In this paper, we propose a dynamic model and control framework for tendon-driven continuum robots (TDCRs) with multiple segments and an arbitrary number of tendons per segment. Our approach leverages the Clarke transform, the Euler-Lagrange formalism, and the piecewise constant curvature assumption to formulate a dynamic model on a two-dimensional manifold embedded in the joint space that inherently satisfies tendon constraints. We present linear and constraint-informed controllers that operate directly on this manifold, along with practical methods for preventing negative tendon forces without compromising control fidelity. This opens up new design possibilities for overactuated TDCRs with improved force distribution and stiffness without increasing controller complexity. We validate these approaches in simulation and on a physical prototype with one segment and five tendons, demonstrating accurate dynamic behavior and robust trajectory tracking under real-time conditions.

In contrast to conventional robots, accurately modeling the kinematics and statics of continuum robots is challenging due to partially unknown material properties, parasitic effects, or unknown forces acting on the continuous body. Consequentially, state estimation approaches that utilize additional sensor information to predict the shape of continuum robots have garnered significant interest. This article presents a novel approach to state estimation for systems with multiple coupled continuum robots, which allows estimating the shape and strain variables of multiple continuum robots in an arbitrary coupled topology. Simulations and experiments demonstrate the capabilities and versatility of the proposed method, while achieving accurate and continuous estimates for the state of such systems, resulting in average end-effector errors of 3.3 mm and 5.02$^\circ$depending on the sensor setup. It is further shown, that the approach offers fast computation times of below 10 ms, enabling its utilization in quasi-static real-time scenarios with average update rates of 100–200 Hz. An open-source C++ implementation of the proposed state estimation method is made publicly available to the community.

Concentric tube continuum robots utilize nested tubes, which are subject to a set of inequalities. Current approaches to account for inequalities rely on branching methods such as if-else statements. It can introduce discontinuities, may result in a complicated decision tree, has a high wall-clock time, and cannot be vectorized. This affects the behavior and result of downstream methods in control, learning, workspace estimation, and path planning, among others.In this paper, we investigate a mapping to mitigate branching methods. We derive a lower triangular transformation matrix to disentangle the inequalities and provide proof for the unique existence. It transforms the interdependent inequalities into independent box constraints. Further investigations are made for sampling, control, and workspace estimation. Approaches utilizing the proposed mapping are at least 14 times faster (up to 176 times faster), generate always valid joint configurations, are more interpretable, and are easier to extend.

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.

Tendon driven continuum robots are often considered to navigate through and operate in cluttered environments. While their compliance allows them to conform safely to obstacles, it leads them also to buckle under tendon actuation. In this article, we perform for the first time an extensive elastic stability analysis of these robots for arbitrary planar designs. The buckling phenomena are investigated and analyzed using bifurcation diagrams, complementing the current state of the art and adding new knowledge about robots composed of$n$spacer disks. We show the existence of multiple robot configurations with different shapes, achievable with the same actuation inputs. A global stability criterion is also established, which links the critical tendon force, until which the robot is stable to the design parameters. Finally, the buckling phenomena are used to actively soften the robot for a better compromise between compliance and payload. An open loop control strategy is proposed, which can theoretically decrease the stiffness to zero, while maintaining the same robot shape. Experimentally, the robot is made four times more compliant than it is nominally using tendon actuation only.

Tendon-driven parallel continuum robots (PCR) consist of multiple individual continuous kinematic chains, that are actuated in bending utilizing tendons routed along their backbones. This work derives and proposes a Cosserat rod based kinetostatic modeling framework for such parallel structures that allows for efficiently solving the forward, inverse and velocity kinetostatic problems. Using this model, the kinematic properties such as reachable workspace, singularities, manipulability, and compliance of tendon-driven PCR are studied in detail. Experiments are conducted using a real robotic prototype to validate the derived modeling approach. Overall, a median pose accuracy of 4.9 mm, corresponding to 3.4% of the continuum robots' lengths, and 6.2$^\circ$is achieved. The median of the model's computation time results in 0.51 s on standard computing hardware. Fast computations of below 100 ms can be achieved, if an appropriate initial guess for solving the kinetostatic model is available, making the model suitable for a range of different applications including optimization or control.

