Real-Time Projected Adaptive Control for Closed-Chain Co-Manipulative Continuum Robots

Abstract

In co-manipulative continuum robots (CCRs), multiple continuum arms cooperate by grasping a common flexible object, forming a closed-chain deformable mechanical system. The closed-chain coupling induces strong dynamic interactions and internal reaction forces. Moreover, in practical tasks, the flexible object's physical parameters are often unknown and vary between operations, rendering nominal model-based controllers inadequate. This paper presents a projected adaptive control framework for CCRs formulated at the dynamic level. The coupled dynamics are expressed using the Geometric Variable Strain (GVS) representation, yielding a finite-dimensional model that accurately represents the system, preserves the linear-in-parameters structure required for adaptive control, and is suitable for real-time implementation. Closed-chain interactions are enforced through Pfaffian velocity constraints, and an orthogonal projection is used to express the dynamics in the constraint-consistent motion subspace. Based on the projected dynamics, an adaptive control law is developed to compensate online for uncertain dynamic parameters of both the continuum robots and the manipulated flexible object. Lyapunov analysis establishes closed-loop stability and convergence of the task-space tracking errors to zero. Simulation and experiments on a tendon-driven CCR platform validate the proposed framework in task-space regulation and trajectory tracking.

Figures (12)

• Figure 1: Closed-loop configuration of the tendon-driven CCR system manipulating a flexible object, with backbone parameterization and reference frames.
• Figure 2: Task-space regulation of the manipulated object midpoint in simulation. (a)--(c) $x$-, $y$-, and $z$-coordinates of the midpoint versus desired values, (d) Euclidean norm of the regulation error $\|\mathbf{e}\|$.
• Figure 3: Control inputs and adaptive parameter estimates during task-space regulation in simulation. (a) tendon tensions $u_1$, $u_2$, and $u_3$ (with saturation bounds of $[-20,\,20]~\mathrm{N}$). (b) tendon displacements $S_1$, $S_2$, and $S_3$. (c) estimated mass parameters of the two CRs and the manipulated flexible object. (d) estimated inertia parameter $I_x$.
• Figure 4: Task-space tracking performance of the manipulated object midpoint in simulation. (a)--(c) $x$-, $y$-, and $z$-coordinates of the midpoint, (d) norm of the tracking error $\|\mathbf{e}\|$. The inset in (d) shows a zoomed view over $0$--$0.2\,\mathrm{s}$.
• Figure 5: Control inputs and adaptive parameter estimates during task-space tracking in simulation. (a) tendon tensions $u_1$, $u_2$, and $u_3$; (b) tendon displacements $S_1$, $S_2$, and $S_3$; (c) online estimates of the mass parameters; (d) online estimates of the inertia parameter $I_x$.

Theorems & Definitions (4)

• Lemma 1
• Lemma 2
• Theorem 1
• proof