Clarke Transform -- A Fundamental Tool for Continuum Robotics

This article introduces the Clarke transform and Clarke coordinates, which present a solution to the disengagement of an arbitrary number of coupled displacement actuation of continuum and soft robots. The Clarke transform utilizes the generalized Clarke transformation and its inverse to reduce any number of joint values to a two-dimensional space without sacrificing any significant information. This space is the manifold of the joint space and is described by two orthogonal Clarke coordinates. Application to kinematics, sampling, and control are presented. By deriving the solution to the previously unknown forward robot-dependent mapping for an arbitrary number of joints, the forward and inverse kinematics formulations are branchless, closed-form, and singular-free. Sampling is used as a proxy for gauging the performance implications for various methods and frameworks, leading to a branchless, closed-form, and vectorizable sampling method with a 100 percent success rate and the possibility to shape desired distributions. Due to the utilization of the manifold, the fairly simple constraint-informed, two-dimensional, and linear controller always provides feasible control outputs. On top of that, the relations to improved representations in continuum and soft robotics are established, where the Clarke coordinates are their generalizations. The Clarke transform offers valuable geometric insights and paves the way for developing approaches directly on the two-dimensional manifold within the high-dimensional joint space, ensuring compliance with the constraint. While being an easy-to-construct linear map, the proposed Clarke transform is mathematically consistent, physically meaningful, as well as interpretable and contributes to the unification of frameworks across continuum and soft robots.

Clarke Transform and Clarke Coordinates -- A New Kid on the Block for State Representation of Continuum Robots

For almost all tendon-driven continuum robots, a segment is actuated by three or four tendons constrained by its mechanical design. For both cases, methods to account for the constraints are known. However, for an arbitrary number of tendons, a disentanglement method has yet to be formulated. Motivated by this unsolved general case, we explored state representations and exploited the two-dimensional manifold. We found that the Clarke transformation, a mathematical transformation used in vector control, can be generalized to address this problem. We present the Clarke transform and Clarke coordinates, which can be used to overcome the troublesome interdependency between the tendons, simplify modeling, and unify different improved state representations. Further connection to arc parameters leads to the possibility to derive more generalizable approaches applicable to a wider range of robot types.

A Non-Linear Model Predictive Task-Space Controller Satisfying Shape Constraints for Tendon-Driven Continuum Robots

Tendon-Driven Continuum Robots (TDCRs) have the potential to be used in minimally invasive surgery and industrial inspection, where the robot must enter narrow and confined spaces. We propose a Model Predictive Control (MPC) approach to leverage the non-linear kinematics and redundancy of TDCRs for whole-body collision avoidance, with real-time capabilities for handling inputs at 30Hz. Key to our method's effectiveness is the integration of a nominal Piecewise Constant Curvature (PCC) model for efficient computation of feasible trajectories, with a local feedback controller to handle modeling uncertainty and disturbances. Our experiments in simulation show that our MPC outperforms conventional Jacobian-based controller in position tracking, particularly under disturbances and user-defined shape constraints, while also allowing the incorporation of control limits. We further validate our method on a hardware prototype, showcasing its potential for enhancing the safety of teleoperation tasks.

Towards Contact-Aided Motion Planning for Tendon-Driven Continuum Robots

Tendon-driven continuum robots (TDCRs), with their flexible backbones, offer the advantage of being used for navigating complex, cluttered environments. However, to do so, they typically require multiple segments, often leading to complex actuation and control challenges. To this end, we propose a novel approach to navigate cluttered spaces effectively for a single-segment long TDCR which is the simplest topology from a mechanical point of view. Our key insight is that by leveraging contact with the environment we can achieve multiple curvatures without mechanical alterations to the robot. Specifically, we propose a search-based motion planner for a single-segment TDCR. This planner, guided by a specially designed heuristic, discretizes the configuration space and employs a best-first search. The heuristic, crucial for efficient navigation, provides an effective cost-to-go estimation while respecting the kinematic constraints of the TDCR and environmental interactions. We empirically demonstrate the efficiency of our planner-testing over 525 queries in environments with both convex and non-convex obstacles, our planner is demonstrated to have a success rate of about 80% while baselines were not able to obtain a success rate higher than 30%. The difference is attributed to our novel heuristic which is shown to significantly reduce the required search space.

On the Disentanglement of Tube Inequalities in Concentric Tube Continuum Robots

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.

State Estimation for Continuum Multi-Robot Systems on SE(3)

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 paper 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{\deg} 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.

