Non-Holonomic Hydroelastic Model

• Non-Holonomic Hydroelastic Models integrate non-integrable velocity constraints with hydroelastic interactions to capture path-dependent force memory.
• They couple continuum fluid–structure dynamics with compliant contact mechanics, facilitating realistic tactile simulation in manipulation tasks.
• Recent applications demonstrate improved performance in force transmission and contact regulation, reducing errors in tactile sensing and manipulation.

A non-holonomic hydroelastic model denotes a class of formulations at the intersection of constrained mechanics and hydroelastic interaction rather than a single canonical model. In the classical mechanics sense, non-holonomic refers to velocity constraints that are not integrable into configuration constraints, whereas hydroelastic refers to coupled fluid–elastic response or compliant contact. Across the recent literature, these two ingredients often appear separately: many hydroelastic wave, plate, and contact models are not non-holonomic, and many classical nonholonomic fluid-coupled models are not hydroelastic. The term becomes literal only in recent tactile-manipulation work that augments hydroelastic contact with path-dependent distributed force memory and stick–slip state (Matioc et al., 2024, Fedorov et al., 2010, Oller et al., 16 Sep 2025, Dang et al., 28 Feb 2026).

1. Terminology and research scope

The literature represented here falls into three distinct lineages. The first is continuum hydroelasticity, where water-wave or floating-structure dynamics are coupled to plates, shells, or compliant interfaces through free-boundary conditions, bending laws, or pressure continuity. The second is nonholonomic fluid-coupled mechanics, where rigid or low-order bodies evolve under Pfaffian velocity constraints and fluid reactions. The third, and most direct realization of the phrase, is contact-rich tactile simulation in which hydroelastic contact geometry is retained but the force law is made incremental and history dependent.

Lineage	Representative papers	Status relative to the term
Continuum hydroelastic waves, plates, and floating structures	(Matioc et al., 2024, Colomés et al., 2022, Alonso-Orán et al., 29 Mar 2026)	Hydroelastic, but not non-holonomic
Nonholonomic fluid-coupled rigid or reduced-order systems	(Fedorov et al., 2010, Ardister et al., 4 Sep 2025)	Non-holonomic, but not hydroelastic
Tactile/contact models with stateful compliant contact	(Oller et al., 16 Sep 2025, Dang et al., 28 Feb 2026)	Explicitly presented as non-holonomic hydroelastic

In the continuum papers, the active constraints are geometric shell reductions, interface matching, clamped or free-edge conditions, impermeability, and radiation conditions. These are boundary or compatibility conditions, not non-integrable velocity constraints. The two-dimensional periodic hydroelastic wave model with a Cosserat/Kirchhoff shell, for example, explicitly treats Kirchhoff-type shell reductions as holonomic-type geometric constraints rather than non-holonomic ones (Matioc et al., 2024). The monolithic finite-element formulation for very large floating structures likewise solves a coupled potential-flow/beam-or-plate system with interface continuity and elastic-joint laws, but does not introduce Pfaffian constraints of the form $A(q)\dot q=b(q)$ (Colomés et al., 2022).

2. Hydroelastic foundations

The hydroelastic component of the subject spans both continuum fluid–structure interaction and compliant contact. In wave and floating-structure problems, hydroelasticity usually means that water loading and elastic bending are solved together. A thin elastic sheet floating on deep water under a moving perturbation obeys the linear transverse balance

$$B\nabla_{\mathbf r}^{\,4}\zeta - \sigma \nabla_{\mathbf r}^{\,2}\zeta = P + P_\textrm{ext},$$

which, together with linear potential flow, yields the dispersion relation

$$\omega= \sqrt{\frac{Bk^5}{\rho} + \frac{\sigma k^3}{\rho} + gk}.$$

The three restoring contributions are gravity, stretching/tension, and bending (Ono-dit-Biot et al., 2018).

At continuum scale, monolithic hydroelastic formulations solve fluid potential and structural deflection in one coupled variational system. For very large floating structures, the fluid satisfies $\Delta \phi=0$ and the interface kinematic condition $n\cdot \nabla \phi=\eta_t$, while the structure satisfies Euler–Bernoulli or Poisson–Kirchhoff dynamics such as

$$d_0\eta_{tt} + \nabla^2 : (C_\rho : \nabla^2 \eta) + \phi_t + g\eta = 0 \qquad \text{on } \Gamma_s.$$

The resulting formulation is monolithic, energy conserving at the semi-discrete level, and mixed-dimensional, but it remains a holonomic continuum coupling (Colomés et al., 2022).

