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.

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% .

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.

Kinematic parameters of a robot are determined in a calibration method by solving an over-constrained system of equations . 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.

This work implements algebraic elimination methods for an original and general calibration method of parallel robots . 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.

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.

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.

The purpose of our research is to introduce and to study a new branch and bound algorithm called QuadSolver. The essential feature of this algorithm is a global constraint (called Quad) that works on a tight and safe linear relaxation of the polynomial relations of the constraint systems. More precisely, QuadSolver is a branch and prune algorithm that combines Quad, local consistencies and interval methods .

QuadSolver has been evaluated on a variety of benchmarks from kinematics, mechanics and robotics. On these benchmarks, it outperforms classical interval methods as well as CSP solvers and it compares well with state-of-the-art optimization solvers.

The relaxation of nonlinear terms is adapted from the classical the ``Reformulation-Linearization Technique (RLT)'' linearization method. The simplex algorithm is used to narrow the domain of each variable with respect to the subset of the linear set of constraints generated by the relaxation process. The coefficients of these linear constraints are updated with the new values of the bounds of the domains and the process is restarted until no more significant reduction can be done. We have demonstrated that the Quad algorithm yields a more effective pruning of the domains than local consistency filtering algorithms (e.g., 2b–consistency or box–consistency). Indeed, the drawback of classical local consistencies comes from the fact that the constraints are handled independently and in a blind way. For example, when dealing with quadratic constraints, classical local consistencies do not exploit the semantic of quadratic term; for reducing the domains of the variables. Conversely, linear programming techniques do capture most of the semantics of nonlinear terms (e.g., convex and concave envelopes of these particular terms). The extension of Quad for handling any polynomial constraint system requires to replace non-quadratic terms by new variables and to add the corresponding identities to the initial constraint system. However, a complete quadrification would generate a huge number of linear constraints. We have introduced a heuristics based on a good tradeoff between a tight approximation of the non linear terms and the size of the generated constraints system.

A safe rounding process is a key issue for the Quad framework. The simplex algorithm is used to narrow the domain of each variable with respect to the subset of the linear set of constraints generated by the relaxation process but most implementations of the simplex algorithm are numerically unsafe. Moreover, the coefficients of the generated linear constraints are computed with floating point numbers. So, two problems may occur in the Quad-filtering process:

We propose a safe procedure for computing the coefficients of the generated linear constraints. The second problem has recently been addressed by Neumaier and Shcherbina  which have proposed a simple and cheap procedure to get a rigorous upper bound of the objective function. The incorporation of these procedures in the Quad-filtering process allows us to call the simplex algorithm without worrying about safety.

An exclusion scheme allows one to calculate a box around a certified solution that is guaranteed to contain only this solution. Propagation of this box on the search space enables to reduce it, thereby improving the global efficiency of the solver.

For distance constraints we have already provided a specific version of the Kantorovitch scheme that produces a larger box than the general version of the theorem. We have then investigated the inflation approach of Neumaier  based on the H-matrix theory. In its general version this method determines incrementally the largest exclusion box and is relatively computer intensive. We have shown that for the specific case of distance constraints we may compute directly the width of the exclusion box without relying on an iterative scheme. This calculation has been implemented in the specific distance constraints solver of ALIAS.

Most of the solver of numerical CSP are based on a relaxation of local consistencies. These techniques prune the domains of the variables by computing an external approximation of the solution space. However, the resulting domains may still contain a huge number of locally inconsistent values, especially when the solution space is composed of disjoint subspaces. Splitting techniques are often used to isolate solutions. However, splitting is ineffective in the presence of continuous subspaces of solutions. Moreover classical splitting methods do not take advantage of the properties of the constraints, especially and more particularly the properties of geometrical constraints.

We have introduced in  a domain decomposition, based on the semantic properties of distance constraints. This method splits and reduces the domains of two points w.r.t. their distance relation, using the monotonic and convex parts of the canonical form of the equations. We halve also proposed a new pruning technique, named LDF (Local Decomposition Filtering), propagates this decomposition over the whole set of constraints, following the standard approximation scheme. The disjoint subspaces of solutions are represented by a graph of sub-domains which has the same structural properties as the arc-consistency graph of a CSP on finite domains. Thus, classical search algorithms, such as MAC (Maintaining Arc Consistency) or FC (Forward Checking), may be used to isolate potential solution subspaces. We have demonstrated that LDF achieves a better pruning than 2B-consistency, and is more efficient than a combination of 2B-consistency and splitting, for small but interesting problems.

