Dynamic Resistive Force Theory (DRFT)

• DRFT is a granular intrusion model that decomposes a body's surface into elements to assign local traction based on depth, orientation, and motion.
• It integrates classical quasistatic resistive forces with dynamic inertial (∝ρv²) and structural free-surface corrections to capture rate-dependent behaviors.
• The framework offers a computationally efficient tool for predicting forces in applications like robotic locomotion and legged or wheeled intrusion in granular media.

Searching arXiv for recent and foundational papers on Dynamic Resistive Force Theory and closely related resistive-force frameworks. Dynamic Resistive Force Theory (DRFT) is a reduced-order framework for predicting forces on bodies moving through granular media across quasistatic and strongly dynamic regimes. Its core construction inherits the local superposition structure of granular Resistive Force Theory (RFT): an intruder is decomposed into small surface elements, each element is assigned a local traction determined by depth, orientation, and motion direction, and the net force is obtained by integration over the intruder surface. DRFT extends that quasistatic picture by introducing dynamic corrections motivated by continuum frictional-plasticity with free-surface separation, especially an inertial term proportional to $\rho v^2$ and a structural correction associated with changes in the free surface around the intruder (Agarwal et al., 2020). In the broader literature, the term is sometimes used narrowly for this continuum-informed dynamic granular model and sometimes more loosely for DRFT-like local superposition models embedded in robotic locomotion studies (Li et al., 2019).

1. Concept and scope

In its most explicit formulation, DRFT is a granular intrusion model that preserves the locality and additivity of classical RFT while extending it into rate-dependent regimes (Agarwal et al., 2020). The intended domain includes wheel locomotion, plate intrusion, and legged locomotion near free surfaces, where grains display solid-like and fluid-like behavior together with an ejected gas-like phase. The framework is therefore neither a grain-resolved simulation nor a full constitutive theory; it is a reduced-order force law designed to capture the dominant mechanisms that determine intrusion forces.

The quasistatic ancestor of DRFT is granular RFT. For a quasi-2D intruder surface subset $S$, that formulation writes the resistive force as

$$(f_x,f_z)=\int_S \big(\alpha_x(\beta,\gamma),\ \alpha_z(\beta,\gamma)\big)\,H(z)\,z\,dS,$$

where $\beta$ is local surface orientation, $\gamma$ is velocity direction, $H(z)$ suppresses resistance above the free surface, and the depth factor reflects gravity-scaled resistance (Askari et al., 2015). DRFT retains this local decomposition but supplements it with dynamic corrections that arise when the momentum balance term $\rho \dot v_i$ becomes consequential even though the constitutive law remains rate-independent (Agarwal et al., 2020).

The literature summarized here also shows that the name is not used uniformly. A resistive force model for legged locomotion on granular media adopts the same segment-by-segment superposition logic and is described as being inspired by earlier resistive force models for sand-swimming, but it is not called DRFT in the paper (Li et al., 2019). A later MuJoCo implementation explicitly distinguishes its quasistatic 3D-RFT core from full Dynamic RFT because it omits the inertial term (Brown et al., 17 Jun 2026). Outside granular media, related generalizations of RFT appear under different motivations, such as load-dependent coefficients for helical filaments and nonlocal operators converging to RFT in Stokes flow (Htet et al., 26 Mar 2025, Ohm, 10 Apr 2026).

2. Quasistatic foundations and continuum rationale

A central question in the DRFT literature is why a local force superposition should work at all in granular media. A continuum-mechanical answer is provided by a frictional-plasticity model with a Drucker–Prager/Mohr–Coulomb-type yield law, incompressible plastic flow during yielding, and an opening rule that sets stress to zero where density drops below a critical packing $\rho_c$ (Askari et al., 2015). In dense regions,

$$\sigma_{ij} = -P \delta_{ij} + 2\mu_c P \, D'_{ij}/\dot{\gamma},$$

with

$$D_{ij}=\frac{\partial v_i/\partial x_j+\partial v_j/\partial x_i}{2},\qquad D'_{ij}=D_{ij}-\delta_{ij}D_{kk}/3,\qquad \dot{\gamma}=\sqrt{2D'_{ij}D'_{ij}},$$

while the open-state rule is

$S$0

The momentum balance is

$S$1

Within this continuum model, the local RFT force law emerges naturally rather than empirically. For a granular “garden hoe” geometry, dimensional analysis yields

$S$2

or, after absorbing material constants,

$S$3

For a small plate element this implies

$S$4

Integrating those local laws reproduces the same scaling and structure as the full continuum solution for that class of geometries (Askari et al., 2015).

