In analogy to flexible quadrilateral bipyramids, also known as Bricard octahedra, we study flexible couplings of two Bennett 4R mechanisms. The resulting bi-Bennett structures can be used as building blocks of flexible tubes with quadrilateral cross-section that possess skew faces. We present three 4-dimensional families of flexibly arranged Bennett tubes and discuss some of their properties as well as their relation to flexible bipyramids and biprisms, respectively. As a by-product, we also obtain a novel class of mobile 6R loops. • In analogy to flexible bipyramids we study flexible couplings of two Bennett 4R loops. • These bi-Bennett can be used as building blocks of flexible tubes with skew faces. • Three 4-dim families of bi-Bennetts are presented and their properties are discussed. • As a by-product, we also obtain a novel class of mobile 6R loops.

We introduce a novel class of polyhedral tori (PQ-toroids) that snap between two stable configurations - a flat state and a deployed one separated by an energy barrier. Being able to create PQ-toroids from any set of given planar bottom and side faces opens the possibility to assemble the bistable blocks into a thick freeform curved shell structure to follow a planar quadrilateral (PQ) net with coplanar adjacent offset directions. A design pipeline is developed and presented for inversely computing PQ-toroid modules using conjugate net decompositions of a given surface. We analyze the snapping behavior and energy barriers through simulation and build physical prototypes to validate the feasibility of the proposed system. This work expands the geometric design space of multistable origami for lightweight modular structures and offers practical applications in architectural and deployable systems.

Avoiding singularities is a crucial task in robotics and path planning. This paper proposes a novel algorithm for detecting the closest singularity to a given pose for nine interpretations of the 3-RPR manipulator. The algorithm utilizes intrinsic metrics based on the framework's total elastic strain energy density, employing the physical concept of Green-Lagrange strain. The constrained optimization problem for detecting the closest singular configuration with respect to these metrics is solved globally using tools from numerical algebraic geometry implemented in the software package Bertini. The effectiveness of the proposed algorithm is demonstrated on a 3-RPR manipulator executing a one-parametric motion. Additionally, the obtained intrinsic singularity distances are compared with extrinsic metrics. Finally, the paper illustrates the advantage of employing a well-defined metric for identifying the closest singularities in comparison with the existing methods in the literature, highlighting its application in design optimization.

Voss nets are surface parametrizations whose parameter lines follow a conjugate network of geodesics. Their discrete counterparts, so called V-hedra, are flexible quadrilateral meshes with planar faces such that opposite angles at every vertex are equal; by replacing this equality condition with the closely related constraint that opposite angles are supplementary, we get so-called anti-V-hedra. In this paper, we study the problem of constructing and manipulating (anti-)V-hedra. First, we present a V-hedra generator that constructs a modifiable (anti-)V-hedron in a geometrically exact way, from a set of simple conditions already proposed by Sauer in 1970; our generator can compute and visualize the flexion of the (anti-)V-hedron in real time. Second, we present an algorithm for the design and interactive exploration of V-hedra using a handle-based deformation approach; this tool is capable of simulating the one-parametric isometric deformation of an imperfect V-hedron via a quad soup approach. Moreover, we evaluate the performance and accuracy of our tools by applying the V-hedra generator to constraints obtained by numerical optimization. In particular, we use example surfaces that originate from one-dimensional families of smooth Voss surfaces – each spanned by two isothermal conjugate nets – for which an explicit parametrization is given. This allows us to compare the isometric deformation of the smooth target surface with the rigid folding of the optimized (imperfect) V-hedron.

Abstract We present an interactive tool compatible with existing software (Rhino/Grasshopper) to design ring structures with a paradoxic mobility, which are self‐collision‐free over the complete motion cycle. Our computational approach allows non‐expert users to create these invertible paradoxic loops with six rotational joints by providing several interactions that facilitate design exploration. In a first step, a rational cubic motion is shaped either by means of a four pose interpolation procedure or a motion evolution algorithm. By using the representation of spatial displacements in terms of dual‐quaternions, the associated motion polynomial of the resulting motion can be factored in several ways, each corresponding to a composition of three rotations. By combining two suitable factorizations, an arrangement of six rotary axes is achieved, which possesses a 1‐parametric mobility. In the next step, these axes are connected by links in a way that the resulting linkage is collision‐free over the complete motion cycle. Based on an algorithmic solution for this problem, collision‐free design spaces of the individual links are generated in a post‐processing step. The functionality of the developed design tool is demonstrated in the context of an architectural and artistic application studied in a master‐level studio course. Two results of the performed design experiments were fabricated by the use of computer‐controlled machines to achieve the necessary accuracy ensuring the mobility of the models.