Further works concern the integration of this semantic decomposition in a general splitting algorithm for solving a larger class of problems, such as classical robotic benchmarks.

We have studied how to manage small uncertainties in the parameters of a system of distance equations. The problem has no more isolated solution points, but isolated sub-spaces of solutions and we cannot directly apply classical interval solvers. We have proposed the following method: we first compute the solutions of the system without uncertainties and then we use these solutions to determine a dynamic splitting policy of the domains. Filtering algorithms (for example 3B-consistency) are then applied to each subspace to obtain boxes containing the solutions of the system with uncertainties. We have studied the limitations of this method. It is complete (no region with solutions is lost) but the regions finally found do not always correspond to an extension around the solution points without uncertainties. If new regions with solutions appear when the uncertainties are taken into account, the method is not able to isolate them. When regions overlap, the splitting will separate two regions that will be each smaller than the extensions of solution points. Finally, we have started to extend our method to other types of equations.

Geometric constraint satisfaction problems (GCSPs) are ubiquitous in applications like CAD, robotics or molecular biology . They consist in searching positions, orientations and dimensions of geometric objects bound by geometric constraints. The goal of the thesis  was to find an efficient and complete solving method for GCSPs.

We have compared solving methods and decomposition techniques, and we have focused on Hoffmann's decomposition and interval solving methods. We define a general framework for the study of rigidity in GCSPs, a concept used in all the geometric decomposition methods .

Hoffmann's method has limits inherent to all the structural geometric approaches. We propose the degree of rigidity concept to overcome some of these limits together with a new decomposition and its combination with interval solving techniques .

This work is performed in collaboration with Marta Wilczkowiak from the MOVI project at INRIA Rhônes-Alpes. Marta Wilczkowiak's PhD thesis is supervised by Edmond Boyer and Peter Sturm.

We present a new approach to 3D scene modeling based on geometric constraints . Contrary to the existing methods, we can quickly obtain 3D scene models that respect the given constraints exactly. Our system can handle a large variety of linear and non-linear constraints in a flexible way.

To deal with the constraints, we decided to exploit the properties of the GPDOF algorithm developed by Trombettoni . The approach is based on a dictionary of so-called r-methods, relying on geometry theorems, which can solve a subset of geometric constraints in a very efficient way. GPDOF is used to find, in polynomial-time, a reduced parameterization of a scene, and to decompose the equation system induced by constraints into a sequence of r-methods. We have validated our approach in reconstructing 3D models of building from images, using linear and quadratic geometric constraints.

We have studied a technique, called inter-block backtracking (IBB), which improves the solving by interval techniques of decomposed systems of non-linear equations over the reals .

This approach, introduced in 1998 by Bliek, Neveu and Trombettoni , handles a system of equations previously decomposed into a set of (small) $k\times k$ sub-systems, called blocks. A solution is obtained by combining the solutions computed in the different blocks. The approach seems particularly suitable for improving interval solving techniques.

In this paper, we analyze into the details the different variants of IBB which differ in their backtracking and filtering strategies. We also introduce IBB-GBJ, a new variant based on Dechter's graph-based back-jumping.

An extensive comparison on a sample of 8 CSPs has allowed us to better understand the behavior of IBB. In particular, we clearly show that limiting the scope of the filtering to the current block is very fruitful. For all the tested instances, IBB efficiency is larger by several orders of magnitude as compared to global solving.

In his PhD thesis , Blaise Madeline has addressed the use of evolutionary algorithms (EA) to solve constraint satisfaction problems in finite domains, without any particular specialization nor hybridization. A comparison between tree search methods and meta-heuristics on over-constrained graph coloring, in a context of minimal tuning of the parameters was proposed. The search landscape was studied for understanding why the various methods have so different efficiency. Finally, a new genetic operators (crossover, mutation, diversification) with a parameter setting simpler than the classical operators was introduced.