Morphology-Specific Closed-Loop Control of Logarithmic-Spiral Continuum Arms via Online Jacobian Error Compensation
Authors: Partha Datta, Yi Jin, Wei Lin, C. Chase Cao
Organizations: Laboratory for Soft Machines and Electronics, Department of Mechanical and Aerospace Engineering, Case Western Reserve University, Cleveland, OH, USA · Department of Electrical, Computer and Systems Engineering, Case Western Reserve University, Cleveland, OH, USA · Laboratory for Soft Machines and Electronics, Department of Mechanical and Aerospace Engineering, Case Western Reserve University, Cleveland, OH 44106, USA · Department of Electrical, Computer and Systems Engineering, Case Western Reserve University, Cleveland, OH 44106, USA · Advanced Platform Technology · 3Advanced Platform Technology (APT) Center, Louis Stokes Cleveland VA Medical Center, Cleveland, OH 44106, USA
Logarithmic spirals are ubiquitous in biological appendages and provide an attractive morphology for continuum manipulators capable of reaching, wrapping, and grasping. Recently reported logarithmic-spiral robots demonstrated scalable fabrication and versatile grasping but lacked inverse kinematics and closed-loop control. This work presents the first morphology-specific closed-loop task-space control framework for logarithmic-spiral continuum arms. A segmented tendon-driven model with a centerline backbone and equilateral tendon routing is developed in MuJoCo to capture tapered compliance and contact dynamics. An analytical task-space Jacobian is derived directly from the logarithmic-spiral kinematics and combined with online Jacobian error compensation using a Broyden secant update and Kalman-filter estimation. The resulting controller continuously corrects modeling errors arising from nonlinear deformation, contact, and geometric mismatch. The framework is validated through planar and spatial simulations, including trajectory tracking, attitude regulation, disturbance rejection, three-dimensional position tracking, and simultaneous position-orientation control. Compared with a piecewise-constant-curvature (PCC) baseline, the proposed method consistently reduces tracking errors, suppresses attitude drift, and maintains a bounded Jacobian estimation error. The controller is further applied to morphology-enabled manipulation tasks, including obstacle-assisted reach-wrap-release motions, adaptive whole-arm grasping, and cooperative multi-arm object handling. Results demonstrate that combining logarithmic-spiral morphology with online Jacobian compensation enables accurate, robust, and scalable control of highly underactuated continuum manipulators. The proposed framework establishes a physics-grounded baseline for future hardware implementation and learning-augmented soft robotic control.
Reconstructing the shape of continuum manipulators from sparse, noisy sensor data is a challenging task, owing to the infinite-dimensional nature of such systems. Existing approaches broadly trade off between parametric methods that yield compact state representations but lack probabilistic structure, and Cosserat rod inference on factor graphs, which provides principled uncertainty quantification at the cost of a state dimension that grows with the spatial discretization. This letter combines the strength of both paradigms by estimating the coefficients of a low-dimensional Geometric Variable Strain (GVS) parameterization within a factor graph framework. A novel kinematic factor, derived from the Magnus expansion of the strain field, encodes the closed-form rod geometry as a prior constraint linking the GVS strain coefficients to the backbone pose variables. The resulting formulation yields a compact state vector directly amenable to model-based control, while retaining the modularity, probabilistic treatment and computational efficiency of factor graph inference. The proposed method is evaluated in simulation on a 0.4 m long tendon-driven continuum robot under three measurement configurations, achieving mean position errors below 2 mm for all three scenarios and demonstrating a sixfold reduction in orientation error compared to a Gaussian process regression baseline when only position measurements are available.
Lorenzo Ticozzi, Patricio A. Vela, Panagiotis Tsiotras
Continuum robots are well suited for navigating confined and fragile environments, such as vascular or endoluminal anatomy, where contact with surrounding structures is often unavoidable. While controlled contact can assist motion, unfavorable contact can degrade controllability, induce kinematic singularities, or introduce safety risks. We present a contact-aware planning approach that evaluates contact quality, penalizing hazardous interactions, while permitting benign contact. The planner produces kinematically feasible trajectories and contact-aware Jacobians which can be used for closed-loop control in hardware experiments. We validate the approach by testing the integrated system (planning, control, and mechanical design) on anatomical models from patient scans. The planner generates effective plans for three common anatomical environments, and, in all hardware trials, the continuum robot was able to reach the target while avoiding dangerous tip contact (100% success). Mean tracking errors were 1.9 +/- 0.5 mm, 1.2 +/- 0.1 mm, and 1.7 +/- 0.2 mm across the three different environments. Ablation studies showed that penalizing end-of-continuum-segment (ECS) contact improved manipulability and prevented hardware failures. Overall, this work enables reliable, contact-aware navigation in highly constrained environments.
Tendon-Driven Continuum Robots (TDCRs) pose significant control challenges due to their highly nonlinear, path-dependent dynamics and non-Markovian characteristics. Traditional Jacobian-based controllers often struggle with hysteresis-induced oscillations, while conventional learning-based approaches suffer from poor generalization to out-of-distribution trajectories. This paper proposes a reference-augmented offline learning framework for precise 6-DOF tracking control of TDCRs. By leveraging a differentiable RNN-based dynamics surrogate as a gradient bridge, we optimize a control policy through an augmented reference distribution. This multi-scale augmentation scheme incorporates stochastic bias, harmonic perturbations, and random walks, forcing the policy to internalize diverse tracking error recovery mechanisms without additional hardware interaction. Experimental results on a three-section TDCR platform demonstrate that the proposed policy achieves a 50.9% reduction in average position error compared to non-augmented baselines and significantly outperforms Jacobian-based methods in both precision and stability across various speeds. For implementation details and source code, please refer to https://github.com/ZiqingZou/ContinuumControl.