Actuation space reduction to facilitate insightful shape matching in a novel reconfigurable tendon driven continuum manipulator
Authors: Sabyasachi Dash, John Golden, Girish Krishnan
Organizations: Department of Industrial and Enterprise Systems Engineering, University of Illinois Urbana-Champaign · Department of Mechanical Science and Engineering, University of Illinois, Urbana-Champaign
In tendon driven continuum manipulators (TDCMs), reconfiguring the tendon routing enables tailored spatial deformation of the backbone. This work presents a design in which tendons can be rerouted either prior to or after actuation by actively rotating the individual spacer disks. Each disk rotation thus adds a degree of freedom to the actuation space, complicating the mapping from a desired backbone curve to the corresponding actuator inputs. However, when the backbone shape is projected into an intermediate space defined by curvature and torsion (C-T), patterns emerge that highlight which disks are most influential in achieving a global shape. This insight enables a simplified, sequential shape-matching strategy: first, the proximal and intermediate disks are rotated to approximate the global shape; then, the distal disks are adjusted to fine-tune the end-effector position with minimal impact on the overall shape. The proposed actuation framework offers a model-free alternative to conventional control approaches, bypassing the complexities of modeling reconfigurable TDCMs.
Figures & tables
Fig. 1: Conventional TDCMs (a) rely on fixed tendon routing which limits the range of achievable spatial defomration modes. The proposed Reconfigurable TDCM (RTDCM) design (b) enables local tendon rerouting via controlled rotation of intermediate spacer disks, allowing more versatile backbone spatial configurations. (c) The geometric Curvature-Torsion (C-T) space provides an intermediate mapping between the manipulator’s spatial deformation and the corresponding actuation space.
Fig. 2: (a) RTDCM design with each disk assembly capable of rotating the tendon locally. The servo motor (1) sits on the fixed mount (2) which is attached to the backbone using epoxy. The fixed mount is connected to the rotating disk (5) through a bearing (4), and a gear (3) attached to the servo motor-head rotates (5) which holds the tendon. (b) Experimental setup with stepper motor (1) to actuate the tendon, and a magnetic stylus (3) with reference at (2) to retrieve track coordinates of the backbone.
Fig. 3: Continuous rotation of (a) second, (b) fourth, (c) eighth disk from 90∘ clockwise to 90∘ anticlockwise direction, where (a) proximal actuation yields behavior similar to entire base rotation, (b) intermediate actuation induces a more pronounced global shape change of the backbone, and (c) distal actuation enables local tip maneuvering with minimal effect on the overall shape
Fig. 4: (a) Parallel tendon routing yields a planar deformation profile of the backbone, which is signified by zero torsion throughout, and a rising curvature due to gravity. (b) Spatial deformation induced by local rotation of the fifth disk in (I) clockwise and (II) counterclockwise direction. Torsion plot demonstrates a sign change near the rotated disk, with the signs flipped between the clockwise and anticlockwise case. (c) Torsion plots for 90∘ clockwise rotation of (I) third through (V) seventh disk. The torsion flips sign from positive to negative in close proximity of each rotated disk. (d) Torsion plots for combination of two disks (third and sixth) rotated in opposite directions demonstrate two distinct sign changes. (e) Increasing tendon displacement directly correlates with a steeper curvature for both (I) parallel routing, and (II) routing with a rotated disk, while the peak curvature and slope reduce for the latter case. (f) For a fixed tendon displacement, the fifth disk is rotated incrementally from 30∘ to 90∘ , both in (I) clockwise and (II) anticlockwise directions. The resulting curvature profiles remain close to each other in both cases.
Fig. 5: Sequential Actuation Framework for Target Shape Matching
Fig. 6: (a) Target shape (I) with its corresponding torsion (II) profile demonstrating clear sign reversal from negative to positive near the fifth disk implying a counterclockwise rotation. (b) (I) With the fifth disk rotated to 90∘ , a tendon pull of 103 mm minimizes error from target curvature. (II) An exact angle of 78∘ in the fifth disk minimizes the global shape error, and (III) a counterclockwise rotation of the eighth disk to 10∘ minimizes the end-effector error. (c) Overlay of spatial curves obtained when Disk 5 is rotated iteratively till error with target shape is minimized, and Disk 8 is adjusted to match the tip. (d) A second target shape (I) with its torsion profile in (II), demonstrates a significant sign change near the fourth disk, and a minor sign change around the sixth/seventh disk. (e) (I) Both the fourth and sixth disks are rotated to 90∘ clockwise and counterclockwise directions respectively, and a tendon displacement of 131 mm minimizes the curvature error. Required disks are rotated sequentially: (II) fourth disk first to 87∘ clockwise, and (III) sixth disk next 55∘ counterclockwise to minimize overall shape error. (IV) End effector is adjusted by rotating the eighth disk clockwise by 16∘ . (f) Overlay of Target shape with shapes obtained after rotation of fourth disk, fourth + sixth disk, and fourth + sixth + end effector.
