Project : coprin

Section: New Results

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

Robotics and mechanism theory

The core of our activity in robotics and mechanism theory is the optimal design of mechanism and the analysis of parallel robots [17] [6]. This year we have focused on:

• the dimensional synthesis of 3-dof positioning device: last year we have solved the problem of dimensional synthesis of 3R mechanism (for which no solution was known): 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 started this year a preliminary study of the same problem for others mechanical architectures (such as RPS, CPS, $...$ ). This study has shown that while we were able to determine some solutions to the problem, it was difficult to find all the solutions. As for the 3R structure it seems that a solving algorithm combining various formulations of the problem must be developed

• exact determination of a robot dexterity over a given workspace

• optimal geometry of modular parallel robots

• the determination of the location of the measurement poses for optimal calibration of robots with a local method

• calibration of parallel robots based either on an algebraic approach or on interval analysis

Exact calculation on the dexterity of a robot

Participant : Jean-Pierre Merlet.

The minimal and maximal values of the eigenvalues of the Jacobian matrix of a robot, together with the condition number of this matrix, are the usual indexes to characterize the dexterity of the robot. Although these values may be computed easily for a given pose of the robot, it is much more difficult to determine exactly the minimum and maximum of these values over a given workspace. We have developed a generic algorithm based on interval analysis that allows one to determine these quantities up to an arbitrary accuracy. As an example, the case of the 3-dof Orthoglide robot of IRCCYN (Nantes) has been treated. This work should have been presented at the IFToMM World Congress that has been postponed due to the SARS virus.

Modular parallel robots

Participant : Jean-Pierre Merlet.

A modular parallel robot has a geometry that may be modified to adjust the performances of the robot to the task. Although prototypes of such robot exist, there is no known algorithm that allows one to determine the best geometry of the robot from given constraints on the task.

We have considered modular robots for which the location of the anchor points on the base may slide along a given direction. We have shown that it was possible to determine all the locations of the anchor points so that a given trajectory of the robot (defined by arbitrary analytical time functions) will be fully included in the workspace of the robot. Furthermore as the set of possible locations is defined as a set of ranges we are still able to optimize a secondary criteria. For example we have shown that it was possible to maximize the minimal value of the stiffness of the robot along the normal of a planar trajectory: typically a gain of 20% on the minimal stiffness is obtained while the average stiffness on the trajectory may be improved by 40% [29].

Micro-robot

Participant : Jean-Pierre Merlet.

In the last years we have developed a parallel 3-dof micro-robot to be used for endoscopy in collaboration with the LMARC laboratory of Besançon University, the Technion of Haifa and the company DG Créations. Two prototypes have been developed with a diameter of respectively 7mm for a length of 28 mm and 8.6 mm with a similar length. Motions have been tested but we have been confronted with a major problem: the small electrical motors that are used in the prototypes (from the companies MicroMotor and RMB) will usually fail after a cumulative time of use of about 10 minutes. It seems that the gear box provided with these motors (necessary as their rotation speed is over 100 000 rpm) are very delicate (and cannot be repaired on site). We are investigating how to solve this problem together with considering the use of alternate actuators. This problem has evidently postponed the first clinical trial that should have taken place this year.

Determination of measurement configurations for robot calibration by local convergence method associated to meta-heuristic methods

Participants : David Daney, Blaise Madeline.

Kinematic parameters of a robot are determined in a calibration method by solving an over-constrained system of equations [16]. This system is a function of informations obtained by controlling and measuring the robot state in different poses. An optimal choice of these poses within the robot workspace allows one to improve the numerical quality of the system of equations and increase the robustness of identification with respect to measurement noise. We propose to determine these optimal poses by a numerical optimization algorithm associated to meta-heuristic methods to decrease the sensibility to local minima. We show that our algorithm allows us to divide by twenty the identification error compared to randomly chosen poses when using the simplest method of parallel robot calibration.

Identification of parallel robot kinematic parameters under uncertainties by using algebraic methods

Participants : David Daney, Ioannis Emiris.

This work implements algebraic elimination methods for an original and general calibration method of parallel robots [20]. It focuses on two approaches, namely algebraic variable elimination and monomial linearization, both being based on sparse resultant combined with numerical linear algebra. Several experiments have allowed us to compare these two methods together with a classical numerical optimization method in the presence of measurement errors. Our main conclusion is that elimination methods offer an interesting alternative to more well-established methods for parallel robot calibration by satisfying the goals of accuracy and robustness. Moreover, our methods require no initial estimate and no hypothesis on the noise distribution and allow one to derive a quality index for the solution.

Guaranteed parallel robot calibration with measurement errors

Participants : David Daney, Arnold Neumaier, Yves Papegay.

The identification of robot kinematic parameters is difficult to certify due to the errors in the measurements needed by a calibration process. If these errors are taken into account the kinematic parameters cannot be determined exactly and we propose to bound the possible values of the parameters without assuming any knowledge on measurement distribution. For that purpose, the measurement errors are introduced inside the constraint equations as additional variables with an interval representation. The problem is then to solve a parametric system of equations by interval analysis and constraint programming methods. We have already general purpose algorithms allowing to get an approximation of the set of solutions. But these algorithms must be adapted to deal more efficiently with this specific problem.

Influence of joint tolerances on closed-loop kinematic chains

Participants : David Daney, Yves Papegay, Jean-Pierre Merlet.

The manufacturing tolerances associated to the joints of a kinematic chain modify the theoretical properties of the mechanism (accuracy, number of degrees of freedom ...). The values of theses tolerances are constrained to lie within ranges and we propose to propagate their intervals to verify their influences on the robot properties. This work is difficult for closed-loop kinematic chains as the closure equations become a system of nonlinear equations parameterized by intervals but we have started to develop a generic framework that may allow to analyze the most important robot properties.