, rotation, and size of the object. However, due to the recursive nature of the calculations and the limited speed of microprocessors, the moments were not computable in real-time. In this paper we present fast invariant moment computations using the “Delta Method”, as a means of scene representation. This method computes moments of contiguous images by using the contribution of each line rather than individual pixel.

Singularity analysis of planar parallel manipulators

Abstract With regard to planar parallel-manipulators, a general classification of singularities into three groups is introduced. The classification scheme relies on the properties of the Jacobian matrices of the manipulator at hand. The Jacobian matrices of two classes of planar parallel manipulators are derived and the three types of singularities are identified for them. The first class contains 20 manipulators constructed with three different combinations of legs of the PRR, PPR, RRR and RPR types, P and R representing prismatic and revolute pairs, respectively. The second class consists of 4 manipulators constructed with three legs of the PRP and RRP types. Finally, one example for each class is included. Contrary to earlier claims, we show that the third type of singularity is not necessarily architecture-dependent.

Direct and specific least-square fitting of hyperbolæ and ellipses

A new method based on quadratic constrained least- mean-square fitting to simultaneously determine both the best hy- perbolic and elliptical fits to a set of scattered data is presented. Thus a linear solution to the problem of hyperbola-specific fitting is revealed for the first time. Pilus method to fit an ellipse (with respect to distance) to observed data points is extended to select, without prejudice, both ellipses and hyperbolae as well as their degenerate forms as indicated by optimality with respect to the algebraic dis- tance. This novel method is numerically efficient and is suitable for fitting to dense datasets with low noise. Furthermore, it is deemed highly suited to initialize a better but more computationally costly least-square minimization of orthogonal distance. Moreover, Grass- mannian coordinates of the hyperbolae are introduced, and it is shown how these apply to fitting a prototypical hyperbola. Two new theorems on fitting hyperbolae are presented together with rigorous proofs. A new method to determine the spatial uncertainty of the fit from the eigen or singular values is derived and used as an indicator for the quality of fit. All proposed methods are verified using numeri- cal simulation, and working MATLAB ® programs for the implemen- tation are made available. Further, an application of the methods to automatic industrial inspection is presented.

On the workspace determination of spherical serial and platform mechanisms

Abstract A technique based on the Euler–Rodrigues parameters of the rotation of a rigid body is developed to determine the workspace of spherical platform mechanisms. The workspace boundary is characterized by the occurrence of manipulator singularity. The singularity locus is obtained both as a set of two implicit functions of the four Euler–Rodrigues parameters, which thus leads to a two-parameter manifold, and as a set of two-parameter explicit functions of joint variables that yield the four Euler–Rodrigues parameters of the rotation of the moving platform at a singular posture. The technique is illustrated with various wrists and spherical platform examples.

A Special Type of Singular Stewart-Gough Platform

A Stewart-Gough platform, whose base attachment points occupy a particular cubic surface, may exhibit continuous motion while all six prismatic actuators are locked. Line geometric analysis reveals that, during such motion, the six leg axes remain in a specific linear complex, congruence or hyperboloidal ruled surface. Furthermore the pose or direct kinematics of any platform, five of whose leg base attachment points lie in such a cubic surface, is readily obtained and admits no more than four real solutions.

Singularity Analysis of a General Class of Planar Parallel Manipulators

With regard to planar parallel manipulators, a general classification of singularities into three groups is introduced. The classification scheme relies on the properties of the Jacobian matrices of the manipulator at hand. The Jacobian matrices of a general class of planar parallel manipulators are derived and the three types of singularities are identified for them. This class contains 20 manipulators constructed with three different combinations of legs of the PRR, PPR, RRR and RPR types, P and R representing prismatic and revolute pairs, respectively.

The Kinematics of 3-DOF Planar and Spherical Double-Triangular Parallel Manipulators

Two double-triangular mechanisms are introduced here. These are planar and spherical three-degree-of-freedom mechanisms that consist of two triangles moving with respect to each other. Moreover, each side of the moving triangle intersects one corresponding side of the fixed one at a given point defined over this side. The direct kinematic analysis of the mechanisms leads to a quadratic equation for the planar and a polynomial of 16th degree for the spherical mechanism. Numerical examples are included that admit two real solutions for the former and four real solutions for the latter, among which only two positive values are acceptable. All solutions, both real and complex, are listed.

Binary-Decision-Based Programmable Controllers

In certain applications, a microcomputer system employing one or more peripheral binary-decision-based controllers can react more quickly than a system using conventional microprocessor-based controllers.

The synthesis of smooth trajectories for pick-and-place operations

A spline-based method of programming smooth trajectories for pick-and-place operations is introduced. Unlike continuous-path operations, which impose a unique Cartesian trajectory, an infinite number of smooth trajectories can be described between any given pick and its corresponding place configuration. The method begins with the mapping of the pick and the place configuration in Cartesian space into joint-coordinate space, using a general-purpose inverse kinematics package that handles singularities and redundancies. Next, a trajectory, composed of a C/sup 2/-continuous, periodic cubic spline segment, is defined between the pick and the place configurations in the joint-coordinate space. It is demonstrated that C/sup 2/-continuity will prevail in Cartesian space as well. The software implementing this method includes a graphics package as well as an interface to an offline programming system to realize the synthesis of the actual robot motion. Details of the procedure are illustrated with a numerical example applied to a commercial industrial robot. >

Unified Kinematic Analysis of General Planar Parallel Manipulators

A kinematic mapping of planar displacements is used to derive generalized constraint equations having the form of ruled quadric surfaces in the image space. The forward kinematic problem for all three-legged, three-degree-of-freedom planar parallel manipulators thus reduces to determining the points of intersection of three of these constraint surfaces, one corresponding to each leg. The inverse kinematic solutions, though trivial, are implicit in the formulation of the constraint surface equations. Herein the forward kinematic solutions of planar parallel robots with arbitral, mixed leg architecture are exposed completely, and in a unified way, for the first time. Copyright