This continuum derivation also explains why granular RFT is stronger than its viscous-fluid analogue. In Stokes flow, the force scales as

$S$5

whereas a naive RFT construction over a patch of size $S$6 gives

$S$7

The unavoidable dependence on the arbitrary length $S$8 weakens strong superposition in viscous flow. By contrast, granular frictional plasticity, hydrostatic pressure dependence, and the inability to sustain tension create a local, depth-scaled response that is especially favorable to RFT-style reduction (Askari et al., 2015).

A plausible implication is that DRFT should be expected to work best when those quasistatic premises remain meaningful locally even as macro-inertial effects alter the global intrusion response. That is precisely the regime formalized in the dynamic extension.

3. DRFT formulation and force decomposition

The 2020 dynamic intrusion paper identifies three force contributions that together define DRFT: a static quasistatic resistive term, a dynamic inertial correction, and a dynamic structural correction (Agarwal et al., 2020). The static term is the classical local traction

$S$9

where $$(f_x,f_z)=\int_S \big(\alpha_x(\beta,\gamma),\ \alpha_z(\beta,\gamma)\big)\,H(z)\,z\,dS,$$0 is the empirical traction-per-depth function and $$(f_x,f_z)=\int_S \big(\alpha_x(\beta,\gamma),\ \alpha_z(\beta,\gamma)\big)\,H(z)\,z\,dS,$$1 is height relative to the free surface.

The dynamic inertial correction is a normal traction

$$(f_x,f_z)=\int_S \big(\alpha_x(\beta,\gamma),\ \alpha_z(\beta,\gamma)\big)\,H(z)\,z\,dS,$$2

where $$(f_x,f_z)=\int_S \big(\alpha_x(\beta,\gamma),\ \alpha_z(\beta,\gamma)\big)\,H(z)\,z\,dS,$$3 is the outward normal, $$(f_x,f_z)=\int_S \big(\alpha_x(\beta,\gamma),\ \alpha_z(\beta,\gamma)\big)\,H(z)\,z\,dS,$$4 is the normal component of the surface velocity, $$(f_x,f_z)=\int_S \big(\alpha_x(\beta,\gamma),\ \alpha_z(\beta,\gamma)\big)\,H(z)\,z\,dS,$$5 is the effective bulk density, and $$(f_x,f_z)=\int_S \big(\alpha_x(\beta,\gamma),\ \alpha_z(\beta,\gamma)\big)\,H(z)\,z\,dS,$$6 is an $$(f_x,f_z)=\int_S \big(\alpha_x(\beta,\gamma),\ \alpha_z(\beta,\gamma)\big)\,H(z)\,z\,dS,$$7 fitting constant. This term is the momentum-flux contribution that emerges from the inertial term in the momentum equation.

The dynamic structural correction modifies the depth entering the quasistatic law: $$(f_x,f_z)=\int_S \big(\alpha_x(\beta,\gamma),\ \alpha_z(\beta,\gamma)\big)\,H(z)\,z\,dS,$$8 The full DRFT traction law is then

$$(f_x,f_z)=\int_S \big(\alpha_x(\beta,\gamma),\ \alpha_z(\beta,\gamma)\big)\,H(z)\,z\,dS,$$9

For an intruder surface $\beta$0, the net force is

$\beta$1

Contribution	Expression	Physical role
Static quasistatic term	$\beta$2	Depth- and orientation-dependent resistance
Dynamic inertial correction	$\beta$3	Momentum transfer to grains
Dynamic structural correction	$\beta$4	Free-surface modification around the intruder

The limiting cases are explicit. In the quasistatic limit, $\beta$5 and $\beta$6, so DRFT reduces to classical RFT. If $\beta$7, one obtains a static-plus-$\beta$8 law. If the free surface changes strongly with speed, the structural correction can dominate the rate effect (Agarwal et al., 2020).

The reduction from continuum mechanics to DRFT is physics-guided rather than a strict asymptotic derivation. The continuum analysis shows that if a slow quasistatic solution is scaled to a faster one with similar flow topology, then the inertial term implies an added interface stress proportional to $\beta$9. The structural correction is introduced because the quasistatic term depends on depth below the free surface; once the free surface itself is altered by intrusion, the local resistive force changes even if the constitutive law remains rate-independent (Agarwal et al., 2020).

