RealCOD-Rigid: Rigid Body Research and Applications

• RealCOD-Rigid involves computational descriptions of rigid objects, focusing on geometry, dynamics, and perception for applications in engineering and robotics without a single official framework.
• Rigid-body research under RealCOD-Rigid explores geometric constraint systems, such as JACOB measure and CAD-oriented systems, where rigid components are analyzed through metrics like global and infinitesimal rigidity and dependency structures known as Maxwell conditions.
• Advanced simulations and models in RealCOD-Rigid include the use of stress matrices and combined constraint-geometry methods to ascertain the rigidity of complex multi-body systems and integrate fluid-dynamic and contact mechanics interactions.

RealCOD-Rigid is an editorial umbrella term for research concerning rigid geometric objects, their constraints, dynamics, localization, manipulation, perception, simulation, registration, and interactions with deformable or ambient media. The designation is not an official name for a single framework: one paper states that “RealCOD-Rigid” does not occur in its terminology (Jaén, 2018), while several others use it only as an external shorthand for related objectives. Across the cited literature, the common technical themes are rigid-body configuration in $\mathrm{SE}(3)$, constraint Jacobians and rigidity matrices, global and infinitesimal rigidity, geometric reconstruction, physically consistent simulation, and the distinction between ideal rigid abstractions and materially compliant systems.

1. Terminology and conceptual scope

“RealCOD-Rigid” does not denote a unified established discipline, benchmark, or software specification in the cited literature. Its interpretation depends on the associated research problem. In geometric constraint systems, it refers to a potential CAD-oriented system involving rigid parts, pairwise geometric constraints, redundant constraints, and detection of rigid components. In relativistic coordinate theory, it is used only as an external shorthand for rigid covariance and Fermi rigid coordinates (Jaén, 2018). In robotic manipulation, it denotes a possible system for rigid cooperative manipulation, tensegrity dynamics, or rigid-object transport through ropes (Verginis et al., 2019, Luo et al., 2022, Wang et al., 2023, Wang et al., 2023). In simulation, it may refer to immersed rigid bodies, rigid-body fluid interaction, or the discrepancy between rigid and deformable models (Wallis et al., 2021, Ramakrishnan et al., 8 Feb 2025, Padilla et al., 20 Jan 2026).

A plausible editorial interpretation is that RealCOD-Rigid concerns realistic computational descriptions of rigid objects: representations that preserve geometric constraints, dynamic coupling, sensing uncertainty, contact mechanics, and the limitations of ideal rigidity. Under this interpretation, the term covers several layers:

• Geometric representation: points, lines, planes, frames, rigid bodies, bars, meshes, landmarks, and $\mathrm{SE}(3)$ configurations.
• Constraint structure: coincidence, distance, angular, bearing, grasp, joint, periodic, and cable constraints.
• Rigidity analysis: rank, nullspaces, infinitesimal motions, minimality, global uniqueness, and stress certificates.
• Forward dynamics: rigid-body equations, constrained DAEs, added mass, buoyancy, fluid pressure, contact, and cable forces.
• Inference and control: localization, registration, cooperative manipulation, learned action selection, and internal-force regulation.
• Model validity: compliance, deformability, numerical discretization, perceptual ambiguity, and sim-to-real discrepancies.

The literature therefore distinguishes mathematical rigidity from physical rigidity. A body may be geometrically rigid while its material compliance remains dynamically significant, or it may be perceived as rigid despite ambiguous image motion (Maruya et al., 2023, Ramakrishnan et al., 8 Feb 2025).

2. Geometric constraint systems and rigidity matrices

The body-and-CAD model provides a formal foundation for three-dimensional rigid geometric constraint systems (Haller et al., 2010). A structure consists of rigid bodies with frames $T_i\in SE(3)$ and attached geometric elements: points, lines, and planes. A cad graph is an edge-colored multigraph whose vertices represent bodies and whose edges represent pairwise constraints. The model identifies 21 constraint types, including point-point coincidence, point-point distance, point-line coincidence, line-line coincidence, angular relations, plane coincidence, and plane distance.

