Discrete Bending Research Articles

Stable Discrete Bending by Analytic Eigensystem and Adaptive Orthotropic Geometric Stiffness

In this paper, we address two limitations of dihedral angle based discrete bending (DAB) models, i.e. the indefiniteness of their energy Hessian and their vulnerability to geometry degeneracies. To tackle the indefiniteness issue, we present novel analytic expressions for the eigensystem of a DAB energy Hessian. Our expressions reveal that DAB models typically have positive, negative, and zero eigenvalues, with four of each, respectively. By using these expressions, we can efficiently project an indefinite DAB energy Hessian as positive semi-definite analytically. To enhance the stability of DAB models at degenerate geometries, we propose rectifying their indefinite geometric stiffness matrix by using orthotropic geometric stiffness matrices with adaptive parameters calculated from our analytic eigensystem. Among the twelve motion modes of a dihedral element, our resulting Hessian for DAB models retains only the desirable bending modes, compared to the undesirable altitude-changing modes of the exact Hessian with original geometric stiffness, all modes of the Gauss-Newton approximation without geometric stiffness, and no modes of the projected Hessians with inappropriate geometric stiffness. Additionally, we suggest adjusting the compression stiffness according to the Kirchhoff-Love thin plate theory to avoid over-compression. Our method not only ensures the positive semidefiniteness but also avoids instability caused by large bending forces at degenerate geometries. To demonstrate the benefit of our approaches, we show comparisons against existing methods on the simulation of cloth and thin plates in challenging examples.

Pneumatic Bending Soft Actuator Coupling With Revolute Joint With Different Boundary Constraints

This article presents a study on pneumatic bending soft actuator (SA) coupling with revolute joint (RJ) with different boundary constraints (BCs), aimed to discover the bending performance of SA wearing on physical joint of human body as body joint actuators with different installation methods. First, a new pneumatic bidirectional bending SA is designed and fabricated, and its mechanical characteristic is obtained by experimental tests and fitting methods. Then, the mechanical characteristic is applied to the discrete bending model that approximates the continuous bending of the SA, and the kinetostatic analysis based on the principle of virtual work is conducted on the SA-RJ coupling system (SA-RJ CS) with fixing BC (FBC), hinging BC (HBC), and sliding BCs (SBCs). The kinetostatic equations are solved numerically, and thus the bending performance of the SA-RJ CS with different BCs is obtained. Results show the SA-RJ CS with SBCs performs better in bending angle and actuating torque than that with FBC or HBC. Therefore, it is suggested to install the SA of joint actuator on human body by SBCs. This study guides the application of pneumatic bending SA in rehabilitation exoskeleton. At present, we have used the axial SBC in rehabilitation exoskeleton.

Efficient cloth simulation based on the material point method

AbstractIn this article, we propose a novel cloth simulation method based on the material point method (abbr. MPM). The response to the cloth strain is characterized by three separated models: the continuous in‐plane hyperelasticity, the discrete bending elastoplasticity, and the continuous frictional contact elastoplasticity. We make use of the moving least squares MPM (abbr. MLS‐MPM) transfer scheme and a discrete bending model to reduce computation overhead. We further extend the hyperelastic bending model to be elastoplastic, which can be further used for the simulation of a wide range of thin shell materials, such as thin metal sheets. A fast and approximate signed distance field generation method for humanoid animation is also proposed. Experimental results demonstrate that our method can efficiently and robustly simulate high‐resolution challenging scenes with hundreds of thousands of triangles.

Variational convergence of discrete elasticae

AbstractWe discuss a discretization of the Euler–Bernoulli bending energy and of Euler elasticae under clamped boundary conditions by polygonal lines. We show Hausdorff convergence of the set of almost minimizers of the discrete bending energy to the set of smooth Euler elasticae under mesh refinement in (i) the $W^{1,\infty }$-topology for piecewise-linear interpolation; and in (ii) the $W^{2,p}$-topology, $p \in [2,\infty [$, using a suitable smoothing operator to create $W^{2,p}$-curves from polygons.

