Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2608.24280

The Flat Earth Error: Differential Geometry in Vehicle Dynamics

David J. N. Limebeer·Charl van de Merwe
View original source ↗Read source document ↗Export metadata ↓

Abstract

The widespread idealization of road surfaces as horizontal planes can introduce significant inaccuracies into vehicle dynamics simulations, a phenomenon termed the ``Flat Earth Error.'' This article provides an expository guide to the use of classical differential geometry to model vehicular motion on curved surfaces. The influence of road curvature is characterized using the metric tensor, the second fundamental form, the shape operator, and the Christoffel symbols. Reproducible MATLAB scripts using an elliptic cone as a benchmark example illustrate the construction of geodesic curves and the simulation of particle dynamics on curved surfaces. The resulting geometric structures are integrated into a single-track vehicle-and-track model within a pseudospectral optimal control framework. Trajectory optimization results over a high-density mobile LiDAR profile of Darlington Raceway are used to generate a high-fidelity road-surface model. This model is used within an hp-adaptive collocation framework to investigate minimum lap time optimal control vehicular trajectories. These computations capture the non-smooth traction saturation limits of the tyres alongside position-dependent variations in gravitational forcing. Integrating differential geometry, multibody mechanics, and optimal control is essential for high-fidelity driven and autonomous vehicle-dynamics simulations. The optimized velocity and tyre slip profiles show that real-world racing track geometries induce dynamically significant three-dimensional effects. These results are particularly relevant to performance-limited simulations on highly banked track surfaces such as NASCAR ovals.

Explore related subjects

eess.SYcs.SY

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

David J. N. Limebeer, Charl van de Merwe. 2026-08-25. The Flat Earth Error: Differential Geometry in Vehicle Dynamics. https://arxiv.org/abs/2608.24280

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Failure-Aware Iterative Learning of State-Control Invariant Sets

In this paper, we address the problem of computing maximal state-control invariant sets for deterministic linear systems using failing trajectories. We introduce the concept of state-control invariance, which extends control invariance from the state space to the joint state-control space. The maximal state-control invariant (MSCI) set simultaneously encodes the maximal control invariant set (MCI) and, for each state in the MCI, the set of control inputs that preserve invariance. We prove that the state projection of the MSCI is the MCI and the state-dependent sections of the MSCI are the admissible invariance-preserving inputs. Building on this framework, we develop a Failure-Aware Iterative Learning (FAIL) algorithm for deterministic linear time-invariant systems with polytopic constraints. The algorithm iteratively updates a constraint set in the state-control space by learning predecessor halfspaces from one-step failing state-input pairs, without knowing the dynamics. For each failure, FAIL learns the violated halfspaces of the predecessor of the constraint set by a regression on failing trajectories. We prove that the learned constraint set converges monotonically to the MSCI. Numerical experiments on a double integrator system validate the proposed approach.

eess.SY↗
arXiv2026

Consensus and Synchronization of Multi-agent Systems over Finite Fields - Graph Topologies

This paper presents cooperative protocols for multi-agent systems with agents having a finite state-space. Both scalar single-integrator consensus and general LTI system synchronization are considered. Systems having a finite state-space describe agents with minimal memory capacity processing only a finite alphabet. Such systems are remarkably resilient to communication noise. The crucial problem, however, is to construct the admissible communication topology, which is NP-hard. We address this by efficiently exploring the subsets of admissible graph matrices and propose two new algorithms to generate them. Simulations validate the proposed approach.

eess.SY↗
arXiv2026

Extracting Exact Lie Derivatives Without Backpropagation: A Dual Compiler for Neural Control Barrier Functions

A safety filter based on a neural control barrier function (CBF) deployed in an embedded control loop evaluates, at each control cycle, the trained network and its Lie derivatives along the system vector fields, under the memory and worst-case execution time (WCET) constraints that safety-oriented coding standards impose. Reverse-mode automatic differentiation, by which training frameworks obtain these derivatives, retains an activation cache whose size grows with the sum of the layer widths, and general-purpose differentiation runtimes allocate the computational graph from the heap at each call. This paper presents a compiler that evaluates a neural CBF and its exact Lie derivatives by forward-mode dual-number arithmetic. The compiler emits self-contained C++ code in which a single forward pass, without backpropagation, returns the barrier value and its exact Lie derivative along a given vector field; the drift and input Lie derivatives of the safety constraint are obtained from one such pass per vector field, and a second-order extension based on hyper-dual numbers returns the exact second-order Lie derivatives required by CBFs of relative degree two. The dual forward pass requires a workspace bounded by four times the widest layer, independent of network depth, and the emitted code contains no allocation call sites, so the absence of dynamic allocation is verifiable by inspection of the code. On an ESP32-S3 microcontroller, the compiled filter assembles the complete safety constraint in under one millisecond from statically allocated buffers of at most 768 bytes, and the maximum execution time over 1000 evaluations lies within 5% of the median in all three examples, whereas a heap-allocating reverse-mode baseline shows maxima 33% and 70% above its median in the two first-order examples. The compiler and the embedded experiments are released as open-source software.

eess.SY↗