Team Coprin

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Inria / Raweb 2004
Project: Coprin

Project : coprin

Section: New Results


Keywords : robotics, calibration, robot accuracy, mechanism theory, parallel robots.

Parallel robots

The core of our activity in robotics and mechanism theory is the optimal design of mechanism and the analysis of parallel robots. The following points have been addressed this year:

Optimal design of parallel robots with respect to workspace and accuracy requirements

Keywords : optimal design, parallel robots.

Participants : Yuan Cheng, David Daney, Fang Hao, Jean-Pierre Merlet.

Our methodology for optimal design is to determine an approximation of all the possible values of the n design parameters so that a design requirements (or a set of them) is satisfied. Such approximation is obtained as a set of boxes in the parameters space, an n dimensional space in which each frame axis represent the value of one design parameters. If an approximation Ai may be obtained for any design requirements Ri in the set of requirements {R1, ..., Rm}, then the possible design parameters values will be obtained as the intersection of all the Ri. This approach has the following advantages over more classical approaches:

The difficulty in this approach is to calculate the approximation Ri. Interval analysis is a tool of choice for such problem. We have shown that it was possible to compute the Ri in a 6-dimensional parameters space so that the robot workspace will include an arbitrary large set of pre-defined poses, while the positioning errors of the robot at these poses will not be larger than a pre-defined threshold [12].

Recently we have also shown that it was possible to compute the Ri in a 26-dimensional parameters space so that the positioning errors of the robot over a given workspace are lower than pre-defined threshold.

Forward kinematics

Keywords : forward kinematics, parallel robots.

Participant : Jean-Pierre Merlet.

It may be thought that this "old" but difficult problem has been solved but this is not the case in practice. Indeed, with one exception (the combination of the FgB and RS software of Faugère and Rouillier of the SALSA project) the proposed algorithms either deal with special geometries of the robot or do not provide certified answers (solutions may be either wrong or lost) or are not fully automated. Furthermore it must be reminded that the real problem is not to determine all the solutions but only the one corresponding to the actual pose of the robot (and there is no known algorithm for determining which is the current pose among the set of all solutions).

We have developed within the ALIAS library a distance equations solver that may be used for this problem. Most of the theorems we are using within our general purpose solvers have been revisited for distance equations, allowing to get stronger versions of these theorems (for example the exclusion regions obtained with the Kantorovitch theorem or Neumaier exclusion theorem [43], that are guaranteed to include a unique solution, are usually about 20 times larger than obtained with the general purpose version).

Our tests [15] have shown that while providing certified solutions (no solution will be lost and the solutions can be usually calculated with an arbitrary accuracy) our algorithm is just outperformed by FgB and RS when looking for all solutions. When the search space is restricted (as this will be the case in practice) the algorithm is the fastest available. In particular for real-time application the algorithm is almost as fast as the Newton scheme while ensuring to provide either the exact current pose (and this is not the case of the Newton scheme) or an emergency signal indicating that multiple solutions exist, in which case it is necessary to stop the robot as it is no more possible to control it.

Certification of parallel robot calibration: experimental results

Keywords : robotics, parallel robot, calibration, parametric system of equation solving, interval analysis, constraint programming, experimental results.

Participants : David Daney, Yves Papegay, Gilles Chabert.

Accurate identification of the kinematic parameters of a robot is difficult due to two main problems :

We propose to use interval methods (2B, 3B and a specific interval Newton method) which permit to bound the set of solutions. Additionally, this approach permits to check the validity of the calibration equations and also proposes possible correction for the robot modeling. This method has been successfully verified experimentally on a Deltalab Gough platform. This work was performed within the framework of the national Robea/MP2 project with the collaboration of N. Andreff of IFMA.

Algebraic methods for the simplification of calibration equations

Keywords : calibration, parallel robot, algebraic geometry.

Participants : David Daney, Ioannis Emiris.

An interesting measurement device for calibration is the double-ball bar mechanism (DBB) constituted of two ball-and-socket joints connected by an extensible leg whose length is measured. This device may be used for control or calibration of a parallel robot by linking the joints to the base and platform of the robot. Usually the internal state of a n d.o.f. parallel robots is measured by exactly n sensors. Introducing at least one additional measurement (for example with a DBB) allows theoretically the calibration of the robot by using only its internal sensors. But with only one additional measurement it is difficult to obtain a numerically stable system of calibration equations. We have proposed algebraic methods to construct such a system. Moreover we have also studied the influence of the number of additional DBB on the numerical stability with regards to measurement noise. The aim of this work is to propose a simple and robust calibration scheme adapted to a deployable robot.


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