Utilization of 2N + 1 Units for 2-DOF Discrete Bending Joint to Transmit Perfect Axial Rotation for Laparoscopic Instruments

Laparoscopic instruments, which have traditionally been long and straight, have been equipped with wrist joints for executing complex procedures. Among the many methods for constructing a cable-driven wrist joint for such instruments, the use of pulleyless rolling unit joints has recently been demonstrated to be effective and applied to various surgical robots. Although it is a common practice to alternately stack 2N one-degree-of-freedom (1-DOF) units to form a 2-DOF bending joint, it is cumbersome to determine the optimal stacking sequence, and the determined sequence might not satisfy performance requirements. Therefore, in this paper, we propose a new method to construct a 2-DOF discrete bending joint by using 2N + 1 units. The advantages of the proposed method over the previous method of determining the optimal structure by using 2N units are demonstrated quantitatively through kinematic modeling and calculation. As a result, the behavior of the proposed discrete bending joint becomes increasingly similar to that of a continuum joint as the number of units employed in the construction increases. In addition, it is confirmed that the 2N + 1 model can accurately transfer proximal rotation to the distal tip, similarly to a constant velocity joint. It is expected that the proposed simple method for constructing bending joints will be applied to develop various surgical robot structures in the future.

Design of a Soft Composite Finger with Adjustable Joint Stiffness

This article presents the design of a soft composite finger with tunable joint stiffness. The composite finger, made of two different types of silicone, uses hybrid actuation by combining tendon and pneumatic actuation schemes. Tendons control the finger shape in a prescribed direction to demonstrate discrete bending behavior due to different material moduli, similar to that of a human's finger. The pneumatic actuation changes the stiffness of joints using air chambers. The feasibility of adjustable stiffness joints is proven using both the parallel spring model and experiments that demonstrate the stiffening effect when pressurized. A set of experiments were also conducted on fingers with four different chamber shapes to observe the effect of chamber shape on stiffening and the discrete bending capability of the finger. The stiffness control can tune the structural softness of the finger, which leads to firm grasp during higher acceleration object manipulation.

Complex bend: Strong-focusing magnet for low-emittance synchrotrons

Modern synchrotron light sources are competing intensively to increase x-ray brightness and, eventually, approach the diffraction limit, which sets the final goal of lattice emittance. All recent synchrotron facility upgrades follow the multibend achromat approach by arranging small horizontal beta function and dispersion inside discrete bending dipoles. In this paper we propose a concept of a lattice element that we call ``complex bend,'' which has the potential of becoming a main building block for low emittance lattices. The complex bend is a sequence of dipole poles interleaved with strong alternate focusing so as to maintain the beta function and dispersion oscillating at low values. Comprising the ring lattice with complex bends, instead of regular dipoles, will minimize the $H$-function and reduce horizontal emittance while localizing bending to a small fraction of the storage ring circumference, which should provide more space for insertion devices. In this paper we present the details of the complex bend, considerations regarding the choice of optimal parameters, and thoughts for its practical realization. We focus here on complex bend physics and engineering design, rather than integration of this complex bend into a specific ring lattice.

Harmonious Cable Actuation Mechanism for Soft Robot Joints Using a Pair of Noncircular Pulleys

Various slim robots, such as surgical robots or humanoid robot fingers, are remotely actuated using transmission cables. If pull–pull drive is applied to actuate them using circular driving spools regardless of the shape of the joints, the tension of the driving cable becomes difficult to be maintained properly. Fortunately, it is possible to solve such a cable slack problem by providing an appropriate cable actuation length to the joint structure of the robot from the cable driving unit. Therefore, we propose a harmonious nonlinear cable actuation mechanism suitable for driving noncircular shaped joints. The proposed cable driver can mechanically provide the required cable actuation length to suit the angle change of the target joint using a pair of noncircular pulleys without increasing the number of actuators. In this paper, a design methodology of a noncircular pulley that can be applied to pulleyless rolling joints (PR joints) as well as pulleyless hinge joints is shown. Moreover, a practical cable driver is designed for actuating a hyper-redundant discrete bending joint composed of PR joints, and its effectiveness is verified through experiments. This novel cable actuation mechanism using noncircular pulleys or gears is expected to be applicable to various miniature robots such as surgical robots and animal robots of continuum structure in the future.