Each body has six instantaneous degrees of freedom: three translations and three rotations. An infinitesimal motion is represented by a screw

$s=(-\omega,v)\in\mathbb R^6,$

with point velocity

$p'=\omega\times p+v.$

For a pairwise constraint between bodies $i$ and $j$, the relevant instantaneous quantity is the relative screw $s_i-s_j$. Linearizing every primitive relation produces a rigidity matrix $R$ with six columns per body. Its kernel contains the six-dimensional space of global rigid motions: $s_i=s_j\qquad\text{for all }i,j.$ Consequently,

$\mathrm{SE}(3)$0

The framework is infinitesimally rigid when

$\mathrm{SE}(3)$1

The paper separates primitive constraints into angular and blind classes. Angular rows constrain only rotational variables and have zero translational entries. Blind rows can constrain both translation and rotation. This distinction produces a stronger combinatorial condition than the ordinary Maxwell count. After replacing compound CAD constraints with primitive angular and blind edges, the primitive cad graph must satisfy $\mathrm{SE}(3)$2-nested tightness as a necessary condition for generic minimal rigidity: $\mathrm{SE}(3)$3 For the full graph,

$\mathrm{SE}(3)$4

Nested sparsity is not sufficient for rigidity. The cited counterexample is a three-body structure whose primitive graph is $\mathrm{SE}(3)$5-nested tight but retains an infinitesimal translation of one body. The failure results from geometric dependencies that graph counts do not encode. Thus a suitable RealCOD-Rigid analysis must combine combinatorial preprocessing with numerical or symbolic rank analysis rather than treating sparsity as a rigidity certificate.

The same distinction appears in classical body-and-bar theory. A body-and-bar bar is exactly a point-point distance constraint: $\mathrm{SE}(3)$6 It contributes one rigidity-matrix row. Body-and-CAD systems generalize this model by permitting compound geometric relations that contribute between one and four primitive rows.

3. Global rigidity, stresses, and uniqueness certificates

Infinitesimal rigidity concerns local motions, whereas global rigidity concerns uniqueness of a realization up to Euclidean congruence. For a framework $\mathrm{SE}(3)$7 in $\mathrm{SE}(3)$8, global rigidity requires every realization with the same edge lengths to be congruent to $\mathrm{SE}(3)$9. Universal rigidity strengthens this requirement by allowing competing realizations in arbitrary dimensions (Connelly et al., 2016).

The hierarchy is

$T_i\in SE(3)$0

A framework is infinitesimally rigid when

$T_i\in SE(3)$1

An equilibrium stress assigns edge weights $T_i\in SE(3)$2 satisfying

$T_i\in SE(3)$3

at every vertex. The associated stress matrix $T_i\in SE(3)$4 satisfies

$T_i\in SE(3)$5

and therefore

$T_i\in SE(3)$6

A framework is super stable if it has a positive-semidefinite equilibrium stress matrix of rank $T_i\in SE(3)$7 and its edge directions do not lie on a conic at infinity. Super stability implies universal rigidity. The cited theorem proves that every graph generically globally rigid in $T_i\in SE(3)$8 has a realization that is simultaneously generic, infinitesimally rigid, super stable, and universally rigid.

The construction uses a general-position orthogonal representation obtained through the Lovász–Saks–Schrijver method. For a generically globally rigid graph, $T_i\in SE(3)$9-connectivity yields an orthogonal representation in dimension

$s=(-\omega,v)\in\mathbb R^6,$0

Its Gram matrix is converted, through diagonal centering, into a positive-semidefinite maximal-rank stress matrix. A dimension argument then establishes that at least one such stress corresponds to an infinitesimally rigid framework. Openness of maximal stress rank, super stability, and infinitesimal rigidity permits selection of a generic realization in a neighborhood of that framework.

This result supplies a certificate relevant to geometry-based RealCOD-Rigid systems. A candidate realization can be certified using

$s=(-\omega,v)\in\mathbb R^6,$1

together with

$s=(-\omega,v)\in\mathbb R^6,$2

The resulting certificate is existential rather than a complete polynomial-time recognition algorithm for generically globally rigid graphs. It also does not imply that every generic realization is universally rigid.

4. Rigid-body dynamics, DAEs, and heterogeneous mechanisms

Rigid mechanisms containing arbitrary bodies, bars, joints, cables, and moving supports can be formulated using nonminimal natural coordinates (Luo et al., 2022). A three-dimensional rigid body is represented by basic points and base vectors, while a slender rigid bar is represented without assigning it a fictitious torsional degree of freedom. Generic body points are affine-linear functions of the natural coordinates: $s=(-\omega,v)\in\mathbb R^6,$3

For a rigid body, six quadratic intrinsic constraints preserve the lengths and mutual dot products of the body-attached basis vectors. For a rigid bar, a single length constraint is sufficient. The body coordinates may have 12 components but only six physical degrees of freedom; a spatial bar has six coordinates and one constraint, corresponding to five physical degrees of freedom.

