Conformational Analysis of Stiff Chiral Polymers with End-Constraints

Optical Fiber-Based Needle Shape Sensing in Real Tissue: Single Core vs. Multicore Approaches

Flexible needle insertion procedures are common for minimally-invasive surgeries for diagnosing and treating prostate cancer. Bevel-tip needles provide physicians the capability to steer the needle during long insertions to avoid vital anatomical structures in the patient and reduce post-operative patient discomfort. To provide needle placement feedback to the physician, sensors are embedded into needles for determining the real-time 3D shape of the needle during operation without needing to visualize the needle intra-operatively. Through expansive research in fiber optics, a plethora of bio-compatible, MRI-compatible, optical shape-sensors have been developed to provide real-time shape feedback, such as single-core and multicore fiber Bragg gratings. In this paper, we directly compare single-core fiber-based and multicore fiber-based needle shape-sensing through identically constructed, four-active area sensorized bevel-tip needles inserted into phantom and ex-vivo tissue on the same experimental platform. In this work, we found that for shape-sensing in phantom tissue, the two needles performed identically with a p -value of 0.164 > 0.05, but in ex-vivo real tissue, the single-core fiber sensorized needle significantly outperformed the multicore fiber configuration with a p -value of 0.0005 < 0.05. This paper also presents the experimental platform and method for directly comparing these optical shape sensors for the needle shape-sensing task, as well as provides direction, insight and required considerations for future work in constructively optimizing sensorized needles.

Lie-Group Theoretic Approach to Shape-Sensing Using FBG-Sensorized Needles Including Double-Layer Tissue and S-Shape Insertions

Flexible bevel-tipped needles are often used for needle insertion in minimally-invasive surgical techniques due to their ability to be steered in cluttered environments. Shapesensing enables physicians to determine the location of needles intra-operatively without requiring radiation of the patient, enabling accurate needle placement. In this paper, we validate a theoretical method for flexible needle shape-sensing that allows for complex curvatures, extending upon a previous sensor-based model. This model combines curvature measurements from fiber Bragg grating (FBG) sensors and the mechanics of an inextensible elastic rod to determine and predict the 3D needle shape during insertion. We evaluate the model's shape sensing capabilities in C- and S-shape insertions in single-layer isotropic tissue, and C-shape insertions in two-layer isotropic tissue. Experiments on a four-active area, FBG-sensorized needle were performed in varying tissue stiffnesses and insertion scenarios under stereo vision to provide the 3D ground truth needle shape. The results validate a viable 3D needle shape-sensing model accounting for complex curvatures in flexible needles with mean needle shape sensing root-mean-square errors of 0.160 ± 0.055 mm over 650 needle insertions.

Toward FBG-Sensorized Needle Shape Prediction in Tissue Insertions

Complex needle shape prediction remains an issue for planning of surgical interventions of flexible needles. In this paper, we validate a theoretical method for flexible needle shape prediction allowing for non-uniform curvatures, extending upon a previous sensor-based model which combines curvature measurements from fiber Bragg grating (FBG) sensors and the mechanics of an inextensible elastic rod to determine and predict the 3D needle shape during insertion. We evaluate the model's effectiveness in single-layer isotropic tissue for shape sensing and shape prediction capabilities. Experiments on a four-active area, FBG-sensorized needle were performed in varying single-layer isotropic tissues under stereo vision to provide 3D ground truth of the needle shape. The results validate a viable 3D needle shape prediction model accounting for non-uniform curvatures in flexible needles with mean needle shape sensing and prediction root-mean-square errors of 0.479 mm and 0.892 mm, respectively.

Toward FBG-Sensorized Needle Shape Detection in Real Tissue Insertions

The determination of flexible needle shape during insertion is critical for planning and validation in minimally invasive surgical percutaneous procedures. In this paper, we validate a needle shape-sensing method using fiber Bragg grating (FBG) sensors over sequential needle insertion lengths in gel phantom and real tissue. Experiments on a four-active area, FBG-sensorized needle were performed in both isotropic simulated tissue and inhomogeneous animal tissue with computed tomography (CT) as the ground truth of the needle shape. The results show that the needle shape obtained from the FBG sensors has an overall consistent accuracy in real tissue in comparison to the phantom gel. The results validate a viable 3D needle shape-sensing model and reconstruction method over various insertion depths in comparison to the needle shapes determined from CT in both gel phantom and real tissue.

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.

Trajectory Generation of FBG-Sensorized Needles for Insertions into Multi-Layer Tissue

Several models incorporate needle shape prediction, however prediction in multi-layer tissue for complex needle shape remains an issue. In this work, we present a method for trajectory generation of flexible needles that allows for complex curvatures, extending upon a previous sensor-based model. This model combines curvature measurements from fiber Bragg grating (FBG) sensors and the mechanics of an inextensible elastic rod for shape-sensing. We evaluate the method's effectiveness in single- and double-layer isotropic tissue prediction. The results illustrate a valid trajectory generation method accounting for complex curvatures in flexible needles.

