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.

翻译：缆绳/绳索元件在可变形物体操纵中普遍存在，常作为可变形力传递介质，其布线与接触决定了力矩的传递方式。在缆绳牵引操纵中，传递是单向且混合的：牵引索仅在绷紧时可施加拉力，松弛时则无力作用；实践中，牵引索还可能接触物体边界并在边缘发生自缠绕，这不仅是碰撞避免，更是通过改变有效作用点和力臂来改变力矩传递通道，从而将布线几何与刚体运动及张紧过程耦合。我们将自缠绕牵引建模为一个布线感知、张力隐式的轨迹优化问题，该问题耦合了（i）张力隐式的绷紧/松弛约束和（ii）布线条件化的有效长度与力矩传递映射，并构建了从严格模式条件化基准到三个可处理松弛形式的层次结构：全模式松弛、二值模式松弛和隐式模式松弛。在平面牵引任务中，我们发现将布线作为显式决策通常会产生保守解，使其保持在切换边界附近，而隐式模式松弛则通过状态演化诱导自缠绕，并在转向需要时利用重定向的扭矩通道。