Soft and continuum robots have unique advantages and capabilities useful in medical applications and confined environments due to their highly flexible structure. However, their underactuation and theoretically infinite degrees-of-freedom present significant challenges, particularly in control and state estimation. Prior work showed that a 2D discrete rod mechanics model, derived from the continuous partial differential equations of a Kirchhoff rod and expressed in maximal coordinates, can facilitate continuum robot state estimation and feedback-linearization-based control without conversion to a classical minimal robot dynamics form. We here extend that maximal-coordinate modeling approach to 3D, in a symmetric, constrained-Lagrangian form, using quaternions and Baumgarte constraint stabilization. We also formulate a physically intuitive, model-based observer to estimate the full state of a continuum robot by introducing virtual forces and moments. We validate both the open-loop model and the observer through optical tracking in experiments on a large tendon-driven continuum robot prototype, demonstrating the ability to accurately estimate the robot’s dynamic state during actuation.

Soft tissue simulation can play an essential role in the automation of robotic surgery by providing contextual information during surgery and generating datasets for training. Any time tissue deformations are simulated, computational speed, accuracy, and stability are key concerns. State-of-the-art tissue simulation resolves inertial dynamics solutions using position-based computational methods. However, existing methods fail to efficiently resolve steady-state solutions at surgical size scales because of transient inertial dynamics and the small time step required for stability at such size scales. We propose a position-based tissue simulation framework which is based on large-deformation Neo–Hookean elasticity and enables fast resolution to steady-state for efficient simulation. Our method replaces the inertial terms in the model with a virtual viscous damping term. This enables realistic tissue motion while eliminating the transient vibrations that require more computation. It also enables smooth and stable dynamic transitions between disparate static states. Using our method, we develop an interactive simulator capable of stable, real-time tissue manipulation with a deformable concentric tube robot (CTR) model. Stable collision and simulator realism are achieved through the inclusion of local iterations of collision areas and a novel hydrostatic strain energy formulation.

Equation (9) in [1] was calculated from $\mathbf {J}=\mathbf {U}^\top \mathbf {U}$ and is thus inconsistent with (7) in [1], which defines the decomposition as $\mathbf {J}=\mathbf {U}\mathbf {U}^\top$. The correct parameter mapping $\boldsymbol{\pi }=\mathbf {f}(\boldsymbol{\theta })$ consistent with (7) is \begin{equation*} \boldsymbol{\pi }= e^{2\alpha } \begin{bmatrix}1 \\ t_{1} \\ t_{2} \\ t_{3} \\ s_{23}^{2}+t_{2}^{2}+t_{3}^{2}+e^{2d_{2}}+e^{2d_{3}}\\ s_{12}^{2}+s_{13}^{2}+t_{1}^{2}+t_{3}^{2}+e^{2d_{1}}+e^{2d_{3}}\\ s_{12}^{2}+s_{13}^{2}+s_{23}^{2}+t_{1}^{2}+t_{2}^{2}+e^{2d_{1}}+e^{2d_{2}}\\ -s_{13}s_{23}-t_{1}t_{2}-s_{12}e^{d_{2}}\\ -t_{2}t_{3}-s_{23}e^{d_{3}}\\ -t_{1}t_{3}-s_{13}e^{d_{3}} \end{bmatrix} \tag{1} \end{equation*}

In this paper, we introduce the Soft Lamprey-Inspired Dual Environment Robot (SLIDER) and a proper modeling and optimization procedure employed to design the robot. We represent the primary fluid environment actions - inertial effects, vortex forces, and viscous dissipation - using Lighthill's theory for large-amplitude elongated bodies. For structural design parameters such as internal pressure, tail size, and body stiffness, a fast, geometrically and materially nonlinear model is developed and validated. The fluid-structure interaction equations are solved implicitly with an efficient second-order box method. A pneumatic manifold robotic system is employed to actuate SLIDER in a quiescent water tank environment, allowing cross-comparison of computational and experimental results. We find that low-frequency swimming is dominated by resistant environmental forces, whereas higher-frequency swimming is primarily affected by inertial fluid forces. Using our efficient model alongside a genetic algorithm, we co-optimize a swimming control pattern and caudal fin design (subject to SLIDER's climbing morphology) to achieve a tethered swimming speed of 21.7 +/- 0.4 cm/s (0.59 Bl/s). Furthermore, we investigate the optimization procedure for a multimodal robot performing both swimming and climbing tasks.