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.

Kinematics and Manipulability Analysis of a Highly Articulated Soft Robotic Manipulator

Recently, the idea of applying “jamming” of appropriate media has been exploited for a novel continuum robot design. It is completed by applying vacuum in a robot structure filled with granular media. The backbone deformation and motion are achieved by controlling the fluid pressure. A jammable robotic manipulator does not certainly follow constant curvature during bending, that is, gravitational loads cause section sag. The kinematics describes the deformation of continuum manipulators. This formulation is expected to facilitate additional synthesis and analysis on workspace. This paper presents a Jacobian-based approach to obtain the forward kinematics solution. The proposed kinematic formulation in this paper tries to combine the key advantages of the techniques in constant curvature and variable curvature models. Hence, the deformation of any arbitrary bending is modeled. The workspace synthesis is continued by kinematic analysis, and in this regard, the manipulability measure is computed. This is an important improvement when compared with existing work for this kind of manipulators. It shows how manipulability measure can determine the workspace quality, where usually reachability is used for robot’s capabilities representation. As a result, the forward kinematics and manipulability analysis based on a piecewise-constant-curvature approximation are discussed in the simulation. The simulation has been carried out according to the fabricated experimental robot.

Snakes partition their body to traverse large steps stably

Many snakes live in deserts, forests and river valleys and traverse challenging 3-D terrain such as rocks, felled trees and rubble, with obstacles as large as themselves and variable surface properties. By contrast, apart from branch cantilevering, burrowing, swimming and gliding, laboratory studies of snake locomotion have focused on locomotion on simple flat surfaces. Here, to begin to understand snake locomotion in complex 3-D terrain, we studied how the variable kingsnake, a terrestrial generalist, traversed a large step of variable surface friction and step height (up to 30% snout-vent length). The snake traversed by partitioning its body into three sections with distinct functions. Body sections below and above the step oscillated laterally on horizontal surfaces for propulsion, whereas the body section in between cantilevered in a vertical plane to bridge the large height increase. As the animal progressed, these three sections traveled down its body, conforming overall body shape to the step. In addition, the snake adjusted the partitioned gait in response to increase in step height and decrease in surface friction, at the cost of reduced speed. As surface friction decreased, body movement below and above the step changed from a continuous lateral undulation with little slip to an intermittent oscillatory movement with much slip, and initial head lift-off became closer to the step. Given these adjustments, body partitioning allowed the snake to be always stable, even when initially cantilevering but before reaching the surface above. Such a partitioned gait may be generally useful for diverse, complex 3-D terrain.

Inverse kinematic solutions of 6-D.O.F. biopolymer segments

We present two methods to find all the possible conformations of short six degree-of-freedom segments of biopolymers which satisfy end constraints in position and orientation. One of our methods is motivated by inverse kinematic solution techniques which have been developed for “general” 6R serial robotic manipulators. However, conventional robot kinematics methods are not directly applicable to the geometry of polymers, which can be treated as a degenerate case where all the “link lengths” are zero. Here, we propose a method which extends the elimination method of Kohli and Osvatic. This method can be applied directly to the geometry of biopolymers. We also propose a heuristic method based on a Lie-group-theoretic description. In this method, we utilize inverse iterations of the Jacobian matrix to obtain all conformations which satisfy end constraints. This can be easily implemented for both the general 6R manipulator and polymers. Although the extended elimination method is computationally faster than the Jacobian method, in cases where some of the joint angles are 180° (i.e., where the elimination method fails), we combine these two methods effectively to obtain the full set of inverse kinematic solutions. We demonstrate our approach with several numerical examples.

Conformational Modeling of Continuum Structures in Robotics and Structural Biology: A Review

Hyper-redundant (or snakelike) manipulators have many more degrees of freedom than are required to position and orient an object in space. They have been employed in a variety of applications ranging from search-and-rescue to minimally invasive surgical procedures, and recently they even have been proposed as solutions to problems in maintaining civil infrastructure and the repair of satellites. The kinematic and dynamic properties of snakelike robots are captured naturally using a continuum backbone curve equipped with a naturally evolving set of reference frames, stiffness properties, and mass density. When the snakelike robot has a continuum architecture, the backbone curve corresponds with the physical device itself. Interestingly, these same modeling ideas can be used to describe conformational shapes of DNA molecules and filamentous protein structures in solution and in cells. This paper reviews several classes of snakelike robots: (1) hyper-redundant manipulators guided by backbone curves; (2) flexible steerable needles; and (3) concentric tube continuum robots. It is then shown how the same mathematical modeling methods used in these robotics contexts can be used to model molecules such as DNA. All of these problems are treated in the context of a common mathematical framework based on the differential geometry of curves, continuum mechanics, and variational calculus. Both coordinate-dependent Euler-Lagrange formulations and coordinate-free Euler-Poincaré approaches are reviewed.

