Searcharxiv⌕ Search

arXiv subjects

Philippe Nadeau

Publications and source records attributed to Philippe Nadeau.

At least 19 recordsLinked to original sources
arXiv2026

Grove polynomials and $K$-theoretic quasisymmetry

We define the grove polynomials, a set-valued extension of forest polynomials. We show that they are $K$-theoretically dual to the quasisymmetric Schubert cells which pave the quasisymmetric flag variety, in the same way that Grothendieck polynomials are dual to Schubert cells in the complete flag variety. As a consequence, the finite truncations of the multi-fundamental quasisymmetric functions of Lam-Pylyavskyy acquire a geometric interpretation as $K$-theoretic representatives of quasisymmetric Schubert cells.

math.CO↗
arXiv2026

Bijections for rhombic alternative tableaux

We generalize well-known bijections between alternative tableaux and permutations to bijections between rhombic alternative tableaux (RAT) and assemblées of permutations. We show how these various bijections are connected. As a consequence, we find a refined enumeration formula for RAT. One of our bijections carries many statistics from RAT to assemblées; notably, it sends the number of free cells to the number of crossings, which answers a question of Mandelshtam and Viennot. We also find an $r!$-to-$1$ map from marked Laguerre histories to assemblées, answering a question of Corteel and Nunge.

math.CO↗
arXiv2026

Bilateral parking procedures

We introduce the class of bilateral parking procedures on the integer line. While cars try to park in the nearest available spot to their right in the classical case, we consider more general parking rules that allow cars to use the nearest available spot to their left. We show that for a natural subclass of local procedures, the number of corresponding parking functions of length $r$ is always equal to $(r+1)^{r-1}$. The setting can be extended to probabilistic procedures, in which the decision to park left or right is random. We finally describe how bilateral procedures can naturally be encoded by certain labeled binary forests, whose combinatorics shed light on several results from the literature.

math.CO↗
arXiv2026

Quasisymmetric divided differences

We develop a quasisymmetric analogue of the combinatorial theory of Schubert polynomials and the associated divided difference operators. Our counterparts are "forest polynomials", and a new family of linear operators, whose theory of compositions is governed by forests and the "Thompson monoid". Our approach extends naturally to $m$-colored quasisymmetric functions. We then give several applications of our theory to fundamental quasisymmetric functions, the study of quasisymmetric coinvariant rings and their associated harmonics, and positivity results for various expansions. In particular we resolve a conjecture of Aval-Bergeron-Li regarding quasisymmetric harmonics.

math.CO↗
arXiv2025

Generating Stable Placements via Physics-guided Diffusion Models

Stably placing an object in a multi-object scene is a fundamental challenge in robotic manipulation, as placements must be penetration-free, establish precise surface contact, and result in a force equilibrium. To assess stability, existing methods rely on running a simulation engine or resort to heuristic, appearance-based assessments. In contrast, our approach integrates stability directly into the sampling process of a diffusion model. To this end, we query an offline sampling-based planner to gather multi-modal placement labels and train a diffusion model to generate stable placements. The diffusion model is conditioned on scene and object point clouds, and serves as a geometry-aware prior. We leverage the compositional nature of score-based generative models to combine this learned prior with a stability-aware loss, thereby increasing the likelihood of sampling from regions of high stability. Importantly, this strategy requires no additional re-training or fine-tuning, and can be directly applied to off-the-shelf models. We evaluate our method on four benchmark scenes where stability can be accurately computed. Our physics-guided models achieve placements that are 56% more robust to forceful perturbations while reducing runtime by 47% compared to a state-of-the-art geometric method.

cs.RO↗
arXiv2025

Robustness Assessment of Assemblies in Frictional Contact

This work establishes a solution to the problem of assessing the capacity of multi-object assemblies to withstand external forces without becoming unstable. Our physically-grounded approach handles arbitrary structures made from rigid objects of any shape and mass distribution without relying on heuristics or approximations. The result is a method that provides a foundation for autonomous robot decision-making when interacting with objects in frictional contact. Our strategy relies on a contact interface graph representation to reason about instabilities and makes use of object shape information to decouple sub-problems and improve efficiency. Our algorithm can be used by motion planners to produce safe assembly transportation plans, and by object placement planners to select better poses. Compared to prior work, our approach is more generally applicable than commonly used heuristics and more efficient than dynamics simulations.