Surface reconstruction by means of a flexible sensor array

In recent years, an increasing popularity of flexible sensor systems has been observed, which can largely be attributed to their ability to continuously adapt the shape to deformable bodies with non-planar surfaces without losing functionality. In this paper, we present a self-sensing, ultra-thin and flexible sensor array foil, which allows for determining its actual shape by analyzing signals from 6×6 sensors. Raw sensor signals clearly show the dependence from strength and direction of bending. The local bending vector is determined from signals of sensors oriented in different directions using rules which are already applied for strain gauge rosettes. The algorithm for the surface reconstruction divides the sensor foil into discrete bending segments for which the bending and subsequently new coordinates of segment edges are determined. A sensor diagnostics routine intercepts failure of the complete system due to the failure of single sensors. The functionality of the sensor array and the surface reconstruction is demonstrated for a foil subsequently adapting to a tube in different orientations. The obtained surface reconstruction clearly correlates with the visually observed bending. Such surface reconstruction could provide diagnostic information and potentially be used to detect diseases like pneumothorax. It could not only help to improve medical treatments but also to monitor the structural health of technical constructions.

Euler elastica as a Γ-limit of discrete bending energies of one-dimensional chains of atoms

In the 1920s, Hencky proposed a discrete elastica model describing a chain of identical rigid bars connected by torsional springs. Hencky observed that this discrete elastica model converges to Euler’s elastica as the number of bars increases while their lengths decrease, and Hencky’s bar-chain model has been used primarily as an approximation of Euler’s elastica. A Hencky-type bar-chain model can also be incorporated into a Frenkel–Kontorova-type discrete atomistic model, where the joints and bars represent the atoms and interatomic bonds, respectively, while the entire chain of atoms interacts with either a substrate or other chains. The energy of a continuum system corresponding to this Frenkel–Kontorova-type model can then be recovered by taking an appropriate discrete-to-continuum limit. Developing a correct limiting procedure for the discrete elastica establishes the bending component of this continuum energy. In this paper we use Γ-convergence to rigorously show that as the bar length in the discrete elastica model we consider goes to 0, the bending energies of the chain Γ-converge to the continuum bending energy associated with Euler’s elastica.

Design of a discrete bending joint using multiple unit PREF joints for isotropic 2-DOF motion

Miniature steerable robots are required for various medical applications. Although a typical continuum joint mechanism has a great advantage of miniaturization, it is difficult to have a small bending radius of curvature. With a discrete joint, it is easy to secure definite bending with strength; however, a coarsely structured joint cannot provide a stable distal rolling motion to the end-effector. This paper proposes a method to construct a 2-DOF discrete bending joint using multiple pulleyless rolling joints with an example of a 4-segmental joint. The effects of the stacking sequence on its performance are analyzed. Then, three evaluation criteria are established, and the best stacking sequences are determined. The proposed design method is valid for various numbers of unit joints, and it can be easily applied to the structural design of soft robots resembling snakes or elephant trunks.

Roll levelling semi-analytical model for process optimization

Roll levelling is a primary manufacturing process used to remove residual stresses and imperfections of metal strips in order to make them suitable for subsequent forming operations. In the last years the importance of this process has been evidenced with the apparition of Ultra High Strength Steels with strength > 900 MPa. The optimal setting of the machine as well as a robust machine design has become critical for the correct processing of these materials. Finite Element Method (FEM) analysis is the widely used technique for both aspects. However, in this case, the FEM simulation times are above the admissible ones in both machine development and process optimization. In the present work, a semi-analytical model based on a discrete bending theory is presented. This model is able to calculate the critical levelling parameters i.e. force, plastification rate, residual stresses in a few seconds. First the semi-analytical model is presented. Next, some experimental industrial cases are analyzed by both the semi-analytical model and the conventional FEM model. Finally, results and computation times of both methods are compared.

Evaluation of Curve Fitting Techniques in Deriving p–y Curves for Laterally Loaded Piles

The p–y method is one of the most popular methods for the analysis and design of laterally loaded piles. The mathematical relationship it provides between the bending moment, which can be easily measured at strain gauges along the pile, and the soil resistance and lateral pile displacement, facilitates the construction of p–y curves. Numerical techniques are required to fit smooth continuous curves to the discrete bending moment data in order to improve the accuracy of subsequent differentiation and integration operations. Due to the lack of guidance on the optimum positioning of strain gauges and the reliability and accuracy of curve fitting methods, a unifying study, inclusive of small (0.61 m) and large (3.8 and 7.5 m) diameter piles in clay, was carried out using 18 strain gauge layouts and cubic spline, cubic to quintic B-spline and 3rd to 10th degree global polynomial techniques. Bending moment data was obtained using 3D finite element analysis. Through a comprehensive evaluation, the cubic and cubic B-spline methods were found to be consistently accurate in deriving p–y curves for both the small and large diameter piles.

