Endoscopic Submucosal Dissection (ESD) is an effective minimally invasive approach to removing colon cancer, yet it is underutilized, since it is challenging to learn and perform. To promote the adoption of ESD by making it easier, we propose a system in which two small, flexible robotic manipulators are delivered through a colonoscope. Our system differs from prior robotic systems aimed at this application in that our manipulators are small enough to fit through a clinically used colonoscope. By not re-engineering the colonoscope, we maintain overall system diameter at the current clinical gold standard, and streamline the path to eventual clinical deployment. Our concentric push-pull robot (CPPR) manipulators offer dexterity and simultaneously provide a conduit for grasper or cutting tool deployment. Each manipulator in our system consists of two push-pull tube pairs, and we describe how they are actuated. We describe for the first time our approach to compensating for undesirable CPPR tip motion induced by differences in the tubes' transmission stiffness. We also evaluate the workspace of the manipulators and demonstrate teleoperation in a point-touching experiment. Lastly, we demonstrate the ability of the system to resect tissue via ex vivo animal experiments.

Concentric push-pull robots (CPPR) operate through the mechanical interactions of concentrically nested, laser-cut tubes with offset stiffness centers. The distal tips of the tubes are attached to each other, and relative displacement of the tube bases generates bending in the CPPR. Previous CPPR kinematic models assumed two tubes, planar shapes, no torsion, and no external loads. In this paper, we develop a new, more general CPPR model accounting for any number of tubes, describing their variable-curvature 3D shape when actuated, including the effects of torsion and external loads. To accomplish this, we employ a modified Kirchhoff rod model for each tube (with offset stiffness center) and embed the constraints of concentricity. We use an energy method to determine robot shape as a function of actuation and external loading. We experimentally validate this kinetostatic model on prototype CPPRs with two tubes and three tubes and non-constant laser-cut patterns that create variable curvature and stiffness. Experimental results agree with the model, paving the way for use of this model in design optimization, planning, and control of CPPRs.

Continuum robots navigate narrow, winding passageways while safely and compliantly interacting with their environments. Sensing the robot's shape under these conditions is often done indirectly, using a few coarsely distributed (e.g. strain or position) sensors combined with the robot's mechanics-based model. More recently, given high-fidelity shape data, external interaction loads along the robot have been estimated by solving an inverse problem on the mechanics model of the robot. In this paper, we argue that since shape and force are fundamentally coupled, they should be estimated simultaneously in a statistically principled approach. We accomplish this by applying continuous-time batch estimation directly to the arclength domain. A general continuum robot model serves as a statistical prior which is fused with discrete, noisy measurements taken along the robot's backbone. The result is a continuous posterior containing both shape and load functions of arclength, as well as their uncertainties. We first test the approach with a Cosserat rod, i.e. the underlying modeling framework that is the basis for a variety of continuum robots. We verify our approach numerically using distributed loads with various sensor combinations. Next, we experimentally validate shape and external load errors for highly concentrated force distributions (point loads). Finally, we apply the approach to a tendon-actuated continuum robot demonstrating applicability to more complex actuated robots.

Parallel-continuum robots combine the advantages of both parallel and continuum robotics. They offer a compromise between the inherent compliance and slenderness of continuum robots and the high precision and strength of rigid-link parallel robots. Throughout recent years there has been an increasing research interest in these novel architectures, which form closed kinematic chains that feature flexible, continuous links undergoing elastic deformations. As the number of publications in this emerging research field is steadily increasing, this survey article summarizes and reviews the state of the art in parallel-continuum robots, discussing their design and modeling. A definition and notation for parallel-continuum robots is introduced, allowing for a clear classification. In conclusion, current open research questions and possible applications for such robots are discussed.

Underactuation is a core challenge associated with controlling soft and continuum robots, which possess theoreti-cally infinite degrees of freedom, but few actuators. However,$m$actuators may still be used to control a dynamic soft robot in an m-dimensional output task space. In this paper we develop a task-space control approach for planar continuum robots that is robust to modeling error and requires very little sensor information. The controller is based on a highly underactuated discrete rod mechanics model in maximal coordinates and does not require conversion to a classical robot dynamics model form. This promotes straightforward control design, implementation and efficiency. We perform input-output feedback linearization on this model, apply sliding mode control to increase robustness, and formulate an observer to estimate the full state from sparse output measurements. Simulation results show exact task-space reference tracking behavior can be achieved even in the presence of significant modeling error, inaccurate initial conditions, and output-only sensing.