Rotational degrees of freedom in Extended Position-Based Dynamics (XPBD) require computations on the nonlinear manifold of 3D rotations. We show that Lie theory provides a clean, unified framework for expressing rotations, constraints, interpolation, and differentiation in XPBD, enabling both improved rigid-body constraints and higher-order finite-element Cosserat rods. We derive explicit Lie-theoretic constraint formulations and their gradients for rigid-body simulation, improving the dynamic consistency of constrained rigid-body simulations in XPBD by a factor of over 10^4 compared to the state-of-the-art. Our framework naturally extends to finite-element Cosserat rods by enabling on-manifold interpolation of nodal rotations. Linear finite elements outperform the conventional chain-of-rigid-bodies discretization, while higher-order basis functions provide even smoother solutions and faster convergence. Utility is demonstrated in a variety of examples with large deformations and contact.

This paper presents the design, modeling, and experimental verification of a deflection-based force-sensing probe in a concentric-tube robot for palpation during robotic surgery. The proposed method leverages the inherent elastic compliance of the robot and a quasi-static Cosserat rod model to estimate contact forces from real-time tip deflections measured by a 5-DOF magnetic tracker, eliminating the need for distal force sensors. The system was experimentally validated on the Virtuoso Endoscopy System (Virtuoso Surgical, Nashville, TN) across multiple palpation locations and directions, including experiments on compliant tissue phantoms. The proposed method achieved a mean absolute force error of approximately 0.05 N, evaluated using a leave-one-group-out cross-validation scheme across palpation locations. Simulation studies using a real-time deformable-tissue model further demonstrate the tool's potential for automated tissue stiffness mapping, which is further demonstrated by stiffness mapping a tissue phantom to localize an embedded simulated tumor. These results establish a compact, sensor-efficient, and accurate framework for model-based force sensing in continuum robotic palpation, providing a foundation for future clinical applications in minimally invasive surgery.

Both the feasibility and clinical benefits of endoscopic interventions for colon lesions have been demonstrated by pioneering surgeons. Yet endoscopic approaches have not been widely deployed clinically because they are challenging to learn and perform with conventional endoscopes. Motivated by this, several robotic platforms have been created over the past few years with the goal of enhancing surgeon dexterity. To date, all such robots have used custom-made endoscopes with integrated manipulators. This typically results in robots significantly larger than conventional endoscopes, and the systems proposed would ultimately require hospitals to completely replace their conventional endoscopes with a new robotic solution, leading to undesirable cost and workflow ramifications. It has never before been possible to use conventional endoscopes directly with this type of system, because there were no robotic manipulators small, dexterous, and strong enough to pass through existing endoscope ports and enable the maneuvers needed to perform the surgery. In this paper, we show that the recent invention of steerable sheaths based on the concentric push-pull principle makes it possible to convert conventional endoscopes into multi-arm robotic platforms for colon procedures. We demonstrate our system in endoscopic submucosal dissection (ESD) in ex vivo porcine feasibility experiments. Even users with no prior experience in endoscopic submucosal dissection were able to successfully perform the procedure using our system.

Continuum robots offer unique advantages for applications such as minimally invasive surgery, navigation through confined environments, and safe human-robot interaction. However, while most continuum robot segments are designed to exhibit constant curvature over their length, they passively deform into a non-constant curvature s-shape when holding payloads at the tip, and their dynamic movement is often subject to unwanted vibration of the passive non-constant curvature modes. In this paper, we propose a simple solution to dramatically improve these issues: a continuum robot segment design that utilizes a diagonal backbone and flexible push-pull actuation rods. This simple modification to common continuum-robot construction enables us to eliminate the passive s-shaped mode, creating a bending segment that can handle large loads without significant deformation or vibration while requiring no more actuation force than conventional designs. We show that a modified version of 1-DOF constant-curvature kinematics accurately describes the structure when actuator translations are equal and opposite. We also develop and validate a 2-DOF model that predicts tip position and orientation resulting from more general actuation inputs. The models and increased output stiffness were verified experimentally and the concept was demonstrated on a multi-segment robot following a 3D trajectory with minimal disturbance from added loads.

