Curvature-Based Continuous Steering of Stiffness-Dominant Concentric Tube Robots

Existing works on controlling a concentric tube robot (CTR) mostly focus on the trajectory of its tip position or pose. In order to safely send CTRs in a confined lumen space, we propose to continuously steer the CTRs so that its entire shape will always attempt to approximate target curves over time. We focus on stiffness-dominant CTRs. Considering the differential geometry of such CTR shapes, we propose to work on the curvature domain to reduce the computational cost in searching the configuration of the CTRs. With our formulation, we model the curvature control of the CTR to find the optimal translation of each tube and then search for the rotation of the tubes to fit the target shapes. We demonstrate our method using sets of different target paths. The computational time per frame, ranging between 0.1 to 0.3 seconds across all experiments, highlights the efficiency of our approach in aligning the complete shape of the CTR with specified paths. Notably, for time-varying trajectories that could be reproduced by the CTR with its maximum deployment length reaching 150 mm, the root mean square error and median error were 0.98mm and 0.46mm, respectively. Note to Practitioners—This paper presents a novel approach for controlling stiffness-dominant CTRs, such that their entire shape consistently attempts to approximate target curves over time. This is crucial for applications such as navigation in confined anatomical spaces. Considering the differential geometry of these CTR shapes, we propose operating in the curvature domain. We model the curvature control of the CTR to determine the optimal translation for each segment, followed by searching for the rotation of the segments to fit the target shape. Our method avoids costly and computationally intensive full morphology evaluations while also achieving superior control accuracy, making it feasible for practical use in robotic surgery. In fact, this solution could be implemented in robot-assisted minimally invasive surgical platforms. Future work could explore adaptive modeling to address more complex clinical conditions, such as torsional deformations or variable tissue stiffness, or incorporate shape sensing for closed-loop control.