A Sliding-Window Filter for Online Continuous-Time Continuum Robot State Estimation

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.

A Stochastic Framework for Continuous-Time State Estimation of Continuum Robots

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.

Estimating Dynamic Soft Continuum Robot States From Boundaries

Accurate state estimation is essential for effective control of robots. For soft robots, this task is particularly challenging because their states are inherently infinite-dimensional functions due to the robots' continuous deformability. Traditional sensing techniques, however, can only provide discrete measurements. Recently, a dynamic state estimation method known as a boundary observer was introduced, which leverages Cosserat rod theory to recover all infinite-dimensional states by measuring only the velocity twist at the robot's tip. In this work, we present a novel boundary observer that can also recover infinite-dimensional dynamic states, but instead relies on measuring the internal wrench at the robot's base. This design exploits the duality between the velocity twist at the tip and the internal wrench at the base, with both types of boundary observers being inspired by principles of energy dissipation. Despite the mathematical duality, the proposed approach offers a distinct advantage: it requires only a 6-axis force/torque sensor embedded at the base, eliminating the need for external sensing systems such as motion capture cameras. Moreover, combining both tip- and base-based techniques enhances energy dissipation, accelerates convergence, and improves estimation accuracy. We validate the proposed algorithms through both simulation studies and experiments based on tendon-driven continuum robots. Our results demonstrate that all boundary observers converge to the ground truth within 3 seconds, even with significantly deviated initial conditions. Furthermore, they recover from unknown perturbations and effectively track high-frequency vibrations. We also show that combining the dual techniques further improves convergence speed and accuracy. Finally, the computational efficiency of these algorithms indicates their feasibility for real-time state estimation.

Toward Dynamic Control of Tendon-Driven Continuum Robots using Clarke Transform

In this paper, we propose a dynamic model and control framework for tendon-driven continuum robots 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 controllers that operate directly on this manifold, along with practical methods for preventing negative tendon forces without compromising control fidelity. 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.

Displacement-Actuated Continuum Robots: A Joint Space Abstraction

The displacement-actuated continuum robot as an abstraction has been shown as a key abstraction to significantly simplify and improve approaches due to its relation to the Clarke transform. To highlight further potentials, we revisit and extend this abstraction that features an increasingly popular length extension and an underutilized twisting. For each extension, the corresponding mapping from the joint values to the local coordinates of the manifold embedded in the joint spaces is provided. Each mapping is characterized by its compactness and linearity.

Clarke Coordinates Are Generalized Improved State Parametrization for Continuum Robots

In this letter, we demonstrate that previously proposed improved state parameterizations for soft and continuum robots are specific cases of Clarke coordinates. By explicitly deriving these improved parameterizations from a generalized Clarke transformation matrix, we unify various approaches into one comprehensive mathematical framework. This unified representation provides clarity regarding their relationships and generalizes them beyond existing constraints, including arbitrary joint numbers, joint distributions, and underlying modeling assumptions. This unification consolidates prior insights and establishes Clarke coordinates as a foundational tool, enabling systematic knowledge transfer across different subfields within soft and continuum robotics.

Using Clarke Transform to Create a Framework on the Manifold: From Sampling via Trajectory Generation to Control

We present a framework based on Clarke coordinates for spatial displacement-actuated continuum robots with an arbitrary number of joints. This framework consists of three modular components, i.e., a planner, trajectory generator, and controller defined on the manifold. All components are computationally efficient, compact, and branchless, and an encoder can be used to interface existing framework components that are not based on Clarke coordinates. We derive the relationship between the kinematic constraints in the joint space and on the manifold to generate smooth trajectories on the manifold. Furthermore, we establish the connection between the displacement constraint and parallel curves. To demonstrate its effectiveness, a demonstration in simulation for a displacement-actuated continuum robot with four segments is presented.

Clarke Transform and Encoder-Decoder Architecture for Arbitrary Joints Locations in Displacement-Actuated Continuum Robots

In this paper, we consider an arbitrary number of joints and their arbitrary joint locations along the center line of a displacement-actuated continuum robot. To achieve this, we revisit the derivation of the Clarke transform leading to a formulation capable of considering arbitrary joint locations. The proposed modified Clarke transform opens new opportunities in mechanical design and algorithmic approaches beyond the current limiting dependency on symmetric arranged joint locations. By presenting an encoder-decoder architecture based on the Clarke transform, joint values between different robot designs can be transformed enabling the use of an analogous robot design and direct knowledge transfer. To demonstrate its versatility, applications of control and trajectory generation in simulation are presented, which can be easily integrated into an existing framework designed, for instance, for three symmetric arranged joints.

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.