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Singularities of Nonredundant Manipulators: A Short Account and a Method for their Computation in the Planar Case
, 2013
"... The study of the singularity set is of utmost utility in understanding the local and global behavior of a manipulator. After reviewing the mathematical conditions that characterize this set, and their significance, this paper shows how these conditions can be formulated in an amenable manner in the ..."
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Cited by 5 (3 self)
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The study of the singularity set is of utmost utility in understanding the local and global behavior of a manipulator. After reviewing the mathematical conditions that characterize this set, and their significance, this paper shows how these conditions can be formulated in an amenable manner in the planar case, allowing to define a conceptuallysimple method for isolating the set exhaustively, even in higherdimensional cases. As a result, the method delivers a collection of boxes bounding the location of all points of the set, whose accuracy can be adjusted through a threshold parameter. Such boxes can then be projected to the input or output coordinate spaces, obtaining informative diagrams, or portraits, on the global motion capabilities of the manipulator. Examples are included that show the application of the method to simple manipulators, and to a complex mechanism that would be difficult to analyze using commonpractice procedures.
SingularityInvariant Families of LinePlane 5SPU Platforms
, 2011
"... A 5SPU robot with collinear universal joints is well suited to handling an axisymmetric tool, since it has 5 controllable DoFs and the remaining one is a free rotation around the tool. The kinematics of such a robot having also coplanar spherical joints has previously been studied as a rigid suba ..."
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Cited by 4 (1 self)
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A 5SPU robot with collinear universal joints is well suited to handling an axisymmetric tool, since it has 5 controllable DoFs and the remaining one is a free rotation around the tool. The kinematics of such a robot having also coplanar spherical joints has previously been studied as a rigid subassembly of a StewartGough platform, it being denoted a lineplane component. Here we investigate how to move the leg attachments in the base and the platform without altering the robot’s singularity locus. By introducing the socalled 3D space of leg attachments, we prove that there are only three general topologies for the singularity locus corresponding to the families of quartically, cubically and quadraticallysolvable 5SPU robots. The members of the last family have only 4 assembly modes, which are obtained by solving two quadratic equations. Two practical features of these quadraticallysolvable robots are the large manipulability within each connected component and the fact that, for a fixed orientation of the tool, the singularity locus reduces to a plane.
Architectural singularities of a class of pentapods
 Mech. Mach. Theor
, 2011
"... A pentapod is usually defined as a 5degreesoffreedom fullyparallel manipulator with an axial spindle as moving platform. This kind of manipulators has revealed as an interesting alternative to serial robots handling axisymmetric tools. Their particular geometry permits that, in one tool axis, hi ..."
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A pentapod is usually defined as a 5degreesoffreedom fullyparallel manipulator with an axial spindle as moving platform. This kind of manipulators has revealed as an interesting alternative to serial robots handling axisymmetric tools. Their particular geometry permits that, in one tool axis, high inclination angles could be attained, thus overcoming the orientation limits of the classical StewartGough platform. This paper deals with pentapods with coplanar base attachments. In previous works changes in the location of the leg attachments that do not modify the singularity locus of the pentapod were studied. Such leg rearrangements reveal here as a powerful tool to shed light on the geometric structure of the singularity locus and, in particular, on architectural singularities. Indeed, a complete analysis of such singularities is carried out, providing both algebraic conditions, which complete previous results found in literature, and a geometrical interpretation that permits defining a measure of distance to architectural singularities. Such measure can be used as a index in the design process to obtain manipulators as far as possible from architectural singularities, leading to a better global behavior.
On the Primal and Dual Forms of the Stewart Platform Pure Condition
 SUBMITTED TO THE IEEE TRANSACTIONS ON ROBOTICS
, 2012
"... The algebraic characterization of the singularities of a Stewart platform is usually presented as a 6 × 6 determinant, whose rows correspond to the line coordinates of its legs, equated to zero. This expression can be rewritten in a more amenable way, known as the pure condition, as sums and product ..."
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Cited by 3 (2 self)
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The algebraic characterization of the singularities of a Stewart platform is usually presented as a 6 × 6 determinant, whose rows correspond to the line coordinates of its legs, equated to zero. This expression can be rewritten in a more amenable way, known as the pure condition, as sums and products of 4 ×4 determinants whose rows correspond to the point coordinates of the legs attachments. Researchers usually rely on one of these two expressions to find the geometric conditions associated with the singularities of a particular Stewart platform. Although both are equivalent, it is advantageous to use either line or point coordinates depending on the platform topology. In this context, an equivalent expression involving only plane coordinates, a dual expression to that using point coordinates, seems to be missing. This paper is devoted to its derivation and to show how its use is advantageous in many practical cases mainly because of its surprising simplicity: it only involves the addition of 4 × 4 determinants whose rows are plane coordinates defined by sets of three attachments.
