Group-theoretic analysis of symmetry-preserving deployable structures and metamaterials.

Many deployable structures in nature, as well as human-made mechanisms, preserve symmetry as their configurations evolve. Examples in nature include blooming flowers, dilation of the iris within the human eye, viral capsid maturation and molecular and bacterial motors. Engineered examples include opening umbrellas, elongating scissor jacks, variable apertures in cameras, expanding Hoberman spheres and some kinds of morphing origami structures. In these cases, the structures either preserve a discrete symmetry group or are described as an evolution from one discrete symmetry group to another of the same type as the structure deploys. Likewise, elastic metamaterials built from lattice structures can also preserve symmetry type while passively deforming and changing lattice parameters. A mathematical formulation of such transitions/deployments is articulated here. It is shown that if [Formula: see text] is Euclidean space, [Formula: see text] is a continuous group of motions of Euclidean space and [Formula: see text] is the type of the discrete subgroup of [Formula: see text] describing the symmetries of the deploying structure, then the symmetry of the evolving structure can be described by time-dependent subgroups of [Formula: see text] of the form [Formula: see text], where [Formula: see text] is a time-dependent affine transformation. Then, instead of considering the whole structure in [Formula: see text], a 'sector' of it that lives in the orbit space [Formula: see text] can be considered at each instant in time, and instead of considering all motions in [Formula: see text], only representatives from right cosets in the space [Formula: see text] need to be considered. This article is part of the theme issue 'Current developments in elastic and acoustic metamaterials science (Part 1)'.

A Mosquito Pick-and-Place System for PfSPZ-based Malaria Vaccine Production.

The treatment of malaria is a global health challenge that stands to benefit from the widespread introduction of a vaccine for the disease. A method has been developed to create a live organism vaccine using the sporozoites (SPZ) of the parasite Plasmodium falciparum (Pf), which are concentrated in the salivary glands of infected mosquitoes. Current manual dissection methods to obtain these PfSPZ are not optimally efficient for large-scale vaccine production. We propose an improved dissection procedure and a mechanical fixture that increases the rate of mosquito dissection and helps to deskill this stage of the production process. We further demonstrate the automation of a key step in this production process, the picking and placing of mosquitoes from a staging apparatus into a dissection assembly. This unit test of a robotic mosquito pick-and-place system is performed using a custom-designed micro-gripper attached to a four degree of freedom (4-DOF) robot under the guidance of a computer vision system. Mosquitoes are autonomously grasped and pulled to a pair of notched dissection blades to remove the head of the mosquito, allowing access to the salivary glands. Placement into these blades is adapted based on output from computer vision to accommodate for the unique anatomy and orientation of each grasped mosquito. In this pilot test of the system on 50 mosquitoes, we demonstrate a 100% grasping accuracy and a 90% accuracy in placing the mosquito with its neck within the blade notches such that the head can be removed. This is a promising result for this difficult and non-standard pick-and-place task. NOTE TO PRACTITIONERS­: Automated processes could help increase malaria vaccine production to global scale. Currently, production requires technicians to manually dissect mosquitoes, a process that is slow, tedious, and requires a lengthy training regimen. This paper presents an an improved manual fixture and procedure that reduces technician training time. Further, an approach to automate this dissection process is proposed and the critical step of robotic manipulation of the mosquito with the aid of computer vision is demonstrated. Our approach may serve as a useful example of system design and integration for practitioners that seek to perform new and challenging pick-and-place tasks with small, non-uniform, and highly deformable objects.

Continuous body 3-D reconstruction of limbless animals.

Limbless animals such as snakes, limbless lizards, worms, eels and lampreys move their slender, long bodies in three dimensions to traverse diverse environments. Accurately quantifying their continuous body's 3-D shape and motion is important for understanding body-environment interactions in complex terrain, but this is difficult to achieve (especially for local orientation and rotation). Here, we describe an interpolation method to quantify continuous body 3-D position and orientation. We simplify the body as an elastic rod and apply a backbone optimization method to interpolate continuous body shape between end constraints imposed by tracked markers. Despite over-simplifying the biomechanics, our method achieves a higher interpolation accuracy (∼50% error) in both 3-D position and orientation compared with the widely used cubic B-spline interpolation method. Beyond snakes traversing large obstacles as demonstrated, our method applies to other long, slender, limbless animals and continuum robots. We provide codes and demo files for easy application of our method.

Rate of Entropy Production in Stochastic Mechanical Systems.

