Nicolas Perrin

Fast Humanoid Robot Collision-Free Footstep Planning Using Swept Volume Approximations

In this paper, we propose a novel and coherent framework for fast footstep planning for legged robots on a flat ground with 3-D obstacle avoidance. We use swept volume approximations that are computed offline in order to considerably reduce the time spent in collision checking during the online planning phase, in which a rapidly exploring random tree variant is used to find collision-free sequences of half-steps (which are produced by a specific walking pattern generator). Then, an original homotopy is used to smooth the sequences into natural motions, gently avoiding the obstacles. The results are experimentally validated on the robot HRP-2.

ON THE GEOMETRY OF SPHERICAL VARIETIES

This is a survey article on the geometry of spherical varieties.

On the quantum cohomology of adjoint varieties

We study the quantum cohomology of quasi-minuscule and quasi-cominuscule homogeneous spaces. The product of any two Schubert cells does not involve powers of the quantum parameter higher than 2. With the help of the quantum to classical principle we give presentations of the quantum cohomology algebras. These algebras are semi-simple for adjoint non coadjoint varieties and some properties of the induced strange duality are shown.

Small resolutions of minuscule Schubert varieties

Let X be a minuscule Schubert variety. In this paper, we associate a quiver with X and use the combinatorics of this quiver to describe all relative minimal models � : � X → X. We prove that all the morphismsare small and give a combinatorial criterion for � X to be smooth and thus a small resolution of X. We describe in this way all small resolutions of X. As another application of this description of relative minimal models, we obtain the following more intrinsic statement of the main result of Perrin, J. Algebra 294 (2005), 431-462. Let α ∈ A1(X) be an effective 1-cycle class. Then the irreducible components of the scheme Homα(p 1 ,X ) of morphisms from P 1 to X and of class α are indexed by the set: ne(α )= {β ∈ A1( � X) | β is effective and � π∗β = α} which is independent of the choice of a relative minimal model � X of X.

Quantum cohomology of minuscule homogeneous spaces III Semi-simplicity and consequences

We prove that the quantum cohomology ring of any minuscule or cominuscule homogeneous space, specialized at q=1, is semisimple. This implies that complex conjugation defines an algebra automorphism of the quantum cohomology ring localized at the quantum parameter. We check that this involution coincides with the strange duality defined in a previous paper. We deduce Vafa-Intriligator type formulas for the Gromov-Witten invariants.

Quantum cohomology of minuscule homogeneous spaces II : Hidden symmetries

We prove that the quantum cohomology ring of any minuscule or cominuscule homogeneous space, once localized at the quantum parameter, has a non trivial involution mapping Schubert classes to multiples of Schubert classes. This can be stated as a strange duality property for the Gromov-Witten invariants, which turn out to be very symmetric.

RATIONALITY OF SOME GROMOV–WITTEN VARIETIES AND APPLICATION TO QUANTUM K-THEORY

We show that for any minuscule or cominuscule homogeneous space X, the Gromov-Witten varieties of degree d curves passing through three general points of X are rational or empty for any d. Applying techniques of A. Buch and L. Mihalcea to constructions of the authors together with L. Manivel, we deduce that the equivariant K-theoretic three points Gromov-Witten invariants are equal to classical equivariant K-theoretic invariants on auxilliary spaces.

Compliant attitude control and stepping strategy for balance recovery with the humanoid COMAN

In this paper we describe an approach for hu-manoid robot balance recovery that combines a novel attitude control algorithm adding compliance to the robots behavior and increasing the smoothness of its motion, and an omnidirectional stepping strategy that can trigger one or two steps based on a measured disturbance vector. The proposed method is validated through experiments with the inherently compliant humanoid COMAN.

Effective Generation of Dynamically Balanced Locomotion with Multiple Non-coplanar Contacts

Studies of computationally and analytically convenient approximations of rigid body dynamics have brought valuable insight into the field of humanoid robotics. Additionally, they facilitate the design of effective walking pattern generators. Going further than the classical Zero Moment Point-based methods, this paper presents two simple and novel approaches to solve for 3D locomotion with multiple non-coplanar contacts. Both formulations use model predictive control to generate dynamically balanced trajectories with no restrictions on the center of mass height trajectory. The first formulation treats the balance criterion as an objective function, and solves the control problem using a sequence of alternating convex quadratic programs. The second formulation considers the criterion as constraints, and solves a succession of convex quadratically constrained quadratic programs.

Lyapunov Stability Margins for humanoid robot balancing

This work introduces a novel balance monitoring strategy for humanoid robots. The proposed method addresses the problem of ensuring the balance maintenance of a humanoid robot, through the online monitoring of its state of balance by means of a Lyapunov (energy) function. The proposed method involves the use of dynamical models accounting for both the link and motor states. Energy limits corresponding to the front and rear edges of the support polygon are computed using a closed-loop Lyapunov function. Therefore, this method focuses on the resolution of two issues through a single control scheme, namely, guaranteeing asymptotical stability of the robot at the joint level, in addition to ensuring that it maintains its dynamical balance. A mathematical proof of the previous claims, as well as of the methods validity, is provided in the paper, whereby a direct relationship between the CoP and the systems energy has been established for the first time. Experimental results of step recovery and walking tests performed on the COmpliant huMANoid (COMAN) corroborate the methods applicability and performance as a balance monitor.