# 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:

*dimensional synthesis of the 3-dof 3R positioning device*: a set of poses that have to be reached by the wrist of the robot are specified and the problem is to determine the geometries of all the robots that can reach all the poses in the set. We have proposed an improved algorithm to solve this problem that have no known solution up to now [14]*trajectory planner for parallel robot*: we have improved our trajectory planner that allows one to verify if a given almost arbitrary trajectory fully lie within the workspace of a parallel robot. This planner takes into account the uncertainties in the trajectory execution and in the robot modeling [28]*Jacobian matrix of parallel robot*: we have improved our algorithm that allows one to determine the minimal and maximal eigenvalues of the Jacobian matrix of a robot whose pose is constrained to lie within a given workspace. These values give precious indication on the maximal positioning error of the robot [27]*optimal design of a parallel with respect to workspace and accuracy requirements*: we have developed a design algorithm that allow one to compute almost all design solution for given workspace and accuracy requirement*forward kinematics of Gough platform with general geometry*: we propose a fast and efficient algorithm to solve this difficult problem*wire interference in parallel robots*: we have developed an algorithm to take into account wire interference for the workspace analysis of wire robots*parallel robot calibration*: we have proposed algorithms based in interval analysis to solve and certify calibration equations while algebraic geometry has been used to provide more robust calibration equations

#### 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 A_{i} may be obtained for any design
requirements R_{i} in the set of requirements {R_{1}, ..., R_{m}},
then the possible design parameters values will be
obtained as the intersection of all the R_{i}. This approach has the
following advantages over more classical approaches:

it allows to deal with

*imperative*requirements i.e. requirements that must absolutely be satisfied by a design solutionit offers all the possible compromises between requirements that are antagonistic

it allow to deal with uncertainties: indeed the physical instance of a theoretical solution will differ from it due to manufacturing tolerances. In our approach the approximation includes only boxes whose width is at least twice the manufacturing tolerances

_{i}. For example if a solution is provided for the design parameter D_{1}as the range [a_{1}, b_{1}], then we may choose as manufacturing solution any value in the range [a_{1}+_{1}, b_{1}-_{1}] so that we can guarantee that the physical instance of D_{1}will lie in the range [a_{1}, b_{1}]

The difficulty in this approach is to calculate the approximation
R_{i}. Interval analysis is a tool of choice for such problem. We have
shown that it was possible to compute the R_{i} 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 R_{i}
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 :

taking into account the influence of measurement noise on calibration results, if the error distribution is not given,

choosing an appropriate geometrical model of the robot that is sufficiently simplified for providing manageable calibration equations while still describing realistically the robot behavior

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.