Entropy production in stochastic mechanical systems is examined here with strict bounds on its rate. Stochastic mechanical systems include pure diffusions in Euclidean space or on Lie groups, as well as systems evolving on phase space for which the fluctuation-dissipation theorem applies, i.e., return-to-equilibrium processes. Two separate ways for ensembles of such mechanical systems forced by noise to reach equilibrium are examined here. First, a restorative potential and damping can be applied, leading to a classical return-to-equilibrium process wherein energy taken out by damping can balance the energy going in from the noise. Second, the process evolves on a compact configuration space (such as random walks on spheres, torsion angles in chain molecules, and rotational Brownian motion) lead to long-time solutions that are constant over the configuration space, regardless of whether or not damping and random forcing balance. This is a kind of potential-free equilibrium distribution resulting from topological constraints. Inertial and noninertial (kinematic) systems are considered. These systems can consist of unconstrained particles or more complex systems with constraints, such as rigid-bodies or linkages. These more complicated systems evolve on Lie groups and model phenomena such as rotational Brownian motion and nonholonomic robotic systems. In all cases, it is shown that the rate of entropy production is closely related to the appropriate concept of Fisher information matrix of the probability density defined by the Fokker-Planck equation. Classical results from information theory are then repurposed to provide computable bounds on the rate of entropy production in stochastic mechanical systems.

Black-Scholes Theory and Diffusion Processes on the Cotangent Bundle of the Affine Group.

The Black-Scholes partial differential equation (PDE) from mathematical finance has been analysed extensively and it is well known that the equation can be reduced to a heat equation on Euclidean space by a logarithmic transformation of variables. However, an alternative interpretation is proposed in this paper by reframing the PDE as evolving on a Lie group. This equation can be transformed into a diffusion process and solved using mean and covariance propagation techniques developed previously in the context of solving Fokker-Planck equations on Lie groups. An extension of the Black-Scholes theory with coupled asset dynamics produces a diffusion equation on the affine group, which is not a unimodular group. In this paper, we show that the cotangent bundle of a Lie group endowed with a semidirect product group operation, constructed in this paper for the case of groups with trivial centers, is always unimodular and considering PDEs as diffusion processes on the unimodular cotangent bundle group allows a direct application of previously developed mean and covariance propagation techniques, thereby offering an alternative means of solution of the PDEs. Ultimately these results, provided here in the context of PDEs in mathematical finance may be applied to PDEs arising in a variety of different fields and inform new methods of solution.

Mathematical aspects of molecular replacement. V. Isolating feasible regions in motion spaces.

This paper mathematically characterizes the tiny feasible regions within the vast 6D rotation-translation space in a full molecular replacement (MR) search. The capability to a priori isolate such regions is potentially important for enhancing robustness and efficiency in computational phasing in macromolecular crystallography (MX). The previous four papers in this series have concentrated on the properties of the full configuration space of rigid bodies that move relative to each other with crystallographic symmetry constraints. In particular, it was shown that the configuration space of interest in this problem is the right-coset space Γ\G, where Γ is the space group of the chiral macromolecular crystal and G is the group of rigid-body motions, and that fundamental domains FΓ\G can be realized in many ways that have interesting algebraic and geometric properties. The cost function in MR methods can be viewed as a function on these fundamental domains. This, the fifth and final paper in this series, articulates the constraints that bodies packed with crystallographic symmetry must obey. It is shown that these constraints define a thin feasible set inside a motion space and that they fall into two categories: (i) the bodies must not interpenetrate, thereby excluding so-called `collision zones' from consideration in MR searches; (ii) the bodies must be in contact with a sufficient number of neighbors so as to form a rigid network leading to a physically realizable crystal. In this paper, these constraints are applied using ellipsoidal proxies for proteins to bound the feasible regions. It is shown that the volume of these feasible regions is small relative to the total volume of the motion space, which justifies the use of ellipsoids as proxies for complex proteins in MR searches, and this is demonstrated with P1 (the simplest space group) and with P212121 (the most common space group in MX).

Lateral Oscillation and Body Compliance Help Snakes and Snake Robots Stably Traverse Large, Smooth Obstacles.

