Deriving The Fundamental Equation of Earthmoving and Configuring Vortex Studio Earthmoving Simulation for Soil Property Estimation Experimentation

TL;DR

This work derives the fundamental equation of earthmoving (FEE) for a flat blade in sloped soil under Mohr-Coulomb strength, yielding $F = c d N_c + d^2 \gamma N_\gamma + Q N_Q + c_a d N_{c_a}$ and detailing the beta-like minimization $\beta^*$ that minimizes $N_\gamma$; it also notes a flat-terrain reduction to McKyes theory. It then describes the Vortex Studio soil–tool interaction simulation, which uses a hybrid heightfield–DEM model driven by the FEE to predict cutting forces, with soil properties evolving via relative density $I_d$ and surcharge $Q$; this includes a pipeline for accessing and configuring soil parameters through default presets (e.g., Clay, Loam, Sand, Gravel) and a Python API. The paper provides practical guidance for configuring Vortex parameters, density and geometry mappings, FEE strength coefficients, hyperparameters, particles, compaction/erosion, deformer contacts, and heuristic FEE modifications, with references to Haeri et al. (2020) and related CM Labs tooling for validation against experimental data. A tuning protocol is presented to generate an FEE-only behavior dataset for soil-property estimation experiments, including disabling particle spawning and deformer contacts, adjusting slope-determination, and constraining blade geometry to isolate FEE-driven forces. The combined analytic-empirical framework and the detailed Vortex configuration guidance enable rigorous in silico experimentation for in situ soil-property estimation and earthmoving autonomy research.

Abstract

This document serves as supplementary material for two International Society for Terrain-Vehicle Systems conference publications regarding in situ soil property estimation by Wagner et al. in 2023 and 2025. It covers the derivation of the fundamental equation of earthmoving for a flat blade moving through sloped soil and provides some information regarding the advanced configuration of Vortex Studio's soil-tool interaction simulation.

Figures (6)

• Figure 1: Mohr's circle diagram reproduced from Mckyes1989_ch2.
• Figure 2: The failure strength criterion line is tangent to Mohr's circle due to our assumption of no acceleration. If the line did not touch the circle, then the failure condition would not be met. If the line touched the circle at more than one point, then the the stress at some plane within the material would be greater than the failure stress meaning that the soil would have to be accelerating due to a force imbalance. The angle $\sigma_f$ is the angle of the soil failure plane from the principal stress plane. Reproduced from Mckyes1989_ch2.
• Figure 3: FEE forces nomenclature diagram
• Figure 4: FEE geometry nomenclature diagram
• Figure 5: Accessing advanced soil properties within Vortex Studio Soil Materials Extension