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.
[Show abstract][Hide abstract] 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(3):221-235. DOI:10.1177/0278364904039806 · 2.54 Impact Factor
[Show abstract][Hide abstract] 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.
[Show abstract][Hide abstract] ABSTRACT: In this paper, we consider the direct kinematicproblem of a parallel robot (called the Stewart platformor left hand). We want to show how the useof formal tools help us to guess the solution of thisproblem and then to establish it. We do not tryhere to give real time and numerical solutions tothe problem of inverse images but focus on tools ofeffective algebra, which can help us to know a littlemore about the geometric aspects of the question.In a first part, we describe...
Proceedings of the 1993 international symposium on Symbolic and algebraic computation; 01/1993
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