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Toulouse, France

The Laboratory for analysis and architecture of systems is a laboratory depending from the Centre national de la recherche scientifique. This facility is located near to other important higher education facilities in Toulouse, France: the Paul Sabatier University, SUPAERO, the ENAC, the INSA, as well as other research centers . Wikipedia.


Sola J.,CNRS Laboratory for Analysis and Architecture of Systems
Proceedings - IEEE International Conference on Robotics and Automation | Year: 2010

We benchmark in this article three different landmark parametrizations in monocular 6DOF EKF-SLAM. These parametrizations are homogeneous points (HP), inverse-distance points (IDP, better known as inverse-depth), and the new anchored homogeneous points (AHP). The discourse used for describing them is chosen to highlight their differences and similarities, showing that they are just incremental variations of ones with respect to the others. We show for the first time a complete comparison of HP against IDP, two methods that are getting popular, and introduce also for the first time AHP, whose description falls precisely between the other two. The benchmarking is done by running all algorithms on the same data and by using the well-established NEES consistency analysis. Our conclusion is that the new AHP parametrization is the most interesting one for monocular EKF-SLAM (followed by IDP and then HP) because it greatly postpones the apparition of EKF inconsistency. ©2010 IEEE. Source


Lasserre J.B.,CNRS Laboratory for Analysis and Architecture of Systems
SIAM Journal on Optimization | Year: 2016

In the family of unit balls with constant volume we look at the ones whose algebraic representation has some extremal property. We consider the family of nonnegative homogeneous polynomials of even degree p whose sublevel set G = -x : g(x) ≤ 1} (a unit ball) has the same fixed volume and want to find in this family the polynomial that minimizes either the parsimony-inducing ℓ1-norm or the ℓ2-norm of its vector of coefficients. Equivalently, among all degree-p polynomials of constant ℓ1- or ℓ2-norm, which one minimizes the volume of its level set G? We first show that in both cases this is a convex optimization problem with a unique optimal solution g∗1 or g∗2, respectively. We also show that g∗1 is the Lp-norm polynomial x→Σin=1 xip, thus recovering a parsimony property of the Lp-norm polynomial via ℓ1-norm minimization. This once again illustrates the power and versatility of the ℓ1-norm relaxation strategy in optimization when one searches for an optimal solution with parsimony properties. Next we show that g∗2 is not sparse at all (and thus differs from g∗1) but is still a sum of p-powers of linear forms. In fact, and surprisingly, for p = 2, 4, 6, 8, we show that g∗2 = (Σi xi2)p/2, whose level set is the Euclidean (i.e., the L2-norm) ball. We also characterize the unique optimal solution of the same problem where one searches for a sum of squares homogeneous polynomial that minimizes the (parsimony-inducing) nuclear norm of its associated (positive semidefinite) Gram matrix, hence aiming at finding a solution which is a sum of a few squares only. Again for p = 2, 4 the optimal solution is (Σi xi2)p/2, whose level set is the Euclidean ball, and when p ∈ 4ℕ, this is also true when n is sufficiently large. Finally, we also extend these results to generalized homogeneous polynomials, which include Lp-norms when 0 < p is rational. © 2016 Society for Industrial and Applied Mathematics. Source


Lasserre J.B.,CNRS Laboratory for Analysis and Architecture of Systems
Journal of Global Optimization | Year: 2011

We consider the robust (or min-max) optimization problem J*:=max y∈Ω min x{f(x,y): (x,y) ∈ Δ} where f is a polynomial and Δ ⊂ R n × R p as well as Ω ⊂ R p are compact basic semi-algebraic sets. We first provide a sequence of polynomial lower approximations (J i) ⊂ R[y] of the optimal value function J(y):= min x{f(x,y): (x,y) ∈ Δ. The polynomial J i ∈ R[y] is obtained from an optimal (or nearly optimal) solution of a semidefinite program, the ith in the "joint + marginal" hierarchy of semidefinite relaxations associated with the parametric optimization problem y → J(y), recently proposed in Lasserre (SIAM J Optim 20, 1995-2022, 2010). Then for fixed i, we consider the polynomial optimization problem J* i:= max y{J i(y): y ∈ Ω and prove that J^* i(:= max l=1,...,iJ* l) converges to J*as i → ∞. Finally, for fixed l ≤ i, each J^* l (and hence J^* i) can be approximated by solving a hierarchy of semidefinite relaxations as already described in Lasserre (SIAM J Optim 11, 796-817, 2001; Moments, Positive Polynomials and Their Applications. Imperial College Press, London 2009). © 2010 Springer Science+Business Media, LLC. Source


Lasserre J.B.,CNRS Laboratory for Analysis and Architecture of Systems
SIAM Journal on Optimization | Year: 2010

Given a compact parameter set Y ⊂ ℝp, we consider polynomial optimization problems (Py) on Rn whose description depends on the parameter y ∈ Y. We assume that one can compute all moments of some probability measure φ on Y, absolutely continuous with respect to the Lebesgue measure (e.g., Y is a box or a simplex and φ is uniformly distributed). We then provide a hierarchy of semidefinite relaxations whose associated sequence of optimal solutions converges to the moment vector of a probability measure that encodes all information about all global optimal solutions x*(y) of Py, as y ∈ Y. In particular, one may approximate as closely as desired any polynomial functional of the optimal solutions like, e.g., their φ-mean. In addition, using this knowledge on moments, the measurable function y-→ x*k(y) of the kth coordinate of optimal solutions, can be estimated, e.g., by maximum entropy methods. Also, for a boolean variable xk, one may approximate as closely as desired its persistency φ({y : x* k(y) = 1}, i.e., the probability that in an optimal solution x* (y), the coordinate x* k(y) takes the value 1. Last but not least, from an optimal solution of the dual semidefinite relaxations, one provides a sequence of polynomial (resp., piecewise polynomial) lower approximations with L1(φ) (resp., φ-almost uniform) convergence to the optimal value function. Copyright © 2010, Society for Industrial and Applied Mathematics. Source


Escande A.,Japan National Institute of Advanced Industrial Science and Technology | Mansard N.,CNRS Laboratory for Analysis and Architecture of Systems | Wieber P.-B.,French Institute for Research in Computer Science and Automation
International Journal of Robotics Research | Year: 2014

Hierarchical least-square optimization is often used in robotics to inverse a direct function when multiple incompatible objectives are involved. Typical examples are inverse kinematics or dynamics. The objectives can be given as equalities to be satisfied (e.g. point-to-point task) or as areas of satisfaction (e.g. the joint range). This paper proposes a complete solution to solve multiple least-square quadratic problems of both equality and inequality constraints ordered into a strict hierarchy. Our method is able to solve a hierarchy of only equalities 10 times faster than the iterative-projection hierarchical solvers and can consider inequalities at any level while running at the typical control frequency on whole-body size problems. This generic solver is used to resolve the redundancy of humanoid robots while generating complex movements in constrained environments. © The Author(s) 2014. Source

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