Cable-driven continuum robots, snake-arm robots, use flexible backbones actuated by tendons or cables. These robots are ideal for constrained environments where traditional rigid-link arms would fail, such as inside pipes, around obstacles, or in industrial machinery. However, controlling them is difficult because of several key challenges: cables can only pull (no pushing), the system is highly nonlinear, and actuation is often redundant, multiple cables can influence a single bending direction.

To make the system more manageable, we adopt a piecewise-constant-curvature (PCC) model. In this framework, each segment of the robot is modeled as a circular arc defined by a curvature κi and a bending-plane orientation ϕi, which reduces the full continuum model to a lower-dimensional configuration vector. This makes it feasible to derive the system’s forward kinematics and dynamics.

We model the robot dynamics using standard Euler-Lagrange mechanics, resulting in a control-affine system where the generalized joint torques arise from the cable tensions via a geometry-dependent Jacobian. To control this system, we propose a robust scheme that combines sliding- mode control and adaptive parameter estimation . Sliding-mode control offers robustness to disturbances and model uncertainty, while the adaptive component updates unknown system parameters in real time.

A practical challenge is that the robot’s actuation is redundant and unidirectional. Since cables can only pull and actuator effort is bounded, not every desired torque can be directly realized. To resolve this, we solve a quadratic program (QP) at each time step that distributes the joint torques across the available cables in a way that respects tension constraints . This makes the control law both effective and implementable on hardware.

We analyze the stability of the closed-loop system using a Lyapunov-based approach. In the disturbance-free case, the tracking error and parameter error converge asymptotically. Under bounded disturbances, we prove the error remains bounded inside a “sliding tube.” We then ap- ply contraction analysis to quantify the system’s robustness and exponential convergence behavior globally.

In summary, this paper contributes a reduced-order dynamic model for cable-driven continuum arms based on the PCC assumption. We derive a cable Jacobian that maps joint torques to nonnegative cable tensions. A robust adaptive controller combining sliding-mode control and online parameter estimation then with formal stability analysis using Lyapunov and contraction theory.