Because the point map is linear,

$s=(-\omega,v)\in\mathbb R^6,$4

the kinetic energy has the form

$s=(-\omega,v)\in\mathbb R^6,$5

The mass matrix is constant. After assembly, the global system is governed by a constrained differential-algebraic equation: $s=(-\omega,v)\in\mathbb R^6,$6 subject to

$s=(-\omega,v)\in\mathbb R^6,$7

Massless cables contribute unilateral elastic-viscous tension. For a cable with length $s=(-\omega,v)\in\mathbb R^6,$8 and rest length $s=(-\omega,v)\in\mathbb R^6,$9,

$p'=\omega\times p+v.$0

with tension set to zero when the cable is slack. The resulting generalized cable force is linear in the cable force-density vector, although the geometry and constitutive law remain nonlinear.

Linearization around equilibrium produces a reduced system

$p'=\omega\times p+v.$1

The stiffness contains a geometric-stiffness contribution from constraint reactions, which is essential for prestressed tensegrities. A modified symplectic integration scheme enforces position-level constraints while accommodating damping, external loads, prescribed coordinates, moving boundaries, and changing cable rest lengths.

A related cooperative-manipulation formulation represents robotic end effectors and an object in $p'=\omega\times p+v.$2 (Verginis et al., 2019). Rigid grasping imposes

$p'=\omega\times p+v.$3

where $p'=\omega\times p+v.$4 is the grasp matrix. Since $p'=\omega\times p+v.$5 has full row rank,

$p'=\omega\times p+v.$6

The nullspace consists of contact-wrench distributions that generate no net object wrench. Under a nondegenerate complete formation,

$p'=\omega\times p+v.$7

so internal contact forces correspond to the transpose-range of the formation rigidity matrix.

The dynamically relevant internal-force projection is

$p'=\omega\times p+v.$8

The associated internal-force-free desired distribution for an object wrench $p'=\omega\times p+v.$9 is

$i$0

Unlike the Moore–Penrose allocation, this distribution is inertia-weighted. The framework assumes rigid maintained grasps, positive-definite task-space inertia, full-rank grasp matrices, and nondegenerate rigidity configurations; friction cones, unilateral contact, actuator limits, and contact failure require additional modeling.

5. Perception, localization, registration, and learned manipulation

Rigid-object inference can be based on range geometry, image structure, or learned action-conditioned prediction.

The Egoistic MDS-based rigid-body localization method estimates the relative translation and rotation of an unknown target body using cross-body distances and known sensor geometry on an observing body (Führling et al., 20 Jan 2025). The target may have different dimensions, shape, and number of landmarks. With conformation matrix $i$1, rigid transformations are represented as

$i$2

The method forms an approximate complete Euclidean distance matrix by estimating the unknown target block through a Nyström approximation: $i$3 Double centering,

$i$4

produces a centered Gram matrix, whose dominant positive eigenspace yields reconstructed coordinates. Procrustes alignment with the known observing-body geometry resolves the arbitrary MDS frame. The approach requires indexed sensor correspondences, sufficiently informative three-dimensional landmark configurations, and generally favors an observing body with at least as many informative sensors as the target.

Learnable edge kernels provide a different route to rigid or affine registration in multimodal medical images (Siyal et al., 1 Dec 2025). The method initializes convolutional filters from a Laplacian edge kernel, perturbs them with multiplicative Gaussian noise, and then learns them through the registration objective. Its so-called rigid model actually predicts a 12-degree-of-freedom affine transformation: three translations, three rotations, three scale parameters, and three shear parameters. The affine map is

$i$5

Four variants compare fixed Laplacian filtering, separate learnable edge modules, early edge extraction, and multiscale edge processing. The multiscale variant obtains the best reported WM-GM Dice on the institutional dataset: $i$6 with skull intact and $i$7 after skull removal. These results are methodological evidence for edge-aware multimodal registration, not direct RealCOD-Rigid benchmark results.

Visual rigidity perception illustrates that physical rigidity and perceived rigidity are not equivalent (Maruya et al., 2023). For rigidly linked rings, observers may report rigid rotation at slow speeds but wobbling at moderate and high speeds. Motion-energy mechanisms primarily produce contour-normal vectors and can favor a wobbling interpretation, whereas feature tracking and MT-like pattern integration can recover motion more consistent with rotation. The integrated model combines motion-energy and feature-tracking fields with speed-dependent weights and shape priors concerning symmetry and feature strength. Its reported fit is

$i$8

The result concerns perceptual inference rather than mechanical rigidity.