Establishing a physics-based model capturing the kinetostatic behavior of concentric tube continuum robots is challenging as elastic interactions between the flexible tubes constituting the robot result in a highly non-linear problem. The Goldstandard physics-based model using the Cosserat theory of elastic rods achieves reasonable approximations with 1.5 - 3 % with respect to the robot's length, if well-calibrated. Learning-based models of concentric tube continuum robots have been shown to outperform the Goldstandard model with approximation errors below 1 %. Yet, the merits of learning-based models remain largely unexplored as no common dataset and benchmark exist. In this paper, we present a dataset captured from a three-tube concentric tube continuum robot for use in learning-based kinematics research. The dataset consists of 100 000 joint configurations and the corresponding four 6 dof sensors in SE(3) measured with an electromagnetic tracking system (github.com/ContinuumRoboticsLab/CRL-Dataset-CTCR-Pose). With our dataset, we empower the continuum robotics and machine learning community to advance the field. We share our insights and lessons learned on joint space representation, shape representation in task space, and sampling strategies. Furthermore, we provide benchmark results for learning the forward kinematics using a simple, shallow feedforward neural network. The benchmark results for the tip error are 0.74 mm w.r.t. position (0.4 % of total robot length) and 6.49° w.r.t. orientation.

Tendon-driven continuum robots show promise for use in surgical applications as they can assume complex configurations to navigate along tortuous paths. However, to achieve these complex robot shapes, multiple segments are required as each robot segment can bend only with a single constant curvature. To actuate these additional robot segments, multiple tendons must typically be added on-board the robot, complicating their integration, robot control, and actuation. This work presents a method of achieving two curvatures in a single tendon-driven continuum robot segment through use of a novel magnetic locking mechanism. Thus, the need for additional robot segments and actuating tendons is eliminated. The resulting two curvatures in a single segment are demonstrated in two and three dimensions. Furthermore, the maximum magnetic field required to actuate the locking mechanism for different robot bending angles is experimentally measured to be 6.1 mT. Additionally, the locking mechanism resists unintentional unlocking unless the robot assumes a 0° bending angle and a magnetic field of 18.1 mT is applied, conditions which are not typically reached during routine use of the system. Finally, addressable actuation of two locking mechanisms is achieved, demonstrating the capability of producing multiple curvatures in a single robot segment.

We introduce a methodology to compute the inverse kinematics for concentric tube continuum robots from a desired shape as input. We demonstrate that it is possible to accurately learn joint parameters using neural networks for a discrete point-wise shape representation with different discretization. In comparison to a vanilla numerical method, the learning-based method is preferred in terms of accuracy in joint space and computation. Representing the shape with up to 20 equidistant points, a shape-to-joint inverse kinematics with errors of 2.22° and 1.45 mm is obtained. Further, we extend the shape-to-joint inverse kinematics to image-to-joint inverse kinematics utilizing multi-view images as shape representation. This image-based method achieves errors of 6.02° and 2.76 mm. Both approaches, i.e., shape-to-joint and image-to-joint, result in higher accuracy compared to the learning-based state-of-the-art approach which only considers the tip pose.

Due to the continuous and flexible nature of continuum robot backbones and the infinite number of parameters required to represent them in configuration space, modeling them accurately and in real-time is challenging. While the constant curvature assumption provides a simple alternative, it is limited in its capabilities as it cannot account for external tip forces. In cases where the backbone deviates from the constant curvature backbone, Euler curves are an interesting alternative for modeling continuum robots. In this paper, we show that a linear approximation of the backbone curvature is sufficiently accurate for estimating the shape of a robot subject to external tip forces. Next, we propose a numerical static model for tendon-driven continuum robots experiencing in-plane external tip forces. In this model, we use Euler arc splines to circumvent the limitations of standard numerical integration schemes required to calculate these curves. The system reduces to solving two nonlinear equations, allowing fast approximation of the backbone shape. The proposed model is validated experimentally on a robot prototype. Average tip error of 3.07% of the robot length is obtained for an average computation time of 0.51 ms.

