Trajectory Planning of Cable-Suspended Parallel Robots Using Interval Positive-Definite Polynomials

In this paper, the dynamic point-to-point trajectory planning of cable-suspended robots is investigated. A simple planar two-degree-of-freedom (2-dof) robot is used to demonstrate the technique. In order to maintain the cables’ positive tension, a set of algebraic inequalities is derived from the dynamic model of the 2-dof robot. The trajectories are defined using parametric polynomials with the coefficients determined by the prescribed initial and final states, and the variable time duration. With the polynomials substituted into the inequality constraints, the planning problem is then converted into an algebraic investigation on how the coefficients of the polynomials will affect the number of real roots over a given interval. An analytical approach based on a polynomial’s Discrimination Matrix and Discriminant Sequence is proposed to solve the problem. It is shown that, by adjusting the time duration within appropriate ranges, it is possible to find positive-definite polynomials such that the polynomial-based trajectories always satisfy the inequality constraints of the dynamic system. Feasible dynamic trajectories that are able to travel both beyond and within the static workspace will exploit more potential of the cable-suspended robotic platform.

Quasi-subtractive varieties

Varieties like groups, rings, or Boolean algebras have the property that, in any of their members, the lattice of congruences is isomorphic to a lattice of more manageable objects, for example normal subgroups of groups, two-sided ideals of rings, filters (or ideals) of Boolean algebras. Abstract algebraic logic can explain these phenomena at a rather satisfactory level of generality: in every member A of a τ-regular variety the lattice of congruences of A is isomorphic to the lattice of deductive filters on A of the τ-assertional logic of . Moreover, if has a constant 1 in its type and is 1-subtractive, the deductive filters on A ∈ of the 1-assertional logic of coincide with the -ideals of A in the sense of Gumm and Ursini, for which we have a manageable concept of ideal generation.However, there are isomorphism theorems, for example, in the theories of residuated lattices, pseudointerior algebras and quasi-MV algebras that cannot be subsumed by these general results. The aim of the present paper is to appropriately generalise the concepts of subtractivity and τ-regularity in such a way as to shed some light on the deep reason behind such theorems. The tools and concepts we develop hereby provide a common umbrella for the algebraic investigation of several families of logics, including substructural logics, modal logics, quantum logics, and logics of constructive mathematics.

The lattice of modal logics: an algebraic investigation

Modal logics are studied in their algebraic disguise of varieties of so-called modal algebras. This enables us to apply strong results of a universal algebraic nature, notably those obtained by B. Jónsson. It is shown that the degree of incompleteness with respect to Kripke semantics of any modal logic containing the axiom □p→P or containing an axiom of the form □mp↔□m+1p for some natural number m is . Furthermore, we show that there exists an immediate predecessor of classical logic (axiomatized by p↔□p) which is not characterized by any finite algebra. The existence of modal logics having immediate predecessors is established. In contrast with these results we prove that the lattice of extensions of S4 behaves much better: a logic extending S4 is characterized by a finite algebra iff it has finitely many extensions and any such logic has only finitely many immediate predecessors, all of which are characterized by a finite algebra.

A Purity Criterion for Pairs of Linear Transformations

In connection with the study of perturbation methods for differential eigenvalue problems, Aronszajn put forth a theory of systems (X, Y; A, B) consisting of a pair of linear transformations A, B:X → Y (see [1]; cf. also [2]). Here X and Y are complex vector spaces, possibly of infinite dimension. The algebraic aspects of this theory, where no restrictions of topological nature are imposed, where developed in [3] and [5]. We hasten to point out that the category of C2-systems (definition in § 1) in which this algebraic investigation takes place is equivalent to the category of all right modules over the ring of matrices of the form

On the aberrations of compound lenses and object-glasses

To those mathematicians who have investigated the theory of the refracting telescope, it has often, says Mr. Herschel, been objected, that little practical benefit has resulted from their speculations. Although the simplest considerations suffice for correcting that part of the aberration which arises from the different refrangibility of the different coloured rays, yet in the more difficult part of the theory of optical instruments which relates to the correction of the spherical aberration, the necessity of algebraic investigation has always been , acknowledged; although, however, the subject is confessedly within its reach, a variety of causes have interfered with its successful prosecution, and the best artists are content to work their glasses by empirical rules. In the investigations detailed in this paper, the author’s object is, first to present, under a general and uniform analysis, the whole theory of the aberration of spherical surfaces; and then to furnish practical results of easy computation to the artist, and applicable, by the simplest interpolations, to the ordinary materials on which he works. In pursuing these ends he has found it necessary somewhat to alter the usual language employed by optical writers;—thus, instead of speaking of the focal length of lenses, or the radii of their surfaces , he speaks of their powers and curvatures ; designating, by the former expression, the quotient of unity by the number of parts of any scale which the focal length is equal to; and by the latter, the quotient similarly derived from the radius in question. After adverting to some other parts of the subject of this paper, more especially to the problem of the destruction of the spherical aberration in a double or multiple lens, and to the difficulties which it involves, Mr. Herschel observes, that one condition, hitherto unaccountably overlooked, is forced upon our attention by the nature of the formulæ of aberration given in this paper; namely, its destruction not only from parallel rays, but also from rays diverging from a point at any finite distance, and which is required in a perfect telescope for land objects, and is of considerable advantage in those for astronomical use: 1st, The very moderate curvatures required for the surfaces; 2nd, That in this construction the curvatures of the two exterior surfaces of the compound lens of given focal length vary within very narrow limits, by any variation in either the refractive or dispersive powers at all likely to occur in practice; 3rd, That the two interior surfaces always approach so nearly to coincidence, that no considerable practical error can arise from neglecting their difference, and figuring them on tools of equal radii.