Lluís Ros

Revisiting trilateration for robot localization

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.

An ellipsoidal calculus based on propagation and fusion

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.

Synthesizing grasp configurations with specified contact regions

In this paper we present a new method to solve the configuration problem on robotic hands: determining how a hand should be configured so as to grasp a given object in a specific way, characterized by a number of hand—object contacts to be satisfied. In contrast to previous algorithms given for the same purpose, the method presented here allows such contacts to be specified between free-form regions on the hand and object surfaces, and always returns a solution whenever one exists. The method is based on formulating the problem as a system of polynomial equations of special form, and then exploiting this form to isolate the solutions, using a numerical technique based on linear relaxations. The approach is general, in the sense that it can be applied to any grasping mechanism involving lower-pair joints, and it can accommodate as many hand—object contacts as required. Experiments are included that illustrate the performance of the method in the particular case of the Schunk Anthropomorphic hand.

A Linear Relaxation Technique for the Position Analysis of Multiloop Linkages

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.

Complete maps of molecular‐loop conformational spaces

This paper presents a numerical method to compute all possible conformations of distance‐constrained molecular loops, i.e., loops where some interatomic distances are held fixed, while others can vary. The method is general (it can be applied to single or multiple intermingled loops of arbitrary topology) and complete (it isolates all solutions, even if they form positive‐dimensional sets). Generality is achieved by reducing the problem to finding all embeddings of a set of points constrained by pairwise distances, which can be formulated as computing the roots of a system of Cayley–Menger determinants. Completeness is achieved by expressing these determinants in Bernstein form and using a numerical algorithm that exploits such form to bound all root locations at any desired precision. The method is readily parallelizable, and the current implementation can be run on single‐ or multiprocessor machines. Experiments are included that show the methods performance on rigid loops, mobile loops, and multiloop molecules. In all cases, complete maps including all possible conformations are obtained, thus allowing an exhaustive analysis and visualization of all pseudo‐rotation paths between different conformations satisfying loop closure.

A Complete Method for Workspace Boundary Determination on General Structure Manipulators

This paper introduces a new method for workspace boundary determination on general structure manipulators. The method uses a branch-and-prune technique to isolate a set of output singularities and then classifies the points on such a set according to whether they correspond to motion impediments in the workspace. A detailed map of the workspace is obtained as a result, where all interior and exterior regions, together with the singularity and barrier sets that separate them, get clearly identified. The method can deal with open- or closed-chain manipulators, whether planar or spatial, and is able to take joint limits into account. Advantages over previous general methods based on continuation include the ability to converge to all boundary points, even in higher dimensional cases, and the fact that manual guidance with a priori knowledge of the workspace is not required. Examples are included that show the performance of the method on benchmark problems documented in the literature, as well as on new ones unsolved so far.

A branch-and-prune solver for distance constraints

Given some geometric elements such as points and lines in R/sup 3/, subject to a set of pairwise distance constraints, the problem tackled in this paper is that of finding all possible configurations of these elements that satisfy the constraints. Many problems in robotics (such as the position analysis of serial and parallel manipulators) and CAD/CAM (such as the interactive placement of objects) can be formulated in this way. The strategy herein proposed consists of looking for some of the a priori unknown distances, whose derivation permits solving the problem rather trivially. Finding these distances relies on a branch-and-prune technique, which iteratively eliminates from the space of distances entire regions which cannot contain any solution. This elimination is accomplished by applying redundant necessary conditions derived from the theory of distance geometry. The experimental results qualify this approach as a promising one.

Box Approximations of Planar Linkage Configuration Spaces

This paper presents a numerical method able to compute all possible configurations of planar linkages. The procedure is applicable to rigid linkages (i.e., those that can only adopt a finite number of configurations) and to mobile ones (i.e., those that exhibit a continuum of possible configurations). The method is based on the fact that this problem can be reduced to finding the roots of a polynomial system of linear, quadratic, and hyperbolic equations, which is here tackled with a new strategy exploiting its structure. The method is conceptually simple and easy to implement, yet it provides solutions of the desired accuracy in short computation times. Experiments are included that show its performance on the double butterfly linkage and on larger linkages formed by the concatenation of basic patterns.

Finding all valid hand configurations for a given precision grasp

Planning a precision grasp for a robot hand is usually decomposed into two main steps. First, a set of contact points over the object surface must be determined, ensuring they allow a stable grasp. Second, the inverse kinematics of the robot hand must be solved to verify whether the contact points can actually be reached. Whereas the first problem has been largely solved in a general posing, the second one has only been tackled with local convergence methods. These methods only provide one solution to the problem, even if many are possible, and depending on the initial estimation they use, they may fail to converge, which results in grasp re-planning in situations where it could be avoided. This paper overcomes both issues by providing a complete method to solve the kinematics of human-like hands. The method is able to find all possible configurations that reach the specified contact points, even when positive-dimensional sets of such configurations are possible.

Motion planning for 6-D manipulation with aerial towed-cable systems

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 cable-driven 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.