Real-time motion generation for tendon-driven continuum robots requires accurate modeling of nonuniform bending and whole-body collision avoidance. This paper presents a unified actuation-space framework for planar multi-segment tendon-driven continuum robots. An energy-based variable-curvature model captures spatially varying tendon spacing and bending stiffness and provides analytical Jacobians for differential inverse kinematics and safety monitoring. A multipoint CBF-QP enforces backbone clearance under obstacle motion and actuation-velocity bounds, while its decision dimension depends only on the number of independently actuated segments. The model closely agrees with GVS references, with a maximum curvature error of 5.223×10−2m−1. Over 100 MuJoCo trials, the proposed method achieves collision-free success rates of 96% and 100% in static and dynamic scenarios, respectively, compared with approximately 60% and 80% without CBF constraints. Hole-traversal tests further demonstrate safe motion in constrained environments. With 600 backbone monitoring points, the mean control-step time is 6.66 ms, demonstrating real-time whole-body safe motion generation.
Figures & tables
Fig. 1: Real-time whole-body safe motion generation for continuum robots. (A) Robot and kinematics. (B) Whole-body collision avoidance. (C) Real-time CBF-QP control framework.
Parameter
Symbol
Value or profile
Unit
Number of segments
N
3
–
Total backbone length
L
0.30
m
Segment lengths
Li
[0.12,0.10,0.08]
m
Young’s moduli
Ei
[6.0,7.0,5.5]
MPa
Diameter profile
D(s)
6.4×10−3(1−0.5s/L)
m
Second moment of area
I(s)
πD4(s)/64
m4
TABLE I: Simulation Parameters
Fig. 2: Kinematic validation against GVS: shape comparison and segment-wise curvature errors.
Fig. 3: Sim-to-sim validation of real-time whole-body safe motion generation in MuJoCo: static and dynamic obstacle avoidance and hole traversal.
CBF sampling points
Δs[m]
Mean control-step time (ms)
100
0.00758
1.1692
200
0.00377
1.9870
300
0.00251
2.9671
400
0.00188
3.6831
500
0.00150
5.0753
600
0.00125
6.6641
TABLE II: CBF Sampling Density and Computational Cost
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.
Continuum robots provide intrinsic compliance, high dexterity, and safe physical interaction, enabling navigation and manipulation in confined and unstructured environments. Despite recent advances in sensing and control, heightening the need for precise motion generation, most widely used planning libraries are grounded in rigid-body assumptions, creating a critical gap for fast and practical tools for continuum robots. To address this, we present CR-Solver, a two-stage, optimization-based solver for the motion generation of tendon-driven continuum robots. Our method unifies inverse kinematics, path following, and trajectory planning within a single constrained nonlinear optimization framework. Leveraging GPU-accelerated parallel optimization, CR-Solver delivers fast, accurate, and constraint-aware solutions. We validate our approach on three tasks, demonstrating significant speedups over traditional CPU-based solvers while achieving a consistently high success rate above 95% and millimeter-level accuracy. The solver is implemented in pure Python, reducing the barrier to adoption and offering a practical, extensible foundation for continuum robots' high-performance motion planning.
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.