Cosserat rod models are widely used to simulate, design, and control soft robots. The Cosserat framework accounts for bending, torsion, transverse shear, and elongation of a long, slender structure and correctly handles large rotations and deflections in 3D, while being far less computationally expensive than full 3D elasticity models using finite elements. However, the Cosserat model is not always appropriate for soft robotic structures since it assumes the cross sections never change size or shape. In this letter, we extend the standard Cosserat rod model to include cross-sectional deformation while retaining much of its simplicity. We add to the Cosserat model additional degrees of freedom that parameterize stretch and shear in the cross-sectional plane and their rates of change along the rod length. We then formulate several possible constitutive laws on the state variables (one linear and one non-linear) and compare them to the standard Cosserat energy expressions to gain insight. We further show how fluidic actuation and tendon actuation can be incorporated into the model, and we compare the extended Cosserat models to 3D nonlinear finite-element simulations with good agreement. Finally, we demonstrate use of this model in a robotics context to control the path-following gait of a peristaltic worm-inspired soft robot.

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.

Prior models of continuously flexible robots typically assume uniform stiffness, and in this paper we relax this assumption. Geometrically varying stiffness profiles provide additional design freedom to influence the motions and workspaces of continuum robots. These results are timely, because with recent rapid advancements in multimaterial additive manufacturing techniques, it is now straightforward to create more complex stiffness profiles in robots. The key insight of this paper is to project forces and moments applied to the robot onto its center of stiffness (i.e. the Young’s modulus-weighted center of each cross section). We show how the center of stiffness can be thought of as analogous to a “precurved backbone” in a robot with uniform stiffness. This analogy enables a large body of prior work in Cosserat Rod modeling of such robots to be applied directly to those with stiffness variations. We experimentally validate this approach using multimaterial, soft, tendon-actuated robots. Lastly, to illustrate how these results can be used in practice, we investigate how stiffness variation can improve performance in a neurosurgical task.

Concentric push-pull robots delivered through flexible endoscopes work best if their laser-cut transmission tubes have high axial stiffness, high torsional stiffness, and low bending stiffness. This paper simultaneously addresses all three output stiffness values in the transmission design problem, explicitly considering axial stiffness, whereas prior work on laser-cut tube design has focused on the bending/torsional stiffness ratio. We demonstrate an inherent trade-off present in existing laser-cut patterns: it is difficult to simultaneously achieve high axial stiffness and low bending stiffness because these properties are very tightly correlated. To break this correlation and design all three stiffness independently, we propose a new type of laser material removal pattern that leverages local stiffness asymmetry ( E I x ≠ E I y ) in discrete bending segments separated by segments of solid tube. These discrete asymmetric segments are then rifled down the tube to achieve global stiffness symmetry. We parameterize the design and provide a study of the properties through finite-element analysis. We also consider the effect of interference between the tubes when the discrete segments are not aligned. Results show that our discrete asymmetric segment concept can achieve high axial stiffness and torsional stiffness better than previously suggested laser patterns while maintaining equally low bending stiffness. We also experimentally validated the proposed design's properties and actuation performance with professionally manufactured prototype Nitinol tubes for use in an endoscopic robot system.

Soft tissue simulation can play an essential role in the automation of robotic surgery by providing contextual information during surgery and generating datasets for training. Any time tissue deformations are simulated, computational speed, accuracy, and stability are key concerns. State-of-theart tissue simulation resolves inertial dynamics solutions using position-based computational methods. However, existing methods fail to efficiently resolve steady-state solutions at surgical size scales because of transient inertial dynamics and the small time step required for stability at such size scales. We propose a position-based tissue simulation framework which is based on large-deformation Neo-Hookean elasticity and enables fast resolution to steady-state for efficient simulation. Our method replaces the inertial terms in the model with a virtual viscous damping term. This enables realistic tissue motion while eliminating the transient vibrations that require more computation. It also enables smooth and stable dynamic transitions between disparate static states. We detail the selection of parameters and step sizes for efficient steady-state simulation. We further compare our approach to a state-of-the-art position-based method and show significant improvements in stability and realtime performance at surgical size scales.

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 article, we argue that since shape and force are fundamentally coupled, they should be estimated simultaneously using 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 that 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.