A different hydroelastic tradition arises in compliant contact simulation. In HydroelasticTouch, each body is assigned a precomputed scalar pressure field $p_O(x)$, maximal at the center and vanishing at the boundary. When two bodies overlap, contact is represented not by isolated points but by an equal-pressure isosurface

$p_e(x)=p_A(x)=p_B(x),$

triangulated into a distributed contact surface. The normal elastic force on a triangle of area $A$ and centroid $x_c$ is approximated by

$$B\nabla_{\mathbf r}^{\,4}\zeta - \sigma \nabla_{\mathbf r}^{\,2}\zeta = P + P_\textrm{ext},$$0

and contact moments follow from summing triangle-level wrench contributions. This produces smooth distributed normal loads over possibly non-convex and disconnected contact patches, but tangential traction is not derived as a distributed hydroelastic shear law (Leins et al., 14 Jan 2025).

Weakly nonlinear hydroelastic interface reduction offers yet another baseline. For deep water coupled to a nonlinear viscoelastic plate, one reduced bidirectional model has the structure

$$B\nabla_{\mathbf r}^{\,4}\zeta - \sigma \nabla_{\mathbf r}^{\,2}\zeta = P + P_\textrm{ext},$$1

where $$B\nabla_{\mathbf r}^{\,4}\zeta - \sigma \nabla_{\mathbf r}^{\,2}\zeta = P + P_\textrm{ext},$$2 is the deep-water Dirichlet–Neumann symbol, $$B\nabla_{\mathbf r}^{\,4}\zeta - \sigma \nabla_{\mathbf r}^{\,2}\zeta = P + P_\textrm{ext},$$3 encodes plate inertia, $$B\nabla_{\mathbf r}^{\,4}\zeta - \sigma \nabla_{\mathbf r}^{\,2}\zeta = P + P_\textrm{ext},$$4 bending, and $$B\nabla_{\mathbf r}^{\,4}\zeta - \sigma \nabla_{\mathbf r}^{\,2}\zeta = P + P_\textrm{ext},$$5 Kelvin–Voigt damping. The model is hydroelastic and quasilinear, but not non-holonomic (Alonso-Orán et al., 29 Mar 2026).

3. Non-holonomic precedents without hydroelasticity

Classical nonholonomic fluid-coupled mechanics enters through rigid or low-order systems. The hydrodynamic Chaplygin sleigh is a rigid body in ideal potential flow with a knife-edge-type nonholonomic constraint suppressing lateral slip. In planar body coordinates, the constraint is

$$B\nabla_{\mathbf r}^{\,4}\zeta - \sigma \nabla_{\mathbf r}^{\,2}\zeta = P + P_\textrm{ext},$$6

and the reduced equations become

$$B\nabla_{\mathbf r}^{\,4}\zeta - \sigma \nabla_{\mathbf r}^{\,2}\zeta = P + P_\textrm{ext},$$7

The fluid enters through added inertia, and the off-diagonal hydrodynamic coupling $$B\nabla_{\mathbf r}^{\,4}\zeta - \sigma \nabla_{\mathbf r}^{\,2}\zeta = P + P_\textrm{ext},$$8 changes the asymptotic behavior from straight-line motion to circular motion. The system is genuinely non-holonomic, but it is not hydroelastic because the body is rigid and the fluid is represented only through added mass (Fedorov et al., 2010).

A related reduced-order swimmer model imposes a Pfaffian constraint on a two-rigid-body system: the tail effective point velocity must align with the tail orientation,

$$B\nabla_{\mathbf r}^{\,4}\zeta - \sigma \nabla_{\mathbf r}^{\,2}\zeta = P + P_\textrm{ext},$$9

The head translates along a straight line, the tail angle is prescribed by $$\omega= \sqrt{\frac{Bk^5}{\rho} + \frac{\sigma k^3}{\rho} + gk}.$$0, and the equations are derived with Lagrange multipliers. CFD validation shows that an effective period-averaged nonholonomic location can reproduce kinematics and normal force well, but the model contains no distributed elasticity, no bending energy, and no hydroelastic constitutive law (Ardister et al., 4 Sep 2025).

These precedents are important because they supply the non-holonomic half of the subject: multiplier-enforced velocity restrictions, effective constraint locations, and reduced reaction forces. What they do not supply is hydroelastic deformation. The later tactile models inherit the geometric efficiency of hydroelastic contact and the statefulness of nonholonomic mechanics, but in a different constitutive setting.

4. Explicit non-holonomic hydroelastic models in tactile manipulation

The clearest self-identified non-holonomic hydroelastic models are recent tactile-contact formulations for compliant manipulation. Their central move is to treat the current distributed force field as part of the state, so that identical instantaneous poses can produce different contact tractions depending on how the configuration was reached.

