Abstract-Locating a robot from its distances, or range measurements, to three other known points or stations is a common operation, known as trilateration. This problem has been traditionally solved either by algebraic or numerical methods. An approach that avoids the direct algebrization of the problem is proposed here. Using constructive geometric arguments, a coordinate-free formula containing a small number of Cayley-Menger determinants is derived. This formulation accommodates a more thorough investigation of the effects caused by all possible sources of error, including round-off errors, for the first time in this context. New formulas for the variance and bias of the unknown robot location estimation, due to station location and range measurements errors, are derived and analyzed. They are proved to be more tractable compared with previous ones, because all their terms have geometric meaning, allowing a simple analysis of their asymptotic behavior near singularities.

Abstract-This paper presents a new method to isolate all configurations that a multiloop linkage can adopt. The problem is tackled by means of formulation and resolution techniques that fit particularly well together. The adopted formulation yields a system of simple equations (only containing linear, bilinear, and quadratic monomials, and trivial trigonometric terms for the helical pair only) whose structure is later exploited by a branch-and-prune method based on linear relaxations. The method is general, as it can be applied to linkages with single or multiple loops with arbitrary topology, involving lower pairs of any kind, and complete, as all possible solutions get accurately bounded, irrespective of whether the linkage is rigid or mobile.

Abstract-This paper presents an Ellipsoidal Calculus based solely on two basic operations: propagation and fusion. Propagation refers to the problem of obtaining an ellipsoid that must satisfy an affine relation with another ellipsoid, and fusion to that of computing the ellipsoid that tightly bounds the intersection of two given ellipsoids. These two operations supersede the Minkowski sum and difference, affine transformation and intersection tight bounding of ellipsoids on which other ellipsoidal calculi are based. Actually, a Minkowski operation can be seen as a fusion followed by a propagation and an affine transformation as a particular case of propagation. Moreover, the presented formulation is numerically stable in the sense that it is immune to degeneracies of the involved ellipsoids and/or affine relations.Examples arising when manipulating uncertain geometric information in the context of the spatial interpretation of line drawings are extensively used as a testbed for the presented calculus.Index Terms-Ellipsoidal bounds, ellipsoidal calculus, set-membership uncertainty description.

Abstract-Performing aerial 6-dimensional manipulation using flying robots is a challenging problem, to which only little work has been devoted. This paper proposes a motion planning approach for the reliable 6-dimensional quasi-static manipulation with an aerial towed-cable system. The novelty of this approach lies in the use of a cost-based motion-planning algorithm together with some results deriving from the static analysis of cabledriven manipulators. Based on the so-called wrench-feasibility constraints applied to the cable tensions, as well as thrust constraints applied to the flying robots, we formally characterize the set of feasible configurations of the system. Besides, the expression of these constraints leads to a criterion to evaluate the quality of a configuration. This allows us to define a cost function over the configuration space, which we exploit to compute good-quality paths using the T-RRT algorithm. As part of our approach, we also propose an aerial towed-cable system that we name the FlyCrane. It consists of a platform attached to three flying robots using six fixed-length cables. We validate the proposed approach on two simulated 6-D quasi-static manipulation problems involving such a system, and show the benefit of taking the cost function into account for such motion planning tasks.