arXiv subjects

Richard Schubert

Publications and source records attributed to Richard Schubert.

17 recordsLinked to original sources

arXiv2026

Equalized Coverage in Motion Control Performance Prediction for Self-Adaptive Road Vehicles

Automated driving systems require monitoring mechanisms to ensure operation as intended, especially when system elements degrade and/or fail. Hence, capability monitoring is crucial in order to evaluate the system's remaining performance and implement capability-based behavior. In this paper, we investigate the dynamics of a highly over-actuated automated vehicle under actuator degradations and failures, affecting the vehicle's motion control capabilities. We propose a lightweight prediction model based on conformalized quantile regression that predicts whether an automated vehicle can be controlled with sufficiently low lateral deviation from a planned trajectory under nominal, degraded, and failed actuator conditions. We recognize that statistical guarantees should hold not only across all data (marginal coverage) but also for different regimes within the data (conditional coverage). We therefore employ equalized coverage methods to address this challenge. During runtime behavior generation our predictor can provide a heuristic for determining the admissible action space. Its application and limitations are discussed in this paper.

eess.SY↗

arXiv2026

Coordinating Stakeholders in the Consideration of Performance Indicators and Respective Interface Requirements for Automated Vehicles

This paper presents a process for coordinating stakeholders in their consideration of performance indicators and respective interface requirements for automated vehicles. These performance indicators are obtained and processed based on the system's self-perception and enable the realization of self-aware and self-adaptive vehicles. This is necessary to allow SAE Level 4 vehicles to handle external disturbances as well as internal degradations and failures at runtime. Without such a systematic process for stakeholder coordination, architectural decisions on realizing self-perception become untraceable and effective communication between stakeholders may be compromised. Our process-oriented approach includes necessary ingredients, steps, and artifacts that explicitly address stakeholder communication, traceability, and knowledge transfer through clear documentation. Our approach is based on the experience gained from applying the process in the autotech.agil project, from which we further present lessons learned, identified gaps, and steps for future work.

cs.SE↗

arXiv2025

Approaching Safety-Argumentation-by-Design: A Requirement-based Safety Argumentation Life Cycle for Automated Vehicles

Despite the growing number of automated vehicles on public roads, operating such systems in open contexts inevitably involves incidents. Developing a defensible case that the residual risk is reduced to a reasonable (societally acceptable) level is hence a prerequisite to be prepared for potential liability cases. A "safety argumentation" is a common means to represent this case. In this paper, we contribute to the state of the art in terms of process guidance on argumentation creation and maintenance - aiming to promote a safety-argumentation-by-design paradigm, which mandates co-developing both the system and argumentation from the earliest stages. Initially, we extend a systematic design model for automated driving functions with an argumentation layer to address prevailing misconceptions regarding the development of safety arguments in a process context. Identified limitations of this extension motivate our complementary design of a dedicated argumentation life cycle that serves as an additional process viewpoint. Correspondingly, we define literature- and expert-based process requirements. To illustrate the safety argumentation life cycle that we propose as a result of implementing these consolidated requirements, we demonstrate principles of the introduced process phases (baselining, evolution, continuous maintenance) by an argumentation example on an operational design domain exit response.

eess.SY↗

arXiv2025

Propagation of chaos for first-order mean-field systems with non-attractive moderately singular interaction

We consider particle systems that evolve by inertialess binary interaction through general non-attractive kernels of singularity $|x|^{-\alpha}$ with $\alpha<d-1$. We prove a quantitative mean-field limit in terms of Wasserstein distances under certain conditions on the initial configuration while maintaining control of the particle configuration in the form of the minimal distance and certain singular sums of the particle distances. As a corollary, we show propagation of chaos for $\alpha<\frac{d-1}{2}$ for $d\ge 3$ and $\alpha<\frac 13=\frac{2d-3}{3}$ for $d=2$. This extends the results of Hauray (https://doi.org/10.1142/S0218202509003814), which yield propagation of chaos for $\alpha < \frac{d-2}{2}$ without an assumption on the sign of the interaction. The main novel ingredient is that due to the non-attraction property it is enough to control the distance to the next-to-nearest neighbour particle.

math.AP↗

arXiv2024

Conformal Prediction of Motion Control Performance for an Automated Vehicle in Presence of Actuator Degradations and Failures

