Minimum-Violation Temporal Logic Planning for Heterogeneous Robots under Robot Skill Failures

TL;DR

A reactive LTL planning algorithm that adapts to unexpected failures during deployment that strategically prioritizes the most crucial sub-tasks and locally revises the team's plans, as per user-specified priorities, to minimize mission violations.

Abstract

In this paper, we consider teams of robots with heterogeneous skills (e.g., sensing and manipulation) tasked with collaborative missions described by Linear Temporal Logic (LTL) formulas. These LTL-encoded tasks require robots to apply their skills to specific regions and objects in a temporal and logical order. While existing temporal logic planning algorithms can synthesize correct-by-construction plans, they typically lack reactivity to unexpected failures of robot skills, which can compromise mission performance. This paper addresses this challenge by proposing a reactive LTL planning algorithm that adapts to unexpected failures during deployment. Specifically, the proposed algorithm reassigns sub-tasks to robots based on their functioning skills and locally revises team plans to accommodate these new assignments and ensure mission completion. The main novelty of the proposed algorithm is its ability to handle cases where mission completion becomes impossible due to limited functioning robots. Instead of reporting mission failure, the algorithm strategically prioritizes the most crucial sub-tasks and locally revises the team's plans, as per user-specified priorities, to minimize mission violations. We provide theoretical conditions under which the proposed framework computes the minimum-violation task reassignments and team plans. We provide numerical and hardware experiments to demonstrate the efficiency of the proposed method.

Figures (11)