Rerouting the tendon path in tendon driven continuum manipulators (TDCMs) enables a broad range of deformation modes. This work presents a Reconfigurable TDCM design which allows independent rotation of intermediate spacer disks, thereby locally rerouting the tendon and achieving non-trivial backbone spatial deformations. Two such designs, (a) Manual Disk Locked (MDL) and (b) Continuous Disk Rotor (CDR) manipulators are presented to achieve disk rotations before and during operation, respectively. A predictive static model based on the piecewise constant strain (PCS) assumption is developed within a potential energy minimization framework, incorporating (a) disk rotations, (b) discrete tendon paths between disk segments, (c) rigid thickness of spacer disks, and (d) elasticity of the tendons. The model is validated against experimental results, demonstrating an average tip error of 1.2% of the manipulator's total length for parallel tendon routing and around 3% for the case when multiple disks are rotated. The computation time is an order of magnitude lower than the state of the art Cosserat rod solver.
Sabyasachi Dash, Yangkun Liu, Will Hunter +2
Department of Industrial & Enterprise Systems Engineering, University of Illinois Urbana-Champaign · Department of Mechanical Science & Engineering, University of Illinois Urbana-Champaign
Many planning and control tasks for tendon-driven continuum robots (TDCRs) require the complete Cartesian backbone geometry. We present a reduced Cartesian framework for planar, axially compressible TDCRs that propagates equilibrium configurations along prescribed tendon-force and tendon-displacement trajectories. The backbone is represented by two global position fields. Following exact variation, a Taylor-Galerkin reduction condenses prescribed spatial properties and distributed loads into offline moment vectors, yielding analytic reduced residuals and Jacobians without online spatial quadrature or numerical differentiation. Analytical differentiation and residual correction yield first-order rate systems requiring one fixed-dimensional linear solve per rate evaluation after initial equilibrium alignment on a regular branch. Across four simulated cases covering variable tendon routing, nonuniform geometry, axial compression, and their combined effects, the propagated Cartesian shapes and distributed strains closely match pointwise geometrically variable-strain (GVS) equilibrium solutions. Residual correction suppresses propagation drift across the tested step sizes while adding only about 0.98% to the mean update time of uncorrected Euler. The proposed method requires 0.508 ms per update on average, approximately 11 times faster than pointwise GVS solves. Displacement-driven experiments yield a maximum normalized mean backbone position error of 1.02% and a maximum end-effector position error of 0.28%. These results support efficient and accurate Cartesian full-shape prediction along prescribed actuation paths.
Predicting the shape of tendon driven continuum robots (TDCRs) at steady state from actuation remains challenging due to continuous deformation, complex tendon routing, compliance, friction, and fabrication variability. In this paper, we address this problem as kinematic self modeling conditioned on action. We present a lightweight 3D printed TDCR hardware platform and an RGB-D data collection pipeline with multiple cameras, and we learn a point cloud flow matching model that maps motor actuation states to the robot's settled 3D geometry. The model is trained from randomly sampled quasi static configurations and evaluated on test motor commands within the same TDCR design family and actuation range. We compare against prior 3D deformable object and robot self modeling approaches in both MuJoCo simulation and real hardware experiments. Experiments on simulated 2-, 3-, and 5-module TDCRs and real 2- and 3-module robots show improved shape prediction accuracy under CD and EMD metrics. We further show in simulation that the same conditional formulation generalizes to tip payload as a conditioning input, enabling payload conditioned steady-state shape prediction. These results demonstrate a data driven self modeling framework for quasi static TDCR geometry prediction.
Jiong Lin, Jinchen Ruan, Hod Lipson
Department of Mechanical Engineering, Columbia University, New York, NY, USA