Inexact trajectory planning and inverse problems in the Hamilton-Pontryagin framework

We study a trajectory-planning problem whose solution path evolves by means of a Lie group action and passes near a designated set of target positions at particular times. This is a higher-order variational problem in optimal control, motivated by potential applications in computational anatomy and quantum control. Reduction by symmetry in such problems naturally summons methods from Lie group theory and Riemannian geometry. A geometrically illuminating form of the Euler-Lagrange equations is obtained from a higher-order Hamilton-Pontryagin variational formulation. In this context, the previously known node equations are recovered with a new interpretation as Legendre-Ostrogradsky momenta possessing certain conservation properties. Three example applications are discussed as well as a numerical integration scheme that follows naturally from the Hamilton-Pontryagin principle and preserves the geometric properties of the continuous-time solution.

Equilibrium Conformations of Concentric-tube Continuum Robots

Robots consisting of several concentric, preshaped, elastic tubes can work dexterously in narrow, constrained, and/or winding spaces, as are commonly found in minimally invasive surgery. Previous models of these "active cannulas" assume piecewise constant precurvature of component tubes and neglect torsion in curved sections of the device. In this paper we develop a new coordinate-free energy formulation that accounts for general preshaping of an arbitrary number of component tubes, and which explicitly includes both bending and torsion throughout the device. We show that previously reported models are special cases of our formulation, and then explore in detail the implications of torsional flexibility for the special case of two tubes. Experiments demonstrate that this framework is more descriptive of physical prototype behavior than previous models; it reduces model prediction error by 82% over the calibrated bending-only model, and 17% over the calibrated transmissional torsion model in a set of experiments.

Group theory and biomolecular conformation: I. Mathematical and computational models

Biological macromolecules, and the complexes that they form, can be described in a variety of ways ranging from quantum mechanical and atomic chemical models, to coarser grained models of secondary structure and domains, to continuum models. At each of these levels, group theory can be used to describe both geometric symmetries and conformational motion. In this survey, a detailed account is provided of how group theory has been applied across computational structural biology to analyze the conformational shape and motion of macromolecules and complexes.

Design and Kinematic Modeling of Constant Curvature Continuum Robots: A Review

Continuum robotics has rapidly become a rich and diverse area of research, with many designs and applications demonstrated. Despite this diversity in form and purpose, there exists remarkable similarity in the fundamental simplified kinematic models that have been applied to continuum robots. However, this can easily be obscured, especially to a newcomer to the field, by the different applications, coordinate frame choices, and analytical formalisms employed. In this paper we review several modeling approaches in a common frame and notational convention, illustrating that for piecewise constant curvature, they produce identical results. This discussion elucidates what has been articulated in different ways by a number of researchers in the past several years, namely that constant-curvature kinematics can be considered as consisting of two separate submappings: one that is general and applies to all continuum robots, and another that is robot-specific. These mappings are then developed both for the single-section and for the multi-section case. Similarly, we discuss the decomposition of differential kinematics (the robot’s Jacobian) into robot-specific and robot-independent portions. The paper concludes with a perspective on several of the themes of current research that are shaping the future of continuum robotics.

Optimal plane beams modelling elastic linear objects

This paper discusses analytical and deterministic models for a plane curve with minimum deformation that may be utilized in planning the motion of elastic linear objects and investigating the inverse kinematics of a hyper-redundant robot. It usually requires intensive computation to determine the configuration of elastic linear objects. In addition, conventional optimization-based numerical techniques that identify the shape of elastic linear objects in equilibrium involve non-deterministic aspects. Several analytical models that produce the configuration of elastic linear objects in an efficient and deterministic manner are presented in this paper. To develop the analytical expressions for elastic linear objects, we consider a cantilever beam where the deflections are determined according to the Euler–Bernoulli beam theory. The deflections of the cantilever beam are determined for prescribed constraints imposed on the deflections at the free end to replicate various elastic linear objects. Deflections of a cantilever beam with roller supports are explored to replicate elastic linear objects in contact with rigid objects. We verify the analytical models by comparing them with exact beam deflections. The analytical model is precisely accurate for beams with small deflections as it is developed on the basis of the Euler–Bernoulli beam theory. Although it is applied to beams undergoing large deflections, it is still reasonably accurate and at least as precise as the conventional pseudo-rigid-body model. The computational demand involved in using the analytical models is negligible. Therefore, efficient motion planning for elastic linear objects can be realized when the proposed analytical models are combined with conventional motion planning algorithms. We also demonstrate that the analytical model solves the inverse kinematics problem in an efficient and robust manner through numerical simulations.