Continuum robots are not constructed with discrete joints but, instead, change shape and position their tip by flexing along their entire length. Their narrow curvilinear shape makes them well suited to passing through body lumens, natural orifices, or small surgical incisions to perform minimally invasive procedures. Modeling and controlling these robots are, however, substantially more complex than traditional robots comprised of rigid links connected by discrete joints. Furthermore, there are many approaches to achieving robot flexure. Each presents its own design and modeling challenges, and to date, each has been pursued largely independently of the others. This article attempts to provide a unified summary of the state of the art of continuum robot architectures with respect to design for specific clinical applications. It also describes a unifying framework for modeling and controlling these systems while additionally explaining the elements unique to each architecture. The major research accomplishments are described for each topic and directions for the future progress needed to achieve widespread clinical use are identified.

Continuum and soft robots can leverage complex actuator shapes to take onuseful shapes while actuating only a few of their many degrees of freedom. Continuum robotsthat alsogrow increasethe range of potential shapes that can be actuated and enable easier access to constrained environments. Existing models for describing the complex kinematics involved in general actuation of continuum robots rely on simulation or well-behaved stress–strain relationships, but the nonlinear behavior of the thin-walled inflated-beams used in growing robots makes these techniques difficult to apply. Here, we derive kinematic models of single, generally routed tendon paths on a soft pneumatic backbone of inextensible but flexible material from geometric relationships alone. This allows for forward modeling of the resulting shapes with only knowledge of the geometry of the system. We show that this model can accurately predict the shape of the whole robot body and how the model changes with actuation type. We also demonstrate the use of this kinematic model for inverse design, where actuator designs are found based on desired final robot shapes. We deploy these designed actuators on soft pneumatic growing robots to show the benefits of simultaneous growth and shape change.

Concentric push–pull robots (CPPR) combine the simplicity, miniaturization potential, and open lumen of concentric-tube robots with the kinematic advantages and stability of push–pull multibackbone designs. A CPPR segment is made from a pair concentric tubes with notches asymmetrically cut into their sides in opposing directions. The two tubes are attached to one another at their tips, and push–pull translation of the tube bases relative to each other changes the curvature along the length of the combined tube pair. Custom, variable-curvature shapes are possible by varying the notch parameters along the tubes. In this article, we present a planar, variable-curvature mechanics model for the actuated segment shape and a method for designing the notch pattern to achieve a desired planar, variable-curvature shape with maximal stiffness within specified strain limits. Experiments validate accuracy for various shapes, materials, and cross sections, showing that the design method achieves a variety of desired shapes. We also demonstrate a multisegment robot made from multiple tube pairs that can independently rotate and actuate, increasing the robot DOF.

Existing static and kinematic models of concentric tube robots are based on the ordinary differential equations of a static Cosserat rod. In this paper, we provide the first dynamic model for concentric tube continuum robots by adapting the partial differential equations of a dynamic Cosserat rod to describe the coupled inertial dynamics of precurved concentric tubes. This generates an initial-boundary-value problem that can capture robot vibrations over time. We solve this model numerically at high time resolutions using implicit finite differences in time and arc length. This approach is capable of resolving the high-frequency torsional dynamics that occur during unstable "snapping" motions and provides a simulation tool that can track the true robot configuration through such transitions. Further, it can track slower oscillations associated with bending and torsion as a robot interacts with tissue at real-time speeds. Experimental verification of the model shows that this wide range of effects is captured efficiently and accurately.

Mechanics-based models of thin elastic structures are prevalent in robotics research, both in soft/continuum robot modeling, and in robotic manipulation of strings, sutures, needles, and endoscopes. In all these applications, distributed loads along the device's length can affect its shape in space. Estimation of the distributed loading based on observation of the object's shape constitutes a classical mechanics inverse problem that would be useful in many applications, but this problem has received relatively little attention to date. In this paper, we propose methods to estimate distributed loads on an elastic rod using a large-deflection Cosserat-rod model and constrained nonlinear optimization. We perform experiments that illustrate the feasibility of using these methods to locate regions of high contact force along the rod, and to estimate magnitudes of the forces that are applied. Results show that overall force magnitudes and locations can be estimated with average error of 0.29 N (6.7% of average resultant magnitude) and 4 mm (2% of rod length) for complex double-bend shapes, and the shape approximation has near-zero error.

