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Cuspidal robots can travel from one inverse kinematic solution to another without meeting a singularity. The name cuspidal was coined based on the existence of a cusp point in the workspace of 3R serial robots. The existence of a cusp point was proved to be a necessary and sufficient condition for orthogonal robots to be cuspidal, but it was not possible to extend this condition to non-orthogonal robots. The goal of this paper is to prove that this condition stands for any generic 3R robot. This result would give the designer more flexibility. In the presented work, the geometrical interpretation of the inverse kinematics of 3R robots is revisited and important observations on the nonsingular change of posture are noted. The paper presents a theorem regarding the existence of reduced aspects in any generic 3R serial robot. Based on these observations and on this theorem, we prove that the existence of a cusp point is a necessary and sufficient condition for any 3R generic robot to be cuspidal.

Answering connectivity queries in semi-algebraic sets is a long-standing and challenging computational issue with applications in robotics, in particular for the analysis of kinematic singularities. One task there is to compute the number of connected components of the complementary of the singularities of the kinematic map. Another task is to design a continuous path joining two given points lying in the same connected component of such a set. In this paper, we push forward the current capabilities of computer algebra to obtain computer-aided proofs of the analysis of the kinematic singularities of various robots used in industry. We first show how to combine mathematical reasoning with easy symbolic computations to study the kinematic singularities of an infinite family (depending on paramaters) modelled by the UR-series produced by the company ``Universal Robots''. Next, we compute roadmaps (which are curves used to answer connectivity queries) for this family of robots. We design an algorithm for ``solving'' positive dimensional polynomial system depending on parameters. The meaning of solving here means partitioning the parameter's space into semi-algebraic components over which the number of connected components of the semi-algebraic set defined by the input system is invariant. Practical experiments confirm our computer-aided proof and show that such an algorithm can already be used to analyze the kinematic singularities of the UR-series family. The number of connected components of the complementary of the kinematic singularities of generic robots in this family is $8$.

We develop an algorithm that solves the inverse kinematics of general serial 2RP3R, 2R2P2R, 3RP2R and 6R manipulators based on the HuPf algorithm. We identify the workspaces of the 3-subchains of the manipulator with a quasi-projective variety in $\mathbb{P}^7$ via dual quaternions. This allows us to compute linear forms that describe linear spaces containing the workspaces of these 3-subchains. We present numerical examples that illustrate the algorithm and show the real solutions., Comment: 11 pages, 9 figures

There exist homogeneous polynomials $f$ with $\mathbb Q$-coefficients that are sums of squares over $\mathbb R$ but not over $\mathbb Q$. The only systematic construction of such polynomials that is known so far uses as its key ingredient totally imaginary number fields $K/\mathbb Q$ with specific Galois-theoretic properties. We first show that one may relax these properties considerably without losing the conclusion, and that this relaxation is sharp at least in a weak sense. In the second part we discuss the open question whether any $f$ as above necessarily has a (non-trivial) real zero. In the minimal open cases $(3,6)$ and $(4,4)$, we prove that all examples without a real zero are contained in a thin subset of the boundary of the sum of squares cone.

Laman graphs model planar frameworks which are rigid for a general choice of distances between the vertices. There are finitely many ways, up to isometries, to realize a Laman graph in the plane. In a recent paper we provide a recursion formula for this number of realizations using ideas from algebraic and tropical geometry. Here, we present a concise summary of this result focusing on the main ideas and the combinatorial point of view., Comment: Extended abstract; the long version is arxiv:1701.05500

Laman graphs model planar frameworks that are rigid for a general choice of distances between the vertices. There are finitely many ways, up to isometries, to realize a Laman graph in the plane. Such realizations can be seen as solutions of systems of quadratic equations prescribing the distances between pairs of points. Using ideas from algebraic and tropical geometry, we provide a recursive formula for the number of complex solutions of such systems., Comment: 36 pages

In this work we attempt to generalize our result in [6] [7] for real rings (not just von Neumann regular real rings). In other words we attempt to characterize and construct real closure * of commutative unitary rings that are real. We also make some very interesting and significant discoveries regarding maximal partial orderings of rings, Baer rings and essentail extension of rings. The first Theorem itself gives us a noteworthy bijection between maximal partial orderings of two rings by which one is a rational extension of the other. We characterize conditions when a Baer reduced ring can be integrally closed in its total quotient ring. We prove that Baer hulls of rings have exactly one automorphism (the identity) and we even prove this for a general case (Lemma 12). Proposition 14 allows us to study essential extensions of rings and their relation with minimal prime spectrum of the lower ring. And Theorem 15 gives us a construction of the real spectrum of a ring generated by adjoining idempotents to a reduced commutative subring (for instance the construction of Baer hull of reduced commutative rings). From most of the above interesting theories we prove that there is a bijection between the real closure * of real rings (upto isomorphisms) and their maximal partial orderings. We then attempt to develop some topological theories for the set of real closure * of real rings (upto isomorphism) that will help us give a topological characterization in terms of the real and prime spectra of these rings. The topological characterization will be revealed in a later work. It is noteworthy to point out that we can allow ourself to consider mostly the minimal prime spectrum of the real ring in order to develop our topological theories., Comment: 17pages. Part of PhD dissertation

It was pointed out in my last paper that there are rings whose real closure * are not unique. In [4] we also discussed some example of rings by which there is a unique real closure * (mainly the real closed rings). Now we want to determine more classes of rings by which real closure * is unique. The main results involve characterisations of domains and Baer regular rings having unique real closure *, and an example showing that regular rings need not be f-rings in order to have a unique real closure *. The main objective here is to find characterisation for uniqueness of real closure * for real regular rings that will primarily only require information of the prime spectrum and the real spectrum of the ring.

Here we try to distinguish and compare different notions of real closedness mainly one developed by N. Schwartz in his Habilitationschrift and the other developed by A. Sankaranand K. Varadarajan which we shall call real closed *. We stick to the definition of real closed rings as defined and characterized N. Schwartz and we try to determine and characterize real closed rings that are real closed *. The main result is that real closed rings have unique real closure * and that real closure of real closed * rings arent necessarily real closed *.