arXiv subjects

Bernd Schulze

Publications and source records attributed to Bernd Schulze.

At least 19 recordsLinked to original sources

arXiv2026

Formation control from the generic combinatorial viewpoint: edge dynamics and directed sensing

We develop a geometric framework for distance-based formation control that separates the evolution of inter-agent distances from its realization by compatible node motions, reducing the stability problem to the edge space. We show that local exponential convergence of the edge dynamics implies local exponential convergence of the formation, and that stability is certified by spectral properties of a linear edge operator. We introduce a hierarchy of generic spectral properties --- weak admissibility, admissibility, and strong admissibility --- that provide necessary conditions for local exponential stability. Specializing to directed sensing, we obtain a necessary and sufficient spectral condition for local stability at an arbitrary target, together with a quadratic sufficient certificate. These conditions reveal that stability depends jointly on the graph orientation and target geometry, and show that persistence is neither necessary nor sufficient for local convergence. We show that every generically rigid graph admits an admissible orientation and, for acyclic orientations, we give an exact combinatorial characterization of admissibility. Finally, the quadratic certificate leads to a semidefinite program for synthesizing stabilizing edge gains.

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arXiv2026

On weavings, grillages, tensegrities, and frameworks

We investigate the stability of discrete structures comprising of woven elastic beams in a plane, exposing a connection between admissible over / under patterns and the first-order and static rigidity of an associated tensegrity that is related through a natural polarity transformation. The relationship between the Airy and Whiteley stress functions for the tensegrity and weaving structures is explored. Our results lead to an efficient method for finding such an over / under pattern. The method is illustrated through a worked example modelled on the complete bipartite graph $K_{4,4}$.

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arXiv2026

Rigidity of polytopes with edge length and coplanarity constraints

We investigate a novel setting for polytope rigidity, where a flex must preserve edge lengths and the planarity of faces, but is allowed to change the shapes of faces. For instance, the regular cube is flexible in this notion. We present techniques for constructing flexible polytopes and find that flexibility seems to be an exceptional property. Based on this observation, we introduce a notion of generic realizations for polytopes and conjecture that convex polytopes are generically rigid in dimension $d\geq 3$. We prove this conjecture in dimension $d=3$. Motivated by our findings we also pose several questions that are intended to inspire future research into this notion of polytope rigidity.

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arXiv2026

Non-Euclidean Crystallographic Rigidity

This paper establishes combinatorial characterisations of forced-symmetric and forced-periodic rigidity (under a fixed lattice) of bar-joint frameworks in non-Euclidean normed planes. In $\ell_q$-planes for $q\in(1,\infty)\backslash\{2\}$, we prove characterisations for forced-periodic rigidity and forced-reflectionally-symmetric rigidity. We also characterise forced-symmetric rigidity in this space with respect to the orientation-reversing wallpaper group $\mathbb{Z}^2\rtimes\mathcal{C}_s$, otherwise known as $pm$ in crystallography. In the $\ell_1$ and $\ell_\infty$-planes, we provide characterisations for forced-periodic rigidity and forced-$\mathbb{Z}^2\rtimes\mathcal{C}_s$-symmetric rigidity. All of these characterisations are proved by inductive constructions involving Henneberg-type graph operations.

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arXiv2025

Uniquely realizable crystalline structures

We construct infinite periodic versions of the stress matrix and establish sufficient conditions for periodic tensegrity frameworks to be globally rigid in $\mathbb{R}^d$ in the cases when the lattice is either fixed, fully flexible, or flexible with a volume constraint for the fundamental domain. For the fixed and fully flexible lattice variants, we also establish necessary and sufficient conditions for generic infinite periodic bar-joint frameworks to be globally rigid in $\mathbb{R}^d$. These results provide periodic versions of the fundamental results of Connelly, as well as Gortler, Healy and Thurston on the global rigidity of generic finite bar-joint frameworks.

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arXiv2025

Counting for rigidity under projective transformations in the plane

Let $P$ be a set of points and $L$ a set of lines in the (extended) Euclidean plane, and $I \subseteq P\times L$, where $i =(p,l) \in I$ means that point $p$ and line $l$ are incident. The incidences can be interpreted as quadratic constraints on the homogeneous coordinates of the points and lines. We study the space of incidence preserving motions of the given incidence structure by linearizing the system of quadratic equations. The Jacobian of the quadratic system, our projective rigidity matrix, leads to the notion of independence/dependence of incidences. Column dependencies correspond to infinitesimal motions. Row dependencies or self-stresses allow for new interpretations of classical geometric incidence theorems. We show that self-stresses are characterized by a 3-fold balance. As expected, infinitesimal (first order) projective rigidity as well as second order projective rigidity imply projective rigidity but not conversely. Several open problems and possible generalizations are indicated.

