Model-based control of a continuum manipulator with online Jacobian error compensation using Kalman filtering

Recent research has yielded significant progress in the modeling, control, and application of continuum robots, which, inspired by natural appendages like elephant trunks, offer inherent compliance and adaptability for tasks in confined spaces, robotic surgery, and collaborative manipulation. Nevertheless, achieving precise and efficient control of these robots remains a critical challenge due to their intrinsic nonlinearities, large deformations, and frictional effects—issues often exacerbated by the trade-off between modeling accuracy and computational efficiency in traditional methods. “By integrating a model-based framework with online data-driven error compensation, we eliminate the need for time-consuming offline dataset collection and training, while enhancing control precision,” explained study author Yujia Zhai from the Department of Biomedical Engineering at the City University of Hong Kong. The proposed hybrid method leverages three core components: (a) the piecewise constant curvature (PCC) model for efficient kinematic modeling of the continuum robot’s segments, (b) a Kalman filter for real-time estimation and compensation of Jacobian errors arising from PCC model simplifications, and (c) additional constraints on consecutive Jacobian estimates to reduce numerical oscillations and improve stability. “This integrated solution addresses the limitations of pure model-based or data-driven approaches, making high-precision control of tendon-driven continuum robots more accessible in practical applications,” noted the study team. Thus, they developed a 3-segment tendon-driven continuum manipulator, featuring a lightweight design (8.4 g total mass) with 3D-printed nylon disks and NiTi elastic backbones, controlled via the hybrid framework to optimize end-effector pose tracking.

Continuum robots can be actuated through various mechanisms, from tendon-driven systems to pneumatic actuation, with fabrication processes ranging from 3D printing for structural components to precision assembly of tendon-motor interfaces. Tendon-driven designs, as adopted in this study, offer distinct advantages in compactness and controllability—each segment is actuated by two antagonistic tendons, enabling precise bending adjustments via independent motor-driven capstans. “The combination of 3D printing (for disks and capstans) and standard nylon tendons simplifies fabrication, reduces costs, and allows for rapid prototyping of the manipulator’s structure,” said Jihao Xu, a co-author of the study. Moreover, the integration of an infrared camera for real-time pose feedback (via three reflective markers on the end-effector) further streamlines the experimental setup, enabling closed-loop control at 20 Hz without complex sensor integration.

The study authors demonstrated that the hybrid control method achieves superior trajectory tracking performance compared to the PCC model alone, with reduced errors in both position and attitude. In experiments involving three distinct trajectories—pure position tracking, pure attitude tracking and hybrid position-attitude tracking—the proposed method yielded root mean square errors (RMSEs) as low as 0.6 mm (x-direction), 0.8 mm (y-direction), and 1.1° (attitude), outperforming the PCC model by 30–40% in key metrics. The ON–OFF controller, augmented with a damped pseudoinverse of the constrained Jacobian, maintained stable end-effector pose even when subjected to unexpected payloads (up to 17.5 g, twice the manipulator’s mass). The continuum robot exhibited dexterity in following pre-defined trajectories and resilience to external disturbances, with temporary deviations quickly corrected by the online error compensation loop.

“The application of Kalman filtering to Jacobian error estimation ensures that the model adapts to real-time measurements, while the constraints on Jacobian variation prevent sudden fluctuations in control signals,” said Dong Sun, the corresponding author of the study and a professor at the City University of Hong Kong. “Nevertheless, the method currently has limitations: its quasi-static assumption means tracking accuracy degrades with shorter trajectory periods—for example, reducing the period from 40 s to 5 s led to a fivefold increase in y-direction RMSE (from 1.9 mm to 15.6 mm)—and it neglects dynamic effects, restricting its use to low-velocity operations.” Additionally, while the manipulator successfully recovered from payload disturbances, heavier objects (e.g., the 17.5 g USB converter) caused larger temporary deviations (up to 38.9 mm in the y-direction), highlighting the need to incorporate dynamic modeling in future work. Since the continuum robot’s kinematics are highly coupled—with tendon length changes, segment bending angles, and end-effector pose mutually dependent—future research will focus on extending the framework to include dynamic effects, as well as standardizing testing benchmarks for disturbance resilience and high-velocity tracking. Totally, this hybrid control method offers a cost-effective and efficient solution for continuum robot control, avoiding the drawbacks of offline training and over-reliance on complex mechanics-based models, and paves the way for broader applications in precision robotics.

Authors of the paper include Yujia Zhai, Jihao Xu, Hangjie Mo, Chunqi Zhang, and Dong Sun.

This work was supported in part by grants from the Research Grants Council of the Hong Kong Special Administrative Region, China (nos. CityU C1134-20G and CityU 11211421).

The paper, “Model-Based Control of a Continuum Manipulator with Online Jacobian Error Compensation Using Kalman Filtering” was published in the journal Cyborg and Bionic Systems on Aug 7, 2025, at DOI: 10.34133/cbsystems.0339