4. Robotic locomotion and intrusion applications

DRFT and DRFT-like models have been used across wheels, articulated legs, bipedal feet, and full multibody robot simulators. The most direct dynamic application in the summarized literature is grousered wheel locomotion. In that setting, a purely quasistatic RFT captures low-speed behavior but fails at higher spin rates, where translation saturates, slip increases, and sinkage increases. Adding only the inertial $\gamma$0 term improves the model little and can predict the wrong sinkage trend; the key missing mechanism is the structural correction, because high spin expels material behind the wheel, lowers the rear free surface, and reduces support in the trailing shear zone. For the wheel case, the free-surface drop is approximated as

$\gamma$1

applied to the rear zone, and with $\gamma$2 the resulting DRFT reproduces the observed velocity and sinkage trends well (Agarwal et al., 2020).

A closely related but explicitly quasistatic application is rigid-wheel interaction with dry granular media under forced-slip conditions. There, granular RFT is implemented as a local surface-stress superposition model in which each wheel surface element contributes according to depth, orientation, and movement direction, subject to a leading-edge hypothesis. The study concludes that, for the range of inputs considered, RFT can be reliably used to predict rigid wheel granular media interactions with accuracy exceeding that of traditional terramechanics methodology in several circumstances, while still remaining fundamentally quasistatic rather than a full DRFT law (Agarwal et al., 2019).

For legged locomotion on granular media, a resistive force model in the vertical plane applies the same local-superposition principle to complex limbs. The leg is divided into small segments, each segment is assigned a local force determined by depth $\gamma$3, angle of attack $\gamma$4, and angle of intrusion $\gamma$5, and the net force is obtained by integration: $\gamma$6 In loosely packed poppy seeds of packing fraction about $\gamma$7, with a model aluminum plate of area $\gamma$8 moved at $\gamma$9, both $H(z)$0 and $H(z)$1 were found to be approximately linear in depth. Applied to an L-leg and a reversed L-leg, the model predicts $H(z)$2 and $H(z)$3 without any fitting parameters, matching measured forces to within about $H(z)$4. Embedded into a multi-body dynamics simulation using MBDyn for a robot about $H(z)$5 long and $H(z)$6 in mass, the simulated average forward speed matches experiments within $H(z)$7 for L-legs across the tested frequencies and within $H(z)$8 for reversed L-legs up to about $H(z)$9 (Li et al., 2019).

A more explicit DRFT extension appears in the foot-shape-dependent resistive force model for bipedal walkers on sand. That model adds both inertial drag and an effective intrusion depth correction. After discretizing the foot into plates, the dynamic contribution is

$\rho \dot v_i$0

while the corrected depth is

$\rho \dot v_i$1

The full plate-level force is

$\rho \dot v_i$2

On fine sand with particle size $\rho \dot v_i$3–$\rho \dot v_i$4, calibrated values include $\rho \dot v_i$5, $\rho \dot v_i$6, and $\rho \dot v_i$7. Across slow, medium, and high gait conditions, the corrected model reduces RMSE relative to conventional 3D-RFT for $\rho \dot v_i$8, $\rho \dot v_i$9, and $\rho_c$0, with especially large improvements at medium and high gait speeds. The same study reports that the elliptical foot is better for high-speed walking because it required less work than flat foot in fast gait conditions, whereas the flat foot is better for slow walking because it saved energy under slow motion (Chen et al., 2024).

A recent open-source simulator integrates 3D granular RFT into MuJoCo by replacing rigid-contact ground forces with a distributed granular contact model. Each body is discretized into plate elements, only leading surfaces contribute, and forces are applied at plate centroids using MuJoCo sites and mj.applyFT. The implementation uses

$\rho_c$1

with depth-, orientation-, and motion-direction-dependent empirical terms, and it introduces single exponential moving average smoothing,

$\rho_c$2

to stabilize low-speed dynamics. The paper explicitly states that it did not implement the inertial term of Dynamic RFT because simulated speeds were below about $\rho_c$3 tip speed and below the cited threshold of about $\rho_c$4. Even so, the resulting RFT-SiM predicts walking distance and foot sinkage of a 12-degree-of-freedom hexapod robot within $\rho_c$5 of experiments in sand in the abstract, while the detailed validation reports walking-distance underestimation of about $\rho_c$6 to $\rho_c$7 and sinkage overestimation by about $\rho_c$8 (Brown et al., 17 Jun 2026).