Recent physics-based models of concentric tube continuum robots are able to describe pose of the tip, given the preformed translation and rotation in joint space of the robot. However, such model-based approaches are associated with high computational load and highly non-linear modeling effort. A data-driven approach for computationally fast estimation of the kinematics without requiring the knowledge and the uncertainties in the physics-based model would be an asset. This paper introduces an approach to solve the forward kinematics as well as the inverse kinematics of concentric tube continuum robots with 6-DOF in three dimensional space SE(3). Two artificial neural networks with ReLU (rectified linear unit) activation functions are designed in order to approximate the respective kinematics. Measured data from a robot prototype are used in order to train, validate, and test the proposed approach. We introduce a representation of the rotatory joints by trigonometric functions that improves the accuracy of the approximation. The results with experimental measurements show higher accuracy for the forward kinematics compared to the state of the art mechanics modeling. The tip error is less then 2.3 mm w.r.t. position (1 % of total robot length) and 1.1° w.r.t. orientation. The single artificial neural network for the inverse kinematics approximation achieves a translation and rotation actuator error of 4.0 mm and 8.3 0, respectively.

Continuum manipulators possess the ability to travel on nonlinear paths and avoid obstacles in confined environments. A variety of designs were proposed for several applications, such as minimally invasive surgery or inspections and maintenance in hazardous spaces. While hyperredundant robots can follow a path on the large scale, path following behavior on small the scale is still a challenge. Only a limited number of paths can be followed by small continuum robots due to design constraints such as fixed curvatures or fixed segment lengths. For most applications, continuum robot parameters have to be optimized and selected according to specific task and design constraints. This is time consuming and limits the utilization of the manipulator to specific application scenarios. In this work, we propose a hybrid continuum robot design to overcome these disadvantages and offer a universal tool which is able to follow any constant curvature path. The design is comprised of a telescopic composition of several tendon driven patterned elastic tubes. Each one can be translated and bent independently. With this design we achieve improved path deviation errors (max 0.6 mm) in comparison to previously proposed continuum robots in simulation. We prove the path following behavior on random paths with a two segment prototype robot with overall outer diameter of 6 mm in an experimental setup with average path deviation errors lower than 5 mm.

Tendon-driven continuum robots possess versatile application capabilities and have a robust design. The actuation of such robots with non-straight tendons that wrap around the backbone, described by a variable function, offers a lot of untapped potentials. While it has been shown that these continuum robots are able to take up complex shapes using only one actuated segment, the merits of non-straight tendon routing have not been quantified in terms of workspace and motion. In this paper, we show that one additional helically routed tendon can greatly benefit the robot's reachable workspace. For instance, the reachable workspace of a one-segment robot with 3 conventional straight tendons increases by 400 % by adding one helically routed tendon. Furthermore, the dexterity of such a continuum robot is improved, i.e. motion sequences to avoid obstacles or to twine an object for grasping. For the first time, the potential of tendon-driven continuum robots with two segments and helically routed tendons is investigated. The general findings on the merits of helical tendon routing are supported with both simulation and experimental results.

Path following and follow-the-leader motion is particularly desirable for minimally-invasive surgery in confined spaces which can only be reached using tortuous paths, e.g. through natural orifices. While path following and followthe- leader motion can be achieved by hyper-redundant snake robots, their size is usually not applicable for medical applications. Continuum robots, such as tendon-driven or concentric tube mechanisms, fulfill the size requirements for minimally invasive surgery, but yet follow-the-leader motion is not inherently provided. In fact, parameters of the manipulator's section curvatures and translation have to be chosen wisely a priori. In this paper, we consider a tendon-driven continuum robot with extensible sections. After reformulating the forward kinematics model, we formulate prerequisites for follow-the-leader motion and present a general approach to determine a sequence of robot configurations to achieve follow-the-leader motion along a given 3D path. We evaluate our approach in a series of simulations with 3D paths composed of constant curvature arcs and general 3D paths described by B-spline curves. Our results show that mean path errors <;0.4mm and mean tip errors <;1.6mm can theoretically be achieved for constant curvature paths and <;2mm and <;3.1mm for general B-spline curves respectively.