Snakes can move through almost any terrain. Similarly, snake robots hold the promise as a versatile platform to traverse complex environments such as earthquake rubble. Unlike snake locomotion on flat surfaces which is inherently stable, when snakes traverse complex terrain by deforming their body out of plane, it becomes challenging to maintain stability. Here, we review our recent progress in understanding how snakes and snake robots traverse large, smooth obstacles such as boulders and felled trees that lack "anchor points" for gripping or bracing. First, we discovered that the generalist variable kingsnake combines lateral oscillation and cantilevering. Regardless of step height and surface friction, the overall gait is preserved. Next, to quantify static stability of the snake, we developed a method to interpolate continuous body in three dimensions (3D) (both position and orientation) between discrete tracked markers. By analyzing the base of support using the interpolated continuous body 3-D kinematics, we discovered that the snake maintained perfect stability during traversal, even on the most challenging low friction, high step. Finally, we applied this gait to a snake robot and systematically tested its performance traversing large steps with variable heights to further understand stability principles. The robot rapidly and stably traversed steps nearly as high as a third of its body length. As step height increased, the robot rolled more frequently to the extent of flipping over, reducing traversal probability. The absence of such failure in the snake with a compliant body inspired us to add body compliance to the robot. With better surface contact, the compliant body robot suffered less roll instability and traversed high steps at higher probability, without sacrificing traversal speed. Our robot traversed large step-like obstacles more rapidly than most previous snake robots, approaching that of the animal. The combination of lateral oscillation and body compliance to form a large, reliable base of support may be useful for snakes and snake robots to traverse diverse 3-D environments with large, smooth obstacles.

A simple insertion technique to reduce the bending of thinbevel-point needles.

Objective: Needle insertion is a common component of most diagnostic and therapeutic interventions. Needles with asymmetrically sharpened points such as the bevel point are ubiquitous. Their insertion path is typically curved due to the rudder effect at the point. However, the common planned path is straight, leading to targeting errors. We present a simple technique that may substantially reduce these errors. The method was inspired by practical experience, conceived mathematically, and refined experimentally. Methods: Targeting errors are reduced by flipping the bevel on the opposite side (rotating the needle 180° about its axis), at a certain depth during insertion. The ratio of the flip depth to the full depth of insertion is defined as the flip depth ratio (FDR). Based on a model, FDR is constant 0.3. Results: Experimentally, the ratio depends on the needle diameter, 0.35 for 20Ga and 0.45 for 18Ga needles. Thinner needles should be flipped a little shallower, but never less than 0.3. Conclusion: Practically, a physician may expect to reduce ∼80% of needle deflection errors by simply flipping the needle. The technique may be used by hand or with guidance devices.

Quantizing Euclidean Motions via Double-Coset Decomposition.

Concepts from mathematical crystallography and group theory are used here to quantize the group of rigid-body motions, resulting in a "motion alphabet" with which robot motion primitives are expressed. From these primitives it is possible to develop a dictionary of physical actions. Equipped with an alphabet of the sort developed here, intelligent actions of robots in the world can be approximated with finite sequences of characters, thereby forming the foundation of a language in which robot motion is articulated. In particular, we use the discrete handedness-preserving symmetries of macromolecular crystals (known in mathematical crystallography as Sohncke space groups) to form a coarse discretization of the space SE(3) of rigid-body motions. This discretization is made finer by subdividing using the concept of double-coset decomposition. More specifically, a very efficient, equivolumetric quantization of spatial motion can be defined using the group-theoretic concept of a double-coset decomposition of the form Γ\SE(3)/Δ, where Γ is a Sohncke space group and Δ is a finite group of rotational symmetries such as those of the icosahedron. The resulting discrete alphabet is based on a very uniform sampling of SE(3) and is a tool for describing the continuous trajectories of robots and humans. An efficient coarse-to-fine search algorithm is presented to round off any motion sampled from the continuous group of motions to the nearest element of our alphabet. It is shown that our alphabet and this efficient rounding algorithm can be used as a geometric data structure to accelerate the performance of other sampling schemes designed for desirable dispersion or discrepancy properties. Moreover, the general "signals to symbols" problem in artificial intelligence is cast in this framework for robots moving continuously in the world.

Sensorless Pose Determination Using Randomized Action Sequences.

This paper is a study of 2D manipulation without sensing and planning, by exploring the effects of unplanned randomized action sequences on 2D object pose uncertainty. Our approach follows the work of Erdmann and Mason's sensorless reorienting of an object into a completely determined pose, regardless of its initial pose. While Erdmann and Mason proposed a method using Newtonian mechanics, this paper shows that under some circumstances, a long enough sequence of random actions will also converge toward a determined final pose of the object. This is verified through several simulation and real robot experiments where randomized action sequences are shown to reduce entropy of the object pose distribution. The effects of varying object shapes, action sequences, and surface friction are also explored.

Phantom with multiple active points for ultrasound calibration.

