CDR - Cinemàtica i Disseny de Robotshttp://hdl.handle.net/2117/797172022-01-21T09:54:51Z2022-01-21T09:54:51ZA distance geometry approach to the singularity analysis of 3R robotsThomas, Federicohttp://hdl.handle.net/2117/826022020-07-23T21:26:57Z2016-02-05T09:51:05ZA distance geometry approach to the singularity analysis of 3R robots
Thomas, Federico
This paper shows how the computation of the singularity locus of a 3R robot can be reduced to the analysis of the relative position of two coplanar ellipses. Since the relative position of two conics is a projective invariant, and the basic projective geometric invariants are determinants, it is not surprising that, using Distance Geometry, the computation of the singularity locus of a 3R robot can be fully expressed in terms of determinants. Geometric invariants have the benefit of simplifying symbolic manipulations. This paper shows how their use leads to a simpler characterization, compared to previous approaches, of the cusps and nodes in the singularity loci of 3R robots.
2016-02-05T09:51:05ZThomas, FedericoThis paper shows how the computation of the singularity locus of a 3R robot can be reduced to the analysis of the relative position of two coplanar ellipses. Since the relative position of two conics is a projective invariant, and the basic projective geometric invariants are determinants, it is not surprising that, using Distance Geometry, the computation of the singularity locus of a 3R robot can be fully expressed in terms of determinants. Geometric invariants have the benefit of simplifying symbolic manipulations. This paper shows how their use leads to a simpler characterization, compared to previous approaches, of the cusps and nodes in the singularity loci of 3R robots.A unified position analysis of the Dixon and the generalized Peaucellier linkagesRojas Libreros, Nicolás EnriqueDollar, Aaron M.Thomas, Federicohttp://hdl.handle.net/2117/793492020-07-23T21:43:08Z2015-11-17T10:22:46ZA unified position analysis of the Dixon and the generalized Peaucellier linkages
Rojas Libreros, Nicolás Enrique; Dollar, Aaron M.; Thomas, Federico
This paper shows how, using elementary Distance Geometry, a closure polynomial of degree 8 for the Dixon linkage can be derived without any trigonometric substitution, variable elimination, or artifice to collapse mirror configurations. The formulation permits the derivation of the geometric conditions required in order for each factor of the leading coefficient of this polynomial to vanish. These conditions either correspond to the case in which the quadrilateral defined by four joints is orthodiagonal, or to the case in which the center of the circle defined by three joints is on the line defined by two other joints. This latter condition remained concealed in previous formulations. Then, particular cases satisfying some of the mentioned geometric conditions are analyzed. Finally, the obtained polynomial is applied to derive the coupler curve of the generalized Peaucellier linkage, a linkage with the same topology as that of the celebrated Peaucellier straight-line linkage but with arbitrary link lengths. It is shown that this curve is 11-circular of degree 22 from which the bicircular quartic curve of the Cayley's scalene cell is derived as a particular case.
2015-11-17T10:22:46ZRojas Libreros, Nicolás EnriqueDollar, Aaron M.Thomas, FedericoThis paper shows how, using elementary Distance Geometry, a closure polynomial of degree 8 for the Dixon linkage can be derived without any trigonometric substitution, variable elimination, or artifice to collapse mirror configurations. The formulation permits the derivation of the geometric conditions required in order for each factor of the leading coefficient of this polynomial to vanish. These conditions either correspond to the case in which the quadrilateral defined by four joints is orthodiagonal, or to the case in which the center of the circle defined by three joints is on the line defined by two other joints. This latter condition remained concealed in previous formulations. Then, particular cases satisfying some of the mentioned geometric conditions are analyzed. Finally, the obtained polynomial is applied to derive the coupler curve of the generalized Peaucellier linkage, a linkage with the same topology as that of the celebrated Peaucellier straight-line linkage but with arbitrary link lengths. It is shown that this curve is 11-circular of degree 22 from which the bicircular quartic curve of the Cayley's scalene cell is derived as a particular case.