Rotate Disks to Reach Farther: Design and Modeling of a Novel Reconfigurable Tendon Driven Manipulator
Authors: Sabyasachi Dash, Yangkun Liu, Will Hunter, John Golden, Girish Krishnan
Organizations: Department of Industrial & Enterprise Systems Engineering, University of Illinois Urbana-Champaign · Department of Mechanical Science & Engineering, University of Illinois Urbana-Champaign
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.
Figures & tables
Fig. 1: Conventional TDCMs (a) rely on fixed predefined parallel tendon routing which constrains their attainable deformation modes. The proposed RTDCM design (b) leverages intermediate spacer disk rotation to locally reroute the tendon allowing more versatile backbone spatial configurations. Though the backbone deforms as a smooth continuous curve, the tendon follows a discrete linear path between the rotated disk segments (II).
Fig. 2: The spacer disk for the MDL manipulator (a) is split into two main components, with the top bushing (1) attached to the backbone using epoxy and the bottom disk (3) freely rotating on the bearing (2) to locally reroute the tendon. A desired tendon configuration is locked by inserting the cotter pin (4). The CDR manipulator (b) enables active tendon rerouting using the servo motor (1), which sits on the larger top bushing (2) attached to the backbone and is meshed with the rotating disk (5) via the spur gear (3).
Parameter
Symbol
MDL
CDR
Unit
Number of segments
n
10
8
–
Segment length
Lseg
27
69
mm
Backbone diameter
db
1.0
1.5
mm
Backbone density
ρ
6450
6450
kg/m 3
Tendon Young’s modulus
Et
4.5
4.5
GPa
Tendon radial offset
Rrad
8
34
mm
TABLE I: Manipulator Design Parameters
Fig. 3: Representative manipulator description (left) with piecewise constant strain segment (right) illustrating the various parameters and notations used in the model. p0 and R0 correspond to the base frame pose.
Routing
Avg. RMSE (cm)
Avg. Tip Error (cm)
Avg. Time PCS (s)
Avg. Time VC (s)
Parallel
0.0144
0.0309
0.7925
7.65
D6 90∘ CW
0.0778
0.1476
0.7769
9.75
D6 90∘ CW + D9 90∘ CCW
0.1737
0.3022
0.7761
15.7362
TABLE II: Model Comparison
Fig. 4: The MDL manipulator (4) is suspended within an enclosure, with the tendon being actuated by the stepper motor (1). Two cameras (2,3) capture the manipulator’s deformation by reconstructing the spatial coordinates of the spacer disks.
Fig. 5: The PCS model is validated against experimental results in three different tendon routing scenarios (a)-I: parallel, (b)-I: disk 6 rotated 90∘ CW, and (c)-I: disk 6 and 9 rotated 90∘ in opposite directions. The second row (II) demonstrates the model deformations for different tendon actuation using smooth curves, compared with black filled circles denoting experimental data, for each of the three routing scenarios (a)-(c). The RMSE and tip errors are plotted for each routing scenario in row (III).
Fig. 6: CDR manipulator demonstrating tendon rerouting on-the-fly achieving a helical deformation profile from an initial parallel configuration
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.
Sabyasachi Dash, John Golden, Girish Krishnan
Department of Industrial and Enterprise Systems Engineering, University of Illinois Urbana-Champaign · Department of Mechanical Science and 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.
Forward kinetostatics of spatial tendon-driven continuum robots typically requires a nonlinear equilibrium solve for each actuation input. This paper develops a force-to-Cartesian-configuration model with a closed-form solution in quadratures for spatial multi-segment robots under tendon actuation. The Cartesian backbone centerline and accumulated material twist serve as generalized coordinates, from which the strain measures and tendon geometry are derived. Variational equilibrium yields explicit axial and bending relations and establishes zero equilibrium material twist within the proposed model for admissible longitudinal non-helical routing. The solution is propagated segment by segment without an iterative equilibrium solve, while retaining axial deformation, spatially varying axial and bending stiffnesses and tendon-routing diameter, and segment-dependent tendon participation. Numerical comparisons with a full-strain geometric variable-strain model (GVS) yield maximum length-normalized tip-position discrepancies of 8.91 x 10^-6 and 1.01 x 10^-5 for the single- and three-segment robots, respectively. Mean evaluation times of 1.52 μs and 2.94 μs, with corresponding speedups of approximately 1864x and 3348x over the baseline, demonstrate the computational advantage of the explicit force-to-configuration mapping in the reported benchmark.