Trajectory Optimization for Self-Wrap-Aware Cable-Towed Planar Object Manipulation under Implicit Tension Constraints

TL;DR

Self-wrap towing is formulated as a routing-aware, tensioning-implicit trajectory optimization (TITO) problem that couples a tensioning-implicit taut/slack constraint and routing-conditioned transmission maps for effective length and wrench, and a relaxation hierarchy is built from a strict mode-conditioned reference to three tractable relaxations

Abstract

Cable/rope elements are pervasive in deformable-object manipulation, often serving as a deformable force-transmission medium whose routing and contact determine how wrenches are delivered. In cable-towed manipulation, transmission is unilateral and hybrid: the tether can pull only when taut and becomes force-free when slack; in practice, the tether may also contact the object boundary and self-wrap around edges, which is not merely collision avoidance but a change of the wrench transmission channel by shifting the effective application point and moment arm, thereby coupling routing geometry with rigid-body motion and tensioning. We formulate self-wrap towing as a routing-aware, tensioning-implicit trajectory optimization (TITO) problem that couples (i) a tensioning-implicit taut/slack constraint and (ii) routing-conditioned transmission maps for effective length and wrench, and we build a relaxation hierarchy from a strict mode-conditioned reference to three tractable relaxations: Full-Mode Relaxation (FMR), Binary-Mode Relaxation (BMR), and Implicit-Mode Relaxation (IMR). Across planar towing tasks, we find that making routing an explicit decision often yields conservative solutions that stay near switching boundaries, whereas IMR induces self-wrap through state evolution and exploits the redirected torque channel whenever turning requires it.

Figures (6)

• Figure 1: Illustration of cable-towed manipulation of a box by a mobile robot at three phases: (i) slack cable, (ii) taut cable without self-wrap contact, and (iii) taut cable with self-wrap contact with the object, where the application point of the wrench changes, as detailed in Fig. \ref{['fig:inst_square']} for projected planar interpretation.
• Figure 2: Illustration of cable-towed manipulation of a box in 2D (top view) at two modes: (a) direct routing mode, and (b) self-wrap routing mode.
• Figure 3: Representative planar towing rollouts for three scenes (columns) and three formulations (rows: FMR, BMR, IMR). Dashed: reference box CoM path. Solid: actual box CoM path with wrap-active segments highlighted. Snapshots show box poses; the cable and the active redirection vertex are overlaid when applicable. The results are shown with the same initial condition (e.g. $x^g_{k=0}, T_{k=0}$, etc.) and model parameters.
• Figure 4: Time histories for the rollouts in Fig. \ref{['fig:spatial-xy-traj']} (columns). From top: tracking error $\|e(x)\|$, tension $T$, taut/slack state (from gap $g$; cf. \ref{['eq:abs_gap']}, \ref{['eq:ref_ocp_slackcomp']}), effective torque-channel difference $\Delta\tau_{\mathrm{eff}}$, control action $u$, and routing mode.
• Figure 5: Parameter sweep on the slalom benchmark for BMR and IMR. Top: tracking RMSE (gray indicates failed runs). Bottom: planning of direct/redirected (wrap-active) mode.