Soft and continuum robots driven by tendons or cables have wide-ranging applications, and many mechanics-based models for their behavior have been proposed. In this paper, we address the unsolved problem of predicting robot deflection and stiffness with respect to environmental loads where the axial displacements of the tendon ends are held constant. We first solve this problem analytically for a tendon-embedded Euler-Bernoulli beam. Nondimensionalized equations and plots describe how tendon stretch and routing path affect the robot's output stiffness at any point. These analytical results enable stiffness analysis of candidate robot designs without extensive computational simulations. Insights gained through this analysis include the ability to increase robot stiffness by using converging tendon paths. Generalizing to large deflections in three dimensions (3-D), we extend a previous nonlinear Cosserat-rod-based model for tendon-driven robots to handle prescribed tendon displacements, tendon stretch, pretension, and slack. We then provide additional dimensionless plots in the actuated case for loads in 3-D. The analytical formulas and numerically computed model are experimentally validated on a prototype robot with good agreement.

Parallel continuum robots (PCRs) combine the compactness, simplicity, and compliance of continuum robots with the precision and strength of rigid-link parallel robots. In this paper, we provide a generalized Cosserat-rod-based kinetostatic model framework that accommodates various joint types and problem formulations (e.g., forward and inverse kinematics under loads, and deflection-based and actuation-based force sensing) useful for simulation and control. Linearization of this general model provides the manipulator Jacobian, end-effector compliance, input stiffness, and wrench reflectivity matrices, which allow us to examine the effect of design parameters on dexterity, force application, and force-sensing ability. Using ellipsoids based on the matrices, we provide a set of design simulations and graphically depict the relationships between pose, actuation, and forces. We further provide a nondimensional analysis of the compliance of PCRs. Finally, we experimentally demonstrate and validate actuation-based force sensing on a prototype six-degree-of-freedom PCR, demonstrating 3-D force sensing with a median magnitude and a directional error of 0.23 N (8% of actual load) and 12°, respectively.

Parallel continuum robots consist of a parallel arrangement of flexible legs and are dexterous, compliant, and easily miniaturized for minimally invasive surgery. By design, parallel continuum robots exhibit large, nonlinear deformations in their legs to achieve multi-DOF end effector articulation, but excess leg bowing can limit their reachable workspace, especially for long slender designs. In this paper, we investigate a parallel continuum robot design with a passive spring backbone carrying disks that constrain the legs at intermediate points. The constraints route the legs in helical paths around the backbone and prevent large divergence of the legs, expanding the reachable workspace for slender form factors while preserving the manipulator's six degrees of freedom. We present a novel forward and inverse kinematics model, based on Cosserat rod theory, that accommodates general leg routing paths and any number of intermediate constraint disks. We also explore manipulator workspace with experiments and simulations, demonstrating that intermediate constraints expand the reachable workspace of slender parallel continuum robots.

The large dynamic deflections of continuum robots, soft robots, and slender elastic objects can be accurately modeled with classical rod theories in nonlinear elasticity. In this paper, we propose a real-time computational approach for solving the partial differential equations of a dynamic Kirchhoff rod. Our approach is based on implicit time discretization of the Kirchhoff equations and subsequent solution of the resulting continuous spatial boundary value problem at each time step. This modular approach can exhibit low numerical damping, handle arbitrarily large time steps, and provide an accurate, high-order representation of the rod shape in steady-state. We experimentally validated the method by capturing footage of a dynamic rod with a high speed camera and comparing this experimental data with simulations using the proposed approach. Soft-real-time performance is achieved, and the relationship between time step and real-time performance is explored in a plot.