In Hydrosoft, a rigid object surface is discretized into points $$\omega= \sqrt{\frac{Bk^5}{\rho} + \frac{\sigma k^3}{\rho} + gk}.$$1, each with area $$\omega= \sqrt{\frac{Bk^5}{\rho} + \frac{\sigma k^3}{\rho} + gk}.$$2, while the passive compliant tactile sensor is represented as a hydroelastic body with signed-distance function $$\omega= \sqrt{\frac{Bk^5}{\rho} + \frac{\sigma k^3}{\rho} + gk}.$$3. The state is explicitly augmented as

$$\omega= \sqrt{\frac{Bk^5}{\rho} + \frac{\sigma k^3}{\rho} + gk}.$$4

where $$\omega= \sqrt{\frac{Bk^5}{\rho} + \frac{\sigma k^3}{\rho} + gk}.$$5 is object pose, $$\omega= \sqrt{\frac{Bk^5}{\rho} + \frac{\sigma k^3}{\rho} + gk}.$$6 the hydroelastic-body pose, and $$\omega= \sqrt{\frac{Bk^5}{\rho} + \frac{\sigma k^3}{\rho} + gk}.$$7 the distributed contact-force state. Normal contact in the baseline hydroelastic layer is

$$\omega= \sqrt{\frac{Bk^5}{\rho} + \frac{\sigma k^3}{\rho} + gk}.$$8

but the model then replaces memoryless tangential contact by an incremental update based on the fraction of each displacement step spent inside the compliant body. With in-body displacement $$\omega= \sqrt{\frac{Bk^5}{\rho} + \frac{\sigma k^3}{\rho} + gk}.$$9, normal increment $\Delta \phi=0$0, and tangential increment $\Delta \phi=0$1, the force update is

$\Delta \phi=0$2

The updated force is projected onto the Coulomb cone,

$\Delta \phi=0$3

and reset when contact disappears. The paper is explicit that, in this usage, “non-holonomic” does not mean rolling-without-slipping kinematics; it means that the current force distribution cannot be recovered from instantaneous pose alone and instead depends on the prior force state and motion history (Oller et al., 16 Sep 2025).

HydroShear develops an analogous idea for vision-based tactile sensors such as GelSight Minis. The tactile field is defined by a map

$\Delta \phi=0$4

so the current shear field depends on the full pose history $\Delta \phi=0$5. The field is decomposed into dilation and shear,

$\Delta \phi=0$6

and object geometry is represented by SDFs. Surface points $\Delta \phi=0$7 are transported by the rigid motion $\Delta \phi=0$8, and the recursive force tracker

$\Delta \phi=0$9

accumulates normal and tangential load over time. The in-contact fraction $n\cdot \nabla \phi=\eta_t$0 is computed from the elastomer SDF, the displacement is decomposed into normal and tangential parts, tangential force is clipped by Coulomb friction, and the projected elastomer contact point shifts under slip. This yields path-dependent shear buildup, stick–slip transitions, and full $n\cdot \nabla \phi=\eta_t$1 tactile interactions from arbitrary watertight geometries (Dang et al., 28 Feb 2026).

5. Constitutive memory and the meaning of non-holonomy

The main conceptual difficulty is terminological. In classical nonholonomic mechanics, the defining object is a non-integrable velocity constraint, typically written

$n\cdot \nabla \phi=\eta_t$2

or, more generally, $n\cdot \nabla \phi=\eta_t$3. The hydrodynamic Chaplygin sleigh and the two-body swimmer fall squarely in that category (Fedorov et al., 2010, Ardister et al., 4 Sep 2025).

In the tactile hydroelastic literature, the word is used differently. The decisive feature is constitutive path dependence: the force law is incremental rather than state determined by instantaneous geometry alone. In Hydrosoft, the current distributed force requires the previous force state $n\cdot \nabla \phi=\eta_t$4, and in HydroShear the current tactile field depends on the entire pose path $n\cdot \nabla \phi=\eta_t$5 (Oller et al., 16 Sep 2025, Dang et al., 28 Feb 2026). This usage is close to internal-variable or hysteretic constitutive modeling: memory persists while contact remains active, builds under stick, saturates under slip, and is reset on detachment.

This distinction explains why many hydroelastic models are not non-holonomic even when they contain constraints or implicit operators. HydroelasticTouch computes distributed normal pressure from overlap geometry and optionally adds Hunt–Crossley damping, but tangential behavior is delegated to simulator friction coefficients and there is no distributed shear-stress integration, rolling law, or non-integrable contact kinematics (Leins et al., 14 Jan 2025). Likewise, weakly nonlinear hydroelastic wave equations may contain a nonlinear operator acting on acceleration, but that is a quasilinear evolution law rather than an admissibility constraint on velocities (Alonso-Orán et al., 29 Mar 2026).

A non-holonomic hydroelastic model in the strict classical sense would therefore require both ingredients simultaneously: a hydroelastic normal or distributed compliant-contact layer, and an independent non-integrable kinematic restriction or internal state that cannot be eliminated into a configuration-only law. Recent tactile models realize the second ingredient through internal force memory rather than Pfaffian rolling constraints.

