Entanglement Definitions for Tethered Robots: Exploration and Analysis

TL;DR

This work addresses the risk of tether entanglement in tethered robots by proposing a broad, topology-informed set of entanglement definitions that are agnostic to tether type, environment, and dimension. It systematically reviews existing definitions, introduces new criteria (including Obstacle-free Convex Hull, Obstacle-free Linear Homotopy, Path Homotopy to Safe Set, Local Visibility Homotopy, and their relaxations), and analyzes their properties and interrelationships through formal topology tools such as homotopy and the homotopic Fréchet distance. The authors characterize the non-entangled reachable workspace under each definition, establish relationships among the definitions, and validate the approach qualitatively with 12 tethered-robot experts across diverse scenarios. The results indicate that the new definitions can cover broader situations while preserving safety guarantees, guiding the development of entanglement-aware motion planning and enabling safer, more robust tethered operations.

Abstract

In this article we consider the problem of tether entanglement for tethered mobile robots. One of the main risks of using a tethered connection between a mobile robot and an anchor point is that the tether may get entangled with the obstacles present in the environment or with itself. To avoid these situations, a non-entanglement constraint can be considered in the motion planning problem for tethered robots. This constraint is typically expressed as a set of specific tether configurations that must be avoided. However, the literature lacks a generally accepted definition of entanglement, with existing definitions being limited and partial in the sense that they only focus on specific instances of entanglement. In practice, this means that the existing definitions do not effectively cover all instances of tether entanglement. Our goal in this article is to bridge this gap and to provide new definitions of entanglement, which, together with the existing ones, can be effectively used to qualify the entanglement state of a tethered robot in diverse situations. The new definitions find application in motion planning for tethered robots, where they can be used to obtain more safe and robust entanglement-free trajectories.

Figures (9)

• Figure 1: Example of a motion planning problem for a tethered robot in a 2D environment. The robot must reach the location $\bm{x_\mathrm{target}}$ from its current location $\bm{x_\mathrm{r}}$. Three possible paths $\bm{\lambda_1, \lambda_2, \lambda_3}$ are depicted in the image. Given the initial tether configuration $\bm{\gamma_0}$, the three possible paths would result in three different tether configurations after the motion of the robot, which are represented by the gray dashed lines. Each of the resulting tether configuration can have a different entanglement state depending on the entanglement definition being used.
• Figure 2: Example of entanglement with respect to Definition \ref{['def:risk_of_entanglement']}. Two robots and their respective tethers are shown in an obstacle-free 3D environment. If the robots continue moving along the dashed lines, the tethers will come into contact, restricting the motion capabilities of the two robots. The image is adapted from cao2023neptune.
• Figure 3: Example of the Obstacle-free Convex Hull and Obstacle-free Linear Homotopy non-entanglement definitions (Definition \ref{['def:obstacle_free_conv_hull']} and \ref{['def:obstacle_free_linear_homotopy']}) applied to a tether configuration $\bm{\gamma}$. The blue shaded region represents the set of points covered by the linear homotopic transformation $\bm{H}$ from $\bm{\gamma}$ and to $\bm{x_\mathrm{a}}$. The blue shaded region does not intersect with the obstacle $\bm{O_1}$, so the configuration is not entangled with respect to the Obstacle-free Linear Homotopy definition (Definition \ref{['def:obstacle_free_linear_homotopy']}). However, the same configuration is entangled with respect to the Obstacle-free Convex Hull definition (Definition \ref{['def:obstacle_free_conv_hull']}). In fact, the light-orange shaded area, which corresponds to $\bm{\textrm{conv}(\gamma)}$, intersects with $\bm{O_1}$.
• Figure 4: Example of the application of the Path Homotopy to Safe Set definition. The set of safe configurations $\bm{\Gamma_{x_\mathrm{a}}}^\textrm{safe}$ is visualized as the set of all points that are reachable through at least one configuration that is not entangled with respect to the Obstacle-free Convex Hull definition (Definition \ref{['def:obstacle_free_conv_hull']}), and is represented as the blue shaded area. The sets of paths $\bm{\Lambda_{x_{\textrm{r}_i}, \mathrm{maxlen}}}, i=1,2$ are defined as the sets of all straight line paths starting from $\bm{x_{r_i}}, i=1,2$ and having length less than or equal to $\bm{d_\textrm{max}}$. The sets $\bm{\Lambda_{x_{\textrm{r}_1},\mathrm{maxlen}}}$ and $\bm{\Lambda_{x_{\textrm{r}_2},\mathrm{maxlen}}}$ are visualized by the orange shaded areas.
• Figure 5: Example of the application of the Local Visibility Homotopy non-entanglement definition for two different tether configurations. Configuration $\bm{\gamma_{1}}$ is not entangled. On the contrary, $\bm{\gamma_{2}}$ is entangled. In fact, the part of the path $\bm{\gamma_{2}}$ between $\bm{x_{1}}$ and $\bm{x_{2}}$ is not homotopic to the straight line $\bm{l_{x_{1}, x_{2}}}$ between the points $\bm{x_{1}}$ and $\bm{x_{2}}$ due to the presence of obstacle $\bm{O_{2}}$.

Theorems & Definitions (52)

• Definition 1: Taut Tether Contact with Obstacle
• Definition 2: Taut Tether Contact with Other Tethers
• Definition 3: Entanglement between Slack Tethers
• Definition 4: 2D Tether Loop around Obstacle
• Definition 5: Closed Tether Homotopy to Constant Map
• Definition 6: Obstacle-free Convex Hull
• Definition 7: Obstacle-free Linear Homotopy
• Definition 8: Path Homotopy to Safe Set
• Example 1
• Definition 9: Local Visibility Homotopy