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arXiv2025

Projective rigidity of point-line configurations in the plane

In this paper, we establish a general setup for studying incidence-preserving motions of projective geometric configurations of points and lines via a "projective rigidity matrix". The spaces of infinitesimal motions of a point-line configuration and dependencies amongst the point-line incidences can be interpreted as the kernel and co-kernel of this projective rigidity matrix, respectively. We also introduce a symmetry-adapted projective rigidity matrix for analysing symmetric configurations and their symmetry-preserving motions. The symmetry may be a point group or a more general symmetry, such as an autopolarity.

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arXiv2025

Equilibrium stresses in frameworks via symmetric averaging

For a bar-joint framework $(G,p)$, a subgroup $Γ$ of the automorphism group of $G$, and a subgroup of the orthogonal group isomorphic to $Γ$, we introduce a symmetric averaging map which produces a bar-joint framework on $G$ with that symmetry. If the original configuration is ``almost symmetric", then the averaged one will be near the original configuration. With a view on structural engineering applications, we then introduce a hierarchy of definitions of ``localised" and ``non-localised" or ``extensive" self-stresses of frameworks and investigate their behaviour under the symmetric averaging procedure. Finally, we present algorithms for finding non-degenerate symmetric frameworks with many states of self-stress, as well as non-symmetric and symmetric frameworks with extensive self-stresses. The latter uses the symmetric averaging map in combination with symmetric Maxwell-type character counts and a procedure based on the pure condition from algebraic geometry. These algorithms provide new theoretical and computational tools for the design of engineering structures such as gridshell roofs.

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arXiv2025

Orientation-Reversing Crystallographic Rigidity

This paper provides a combinatorial characterisation for generic forced symmetric rigidity of bar-joint frameworks in the Euclidean plane that are symmetric with respect to the orientation-reversing wallpaper group $\mathbb{Z}^2\rtimes\mathcal{C}_s$, also known as $pm$ in crystallography, under a fixed lattice representation. Corresponding results for the wallpaper groups $cm$ and $pg$ follow directly from this. The method used also provides an inductive construction for the corresponding gain graphs, in terms of Henneberg-type graph operations.

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arXiv2024

Klein-Arnold tensegrities

In this paper, we introduce new classes of infinite and combinatorially periodic tensegrities, derived from algebraic multidimensional continued fractions in the sense of F. Klein. We describe the stress coefficients on edges through integer invariants of these continued fractions, as initiated by V.I. Arnold, thereby creating a novel connection between geometric rigidity theory and the geometry of continued fractions. Remarkably, the new classes of tensegrities possess rational self-stress coefficients. To establish the self-stressability of the frameworks, we present a projective version of the classical Maxwell-Cremona lifting principle, a result of independent interest.

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arXiv2024

Generic infinitesimal rigidity for rotational groups in the plane

In this paper we establish combinatorial characterisations of symmetry-generic infinitesimally rigid frameworks in the Euclidean plane for rotational groups of order 4 and 6, and of odd order between 5 and 1000, where a joint may lie at the centre of rotation. This extends the corresponding results for these groups in the free action case obtained by R. Ikeshita and S. Tanigawa in 2015, and our recent results for the reflection group and the rotational groups of order 2 and 3 in the non-free action case. The characterisations are given in terms of sparsity counts on the corresponding group-labelled quotient graphs, and are obtained via symmetry-adapted versions of recursive Henneberg-type graph constructions. For rotational groups of even order at least 8, we show that the sparsity counts alone are not sufficient for symmetry-generic infinitesimal rigidity.

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arXiv2024

Forced Symmetric Formation Control

This work considers the distance constrained formation control problem with an additional constraint requiring that the formation exhibits a specified spatial symmetry. We employ recent results from the theory of symmetry-forced rigidity to construct an appropriate potential function that leads to a gradient dynamical system driving the agents to the desired formation. We show that only $(1+1/|Γ|)n$ edges are sufficient to implement the control strategy when there are $n$ agents and the underlying symmetry group is $Γ$. This number is considerably smaller than what is typically required from classic rigidity-theory based strategies ($2n-3$ edges). We also provide an augmented control strategy that ensures the agents can converge to a formation with respect to an arbitrary centroid. Numerous numerical examples are provided to illustrate the main results.

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arXiv2024

Mobility of geometric constraint systems with extrusion symmetry

If we take a (bar-joint) framework, prepare an identical copy of this framework, translate it by some vector $τ$, and finally join corresponding points of the two copies, then we obtain a framework with `extrusion' symmetry in the direction of $τ$. This process may be repeated $t$ times to obtain a framework whose underlying graph has $\mathbb{Z}_2^t$ as a subgroup of its automorphism group and which has `$t$-fold extrusion' symmetry. We show that while $t$-fold extrusion symmetry is not a point-group symmetry, the rigidity matrix of a framework with $t$-fold extrusion symmetry can still be transformed into a block-decomposed form in the analogous way as for point-group symmetric frameworks. This allows us to use Fowler-Guest-type character counts to analyse the mobility of such frameworks. We show that this entire theory also extends to the more general point-hyperplane frameworks with $t$-fold extrusion symmetry. Moreover, we show that under suitable regularity conditions the infinitesimal flexes we detect with our symmetry-adapted counts extend to finite (continuous) motions. Finally, we establish an algorithm that checks for finite motions via linearly displacing framework points along velocity vectors of infinitesimal motions.