Classic theories in nonlinear elasticity have increasingly been used to obtain accurate and efficient models for continuum robots and other elastic structures. Numerically computed solutions of these models typically satisfy the first-order conditions necessary for equilibrium, but do not provide any information about the elastic stability of the solution. The inability to detect or avoid physically unstable model solutions poses a major hindrance to reliable model-based simulation, planning, design, and control. In this paper, we adapt results from optimal control to determine the stability of Kirchhoff rods and Cosserat rods subject to general end constraints, including coupled multirod models which describe parallel continuum robots. We formulate a sufficient condition for the stability of a solution, a numerical test for evaluating this condition, and a heuristic stability metric. We verify that our numerical stability test agrees with the classical results for the buckling of single columns with various end constraints and for multicolumn frames. We then validate our approach experimentally on a six degree-of-freedom parallel continuum robot.

Parallel continuum robots have the potential to provide multi-degree-of-freedom articulation using a structure that is simple, compact, compliant, and highly scalable. These characteristics may be useful in micromanipulation, endoscopic robotic-assisted surgery, and human-robot interaction. Our prior work formulated a kinematic model which treats a parallel continuum robot as a set of multiple Cosserat rods with coupled boundary conditions. In this paper, we detail methods for the efficient numerical solution of this model at rates that enable real-time interactive simulation, motion planning, design optimization, and control. Exploitation of the model structure enables a significant reduction in the number of integrations required to evaluate the boundary value Jacobian matrix used in a shooting method. Our approach is used to teleoperate a prototype robot using real-time inverse kinematics solutions, and simulation tests show that inverse kinematics solutions are consistently computed at rates of several kilohertz using standard desktop computing hardware.

In this paper, we describe the state of the art in continuum robot manipulators and systems intended for application to interventional medicine. Inspired by biological trunks, tentacles, and snakes, continuum robot designs can traverse confined spaces, manipulate objects in complex environments, and conform to curvilinear paths in space. In addition, many designs offer inherent structural compliance and ease of miniaturization. After decades of pioneering research, a host of designs have now been investigated and have demonstrated capabilities beyond the scope of conventional rigid-link robots. Recently, we have seen increasing efforts aimed at leveraging these qualities to improve the frontiers of minimally invasive surgical interventions. Several concepts have now been commercialized, which are inspiring and enabling a current paradigm shift in surgical approaches toward flexible access routes, e.g., through natural orifices such as the nose. In this paper, we provide an overview of the current state of this field from the perspectives of both robotics science and medical applications. We discuss relevant research in design, modeling, control, and sensing for continuum manipulators, and we highlight how this work is being used to build robotic systems for specific surgical procedures. We provide perspective for the future by discussing current limitations, open questions, and challenges.

In this paper, we investigate continuum manipulators that are analogous to conventional rigid-link parallel robot designs. These “parallel continuum manipulators” have the potential to inherit some of the compactness and compliance of continuum robots while retaining some of the precision, stability, and strength of rigid-link parallel robots, yet they represent a relatively unexplored area of the broad manipulator design space. We describe the construction of a prototype manipulator structure with six compliant legs connected in a parallel pattern similar to that of a Stewart-Gough platform. We formulate the static forward and inverse kinematics problems for such manipulators as the solution to multiple Cosserat-rod models with coupled boundary conditions, and we test the accuracy of this approach in a set of experiments, including the prediction of leg buckling. An inverse kinematics simulation of slices through the 6 degree-of-freedom (DOF) workspace illustrates the kinematic mapping, range of motion, and force required for actuation, which sheds light on the potential advantages and tradeoffs that parallel continuum manipulators may bring. Potential applications include miniature wrists and arms for endoscopic medical procedures, and lightweight compliant arms for safe interaction with humans.

In open abdominal image-guided liver surgery, sparse measurements of the organ surface can be taken intraoperatively via a laser-range scanning device or a tracked stylus with relatively little impact on surgical workflow. We propose a novel nonrigid registration method which uses sparse surface data to reconstruct a mapping between the preoperative CT volume and the intraoperative patient space. The mapping is generated using a tissue mechanics model subject to boundary conditions consistent with surgical supportive packing during liver resection therapy. Our approach iteratively chooses parameters which define these boundary conditions such that the deformed tissue model best fits the intraoperative surface data. Using two liver phantoms, we gathered a total of five deformation datasets with conditions comparable to open surgery. The proposed nonrigid method achieved a mean target registration error (TRE) of 3.3 mm for targets dispersed throughout the phantom volume, using a limited region of surface data to drive the nonrigid registration algorithm, while rigid registration resulted in a mean TRE of 9.5 mm. In addition, we studied the effect of surface data extent, the inclusion of subsurface data, the trade-offs of using a nonlinear tissue model, robustness to rigid misalignments, and the feasibility in five clinical datasets.