Tubular continuum robots are particularly useful for minimally invasive surgery due to their small size and ability to dexterously manipulate in constrained environments. The robot can be used teleoperated or operated (semi-)autonomously. In both cases, safe manipulation has to be ensured considering constraints from the patient's unique anatomy and pathology to prevent injury of nerves, vessels, and soft tissue. While probabilistic path planning methods have been proposed for tubular continuum robots which generate sequences of collision free configurations in obstructed environments, these methods do usually not explicitly consider different trajectory generations. In this paper, we evaluate four trajectory generation strategies in terms of the resulting Cartesian paths and spatial extent of the course of motion. We demonstrate the implication of the different strategies on both collision free path planning and an example medical scenario. In conclusion, consideration of trajectory generation strategies is indispensable, if the distance between intermediate configurations increases - even though those intermediate configurations might be collision free.

Several robotic systems have been proposed for removing blood from the brain in patients who have undergone a hemorrhagic stroke. In this paper we explore the use of imagebased feedback to address tissue deformation when aspirating a hemorrhage in a phantom model. This is the first time intraoperative image feedback has been used with a concentric tube robot in this application. We describe a layer by layer approach to motion planning. Computed tomography (CT) images are collected periodically during hemorrhage removal. After each CT scan, the robot's tip path is re-planned to account for the tissue deformation that has occurred since the previous scan. We compare open loop hemorrhage removal to our sequential imaging-replanning approach, illustrating that the latter has the potential to enhance the safety and efficacy of the procedure.

Tendon-driven continuum robots offer increased dexterity and manipulability in comparison to conventional rigid link serial manipulators. Being able to conform to complex curves in 3D space, continuum robots are in particular useful for applications in restricted and hardly accessible environments. The workspace of a tendon-driven continuum robot depends on the number of sections, as well as the length of each section and its range of bending radii. Common tendon-driven robot designs have fixed section lengths such that deployment along a tortuous paths requires additional linear translation of the whole robot. In this paper, we propose a novel tendon-driven continuum robot with extensible sections. A telescoping backbone allows control of the section length during operation in addition to bending through tendon actuation. Thus, the arc length of a section can vary and the range of bending radii is enlarged. As a result, the novel robot design inherently allows for deployment along tortuous paths in a follow-the-leader fashion. We suggest the use of spacer disks equipped with permanent magnets with alternating pole orientation. The magnetic repulsion forces enable equidistant spacing of the disks at any lengths of a robot section. We prove the concepts of our novel design with experiments using a first prototype.

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.

Concentric tube robots exhibit complex workspaces due to the way their component tubes bend and twist as they interact with one another. This paper explores ways to compute and characterize their workspaces. We use Monte Carlo random samples of the robot's joint space and a discrete volumetric workspace representation, which can describe both reachability and redundancy. Experiments on two physical prototypes are provided to illustrate the proposed approach.

Steerable needles have the potential to improve the effectiveness of needle-based clinical procedures such as biopsy and drug delivery by improving targeting accuracy and reaching previously inaccessible targets that are behind sensitive or impenetrable anatomical regions. We present a new needle steering system capable of automatically reaching targets in 3-D environments while avoiding obstacles and compensating for real-world uncertainties. Given a specification of anatomical obstacles and a clinical target (e.g., from preoperative medical images), our system plans and controls needle motion in a closed-loop fashion under sensory feedback to optimize a clinical metric. We unify planning and control using a new fast algorithm that continuously replans the needle motion. Our rapid replanning approach is enabled by an efficient sampling-based rapidly exploring random tree (RRT) planner that achieves orders-of-magnitude reduction in computation time compared with prior 3-D approaches by incorporating variable curvature kinematics and a novel distance metric for planning. Our system uses an electromagnetic tracking system to sense the state of the needle tip during the procedure. We experimentally evaluate our needle steering system using tissue phantoms and animal tissue ex vivo. We demonstrate that our rapid replanning strategy successfully guides the needle around obstacles to desired 3-D targets with an average error of less than 3 mm.