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arXiv2024

Rigidity of symmetric frameworks with non-free group actions on the vertices

For plane frameworks with reflection or rotational symmetries, where the group action is not necessarily free on the vertex set, we introduce a phase-symmetric orbit rigidity matrix for each irreducible representation of the group. We then use these generalised orbit rigidity matrices to provide necessary conditions for infinitesimal rigidity for frameworks that are symmetric with a cyclic group that acts freely or non-freely on the vertices. Moreover, for the reflection, the half-turn, and the three-fold rotational group in the plane, we establish complete combinatorial characterisations of symmetry-generic infinitesimally rigid frameworks. This extends well-known characterisations for these groups to the case when the group action is not necessarily free on the vertices. The presence of vertices that are fixed by non-trivial group elements requires the introduction of generalised versions of group-labelled quotient graphs leads to more refined types of combinatorial sparsity counts for characterising symmetry-generic infinitesimal rigidity.

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arXiv2024

Homology of Moment Frames

Using homological techniques we show that a pin-anchored frame that involves only moments and shears provides a conceptual bridge between the statics of moment frames and the kinematics of pin-jointed trusses. One immediate result is a long exact sequence whose alternating sum of dimensions gives a novel counting rule for self-stresses and mechanisms. This combines the Maxwell-Calladine count for pin-jointed trusses with the circuit rank (first Betti number) associated with self-stresses in moment frames. These relations apply to frames in 2, 3 or any dimensions. This work heralds a shift towards a deeper study of the relationships and dualities that exist between structural equilibria and kinematics.

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arXiv2024

Equivariant Cosheaves and Finite Group Representations in Graphic Statics

This work extends the theory of reciprocal diagrams in graphic statics to frameworks that are invariant under finite group actions by utilizing the homology and representation theory of cellular cosheaves, recent tools from applied algebraic topology. By introducing the structure of an equivariant cellular cosheaf, we prove that pairs of self-stresses and reciprocal diagrams of symmetric frameworks are classified by the irreducible representations of the underlying group. We further derive the symmetry-aligned Euler characteristics of a finite dimensional equivariant chain complex, which for the force cosheaf yields a new formulation of the symmetry-adapted Maxwell counting rule for detecting symmetric self-stresses and kinematic degrees of freedom in frameworks. A freely available program is used to implement the relevant cosheaf homologies and illustrate the theory with examples.

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arXiv2023

Rigidity of symmetric linearly constrained frameworks in the plane

A bar-joint framework $(G,p)$ is the combination of a finite simple graph $G=(V,E)$ and a placement $p:V\rightarrow \mathbb{R}^d$. The framework is rigid if the only edge-length preserving continuous motions of the vertices arise from isometries of the space. Motivated by applications where boundary conditions play a significant role, one may generalise and consider linearly constrained frameworks where some vertices are constrained to move on fixed affine subspaces. Streinu and Theran characterised exactly which linearly constrained frameworks are generically rigid in 2-dimensional space. In this article we extend their characterisation to symmetric frameworks. In particular necessary combinatorial conditions are given for a symmetric linearly constrained framework in the plane to be isostatic (i.e. minimally infinitesimally rigid) under any finite point group symmetry. In the case of rotation symmetry groups whose order is either 2 or odd, these conditions are then shown to be sufficient under suitable genericity assumptions, giving precise combinatorial descriptions of symmetric isostatic graphs in these contexts.

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arXiv2023

Rigidity of symmetric frameworks on the cylinder

A bar-joint framework $(G,p)$ is the combination of a finite simple graph $G=(V,E)$ and a placement $p:V\rightarrow \mathbb{R}^d$. The framework is rigid if the only edge-length preserving continuous deformations of the vertices arise from isometries of the space. This article combines two recent extensions of the generic theory of rigid and flexible graphs by considering symmetric frameworks in $\mathbb{R}^3$ restricted to move on a surface. In particular necessary combinatorial conditions are given for a symmetric framework on the cylinder to be isostatic (i.e. minimally infinitesimally rigid) under any finite point group symmetry. In every case when the symmetry group is cyclic, which we prove restricts the group to being inversion, half-turn or reflection symmetry, these conditions are then shown to be sufficient under suitable genericity assumptions, giving precise combinatorial descriptions of symmetric isostatic graphs in these contexts.

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