Automated driving systems require monitoring mechanisms to ensure safe operation, especially if system components degrade or fail. Their runtime self-representation plays a key role as it provides a-priori knowledge about the system's capabilities and limitations. In this paper, we propose a data-driven approach for deriving such a self-representation model for the motion controller of an automated vehicle. A conformalized prediction model is learned and allows estimating how operational conditions as well as potential degradations and failures of the vehicle's actuators impact motion control performance. During runtime behavior generation, our predictor can provide a heuristic for determining the admissible action space.

cs.RO↗

arXiv2024

A Prototypical Expert-Driven Approach Towards Capability-Based Monitoring of Automated Driving Systems

Supervising the safe operation of automated vehicles is a key requirement in order to unleash their full potential in future transportation systems. In particular, previous publications have argued that SAE Level 4 vehicles should be aware of their capabilities at runtime to make appropriate behavioral decisions. In this paper, we present a framework that enables the implementation of an online capability monitor. We derive a graphical system model that captures the relationships between the quality of system elements across different architectural views. In an expert-driven approach, we parameterize Bayesian Networks based on this structure using Fuzzy Logic. Using the online monitor, we infer the quality of the system's capabilities based on technical measurements acquired at runtime.

eess.SY↗

arXiv2023

A variational approach to the Navier-Stokes equations with shear-dependent viscosity

We present a variational approach for the construction of Leray-Hopf solutions to the non-Newtonian Navier-Stokes system. Inspired by the work [42] on the corresponding Newtonian problem, we minimise certain stabilised Weighted Inertia-Dissipation-Energy (WIDE) functionals and pass to the limit of a vanishing parameter in order to recover a Leray-Hopf solution of the non-Newtonian Navier-Stokes equations. The investigation of the non-Newtonian Navier-Stokes system via this variational approach is particularly well suited to gain insights into weak, respectively strong convergence properties of approximating sequences for different flow-behaviour exponents. With this analysis we extend the results of [4] to power-law exponents $\tfrac{2d}{d+2} < p < \tfrac{3d+2}{d+2}$, where weak solutions do not satisfy the energy equality and the involved convergence is genuinely weak. Key of the argument is to pass to the limit in the nonlinear viscosity term in the time-dependent setting. For this we provide an elliptic-parabolic solenoidal Lipschitz truncation that might be of independent interest.

math.AP↗

arXiv2023

Sedimentation of particles with very small inertia II: Derivation, Cauchy problem and hydrodynamic limit of the Vlasov-Stokes equation

We consider the sedimentation of $N$ spherical particles with identical radii $R$ in a Stokes flow in $\mathbb R^3$. The particles satisfy a no-slip boundary condition and are subject to constant gravity. The dynamics of the particles is modeled by Newton's law but with very small particle inertia as $N$ tends to infinity and $R$ to $0$. In a mean-field scaling we show that the evolution of the $N$-particle system is well approximated by the Vlasov-Stokes equation. In contrast to the transport-Stokes equation considered in the first part of this series, \cite{HoferSchubert23}, the Vlasov-Stokes equation takes into account the (small) inertia. Therefore we obtain improved error estimates. We also improve previous results on the Cauchy problem for the Vlasov-Stokes equation and on its convergence to the transport-Stokes equation in the limit of vanishing inertia. The proofs are based on relative energy estimates. In particular, we show new stability estimates for the Vlasov-Stokes equation in the $2$-Wasserstein distance. By combining a Lagrangian approach with a study of the energy dissipation, we obtain uniform stability estimates for arbitrary small particle inertia. We show that a corresponding stability estimate continues to hold for the empirical particle density which formally solves the Vlasov-Stokes equation up to an error. To this end we exploit certain uniform control on the particle configuration thanks to results in the first part \cite{HoferSchubert23}.

math.AP↗

arXiv2023

Convergence to the planar interface for a nonlocal free-boundary evolution

We capture optimal decay for the Mullins-Sekerka evolution, a nonlocal, parabolic free boundary problem from materials science. Our main result establishes convergence of BV solutions to the planar profile in the physically relevant case of ambient space dimension three. Far from assuming small or well-prepared initial data, we allow for initial interfaces that do not have graph structure and are not connected, hence explicitly including the regime of Ostwald ripening. In terms only of initially finite (not small) excess mass and excess surface energy, we establish that the surface becomes a Lipschitz graph within a fixed timescale (quantitatively estimated) and remains trapped within this setting. To obtain the graph structure, we leverage regularity results from geometric measure theory. At the same time, we extend a duality method previously employed for one-dimensional PDE problems to higher dimensional, nonlocal geometric evolutions. Optimal algebraic decay rates of excess energy, dissipation, and graph height are obtained.

