--Soft continuum robots achieve complex deformation through elastic equilibrium, making their reachable motions governed jointly by structural design and actuation-induced mechanics. This work develops a general formulation that integrates equilibrium computation with kinematic performances by evaluating Riemannian Jacobian spectra on the equilibrium manifold shaped by internal/external loading. The resulting framework yields a global performance functional that directly links structural parameters, actuation inputs, and the induced configuration space geometry. We apply this general framework to magnetic actuation. Analytical characterization is obtained under weak uniform fields, revealing optimal placement and orientation of the embedded magnet with invariant scale properties. T o address nonlinear deformation and spatially varying fields, a two-level optimization algorithm is developed that alternates between energy based equilibrium search and gradient based structural updates. Simulations and physical experiments across uniform field, dipole field, and multi-magnet configurations demonstrate consistent structural tendencies: aligned moments favor distal or mid-distal solutions through constructive torque amplification, whereas opposing moments compress optimal designs toward proximal regions due to intrinsic cancellation zones. OFT continuum robots have gained growing attention for tasks involving compliant interaction, dexterous access, and safe manipulation in complex or confined environments. Their ability to realize smooth, multi-segment deformation without rigid joints supports applications in minimally invasive navigation, inspection, and human-centered tasks.

Tendon-driven continuum robots offer intrinsically safe and contact-rich interactions owing to their kinematic redundancy and structural compliance. However, their perception often depends on external sensors, which increase hardware complexity and limit scalability. This work introduces a unified multi-dynamics modeling framework for tendon-driven continuum robotic systems, exemplified by a spiral-inspired robot named Spirob. The framework integrates motor electrical dynamics, motor-winch dynamics, and continuum robot dynamics into a coherent system model. Within this framework, motor signals such as current and angular displacement are modeled to expose the electromechanical signatures of external interactions, enabling perception grounded in intrinsic dynamics. The model captures and validates key physical behaviors of the real system, including actuation hysteresis and self-contact at motion limits. Building on this foundation, the framework is applied to environmental interaction: first for passive contact detection, verified experimentally against simulation data; then for active contact sensing, where control and perception strategies from simulation are successfully applied to the real robot; and finally for object size estimation, where a policy learned in simulation is directly deployed on hardware. The results demonstrate that the proposed framework provides a physically grounded way to interpret interaction signatures from intrinsic motor signals in tendon-driven continuum robots.

Accurate shape reconstruction is essential for precise control and reliable operation of soft robots. Compared to sensor-based approaches, vision-based methods offer advantages in cost, simplicity, and ease of deployment. However, existing vision-based methods often rely on complex camera setups, specific backgrounds, or large-scale training datasets, limiting their practicality in real-world scenarios. In this work, we propose a vision-based, markerless, and training-free framework for soft robot shape reconstruction that directly leverages the robot's natural surface appearance. These surface features act as implicit visual markers, enabling a hierarchical matching strategy that decouples local partition alignment from global kinematic optimization. Requiring only an initial 3D reconstruction and kinematic alignment, our method achieves real-time shape tracking across diverse environments while maintaining robustness to occlusions and variations in camera viewpoints. Experimental validation on a continuum soft robot demonstrates an average tip error of 2.6% during real-time operation, as well as stable performance in practical closed-loop control tasks. These results highlight the potential of the proposed approach for reliable, low-cost deployment in dynamic real-world settings.

Stochastic state estimation methods for continuum robots (CRs) often struggle to balance accuracy and computational efficiency. While several recent works have explored sliding-window formulations for CRs, these methods are limited to simplified, discrete-time approximations and do not provide stochastic representations. In contrast, current stochastic filter methods must run at the speed of measurements, limiting their full potential. Recent works in continuous-time estimation techniques for CRs show a principled approach to addressing this runtime constraint, but are currently restricted to offline operation. In this work, we present a sliding-window filter (SWF) for continuous-time state estimation of CRs that improves upon the accuracy of a filter approach while enabling continuous-time methods to operate online, all while running at faster-than-real-time speeds. This represents the first stochastic SWF specifically designed for CRs, providing a promising direction for future research in this area.

Abstract-- T endon-actuated concentric tube mechanisms combine the advantages of tendon-driven continuum robots and concentric tube robots while addressing their respective limitations. They overcome the restricted degrees of freedom often seen in tendon-driven designs, and mitigate issues such as snapping instability associated with concentric tube robots. However, a complete and general mechanical model for these systems remains an open problem. The model allows each tube to twist and elongate while enforcing a shared centerline for bending. We validate the proposed framework through experiments with two-tube and three-tube assemblies under various tendon routing configurations, achieving tip prediction errors < 4% of the robot's total length. We further demonstrate the model's generality by applying it to existing robots in the field, where maximum tip deviations remain around 5% of the total length. This model provides a foundation for accurate shape estimation and control of advanced tendon-actuated concentric tube robots. Minimally invasive surgical interventions have revolutionized modern medicine by reducing patient trauma, shortening recovery times, and improving procedural outcomes. However, accessing deep-seated anatomical targets, such as the spine, brain, or vasculature, poses significant challenges due to the confined, and deformable nature of biological tissues. While highly accurate in structured environments, traditional rigid-link robotic systems often lack the flexibility and compliance required to safely navigate these constrained anatomical spaces.