6. Validation, applications, and current limitations

The modern motivation for these models is contact-rich manipulation, tactile sim-to-real transfer, and force-sensitive planning under compliant contact. Reported validation results differ sharply across the three lineages.

Model	Validation setting	Reported result
HydroelasticTouch (Leins et al., 14 Jan 2025)	zero-shot sim-to-real object orientation estimation from tactile data	average angular errors around $n\cdot \nabla \phi=\eta_t$6 rad on real unseen objects
Hydrosoft (Oller et al., 16 Sep 2025)	real-world closed-loop tactile manipulation	planar pushing $n\cdot \nabla \phi=\eta_t$7 mm and rolling $n\cdot \nabla \phi=\eta_t$8 mm, versus $n\cdot \nabla \phi=\eta_t$9 mm and $$d_0\eta_{tt} + \nabla^2 : (C_\rho : \nabla^2 \eta) + \phi_t + g\eta = 0 \qquad \text{on } \Gamma_s.$$0 mm for PF
HydroShear (Dang et al., 28 Feb 2026)	zero-shot sim-to-real RL on peg insertion, bin packing, book shelving, and drawer pulling	$$d_0\eta_{tt} + \nabla^2 : (C_\rho : \nabla^2 \eta) + \phi_t + g\eta = 0 \qquad \text{on } \Gamma_s.$$1 average success rate, versus $$d_0\eta_{tt} + \nabla^2 : (C_\rho : \nabla^2 \eta) + \phi_t + g\eta = 0 \qquad \text{on } \Gamma_s.$$2 for tactile images and $$d_0\eta_{tt} + \nabla^2 : (C_\rho : \nabla^2 \eta) + \phi_t + g\eta = 0 \qquad \text{on } \Gamma_s.$$3 for alternative shear simulation methods

Hydrosoft also reports wrench-transmission RMSE improvements across multiple object geometries, such as $$d_0\eta_{tt} + \nabla^2 : (C_\rho : \nabla^2 \eta) + \phi_t + g\eta = 0 \qquad \text{on } \Gamma_s.$$4 N versus $$d_0\eta_{tt} + \nabla^2 : (C_\rho : \nabla^2 \eta) + \phi_t + g\eta = 0 \qquad \text{on } \Gamma_s.$$5 N and $$d_0\eta_{tt} + \nabla^2 : (C_\rho : \nabla^2 \eta) + \phi_t + g\eta = 0 \qquad \text{on } \Gamma_s.$$6 N for plane contact, and $$d_0\eta_{tt} + \nabla^2 : (C_\rho : \nabla^2 \eta) + \phi_t + g\eta = 0 \qquad \text{on } \Gamma_s.$$7 N versus $$d_0\eta_{tt} + \nabla^2 : (C_\rho : \nabla^2 \eta) + \phi_t + g\eta = 0 \qquad \text{on } \Gamma_s.$$8 N and $$d_0\eta_{tt} + \nabla^2 : (C_\rho : \nabla^2 \eta) + \phi_t + g\eta = 0 \qquad \text{on } \Gamma_s.$$9 N for triangle contact, when compared against PF and PFF baselines. The same model reports strong closed-loop real-world performance across planar pushing, planar rotation, rolling, and bi-manual in-hand rotation, with the largest gains in tasks requiring sustained tangential loading and contact-patch regulation (Oller et al., 16 Sep 2025).

These gains come with substantial limitations. HydroelasticTouch is explicitly a normal-pressure model; it does not provide a distributed tangential traction law, and its raycasting-based tactile rendering scales poorly to larger sensor arrays or high sampling rates, with the effort described as growing quadratically (Leins et al., 14 Jan 2025). Hydrosoft makes the distributed force state explicit, so state dimension grows with the number of discretized contact elements and compliant bodies; it is quasi-dynamic rather than fully dynamic, and its smoothing for differentiability introduces approximation artifacts (Oller et al., 16 Sep 2025). HydroShear assumes a flat elastomer membrane, relies on SDF-based surface-point tracking, and identifies batched SDF computation as a runtime bottleneck (Dang et al., 28 Feb 2026).

In the broader hydroelastic literature, a further limitation is conceptual rather than numerical. Continuum hydroelastic wave and floating-structure models remain overwhelmingly holonomic in structure: they solve pressure continuity, free-boundary kinematics, bending, damping, or added mass, but they do not formulate non-holonomic constraints in the classical sense (Matioc et al., 2024, Colomés et al., 2022, Alonso-Orán et al., 29 Mar 2026). The present state of the field therefore consists less of a single unified theory than of a convergence of ideas: distributed hydroelastic loading from contact or waves, nonholonomic reasoning from constrained mechanics, and internal-state memory from compliant tactile manipulation.