cs.RO↗
arXiv2025

Stable Object Placement Planning From Contact Point Robustness

We introduce a planner designed to guide robot manipulators in stably placing objects within intricate scenes. Our proposed method reverses the traditional approach to object placement: our planner selects contact points first and then determines a placement pose that solicits the selected points. This is instead of sampling poses, identifying contact points, and evaluating pose quality. Our algorithm facilitates stability-aware object placement planning, imposing no restrictions on object shape, convexity, or mass density homogeneity, while avoiding combinatorial computational complexity. Our proposed stability heuristic enables our planner to find a solution about 20 times faster when compared to the same algorithm not making use of the heuristic and eight times faster than a state-of-the-art method that uses the traditional sample-and-evaluate approach. Our proposed planner is also more successful in finding stable placements than the five other benchmarked algorithms. Derived from first principles and validated in ten real robot experiments, our planner offers a general and scalable method to tackle the problem of object placement planning with rigid objects.

cs.RO↗
arXiv2025

Equivariant quasisymmetry and noncrossing partitions

We introduce a definition of ``equivariant quasisymmetry'' for polynomials in two sets of variables. Using this definition we define quasisymmetric generalizations of the theory of double Schur and double Schubert polynomials that we call double fundamental polynomials and double forest polynomials, where the subset of ``noncrossing partitions'' plays the role of $S_n$. In subsequent work we will show this combinatorics is governed by a new geometric construction we call the ``quasisymmetric flag variety'' which plays the same role for equivariant quasisymmetry as the usual flag variety plays in the classical story.

math.CO↗
arXiv2024

The geometry of quasisymmetric coinvariants

We develop a quasisymmetric analogue of the theory of Schubert cycles, building off of our previous work on a quasisymmetric analogue of Schubert polynomials and divided differences. Our constructions result in a natural geometric interpretation for the ring of quasisymmetric coinvariants.

math.AG↗
arXiv2024

Tamari intervals and blossoming trees

We introduce a simple bijection between Tamari intervals and the blossoming trees (Poulalhon and Schaeffer, 2006) encoding planar triangulations, using a new meandering representation of such trees. Its specializations to the families of synchronized, Kreweras, new/modern, and infinitely modern intervals give a combinatorial proof of the counting formula for each family. Compared to (Bernardi and Bonichon, 2009), our bijection behaves well with the duality of Tamari intervals, enabling also the counting of self-dual intervals.

math.CO↗
arXiv2024

Schubert polynomial expansions revisited

We give an elementary approach utilizing only the divided difference formalism for obtaining expansions of Schubert polynomials that are manifestly nonnegative, by studying solutions to the equation $\sum Y_i\partial_i=\mathrm{id}$ on polynomials with no constant term. This in particular recovers the pipe dream and slide polynomial expansions. We also show that slide polynomials satisfy an analogue of the divided difference formalisms for Schubert polynomials and forest polynomials, which gives a simple method for extracting the coefficients of slide polynomials in the slide polynomial decomposition of an arbitrary polynomial.

math.CO↗
arXiv2024

A Standard Rigid Transformation Notation Convention for Robotics Research

Notation conventions for rigid transformations are as diverse as they are fundamental to the field of robotics. A well-defined convention that is practical, consistent and unambiguous is essential for the clear communication of ideas and to foster collaboration between researchers. This work presents an analysis of conventions used in state-of-the-art robotics research, defines a new notation convention, and provides software packages to facilitate its use. To shed some light on the current state of notation conventions in robotics research, this work presents an analysis of the ICRA 2023 proceedings, focusing on the notation conventions used for rigid transformations. A total of 1655 papers were inspected to identify the convention used, and key insights about trends and usage preferences are derived. Based on this analysis, a new notation convention called RIGID is defined, which complies with the "ISO 80000 Standard on Quantities and Units". The RIGID convention is designed to be concise yet unambiguous and easy to use. Additionally, this work introduces a LaTeX package that facilitates the use of the RIGID notation in manuscripts preparation through simple customizable commands that can be easily translated into variable names for software development.