CIDGIKc: Distance-Geometric Inverse Kinematics for Continuum Robots

The small size, high dexterity, and intrinsic compliance of continuum robots (CRs) make them well suited for constrained environments. Solving the inverse kinematics (IK), that is finding robot joint configurations that satisfy desired position or pose queries, is a fundamental challenge in motion planning, control, and calibration for any robot structure. For CRs, the need to avoid obstacles in tightly confined workspaces greatly complicates the search for feasible IK solutions. Without an accurate initialization or multiple re-starts, existing algorithms often fail to find a solution. We present CIDGIKc (Convex Iteration for Distance-Geometric Inverse Kinematics for Continuum Robots), an algorithm that solves these nonconvex feasibility problems with a sequence of semidefinite programs whose objectives are designed to encourage low-rank minimizers. CIDGIKc is enabled by a novel distance-geometric parameterization of constant curvature segment geometry for CRs with extensible segments. The resulting IK formulation involves only quadratic expressions and can efficiently incorporate a large number of collision avoidance constraints. Our experimental results demonstrate >98% solve success rates within complex, highly cluttered environments which existing algorithms cannot account for.

Open Continuum Robotics -- One Actuation Module to Create them All

Experiments on physical continuum robot are the gold standard for evaluations. Currently, as no commercial continuum robot platform is available, a large variety of early-stage prototypes exists. These prototypes are developed by individual research groups and are often used for a single publication. Thus, a significant amount of time is devoted to creating proprietary hardware and software hindering the development of a common platform, and shifting away scarce time and efforts from the main research challenges. We address this problem by proposing an open-source actuation module, which can be used to build different types of continuum robots. It consists of a high-torque brushless electric motor, a high resolution optical encoder, and a low-gear-ratio transmission. For this letter, we create three different types of continuum robots. In addition, we illustrate, for the first time, that continuum robots built with our actuation module can proprioceptively detect external forces. Consequently, our approach opens untapped and under-investigated research directions related to the dynamics and advanced control of continuum robots, where sensing the generalized flow and effort is mandatory. Besides that, we democratize continuum robots research by providing open-source software and hardware with our initiative called the Open Continuum Robotics Project, to increase the accessibility and reproducibility of advanced methods.

MoSS: Monocular Shape Sensing for Continuum Robots

Continuum robots are promising candidates for interactive tasks in various applications due to their unique shape, compliance, and miniaturization capability. Accurate and real-time shape sensing is essential for such tasks yet remains a challenge. Embedded shape sensing has high hardware complexity and cost, while vision-based methods require stereo setup and struggle to achieve real-time performance. This paper proposes the first eye-to-hand monocular approach to continuum robot shape sensing. Utilizing a deep encoder-decoder network, our method, MoSSNet, eliminates the computation cost of stereo matching and reduces requirements on sensing hardware. In particular, MoSSNet comprises an encoder and three parallel decoders to uncover spatial, length, and contour information from a single RGB image, and then obtains the 3D shape through curve fitting. A two-segment tendon-driven continuum robot is used for data collection and testing, demonstrating accurate (mean shape error of 0.91 mm, or 0.36% of robot length) and real-time (70 fps) shape sensing on real-world data. Additionally, the method is optimized end-to-end and does not require fiducial markers, manual segmentation, or camera calibration. Code and datasets will be made available at https://github.com/ContinuumRoboticsLab/MoSSNet.

Continuum Robot State Estimation Using Gaussian Process Regression on $SE(3)$

Continuum robots have the potential to enable new applications in medicine, inspection, and countless other areas due to their unique shape, compliance, and size. Excellent progess has been made in the mechanical design and dynamic modelling of continuum robots, to the point that there are some canonical designs, although new concepts continue to be explored. In this paper, we turn to the problem of state estimation for continuum robots that can been modelled with the common Cosserat rod model. Sensing for continuum robots might comprise external camera observations, embedded tracking coils or strain gauges. We repurpose a Gaussian process (GP) regression approach to state estimation, initially developed for continuous-time trajectory estimation in $SE(3)$. In our case, the continuous variable is not time but arclength and we show how to estimate the continuous shape (and strain) of the robot (along with associated uncertainties) given discrete, noisy measurements of both pose and strain along the length. We demonstrate our approach quantitatively through simulations as well as through experiments. Our evaluations show that accurate and continuous estimates of a continuum robot's shape can be achieved, resulting in average end-effector errors between the estimated and ground truth shape as low as 3.5mm and 0.016$^\circ$ in simulation or 3.3mm and 0.035$^\circ$ for unloaded configurations and 6.2mm and 0.041$^\circ$ for loaded ones during experiments, when using discrete pose measurements.