math.AP↗

arXiv2023

ODD-Centric Contextual Sensitivity Analysis Applied To A Non-Linear Vehicle Dynamics Model

Advanced driving functions, for assistance or full automation, require strong guarantees to be deployed. This means that such functions may not be available all the time, like now commercially available SAE Level 3 modes that are made available only on some roads and at law speeds. The specification of such restriction is described technically in the Operational Design Domain (ODD) which is a fundamental concept for the design of automated driving systems (ADS). In this work, we focus on the example of trajectory planning and control which are crucial functions for SAE level 4+ vehicles and often rely on model-based methods. Hence, the quality of the underlying models has to be evaluated with respect to the ODD. Mathematical analyses such as uncertainty and sensitivity analysis support the quantitative assessment of model quality in general. In this paper, we present a new approach to assess the quality of vehicle dynamics models using an ODD-centric sensitivity analysis. The sensitivity analysis framework is implemented for a 10-DoF nonlinear double-track vehicle dynamics model used inside a model-predictive trajectory controller. The model sensitivity is evaluated with respect to given ODD and maneuver parameters. Based on the results, ODD-compliant behavior generation strategies with the goal of minimizing model sensitivity are outlined.

eess.SY↗

arXiv2023

Sedimentation of particles with very small inertia I: Convergence to the transport-Stokes equation

We consider the sedimentation of $N$ spherical particles with identical radii $R$ in a Stokes flow in $\mathbb R^3$. The particles satisfy a no-slip boundary condition and are subject to constant gravity. The dynamics of the particles is modeled by Newton's law but with very small particle inertia as $N$ tends to infinity and $R$ to $0$. In a mean-field scaling, we show that the particle evolution is well approximated by the transport-Stokes system which has been derived previously as the mean-field limit of inertialess particles. In particular this justifies to neglect the particle inertia in the microscopic system, which is a typical modelling assumption in this and related contexts. The proof is based on a relative energy argument that exploits the coercivity of the particle forces with respect to the particle velocities in a Stokes flow. We combine this with an adaptation of Hauray's method for mean-field limits to $2$-Wasserstein distances. Moreover, in order to control the minimal distance between particles, we prove a representation of the particle forces. This representation makes the heuristic \enquote{Stokes law} rigorous that the force on each particle is proportional to the difference of the velocity of the individual particle and the mean-field fluid velocity generated by the other particles.

math.AP↗

arXiv2022

Non-existence of mean-field models for particle orientations in suspensions

We consider a suspension of spherical inertialess particles in a Stokes flow on the torus $\mathbb T^3$. The particles perturb a linear extensional flow due to their rigidity constraint. Due to the singular nature of this perturbation, no mean-field limit for the behavior of the particle orientation can be valid. This contrasts with widely used models in the literature such as the FENE and Doi models and similar models for active suspensions. The proof of this result is based on the study of the mobility problem of a single particle in a non-cubic torus, which we prove to exhibit a nontrivial coupling between the angular velocity and a prescribed strain.

math.AP↗

arXiv2022

A data-driven approach to viscous fluid mechanics -- the stationary case

We introduce a data-driven approach to the modelling and analysis of viscous fluid mechanics. Instead of including constitutive laws for the fluid's viscosity in the mathematical model, we suggest to directly use experimental data. Only a set of differential constraints, derived from first principles, and boundary conditions are kept of the classical PDE model and are combined with a data set. The mathematical framework builds on the recently introduced data-driven approach to solid-mechanics [KO16,CMO18]. We construct optimal data-driven solutions that are material model free in the sense that no assumptions on the rheological behaviour of the fluid are made or extrapolated from the data. The differential constraints of fluid mechanics are recast in the language of constant rank differential operators. Adapting abstract results on lower-semicontinuity and $\mathscr{A}$-quasiconvexity, we show a $Γ$-convergence result for the functionals arising in the data-driven fluid mechanical problem. The theory is extended to compact nonlinear perturbations, whence our results apply to both inertialess fluids and flows with finite Reynolds number. Data-driven solutions provide a new relaxed solution concept. We prove that the constructed data-driven solutions are consistent with solutions to the classical PDEs of fluid mechanics if the data sets have the form of a monotone constitutive relation.