Abstract-- This paper presents a learning-based approach for accurately estimating the 3D shape of flexible continuum robots subjected to external loads. The proposed method introduces a spatiotemporal neural network architecture that fuses multi-modal inputs, including current and historical tendon displacement data and RGB images, to generate point clouds representing the robot's deformed configuration. The network integrates a recurrent neural module for temporal feature extraction, an encoding module for spatial feature extraction, and a multi-modal fusion module to combine spatial features extracted from visual data with temporal dependencies from historical actuator inputs. Continuous 3D shape reconstruction is achieved by fitting B ezier curves to the predicted point clouds. Experimental validation demonstrates that our approach achieves high precision, with mean shape estimation errors of 0.08 mm (unloaded) and 0.22 mm (loaded), outperforming state-of-the-art methods in shape sensing for TDCRs.

Robotic systems are increasingly expected to operate in human-centered, unstructured environments where safety, adaptability, and generalization are essential. Vision-Language-Action (VLA) models have been proposed as a language guided generalized control framework for real robots. However, their deployment has been limited to conventional serial link manipulators. Coupled by their rigidity and unpredictability of learning based control, the ability to safely interact with the environment is missing yet critical. In this work, we present the deployment of a VLA model on a soft continuum manipulator to demonstrate autonomous safe human-robot interaction. We present a structured finetuning and deployment pipeline evaluating two state-of-the-art VLA models (OpenVLA-OFT and $π_0$) across representative manipulation tasks, and show while out-of-the-box policies fail due to embodiment mismatch, through targeted finetuning the soft robot performs equally to the rigid counterpart. Our findings highlight the necessity of finetuning for bridging embodiment gaps, and demonstrate that coupling VLA models with soft robots enables safe and flexible embodied AI in human-shared environments.

Abstract-- This paper presents a shape-aware whole-body control framework for tendon-driven continuum robots with direct application to endoluminal surgical navigation. Endo-luminal procedures, such as bronchoscopy, demand precise and safe navigation through tortuous, patient-specific anatomy where conventional tip-only control often leads to wall contact, tissue trauma, or failure to reach distal targets. To address these challenges, our approach combines a physics-informed backbone model with residual learning through an Augmented Neural ODE, enabling accurate shape estimation and efficient Jacobian computation. A task manager further enhances adaptability by allowing real-time adjustment of objectives, such as wall clearance or direct advancement, during tele-operation. Extensive simulation studies demonstrate millimeter-level accuracy across diverse scenarios, including trajectory tracking, dynamic obstacle avoidance, and shape-constrained reaching. Real-robot experiments on a bronchoscopy phantom validate the framework, showing improved lumen-following accuracy, reduced wall contacts, and enhanced adaptability compared to joystick-only navigation and existing baselines. These results highlight the potential of the proposed framework to increase safety, reliability, and operator efficiency in minimally invasive endoluminal surgery, with broader applicability to other confined and safety-critical environments. I. Introduction Continuum robots made from compliant materials can conform to tortuous anatomy while tolerating contact forces [1], [2].

State estimation is one of the fundamental problems in robotics. For soft continuum robots, this task is particularly challenging because their states (poses, strains, internal wrenches, and velocities) are inherently infinite-dimensional functions due to their continuous deformability. Traditional sensing techniques, however, can only provide discrete measurements. Recently, a dynamic state estimation method known as a \textit{boundary observer} was introduced, which uses Cosserat rod theory to recover all infinite-dimensional states by measuring only the tip velocity. In this work, we present a dual design that instead relies on measuring the internal wrench at the robot's base. Despite the duality, this new approach offers a key practical advantage: it requires only a force/torque (FT) sensor embedded at the base and eliminates the need for external motion capture systems. Both observer types are inspired by principles of energy dissipation and can be naturally combined to enhance performance. We conduct a Lyapunov-based analysis to study the convergence rate of these boundary observers and reveal a useful property: as the observer gains increase, the convergence rate initially improves and then degrades. This convex trend enables efficient tuning of the observer gains. We also identify special cases where linear and angular states are fully determined by each other, which further relaxes sensing requirements. Experimental studies using a tendon-driven continuum robot validate the convergence of all observer variants under fast dynamic motions, the existence of optimal gains, robustness against unknown external forces, and the algorithm's real-time computational performance.

Abstract--State estimation techniques for continuum robots (CRs) typically involve using computationally complex dynamic models, simplistic shape approximations, or are limited to quasi-static methods. These limitations can be sensitive to unmodelled disturbances acting on the robot. Inspired by a factor-graph optimization paradigm, this work introduces a continuous-time stochastic state estimation framework for continuum robots. We introduce factors based on continuous-time kinematics that are corrupted by a white noise Gaussian process (GP). By using a simple robot model paired with high-rate sensing, we show adaptability to unmodelled external forces and data dropout. The result contains an estimate of the mean and covariance for the robot's pose, velocity, and strain, each of which can be interpolated continuously in time or space. This same interpolation scheme can be used during estimation, allowing for inclusion of measurements on states that are not explicitly estimated. Our method's inherent sparsity leads to a linear solve complexity with respect to time and interpolation queries in constant time. We demonstrate our method on a CR with gyroscope and pose sensors, highlighting its versatility in real-world systems. Continuum robots (CRs) are jointless, flexible, and easily miniaturizable manipulators capable of bending into contorted spatial shapes. They are often said to be inspired by the animal kingdom, resembling the motion of snakes, elephant trunks, or worms [1]. Their unique properties allow them to navigate in confined and cluttered environments. This makes them particularly suitable for applications such as minimally invasive surgery [2], [3], industrial inspection and repair in hard-to-reach places [4], [5], as well as search and rescue operations in disaster areas [6]. To date, great progress has been made on the modeling of continuum robots [7], using a variety of kinematic, static, and dynamic assumptions and approaches. Such methods aim to accurately predict the robot's shape given its material properties, actuation, and external loads, which is crucial for the aforementioned applications.