Efficient variable-length hanging tether parameterization for marsupial robot planning in 3D environments

TL;DR

The paper tackles the challenge of planning for a tethered marsupial UGV-UAV system in 3D environments by replacing the computationally expensive exact catenary state with an analytic parabola-based surrogate that can be efficiently transformed into a catenary when needed. By formulating a Parabola Decision Problem (PDP) and a Catenary Decision Problem (CDP), the authors develop an iterative, fast pipeline that yields collision-free tether curves and integrates directly into RRT*-based path planning and subsequent trajectory optimization. Three tether-fitting strategies (ByLength, ByFitting, BySampling) are evaluated, with BySampling offering the best trade-off between accuracy and speed, enabling up to two orders-of-magnitude faster planning than prior methods. The trajectory optimization then incorporates tether parameters into the state, solved via Ceres-Solver with explicit length and curve-parameter constraints, resulting in higher feasibility and similar or better obstacle clearance. Overall, the proposed parabola-based parameterization substantially accelerates planning while preserving trajectory quality, enabling near real-time local re-planning in cluttered 3D environments for marsupial UGV-UAV systems.

Abstract

This paper presents a novel approach to efficiently parameterize and estimate the state of a hanging tether for path and trajectory planning of a UGV tied to a UAV in a marsupial configuration. Most implementations in the state of the art assume a taut tether or make use of the catenary curve to model the shape of the hanging tether. The catenary model is complex to compute and must be instantiated thousands of times during the planning process, becoming a time-consuming task, while the taut tether assumption simplifies the problem, but might overly restrict the movement of the platforms. In order to accelerate the planning process, this paper proposes defining an analytical model to efficiently compute the hanging tether state, and a method to get a tether state parameterization free of collisions. We exploit the existing similarity between the catenary and parabola curves to derive analytical expressions of the tether state.

Figures (5)

• Figure 1: Simplified 2D sketch showing an example for motion planning of a tethered UAV-UGV with a hanging tether. (Left) Initial robots and tether configuration, and UAV goal (red circle). (Right) Sequence of robots positions and tether length to reach the given goal. Notice how the goal cannot be reached by means of a taut tether, a hanging tether must be considered in this case.
• Figure 2: Polygons inside the $T$ region, which is delimited by the black trapezoid $\overline{ABF_AF_BA}$. $A$ and $B$ are the suspension points; $F_A$ and $F_B$ are their projections on the floor. The blue polygon is weakly inscribed in the parabolic region $R(v^*)$ and $\overline{AB}$, delimited by the blue parabola.
• Figure 3: Example of a 2D projection, in which each cluster is represented with a set of big dots in a different color. The projection is obtained from a 3D Point Cloud, represented in fine-grained dots with colors from red to green depending on its $z$ coordinate. Figure obtained with RViz rviz.
• Figure 4: Scenarios considered for validation. S1: Open/constrained space with arc as obstacle. S2: Narrow/constrained space with denied access to UGV. S3: Confined space with outlet duct for UAV. S4: Collapsed Fire Station. S5: Open space gas station.
• Figure 5: Violin plots of the distribution of the execution times of the Decision Problem test bench with Parabola and Catenary methods. In blue, the approximate shape of the distribution is represented. The mean value is represented as a red cross, the median as a white dot and the quartiles are linked with a black line.

Theorems & Definitions (8)

• Lemma 1
• proof
• Remark 2
• Lemma 3
• Theorem 4
• Theorem 5
• proof
• Corollary 6