Offline Task Assistance Planning on a Graph:Theoretic and Algorithmic Foundations

TL;DR

This work formalizes task assistance planning (TAP) as a graph-based problem (g-TAP) where an assist robot must strategically time its motion to maximize the duration that a task robot’s path is aided. It introduces the Optimal Timing-Profile (OTP) problem for a fixed path and proves that optimal timing can be found in polynomial time by restricting to vertex-critical times, enabling efficient dynamic programming on a path. The general problem, OPTP, is shown NP-hard, and the authors propose a Branch-and-Bound framework with tight upper bounds and interval-splitting techniques to compute optimal or near-optimal solutions, beating baselines by several orders of magnitude in experiments with planar manipulators and UR robots. The approach provides a practical foundation for TAP, validated in simulation and lab experiments, and points to future work on relaxing roadmap availability, learning assist-paths, and joint planning.

Abstract

In this work we introduce the problem of task assistance planning where we are given two robots Rtask and Rassist. The first robot, Rtask, is in charge of performing a given task by executing a precomputed path. The second robot, Rassist, is in charge of assisting the task performed by Rtask using on-board sensors. The ability of Rassist to provide assistance to Rtask depends on the locations of both robots. Since Rtask is moving along its path, Rassist may also need to move to provide as much assistance as possible. The problem we study is how to compute a path for Rassist so as to maximize the portion of Rtask's path for which assistance is provided. We limit the problem to the setting where Rassist moves on a roadmap which is a graph embedded in its configuration space and show that this problem is NP-hard. Fortunately, we show that when Rassist moves on a given path, and all we have to do is compute the times at which Rassist should move from one configuration to the following one, we can solve the problem optimally in polynomial time. Together with carefully-crafted upper bounds, this polynomial-time algorithm is integrated into a Branch and Bound-based algorithm that can compute optimal solutions to the problem outperforming baselines by several orders of magnitude. We demonstrate our work empirically in simulated scenarios containing both planar manipulators and UR robots as well as in the lab on real robots.

Figures (6)

• Figure 1: TAP in household applications. \ref{['subfig:1a']} Blue manipulator $\mathsf{R}\xspace_{\textbf{task}}$ is tasked with transferring water in a cup from a faucet to a pot (light to dark blue correspond with initial to final configurations). \ref{['subfig:1b']} Green manipulator $\mathsf{R}\xspace_{\textbf{assist}}$ is equiped with a camera that must detect if water is spilled from the cup to initiate clean-up and to ensure that the pot has enough water. Here, the light-green and dark-green robots depict configurations for which the cup is visible and non-visible by $\mathsf{R}\xspace_{\textbf{assist}}$'s point of view, respectively. \ref{['subfig:1c']} A task assistance path maximizing the amount of time the cup is observed by $\mathsf{R}\xspace_{\textbf{assist}}$ (light to dark green correspond with initial to final configurations). Visualization adapted from roberts:2022.
• Figure 2: \ref{['fig:graph']} Toy g-TAP problem. Above each vertex and edge are the intervals for which task assistance can be performed and the edge length, respectively. \ref{['fig:timings']} Two timing profiles for path $\langle v_0, v_1, v_2 \rangle$. Here, each vertex is depicted together with the intervals for which task assistance can be performed. The timing profiles (blue and orange) consist of solid and dotted lines when task assistance can and can't be performed, respectively. Note that the slopes of moving from $v_0$ to $v_1$ and from $v_1$ to $v_2$ are different as the corresponding edge lengths are different ($0.1$ and $0.25$, respectively).
• Figure 3: Reduction graph (all edges are directed from left to right). When omitted, edge length equals zero.
• Figure 3: Reduction graph (all edges are directed from left to right). When omitted, edge length equals zero. (This figure is identical to Fig. \ref{['fig:reduction']} in Sec. \ref{['sec:hardness']} and is added to make the appendix self-contained).
• Figure 4: Simulated environments consisting of a task-robot (orange), assistance-robot (green) and obstacles (yellow). The task-robot end-effector follows a predefined path (blue) and needs to be located in the field of view of a limited-range camera located on the assistance robot's end effector (purple).

Theorems & Definitions (27)

• Definition 1: Timing-profile
• Definition 2: Reward at a vertex
• Definition 3: Reward of a timing-profile
• Definition 4: Vertex-pair critical times
• Definition 5: Vertex-critical times
• Lemma 1
• proof : Proof (sketch)
• Theorem 1
• Theorem 2
• proof : Sketch