Fundamental Equation of Earthmoving

• Fundamental Equation of Earthmoving is a quantitative model that defines soil cutting forces using the Mohr–Coulomb yield criterion and wedge analysis.
• It decomposes resistive forces into weight, cohesion, surcharge, and adhesion components, with dimensionless N-factors capturing geometric and material dependencies.
• The FEE is crucial for simulation frameworks and autonomous control systems, enabling soil property estimation and optimization of excavation strategies.

The Fundamental Equation of Earthmoving (FEE) is an analytical framework that formalizes the prediction of cutting and resistive forces encountered during the mechanical excavation or displacement of soil by rigid tools such as blades and buckets. Rooted in soil failure mechanics, particularly the Mohr–Coulomb model, and wedge equilibrium analysis, the FEE provides a quantitative nexus between soil physical properties, tool geometry, and the energetics of earthmoving, thereby serving as the foundation for both simulation and control in autonomous earthmoving systems.

1. Theoretical Formulation and Derivation

The FEE is founded on the Mohr–Coulomb yield criterion, which states that soil shear strength satisfies

$\tau = c + \sigma_n \tan \phi$

where $c$ is cohesion, $\sigma_n$ is normal stress, and $\phi$ is the internal friction angle (Wagner, 25 May 2025). Considering a rigid blade (or bucket edge) moving through soil, the analysis posits a wedge of failed material bounded by the tool surface, a failure plane (typically planar or spiral), and the free surface. Static equilibrium is imposed by balancing forces in both horizontal and vertical directions, leading to a closed-form for the force $F$ per unit blade width required to cut soil.

The canonical form of the FEE is given as: $$F = \gamma d^2 N_{\gamma} + c d N_c + Q N_Q + c_a d N_a$$ where

• $\gamma$ = moist unit weight of the soil,
• $d$ = cut depth perpendicular to the surface,
• $c$ = soil cohesion,
• $Q$ = surcharge (overburden) force,
• $c_a$ = soil–tool adhesion,
• $N_{\gamma}, N_c, N_Q, N_a$ = dimensionless coefficients (“N-factors”) determined by soil-tool friction $\delta$, tool orientation $\rho$, failure angle $\beta$, slope $\alpha$, and friction angle $\phi$.

The $N$-factors are given by: $$N_{\gamma} = \frac{(\cot \rho + \cot \beta) \sin(\alpha + \phi + \beta)}{2\sin(\delta + \rho + \phi + \beta)}$$

$$N_Q = \frac{\sin(\alpha + \phi + \beta)}{\sin(\delta + \rho + \phi + \beta)}$$

$$N_c = \frac{\cos(\phi)}{\sin(\beta) \sin(\delta + \rho + \phi + \beta)}$$

$$N_a = -\frac{\cos(\rho + \phi + \beta)}{\sin(\rho) \sin(\delta + \rho + \phi + \beta)}$$

The failure angle $\beta$ is determined by minimizing $N_\gamma$ with respect to $\beta$ (Wagner, 25 May 2025, Abdolmohammadi et al., 27 Jun 2025).

2. Physical Interpretation and Key Model Components

The FEE distinctly partitions the contributions to blade resistive force:

• Weight term ($\gamma d^2 N_\gamma$): accounts for the effect of the soil wedge’s self-weight.
• Cohesion term ($c d N_c$): encodes soil cohesion, providing additional shear resistance.
• Surcharge term ($Q N_Q$): models the mass of soil above the cut (e.g., berm loading).
• Adhesion term ($c_a d N_a$): addresses soil–tool interface forces (Wagner, 25 May 2025, Wagner et al., 2023).

The N-factors encapsulate complex geometric dependencies and are typically determined numerically due to their nonlinearity in $\beta$. Key parameters (summarized in the table below) dominate the force calculation:

Parameter	Physical Meaning	Typical Source
$d$	Cut depth	Geometry or sensed excavation depth
$\gamma$	Moist unit weight	Measured, or simulation input
$c$	Cohesion	Calibrated/estimated
$\phi$	Internal friction angle	Calibrated/estimated
$c_a$	Adhesion	Soil-tool calibration or estimate
$Q$	Surcharge force	Dynamic/terrain geometry
$\beta$	Failure angle	Optimization (argmin $N_\gamma$)
$\delta$	Soil–tool friction	Estimated or simulation input

3. Implementation in Simulation and Earthmoving Autonomy

The FEE is computationally efficient and is integrated into high-fidelity earthmoving simulations such as Vortex Studio and AGX Dynamics (Wagner, 25 May 2025, Abdolmohammadi et al., 27 Jun 2025). The simulation workflow typically involves:

• Computing local terrain geometry (cut depth, blade angles, surface slope).
• Using least-squares or geometric fitting to extract $d$, $\alpha$, $\rho$ from elevation grids.
• Solving for $\beta$ and N-factors numerically.
• Applying the FEE to predict interaction forces, which trigger soil element transitions (e.g., from heightfield to discrete particles) when thresholds are exceeded.

