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.

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.

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.

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.

In this paper we introduce a novel point-to-point trajectory planner for serial robotic structures that combines the ability to avoid self-collisions and to respect motion constraints, while complying with the requirement of being C4continuous. The latter property makes our approach also suited for 4th order dynamics flexible joint robots, which gained significant practical relevance recently. In particular, we address the problem of generating a smooth, kinematically nearly time-optimal SE(3) trajectory while simultaneously avoiding potential collisions of the robot end-effector with its base as well as respecting the Cartesian unreachable states induced by the joint limits of the proximal kinematic structure.

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.