1SingularityInvariant Families of LinePlane 5SPU Platforms
"... Abstract — A 5SPU robot with collinear universal joints is well suited to handling an axisymmetric tool, since it has 5 controllable DoFs and the remaining one is a free rotation around the tool. The kinematics of such a robot having also coplanar spherical joints has previously been studied as a r ..."
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Abstract — A 5SPU robot with collinear universal joints is well suited to handling an axisymmetric tool, since it has 5 controllable DoFs and the remaining one is a free rotation around the tool. The kinematics of such a robot having also coplanar spherical joints has previously been studied as a rigid subassembly of a StewartGough platform, it being denoted a lineplane component. Here we investigate how to move the leg attachments in the base and the platform without altering the robot’s singularity locus. By introducing the socalled 3D space of leg attachments, we prove that there are only three general topologies for the singularity locus corresponding to the families of quartically, cubicallyand quadraticallysolvable 5SPU robots. The members of the last family have only 4 assembly modes, which are obtained by solving two quadratic equations. Two practical features of these quadraticallysolvable robots are the large manipulability within each connected component and the fact that, for a fixed orientation of the tool, the singularity locus reduces to a plane. Index Terms — Parallel manipulators, GoughStewart platforms, robot kinematics, kinematics singularities, manipulator design. I.
On QuarticallySolvable Robots
"... Abstract — This paper presents a first attempt at a unified kinematics analysis of all serial and parallel solvable robots, that is, robots whose position analysis can be carried out without relying on numerical methods. The efforts herein are focused on finding a unified formulation for all quartic ..."
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Abstract — This paper presents a first attempt at a unified kinematics analysis of all serial and parallel solvable robots, that is, robots whose position analysis can be carried out without relying on numerical methods. The efforts herein are focused on finding a unified formulation for all quarticallysolvable robots, as all other solvable robots can be seen as particular cases of them. The first part is centered on the quest for the most general quarticallysolvable parallel and serial robots. As a result, representatives of both classes are selected. Then, using Distance Geometry, it is shown how solving the forward kinematics of the parallel representative is equivalent to solve the inverse kinematics of the serial representative, thus providing a unified formulation. Finally, it is shown that the position and singularity analysis of these robots reduces to the analysis of the relative position of two coplanar ellipses. I.
New Geometric Approaches to the Analysis and Design of StewartGough Platforms
"... Abstract — In general, rearranging the legs of a StewartGough platform, i.e., changing the locations of its leg attachments, modifies the platform singularity locus in a rather unexpected way. Nevertheless, some leg rearrangements have been recently found to leave singularities invariant. Identific ..."
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Abstract — In general, rearranging the legs of a StewartGough platform, i.e., changing the locations of its leg attachments, modifies the platform singularity locus in a rather unexpected way. Nevertheless, some leg rearrangements have been recently found to leave singularities invariant. Identification of such rearrangements is useful not only for the kinematic analysis of the platforms, but also as a tool to redesign manipulators avoiding the implementation of multiple spherical joints, which are difficult to construct and have a small motion range. In this work, a summary of these singularityinvariant leg rearrangements is presented, and their practical implications are illustrated with several examples including wellknown architectures. I.
New Geometric Approaches to the Singularity Analysis of Parallel Platforms
"... Abstract — In general, rearranging the legs of a StewartGough platform, i.e., changing the locations of its leg attachments, modifies the platform singularity locus in a rather unexpected way. Nevertheless, some leg rearrangements have been recently found to leave singularities invariant. In this ..."
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Abstract — In general, rearranging the legs of a StewartGough platform, i.e., changing the locations of its leg attachments, modifies the platform singularity locus in a rather unexpected way. Nevertheless, some leg rearrangements have been recently found to leave singularities invariant. In this work, a summary of the some of such singularityinvariant leg rearrangements are presented, and their practical consequences are illustrated with several examples including wellknown architectures. I.
Distance Bound Smoothing under Orientation Constraints
"... Abstract — Distance Bound Smoothing (DBS) is a basic operation originally developed in Computational Chemistry to determine point configurations that are within certain pairwise ranges of distances. This operation consist in the iterative application of filtering processes that reduce the given ran ..."
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Abstract — Distance Bound Smoothing (DBS) is a basic operation originally developed in Computational Chemistry to determine point configurations that are within certain pairwise ranges of distances. This operation consist in the iterative application of filtering processes that reduce the given ranges using triangular and tetrangular inequalities. Standard DBS has a limited range of applications because it does not take into account constraints on the orientations of simplices (triangles or tetrahedra, depending on the dimension of the problem). This paper discuses an extension of DBS that permits incorporating these constraints. This paves the way for the application of DBS techniques to a broad range of problems in Robotics. I.