Steerable needles can potentially increase the accuracy of needle-based diagnosis and therapy delivery, provided they can be adequately controlled based on medical image information. We propose a novel sliding mode control law that can be used to deliver the tip of a flexible asymmetric-tipped needle to a desired point, or to track a desired trajectory within tissue. The proposed control strategy requires no a priori knowledge of model parameters, has bounded input speeds, and requires little computational resources. We show that if the standard nonholonomic model for tip-steered needles holds, then the control law will converge to desired targets in a reachable workspace, within a tolerance that can be defined by the control parameters. Experimental results validate the control law for target points and trajectory following in phantom tissue and ex vivo liver. Experiments with targets that move during insertion illustrate robustness to disturbances caused by tissue deformation.

This paper presents a versatile magnetic resonance imaging (MRI) compatible concentric tube continuum robotic system. The system enables MR image-guided placement of a curved, steerable active cannula. It is suitable for a variety of clinical applications including image-guided neurosurgery and percutaneous interventions, along with procedures that involve accessing a desired image target, through a curved trajectory. This 6 degree-of-freedom (DOF) robotic device is piezoelectrically actuated to provide precision motion with joint-level precision of better than 0.03mm, and is fully MRI-compatible allowing simultaneous robotic motion and imaging with no image quality degradation. The MRI compatibility of the robot has been evaluated under 3 Tesla MRI using standard prostate imaging sequences, with an average signal to noise ratio loss of less than 2% during actuator motion. The accuracy of active cannula control was evaluated in benchtop trials using an external optical tracking system with RMS error in tip placement of 1.00mm. Preliminary phantom trials of three active cannula placements in the MRI scanner showed cannula trajectories that agree with our kinematic model, with a RMS tip placement error of 0.61 - 2.24 mm.

Kinematic models that account for deformation due to applied loads have recently been developed for a variety of continuum robots. In these cases, a set of nonlinear differential equations with boundary conditions must often be solved to obtain the robot shape. Thus, computing manipulator Jacobians and compliance matrices efficiently is not straight forward. In this paper, we propose a method for obtaining an arc length parametrized Jacobian and compliance matrix. Our approach involves obtaining an augmented Jacobian by propagating the necessary partial derivatives through the model equations, resulting in a new set of differential equations. These equations can be solved as an initial value problem, via a single numerical integration. Our method can be generally applied to various continuum robot architectures, regardless of the specific actuation system used. We provide a specific case study using this method to obtain the Jacobian for a concentric-tube robot.

We describe transnasal skull base surgery, including the current clinical procedure and the ways in which a robotic system has the potential to enhance the current standard of care. The available workspace is characterized by segmenting medical images and reconstructing the available 3D geometry. We then describe thin, ¿tentacle-like¿ robotic tools with shafts constructed from concentric tube robots, and an actuation unit designed to robotically control them in a teleoperated setting. Lastly, we discuss the results of a proof-of-concept study in a cadaveric specimen, illustrating the ability of the robot to access clinically relevant skull base targets.

The inherent flexibility of continuum robots allows them to interact with objects in a safe and compliant way. This flexibility also makes it possible to use robot deflection to estimate external forces applied to the robot. This ¿intrinsic force sensing¿ concept is particularly useful for thin continuum robots where application constraints preclude the use of traditional force sensors. This paper describes an Extended Kalman Filter approach to estimate forces applied at the tip of a continuum robot using only uncertain pose measurements and a kinematic-static model of the robot with uncertainty.