math.AP↗

arXiv2021

Optimal relaxation of bump-like solutions of the one-dimensional Cahn-Hilliard equation

REVISED VERSION INCORPORATING THE ERRATUM ON LEMMA 2.1 AND WITH A CORRECTION TO LEMMA 2.8 In this paper we derive optimal relaxation rates for the Cahn-Hilliard equation on the one-dimensional torus and the line. We consider initial conditions with a finite (but not small) $L^1$-distance to an appropriately defined bump. The result extends the relaxation method developed previously for a single transition layer (the ``kink'') to the case of two transition layers (the ``bump''). As in the previous work, the tools include Nash-type inequalities, duality arguments, and Schauder estimates. For both the kink and the bump, the energy gap is translation invariant and its decay alone cannot specify to which member of the family of minimizers the solution converges. Whereas in the case of the kink, the conserved quantity singles out the longtime limit, in the case of a bump, a new argument is needed. On the torus, we quantify the (initially algebraic and ultimately exponential) convergence to the bump that is the longtime limit; on the line, the bump-like states are merely metastable and we quantify the initial algebraic relaxation behavior.

math.AP↗

arXiv2021

The Influence of Einstein's Effective Viscosity on Sedimentation at Very Small Particle Volume Fraction

We investigate the sedimentation of identical inertialess spherical particles in a Stokes fluid in the limit of many small particles. It is known that the presence of the particles leads to an increase of the effective viscosity of the suspension. By Einstein's formula this effect is of the order of the particle volume fraction $ϕ$. The disturbance of the fluid flow responsible for this increase of viscosity is very singular (like $|x|^{-2}$). Nevertheless, for well-prepared initial configurations and $ϕ\to 0$, we show that the microscopic dynamics is approximated to order $ϕ^2 |\log ϕ|$ by a macroscopic coupled transport-Stokes system with an effective viscosity according to Einstein's formula. We provide quantitative estimates both for convergence of the densities in the $p$-Wasserstein distance for all $p$ and for the fluid velocity in Lebesgue spaces in terms of the $p$-Wasserstein distance of the initial data. Our proof is based on approximations through the method of reflections and on a generalization of a classical result on convergence to mean-field limits in the infinite Wasserstein metric by Hauray.

math.AP↗

arXiv2020

Sensitivity Analysis for Vehicle Dynamics Models -- An Approach to Model Quality Assessment for Automated Vehicles

Model-based approaches have become increasingly popular in the domain of automated driving. This includes runtime algorithms, such as Model Predictive Control, as well as formal and simulative approaches for the verification of automated vehicle functions. With this trend, the quality of models becomes crucial for automated vehicle safety. Established tools from model theory which can be applied to assure model quality are uncertainty and sensitivity analysis [1]. In this paper, we conduct sensitivity analyses for a single and double track vehicle dynamics model to gain insights about the models' behavior under different operating conditions. We compare the models, point out the most important findings regarding the obtained parameters sensitivities, and provide examples of possible applications of the gained insights.

eess.SY↗

arXiv2019

A local version of Einstein's formula for the effective viscosity of suspensions

We prove a local variant of Einstein's formula for the effective viscosity of dilute suspensions, that is $μ^\prime=μ(1+\frac 5 2ϕ+o(ϕ))$, where $ϕ$ is the volume fraction of the suspended particles. Up to now rigorous justifications have only been obtained for dissipation functionals of the flow field. We prove that the formula holds on the level of the Stokes equation (with variable viscosity). We consider a regime where the number $N$ of particles suspended in the fluid goes to infinity while their size $R$ and the volume fraction $ϕ=NR^3$ approach zero. We establish $L^\infty$ and $L^p$ estimates for the difference of the microscopic solution to the solution of the homogenized equation. Here we assume that the particles are contained in a bounded region and are well separated in the sense that the minimal distance is comparable to the average one. The main tools for the proof are a dipole approximation of the flow field of the suspension together with the so-called method of reflections and a coarse graining of the volume density.

math.AP↗