Within simulation, the FEE governs when and how soil is converted from bulk to particle representations, and which regions experience plastic failure or cutting (Wagner, 25 May 2025).

For trajectory optimization in autonomous equipment, the FEE or its analogues serve as constraints or cost terms for force/torque minimization subjected to machine and soil limits (Yang et al., 2020). The interaction force function $R$ in joint-space dynamic equations $$T = M(q)\ddot{q} + C(q, \dot{q})\dot{q} + V(q) - J^T(q) R$$ reflects soil resistance computed as a function of machine states and FEE-based force prediction.

4. Soil Property Estimation and Model Inversion

Inversion of the FEE underpins modern in situ soil property estimation frameworks. Physics-infused neural networks (PINNs) are trained to estimate unknown FEE parameters (e.g. $c, \phi, c_a, \gamma, \beta$) from sensor data (kinematics, control commands), either using supervised labels derived from simulation (Wagner et al., 2023, Wagner et al., 30 Jul 2025) or by closing the loop using residual error between predicted and measured cutting force.

The general structure:

1. Observation history is encoded (via convolution and/or attention-based architectures).
2. Network outputs estimated FEE parameters and residuals.
3. Parameters are fed to the closed-form FEE (and augmented by an additional residual force) to synthesize the predicted force.
4. The network is trained to minimize a loss combining mean-squared interaction force error, physical consistency penalties (e.g., optimal $\beta$), and parameter plausibility constraints (Wagner et al., 2023).
5. Estimated uncertainties are propagated via linearization of the FEE; uncertainty-aware loss functions (negative log-likelihood) optimize both accuracy and well-calibrated confidence (Wagner et al., 30 Jul 2025).

This architecture enables real-time soil mapping, with spatial parameter fusion performed Bayesianly as trajectories traverse the site (Wagner et al., 30 Jul 2025).

5. Extensions, Applications, and Adaptations

Beyond terrestrial excavation, the FEE generalizes to lunar regolith manipulation and other planetary site preparation (Lee et al., 2023). In the CraterGrader system, the concept of FEE is extended as a transport cost metric in an optimization-based material movement planner, dynamically updating the soil state and incorporating uncertain geotechnical parameters into cost functions for efficiency and robustness.

Integration with multiscale terrain simulation frameworks further demonstrates the universality of the FEE:

• At the mesoscale, FEE-like force predictions trigger hybridization between continuum and DEM representations (Servin et al., 2020).
• Calibration is achieved through virtual triaxial testing and parametric fitting against simulated or empirical bulk mechanical responses.

For autonomous wheel loaders, cycle-to-cycle adaptation is realized by fitting FEE parameters to data from previous excavation cycles, then using these to predict and control the next action, achieving root-mean-square errors on force predictions of 10–15% (Abdolmohammadi et al., 27 Jun 2025).

6. Practical Challenges, Variants, and Model Limitations

• Ambiguity and Identifiability: The FEE is under-constrained—distinct parameter sets can yield similar force predictions, necessitating regularization or physical priors in soil estimation (Wagner et al., 2023, Wagner et al., 30 Jul 2025).
• Compaction and Residual Effects: Direct FEE predictions may underfit resistive forces from soil compaction or tool dynamics. Augmentation by Bekker–Janosi–Hanamoto sinkage models and explicit residual terms is common (Abdolmohammadi et al., 27 Jun 2025).
• Model-Physics Gaps: Idealized wedge analysis often neglects phenomena such as particle-scale flows, granular avalanching, or cohesive-limited volumetric yield, which can manifest as systematic error, especially in highly compacted or strongly cohesive soils (Servin et al., 2020, Wagner, 25 May 2025).
• Parameter Smoothing and Dynamic Tuning: Modifications such as hyperbolic tangent scaling of friction and adhesion parameters with blade velocity smooth out regime transitions (e.g., between up-slope and down-slope soil flow), improving simulation stability (Wagner, 25 May 2025).

7. Summary and Broader Impact

The Fundamental Equation of Earthmoving provides a unified, parameterized physical model for predicting the force required to mechanically disturb and move soil. Its capability to decompose resistive forces into weight, cohesion, surcharge, and adhesion components—modulated by geometric and material properties—renders it central to a wide spectrum of earthmoving research and technology development. Whether embedded directly in digital twins, serving as the backbone for machine-learning-driven soil property estimation, or acting as an optimization constraint in autonomous planning, the FEE underlies the modern quantitative treatment of earth-tool interactions, with demonstrable accuracy (force prediction within 10–25% of reference models for frictional soils (Servin et al., 2020, Abdolmohammadi et al., 27 Jun 2025)) and computational tractability suitable for real-time control and simulation. Its extensive adoption in simulation environments (e.g., Vortex Studio, AGX Dynamics) and autonomous system architectures has enabled advances in adaptive, robust excavation, terrain mapping, and planetary site preparation.