For manipulation through deformable linear objects, DeRi-Bot uses an Action Prediction Network and a Configuration Prediction Network (Wang et al., 2023). Given depth, segmentation, and target maps, the APN proposes an end-effector action. The CPN predicts the next rigid-object configuration for candidate actions. Additional actions are sampled from

$i$9

The candidate minimizing predicted target distance is executed. DeRi-IGP extends this approach with an Iterative Grasp-Pull primitive: $j$0 where $j$1 is a rope-grasping point and $j$2 is a pull destination (Wang et al., 2023). Its Grasping Point Network, Pulling Point Network, Delta Position Network, and local subgoal planners support repeated regrasping, long-range acquisition, asynchronous multi-agent operation, and human–robot collaboration. The reported random-position simulation success rates are 86% for the geometric-intersection planner and 88% for the linear-greedy planner; removing action sampling reduces success to 48%.

These methods share an important design pattern: rigid-object state is inferred or controlled through observations and constraints without necessarily reconstructing a complete physical model. Their limitations include sensor correspondence, visibility, shape ambiguity, object orientation, unmodeled rope dynamics, quasi-static assumptions, and limited real-world evaluation.

6. Immersed simulation, fluid interaction, and the limits of ideal rigidity

Diffuse-interface methods represent rigid bodies using a scalar field $j$3, with the interface defined approximately by $j$4 (Wallis et al., 2021). Prescribed motion is modeled by

$j$5

Local interface seeding estimates normals from $j$6, probes nearby fluid cells, interpolates a valid state, and reflects the normal velocity relative to the rigid-body velocity. MUSCL-BVD-THINC reconstruction, HLLC or HLLD fluxes, adaptive mesh refinement, and local stencil operations permit simulations involving fluids, elastoplastic solids, reactive mixtures, and complex immersed geometries.

The method treats dynamic rigid-body motion as prescribed. It does not include translational and rotational equations driven by integrated fluid or solid forces. Consequently, a fully coupled RealCOD-Rigid implementation would need force and torque accumulation and rigid-body momentum and energy updates. Without this extension, a moving body can inject or remove energy from the fluid without an explicit body energy balance.

A surface-based rigid-body fluid-interaction framework incorporates ambient-fluid effects without volumetric computational fluid dynamics (Padilla et al., 20 Jan 2026). The method computes a generalized added-mass matrix

$j$7

where $j$8 is obtained from six potential-flow boundary-value problems. The total matrix may contain translation–rotation coupling. Runtime forces are calculated from surface slip velocity, a flow-separation rule, and dynamic pressure: $j$9 The same pressure integration produces drag, lift, Magnus forces, and aerodynamic torques. A separation angle $s_i-s_j$0 determines which surface regions contribute effective slip. The method reports falling-plate fluttering, tumbling, chaotic and steady modes, Magnus drift, golf-ball flight, propeller ascent, buoyant rotation, and American-football precession. Its assumptions include constant density, incompressibility, a quiescent or gently varying ambient fluid, and a confined wake; dominant turbulence, wake memory, and arbitrary external flow fields are outside its stated scope.

The strongest qualification to ideal rigid simulation is supplied by adversarial rigid-body attacks (Ramakrishnan et al., 8 Feb 2025). The method constructs objects with the same external collision geometry, total mass, center of mass, and inertia tensor as a reference object while modifying internal density, stiffness, and topology. The rigid simulator therefore receives effectively identical inputs, but deformable simulations can produce different trajectories. The optimization maximizes a terminal trajectory discrepancy subject to material bounds, occupancy constraints, and mass-moment matching. The examples include a ball that changes from scoring through a hoop to striking the rim, a star with an approximately $s_i-s_j$1 trajectory difference, a bunny whose altered rotation changes bin entry, cubes whose stack falls instead of remaining upright, and a bat that changes the ball trajectory by approximately $s_i-s_j$2.

This result establishes a model-identifiability limitation: collision geometry and low-order rigid-body mass properties do not determine compliance, stress propagation, internal topology, or deformation-dependent contact response. It suggests that RealCOD-Rigid evaluations should distinguish:

• Rigid-model equivalence: identical collision geometry and rigid-body parameters.
• Physical equivalence: similar deformable trajectories, contact events, and energy exchange.
• Perceptual rigidity: an object appearing rigid under observation.
• Numerical equivalence: agreement across discretizations, contact models, and integration schemes.

A robust RealCOD-Rigid system therefore requires more than rigid geometry and $s_i-s_j$3 kinematics. It must combine constraint and rank analysis, physically meaningful dynamics, sensing and registration, contact and fluid interaction, and explicit testing against compliance, degeneracy, uncertainty, and model discrepancy.