Concentric tube continuum robots provide an infinite-dimensional design space, consisting of individual tube space curves and other tube parameters. Even when design choices are made to restrict the design space to a small number of discrete parameters, ad hoc selection of parameter values to achieve coverage of a desired volume, in the presence of geometric workspace constraints, is essentially impossible - even for experienced researchers. General design algorithms proposed to date have focused on reaching a discrete set of specific points, and have made non-physical approximations in the robot model (most significantly assuming infinite torsional rigidity), to speed up model computation. In this paper, we extend prior algorithms to use more accurate models and incorporate volume-based objectives. These extensions are illustrated in a case study on the design of a concentric tube robot for endonasal pituitary surgery. We show that volume-based design optimization increases the reachable percentage of the surgical workspace by an average of approximately 50%, in comparison to various sets of manually selected design parameters. We conclude that volume-based objectives should be included in future multi-objective design optimization procedures for concentric tube continuum robots.

The high incidence of intracerebral hemorrhages, together with a 40% mortality rate, provide strong motivation for enhancements in the treatment methods available to physicians. To minimize the disruption to healthy brain tissue associated with gaining access to the surgical site that is imposed by traditional open or endoscopic surgical intervention, we propose a new minimally-invasive, image-guided, robotic approach that provides articulation within the lesion at the tip of a needle. In this paper we present a biocompatible and sterilizable robot, together with an image-guidance approach designed to deliver the tip of the needle accurately to the blood clot and to move it within the clot, to aspirate it. An experimental evaluation demonstrates removal of 92% of the target clot tissue in a proof-of-concept phantom study.

We propose the use of a haptic touchscreen to convey graphical and mathematical concepts through aural and/or vibratory tactile feedback. We hypothesize that an important application of such a display will be in teaching visually impaired students concepts that are traditionally learned almost entirely visually. This paper describes initial feasibility studies using a commercially available haptic touchscreen to display grids, points, lines, and shapes - some of the first visual graphical entities students encounter in K-12 mathematics education, and from which more complex lessons can be constructed. We conducted user studies designed to evaluate perception of these objects through haptic feedback alone, auditory feedback alone, and combinations of the two. Our results indicate that both sensory channels can be valuable in user perception.

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.

Material processing using a laser has become a widely used method for industrial procedures (e.g. laser welding or cutting). Furthermore the medical laser has become an integral part of dermatology, neurosurgery, ENT, esthetic, plastic and general surgery. Recent publications have shown, that the short pulsed CO2laser is suitable to ablate bony and cartilage tissue and proposes fundamentally new operative techniques in medicine. The obtainable precision in cutting (in the hundred micrometers range) with a laser system can only be reached using means of computer and robot assisted surgery. We established the first robot assisted laser bone ablation setup, comprising a prototype CO2laser system and a six degree of freedom robot. The laser beam is guided through a passive articulated mirror arm to the robots end-effector. The end-effector is composed of a two mirror galvanometric scan head, which deflects the pulsed laser beam onto the tissue. In this paper we present an end-effector calibration and registration method to determine the parameters which are critical in obtaining precise and accurate cutting results.

Robot assisted interventions often require coupling and decoupling of the robot to/from a specific tool. By using manual gripper changing systems these operations are facilitated, but the robot has to approach to and move away from the coupling position. Industrial applications are mostly based on movements which are teached-in, since the working environment is perfectly described (i.e. working cell). Especially in robot assisted surgery we are facing non fixed tools to which the robot has to be coupled (e.g. a holding device attached to a mobilised bone) and restricted working areas with special safety requirements. In this paper we present an automatic end-effector registration method and a semiautomatic coupling procedure exemplarily for robot assisted orthognathic surgery. By using means of an optical localisation system and force- /torque sensing, the coupling procedure is controlled by a multi- sensor data fusion approach. The developed methods can be adapted to any robot assisted intervention.