5. Internal torque, body loading, and generalized resistive-force models

Although DRFT is most often used for net ground-reaction prediction, the same local-force integration logic also determines internal loading. In steady undulatory locomotion through resistive-force-dominated media, the body is prescribed to undergo a traveling curvature wave

$\rho_c$9

with wave speed

$$\sigma_{ij} = -P \delta_{ij} + 2\mu_c P \, D'_{ij}/\dot{\gamma},$$0

Local forces satisfy

$$\sigma_{ij} = -P \delta_{ij} + 2\mu_c P \, D'_{ij}/\dot{\gamma},$$1

with $$\sigma_{ij} = -P \delta_{ij} + 2\mu_c P \, D'_{ij}/\dot{\gamma},$$2 and $$\sigma_{ij} = -P \delta_{ij} + 2\mu_c P \, D'_{ij}/\dot{\gamma},$$3 for a thin cylinder in viscous drag, and the internal torque at body position $$\sigma_{ij} = -P \delta_{ij} + 2\mu_c P \, D'_{ij}/\dot{\gamma},$$4 is obtained by integrating external resistive loads over the anterior body segment: $$\sigma_{ij} = -P \delta_{ij} + 2\mu_c P \, D'_{ij}/\dot{\gamma},$$5 In the small-amplitude approximation,

$$\sigma_{ij} = -P \delta_{ij} + 2\mu_c P \, D'_{ij}/\dot{\gamma},$$6

The main result is that torque typically forms a traveling wave for $$\sigma_{ij} = -P \delta_{ij} + 2\mu_c P \, D'_{ij}/\dot{\gamma},$$7, but the speed ratio $$\sigma_{ij} = -P \delta_{ij} + 2\mu_c P \, D'_{ij}/\dot{\gamma},$$8 decreases from about $$\sigma_{ij} = -P \delta_{ij} + 2\mu_c P \, D'_{ij}/\dot{\gamma},$$9 to $$D_{ij}=\frac{\partial v_i/\partial x_j+\partial v_j/\partial x_i}{2},\qquad D'_{ij}=D_{ij}-\delta_{ij}D_{kk}/3,\qquad \dot{\gamma}=\sqrt{2D'_{ij}D'_{ij}},$$0 as $$D_{ij}=\frac{\partial v_i/\partial x_j+\partial v_j/\partial x_i}{2},\qquad D'_{ij}=D_{ij}-\delta_{ij}D_{kk}/3,\qquad \dot{\gamma}=\sqrt{2D'_{ij}D'_{ij}},$$1 increases from $$D_{ij}=\frac{\partial v_i/\partial x_j+\partial v_j/\partial x_i}{2},\qquad D'_{ij}=D_{ij}-\delta_{ij}D_{kk}/3,\qquad \dot{\gamma}=\sqrt{2D'_{ij}D'_{ij}},$$2 to $$D_{ij}=\frac{\partial v_i/\partial x_j+\partial v_j/\partial x_i}{2},\qquad D'_{ij}=D_{ij}-\delta_{ij}D_{kk}/3,\qquad \dot{\gamma}=\sqrt{2D'_{ij}D'_{ij}},$$3. Near $$D_{ij}=\frac{\partial v_i/\partial x_j+\partial v_j/\partial x_i}{2},\qquad D'_{ij}=D_{ij}-\delta_{ij}D_{kk}/3,\qquad \dot{\gamma}=\sqrt{2D'_{ij}D'_{ij}},$$4, the pattern transitions to a two-wave-like structure, and for $$D_{ij}=\frac{\partial v_i/\partial x_j+\partial v_j/\partial x_i}{2},\qquad D'_{ij}=D_{ij}-\delta_{ij}D_{kk}/3,\qquad \dot{\gamma}=\sqrt{2D'_{ij}D'_{ij}},$$5 more complex patterns appear because phase-shifted force contributions cancel in the distance-weighted integral. The paper explicitly frames this as a resistive-force-theory study of internal torque rather than a new force law, but it shows how distributed resistive loads reorganize internal actuation demands (Ming et al., 2017).

