Multi-query Robotic Manipulator Task Sequencing with Gromov-Hausdorff Approximations

TL;DR

This work tackles the robot task sequencing problem (RTSP) by introducing the Hausdorff Approximation Planner (HAP), whichOffline decomposes the task space into subspaces via $\Cepsilon$-Gromov-Hausdorff approximations, enabling efficient online sequencing while maintaining theoretical suboptimality bounds. The method reduces the RTSP to tractable subproblems by mapping tasks to configuration-space subspaces, solving intra-subspace TSPs, and carefully concatenating subpaths through a stable home configuration; it also provides practical strategies for exploration, smooth transitions, and mobile-base extensions. Empirical evaluations on 6-DOF and 7-DOF manipulators in a bookshelf scenario show up to 3x faster motion planning and up to 5x lower trajectory jerk compared to state-of-the-art baselines, with robust performance in the presence of unknown objects. Collectively, HAP offers a principled, scalable framework for fast, reliable multi-task planning that explicitly accounts for low-level motion in sequencing, enabling more autonomous and agile robotic manipulation in semi-structured environments.

Abstract

Robotic manipulator applications often require efficient online motion planning. When completing multiple tasks, sequence order and choice of goal configuration can have a drastic impact on planning performance. This is well known as the robot task sequencing problem (RTSP). Existing general-purpose RTSP algorithms are susceptible to producing poor-quality solutions or failing entirely when available computation time is restricted. We propose a new multi-query task sequencing method designed to operate in semi-structured environments with a combination of static and non-static obstacles. Our method intentionally trades off workspace generality for planning efficiency. Given a user-defined task space with static obstacles, we compute a subspace decomposition. The key idea is to establish approximate isometries known as $ε$-Gromov-Hausdorff approximations that identify points that are close to one another in both task and configuration space. Importantly, we prove bounded suboptimality guarantees on the lengths of paths within these subspaces. These bounding relations further imply that paths within the same subspace can be smoothly concatenated, which we show is useful for determining efficient task sequences. We evaluate our method with several kinematic configurations in a complex simulated environment, achieving up to 3x faster motion planning and 5x lower maximum trajectory jerk compared to baselines.

Figures (16)

• Figure 1: Example $\epsilon$-Gromov-Hausdorff approximations for a 2-DOF robotic manipulator on a table-top environment with a box obstacle (dark grey). Green and blue regions are subspaces within which end-effector motion does not require large changes in configuration. \ref{['fig:taskspace_subspaces']} Task-space subspace decomposition with overlap shown in dark green. \ref{['fig:configspace_subspaces']} Mapped disjoint subspaces in configuration space.
• Figure 2: A 2-DOF robotic manipulator tasked with moving its end effector to four unordered positions. Positions are shown as coloured dots. The end-effector path is drawn with direction of movement indicated by the arrowhead. \ref{['fig:task1_naive']}-\ref{['fig:task3_naive']} show a sequence of paths produced by a naive planner that only considers task-space distances. \ref{['fig:task1_non_naive']}-\ref{['fig:task3_non_naive']} show the sequence produced by our HAP method. HAP's choice of sequencing exploits short within-subspace paths in \ref{['fig:task1_non_naive']} and \ref{['fig:task3_non_naive']}, whereas the naive planner's choice of sequencing underestimates the true motion cost, resulting in longer paths.
• Figure 3: Overview of problem and approach. \ref{['fig:classic_tsp']} Classic TSP ignores configuration space path costs (nodes are task poses). \ref{['fig:set_tsp']} GTSP considers all possible path sequences (node sets represent task pose IK solutions). \ref{['fig:hap_tsp']} Reduced problem (ours) utilises $\epsilon$-GHAs, only considers subset of IK solutions (green and blue nodes belong to different subspaces) and solves individual TSPs based on subspace assignment.
• Figure 4: Overall process of the HAP framework. On the left is the task decomposition which generates the $\epsilon$-GHAs offline. On the right describes the task-sequencing process given an online scenario which utilises the $\epsilon$-GHAs.
• Figure 5: Example HAP task space decomposition and online scenario. \ref{['fig:offline_grid']} anticipated discretised task space $\hat{T}$ (blue poses) and environment $\hat{m}$. \ref{['fig:offline_edges']} an undirected graph $G$ constructed over $\hat{T}$. \ref{['fig:offline_edges_subspace']} a single $\epsilon$-GHA mapped in task-space represented as a subgraph of $G$, subspaces are searched for by traversing the graph and assigning an unique IK solution to each node such that all connected neighbours are close in configuration space. \ref{['fig:online_scenario']} an example online scenario with the arm in its home configuration. Online, HAP is robust against objects in \ref{['fig:online_scenario']} that are unmodelled in $\hat{m}$ and tasks can differ to those in $\hat{T}$.

Theorems & Definitions (6)

• Definition 1
• Theorem 1: $\theta$ is an $\epsilon$-GHA
• proof
• Definition 2: Geodesics
• Theorem 2: $\epsilon$-GHAs preserve geodesics
• proof