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.
Figures & tables
Fig. 1: The studied tendon-driven continuum robot.
Parameter
Symbol
Value or profile
Unit
Backbone length
L
0.40
m
Young’s modulus
E
2×109
Pa
Diameter profile
D(s)
Case-dependent
m
Area profile
A(s)
πD2(s)/4
m2
Second-moment profile
I(s)
πD4(s)/64
m4
Tendon-routing diameter
W(s)
Case-dependent
m
TABLE I: Simulation Parameters, Numerical Settings, and Validation Cases
Fig. 2: Numerical validation across the four mechanical cases.
Method
Case 1
Case 2
Case 3
Case 4
Mean
Γm
Proposed
0.107
0.153
0.332
1.440
0.508
1.000
GVS
5.347
4.160
4.790
8.045
5.586
10.996
Uncorrected Euler
0.107
0.153
0.328
1.425
0.503
0.990
TABLE II: Online Per-Update Computation Times and Relative Runtime Ratios
Fig. 3: Experimental platform.
Parameter
Symbol
Value or profile
Unit
Backbone length
L
0.4
m
Cross-sectional diameter
D
0.004
m
Young’s modulus
E
2×109
Pa
Tendon-routing diameter
W
0.11
m
Cross-sectional area
A
1.26×10−5
m2
Second moment of area
I
1.26×10−11
m4
TABLE III: Experimental Parameters and Tendon-Displacement Input
Fig. 4: Experimental validation under the linear tendon-displacement input.
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.
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
Tendon-Driven Continuum Robots (TDCRs) pose significant modeling and control challenges due to complex nonlinearities, such as frictional hysteresis and transmission compliance. This paper proposes a differentiable learning framework that integrates high-fidelity dynamics modeling with robust neural control. We develop a GRU-based dynamics model featuring bidirectional multi-channel connectivity and residual prediction to effectively suppress compounding errors during long-horizon auto-regressive prediction. By treating this model as a gradient bridge, an end-to-end neural control policy is optimized through backpropagation, allowing it to implicitly internalize compensation for intricate nonlinearities. Experimental validation on a physical three-section TDCR demonstrates that our framework achieves accurate tracking and superior robustness against unseen payloads, outperforming Jacobian-based methods by eliminating self-excited oscillations. For implementation details and source code, please refer to https://github.com/ZiqingZou/ContinuumControl.
Ziqing Zou, Ke Qiu, Fei Wang +3
Department of Control Science and Engineering, Zhejiang University, Hangzhou, China