In low-Reynolds-number hydrodynamics, generalized resistive-force models introduce different kinds of corrections. For rigid helical filaments with an attached spherical load of radius $$D_{ij}=\frac{\partial v_i/\partial x_j+\partial v_j/\partial x_i}{2},\qquad D'_{ij}=D_{ij}-\delta_{ij}D_{kk}/3,\qquad \dot{\gamma}=\sqrt{2D'_{ij}D'_{ij}},$$6, the local RFT law is written as

$$D_{ij}=\frac{\partial v_i/\partial x_j+\partial v_j/\partial x_i}{2},\qquad D'_{ij}=D_{ij}-\delta_{ij}D_{kk}/3,\qquad \dot{\gamma}=\sqrt{2D'_{ij}D'_{ij}},$$7

but the optimal coefficients are shown to depend on the load through

$$D_{ij}=\frac{\partial v_i/\partial x_j+\partial v_j/\partial x_i}{2},\qquad D'_{ij}=D_{ij}-\delta_{ij}D_{kk}/3,\qquad \dot{\gamma}=\sqrt{2D'_{ij}D'_{ij}},$$8

The resulting coefficients $$D_{ij}=\frac{\partial v_i/\partial x_j+\partial v_j/\partial x_i}{2},\qquad D'_{ij}=D_{ij}-\delta_{ij}D_{kk}/3,\qquad \dot{\gamma}=\sqrt{2D'_{ij}D'_{ij}},$$9 and $S$00 are load dependent rather than universal. For a normal polymorph, using Lighthill’s free-helix coefficients yields only about $S$01 error in the in-plane flow components $S$02 but about $S$03 error in the axial component $S$04, and for larger loads the swimming-speed error reaches nearly $S$05. The parallel resistance coefficient changes by up to about $S$06 as the load grows (Htet et al., 26 Mar 2025). This is a generalized RFT, but the paper itself states that it is not “dynamic” in the time-dependent-memory sense.

A distinct mathematical generalization appears in the nonlocal curve evolution for an immersed elastic filament in a 3D Stokes fluid. There, the RFT operator

$S$07

is compared with a pseudodifferential operator $S$08 built from the exact Fourier symbol of the straight-cylinder slender-body Neumann-to-Dirichlet map. The resulting evolution

$S$09

is globally well posed in the natural energy space, and after the rescaled time $S$10 the solutions converge to RFT dynamics as $S$11 with

$S$12

This establishes, in a rigorous Stokes-flow setting, how local resistive-force dynamics emerge from a more detailed nonlocal model (Ohm, 10 Apr 2026).

6. Validity regime, limitations, and nomenclature

DRFT is best understood as a reduced-order model whose reliability depends on the physical regime and on what is meant by “dynamic.” In the granular continuum foundation, the quasistatic RFT reduction is strongest when the material response is local, frictional, depth-scaled, and effectively decoupled from global geometry. When rate dependence or size effects become important, the local resistive plots acquire explicit dependence on additional groups such as

$S$13

and the simple universal RFT law no longer survives unchanged (Askari et al., 2015).

The 2020 DRFT formulation is also explicit about scope. It assumes additivity of static and dynamic terms, treats the structural correction $S$14 as an effective representation of free-surface change, and is intended as a bridge between continuum simulation, empirical quasistatic RFT, and higher-speed $S$15 intrusion laws. It is not a replacement for continuum simulation, is best suited to shallow intrusions where RFT assumptions still hold, and has been validated primarily for dry, non-cohesive granular media (Agarwal et al., 2020).

A frequent misconception is that any granular resistive-force model embedded in a dynamic simulator is automatically DRFT. Several papers draw a sharper distinction. The MuJoCo implementation states that it uses the quasistatic 3D-RFT core only and explicitly did not implement the inertial term because the motions were below the regime where inertial sand effects become significant. The same paper also lists limitations including no grain-scale effects, no surface deformation, no inertial correction, sensitive calibration, and computational cost about an order of magnitude slower than default MuJoCo contacts due to many force application points (Brown et al., 17 Jun 2026). Similarly, the legged-locomotion model on granular media is DRFT-like in construction but is presented as a resistive force model rather than a named DRFT formalism (Li et al., 2019).

Another nomenclature issue is that related generalized RFT models in fluids need not be “dynamic” in the granular DRFT sense. The load-dependent helical-filament theory modifies local coefficients through global force balance with an attached load rather than through inertial or free-surface corrections (Htet et al., 26 Mar 2025). Conversely, the undulatory-locomotion torque study belongs to the RFT family because torque emerges from integrating local external loads, but its key contribution is an explanation of internal torque-wave formation and transition rather than a new resistive-force law (Ming et al., 2017).

Taken together, these studies define DRFT not as a single universally fixed formula, but as a family of local-additive reduced models whose most developed form in granular media consists of quasistatic RFT plus a momentum-flux correction and a free-surface structural correction. The framework’s central promise is computational economy without abandoning the geometry-aware surface integration that makes resistive-force methods useful across locomotion and intrusion problems (Agarwal et al., 2020).