The performance of concentric push-pull robots passing through endoscopes is best if their laser-cut transmission tubes exhibit high axial stiffness, high torsional stiffness, and low bending stiffness. In this paper we simultaneously consider all three output stiffness values in the design problem, explicitly considering axial stiffness, whereas prior work has focused on the bending/torsional stiffness ratio. We show that it is very challenging for existing laser-cut patterns to simultaneously achieve high axial stiffness and low bending stiffness because these stiffnesses are tightly coupled. To break this coupling and balance all three stiffness factors independently, we propose a new laser material removal design approach that leverages local stiffness asymmetry (EI x ≠ EI y ) in discrete bending segments separated by segments of solid tube. These discrete asymmetric segments are then rifled down the tube to achieve global stiffness symmetry. We parameterize the design and provide a study of the properties through finite-element analysis. We also consider the effect of interference between the tubes when the discrete segments are not aligned. Results show that our discrete asymmetric segment concept can achieve high axial stiffness and torsional stiffness better than previously suggested laser patterns while maintaining equally low bending stiffness.

Recent advances on Concentric Tube Robots (CTRs) enable the construction and analysis of concentric combinations of precurved elastic tubes. These robots are very appropriate for performing Minimally Invasive Surgery (MIS) with a reduction in patient recovery time. In this work, we propose a kinetostatic model for CTRs based on the Geometric Variable-Strain (GVS) approach where the tubes' sliding motion, the distributed external forces along the tubes and concentrated external forces at the tip, are included. Our approach allows us to estimate the shape of CTRs and the tip forces using the displacements of the tubes and the insertion and rotation input forces and torques. Moreover, we propose a modification in the model, which eliminates completely the sliding friction among the tubes. This new approach opens a new way to use CTRs in surgical applications without the need of sensors along the tubes, but only actuation measurements. The simulation results demonstrate the effectiveness of the proposed approach.

Steerable needles are capable of accurately targeting difficult-to-reach clinical sites in the body. By bending around sensitive anatomical structures, steerable needles have the potential to reduce the invasiveness of many medical procedures. However, inserting these needles with curved trajectories increases the risk of tissue damage due to perpendicular forces exerted on the surrounding tissue by the needle’s shaft, potentially resulting in lateral shearing through tissue. Such forces can cause significant tissue damage, negatively affecting patient outcomes. In this work, we derive a tissue and needle force model based on a Cosserat string formulation, which describes the normal forces and frictional forces along the shaft as a function of the planned needle path, friction model and parameters, and tip piercing force. We propose this new force model and associated cost function as a safer and more clinically relevant metric than those currently used in motion planning for steerable needles. We fit and validate our model through physical needle robot experiments in a gel phantom. We use this force model to define a bottleneck cost function for motion planning and evaluate it against the commonly used path-length cost function in hundreds of randomly generated three-dimensional (3D) environments. Plans generated with our force-based cost show a 62% reduction in the peak modeled tissue force with only a 0.07% increase in length on average compared to using the path-length cost in planning. Additionally, we demonstrate planning with our force-based cost function in a lung tumor biopsy scenario from a segmented computed tomography (CT) scan. By directly minimizing the modeled needle-to-tissue force, our method may reduce patient risk and improve medical outcomes from steerable needle interventions.

Colorectal cancer is a pervasive disease: an estimated 4.6% of men and 4.2% of women will suffer from it in their lifetime [1]. Precancerous polyps can be small (<5 mm) medium (6-9 mm) or large (>10 mm) [2]. Small polyps are most frequent, but polyps too large for immediate endoscopic removal during screening occur 135,000 times per year in the US alone [1]. There are two primary options for removing these polyps: endoscopic removal or partial colectomy. Endoscopic procedures, such as endoscopic submucosal dissection (ESD), are less invasive and reduce the risk of infection, reoccurence, and other adverse events [3]. Despite this, approximately 50,000 patients each year undergo partial colectomies for polyps which could have been removed endoscopically [4]. A primary obstacle to wider use of endoscopic pro- cedures is how challenging they are for physicans to perform, due to the limited dexterity of existing trans- endoscopic tools [5]. Currently tools come straight out the tip of the colonscope and moving them requires moving the tip of the colonoscope [6]. To enable tools to move independent of the colonoscope, we propose an endoscopically deployable, flexible robotic system, as shown in Fig. 1. This system deploys a flexible robotic arm through each channel of a standard 2-channel colonoscope. Each arm is composed of a setup sheath followed by a steerable sheath, with each sheath built using a concentric push-pull robot (CPPR) [7]. Each arm has a hollow central lumen through which tools (e.g. forceps, electrosurgery probes, etc.) can be passed. This design adds dexterity and provides the physician with two independent manipulators, with the goal of making ESD easier to perform.