• Figure 1: The small squares below each robot indicate the abilities each robot possesses; the ability to push buttons (blue), retrieve objects (green), take photos (red), and open doors (yellow). The locations are represented by the larger squares, whose color indicates which ability needs to be used at that location as per the LTL-encoded mission $\phi$ discussed in Example \ref{['ex:LTL']}. The mission requires robots $1$, $2$, and $3$ to simultaneously execute their sub-tasks. The penalties for not completing each sub-task is shown in red next to each location. Fig. \ref{['fig:3ra']} shows the plans designed offline and \ref{['fig:3rb']} shows the minimum violation plans, planned after robot $2$ fails (red X on skill).
• Figure 2: Consider in Example \ref{['ex:LTL']} the case where skill $c_3$ of robot $2$ fails, i.e., the failed predicate is $\pi_5$. We present the BFS tree (Alg. \ref{['alg:bfs']}) built to fix $\pi_5$ for the NBA transition enabled by $b_{q_B',q_B"}^{d}=\pi_4\wedge\pi_5\wedge\pi_6\wedge\bar{\pi}_7$. The set ${\mathcal{A}}$ is defined as ${\mathcal{A}}=\{4\}$ and the root of the tree is robot $2$. Robots $3$, and $4$ are adjacent to robot $2$ in ${\mathcal{G}}$. Robot $4$ is not connected to robot $2$ because it does not satisfy $g_{q_B',q_B"}^d(2)\notin V^{d}_{q_B',q_B"}(4)=\{\pi_5,\pi_7,\pi_5\pi_7\}$. Robot $3$ is connected to robot $2$ and subsequently, robot $1$ is connected to robot $3$. Since we did not find a feasible path from $a_{\text{root}}$ to ${\mathcal{A}}$, we pick the node $a^*=3$ which has the lowest predicate penalty, $F(g_{q_B',q_B"}^d(a^*))=15$. The blue dashed arrows show the re-assignment process along the computed path $p$, i.e., robot $3$ will take over the failed predicate and $\pi_4$ will be sacrificed resulting in an assignment cost/penalty of 15. In an alternate case if $\pi_4$ had the lowest penalty, then we would have seen robot $1$ replace robot $3$, and robot $3$ replace failed robot $2$, thus sacrificing $\pi_4$.
• Figure 3: Hypothetical example of online revision of a part of the prefix plan. NBA states $\text{A,B,\dots,J}\in{\mathcal{Q}}_{B}$, where $q_B(t)=q_B(k)=\text{A}$. Fig. \ref{['fig:curNBA']} shows the states in ${\mathcal{P}}_{\text{pre}}^{\tau}$, along with the predicate-robot assignment that enables the transition to each state (the predicates and robot numbers are for illustrative purposes only). Assume some skills of robot $1$ failed rendering some of the transitions in the NBA infeasible (red cross). Fig. \ref{['fig:revNBA']} shows the plan ${\mathcal{P}}_{\text{pre}}^{\text{min}}$ in the NBA that generates the lowest violation cost after fixing the failures. Here the overlap exists from C to F. However, only the transition from D to E will be considered a true overlap. Condition (1) of reusability is satisfied because the most recent predicates done by the robots (R1-$\pi_{\text{D1}}$, R2-$\pi_{\text{D2}}$, and R3-$\pi_{\text{C3}}$) are still the same in both ${\mathcal{P}}_{\text{pre}}^{\text{min}}$ and ${\mathcal{P}}_{\text{pre}}^{\tau}$. Condition (2) is satisfied because the boolean $\pi_{\text{E1}}\wedge\pi_{\text{E2}}$ remains unchanged. The disks in Figs. \ref{['fig:curPath']} and \ref{['fig:revPath']} capture states in $\hat{\tau}_H$. Yellow states $\hat{\tau}_H(k')$ model states for which it holds $q_B(k')\neq q_B(k'-1)$. The part of $\hat{\tau}_H$ connecting NBA states $q_B'$ and $q_B"$, where $e=(q_B',q_B")\in{\mathcal{E}}_{\pi}$ (see Alg. \ref{['alg:RP']}) is marked with a red color and a red 'X' denoting that it requires revision. Note that although the plan from $\hat{\tau}_H(k_1)$ (NBA state C) to $\hat{\tau}_H(k_2)$ (NBA state D) is not marked red, it still needs to be revised as it is not a true overlap. In this example, the planner re-plans a plan from current state $\hat{\tau}_H(k)$ (NBA state A) to $\hat{\tau}_H(k_2)$ (NBA state D) which is the start of a true overlap, and re-plans another plan from the end of the overlap $\hat{\tau}_H(k_3)$ (NBA state E) to the final state $\hat{\tau}_H(k_6)$ (NBA state J). This is done using lines \ref{['re-planner:replanCond']}-\ref{['re-planner:replanAppend']} in Alg. \ref{['alg:re-planner']}. The planner does not plan for the true overlap section (NBA states D to E) and instead re-uses the original plan $\hat{\tau}_H(k_2:k_3)$ to bridge the state $\hat{\tau}_H(k_2)$ to the state $\hat{\tau}_H(k_3)$. In Alg. \ref{['alg:re-planner']}, this is done by lines \ref{['re-planner:reuseCond']}-\ref{['re-planner:reuseAppend']}.
• Figure 4: NBA generated by mission in Example \ref{['ex:LTL']}; $\phi = \Diamond(\pi_1\wedge\pi_2\wedge\pi_3)\wedge\square\Bar{\pi}_4\wedge\Diamond\pi_5$. Assumption \ref{['cond:NBA_self_loop']} is satisfied, since all self-loops are $b_{q_B',q_B'}=\xi_i$, where $\xi_i$ is a Boolean formula defined over atomic predicates of the form \ref{['eq:negpip']}.
• Figure 5: Effect of number of failures on re-planning time: Fewer failures generally means fewer plan changes. Thus while a global replanner will need to replan everything, the local replanner can fix only the necessary changes and generate plans faster. However as number of failures increase, the plan needs to be revised at multiple transitions and the local planner would end up planning globally resulting in times similar to the global planner.

Theorems & Definitions (21)

• Definition 2.1: Penalty Function
• Definition 2.6: NBA
• Example 2.7: Least Violating Transition
• Remark 2.8: Violation Cost Function
• Example 2.9
• Remark 2.10: Assumptions \ref{['as2']}-\ref{['as4']}
• Definition 3.1: Function $V_{q_B',q_B"}^d$
• Definition 3.2: Function $g_{q_B',q_B"}^d$
• Example 3.3: Function $g_{q_B',q_B"}^d$, $V_{q_B',q_B"}^d$ and sets ${\mathcal{R}}_{q_B',q_B"}^d$
• Proposition 4.1: Optimality of Alg. \ref{['alg:bfs']}