cs.RO↗
arXiv2024

Automated Continuous Force-Torque Sensor Bias Estimation

Six axis force-torque sensors are commonly attached to the wrist of serial robots to measure the external forces and torques acting on the robot's end-effector. These measurements are used for load identification, contact detection, and human-robot interaction amongst other applications. Typically, the measurements obtained from the force-torque sensor are more accurate than estimates computed from joint torque readings, as the former is independent of the robot's dynamic and kinematic models. However, the force-torque sensor measurements are affected by a bias that drifts over time, caused by the compounding effects of temperature changes, mechanical stresses, and other factors. In this work, we present a pipeline that continuously estimates the bias and the drift of the bias of a force-torque sensor attached to the wrist of a robot. The first component of the pipeline is a Kalman filter that estimates the kinematic state (position, velocity, and acceleration) of the robot's joints. The second component is a kinematic model that maps the joint-space kinematics to the task-space kinematics of the force-torque sensor. Finally, the third component is a Kalman filter that estimates the bias and the drift of the bias of the force-torque sensor assuming that the inertial parameters of the gripper attached to the distal end of the force-torque sensor are known with certainty.

cs.RO↗
arXiv2024

Smirnov words and the Delta Conjectures

We provide a combinatorial interpretation of the symmetric function $\left.Θ_{e_k}Θ_{e_l}\nabla e_{n-k-l}\right|_{t=0}$ in terms of segmented Smirnov words. The motivation for this work is the study of a diagonal coinvariant ring with one set of commuting and two sets of anti-commuting variables, whose Frobenius characteristic is conjectured to be the symmetric function in question. Furthermore, this function is related to the Delta conjectures. Our work is a step towards a unified formulation of the two versions, as we prove a unified Delta theorem at $t=0$.

math.CO↗
arXiv2023

Fast Object Inertial Parameter Identification for Collaborative Robots

Collaborative robots (cobots) are machines designed to work safely alongside people in human-centric environments. Providing cobots with the ability to quickly infer the inertial parameters of manipulated objects will improve their flexibility and enable greater usage in manufacturing and other areas. To ensure safety, cobots are subject to kinematic limits that result in low signal-to-noise ratios (SNR) for velocity, acceleration, and force-torque data. This renders existing inertial parameter identification algorithms prohibitively slow and inaccurate. Motivated by the desire for faster model acquisition, we investigate the use of an approximation of rigid body dynamics to improve the SNR. Additionally, we introduce a mass discretization method that can make use of shape information to quickly identify plausible inertial parameters for a manipulated object. We present extensive simulation studies and real-world experiments demonstrating that our approach complements existing inertial parameter identification methods by specifically targeting the typical cobot operating regime.

cs.RO↗
arXiv2023

The Sum of Its Parts: Visual Part Segmentation for Inertial Parameter Identification of Manipulated Objects

To operate safely and efficiently alongside human workers, collaborative robots (cobots) require the ability to quickly understand the dynamics of manipulated objects. However, traditional methods for estimating the full set of inertial parameters rely on motions that are necessarily fast and unsafe (to achieve a sufficient signal-to-noise ratio). In this work, we take an alternative approach: by combining visual and force-torque measurements, we develop an inertial parameter identification algorithm that requires slow or 'stop-and-go' motions only, and hence is ideally tailored for use around humans. Our technique, called Homogeneous Part Segmentation (HPS), leverages the observation that man-made objects are often composed of distinct, homogeneous parts. We combine a surface-based point clustering method with a volumetric shape segmentation algorithm to quickly produce a part-level segmentation of a manipulated object; the segmented representation is then used by HPS to accurately estimate the object's inertial parameters. To benchmark our algorithm, we create and utilize a novel dataset consisting of realistic meshes, segmented point clouds, and inertial parameters for 20 common workshop tools. Finally, we demonstrate the real-world performance and accuracy of HPS by performing an intricate 'hammer balancing act' autonomously and online with a low-cost collaborative robotic arm. Our code and dataset are open source and freely available.

cs.RO↗
arXiv2023

Forest polynomials and the class of the permutahedral variety

We study a basis of the polynomial ring that we call forest polynomials. This family of polynomials is indexed by a combinatorial structure called indexed forests and permits several definitions, one of which involves flagged P-partitions. As such, these polynomials have a positive expansion in the basis of slide polynomials. By a novel insertion procedure that may be viewed as a generalization of the Sylvester correspondence we establish that Schubert polynomials decompose positively in terms of forest polynomials. Our insertion procedure involves a correspondence on words which allows us to show that forest polynomials multiply positively. We proceed to show that forest polynomials are a particularly convenient basis in regards to studying the quotient of the polynomial ring modulo the ideal of positive degree quasisymmetric polynomials. This aspect allows us to give a manifestly nonnegative integral description for the Schubert class expansion of the cohomology class of the permutahedral variety in terms of a parking procedure. We study the associated combinatorics in depth and introduce a multivariate extension of mixed Eulerian numbers.

math.CO↗