Tendons are a widely used actuation strategy for continuum robots that enable forces and moments to be transmitted along the robot from base-mounted actuators. Most prior robots have used tendons routed in straight paths along the robot. However, routing tendons through general curved paths within the robot offers potential advantages in reshaping the workspace and enabling a single section of the robot to achieve a wider variety of desired shapes. In this paper, we provide a new model for the statics and dynamics of robots with general tendon routing paths that is derived by coupling the classical Cosserat-rod and Cosserat-string models. This model also accounts for general external loading conditions and includes traditional axially routed tendons as a special case. The advantage of the usage of this coupled model for straight-tendon robots is that it accounts for the distributed wrenches that tendons apply along the robot. We show that these are necessary to consider when the robot is subjected to out-of-plane external loads. Our experimental results demonstrate that the coupled model matches experimental tip positions with an error of 1.7% of the robot length, in a set of experiments that include both straight and nonstraight routing cases, with both point and distributed external loads.

Shape control of continuum robots requires a means of sensing the the curved shape of the robot. Since continuum robots are deformable, they take on shapes that are general curves in space, which are not fully defined by actuator positions. Vision-based shape-estimation provides a promising avenue for shape-sensing. While this is often facilitated by fiducial markers, sometimes fiducials are not feasible due to either the robot's application or its size. To address this, we present a robust and efficient stereo-vision-based, shape-sensing algorithm for continuum robots that does not rely on fiducials or assume orthogonal camera placement. The algorithm employs self-organizing maps to triangulate three-dimensional backbone curves. Experiments with an object with a known shape demonstrate an average accuracy of 1.53 mm on a 239 mm arc length curve.

Intraoperative surface contour sensing can enable the registration of high-resolution three-dimensional preoperative images for precise guidance of surgical robots. This is particularly useful for guiding steerable needles in soft tissues. In this paper we combine a new minimally invasive surface scanning technique based on conoscopic holography with a steerable active cannula robot. We experimentally demonstrate cannula tip placement to multiple physical points inside phantom tissue, which correspond to points specified in preoperative images - the input an eventual clinical system would obtain from the physician. While the image-guided steerable system we propose is broadly applicable to many kinds of surgery, one particular application of interest is in ablating large liver tumors, where it is beneficial for the ablator to be repositioned to multiple locations without being withdrawn from the organ.

Continuum robots made from telescoping precurved elastic tubes enable base-mounted actuators to specify the curved shapes of robots as thin as standard surgical needles. While free space beam mechanics-based models of the shape of these `active cannulas' exist, current models cannot account for external forces and torques applied to the cannula by the environment. In this paper we apply geometrically exact beam theory to solve the statics problem for concentric-tube continuum robots. This yields the equivalent of forward kinematics for an active cannula with general tube precurvature functions and arbitrarily many tubes, under loading from a general wrench distribution. The model achieves average experimental tip errors of less than 3 mm over the workspace of a prototype active cannula subject to various tip forces.

Continuum robots, which are composed of multiple concentric, precurved elastic tubes, can provide dexterity at diameters equivalent to standard surgical needles. Recent mechanics-based models of these "active cannulas" are able to accurately describe the curve of the robot in free space, given the preformed tube curves and the linear and angular positions of the tube bases. However, in practical applications, where the active cannula must interact with its environment or apply controlled forces, a model that accounts for deformation under external loading is required. In this paper, we apply geometrically exact rod theory to produce a forward kinematic model that accurately describes large deflections due to a general collection of externally applied point and/or distributed wrench loads. This model accommodates arbitrarily many tubes, with each having a general preshaped curve. It also describes the independent torsional deformation of the individual tubes. Experimental results are provided for both point and distributed loads. Average tip error under load was 2.91 mm (1.5%-3% of total robot length), which is similar to the accuracy of existing free-space models.

Active cannulas are a relatively new continuum robot subclass characterized by their use of preshaped tubes that transmit bending moments as they slide within one another and are axially rotated. Previous (experimentally vetted) mechanics-based models of active cannula shape assume piecewise constant precurvature of component tubes, and neglect torsion in curved sections of the device. Recently a general, coordinate-free, energy-based framework for active cannula shape has been formulated that relaxes these requirements and includes all prior models as special cases. However, only the 2-tube, constant-precurvature case has thus far been explored in detail using the framework. In this paper we consider the general case of an arbitrary number of component tubes and precurvatures that vary with arc length, deriving a set of differential equations that capture both bending and torsional effects continuously along the active cannula backbone.We then show how to solve these differential equations numerically to describe active cannula shape.