Discrete bending forces and their Jacobians

Computation of bending forces on triangle meshes is required for numerous simulation and geometry processing applications. In particular it is a key component in cloth simulation. A common quantity in many bending models is the hinge angle between two adjacent triangles. This angle is straightforward to compute, and its gradient with respect to vertex positions (required for the forces) is easily found in the literature. However, the Hessian of the bend angle, which is required to compute the associated force Jacobians is not documented in the literature. Force Jacobians are required for efficient numerics (e.g., implicit time stepping, Newton-based energy minimization) and are thus highly desirable. Readily available computations of the force Jacobian, such as those produced by symbolic algebra systems, or by autodifferentiation codes, are expensive to compute and therefore less useful. We present compact, easily reproducible, closed form expressions for the Hessian of the bend angle. Compared to automatic differentiation, we measure up to 7× speedup for the evaluation of the bending forces and their Jacobians.

Tunable stimulus-responsive friction mechanisms of polyelectrolyte films and tube forests

The pH-responsive frictional behavior of layer-by-layer assembled poly(allylamine hydrochloride) and poly(acrylic acid) multilayers is quantified in different geometric forms of a continuous planar film and anisotropic tube forests. A mechanistic change from surface adhesion dominated frictional behavior to visco/poroelasticity-governed shear occurs for the planar film upon pH-stimulus. This pH-dependent friction can be further controlled by the discrete anisotropic geometry of the tube forest, which introduces additional friction due to asymmetric deformation of the discrete bending of the tubes during lateral motion. This study provides important insights into the design of polyelectrolyte-based coatings with a wide range of controllable surface frictional properties, tuned via interactions between the inherent stimulus-responsive material behavior and the microgeometry of the anisotropic tube forest.

Some Results on Minimum Discrete Bending Energy Path in Simple Polygon

Some Results on Minimum Discrete Bending Energy Path in Simple Polygon

First-order phase transition of fixed connectivity surfaces.

We report numerical evidence of the discontinuous transition of a tethered membrane model which is defined within a framework of the membrane elasticity of Helfrich. Two kinds of phantom tethered membrane models are studied via the canonical Monte Carlo simulation on triangulated fixed connectivity surfaces of spherical topology. One surface model is defined by the Gaussian term and the bending energy term, and the other, which is tensionless, is defined by the bending energy term and a hard wall potential. The bending energy is defined by using the normal vector at each vertex. Both models undergo the first-order phase transition characterized by a gap of the bending energy. The phase structure of the models depends on the choice of discrete bending energy.

Time evolution of galactic WARPS

We have followed the time evolution of a warped galactic disk that is initially tilted with respect to an oblate halo potential. If the disk has a discrete normal mode of bending, the disk settles toward the mode shape, from the center outwards: the settled inner warp has a straight line of nodes, while the outer warp eventually twists into a leading spiral. Even if the disk as a whole has no discrete bending mode, the inner part still settles into a coherent warp without spirality. The timescale for settling may be estimated from the group velocity of bending waves; propagation times in the very outer disk exceed a Hubble time. We have run models with parameters corresponding to the observed systems NGC 4013, NGC 4565, NGC 2903, NGC 3198, and NGC 2841; just as with the discrete normal modes, comparing our time-evolving models to observed warps allows us to place constraints on the core radius and on the oblateness of the dark halo.

A new method for derivation of locking‐free plate bending finite elements via mixed/hybrid formulation

AbstractThe shear‐locking phenomenon in discrete bending analysis of Mindlin/Reissner plates is investigated. Mixed/hybrid variational principles are introduced which, unlike the rigorous displacement model, allow systematic derivation of locking‐free finite elements. This is achieved by satisfaction of an auxiliary condition, having the clear physical interpretation of shear‐force elimination on account of equilibrium. An example, using competitive techniques, demonstrates the applicability of the idea.