Accurate tracking and localization of ultrasound (US) images are used in various computer-assisted interventions. US calibration is a preoperative procedure to recover the transformation bridging the tracking sensor and the US image coordinate systems. Although many calibration phantom designs have been proposed, a limitation that hinders the resulted calibration accuracy is US elevational beam thickness. Previous studies have proposed an active-echo (AE)-based calibration concept to overcome this limitation by utilizing dynamic active US feedback from a single PZT element-based phantom, which assists in placing the phantom within the US elevational plane. However, the process of searching elevational midplane is time-consuming and requires dedicated hardware to enable "AE" functionality. Extending this active phantom, we present a US calibration concept and associated mathematical framework enabling fast and accurate US calibration using multiple "active" points. The proposed US calibration can simplify the calibration procedure by minimizing the number of times midplane search is performed and shortening calibration time. This concept is demonstrated with a configuration mechanically tracking a US probe using a robot arm. We validated the concept through simulation and experiment, and achieved submillimeter calibration accuracy. This result indicates that the multiple active-point phantom has potential to provide superior calibration performance for applications requiring high tracking accuracy.

Symmetrical Parameterization of Rigid Body Transformations for Biomolecular Structures.

Assessing preferred relative rigid body position and orientation is important in the description of biomolecular structures (such as proteins) and their interactions. In this article, we extend and apply the "symmetrical parameterization," which we recently introduced in the kinematics community, to address problems in structural biology. We also review parameterization methods that are widely used in structural biology to describe relative rigid body motions (in particular, orientations) as a basis for comparison. The new symmetrical parameterization is useful in describing the relative biomolecular rigid body motions, where the parameters are symmetrical in the sense that the subunits of a complex biomolecular structure are described in the same way for the corresponding motion and its inverse. The properties of this new parameterization, singularity analysis, and inverse kinematics are also investigated in more detail. Finally, parameterization is applied to real biomolecular structures and a potential application to structure modeling of symmetric macromolecules to show the efficacy of the symmetrical parameterization in the field of computational structural biology.

Mathematical aspects of molecular replacement. IV. Measure-theoretic decompositions of motion spaces.

In molecular-replacement (MR) searches, spaces of motions are explored for determining the appropriate placement of rigid-body models of macromolecules in crystallographic asymmetric units. The properties of the space of non-redundant motions in an MR search, called a `motion space', are the subject of this series of papers. This paper, the fourth in the series, builds on the others by showing that when the space group of a macromolecular crystal can be decomposed into a product of two space subgroups that share only the lattice translation group, the decomposition of the group provides different decompositions of the corresponding motion spaces. Then an MR search can be implemented by trading off between regions of the translation and rotation subspaces. The results of this paper constrain the allowable shapes and sizes of these subspaces. Special choices result when the space group is decomposed into a product of a normal Bieberbach subgroup and a symmorphic subgroup (which is a common occurrence in the space groups encountered in protein crystallography). Examples of Sohncke space groups are used to illustrate the general theory in the three-dimensional case (which is the relevant case for MR), but the general theory in this paper applies to any dimension.

Cross-Validation of Data Compatibility Between Small Angle X-ray Scattering and Cryo-Electron Microscopy.

Cryo-electron microscopy (EM) and small angle X-ray scattering (SAXS) are two different data acquisition modalities often used to glean information about the structure of large biomolecular complexes in their native states. A SAXS experiment is generally considered fast and easy but unveils the structure at very low resolution, whereas a cryo-EM experiment needs more extensive preparation and postacquisition computation to yield a three-dimensional (3D) density map at higher resolution. In certain applications, we may need to verify whether the data acquired in the SAXS and cryo-EM experiments correspond to the same structure (e.g., before reconstructing the 3D density map in EM). In this article, a simple and fast method is proposed to verify the compatibility of the SAXS and EM experimental data. The method is based on averaging the two-dimensional correlation of EM images and the Abel transform of the SAXS data. Orientational preferences are known to exist in cryo-EM experiments, and we also consider these effects on our method. The results are verified on simulations of conformational states of large biomolecular complexes.

Computational Analysis of SAXS Data Acquisition.

Small-angle x-ray scattering (SAXS) is an experimental biophysical method used for gaining insight into the structure of large biomolecular complexes. Under appropriate chemical conditions, the information obtained from a SAXS experiment can be equated to the pair distribution function, which is the distribution of distances between every pair of points in the complex. Here we develop a mathematical model to calculate the pair distribution function for a structure of known density, and analyze the computational complexity of these calculations. Efficient recursive computation of this forward model is an important step in solving the inverse problem of recovering the three-dimensional density of biomolecular structures from their pair distribution functions. In particular, we show that integrals of products of three spherical-Bessel functions arise naturally in this context. We then develop an algorithm for the efficient recursive computation of these integrals.

