Chapter

A Geometric Newton-Raphson Method for Gough-Stewart Platforms

DOI: 10.1007/978-3-642-01947-0_23

ABSTRACT A geometric version of the well known Newton-Raphson methods is introduced. This root finding method is adapted to find the
zero of a function defined on the group of rigid body displacements. At each step of the algorithm a rigid displacement is
found that approximates the solution. The method is applied to the forward kinematics problem of the Gough-Stewart platform.

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    ABSTRACT: We consider in this paper a Gough-type parallel robot and we present an efficient algorithm based on interval analysis that allows us to solve the forward kinematics, i.e., to determine all the possible poses of the platform for given joint coordinates. This algorithm is numer- ically robust as numerical round-off errors are taken into account; the provided solutions are either exact in the sense that it will be pos- sible to refine them up to an arbitrary accuracy or they are flagged only as a "possible" solution as either the numerical accuracy of the computation does not allow us to guarantee them or the robot is in a singular configuration. It allows us to take into account physical and technological constraints on the robot (for example, limited motion of the passive joints). Another advantage is that, assuming realis- tic constraints on the velocity of the robot, it is competitive in term of computation time with a real-time algorithm such as the Newton scheme, while being safer. KEY WORDS—forward kinematics, parallel robot
    The International Journal of Robotics Research 03/2004; 23:221-235. DOI:10.1177/0278364904039806 · 2.50 Impact Factor
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    ABSTRACT: In this paper an algorithm for solving the direct kinematics of general Stewart-Gough platforms is introduced. A minimal set of constraint equations obtained by a kinematic mapping to produce ultimately the univariate polynomial of 40th degree is used. The algorithm is illustrated with an example for which the univariate polynomial is computed.ZusammenfassungIn dieser Arbeit wird ein Algorithmus zur Lösung der direkten Kinematik von allgemeinen Stewart-Gough Plattformen vorgestellt. Eine solche Plattform wird geometrisch realisiert durch sechs Punkte im Gangraum und sechs Punkte im Rastraum und durch sechs vorgegebene Beinlängen. Die kinematische Abbildung von Bewegungsvorgängen des dreidimensionalen Raumes— zurückgehen auf Study—wird dazu benutzt, einen Satz von einfachen, durchwegs quadratischen Bedingungsgleichungen zu erhalten. Unter Ausnutzung der speziellen geometrischen Eigenschaften der Bedingungsgleichungen wird eine Elimination der Variablen vorgeführt, was letztendlich auf ein Polynom vierzigsten Grades in einer Variablen führt. Im letzten Abschnitt wird der Algorithmus anhand eines einfachen Beispiels illustriert.
    Mechanism and Machine Theory 05/1996; 31(4-31):365-379. DOI:10.1016/0094-114X(95)00091-C · 1.31 Impact Factor
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    ABSTRACT: Many differential equations of practical interest evolve on Lie groups or on manifolds acted upon by Lie groups. The retention of Lie-group structure under discretization is often vital in the recovery of qualitatively correct geometry and dynamics and in the minimization of numerical error. Having introduced requisite elements of differential geometry, this paper surveys the novel theory of numerical integrators that respect Lie-group structure, highlighting theory, algorithmic issues and a number of applications.
    Acta Numerica 01/2000; 9. DOI:10.1017/S0962492900002154