Framed curves and knotted DNA.

The present mini-review covers the local and global geometry of framed curves and the computation of twist and writhe in knotted DNA circles. Classical inequalities relating the total amount of bending of a closed space curve and associated knot parameters are also explained.

Ultrasound probe and needle-guide calibration for robotic ultrasound scanning and needle targeting.

Image-to-robot registration is a typical step for robotic image-guided interventions. If the imaging device uses a portable imaging probe that is held by a robot, this registration is constant and has been commonly named probe calibration. The same applies to probes tracked by a position measurement device. We report a calibration method for 2-D ultrasound probes using robotic manipulation and a planar calibration rig. Moreover, a needle guide that is attached to the probe is also calibrated for ultrasound-guided needle targeting. The method is applied to a transrectal ultrasound (TRUS) probe for robot-assisted prostate biopsy. Validation experiments include TRUS-guided needle targeting accuracy tests. This paper outlines the entire process from the calibration to image-guided targeting. Freehand TRUS-guided prostate biopsy is the primary method of diagnosing prostate cancer, with over 1.2 million procedures performed annually in the U.S. alone. However, freehand biopsy is a highly challenging procedure with subjective quality control. As such, biopsy devices are emerging to assist the physician. Here, we present a method that uses robotic TRUS manipulation. A 2-D TRUS probe is supported by a 4-degree-of-freedom robot. The robot performs ultrasound scanning, enabling 3-D reconstructions. Based on the images, the robot orients a needle guide on target for biopsy. The biopsy is acquired manually through the guide. In vitro tests showed that the 3-D images were geometrically accurate, and an image-based needle targeting accuracy was 1.55 mm. These validate the probe calibration presented and the overall robotic system for needle targeting. Targeting accuracy is sufficient for targeting small, clinically significant prostatic cancer lesions, but actual in vivo targeting will include additional error components that will have to be determined.

Multiscale modeling of double-helical DNA and RNA: a unification through Lie groups.

Several different mechanical models of double-helical nucleic-acid structures that have been presented in the literature are reviewed here together with a new analysis method that provides a reconciliation between these disparate models. In all cases, terminology and basic results from the theory of Lie groups are used to describe rigid-body motions in a coordinate-free way, and when necessary, coordinates are introduced in a way in which simple equations result. We consider double-helical DNAs and RNAs which, in their unstressed referential state, have backbones that are either straight, slightly precurved, or bent by the action of a protein or other bound molecule. At the coarsest level, we consider worm-like chains with anisotropic bending stiffness. Then, we show how bi-rod models converge to this for sufficiently long filament lengths. At a finer level, we examine elastic networks of rigid bases and show how these relate to the coarser models. Finally, we show how results from molecular dynamics simulation at full atomic resolution (which is the finest scale considered here) and AFM experimental measurements (which is at the coarsest scale) relate to these models.

Mathematical aspects of molecular replacement. II. Geometry of motion spaces.

Molecular replacement (MR) is a well established computational method for phasing in macromolecular crystallography. In MR searches, spaces of motions are explored for determining the appropriate placement of rigid models of macromolecules in crystallographic asymmetric units. In the first paper of this series, it was shown that this space of motions, when endowed with an appropriate composition operator, forms an algebraic structure called a quasigroup. In this second paper, the geometric properties of these MR search spaces are explored and analyzed. This analysis includes the local differential geometry, global geometry and symmetry properties of these spaces.

Mathematical aspects of molecular replacement. I. Algebraic properties of motion spaces.

Molecular replacement (MR) is a well established method for phasing of X-ray diffraction patterns for crystals composed of biological macromolecules of known chemical structure but unknown conformation. In MR, the starting point is known structural domains that are presumed to be similar in shape to those in the macromolecular structure which is to be determined. A search is then performed over positions and orientations of the known domains within a model of the crystallographic asymmetric unit so as to best match a computed diffraction pattern with experimental data. Unlike continuous rigid-body motions in Euclidean space and the discrete crystallographic space groups, the set of motions over which molecular replacement searches are performed does not form a group under the operation of composition, which is shown here to lack the associative property. However, the set of rigid-body motions in the asymmetric unit forms another mathematical structure called a quasigroup, which can be identified with right-coset spaces of the full group of rigid-body motions with respect to the chiral space group of the macromolecular crystal. The algebraic properties of this space of motions are articulated here.